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1011.1991	# Zero dissipation limit to rarefaction wave with vacuum for 1-D compressible Navier-Stokes equations Feimin Huang, Mingjie Li, Yi Wang Institute of Applied Mathematics, AMSS, Academia Sinica, Beijing, China. ###### Abstract It is well-known that one-dimensional isentropic gas dynamics has two elementary waves, i.e., shock wave and rarefaction wave. Among the two waves, only the rarefaction wave can be connected to vacuum. Given a rarefaction wave with one-side vacuum state to the compressible Euler equations, we can construct a sequence of solutions to one-dimensional compressible isentropic Navier-Stokes equations which converge to the above rarefaction wave with vacuum as the viscosity tends to zero. Moreover, the uniform convergence rate is obtained. The proof consists of a scaling argument and elementary energy analysis based on the underlying rarefaction wave structures. Keywords: compressible Navier-Stokes equations, zero dissipation limit, rarefaction wave, vacuum. ## 1 Introduction and main result In this paper, we investigate the zero dissipation limit of the one- dimensional compressible isentropic Navier-Stokes equations $\left\\{\begin{array}[]{ll}\displaystyle\rho_{t}+(\rho u)_{x}=0,&\displaystyle x\in\mathbf{R},t>0,\\\\[5.69054pt] \displaystyle(\rho u)_{t}+\big{(}\rho u^{2}+p(\rho)\big{)}_{x}=\epsilon~{}u_{xx},&\\\\[5.69054pt] \end{array}\right.$ (1.1) where $\rho(t,x)\geq 0$, $u(t,x)$ and $p$ represent the density, the velocity and the pressure of the gas, respectively and $\epsilon>0$ is the viscosity coefficient. Here we assume that the viscosity coefficient $\epsilon$ is a positive constant and the pressure $p$ is given by the $\gamma-$law: $p(\rho)=\frac{\rho^{\gamma}}{\gamma}$ with $\gamma>1$ being the gas constant. Formally, as $\epsilon$ tends to zero, the limit system of the compressible Navier-Stokes equations (1.1) is the following inviscid Euler equations $\left\\{\begin{array}[]{l}\rho_{t}+(\rho u)_{x}=0,\\\\[5.69054pt] \displaystyle(\rho u)_{t}+(\rho u^{2}+p(\rho))_{x}=0.\end{array}\right.$ (1.2) The Euler system (1.2) is a strictly hyperbolic one for $\rho>0$ whose characteristic fields are both genuinely nonlinear, that is, in the equivalent system $\left(\begin{array}[]{l}\displaystyle\rho\\\ \displaystyle u\end{array}\right)_{t}+\left(\begin{array}[]{cc}\displaystyle u&\quad\rho\\\ \displaystyle p^{\prime}(\rho)/\rho&\quad u\end{array}\right)\left(\begin{array}[]{l}\displaystyle\rho\\\ \displaystyle u\end{array}\right)_{x}=0,$ the Jacobi matrix $\left(\begin{array}[]{cc}\displaystyle u&\quad\rho\\\ \displaystyle p^{\prime}(\rho)/\rho&\quad u\end{array}\right)$ has two distinct eigenvalues $\lambda_{1}(\rho,u)=u-\sqrt{p^{\prime}(\rho)},\qquad\lambda_{2}(\rho,u)=u+\sqrt{p^{\prime}(\rho)}$ with corresponding right eigenvectors $r_{i}(\rho,u)=(1,(-1)^{i}\frac{\sqrt{p^{\prime}(\rho)}}{\rho})^{t},\qquad i=1,2,$ such that $r_{i}(\rho,u)\cdot\nabla_{\rho,u}\lambda_{i}(\rho,u)=(-1)^{i}\frac{\rho p^{\prime\prime}(\rho)+2p^{\prime}(\rho)}{2\rho\sqrt{p^{\prime}(\rho)}}\neq 0,\quad i=1,2.$ We can define the $i-$Riemann invariant $(i=1,2)$ by $\Sigma_{i}(\rho,u)=u+(-1)^{i+1}\int^{\rho}\frac{\sqrt{p^{\prime}(s)}}{s}ds$ such that $\nabla_{(\rho,u)}\Sigma_{i}(\rho,u)\cdot r_{i}(\rho,u)\equiv 0,\qquad\forall\rho>0,u.$ The study of the limiting process of viscous flows when the viscosity tends to zero, is one of the important problems in the theory of the compressible fluid. When the solution of the inviscid flow is smooth, the zero dissipation limit can be solved by classical scaling method. However, the inviscid compressible flow contains singularities such as shock and the vacuum in general. Therefore, how to justify the zero dissipation limit to the Euler equations with basic wave patterns and/or the vacuum is a natural and difficult problem. There have been many results on the zero dissipation limit of the compressible fluid with basic wave patterns without vacuum. For the system of the hyperbolic conservation laws with artificial viscosity $u_{t}+f(u)_{x}=\varepsilon u_{xx},$ Goodman-Xin [5] first verified the viscous limit for piecewise smooth solutions separated by non-interacting shock waves using a matched asymptotic expansion method. Later Yu [22] proved it for the corresponding hyperbolic conservation laws with both shock and initial layers. In 2005, important progress made by Bianchini-Bressan[1] justifies the vanishing viscosity limit in BV space even though the problem is still unsolved for the physical system such as the compressible Navier-Stokes equations. For the compressible isentropic Navier-Stokes equations (1.1), Hoff-Liu [6] first proved the vanishing viscosity limit for piecewise constant shock even with initial layer. Later Xin [20] obtained the zero dissipation limit for rarefaction waves without vacuum for both rarefaction wave data and well-prepared smooth data. Then Wang [18] generalized the result of Goodmann-Xin [5] to the isentropic Navier-Stokes equations (1.1). For the full Navier-Stokes equations where the conservation of the energy is also involved, there are also many results on the zero dissipation limit to the corresponding full Euler system with basic wave patterns without vacuum. We refer to Jiang-Ni-Sun [11] and Xin-Zeng [21] for the rarefaction wave, Wang [19] for the shock wave, Ma [14] for the contact discontinuity and Huang-Wang-Yang [9, 10] for the superposition of two rarefaction waves and a contact discontinuity and the superposition of one shock and one rarefaction wave cases. More recently, Chen-Perepelitsa [3] proved the vanishing viscosity to the compressible Euler equations for the compressible Navier-Stokes equations (1.1) by compensated compactness method for the general case if the far field of the initial values of Euler system (1.2) has no vacuums. Note that this result is quite universal since the initial values of the Euler system can contain vacuum states in the interior domain. Huang-Pan-Wang-Wang-Zhai [7] established the corresponding results to the compressible Navier-Stokes equations (1.1) with density-dependent viscosity. Now we turn back to the case of the basic wave patterns with vacuum states. As pointed out by Liu-Smoller [13], among the two nonlinear waves, i.e., shock and rarefaction waves, to the one-dimensional compressible isentropic Euler equations (1.2), only the rarefaction wave can be connected to vacuum. However, to our knowledge, so far there is no any results on the zero dissipation limit of the system (1.1) in the case when the Euler system (1.2) contain the rarefaction wave connected to the vacuum. In this paper, we investigate this fundamental problem and want to obtain the decay rate with respect to the viscosity $\epsilon$. Remark that Perepelitsa [16] consider the time-asymptotic stability of solutions of 1-d compressible Navier-Stokes equations (1.1) toward rarefaction waves connected to vacuum in Lagrangian coordinate and Jiu-Wang-Xin [12] study the large time asymptotic behavior toward rarefaction waves for solutions to the 1-dimensional compressible Navier-Stokes equations (1.1) with density-dependent viscosity for general initial data whose far fields are connected by a rarefaction wave to the corresponding Euler equations with one end state being vacuum. Now we give a description of the rarefaction wave connected to the vacuum to the compressible Euler equations (1.2), see also the references [13] and [17]. For definiteness, 2-rarefaction wave will be considered. If we investigate the compressible Euler system (1.2) with the Riemann initial data $\left\\{\begin{array}[]{rr}\rho(0,x)=0,&x<0,\\\ (\rho,u)(0,x)=(\rho_{+},u_{+}),&x>0,\end{array}\right.$ (1.3) where the left side is the vacuum state and $\rho_{+}>0,u_{+}$ are prescribed constants on the right state, then the Riemann problem (1.2), (1.3) admits a $2-$rarefaction wave connected to the vacuum on the left side. By the fact that along the $2-$rarefaction wave curve, $2-$Riemann invariant $\Sigma_{2}(\rho,u)$ is constant in $(x,t)$, we can get the velocity $u_{-}=\Sigma_{2}(\rho_{+},u_{+})$ being the speed of the fluid coming into the vacuum from the 2-rarefaction wave. This $2-$rarefaction wave connecting the vacuum $\rho=0$ to $(\rho_{+},u_{+})$ is the self-similar solution $(\rho^{r_{2}},u^{r_{2}})(\xi),~{}(\xi=\frac{x}{t})$ of (1.2) defined by $\begin{array}[]{c}\displaystyle\qquad\qquad\qquad\qquad\quad\rho^{r_{2}}(\xi)=0,~{}~{}{\rm if}~{}~{}\xi<\lambda_{2}(0,u_{-})=u_{-},\\\ \lambda_{2}(\rho^{r_{2}}(\xi),u^{r_{2}}(\xi))=\left\\{\begin{array}[]{ll}\displaystyle\displaystyle\xi,&\displaystyle{\rm if}~{}~{}u_{-}\leq\xi\leq\lambda_{2}(\rho_{+},u_{+}),\\\ \displaystyle\lambda_{2}(\rho_{+},u_{+}),&\displaystyle{\rm if}~{}~{}\xi>\lambda_{2}(\rho_{+},u_{+}),\end{array}\right.\end{array}$ (1.4) and $\Sigma_{2}(\rho^{r_{2}}(\xi),u^{r_{2}}(\xi))=\Sigma_{2}(0,u_{-})=\Sigma_{2}(\rho_{+},u_{+}).$ (1.5) Thus we can define the momentum of 2-rarefaction wave by $m^{r_{2}}(\xi)=\left\\{\begin{array}[]{ll}\rho^{r_{2}}(\xi)u^{r_{2}}(\xi),{}{}{}&{\rm if}~{}~{}~{}\rho^{r_{2}}>0,\\\ 0,{}{}{}&{\rm if}~{}~{}~{}\rho^{r_{2}}=0.\end{array}\right.$ (1.6) In the present paper, we want to construct a sequence of solutions $(\rho^{\epsilon},m^{\epsilon})(x,t)$ to the compressible Navier-Stokes equations (1.1) which converge to the 2-rarefaction wave $(\rho^{r}_{2},m^{r}_{2})(x/t)$ defined above as $\epsilon$ tends to zero. The effects of initial layers will be ignored by choosing the well-prepared initial data depending on the viscosity for the Navier-Stokes equations. The main novelty and difficulty of the paper is how to control the degeneracies caused by the vacuum in the rarefaction wave. To overcome this difficulty, we first cut off the 2-rarefaction wave with vacuum along the rarefaction wave curve. More precisely, for any $\mu>0$ to be determined, the cut-off rarefaction wave will connect the state $(\rho,u)=(\mu,u_{\mu})$ and $(\rho_{+},u_{+})$ where $u_{\mu}$ can be obtained uniquely by the definition of the 2-rarefaction wave curve. Then an approximate rarefaction wave to this cut-off rarefaction wave will be constructed through the Burgers equation. Finally, the desired solution sequences to the compressible Navier-Stokes equations (1.1) could be established around the approximate rarefaction wave. The uniform estimates to the perturbation of the solution sequences around the approximate rarefaction wave can be got by the following two observations. One is the fact that the viscosity $\epsilon$ can control the degeneracies caused by the vacuum in rarefaction waves by choosing suitably $\mu=\mu(\epsilon)$. In fact, we choose $\mu=\epsilon^{a}\|\ln\epsilon\|$ with $a$ defined in (3.13) in the present paper. The other observation is that we can carry out the energy estimates under the a priori assumption that the perturbation is suitably small in $H^{1}(\mathbf{R})$ norm with some decay rate with respect to $\epsilon$ as $\epsilon$ tends to zero. See (3.10) in the below for the details. Note that this a priori assumption is natural but is first used in studying zero dissipation limit to our knowledge. With these two observations, we can close the a priori assumption and obtain the desired results. Now our main result is stated as follows. ###### Theorem 1.1. Let $(\rho^{r_{2}},m^{r_{2}})(x/t)$ be the 2-rarefaction wave defined by (1.4)-(1.6) with one-side vacuum state. Then there exists a small positive constant $\epsilon_{0}$ such that for any $\epsilon\in(0,\epsilon_{0})$, we can construct a global smooth solution $(\rho^{\epsilon},m^{\epsilon}=\rho^{\epsilon}u^{\epsilon})(x,t)$ with initial values (3.1) to the compressible Navier-Stokes equation (1.1) satisfying (1) $\begin{array}[]{rl}(\rho^{\epsilon}-\rho^{r_{2}},m^{\epsilon}-m^{r_{2}}),(\rho^{\epsilon},m^{\epsilon})_{x}&\displaystyle\in C^{0}((0,+\infty);L^{2}(\mathbf{R})),\\\ m^{\epsilon}_{xx}&\displaystyle\in L^{2}(0,+\infty;L^{2}(\mathbf{R})).\end{array}$ 2) As viscosity $\epsilon\rightarrow 0$, $(\rho^{\epsilon},m^{\epsilon})(x,t)$ converges to $(\rho^{r_{2}},m^{r_{2}})(x/t)$ pointwisely except the original point $(0,0)$. Furthermore, for any given positive constant $h$, there exists a constant $C_{h}>0$, independent of $\epsilon$, such that $\begin{array}[]{ll}\displaystyle\sup_{t\geq h}\\|\rho^{\epsilon}(\cdot,t)-\rho^{r_{2}}(\frac{\cdot}{t})\\|_{L^{\infty}}\leq C_{h}\epsilon^{a}\|\ln\epsilon\|,\\\ \displaystyle\sup_{t\geq h}\\|m^{\epsilon}(\cdot,t)-m^{r_{2}}(\frac{\cdot}{t})\\|_{L^{\infty}}\leq\left\\{\begin{array}[]{ll}\displaystyle C_{h}\epsilon^{b}\|\ln\epsilon\|^{-\frac{1}{2}},{}{}{}&{\rm if}~{}~{}~{}1<\gamma<3,\\\ \displaystyle C_{h}\epsilon^{\frac{1}{\gamma+4}}\|\ln\epsilon\|,{}{}{}&{\rm if}~{}~{}~{}\gamma\geq 3,\end{array}\right.\end{array}$ (1.7) with the positive constants $a,~{}b$ given by $a=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{6}{}{}{}&{\rm if}~{}~{}~{}1<\gamma\leq 2,\\\ \displaystyle\frac{1}{\gamma+4},{}{}{}&{\rm if}~{}~{}~{}\gamma>2.\end{array}\right.$ (1.8) and $b=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{8}{}{}{}&{\rm if}~{}~{}~{}1<\gamma\leq 2,\\\ \displaystyle\frac{\gamma+1}{4(\gamma+4)},{}{}{}&{\rm if}~{}~{}~{}2<\gamma<3.\\\\[8.53581pt] \end{array}\right.$ (1.9) A few remarks are followed. ###### Remark 1.2. Similar result to Theorem 1.1 is also expected for a one-dimensional compressible Navier-Stokes equation with density-dependent viscosity which reads $\left\\{\begin{array}[]{l}\rho_{t}+(\rho u)_{x}=0,\\\\[5.69054pt] (\rho u)_{t}+\big{(}\rho u^{2}+\rho^{\gamma}\big{)}_{x}=\epsilon\big{(}\rho^{\alpha}u_{x}\big{)}_{x},\\\\[5.69054pt] \end{array}\right.$ (1.10) with suitable $\alpha>0$ and $\gamma>1$. Actually, the system (1.10) can be derived by Chapman-Enskog expansions from the Boltzmann equation where the viscosity of the compressible Navier-Stokes equations depends on the temperature and thus on the density for isentropic flows. Also, the viscous Saint-Venant system for the shallow water, derived from the incompressible Navier-Stokes equation with a moving free surface, is expressed exactly as in (1.10) with $\alpha=1$ and $\gamma=2$, see [2, 4]. In this situation, since viscosity vanishes at vacuum, the convergence rate with respect to $\epsilon$ may become slower than in Theorem 1.1 and may depend on $\alpha$ and $\gamma$. However, this is left to the forthcoming paper. ###### Remark 1.3. Our result and method can also be generalized to the 1-D full compressible Navier-Stokes equations with the conservation of the energy and the Boltzmann equation with slab symmetry. This is left to the forthcoming paper. ###### Remark 1.4. It is also interesting to study the zero dissipation limit of compressible Navier-Stokes equations (1.1) in the case when the Euler system (1.2) has two rarefaction waves with the vacuum states in the middle. However, it is nontrivial to cut off these rarefaction waves with vacuum along the corresponding rarefaction wave curves. In fact, the wave structure containing two rarefaction waves with the medium vacuum is destroyed and some new wave may occur in the cut-off precess, which is quite different from the single rarefaction wave case considered in the present paper. ###### Remark 1.5. It is noted that in the a priori estimates $(\ref{main})$ below , the estimates for $\phi^{2}$ from the potential energy holds with the weight ${\bar{\rho}}^{\gamma-2}$ which is degenerate at vacuum when $\gamma>2$. Therefore, the convergence rate obtained in Lemma 3.2 and thus in Theorem 1.1 depends on $\gamma$ when $\gamma>2$. The rest of the paper is organized as follows. In section 2, we construct a smooth 2-rarefaction wave which approximates the cut-off rarefaction wave based on the inviscid Burgers equation. And the proof the Theorem 1.1 is given in Section 3. Throughout this paper, $H^{l}(\mathbf{R}),l=0,1,2,...$, denotes the $l$-th order Sobolev space with its norm $\\|f\\|_{l}=(\sum^{l}_{j=0}\\|\partial^{j}_{y}f\\|^{2})^{\frac{1}{2}},\quad{\rm and}~{}\\|\cdot\\|:=\\|\cdot\\|_{L^{2}(dy)},$ while $L^{2}(dz)$ means the $L^{2}$ integral over $\mathbf{R}$ with respect to the Lebesgue measure $dz$, and $z=x$ or $y$. For simplicity, we also write $C$ as generic positive constants which are independent of time $t$ and viscosity $\epsilon$ unless otherwise stated. ## 2 Rarefaction waves Since there is no exact rarefaction wave profile for the Navier-Stokes equations (1.1), the following approximate rarefaction wave profile satisfying the Euler equations was motivated by Matsumura-Nishihara [15] and Xin [20]. For the completeness of the presentation, we include its definition and the properties listed in Lemma 2.1. However, Lemma 2.1 is a little different from [20] as stated after Lemma 2.1. Consider the Riemann problem for the inviscid Burgers equation: $\displaystyle\left\\{\begin{array}[]{ll}w_{t}+ww_{x}=0,\\\ w(x,0)=\left\\{\begin{array}[]{ll}w_{-},&x<0,\\\ w_{+},&x>0.\end{array}\right.\end{array}\right.$ (2.5) If $w_{-}<w_{+}$, then the Riemann problem $(\ref{bur})$ admits a rarefaction wave solution $w^{r}(x,t)=w^{r}(\frac{x}{t})$ given by $\displaystyle w^{r}(\frac{x}{t})=\left\\{\begin{array}[]{lr}w_{-},&\frac{x}{t}\leq w_{-},\\\ \frac{x}{t},&w_{-}\leq\frac{x}{t}\leq w_{+},\\\ w_{+},&\frac{x}{t}\geq w_{+}.\end{array}\right.$ (2.9) As in [20], the approximate rarefaction wave to the compressible Navier-Stokes equations (1.1) can be constructed by the solution of the Burgers equation $\displaystyle\left\\{\begin{array}[]{l}\displaystyle w_{t}+ww_{x}=0,\\\ \displaystyle w(0,x)=w_{\delta}(x)=w(\frac{x}{\delta})=\frac{w_{+}+w_{-}}{2}+\frac{w_{+}-w_{-}}{2}\tanh\frac{x}{\delta},\end{array}\right.$ (2.12) where $\delta>0$ is a small parameter to be determined. In fact, we choose $\delta=\epsilon^{a}$ in (3.14) with $a$ given by (3.13) in the following. Note that the solution $w^{r}_{\delta}(t,x)$ of the problem (2.12) is given by $w^{r}_{\delta}(t,x)=w_{\delta}(x_{0}(t,x)),\qquad x=x_{0}(t,x)+w_{\delta}(x_{0}(t,x))t.$ (2.13) And $w^{r}_{\delta}(t,x)$ has the following properties: ###### Lemma 2.1. The problem $(\ref{dbur})$ has a unique smooth global solution $w_{\delta}^{r}(x,t)$ for each $\delta>0$ such that * (1) $w_{-}<w_{\delta}^{r}(x,t)<w_{+},\ \partial_{x}w^{r}_{\delta}(x,t)>0,$ for $x\in\mathbf{R},\ t\geq 0,\ \delta>0.$ * (2) The following estimates hold for all $\ t>0,\ \delta>0$ and p$\in[1,\infty]$: $\displaystyle\\|\partial_{x}w^{r}_{\delta}(\cdot,t)\\|_{L^{p}}\leq$ $\displaystyle\displaystyle C(w_{+}-w_{-})^{1/p}(\delta+t)^{-1+1/p},$ (2.14) $\displaystyle\\|\partial^{2}_{x}w^{r}_{\delta}(\cdot,t)\\|_{L^{p}}\leq$ $\displaystyle\displaystyle C(\delta+t)^{-1}\delta^{-1+1/p},$ (2.15) $\displaystyle\|\frac{\partial^{2}w^{r}_{\delta}(x,t)}{\partial x^{2}}\|\leq\frac{4}{\delta}\frac{\partial w^{r}_{\delta}(x,t)}{\partial x}.$ (2.16) * (3) There exist a constant $\delta_{0}\in(0,1)$ such that for $\delta\in(0,\delta_{0}],t>0$, $\\|w^{r}_{\delta}(\cdot,t)-w^{r}(\frac{\cdot}{t})\\|_{L^{\infty}}\leq\displaystyle C\delta t^{-1}\big{[}\ln(1+t)+\|\ln\delta\|\big{]}.$ The proof of Lemma 2.1 can be found in Xin [20]. However, the description of Lemma 2.1 is equivalent to but a little different from Xin [20]. Take the estimation (2.14) as an example, which is described by $\\|\partial_{x}w^{r}_{\delta}(\cdot,t)\\|_{L^{p}}\leq\displaystyle C\min\\{(w_{+}-w_{-})\delta^{-1+1/p},(w_{+}-w_{-})^{1/p}t^{-1+1/p}\\},$ $None$ in Xin [20]. In fact, two estimations (2.14) and $(\ref{w1})^{\prime}$ are equivalent for fixed wave strength $w_{+}-w_{-}$. However, the advantage of Lemma 2.1 is that the energy estimate can be carried out for all time since there is no singularity to the approximate rarefaction wave even at $t=0$. While in Xin’s paper [20], the energy estimate must be done in two time- scalings, that is, finite time and large time, due to the singularity of the estimations of the approximate rarefaction wave at $t=0$. As mentioned in the introduction, we will cut off the 2-rarefaction wave with vacuum along the wave curve in order to overcome the difficulty caused by the vacuum,. More precisely, for any $\mu>0$ to be determined, we can get a state $(\rho,u)=(\mu,u_{\mu})$ belonging to the 2-rarefaction wave curve. From the fact that 2-Riemann invariant $\Sigma_{2}(\rho,u)$ is constant along the 2-rarefaction wave curve, $u_{\mu}$ can be computed explicitly by $u_{\mu}=\Sigma_{2}(\rho_{+},u_{+})+\frac{2}{\gamma-1}\mu^{\frac{\gamma-1}{2}}$. Now we get a new 2-rarefaction wave $(\rho_{\mu}^{r_{2}},u_{\mu}^{r_{2}})(\xi),~{}(\xi=x/t)$ connecting the state $(\mu,u_{\mu})$ to the state $(\rho_{+},u_{+})$ which can be expressed explicitly by $\displaystyle\lambda_{2}(\rho_{\mu}^{r_{2}},u_{\mu}^{r_{2}})(\xi)=\left\\{\begin{array}[]{ll}\lambda_{2}(\mu,u_{\mu}),&\xi<\lambda_{2}(\mu,u_{\mu}),\\\ \xi,&\lambda_{2}(\mu,u_{\mu})\leq\xi\leq\lambda_{2}(\rho_{+},u_{+}),\\\ \lambda_{2}(\rho_{+},u_{+}),&\xi>\lambda_{2}(\rho_{+},u_{+}).\end{array}\right.$ (2.20) and $\displaystyle\Sigma_{2}(\rho_{\mu}^{r_{2}},u_{\mu}^{r_{2}})=\Sigma_{2}(\mu,u_{\mu})=\Sigma_{2}(\rho_{+},u_{+}).$ (2.21) Correspondingly, we can define the momentum function $m^{r_{2}}_{\mu}=\rho^{r_{2}}_{\mu}u^{r_{2}}_{\mu}.$ It is easy to show that the cut-off 2-rarefaction wave $(\rho_{\mu}^{r_{2}},m_{\mu}^{r_{2}})(x/t)$ converges to the original 2-rarefaction wave with vacuum $(\rho^{r_{2}},m^{r_{2}})(x/t)$ in sup-norm with the convergence rate $\mu$ as $\mu$ tends to zero. More precisely, we have ###### Lemma 2.2. There exist a constant $\mu_{0}\in(0,1)$ such that for $\mu\in(0,\mu_{0}],t>0$, $\\|(\rho_{\mu}^{r_{2}},m_{\mu}^{r_{2}})(\cdot/t)-(\rho^{r_{2}},m^{r_{2}})(\cdot/t)\\|_{L^{\infty}}\leq\displaystyle C\mu.$ The proof of Lemma 2.2 can be obtained directly from the explicit solution formula of rarefaction waves, so we omit it for brevity. Now the approximate rarefaction wave $(\bar{\rho}_{\mu,\delta},\bar{u}_{\mu,\delta})(x,t)$ of the cut-off 2-rarefaction wave $(\rho_{\mu}^{r_{2}},u_{\mu}^{r_{2}})(\frac{x}{t})$ to compressible Navier-Stokes equations $(\ref{ns})$ can be defined by $\displaystyle\left\\{\begin{array}[]{l}\displaystyle w_{+}=\lambda_{2}(\rho_{+},u_{+}),\quad w_{-}=\lambda_{2}(\mu,u_{\mu}),\\\ \displaystyle w_{\delta}^{r}(t,x)=\lambda_{2}(\bar{\rho}_{\mu,\delta},\bar{u}_{\mu,\delta})(t,x),\\\ \displaystyle\Sigma_{2}(\bar{\rho}_{\mu,\delta},\bar{u}_{\mu,\delta})(x,t)=\Sigma_{2}(\rho_{+},u_{+})=\Sigma_{2}(\mu,u_{\mu}),\end{array}\right.$ (2.25) where $w_{\delta}^{r}$ is the solution of Burger’s equation $(\ref{dbur})$ defined in (2.13). From then on, the subscription of $(\bar{\rho}_{\delta,\mu},\bar{u}_{\delta,\mu})(x,t)$ will be omitted as $(\bar{\rho},\bar{u})(x,t)$ for simplicity. Then the approximate cut-off 2-rarefaction wave $(\bar{\rho},\bar{u})$ defined above satisfies $\left\\{\begin{array}[]{ll}\displaystyle\bar{\rho}_{t}+(\bar{\rho}\bar{u})_{x}=0,\\\\[5.69054pt] \displaystyle(\bar{\rho}\bar{u})_{t}+\big{(}\bar{\rho}\bar{u}^{2}+p(\bar{\rho})\big{)}_{x}=0,\\\\[5.69054pt] \end{array}\right.$ (2.26) and the properties of the approximate rarefaction wave $(\bar{\rho},\bar{u})$ is listed without proof in the following Lemma. ###### Lemma 2.3. The approximate cut-off 2-rarefaction wave $(\bar{\rho},\bar{u})$ defined in (2.25) satisfies the following properties: * (i) $\bar{u}_{x}(x,t)=\frac{2}{\gamma+1}(w_{\delta}^{r})_{x}>0,$ for $x\in\mathbf{R},\ t\geq 0;$ $\bar{\rho}_{x}=\bar{\rho}^{\frac{3-\gamma}{2}}\bar{u}_{x}$, and $\bar{\rho}_{xx}=\bar{\rho}^{\frac{3-\gamma}{2}}\bar{u}_{xx}+\frac{3-\gamma}{2}\bar{\rho}^{2-\gamma}(\bar{u}_{x})^{2}$. * (ii) The following estimates hold for all $\ t>0,\ \delta>0$ and p$\in[1,\infty]$: $\begin{array}[]{l}\\|\bar{u}_{x}(\cdot,t)\\|_{L^{p}}\leq C(w_{+}-w_{-})^{1/p}(\delta+t)^{-1+1/p},\\\\[11.38109pt] \\|\bar{u}_{xx}(\cdot,t)\\|_{L^{p}}\leq C(\delta+t)^{-1}\delta^{-1+1/p}.\end{array}$ * (iii) There exist a constant $\delta_{0}\in(0,1)$ such that for $\delta\in(0,\delta_{0}],t>0$, $\begin{array}[]{ll}\\|(\bar{\rho}-\rho^{r_{2}}_{\mu},\bar{u}-u^{r_{2}}_{\mu})(\cdot,t)\\|_{L^{\infty}}\leq\displaystyle C\delta t^{-1}\big{[}\ln(1+t)+\|\ln\delta\|\big{]}.\end{array}$ ## 3 Proof of Theorem 1.1 To prove Theorem 1.1, the solution $(\rho^{\epsilon},u^{\epsilon})$ is constructed as the perturbation around the approximate rarefaction wave $(\bar{\rho},\bar{u})$. Consider the Cauchy problem for (1.1) with smooth initial data $\displaystyle(\rho^{\epsilon},u^{\epsilon})(x,t=0)=(\bar{\rho},\bar{u})(x,0).$ (3.1) Then we introduce the perturbation $(\phi,\psi)(y,\tau)=(\rho^{\epsilon},u^{\epsilon})(x,t)-(\bar{\rho},\bar{u})(x,t),$ (3.2) where $y,\tau$ are the scaled variables as $y=\frac{x}{\epsilon},\quad\tau=\frac{t}{\epsilon},$ (3.3) and $(\rho^{\epsilon},u^{\epsilon})$ is assumed to be the solution to the problem $(\ref{ns})$. For the simplicity of the notation, the superscription of $(\rho^{\epsilon},u^{\epsilon})$ will be omitted as $(\rho,u)$ from now on if there is no confusion of the notation. Substituting $(\ref{pert})$ and $(\ref{tao})$ into $(\ref{ns})$ and using the definition for $(\bar{\rho},\bar{u})$, we obtain $\displaystyle\phi_{\tau}+\rho\psi_{y}+u\phi_{y}=-f,$ (3.4) $\displaystyle\rho\psi_{\tau}+\rho u\psi_{y}+p^{\prime}(\rho)\phi_{y}-\psi_{yy}=-g,$ (3.5) $(\phi,\psi)(y,0)=0,$ (3.6) where $\left\\{\begin{array}[]{l}f={\bar{u}}_{y}\phi+{\bar{\rho}}_{y}\psi,\\\\[5.69054pt] \displaystyle g=-\bar{u}_{yy}+\rho\psi\bar{u}_{y}+\bar{\rho}_{y}\Big{[}p^{\prime}(\rho)-\frac{\rho}{\bar{\rho}}p^{\prime}(\bar{\rho})\Big{]}.\end{array}\right.$ (3.7) We seek a global (in time) solution $(\phi,\psi)$ to the reformulated problem $(\ref{mass})-(\ref{init})$. To this end, the solution space for $(\ref{mass})-(\ref{init})$ is defined by $\displaystyle X(0,\tau_{1})=\Big{\\{}(\phi,\psi)\Big{\|}$ $\displaystyle(\phi,\psi)\in C^{0}([0,\tau_{1}];H^{1}(\mathbf{R})),\quad\phi_{y}\in L^{2}(0,\tau_{1};L^{2}(\mathbf{R})),$ $\displaystyle\psi_{y}\in L^{2}(0,\tau_{1};H^{1}(\mathbf{R}))\Big{\\}}$ with $0<\tau_{1}\leq+\infty$. ###### Theorem 3.1. The problem $(\ref{mass})-(\ref{init})$ admits a unique global-in-time solution $(\phi,\psi)\in X(0,+\infty)$. Furthermore, there exist positive constants $\epsilon_{0}$ and $C$ independent of $\epsilon$, such that if $0<\epsilon\leq\epsilon_{0}$, then $\begin{array}[]{ll}&\displaystyle\sup_{\tau\in[0,+\infty]}\int_{\mathbf{R}}\Big{(}\bar{\rho}\psi^{2}+{\bar{\rho}}^{\gamma-2}\phi^{2}+\phi^{2}_{y}+\psi^{2}_{y}\Big{)}(\tau,y)dy\\\\[11.38109pt] +&\displaystyle\int^{+\infty}_{0}\int_{\mathbf{R}}\Big{[}\psi_{y}^{2}+{\bar{\rho}}^{\gamma-2}\bar{u}_{y}\phi^{2}+\bar{\rho}\bar{u}_{y}\psi^{2}+\bar{\rho}^{\gamma-3}\phi^{2}_{y}+\frac{\psi^{2}_{yy}}{\bar{\rho}}\Big{]}dyd\tau\leq C\epsilon^{(1/2-a)}\|\ln\epsilon\|^{-1/2}.\end{array}$ (3.8) Consequently, $\begin{array}[]{ll}\displaystyle\sup_{0\leq\tau\leq+\infty}\\|\phi(\cdot,\tau)\\|_{L^{\infty}}\leq\left\\{\begin{array}[]{ll}\displaystyle C\epsilon^{1/6}\|\ln\epsilon\|^{-1/4},{}{}{}&{\rm if}~{}~{}~{}1<\gamma\leq 2,\\\ \displaystyle C\epsilon^{\frac{1}{\gamma+4}}\|\ln\epsilon\|^{(1-\gamma)/4},{}{}{}&{\rm if}~{}~{}~{}\gamma>2,\end{array}\right.\\\\[14.22636pt] \displaystyle\sup_{0\leq\tau\leq+\infty}\\|\psi(\cdot,\tau)\\|_{L^{\infty}}\leq\left\\{\begin{array}[]{ll}\displaystyle C\epsilon^{1/8}\|\ln\epsilon\|^{-1/2},{}{}&{\rm if}~{}~{}~{}1<\gamma\leq 2,\\\ \displaystyle C\epsilon^{\frac{\gamma+1}{4(\gamma+4)}}\|\ln\epsilon\|^{-1/2},{}{}&{\rm if}~{}~{}~{}\gamma>2.\end{array}\right.\end{array}$ (3.9) In what follows, the analysis is always carried out under the a priori assumptions $\displaystyle\sup_{0\leq\tau\leq\tau_{1}}\\|\phi(\cdot,\tau)\\|_{L^{\infty}}\leq\epsilon^{a},~{}~{}~{}\sup_{\tau\in[0,\tau_{1}]}\\|\psi_{y}\\|\leq 1,$ (3.10) with $a$ given by $\displaystyle a=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{6},&1<\gamma\leq 2,\\\ \displaystyle\frac{1}{\gamma+4},&\gamma>2.\end{array}\right.$ (3.13) Take $\mu=\epsilon^{a}\|\ln\epsilon\|,\qquad\delta=\epsilon^{a},$ (3.14) in the sequel. Then it follows that $\mu\geq 2\epsilon^{a}$ if $\epsilon\ll 1$. Under the a priori assumption (3.10), we can get $\displaystyle\frac{\bar{\rho}}{2}\leq\rho\leq\frac{3\bar{\rho}}{2}.$ (3.15) In fact, if $\epsilon\ll 1$, then $\displaystyle\rho=\bar{\rho}+\phi\geq\bar{\rho}-\\|\phi\\|_{L^{\infty}}\geq\bar{\rho}-\epsilon^{a}\geq\bar{\rho}-\frac{1}{2}\mu\geq\frac{\bar{\rho}}{2},$ (3.16) $\displaystyle\rho=\bar{\rho}+\phi\leq\bar{\rho}+\\|\phi\\|_{L^{\infty}}\leq\bar{\rho}+\epsilon^{a}\leq\bar{\rho}+\frac{1}{2}\mu\leq\frac{3\bar{\rho}}{2}.$ (3.17) Moreover, under the a priori assumption (3.10), it holds that $\displaystyle C_{1}{\bar{\rho}}^{\gamma-2}\phi^{2}\leq p(\rho)-p(\bar{\rho})-p^{\prime}(\bar{\rho})\phi\leq C_{2}{\bar{\rho}}^{\gamma-2}\phi^{2}.$ (3.18) where $C_{1},C_{2}$ are positive constants independent of $\epsilon$. Since the proof for the local existence of the solution to $(\ref{mass})-(\ref{init})$ is standard, we omit it for brevity. To prove Theorem 3.1, it is sufficient to prove the following a priori estimates. ###### Lemma 3.2. (A priori estimates) Let $\gamma>1$ and $(\phi,\psi)\in X(0,\tau_{1})$ be a solution to the problem $(\ref{mass})-(\ref{init})$. Then under the a priori assumption (3.10), there exist positive constants $\epsilon_{0}$ and $C$ independent of $\epsilon$, such that if $0<\epsilon\leq\epsilon_{0}$, then $\begin{array}[]{ll}&\displaystyle\sup_{\tau\in[0,\tau_{1}]}\int_{\mathbf{R}}\Big{(}\bar{\rho}\psi^{2}+{\bar{\rho}}^{\gamma-2}\phi^{2}+\phi^{2}_{y}+\psi^{2}_{y}\Big{)}(\tau,y)dy\\\\[11.38109pt] +&\displaystyle\int^{\tau_{1}}_{0}\int_{\mathbf{R}}\Big{[}\psi_{y}^{2}+{\bar{\rho}}^{\gamma-2}\bar{u}_{y}\phi^{2}+\bar{\rho}\bar{u}_{y}\psi^{2}+\bar{\rho}^{\gamma-3}\phi^{2}_{y}+\frac{\psi^{2}_{yy}}{\bar{\rho}}\Big{]}dyd\tau\leq C\epsilon^{(1/2-a)}\|\ln\epsilon\|^{-1/2}.\end{array}$ (3.19) Consequently, $\begin{array}[]{ll}\displaystyle\sup_{0\leq\tau\leq\tau_{1}}\\|\phi(\cdot,\tau)\\|_{L^{\infty}}\leq\left\\{\begin{array}[]{ll}\displaystyle C\epsilon^{1/6}\|\ln\epsilon\|^{-1/4},{}{}{}&{\rm if}~{}~{}~{}1<\gamma\leq 2,\\\ \displaystyle C\epsilon^{\frac{1}{\gamma+4}}\|\ln\epsilon\|^{(1-\gamma)/4},{}{}{}&{\rm if}~{}~{}~{}\gamma>2,\end{array}\right.\\\\[14.22636pt] \displaystyle\sup_{0\leq\tau\leq\tau_{1}}\\|\psi(\cdot,\tau)\\|_{L^{\infty}}\leq\left\\{\begin{array}[]{ll}\displaystyle C\epsilon^{1/8}\|\ln\epsilon\|^{-1/2},{}{}&{\rm if}~{}~{}~{}1<\gamma\leq 2,\\\ \displaystyle C\epsilon^{\frac{\gamma+1}{4(\gamma+4)}}\|\ln\epsilon\|^{-1/2},{}{}&{\rm if}~{}~{}~{}\gamma>2.\end{array}\right.\end{array}$ (3.20) Proof of Lemma 3.2: The proof of Lemma 3.2 consists of the following steps. Step 1. First, define $E:=\Phi(\rho,\bar{\rho})+\frac{\psi^{2}}{2},$ where $\Phi(\rho,\bar{\rho}):=\int^{\rho}_{\bar{\rho}}\frac{p(\xi)-p(\bar{\rho})}{\xi^{2}}d\xi=\frac{1}{(\gamma-1)\rho}\big{(}p(\rho)-p(\bar{\rho})-p^{\prime}(\bar{\rho})\phi\big{)}.$ (3.21) Direct computations yield $\begin{array}[]{ll}&\Big{(}\rho E\Big{)}_{\tau}+\Big{[}\rho uE-\psi_{y}\psi+\big{(}p(\rho)-p(\bar{\rho})\big{)}\psi\Big{]}_{y}\\\\[8.53581pt] &+\psi_{y}^{2}+\bar{u}_{y}\Big{(}p(\rho)-p(\bar{\rho})-p^{\prime}(\bar{\rho})\phi\Big{)}+\psi^{2}\rho\bar{u}_{y}=\bar{u}_{yy}\psi.\end{array}$ Then integrating the above equation over ${\mathbf{R}}^{1}\times[0,\tau]$ and using (3.15), (3.18) and (3.21) imply $\begin{array}[]{ll}&\displaystyle\int_{\mathbf{R}}\Big{(}\bar{\rho}\psi^{2}+{\bar{\rho}}^{\gamma-2}\phi^{2}\Big{)}dy+\int^{\tau}_{0}\int_{\mathbf{R}}\Big{(}\psi_{y}^{2}+{\bar{\rho}}^{\gamma-2}\bar{u}_{y}\phi^{2}+\bar{\rho}\bar{u}_{y}\psi^{2}\Big{)}dyd\tau\leq C\displaystyle\int^{\tau}_{0}\int_{\mathbf{R}}\|\bar{u}_{yy}\psi\|dyd\tau.\end{array}$ (3.22) By Sobolev inequality and Lemma 2.3, one has $\begin{array}[]{ll}&\displaystyle\int^{\tau}_{0}\int_{\mathbf{R}}\|\bar{u}_{yy}\psi\|dyd\tau\leq C\int^{\tau}_{0}\\|\bar{u}_{yy}\\|_{L^{1}}\\|\psi\\|^{1/2}\\|\psi_{y}\\|^{1/2}d\tau\\\\[11.38109pt] \leq&\displaystyle C\int^{\tau}_{0}\frac{1}{\tau+\delta/\epsilon}\\|\psi\\|^{1/2}\\|\psi_{y}\\|^{1/2}d\tau\\\ \leq&\displaystyle\frac{1}{8}\int^{\tau}_{0}\\|\psi_{y}\\|^{2}d\tau+C\int^{\tau}_{0}(\frac{1}{\tau+\delta/\epsilon})^{4/3}\\|\psi\\|^{2/3}d\tau\\\\[11.38109pt] \leq&\displaystyle\frac{1}{8}\int^{\tau}_{0}\\|\psi_{y}\\|^{2}d\tau+\frac{1}{8}\sup_{\tau\in[0,\tau_{1}]}\\|\sqrt{\bar{\rho}}\psi\\|^{2}+C\Big{(}\mu^{-1/3}\int^{\infty}_{0}(\frac{1}{\tau+\delta/\epsilon})^{4/3}d\tau\Big{)}^{3/2}\\\\[11.38109pt] \leq&\displaystyle\frac{1}{8}\int^{\tau}_{0}\\|\psi_{y}\\|^{2}d\tau+\frac{1}{8}\sup_{\tau\in[0,\tau_{1}]}\\|\sqrt{\bar{\rho}}\psi\\|^{2}+C\big{(}\frac{\epsilon}{\mu\delta}\big{)}^{1/2}.\end{array}$ (3.23) Combining (3.22) and (3.23) and recalling (3.14) yield $\begin{array}[]{ll}\displaystyle\sup_{\tau\in[0,\tau_{1}]}\int_{\mathbf{R}}\Big{(}\bar{\rho}\psi^{2}+{\bar{\rho}}^{\gamma-2}\phi^{2}\Big{)}(\tau,y)dy+\int^{\tau_{1}}_{0}\int_{\mathbf{R}}\Big{[}\psi_{y}^{2}+{\bar{\rho}}^{\gamma-2}\bar{u}_{y}\phi^{2}+\bar{\rho}\bar{u}_{y}\psi^{2}\Big{]}dyd\tau\\\ \displaystyle\hskip 256.0748pt\leq C\epsilon^{(1/2-a)}\|\ln\epsilon\|^{-1/2}.\end{array}$ (3.24) Step 2. We make use of the idea in [8] with modifications to derive the estimation of $\phi_{y}$. Differentiating $(\ref{mass})$ with respect to $y$ and then multiplying the resulted equation by $\phi_{y}/\rho^{3}$ to get $\begin{array}[]{ll}&\displaystyle(\frac{\phi_{y}^{2}}{2\rho^{3}})_{\tau}+(\frac{u\phi_{y}^{2}}{2\rho^{3}})_{y}+\frac{\psi_{yy}\phi_{y}}{\rho^{2}}=-\frac{\phi_{y}}{\rho^{3}}({\bar{u}}_{yy}\phi+{\bar{\rho}}_{yy}\psi+2{\bar{\rho}}_{y}\psi_{y}).\end{array}$ (3.25) Multiplying $(\ref{mome})$ by $\phi_{y}/\rho^{2}$ gives $\begin{array}[]{ll}&\displaystyle(\frac{\psi\phi_{y}}{\rho})_{\tau}-(\frac{\psi\phi_{\tau}}{\rho}+{\bar{\rho}}_{y}\frac{\psi^{2}}{\rho})_{y}-\psi_{y}^{2}+p^{\prime}(\rho)\frac{\phi_{y}^{2}}{\rho^{2}}-\frac{\psi_{yy}\phi_{y}}{\rho^{2}}\\\\[5.69054pt] &\displaystyle-{\bar{u}}_{y}\frac{\psi_{y}\phi}{\rho}+2{\bar{\rho}}_{y}\frac{\psi\psi_{y}}{\rho}+{\bar{\rho}}_{yy}\frac{\psi^{2}}{\rho}+{\bar{\rho}}_{y}{\bar{u}}_{y}\frac{\psi\phi}{\rho^{2}}-\bar{\rho}{\bar{u}}_{y}\frac{\psi\phi_{y}}{\rho^{2}}=-g\frac{\phi_{y}}{\rho^{2}}.\end{array}$ (3.26) Adding $(\ref{m})$ and $(\ref{v})$ together, then integrating the resulted equation over ${\mathbf{R}}^{1}\times[0,\tau]$ imply $\begin{array}[]{ll}&\displaystyle\int_{\mathbf{R}}\Big{(}\frac{\phi_{y}^{2}}{2\rho^{3}}+\frac{\psi\phi_{y}}{\rho}\Big{)}dy+\int^{\tau}_{0}\int_{\mathbf{R}}p^{\prime}(\rho)\frac{\phi_{y}^{2}}{\rho^{2}}dyd\tau\\\\[14.22636pt] =&\displaystyle\int^{\tau}_{0}\int_{\mathbf{R}}\Big{\\{}\psi_{y}^{2}+{\bar{u}}_{y}\frac{\psi_{y}\phi}{\rho}-2{\bar{\rho}}_{y}\frac{\psi\psi_{y}}{\rho}-{\bar{\rho}}_{yy}\frac{\psi^{2}}{\rho}-{\bar{\rho}}_{y}{\bar{u}}_{y}\frac{\psi\phi}{\rho^{2}}+\bar{\rho}{\bar{u}}_{y}\frac{\psi\phi_{y}}{\rho^{2}}\\\\[14.22636pt] &\displaystyle-\frac{\phi_{y}}{\rho^{3}}\Big{(}{\bar{u}}_{yy}\phi+{\bar{\rho}}_{yy}\psi+2{\bar{\rho}}_{y}\psi_{y}\Big{)}-g\frac{\phi_{y}}{\rho^{2}}\Big{\\}}dyd\tau.\end{array}$ (3.27) The combination of (3.24) and (3.27) leads to $\begin{array}[]{ll}&\displaystyle\int_{\mathbf{R}}\Big{(}\frac{\phi_{y}^{2}}{\bar{\rho}^{3}}+\bar{\rho}\psi^{2}+{\bar{\rho}}^{\gamma-2}\phi^{2}\Big{)}dy+\int^{\tau}_{0}\int_{\mathbf{R}}\Big{(}\psi_{y}^{2}+{\bar{\rho}}^{\gamma-2}\bar{u}_{y}\phi^{2}+\bar{\rho}\bar{u}_{y}\psi^{2}+\bar{\rho}^{\gamma-3}\phi_{y}^{2}\Big{)}dyd\tau\\\\[8.53581pt] &\displaystyle\leq C\int^{\tau}_{0}\int_{\mathbf{R}}\Big{\\{}\|{\bar{u}}_{y}\frac{\psi_{y}\phi}{\bar{\rho}}\|+\|{\bar{\rho}}_{y}\frac{\psi\psi_{y}}{\bar{\rho}}\|+\|{\bar{\rho}}_{yy}\frac{\psi^{2}}{\bar{\rho}}\|+\|{\bar{\rho}}_{y}{\bar{u}}_{y}\frac{\psi\phi}{\bar{\rho}^{2}}\|+\|{\bar{u}}_{y}\frac{\psi\phi_{y}}{\bar{\rho}}\|+\|{\bar{\rho}}_{y}\frac{\phi_{y}}{\bar{\rho}^{3}}\psi_{y}\|\\\\[8.53581pt] &\displaystyle\qquad\qquad\qquad+\|\frac{{\bar{u}}_{yy}}{\bar{\rho}^{3}}\phi_{y}\phi\|+\|\frac{{\bar{\rho}}_{yy}}{\bar{\rho}^{3}}\phi_{y}\psi\|+\|g\frac{\phi_{y}}{\bar{\rho}^{2}}\|\Big{\\}}dyd\tau+C\epsilon^{(1/2-a)}\|\ln\epsilon\|^{-1/2}\\\ &\displaystyle:=\sum_{i=1}^{9}I_{i}+C\epsilon^{(1/2-a)}\|\ln\epsilon\|^{-1/2}.\end{array}$ (3.28) Now we estimate the terms on the right hand side of (3.28) one by one. By Lemma 2.3, (3.15) and Cauchy’s inequality, it holds that $\begin{array}[]{ll}I_{1}&=\displaystyle\int^{\tau}_{0}\int_{\mathbf{R}}\|{\bar{u}}_{y}\frac{\psi_{y}\phi}{\bar{\rho}}\|dyd\tau\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\\|\psi_{y}\\|^{2}d\tau+C\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{\rho}}^{\gamma-2}{\bar{u}}_{y}\phi^{2}\frac{{\bar{u}}_{y}}{{\bar{\rho}}^{\gamma}}dyd\tau\\\\[8.53581pt] &\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\\|\psi_{y}\\|^{2}d\tau+C\mu^{-\gamma}\delta^{-1}\epsilon\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{\rho}}^{\gamma-2}{\bar{u}}_{y}\phi^{2}dyd\tau\\\\[5.69054pt] &\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\\|\psi_{y}\\|^{2}d\tau+\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{\rho}}^{\gamma-2}{\bar{u}}_{y}\phi^{2}dyd\tau\end{array}$ (3.29) where we have used the fact that $C\mu^{-\gamma}\delta^{-1}\epsilon=C\epsilon^{1-a-a\gamma}\|\ln\epsilon\|^{-\gamma}\leq C\epsilon^{\frac{1}{2}}\|\ln\epsilon\|^{-\gamma}\leq\frac{1}{8},\qquad{\rm if}~{}~{}\epsilon\ll 1.$ (3.30) From Lemma 2.3 (i), one has ${\bar{\rho}}_{y}={\bar{\rho}}^{\frac{3-\gamma}{2}}{\bar{u}}_{y},$ (3.31) one can get $\begin{array}[]{ll}I_{2}&\displaystyle=\int^{\tau}_{0}\int_{\mathbf{R}}\|{\bar{\rho}}_{y}\frac{\psi\psi_{y}}{\bar{\rho}}\|dyd\tau\\\\[8.53581pt] &\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\\|\psi_{y}\\|^{2}d\tau+C\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{\rho}}^{-\gamma}{\bar{u}}_{y}(\bar{\rho}{\bar{u}}_{y}\psi^{2})dyd\tau\\\\[8.53581pt] &\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\\|\psi_{y}\\|^{2}d\tau+C\mu^{-\gamma}\delta^{-1}\epsilon\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{\rho}}{\bar{u}}_{y}\psi^{2}dyd\tau\\\ &\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\\|\psi_{y}\\|^{2}d\tau+\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{\rho}}{\bar{u}}_{y}\psi^{2}dyd\tau\end{array}$ (3.32) where in the last inequality we used the fact (3.30). Recalling (2.16) from Lemma 2.1 and the fact $(i)$ in Lemma 2.3, one can arrive at $\displaystyle\begin{array}[]{ll}\|{\bar{\rho}}_{xx}\|&\displaystyle=\|\frac{2}{\gamma+1}{\bar{\rho}}^{\frac{3-\gamma}{2}}\omega^{r}_{\delta xx}+\frac{2(3-\gamma)}{(\gamma+1)^{2}}{\bar{\rho}}^{2-\gamma}(\omega^{r}_{\delta x})^{2}\|\\\ &\displaystyle\leq C\Big{(}{\bar{\rho}}^{\frac{3-\gamma}{2}}\frac{{\bar{u}}_{x}}{\delta}+{\bar{\rho}}^{2-\gamma}{\bar{u}}_{x}^{2}\Big{)}.\end{array}$ (3.35) Thus one has $\begin{array}[]{ll}I_{3}&\displaystyle=\int^{\tau}_{0}\int_{\mathbf{R}}\|{\bar{\rho}}_{yy}\frac{\psi^{2}}{\bar{\rho}}\|dyd\tau\\\\[8.53581pt] &\leq\displaystyle C\epsilon\delta^{-1}\int^{\tau}_{0}\int_{\mathbf{R}}\|\bar{\rho}{\bar{u}}_{y}\psi^{2}{\bar{\rho}}^{-\frac{\gamma+1}{2}}\|dyd\tau+C\epsilon\delta^{-1}\int^{\tau}_{0}\int_{\mathbf{R}}\|{\bar{\rho}}{\bar{u}}_{y}\psi^{2}{\bar{\rho}}^{-\gamma}\|dyd\tau\\\\[8.53581pt] &\leq\displaystyle C\mu^{-\gamma}\delta^{-1}\epsilon\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{\rho}}{\bar{u}}_{y}\psi^{2}dyd\tau\\\ &\leq\displaystyle\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{\rho}}{\bar{u}}_{y}\psi^{2}dyd\tau,\qquad{\rm if}~{}~{}\epsilon\ll 1.\end{array}$ (3.36) It follows from Lemma 2.3 and (3.31) that $\begin{array}[]{ll}I_{4}&=\displaystyle\int^{\tau}_{0}\int_{\mathbf{R}}\|{\bar{\rho}}_{y}{\bar{u}}_{y}\frac{\psi\phi}{\bar{\rho}^{2}}\|dyd\tau\\\\[8.53581pt] &\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\bar{\rho}{\bar{u}}_{y}\psi^{2}dyd\tau+C\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{u}}_{y}{\bar{\rho}}^{\gamma-2}\phi^{2}\frac{{\bar{\rho}}^{2}_{y}}{{\bar{\rho}}^{3+\gamma}}dyd\tau\\\\[11.38109pt] &\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\bar{\rho}{\bar{u}}_{y}\psi^{2}dyd\tau+C\frac{\epsilon^{2}}{\mu^{2\gamma}\delta^{2}}\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{\rho}}^{\gamma-2}{\bar{u}}_{y}\phi^{2}dyd\tau\\\\[11.38109pt] &\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\bar{\rho}{\bar{u}}_{y}\psi^{2}dyd\tau+\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{\rho}}^{\gamma-2}{\bar{u}}_{y}\phi^{2}dyd\tau.\end{array}$ (3.37) Similarly, it holds that $\begin{array}[]{ll}I_{5}&\displaystyle=\int^{\tau}_{0}\int_{\mathbf{R}}\|{\bar{u}}_{y}\frac{\psi\phi_{y}}{\bar{\rho}}\|dyd\tau\\\\[8.53581pt] &\displaystyle\leq\displaystyle\frac{1}{8}\int^{\tau}_{0}\\|\bar{\rho}^{\frac{\gamma-3}{2}}\phi_{y}\\|^{2}d\tau+C\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{\rho}}{\bar{u}}_{y}\psi^{2}\frac{{\bar{u}}_{y}}{{\bar{\rho}}^{\gamma}}dyd\tau\\\\[8.53581pt] &\leq\displaystyle\frac{1}{8}\int^{\tau}_{0}\\|\bar{\rho}^{\frac{\gamma-3}{2}}\phi_{y}\\|^{2}d\tau+C\frac{\epsilon}{\delta\mu^{\gamma}}\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{\rho}}{\bar{u}}_{y}\psi^{2}dyd\tau\\\\[8.53581pt] &\leq\displaystyle\frac{1}{8}\int^{\tau}_{0}\\|\bar{\rho}^{\frac{\gamma-3}{2}}\phi_{y}\\|^{2}d\tau+\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{\rho}}{\bar{u}}_{y}\psi^{2}dyd\tau.\end{array}$ (3.38) By Lemma 2.3, the equality (3.31) and Cauchy’s inequality, one has $\begin{array}[]{ll}I_{6}&\displaystyle=\int^{\tau}_{0}\int_{\mathbf{R}}\|{\bar{\rho}}_{y}\frac{\phi_{y}}{\bar{\rho}^{3}}\psi_{y}\|dyd\tau\\\\[8.53581pt] &\leq\displaystyle\frac{1}{8}\int^{\tau}_{0}\\|\bar{\rho}^{\frac{\gamma-3}{2}}\phi_{y}\\|^{2}d\tau+C\int^{\tau}_{0}\int_{\mathbf{R}}\frac{{\bar{u}}^{2}_{y}}{{\bar{\rho}}^{2\gamma}}\psi_{y}^{2}dyd\tau\\\\[11.38109pt] &\leq\displaystyle\frac{1}{8}\int^{\tau}_{0}\\|\rho^{(\gamma-3)/2}\phi_{y}\\|^{2}d\tau+C\frac{\epsilon^{2}}{\delta^{2}\mu^{2\gamma}}\int^{\tau}_{0}\int_{\mathbf{R}}\psi_{y}^{2}dyd\tau\\\\[11.38109pt] &\leq\displaystyle\frac{1}{8}\int^{\tau}_{0}\\|\rho^{(\gamma-3)/2}\phi_{y}\\|^{2}d\tau+\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\psi_{y}^{2}dyd\tau.\end{array}$ (3.39) Similarly, $I_{7}$ can be estimated as $\begin{array}[]{ll}I_{7}&\displaystyle=\int^{\tau}_{0}\int_{\mathbf{R}}\|\frac{{\bar{u}}_{yy}}{\bar{\rho}^{3}}\phi_{y}\phi\|dyd\tau\\\\[8.53581pt] &\leq\displaystyle\frac{1}{8}\int^{\tau}_{0}\\|\bar{\rho}^{\frac{\gamma-3}{2}}\phi_{y}\\|^{2}d\tau+C\frac{\epsilon}{\delta}\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{\rho}}^{\gamma-2}{\bar{u}}_{y}\phi^{2}\|{\bar{u}}_{yy}\|{\bar{\rho}}^{-1-2\gamma}dyd\tau\\\\[11.38109pt] &\leq\displaystyle\frac{1}{8}\int^{\tau}_{0}\\|\bar{\rho}^{\frac{\gamma-3}{2}}\phi_{y}\\|^{2}d\tau+C\frac{\epsilon^{3}}{\delta^{3}\mu^{1+2\gamma}}\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{u}}_{y}{\bar{\rho}}^{\gamma-2}\phi^{2}dyd\tau\\\\[11.38109pt] &\leq\displaystyle\frac{1}{8}\int^{\tau}_{0}\\|\bar{\rho}^{\frac{\gamma-3}{2}}\phi_{y}\\|^{2}d\tau+\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{u}}_{y}{\bar{\rho}}^{\gamma-2}\phi^{2}dyd\tau.\end{array}$ (3.40) It follows from (3.35) that $\begin{array}[]{ll}I_{8}&=\displaystyle\int^{\tau}_{0}\int_{\mathbf{R}}\|\frac{{\bar{\rho}}_{yy}}{\bar{\rho}^{3}}\phi_{y}\psi\|dyd\tau\\\ &\displaystyle\leq C\epsilon\delta^{-1}\int_{0}^{\tau}\int_{\mathbf{R}}\|\bar{\rho}^{-\frac{3+\gamma}{2}}\bar{u}_{y}\phi_{y}\psi\|dyd\tau+C\epsilon\delta^{-1}\int_{0}^{\tau}\int_{\mathbf{R}}\|\bar{\rho}^{-1-\gamma}\bar{u}_{y}\phi_{y}\psi\|dyd\tau\\\ &\leq\displaystyle\frac{1}{8}\int^{\tau}_{0}\\|\bar{\rho}^{\frac{\gamma-3}{2}}\phi_{y}\\|^{2}d\tau+C\epsilon^{2}\delta^{-2}\int^{\tau}_{0}\int_{\mathbf{R}}(\bar{\rho}^{-2\gamma}+\bar{\rho}^{1-3\gamma})\|\bar{u}_{y}\|^{2}\psi^{2}dyd\tau\\\\[8.53581pt] &\leq\displaystyle\frac{1}{8}\int^{\tau}_{0}\\|\bar{\rho}^{\frac{\gamma-3}{2}}\phi_{y}\\|^{2}d\tau+C\epsilon^{3}\delta^{-3}\mu^{-3\gamma}\int^{\tau}_{0}\int_{\mathbf{R}}\bar{\rho}\bar{u}_{y}\psi^{2}dyd\tau\\\ &\leq\displaystyle\frac{1}{8}\int^{\tau}_{0}\\|\bar{\rho}^{\frac{\gamma-3}{2}}\phi_{y}\\|^{2}d\tau+\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\bar{\rho}\bar{u}_{y}\psi^{2}dyd\tau,\qquad{\rm if}~{}~{}\epsilon\ll 1.\end{array}$ (3.41) Finally, one has $I_{9}=\displaystyle\int^{\tau}_{0}\int_{\mathbf{R}}\|g\frac{\phi_{y}}{\bar{\rho}^{2}}\|dyd\tau\leq\frac{1}{8}\int^{\tau}_{0}\\|\bar{\rho}^{\frac{\gamma-3}{2}}\phi_{y}\\|^{2}d\tau+C\int^{\tau}_{0}\int_{\mathbf{R}}\frac{g^{2}}{\bar{\rho}^{1+\gamma}}dyd\tau.$ (3.42) Recalling that (3.7), (3.15) and (3.31), one can get $\begin{array}[]{ll}\displaystyle\|g\|&\displaystyle\leq\|\bar{u}_{yy}\|+\|\bar{\rho}\bar{u}_{y}\psi\|+C\|\bar{\rho}^{\gamma-2}\bar{\rho}_{y}\phi\|\\\ &\displaystyle\leq\|\bar{u}_{yy}\|+\|\bar{\rho}\bar{u}_{y}\psi\|+C\|\bar{\rho}^{\frac{\gamma-1}{2}}\bar{u}_{y}\phi\|.\end{array}$ (3.43) Thus the last term in (3.42) can be estimated by $\begin{array}[]{ll}&\displaystyle\Big{\|}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{g^{2}}{\bar{\rho}^{1+\gamma}}dyd\tau\Big{\|}\\\\[8.53581pt] \leq&\displaystyle C\int^{\tau}_{0}\int_{\mathbf{R}}\frac{{\bar{u}}^{2}_{yy}}{{\bar{\rho}}^{1+\gamma}}dyd\tau+C\int^{\tau}_{0}\int_{\mathbf{R}}\bar{\rho}^{1-\gamma}{\bar{u}}^{2}_{y}\psi^{2}dyd\tau+\displaystyle C\int^{\tau}_{0}\int_{\mathbf{R}}{\bar{\rho}}^{-2}{\bar{u}}^{2}_{y}\phi^{2}dyd\tau\\\\[8.53581pt] \leq&\displaystyle C\frac{\epsilon^{3}}{\mu^{1+\gamma}}\int_{0}^{\tau}\\|\bar{u}_{xx}\\|_{L^{2}(dx)}^{2}d\tau+C\frac{\epsilon}{\delta\mu^{\gamma}}\int^{\tau}_{0}\int_{\mathbf{R}}\big{(}\bar{\rho}{\bar{u}}_{y}\psi^{2}+{\bar{\rho}}^{\gamma-2}{\bar{u}}_{y}\phi^{2}\big{)}dyd\tau\\\\[8.53581pt] \leq&\displaystyle C\frac{\epsilon}{\delta\mu^{1+\gamma}}\int_{0}^{\tau}(\tau+\frac{\delta}{\epsilon})^{-2}d\tau+C\frac{\epsilon}{\delta\mu^{\gamma}}\int^{\tau}_{0}\int_{\mathbf{R}}\big{(}\bar{\rho}{\bar{u}}_{y}\psi^{2}+{\bar{u}}_{y}{\bar{\rho}}^{\gamma-2}\phi^{2}\big{)}dyd\tau\\\\[8.53581pt] \leq&\displaystyle C\frac{\epsilon^{2}}{\delta^{2}\mu^{1+\gamma}}+\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\big{(}\bar{\rho}{\bar{u}}_{y}\psi^{2}+{\bar{u}}_{y}{\bar{\rho}}^{\gamma-2}\phi^{2}\big{)}dyd\tau,\qquad{\rm if}~{}~{}\epsilon\ll 1.\end{array}$ (3.44) Substituting (3.29)-(3.44) into (3.28) and recalling (3.14), it holds that $\begin{array}[]{ll}\displaystyle\int_{\mathbf{R}}\Big{(}\bar{\rho}\psi^{2}+{\bar{\rho}}^{\gamma-2}\phi^{2}+\phi_{y}^{2}\Big{)}dy+\int^{\tau}_{0}\int_{\mathbf{R}}\Big{(}\psi_{y}^{2}+{\bar{\rho}}^{\gamma-2}\bar{u}_{y}\phi^{2}+\bar{\rho}\bar{u}_{y}\psi^{2}+\bar{\rho}^{\gamma-3}\phi_{y}^{2}\Big{)}dyd\tau\\\\[11.38109pt] \leq\displaystyle C\epsilon^{(1/2-a)}\|\ln\epsilon\|^{-1/2},\qquad{\rm if}~{}~{}\epsilon\ll 1.\end{array}$ (3.45) Step 3. As the last step, we estimate $\displaystyle\sup_{\tau\in[0,\tau_{1}]}\\|\psi_{y}\\|$. For this, multiplying $(\ref{mome})$ by $-\psi_{yy}/\rho$ gives $(\frac{\psi^{2}_{y}}{2})_{\tau}-(\psi_{y}\psi_{\tau}+u\frac{\psi^{2}_{y}}{2})_{y}+u_{y}\frac{\psi_{y}^{2}}{2}-p^{\prime}(\rho)\frac{\phi_{y}\psi_{yy}}{\rho}+\frac{\psi_{yy}^{2}}{\rho}=g\frac{\psi_{yy}}{\rho}.$ (3.46) Integrating the above equation over ${\mathbf{R}}^{1}\times[0,\tau]$ yields $\begin{array}[]{ll}&\displaystyle\int_{\mathbf{R}}\frac{\psi^{2}_{y}}{2}dy+\int^{\tau}_{0}\int_{\mathbf{R}}\Big{(}\frac{{\bar{u}}_{y}\psi_{y}^{2}}{2}+\frac{\psi_{yy}^{2}}{\rho}\Big{)}dyd\tau=\int^{\tau}_{0}\int_{\mathbf{R}}\Big{\\{}p^{\prime}(\rho)\frac{{\psi}_{yy}\phi_{y}}{\rho}+g\frac{\psi_{yy}}{\rho}-\frac{\psi_{y}^{3}}{2}\Big{\\}}dyd\tau.\end{array}$ (3.47) First, one has $\begin{array}[]{ll}&\displaystyle\Big{\|}\int^{\tau}_{0}\int_{\mathbf{R}}p^{\prime}(\rho)\frac{{\psi}_{yy}\phi_{y}}{\rho}dyd\tau\Big{\|}\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\psi_{yy}^{2}}{\rho}dyd\tau+C\int^{\tau}_{0}\\|\bar{\rho}^{\frac{\gamma-3}{2}}\phi_{y}\\|^{2}d\tau.\end{array}$ (3.48) Then it follows from (3.44) and (3.45) that $\begin{array}[]{rl}\displaystyle\Big{\|}\int^{\tau}_{0}\int_{\mathbf{R}}g\frac{\psi_{yy}}{\rho}dyd\tau\Big{\|}\leq&\displaystyle\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\psi^{2}_{yy}}{\rho}dyd\tau+C\Big{\|}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{g^{2}}{\bar{\rho}}dyd\tau\Big{\|}\\\\[8.53581pt] \leq&\displaystyle\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\psi^{2}_{yy}}{\rho}dyd\tau+C\epsilon^{1/2-a}\|\ln\epsilon\|^{-1/2}.\end{array}$ (3.49) Furthermore, we can compute that $\begin{array}[]{ll}\displaystyle\Big{\|}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\psi_{y}^{3}}{2}dyd\tau\Big{\|}&\displaystyle\leq C\int^{\tau}_{0}\\|\psi_{yy}\\|^{\frac{1}{2}}\\|\psi_{y}\\|^{\frac{5}{2}}d\tau\\\ &\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\psi_{yy}^{2}}{\rho}dyd\tau+C\int^{\tau}_{0}\\|\psi_{y}\\|^{\frac{10}{3}}d\tau\\\ &\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\psi_{yy}^{2}}{\rho}dyd\tau+C\sup_{\tau\in[0,\tau_{1}]}\\|\psi_{y}\\|^{\frac{4}{3}}\int^{\tau}_{0}\\|\psi_{y}\\|^{2}d\tau\\\ &\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\psi_{yy}^{2}}{\rho}dyd\tau+C\int^{\tau}_{0}\\|\psi_{y}\\|^{2}d\tau,\end{array}$ (3.50) where in the last inequality we have used the a priori assumption $\sup_{\tau\in[0,\tau_{1}]}\\|\psi_{y}\\|\leq 1.$ (3.51) Substituting (3.48), (3.49) and (3.50) into (3.47) and using (3.15) and (3.45), it holds that $\begin{array}[]{ll}&\displaystyle\int_{\mathbf{R}}\psi_{y}^{2}dy+\int^{\tau}_{0}\int_{\mathbf{R}}\Big{(}{\bar{u}}_{y}\psi_{y}^{2}+\frac{\psi_{yy}^{2}}{\bar{\rho}}\Big{)}dyd\tau\leq\displaystyle C\epsilon^{(1/2-a)}\|\ln\epsilon\|^{-1/2}.\end{array}$ (3.52) Therefore, (3.19) can be derived directly from (3.45) and (3.52) and the a priori assumption (3.51) is verified if $\epsilon$ is suitably small. It follows from (3.19) that if $1<\gamma\leq 2$, then $\begin{array}[]{ll}\displaystyle\sup_{0\leq\tau\leq\tau_{1}}\\|\phi(\cdot,\tau)\\|_{L^{\infty}}&\displaystyle\leq\sqrt{2}\sup_{0\leq\tau\leq\tau_{1}}\\|\phi(\cdot,\tau)\\|^{1/2}\\|\phi_{y}(\cdot,\tau)\\|^{1/2}\\\ &\displaystyle\leq C\sup_{0\leq\tau\leq\tau_{1}}\big{(}\int_{\mathbf{R}}\bar{\rho}^{\gamma-2}\phi^{2}dy\big{)}^{\frac{1}{4}}\big{(}\int_{\mathbf{R}}\phi_{y}^{2}dy\big{)}^{\frac{1}{4}}\\\ &\displaystyle\leq C\epsilon^{1/6}\|\ln\epsilon\|^{-1/4},\end{array}$ (3.53) and if $\gamma>2$, then $\begin{array}[]{ll}\displaystyle\sup_{0\leq\tau\leq\tau_{1}}\\|\phi(\cdot,\tau)\\|_{L^{\infty}}&\displaystyle\leq\sqrt{2}\sup_{0\leq\tau\leq\tau_{1}}\\|\phi(\cdot,\tau)\\|^{1/2}\\|\phi_{y}(\cdot,\tau)\\|^{1/2}\\\ &\displaystyle\leq C\sup_{0\leq\tau\leq\tau_{1}}\big{(}\int_{\mathbf{R}}\mu^{2-\gamma}\bar{\rho}^{\gamma-2}\phi^{2}dy\big{)}^{\frac{1}{4}}\big{(}\int_{\mathbf{R}}\phi_{y}^{2}dy\big{)}^{\frac{1}{4}}\\\ &\displaystyle\leq C\mu^{\frac{1}{2}-\frac{\gamma}{4}}\epsilon^{\frac{1}{4}-\frac{1}{2(\gamma+4)}}\|\ln\epsilon\|^{-\frac{1}{4}}\\\ &\displaystyle\leq C\epsilon^{\frac{1}{\gamma+4}}\|\ln\epsilon\|^{\frac{1-\gamma}{4}}.\end{array}$ (3.54) And we also have $\sup_{\tau\in[0,\tau_{1}]}\\|\psi_{y}\\|\leq C\epsilon^{(\frac{1}{4}-\frac{a}{2})}\|\ln\epsilon\|^{-1/4}\leq 1.$ (3.55) So from (3.53)-(3.55) the a priori assumption (3.10) is verified if $\epsilon\ll 1$. On the other hand, by using Sobolev inequality, one can get $\begin{array}[]{ll}\displaystyle\sup_{0\leq\tau\leq\tau_{1}}\\|\psi(\cdot,\tau)\\|_{L^{\infty}}&\displaystyle\leq\sqrt{2}\sup_{0\leq\tau\leq\tau_{1}}\\|\psi(\cdot,\tau)\\|^{1/2}\\|\psi_{y}(\cdot,\tau)\\|^{1/2}\\\\[8.53581pt] &\displaystyle\leq\sqrt{2}\mu^{-\frac{1}{4}}\sup_{0\leq\tau\leq\tau_{1}}\\|\sqrt{\bar{\rho}}\psi(\cdot,\tau)\\|^{1/2}\\|\psi_{y}(\cdot,\tau)\\|^{1/2}\\\ &\displaystyle\leq\left\\{\begin{array}[]{ll}\displaystyle C\epsilon^{1/8}\|\ln\epsilon\|^{-1/2},{}{}&{\rm if}~{}~{}~{}1<\gamma\leq 2,\\\ \displaystyle C\epsilon^{\frac{\gamma+1}{4(\gamma+4)}}\|\ln\epsilon\|^{-1/2},{}{}&{\rm if}~{}~{}~{}\gamma>2,\end{array}\right.\end{array}$ (3.56) Thus the convergence rate (3.20) is justified and the proof of Lemma 3.2 is completed. $\Box$ Proof of Theorem 1.1: It remains to prove (1.7) with $a,b$ given in (1.8) and (1.9) respectively. From Lemma 2.2, Lemma 2.3 (iii) and Theorem 3.1 and recalling that $\mu=\epsilon^{a}\|\ln\epsilon\|,~{}~{}\delta=\epsilon^{a}$, it holds that for any given positive constant $h$, there exist a constant $C_{h}>0$ which is independent of $\epsilon$ such that $\begin{array}[]{ll}&\displaystyle\sup_{t\geq h}\\|\rho(\cdot,t)-\rho^{r}(\frac{\cdot}{t})\\|_{L^{\infty}}\\\ \leq&\displaystyle\sup_{0\leq\tau\leq+\infty}\\|\phi(\cdot,\tau)\\|_{L^{\infty}}+\sup_{t\geq h}\\|\bar{\rho}(\cdot,t)-\rho^{r}_{\mu}(\frac{\cdot}{t})\\|_{L^{\infty}}+\sup_{t\geq h}\\|\rho^{r}_{\mu}(\frac{\cdot}{t})-\rho^{r}(\frac{\cdot}{t})\\|_{L^{\infty}}\\\ \leq&\displaystyle\left\\{\begin{array}[]{ll}\displaystyle C_{h}\Big{(}\epsilon^{1/6}\|\ln\epsilon\|^{-1/4}+\delta\|\ln\delta\|+\mu\Big{)},{}{}&{\rm if}~{}~{}~{}1<\gamma\leq 2,\\\ \displaystyle C_{h}\Big{(}\epsilon^{\frac{1}{\gamma+4}}\|\ln\epsilon\|^{(1-\gamma)/4}+\delta\|\ln\delta\|+\mu\Big{)},{}{}&{\rm if}~{}~{}~{}\gamma>2,\end{array}\right.\\\\[5.69054pt] \leq&\displaystyle C_{h}\epsilon^{a}\|\ln\epsilon\|,\end{array}$ and $\begin{array}[]{ll}&\displaystyle\sup_{t\geq h}\\|m(\cdot,t)-m^{r}(\frac{\cdot}{t})\\|_{L^{\infty}}\\\ \leq&\displaystyle\sup_{t\geq h}\Big{(}\\|m(\cdot,t)-\bar{m}(\cdot,t)\\|_{L^{\infty}}+\\|\bar{m}(\cdot,t)-m^{r}_{\mu}(\frac{\cdot}{t})\\|_{L^{\infty}}+\\|m^{r}_{\mu}(\frac{\cdot}{t})-m^{r}(\frac{\cdot}{t})\\|_{L^{\infty}}\Big{)}\\\ \leq&\displaystyle C\sup_{0\leq\tau\leq+\infty}\Big{(}\\|\psi(\cdot,\tau)\\|_{L^{\infty}}+\\|\phi(\cdot,\tau)\\|_{L^{\infty}}\Big{)}\\\ &\displaystyle\qquad\qquad\qquad\qquad+\sup_{t\geq h}\Big{(}\\|\bar{m}(\cdot,t)-m^{r}_{\mu}(\frac{\cdot}{t})\\|_{L^{\infty}}+\\|m^{r}_{\mu}(\frac{\cdot}{t})-m^{r}(\frac{\cdot}{t})\\|_{L^{\infty}}\Big{)}\\\ \leq&\displaystyle\left\\{\begin{array}[]{ll}\displaystyle C_{h}\Big{(}\epsilon^{1/8}\|\ln\epsilon\|^{-1/2}+\epsilon^{1/6}\|\ln\epsilon\|^{-1/4}+\delta\|\ln\delta\|+\mu\Big{)},{}{}&{\rm if}~{}~{}~{}1<\gamma\leq 2,\\\ \displaystyle C_{h}\Big{(}\epsilon^{\frac{\gamma+1}{4(\gamma+4)}}\|\ln\epsilon\|^{-1/2}+\epsilon^{\frac{1}{\gamma+4}}\|\ln\epsilon\|^{(1-\gamma)/4}+\delta\|\ln\delta\|+\mu\Big{)},{}{}&{\rm if}~{}~{}~{}\gamma>2,\end{array}\right.\\\\[17.07164pt] \leq&\displaystyle\left\\{\begin{array}[]{ll}\displaystyle C_{h}\epsilon^{b}\|\ln\epsilon\|^{-\frac{1}{2}},{}{}{}&{\rm if}~{}~{}~{}1<\gamma<3,\\\ \displaystyle C_{h}\epsilon^{\frac{1}{\gamma+4}}\|\ln\epsilon\|,{}{}{}&{\rm if}~{}~{}~{}\gamma\geq 3,\end{array}\right.\end{array}$ Thus the proof of Theorem 1.1 is completed. $\Box$ ## Acknowledgments The authors would like to thank the referees for the valuable comments and suggestions which greatly improved the presentation of the paper. The research of F. M. Huang was supported in part by NSFC Grant No. 10825102 for distinguished youth scholar, and National Basic Research Program of China (973 Program) under Grant No. 2011CB808002. The research of Y. Wang was supported by the NSFC Grant No. 10801128. ## References * [1] S. Bianchini, A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2), 161 (2005), pp. 223-342. * [2] D. Bresch, P. Noble, Mathematical derivation of viscous shallow-water equations with zero surface tension, Preprint, 2010. * [3] G. Q. Chen, M. Perepelitsa, Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow, Comm. Pure Appl. Math., 63, (2010), pp. 1469-1504. * [4] J. F. Gerbeau, B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), pp. 89-102. * [5] J. Goodman, Z. P. 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Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), pp. 1-13. * [16] M. Perepelitsa, Asymptotics toward rarefaction waves and vacuum for 1-D compressible Navier-Stokes equations, SIAM J. Math. Anal., 42 (2010), pp. 1404-1412. * [17] J. Smoller, Shock Waves and Reaction-Diffusion Equations. 2nd ed. Grundlehren der Mathematischen Wissenschaften. 258. New York: Springer-Verlag, xxii, 1994. * [18] H. Y. Wang, Viscous limits for piecewise smooth solutions of the p-system, J. Math. Anal. Appl., 299 (2004), pp. 411-432. * [19] Y. Wang, Zero dissipation limit of the compressible heat-conducting Navier-Stokes equations in the presence of the shock, Acta Mathematica Scientia, 28B (2008), pp. 727-748. * [20] Z. P. Xin, Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases, Comm. Pure Appl. Math., 46 (1993), pp. 621-665. * [21] Z. P. Xin, H. H. Zeng, Convergence to the rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations, J. Diff. Eqs., 249 (2010), pp. 827-871. * [22] S. H. Yu, Zero-dissipation limit of solutions with shocks for systems of hyperbolic conservation laws, Arch. Ration. Mech. Anal., 146 (1999), pp. 275-370.	arxiv-papers	2010-11-09T07:29:11	2024-09-04T02:49:14.631724	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Feimin Huang, Mingjie Li and Yi Wang", "submitter": "Yi Wang", "url": "https://arxiv.org/abs/1011.1991" }
1011.2028	11institutetext: Laboratoire d’Astrophysique de Grenoble (LAOG), Université Joseph-Fourier, UMR 5571 CNRS, BP 53, 38041 Grenoble Cedex 09, France 22institutetext: Centro de astrofísica e Faculdade de Ciências, Universidade do Porto, Portugal 33institutetext: Laboratoire Fizeau, OCA/UNS/CNRS UMR6525, Parc Valrose, 06108 Nice cedex 2, France 44institutetext: Université de Lyon, Lyon, F-69003, France; Université Lyon 1, Observatoire de Lyon, 9 avenue Charles André, Saint Genis Laval, F-69230; CNRS, UMR 5574, Centre de Recherche Astrophysique de Lyon; Ecole Normale Supérieure, Lyon, F-69007, France 55institutetext: Georgia State University, P.O. Box 3969, Atlanta GA 30302-3969, USA 66institutetext: CHARA Array, Mount Wilson Observatory, 91023 Mount Wilson CA, USA # The fundamental parameters of the roAp star $\gamma$ Equulei K. Perraut 11 I. Brandão 22 D. Mourard 33 M. Cunha 22 Ph. Bério 33 D. Bonneau 33 O. Chesneau 33 J.M. Clausse 33 O. Delaa 33 A. Marcotto 33 A. Roussel 33 A. Spang 33 Ph. Stee 33 I. Tallon-Bosc 44 H. McAlister 5566 T. ten Brummelaar 66 J. Sturmann 66 L. Sturmann 66 N. Turner 66 C. Farrington and P.J. Goldfinger 6666 (Received …; accepted …) ###### Abstract Context. Physical processes working in the stellar interiors as well as the evolution of stars depend on some fundamental stellar properties, such as mass, radius, luminosity, and chemical abundances. The effective temperature, the surface gravity and the mean density are useful quantities defined from these fundamental properties. Additional physical quantities, like mass loss rate, pulsation period, rotation period, and magnetic field properties are interesting for the study of peculiar evolutionary stages. A classical way to test stellar interior models is to compare the predicted and observed location of a star on theoretical evolutionary tracks in a H-R diagram. This requires the best possible determinations of stellar mass, radius, luminosity and abundances. Aims. To directly and accurately determine its angular diameter and thus derive its fundamental parameters, we observed the well-known rapidly oscillating Ap star, $\gamma$ Equ, using the visible spectro-interferometer VEGA installed on the optical CHARA array. Methods. We recorded data on the W1W2 baseline of the CHARA array in the blue and in the red domains. We computed the calibrated squared visibility and derived the uniform-disk angular diameter and the limb-darkened one. We used the whole energy flux distribution, the parallax and the angular diameter to determine the luminosity and the effective temperature of the star. Results. We obtained a limb-darkened angular diameter of 0.564 $\pm$ 0.017 mas and deduced a radius of $R$ = 2.20 $\pm$ 0.12 ${\rm R_{\odot}}$. Without considering the multiple nature of the system, we derived a bolometric flux of $(3.12\pm 0.21)\times 10^{-7}$ erg cm-2 s-1 and an effective temperature of 7364 $\pm$ 235 K, which is below the effective temperature that has been previously determined. Under the same conditions we found a luminosity of $L$ = 12.8 $\pm$ 1.4 ${\rm L_{\odot}}$. When the contribution of the closest companion to the bolometric flux is considered, we found that the effective temperature and luminosity of the primary star can be, respectively, up to $\sim$ 100 K and up to $\sim$ 0.8 L⊙ smaller than the values mentioned above. Conclusions. For the first time, thanks to the unique capabilities of VEGA, we managed to constrain the angular diameter of a star as small as 0.564 mas with an accuracy of about 3%, and to derive its fundamental parameters. In particular the new values of the radius and effective temperature should bring further constraints on the asteroseismic modelling of the star. ###### Key Words.: Methods: observational - Techniques: high angular resolution - Techniques: interferometric - Stars: individual ($\gamma$Equ) - Stars: fundamental parameters ††offprints: Karine.Perraut@obs.ujf-grenoble.fr ## 1 Introduction Rapidly oscillating Ap (roAp) stars are chemically peculiar main-sequence stars that are characterized by strong and large-scale organized magnetic fields (typically of several kG, and up to 24 kG), abundance inhomogeneities leading to spotted surfaces, small rotational speeds, and pulsations with periods of a few minutes (see, Kochukhov (2009); Cunha (2007), for recent reviews). roAp stars are bright, pulsate with large amplitudes and in high radial orders. Thus they are particularly well-suited for asteroseismic campaigns and they contribute in a unique way to our understanding of the structure and evolution of stars. However, to put constraints on the interior chemical composition, the mixing length parameter, and the amount of convective overshooting, asteroseismic data should be combined with high precision stellar radii (Cunha et al. (2003, 2007)). This radius is generally estimated from the star’s luminosity and effective temperature. But systematic errors are likely to be present in this determination due to the abnormal surface layers of the Ap stars. This well known fact has been corroborated by seismic data on roAp stars (Matthews et al. (1999)), and compromises all asteroseismic results for this class of pulsators. Using long-baseline interferometry to provide accurate angular diameters appears to be a promising approach to overcome the difficulties in deriving accurate global parameters of roAp stars, but is also very challenging due to their small angular size. In fact, except for $\alpha$ Cir, whose diameter is about 1 millisecond of arc (mas) (Bruntt et al. (2008)), all roAp stars have angular diameters smaller than 1 mas. Such a small scale can be resolved only with optical or near- infrared interferometry. This was confirmed again recently by the interferometric study of the second largest (in angular size) roAp star known, namely $\beta$ CrB (Bruntt et al. (2010)). $\gamma$ Equ (HD201601 ; A9p ; $m_{V}$ = 4.7 ; $\pi_{P}$ = 27.55 $\pm$ 0.62 mas (van Leeuwen (2007)) ; v $\sin i\sim$ 10 km/s (Uesugi & Fukuda 1970)) is one of the brightest objects of the class of roAp stars with a period of about 12.3 min (Martinez et al. 1996) in brightness as well as in radial velocity. Despite photometry and spectroscopy of its oscillations obtained over the past 25 years, the pulsation frequency spectrum of $\gamma$ Equ has remained poorly understood. High-precision photometry with the MOST satellite has led to unique mode identifications based on a best model (Gruberbauer et al. (2008)) using a mass of 1.74 $\pm$ 0.03 M⊙, an effective temperature of log $T_{\rm eff}$ = 3.882 $\pm$ 0.011 and a luminosity of log $L/{\rm L_{\odot}}$ = 1.10 $\pm$ 0.03 (Kochukhov & Bagnulo (2006)). As regards to abundance inhomogeneities, Ryabchikova et al. (2002) considered the following stellar parameters ($T_{\rm eff}$ = 7700 K, log $g\leavevmode\nobreak\ =\leavevmode\nobreak\ $4.2, [M/H] = +0.5) to compute synthetic spectra and presented the evidence for abundance stratification in the atmosphere of $\gamma$ Equ: Ca, Cr, Fe, Ba, Si, Na seem to be overabundant in deeper atmospheric layers, but normal to underabundant in the upper layers, which according to the authors agrees well with diffusion theory for Ca and Cr, developed for cool magnetic stars with a weak mass loss of about 2.5 $\times 10^{-15}$ M⊙/yr. Pr and Nd from the rare earth elements have an opposite profile since their abundance is more than 6 dex higher in the upper layers than in the deeper atmospheric ones. Such abundance inhomogeneities clearly lead to a patchy surface, a redistribution of the stellar flux, and a complex atmospheric structure, resulting in biased photometric and spectroscopic determinations of the effective temperature. Guided by these considerations, we have observed $\gamma$ Equ with a spectro- interferometer operating at optical wavelengths, the VEGA spectrograph (Mourard et al. (2009)) installed at the CHARA Array (ten Brummelaar et al. (2005)). The unique combination of the visible spectral range of VEGA and the long baselines of CHARA has allowed us to record accurate squared visibilities at high spatial frequencies (Sect. 2). To derive the fundamental parameters of $\gamma$ Equ, calibrated spectra have been processed to estimate the bolometric flux and to determine the effective temperature (Sect. 3). Finally, we can set the star $\gamma$ Equ in the HR diagram and discuss the derived fundamental parameters (Sect. 4). ## 2 Interferometric observations and data processing ### 2.1 Data Data were collected at the CHARA Array with the VEGA spectropolarimeter recording spectrally dispersed fringes at visible wavelengths thanks to two photon-counting detectors. Two telescopes along the W1W2 baseline were combined. Observations were performed between 570 and 750 nm (according to the detector) at the medium spectral resolution of VEGA (R = 5000). Observations of $\gamma$ Equ were sandwiched with those of a nearby calibration star (HD 195810). The observation log is given in Table 1. Table 1: Journal of $\gamma$ Equ observations on July 29, and August 3 and 5, 2008. Date \| UT (h) \| Star \| B (m) \| PA (∘) ---\|---\|---\|---\|--- 2008-07-29 \| 5.59 \| HD 195810 \| 78.9 \| 106.6 2008-07-29 \| 6.08 \| $\gamma$ Equ \| 76.2 \| 106.4 2008-07-29 \| 6.41 \| HD 195810 \| 92.3 \| 101.9 2008-08-03 \| 8.64 \| HD 195810 \| 107.3 \| 93.0 2008-08-03 \| 8.98 \| $\gamma$ Equ \| 107.8 \| 93.8 2008-08-03 \| 9.31 \| HD 195810 \| 103.7 \| 91.0 2008-08-05 \| 7.68 \| HD 195810 \| 107.3 \| 108.8 2008-08-05 \| 8.14 \| $\gamma$ Equ \| 106.7 \| 95.8 2008-08-05 \| 8.63 \| HD 195810 \| 106.9 \| 92.6 Each set of data was composed of observations following a calibrator-star- calibrator sequence, with 10 files of 3000 short exposures of 15 ms per observation. Each data set was processed in 60 files of 500 short exposures using the $C_{1}$ estimator and the VEGA data reduction pipeline detailed in Mourard et al. (2009). The spectral separation between the two detectors is fixed by the optical design and equals about 170 nm in the medium spectral resolution. The red detector was centered around 750 nm on July 29 and around 640 nm on August 3 and 5. The blue detector was centered around 590 nm on July, 29 and around 470 nm on August 3 and 5. The bluer the wavelength, the more stringent the requirements on seeing. As a consequence the blue data on August 3 and 5 did not have a sufficient signal-to-noise ratio and squared visibilities could not be processed. All the squared visibilities are calibrated using an uniform-disk angular diameter of 0.29 $\pm$ 0.02 mas in the V and R bands for the calibrator HD 195810. This value is determined from the limb-darkened angular diameter provided by SearchCal111http://www.jmmc.fr/searchcal_page.htm(Table LABEL:tab:V2). Table 2: Calibrated squared visibilities of $\gamma$ Equ. Each visibility point corresponds to the average on the 60 blocks of 500 frames. UT (h) \| B (m) \| $\lambda_{0}$ (nm) \| $V^{2}$ ---\|---\|---\|--- 6.08 \| 76.1 \| 745.0 \| 0.84 $\pm$ 0.02 6.08 \| 76.2 \| 582.5 \| 0.72 $\pm$ 0.02 8.98 \| 107.6 \| 640.0 \| 0.62 $\pm$ 0.04 8.14 \| 106.7 \| 640.0 \| 0.61 $\pm$ 0.05 ### 2.2 Angular diameter determination $\gamma$ Equ is the brightest component of a multiple system. The closest component lies at 1.25” $\pm$ 0.04”, it has a magnitude difference with the primary star of $\Delta m$ = 4 and a position angle of PA = 264.6∘ $\pm$ 1.3∘ (Fabricius et al. (2002)). The entrance slit of the spectrograph (height=4” and width=0.2” for these observations) will affect the transmission of the companion flux. Taking into account the seeing during the observations (about 1”), the field rotation during the hour angle range of our observations ([-30∘ ; 0∘]), the position angle of the companion, we determine the throughput efficiency of the VEGA spectrograph slit for this companion. This efficiency varies from 10% for the longer baselines (around 107 m) to 30% for the smaller ones (around 80 m). We use the Visibility Modeling Tool (VMT)222http://www.nexsciweb.ip c.caltech.edu/vmt/vmtWeb to build a composite model including the companion of $\gamma$ Equ. For the longer baselines, the resulting modulation in the visibility is below 2%, which is 3 or 4 times below our accuracy on squared visibilities. We thus neglected the influence of the companion and interpreted our visibility data points in terms of angular diameter (Fig. 1). We performed model fitting with LITpro333http://www.jmmc.fr/litpro_page.htm. This fitting engine is based on a modified Levenberg-Marquardt algorithm combined with the trust regions method (Tallon-Bosc et al. (2008)). The software provides a user-expandable set of geometrical elementary models of the object, combinable as building blocks. The fit of the visibility curve versus spatial frequency leads to a uniform- disk angular diameter of 0.540 $\pm$ 0.016 mas for $\gamma$ Equ. We used the tables of Diaz-Cordoves et al. (1995) to determine the linear limb-darkening coefficient in the R band for 4.0 $\leq\log g\leq$ 4.5 and 7500 K $\leq T_{\rm eff}\leq$ 7750 K. By fixing this limb-darkening coefficient, LITPRO provides a limb-darkened angular diameter in the R band of $\theta_{LD}$ = 0.564 $\pm$ 0.017 mas with a reduced $\chi^{2}$ of 0.37. Figure 1: Squared visibility versus spatial frequency $u$ for $\gamma$ Equ obtained with the VEGA observations. The solid line represents the uniform- disk best model. ## 3 Bolometric flux and effective temperature The effective temperature, $T_{\rm eff}$, of a star can be obtained through the relation, $\sigma T_{\rm eff}^{4}=4f_{bol}/\theta_{LD}^{2},$ (1) where $\sigma$ stands for the Stefan-Boltzmann constant ($5.67\times\,10^{-5}$ erg cm-2 s-1 K-4), $\theta_{LD}$ for the limb-darkened angular diameter, and $f_{bol}$ is the star’s bolometric flux given by, $f_{bol}={\int\limits_{0}^{\infty}\,F(\lambda)\mathrm{d}\lambda}.$ (2) Thus, the effective temperature of $\gamma$ Equ can be computed if we know its angular diameter and its bolometric flux. The angular diameter of $\gamma$ Equ was derived in Sect. 2. To compute the bolometric flux we need a single spectrum that covers the whole wavelength range. This spectrum was obtained by combining photometric and spectroscopic data of $\gamma$ Equ available in the literature, together with ATLAS9 Kurucz models, in the way explained below. ### 3.1 Data We collected two rebinned high resolution spectra ($R$ = 18000 at $\lambda$ = 1400 Å, $R$ = 13000 at $\lambda$ = 2600 Å) from the Sky Survey Telescope obtained at the IUE “Newly Extracted Spectra” (INES) data archive444http://sdc.laeff.inta.es/cgi-ines/IUEdbsMY, covering the wavelength range [1850 Å ; 3350 Å]. The two spectra were obtained with the Long Wavelength Prime camera and the large aperture of 10” $\times$ 20” (Table 3). Based on the quality flag listed in the IUE spectra (Garhart et al. (1997)) we removed all bad pixels from the data, and we also removed the points with negative flux. The mean of the two spectra was then computed to obtain one single spectrum of $\gamma$ Equ in the range 1850 Å $<\lambda<$ 3350 Å. Table 3: UV spectra obtained with IUE. Image \| Date \| Starting time \| Exposure time ---\|---\|---\|--- Number \| \| (UT) \| (s) 06874 \| 08/10/1985 \| 18:55:04 \| 599.531 09159 \| 23/09/1986 \| 20:41:13 \| 539.730 We collected two spectra for $\gamma$ Equ in the visible, one from Burnashev (1985), which is a spectrum from Kharitonov et al. (1978) reduced to the uniform spectrophotometric system of the “Chilean Catalogue”, and one from Kharitonov et al. (1988). We verified that the latter was in better agreement with the Johnson (Morel & Magnenat (1978)) and the Geneva (Rufener (1988)) photometry than the other spectrum. To convert from Johnson and Geneva magnitudes to fluxes we used the calibrations given by Johnson (1966) and Rufener & Nicolet (1988), respectively. For the infrared, we collected the photometric data available in the literature. The calibrated observational photometric fluxes that we considered in this study are given in Table 4. Table 4: Calibrated photometric infrared fluxes for $\gamma$ Equ. Band \| $\lambda_{\rm eff}$ \| Flux \| Source \| Calibration ---\|---\|---\|---\|--- \| (Å) \| ($\times 10^{-12}\,\rm{erg\,cm^{-2}\,s^{-1}\,\AA^{-1}}$) \| \| I \| 9000 \| 15.53 \| 1 \| a J \| 12500 \| 5.949 \| 2 \| b H \| 16500 \| 2.420 \| 2 \| b K \| 22000 \| 0.912 \| 2 \| b L \| 36000 \| 0.140 \| 2 \| b M \| 48000 \| 0.0512 \| 2 \| b J \| 12350 \| 6.090 \| 3 \| c H \| 16620 \| 2.584 \| 3 \| c K \| 21590 \| 1.067 \| 3 \| c Source references: (1) Morel & Magnenat (1978); (2) Groote & Kaufmann (1983); (3) Cutri et al. (2003). Calibration references: (a) Johnson (1966); (b) Wamsteker (1981); (c) Cohen et al. (2003). Figure 2: The whole spectrum obtained for $\gamma$ Equ. Black line corresponds to the average of the IUE spectra and to the Kharitonov et al. (1988)’s spectrum. For wavelengths $\rm\lambda<1854$ Å and $\rm\lambda>7390$ Å , the figure shows the curve obtained using the interpolation method (dark grey line), the Kurucz model that best fits the spectroscopy in the visible and the photometry in the infrared when models are calibrated with the star’s magnitude $m_{V}$ (grey line) and when models are calibrated with the relation $(R/d)^{2}$ (light grey line). The Geneva and infrared photometry from Table 4 (circles) and Johnson UBVRI photometry (triangles) are overplotted to the spectrum. ### 3.2 $f_{bol}$ and $T_{\rm eff}$ determination The spectrum of $\gamma$ Equ was obtained by combining the averaged IUE spectrum between 1854 Å and 3220 Å, the Kharitonov’s (1988) spectrum from 3225 Å to 7375 Å, and, for wavelengths $\rm\lambda<1854$ Å and $\rm\lambda>7390$ Å we considered two cases: (1) we used the synthetic spectrum for the Kurucz model that best fitted both the star’s spectrum in the visible and the star’s photometry in the infrared and, (2) we performed a linear extrapolation between 506 Å and 1854 Å, considering zero flux at 506 Å, a second linear interpolation to the infrared fluxes between 7390 Å and 48000 Å, and a third linear extrapolation from 48000 Å and 1.6 $\times 10^{6}$ Å considering zero flux at 1.6 $\times 10^{6}$ Å. In case (1), when searching for the best Kurucz model we intentionally disregarded the data in the UV, because Kurucz models are particularly unsuitable for modeling that region of the spectra of roAp stars. To find the Kurucz model that best fitted the data in the visible and infrared we ran a grid of models, with different effective temperatures, surface gravities, and metallicities. Since Kurucz models needed to be calibrated (they give the flux of the star, not the value observed on Earth), we tried two different calibrations, namely: (i) the star’s magnitude in the $V$ band, $m_{V}$, (ii) the relation $(R/d)^{2}$, where $R$ is the radius and $d$ the distance to the star. For the $R/d=\theta/2$ we used the limb-darkened angular diameter $\theta_{LD}$ determined in the previous section. The final spectra obtained for $\gamma$ Equ with the two different calibration methods and with the interpolation method are plotted in Fig. 2. The bolometric flux, $f_{bol}$, was then computed from the integral of the spectrum of the star, through Eq. 2 and the effective temperature, $T_{\rm eff}$, was determined using Eq. 1 (Table 5). Table 5: Bolometric flux $f_{bol}$ and effective temperature $T_{\rm eff}$ obtained for $\gamma$ Equ, using three different methods (see text for details). Calibration method \| $f_{bol}$ (erg cm-2 s-1) \| $T_{\rm eff}$ (K) ---\|---\|--- $m_{V}$ \| $(3.09\pm 0.20)\times$10-7 \| 7351 $\pm$ 229 $(R/d)^{2}$ \| $(3.15\pm 0.21)\times$ 10-7 \| 7381 $\pm$ 234 Interpolation \| $(3.11\pm 0.21)\times$ 10-7 \| 7361 $\pm$ 235 The uncertainties in the three values of the bolometric flux given in Table 5 were estimated by considering an uncertainty of $10\%$ on the total flux from the combined IUE spectrum (González-Riestra et al. (2001)), an uncertainty of $4\%$ on the total flux of the low resolution spectrum from Kharitonov et al. (1988), an uncertainty of $20\%$ on the total flux derived from the Kurucz model, and an uncertainty of $20\%$ on the total flux derived from the interpolation. The latter two are somewhat arbitrary. Our attitude was one of being conservative enough to guarantee that the uncertainty in the total flux was not underestimated due to the difficulty in establishing these two values. The corresponding absolute errors were then combined to derive the errors in the flux which are shown in Table 5. Combining these with the uncertainty in the angular diameter, we derived the uncertainty in the individual values of the effective temperature. As a final result we take the mean of the three values and consider the uncertainty to be the largest of the three uncertainties. Thus, the flux and effective temperature adopted for $\gamma$ Equ are, respectively, ($3.12\pm 0.21$) $\times 10^{-7}$ erg cm-2 s-1 and 7364 $\pm$ 235 K. If, instead, we took for the effective temperature an uncertainty such as to enclose the three uncertainties, the result would be $T_{\rm eff}$ = 7364 $\pm$ 250 K. Figure 3: The position of $\gamma$ Equ in the Hertzsprung-Russell diagram. The constraints on the fundamental parameters are indicated by the 1$\sigma$-error box (log $T_{\rm eff}$, log ($L$/L⊙)) and the diagonal lines (radius). The box in solid lines corresponds to the results derived when ignoring the presence of the companion star. The box in dashed lines corresponds to the results derived after subtracting from the total bolometric flux the maximum contribution expected from the companion (see text for details). The box in dotted lines corresponds to the fundamental parameters derived by Kochukhov & Bagnulo (2006) and used by Gruberbauer et al. (2008) in the asteroseismic modelling of $\gamma$ Equ. ### 3.3 Contamination by the companion star In fact, since $\gamma$ Equ is a multiple system and the distance between the primary (hereafter, $\gamma$ Equ A) and the secondary (hereafter, $\gamma$ Equ B) is 1.25”, the bolometric flux of $\gamma$ Equ determined in Sect. 3 contains the contribution of both components. Given its magnitude, one may anticipate that the contribution of $\gamma$ Equ B to the total flux will be small. Although the data available in the literature for this component is very limited, we used them to estimate the impact of $\gamma$ Equ B’s contribution on our determination of the effective temperature of $\gamma$ Equ A. We collected the magnitudes $m_{B}$ = 9.85 $\pm$ 0.03 and $m_{V}$ = 8.69 $\pm$ 0.03 of $\gamma$ Equ B from Fabricius et al. (2002) and determined a value for its effective temperature using the color-$T_{\rm eff}$ calibration from Ramírez & Meléndez (2005). This was done assuming three different arbitrary values and uncertainties for the metallicity, namely $-0.4\pm 0.5$, $0\pm 0.5$ and $0.4\pm 0.5$ dex. The values found for the effective temperature were $T_{\rm eff}$ = 4570, 4686 and 4833 K, respectively, with an uncertainty of $\pm 40$K (Ramírez & Meléndez (2005)). The metallicity, the effective temperature, and the absolute V-band magnitude were used to estimate log $g$, using theoretical isochrones from Girardi et al. (2000)555http://stev.oapd.inaf.it/cgi-bin/param. For the three values of metallicities and $T_{\rm eff}$ mentioned above, we found log $g$ = 4.58, 4.53, and 4.51, respectively. With these parameters we computed three Kurucz models and calibrated each of them in three different ways: (i) using the H${}_{P}=9.054\pm 0.127$ magnitude (Perryman et al. (1997)), (ii) using the $m_{B}$ magnitude, and (iii) using the $m_{V}$ magnitude. To convert from Hipparcos/Tycho magnitudes into fluxes we used the zero points from Bessel & Castelli (private communication). The maximum flux found for $\gamma$ Equ B through the procedure described above was 0.19$\times 10^{-7}\,\rm{erg\,cm^{-2}\,s^{-1}}$, which corresponds to 6% of the total flux. This implies that the effective temperature of $\gamma$ Equ A determined in the previous section may be in excess by up to 111 K due to the contamination introduced by this companion star. ## 4 Discussion ### 4.1 Position in the HR-diagram We derive the radius of $\gamma$ Equ thanks to the formula: $\theta_{LD}=9.305R/d,$ (3) where $\theta_{LD}$ stands for the limb-darkened angular diameter (in mas), $R$ for the stellar radius (in solar radius, ${\rm R_{\odot}}$), and $d$ for the distance (in parsec). We obtain $R$ = 2.20 $\pm$ 0.12 ${\rm R_{\odot}}$. We use the bolometric flux $f_{bol}$ and the parallax $\pi_{P}$ to determine the $\gamma$ Equ’s luminosity from the relation: $L=4\pi f_{bol}\frac{C^{2}}{{\pi_{P}}^{2}},$ (4) where $C$ stands for the conversion factor from parsecs to meters. We obtain $L/{\rm L_{\odot}}$ = 12.8 $\pm$ 1.4 and can set $\gamma$ Equ in the HR diagram (Fig. 3). Recently, seismic data of $\gamma$ Equ obtained with the Canadian-led satellite MOST have been modeled by Gruberbauer et al. (2008) based on the fundamental parameters coming from Kochukhov & Bagnulo (2006) and using a grid of pulsation models including the effect of the magnetic field. A comparison of the HR diagram error-box considered by the authors (see dotted-line box in Figure 3) and our uncertainty regions shows that the regions are considerably different. In fact, even if we do not account for the contribution of the companion, we obtain a lower effective temperature with log $T_{\rm eff}$ = 3.867 $\pm$ 0.014 to be compared to log $T_{\rm eff}$ = 3.882 $\pm$ 0.011 from Gruberbauer et al. (2008). This discrepancy between the uncertainty regions increases if the companion contribution is taken into account. In that case, the overlap between the two regions is very small. As regards to luminosity, our calculation shows that for $\gamma$ Equ (as well as for $\alpha$ Cir) the contributions of the uncertainties in the bolometric flux and parallax to the uncertainty in $L/{\rm L_{\odot}}$ are comparable. This is quite different from the results obtained by Kochukhov & Bagnulo (2006) who found that the dominant contribution to the uncertainty in $L/{\rm L_{\odot}}$ comes from the parallax. The authors mentioned that the bolometric flux that was adopted in their work was that for normal stars. When dealing with peculiar stars, like Ap stars, it may be more adequate to properly compute the bolometric flux. However, it is precisely the difficulty in obtaining the full spectrum of the star that increases the uncertainty in the computed bolometric flux and, hence, in the luminosity and effective temperature. That is well illustrated by the following fact: if the somewhat arbitrary 20$\%$ uncertainties adopted in our work for the total fluxes derived from the Kurucz model and from the interpolation, were replaced by 5$\%$ uncertainties, we would obtain formal uncertainties in $L/{\rm L_{\odot}}$ and $T_{\rm eff}$ comparable and smaller, respectively, to those quoted by Kochukhov & Bagnulo (2006). ### 4.2 Bias due to stellar features We use the whole spectral energy density to determine the bolometric flux. We then deduce the effective temperature from this bolometric flux and the angular diameter. The determination of the angular diameter is based on visibility measurements that are directly linked to the Fourier Transform of the object intensity distribution. For a single circular star, the visibility curve as a function of spatial frequency B/$\lambda$ (where B stands for the interferometric baseline and $\lambda$ for the operating wavelength) is related to the first Bessel function, and contains an ever decreasing series of lobes, separated by nulls, as one observes with an increasing angular resolution. As a rule of thumb, the first lobe of the visibility curve (see Fig. 1 for an example) is sensitive to the size of the object only. As an example, for a star whose angular diameter equals 0.56 mas like $\gamma$ Equ, the difference in squared visibility between a uniform-disk and a limb- darkened one is of the order of 0.5% in the first lobe. The following lobes are sensitive to limb darkening and atmospheric structure but consist of very low visibilities. Finally, departure from circular symmetry (due to stellar spots from instance) requires either interferometric imaging by more than two telescopes or measurement close to the null. As a consequence, our interferometric data collected in the first part of the first lobe are only sensitive to the size of the target and cannot be used to study the potential complex structure of the atmosphere. ## 5 Conclusion Thanks to the unique capabilities of VEGA/CHARA, we present an accurate measurement of the limb-darkened angular diameter of a target as small as 0.564 $\pm$ 0.017 mas. In combination with our estimate of the bolometric flux based on the whole spectral energy density, we determine the effective temperature of $\gamma$ Equ A. Without considering the contribution of the closest companion star ($\gamma$ Equ B) to the bolometric flux, we found an effective temperature 7364 $\pm$ 235 K, which is below the effective temperature that has been previously determined. An estimate of that contribution leads to the conclusion that the above value may still be in excess by up to about 110 K, which increases further the discrepancy between the literature values for the effective temperature of $\gamma$ Equ A and the value derived here. The impact on the seismic analysis of considering the new values of the radius and effective temperature should be considered in future modeling of this star. More generally, this study illustrates the advantages of optical long-baseline interferometry for providing direct and accurate angular diameter measurements and motivates observations of other main-sequence stars to bring constraints on their evolutionary state and their internal structures. Within this context, the operation of VEGA in the visible is very complementary to the similar interferometric studies performed in the infrared range since it allows to study spectral types ranging from B to late-M and thus it opens the new window of the early spectral types (Mourard et al. (2009)). Another promising issue would be to use longer interferometric baselines to be sensitive to the stellar spots and bring constraints on the stellar surface features. ###### Acknowledgements. VEGA is a collaboration between CHARA and OCA/LAOG/CRAL/LESIA that has been supported by the French programs PNPS and ASHRA, by INSU and by the Région PACA. The project has obviously taken benefit from the strong support of the OCA and CHARA technical teams. The CHARA Array is operated with support from the National Science Foundation through grant AST-0908253, the W. M. Keck Foundation, the NASA Exoplanet Science Institute, and from Georgia State University. 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1011.2039	# An algorithm for determining copositive matrices111This research was supported by the National Natural Science Foundation of China(11001228,10901116). Jia Xu j.jia.xu@gmail.com Yong Yao yaoyong@casit.ac.cn College of Computer Science and Technology, Southwest University for Nationalities, Chengdu, Sichuan 610041, China Chengdu Institute of Computer Applications, Chinese Academy of Sciences, Chengdu, Sichuan 610041, China ###### Abstract In this paper, we present an algorithm of simple exponential growth called COPOMATRIX for determining the copositivity of a real symmetric matrix. The core of this algorithm is a decomposition theorem, which is used to deal with simplicial subdivision of $\widehat{T}^{-}=\\{y\in\Delta_{m}\|\beta^{T}y\leq 0\\}$ on the standard simplex $\Delta_{m}$, where each component of the vector $\beta$ is -1, 0 or 1. ###### keywords: copositive matrices , copositive quadratic forms , simplicial subdivision of convex polytope , complete algorithm ###### MSC: 15A48 , 15A57 , 15A63 , 65F30 ††journal: Linear Algebra and its Applications, 435(2011) 2784-2792 ## 1 Introduction QUESTION 1 Let $A$ be a given $n\times n$ real symmetric matrix, ${\mathbb{R}}_{+}$ be the set of nonnegative real numbers, and $Q(X)=X^{T}AX,\ X\neq 0$ be a quadratic form. What conditions should $A$ satisfy for $[\forall X\in{\mathbb{R}}^{n}_{+},\ Q(X)\geq 0\ (>0)]$? If $[\forall X\in{\mathbb{R}}^{n}_{+},\ Q(X)\geq 0\ (>0)]$, then the quadratic form $Q(X)$ is called a (strictly) copositive quadratic form and the corresponding matrix $A$ is called a (strictly) copositive matrix. Copositive matrices have numerous applications in diverse fields of applied mathematics, especially in mathematical programming and graph theory (see [3, 5, 7, 11, 13, 15, 16, 25, 26, 29, 39]). Therefore copositivity has been studied thoroughly since 1950s (see [1, 6, 14, 17, 20, 22, 23, 27, 28, 33, 34, 36, 41, 42, 43, 48, 49]). In general, it is an NP-complete problem to determine whether a given $n\times n$ symmetric matrix is not copositive [37, 38]. This means that every algorithm that solves the problem, in the worst case, will require at least an exponential number of operations, unless P=NP. For that reason, it is still valuable for the existence of so many incomplete algorithms discussing some special kinds of matrices (see [3, 4, 10, 12, 18, 19, 21, 24, 30, 38]). For small values of $n$($\leq 6$), some necessary and sufficient conditions have been constructed (see [1, 14, 30, 49]). From another viewpoint, QUESTION 1 is a typical real quantifier elimination problem [2, 8, 9, 32, 35, 40, 44, 45], which can be solved by standard tools of real quantifier elimination (e.g., using CAD) [2, 8, 9, 46, 47]. Thus, there is a complete algorithm for determining copositive matrices theoretically. Unfortunately, this algorithm is not efficient in practice for the CAD algorithm is of doubly exponential time complexity (see [2, 8, 9]). In this paper, we will construct a complete algorithm with singly exponential time bound. The standard simplex $\Delta_{m}(m\geq 2)$ is defined as the following set $\Delta_{m}=\\{(y_{1},\ldots,y_{m})^{T}\|\ y_{1}+\cdots+y_{m}=1,y_{1}\geq 0,\ldots,y_{m}\geq 0\\}.$ It is well known that the dimension of $\Delta_{m}$ is $m-1$. Denote the vertices of $\Delta_{m}$ as $e_{1},\ldots,e_{m}$, namely, $e_{1}=(1,0,\ldots,0)^{T},\ldots,e_{m}=(0,0,\ldots,1)^{T}$. Let $A\in{\mathbb{R}}^{n\times n}$ be symmetric and be partitioned as $A=[\alpha_{ij}]=\left[\begin{array}[]{cc}\alpha_{11}&\alpha^{T}\\\ \alpha&A_{2}\end{array}\right].$ Define $B=\alpha_{11}A_{2}-\alpha\alpha^{T}$. It is easy to see the following facts (cf.[1]) 1\. If $\alpha_{1i}\geq 0,\ i=2,\ldots,n$, then $A$ is (strictly) copositive $\Longleftrightarrow\alpha_{11}\geq 0\ (>0)$ and $A_{2}$ is (strictly) copositive. 2\. If at least one of $\alpha_{1i}$ is negative, then we need only to focus on the set of points $T^{-}=\\{y\in\Delta_{n-1}\|\ \alpha^{T}y\leq 0\\}$. It is well known that $T^{-}$ is a convex polytope on $\Delta_{n-1}$ (see [1]). The polytope $T^{-}$ can be subdivided into the simplices $S_{1},\ldots,S_{p}$, that is, $T^{-}=\bigcup_{i=1}^{p}S_{i},\ {\rm int}(S_{i})\bigcap{\rm int}(S_{j})=\emptyset,{\rm for}\ i\neq j,$ where ${\rm int}(S_{i})$ denotes the interior of simplex $S_{i}$. The coordinates of the vertices that span the simplex $S_{i}$ constitute a matrix denoted as $W_{i}$. Andersson et al. ([1], p.23) proved the following results. ###### Lemma 1.1 (a) $A$ is copositive iff $\alpha_{11}\geq 0$ and $A_{2},\ W_{1}^{T}BW_{1},\ \ldots,\ W_{p}^{T}BW_{p}$ are all copositive. (b) $A$ is strictly copositive iff $\alpha_{11}>0$ and $A_{2},\ W_{1}^{T}BW_{1},\ \ldots,\ W_{p}^{T}BW_{p}$ are all strictly copositive. In order to formulate the algorithm of Lemma 1.1, we first consider how to obtain the simplicial subdivision of the polytope $T^{-}=\\{y\in\Delta_{n-1}\|\ \alpha^{T}y\leq 0\\}$. For small values of $n(\leq 6)$, Andersson et al.[1] and Yang and Li [49] give the simplicial subdivision of $T^{-}$. However, they do not provide a procedure for a simplicial subdivision of $T^{-}$ for arbitrary values of n. We propose a simplicial subdivision of $T^{-}$ for all values of $n$, and consequently construct a complete algorithm for determining the copositivity of an $n\times n$ matrix. We will adopt a flexible approach. Rather than subdivide $T^{-}$ into simplices (of course our method is also valid for subdividing $T^{-}$ into simplices), we first transform the matrix $A$ into the following matrix called $\widehat{A}$. Let $\alpha=(\alpha_{12},\ldots,\alpha_{1n})^{T}$ and $D={\rm diag}(d_{2},\ldots,d_{n}),$ where $d_{i}=\left\\{\begin{array}[]{ll}1,&{\rm if}\ \alpha_{1i}=0;\\\ 1/\|\alpha_{1i}\|,&{\rm if}\ a_{1i}\neq 0.\end{array}\right.$ Then $\widehat{A}=\left[\begin{array}[]{cc}1&0\\\ 0&D\end{array}\right]A\left[\begin{array}[]{cc}1&0\\\ 0&D\end{array}\right]=\left[\begin{array}[]{cc}\alpha_{11}&\widehat{\alpha}^{T}\\\ \widehat{\alpha}&DA_{2}D\end{array}\right].$ (1) where $\widehat{\alpha}=({\rm sign}(\alpha_{12}),\ldots,{\rm sign}(\alpha_{1n}))^{T}$. Obviously, $A$ is (strictly) copositive $\Longleftrightarrow$$\widehat{A}$ is (strictly) copositive. Apply Lemma 1.1 to $\widehat{A}$. Let $\beta_{1}={\rm sign}(\alpha_{12}),\ldots,\ \beta_{n-1}={\rm sign}(\alpha_{1n}).$ Thus we just need to subdivide $\widehat{T}^{-}$ into simplices, where $\widehat{T}^{-}=\\{y\in\Delta_{n-1}\|(\beta_{1},\ldots,\beta_{n-1})y\leq 0,\ \beta_{i}\in\\{-1,0,1\\}\\}.$ Next we make further simplification: separate -1,0,1 from $\beta_{1},\ldots,\beta_{n-1}$, namely let $\begin{array}[]{c}\beta_{a_{1}}=\cdots=\beta_{a_{s}}=1,\ \beta_{b_{1}}=\cdots=\beta_{b_{t}}=-1,\ \beta_{c_{1}}=\cdots=\beta_{c_{r}}=0.\\\ \\{a_{1},\ldots,a_{s},\ b_{1},\ldots,b_{t},\ c_{1},\ldots,\ c_{r}\\}=\\{1,\ldots,n-1\\},\\\ r,s,t\geq 0,\ t\geq 1,\ \ r+s+t=n-1.\end{array}$ In geometry it is easy to see that the convex polytope $\widehat{T}^{-}$ is the convex hull of its surface $S^{-}$ and its vertices $e_{c_{1}},\ldots,e_{c_{r}}$, that is, $\widehat{T}^{-}={\rm conv}\\{e_{c_{1}},\ldots,e_{c_{r}},S^{-}\\}.$ (2) $\begin{array}[]{l}S^{-}=\\{(y_{1},\ldots,y_{n-1})^{T}\in\Delta_{n-1}\|y_{a_{1}}+\cdots+y_{a_{s}}-y_{b_{1}}-\cdots- y_{b_{t}}\leq 0,\\\ \qquad\quad(y_{a_{1}},\ldots,y_{a_{s}},y_{b_{1}},\ldots,y_{b_{t}})^{T}\in\Delta_{s+t}\\}.\end{array}$ (3) If the simplicial subdivision of $S^{-}$ is known, the simplicial subdivision of $\widehat{T}^{-}$ is directly obtained by (2). So we just need to study the simplicial subdivision of the polytope $S^{-}$. ## 2 A simplicial subdivision algorithm for the convex polytope $S^{-}$ ### 2.1 Fundamental notations The notation $S^{-}$ is simple, but it can not reveal the information of convex polytopes. In order to simplify the descriptions, we will introduce a new notation, which is fundamental to our study. ###### Definition 2.1 Suppose that two sequences of positive integers $[a_{1},a_{2},\ldots,a_{s}]$, $[b_{1},b_{2},\ldots,b_{t}]$ satisfy $\\{a_{1},\ldots,a_{s},\ b_{1},\ldots,b_{t}\\}\subseteq\\{1,2,\ldots,m\\},\ s\geq 0,t\geq 1,m\geq s+t\geq 2,$ where all of $s+t$ elements of $\\{a_{1},\ldots,a_{s},\ b_{1},\ldots,b_{t}\\}$ are distinct. Then the notation $[[a_{1},a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m}$ is defined as the polytope $S^{-}$ in (3). For example, let us compare the polytope $[[2,3],[5]]_{5}$ and the polytope $[[2,3],[5]]_{6}$. $[[2,3],[5]]_{5}$ denotes the polytope $\\{(y_{1},\ldots,y_{5})^{T}\in\Delta_{5}\|y_{2}+y_{3}-y_{5}\leq 0,(y_{2},y_{3},y_{5})^{T}\in\Delta_{3}\\}.$ Here $(y_{2},y_{3},y_{5})^{T}\in\Delta_{3}$ implies that $y_{1}=0,y_{4}=0$. $[[2,3],[5]]_{6}$ indicates the polytope $\\{(y_{1},\ldots,y_{6})^{T}\in\Delta_{6}\|y_{2}+y_{3}-y_{5}\leq 0,(y_{2},y_{3},y_{5})^{T}\in\Delta_{3}\\}.$ Here $(y_{2},y_{3},y_{5})^{T}\in\Delta_{3}$ implies that $y_{1}=0,y_{4}=0,y_{6}=0$. It is clear that $[[2,3],[5]]_{5}$ and $[[2,3],[5]]_{6}$ are congruent, although they are sets in simplices of different dimensions. For $0\leq k\leq m-1$, the polytope $L_{k}^{-}$ is defined as $L_{k}^{-}=\\{(y_{1},\ldots,y_{m})^{T}\in\Delta_{m}\|y_{1}+\cdots+y_{k}-y_{k+1}-\cdots- y_{m}\leq 0\\}.$ $L_{k}^{-}$ is written as $[[1,\ldots,k],[k+1,\ldots,m]]_{m}$ by the notation of Definition 2.1. $L_{k}^{-}$ is a special case of $S^{-}$, but this notation is more convenient for our analysis. In the following we will study the basic geometric properties of convex polytope $[[a_{1},a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m}$. ### 2.2 Geometric properties of $S^{-}$ Let $e_{1},\ldots,e_{m}$ be vertices of the standard simplex $\Delta_{m}$, and $M_{i,j}=(e_{i}+e_{j})/2$ be the midpoint of the line segment $\overline{e_{i}e_{j}}$. The following result is stated in [1] without proof. For completeness, we give a proof. ###### Lemma 2.1 [1] Given a convex polytope $L_{k}^{-}$, then all of its vertices are $V=\\{e_{k+1},\ldots,e_{m},\ M_{i,j},i=1,2,\ldots,k,\ j=k+1,\ldots,m\\}.$ The number of the vertices is $\|V\|=(k+1)(m-k)$. ###### Proof 1 Note that the convex polytope $L_{k}^{-}$ is obtained by cutting the standard simplex $\Delta_{m}$ with the hyperplane $L_{=0}:\ y_{1}+\cdots+y_{k}-y_{k+1}-\cdots-y_{m}=0.$ Therefore the vertices of the polytope $L_{k}^{-}$ come from two parts: one part is vertices of $\Delta_{m}$, that is, $\\{e_{k+1},\ldots,e_{m}\\}$; while the other part is the intersection points of the hyperplane $L_{=0}$ and the edges of standard simplex $\Delta_{m}$. First consider the intersection point of $L_{=0}$ and the edge $ae_{1}+be_{k+1}\ (a,b\geq 0,\ a+b=1)$. Substitute $ae_{1}+be_{k+1}$ into the following equations, $y_{1}+\cdots+y_{k}-y_{k+1}-\cdots-y_{m}=0,\ y_{1}+\cdots+y_{m}=1$ Therefore, the solutions are $a=1/2,b=1/2$, namely, the intersection point is $M_{1,k+1}$. In the same way, we get all intersection points of $L_{=0}$ and the edges of $\Delta_{m}$. They are $\\{M_{i,j},i=1,2,\ldots,k,\ j=k+1,\ldots,m.\\}$. Hence the number of vertices of $L_{k}^{-}$ is $\|V\|=m-k+k(m-k)=(k+1)(m-k)$. Likewise, we can prove the following lemma. ###### Lemma 2.2 Given a convex polytope $[[a_{1},a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m}$, then all of its vertices are $V=\\{e_{b_{1}},\ldots,e_{b_{t}},\ M_{a_{i},b_{j}},i=1,2,\ldots,s,\ j=1,\ldots,t\\}.$ The number of the vertices is $\|V\|=(s+1)t$. We see that the polytope $[[a_{1},a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m}$ and the polytope $L_{k}^{-}$ are similar in many respects, which will be further discussed. ###### Lemma 2.3 The convex polytope $L_{k}^{-}$ is simplicial iff $k=0$, or $k=m-1$. ###### Proof 1 When $k=0$, $L_{0}^{-}=\Delta_{m}$ is simplicial. When $k=m-1$, consider the convex polytope $L_{m-1}^{-}:=\\{(y_{1},\ldots,y_{m})^{T}\in\Delta_{m}\|\ y_{1}+\cdots+y_{m-1}-y_{m}\leq 0\\}.$ By Lemma 2.1, we know that all vertices of $L_{m-1}^{-}$ are $\\{e_{m},\ M_{i,m},i=1,2,\ldots,$ $m-1\\}$. Obviously all the vectors of $\\{M_{i,m}-e_{m},i=1,2,\ldots,m-1\\}$ are linearly independent, so $L_{m-1}^{-}$ is simplicial. Conversely, we know that the dimension of the polytope $L_{k}^{-}$ is $m-1$. If $k\neq 0,m-1$, then by Lemma 2.1, the number of the vertices of $L_{k}^{-}$ is $(k+1)(m-k)\neq m$, so $L_{k}^{-}$ is not simplicial. ###### Lemma 2.4 The convex polytope $[[a_{1},a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m}$ (here the vertices are obtained by Lemma 2.2) is simplicial iff $s=0$, or $t=1$. ###### Lemma 2.5 The dimension of the polytope $[[a_{1},a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m}$ is $(s+t-1)$. If the polytope $[[a_{1},a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m}$ is not a simplex, we will subdivide it into simplices. ###### Lemma 2.6 If the polytope $[[a_{1},a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m}$ is not a simplex, then there are only two $(s+t-2)$-dimensional surfaces that do not include the vertex $M_{a_{1},b_{1}}$. They are $[[a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m},\ [[a_{1},\ldots,a_{s}],[b_{2},\ldots,b_{t}]]_{m}.\ $ (obtained by deleting $a_{1},b_{1}$ from array $[[a_{1},a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m}$ respectively) ###### Proof 1 All the $(s+t-2)$-dimensional surfaces of the convex polytope $[[a_{1},a_{2},\ldots,a_{s}],\ [b_{1},b_{2},\ldots,b_{t}]]_{m}$ are obviously $[[\widehat{a}_{1},a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m},$ $[[a_{1},\widehat{a}_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m},$ $\ldots,$ $[[a_{1},a_{3},\ldots,a_{s}],[b_{1},b_{2},\ldots,\widehat{b}_{t}]]_{m}$ (where the notation $[[\widehat{a}_{1},a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m}$ is the polytope with $a_{1}$ removed) and $\begin{array}[]{l}\\{(y_{1},\ldots,y_{m})^{T}\in\Delta_{m}\|y_{a_{1}}+\cdots+y_{a_{s}}-y_{b_{1}}-\cdots- y_{b_{t}}=0,\ (y_{a_{1}},\ldots,y_{a_{s}},\\\ \quad y_{b_{1}},\ldots,y_{b_{t}})^{T}\in\Delta_{s+t}\\}.\end{array}$ That makes $s+t+1$ $(s+t-2)$-dimensional surfaces in all. We can verify that only $[[a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m},\ [[a_{1},\ldots,a_{s}],[b_{2},\ldots,b_{t}]]_{m}$ do not include the vertex $M_{a_{1},b_{1}}$. Lemma 2.6 leads to the following decomposition theorem. ### 2.3 The decomposition process for the polytope $S^{-}$ ###### Theorem 2.1 (decomposition theorem) If the polytope $[[a_{1},a_{2},\ldots,a_{s}]$, $[b_{1},b_{2},\ldots,b_{t}]]_{m}$ is not simplicial, then it can be decomposed into the union of two convex polytopes (not always simplicial). The expression is $\begin{array}[]{l}\quad[[a_{1},a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m}\\\ ={\rm conv}\\{M_{a_{1},b_{1}},[[a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m}\\}\bigcup\\\ \quad{\rm conv}\\{M_{a_{1},b_{1}},[[a_{1},a_{2},\ldots,a_{s}],[b_{2},\ldots,b_{t}]]_{m}\\}.\end{array}$ Here ${\rm conv}\\{S\\}$ denotes the convex hull of the set $S$ of points . ###### Proof 1 This follows from Lemma 2.6. Based on Theorem 2.1, the polytope $S^{-}$ can be easily subdivided into simplices. ###### Example 1 Show the simplicial subdivision of the following convex polytope $L_{2}^{-}:=\\{(y_{1},\ldots,y_{5})\|\ y_{1}+y_{2}-y_{3}-y_{4}-y_{5}\leq 0,(y_{1},\ldots,y_{5})^{T}\in\Delta_{5}\\}.$ ###### Solution 1 Denote $L_{2}^{-}$ as $[[1,2],[3,4,5]]_{5}$. We know that $[[1,2],[3,4,5]]_{5}$ is not simplicial by Lemma 2.4. Using Theorem 2.1 we have $\begin{array}[]{l}[[1,2],[3,4,5]]_{5}={\rm conv}\\{M_{1,3},[[2],[3,4,5]]_{5}\\}\bigcup{\rm conv}\\{M_{1,3},[[1,2],[4,5]]_{5}\\}.\end{array}$ By Lemma 2.4 we know that both $[[2],[3,4,5]]_{5}$ and $[[1,2],[4,5]]_{5}$ are not simplicial. Therefore we repeatedly apply Theorem 2.1 to them and have $\begin{array}[]{l}[[2],[3,4,5]]_{5}\\\ ={\rm conv}\\{M_{2,3},[[\ ],[3,4,5]]_{5}\\}\bigcup{\rm conv}\\{M_{2,3},[[2],[4,5]]_{5}\\}.\\\ ={\rm conv}\\{M_{2,3},[[\ ],[3,4,5]]_{5}\\}\bigcup{\rm conv}\\{M_{2,3},M_{2,4},[[\ ],[4,5]]_{5}\\}\bigcup\\\ \quad{\rm conv}\\{M_{2,3},M_{2,4},[[2],[5]]_{5}\\}\\\ ={\rm conv}\\{M_{2,3},e_{3},e_{4},e_{5}\\}\bigcup{\rm conv}\\{M_{2,3},M_{2,4},e_{4},e_{5}\\}\bigcup\\\ \quad{\rm conv}\\{M_{2,3},M_{2,4},M_{2,5},e_{5}\\}.\end{array}$ $\begin{array}[]{l}[[1,2],[4,5]]_{5}\\\ ={\rm conv}\\{M_{1,4},[[2],[4,5]]_{5}\\}\bigcup{\rm conv}\\{M_{1,4},[[1,2],[5]]_{5}\\}.\\\ ={\rm conv}\\{M_{1,4},M_{2,4},[[\ ],[4,5]]_{5}\\}\bigcup{\rm conv}\\{M_{1,4},M_{2,4},[[2],[5]]_{5}\\}\bigcup\\\ \quad{\rm conv}\\{M_{1,4},[[1,2],[5]]_{5}\\}\\\ ={\rm conv}\\{M_{1,4},M_{2,4},e_{4},e_{5}\\}\bigcup{\rm conv}\\{M_{1,4},M_{2,4},M_{2,5},e_{5}\\}\bigcup\\\ \quad{\rm conv}\\{M_{1,4},M_{1,5},M_{2,5},e_{5}\\}.\end{array}$ Finally we get the expression of simplicial subdivision of $[[1,2],[3,4,5]]_{5}$, $\begin{array}[]{l}[[1,2],[3,4,5]]_{5}\\\ ={\rm conv}\\{M_{1,3},M_{2,3},e_{3},e_{4},e_{5}\\}\bigcup{\rm conv}\\{M_{1,3},M_{2,3},M_{2,4},e_{4},e_{5}\\}\bigcup\\\ \quad{\rm conv}\\{M_{1,3},M_{2,3},M_{2,4},M_{2,5},e_{5}\\}\bigcup{\rm conv}\\{M_{1,3},M_{1,4},M_{2,4},e_{4},e_{5}\\}\bigcup\\\ \quad{\rm conv}\\{M_{1,3},M_{1,4},M_{2,4},M_{2,5},e_{5}\\}\bigcup{\rm conv}\\{M_{1,3},M_{1,4},M_{1,5},M_{2,5},e_{5}\\}.\end{array}$ So $[[1,2],[3,4,5]]_{5}$ is a union of six 4-dimensional simplices. We summarize the decomposition process of Example 1 into the following algorithm. Algorithm 1 (Vmatrix) Input: The expression of polytope $[[a_{1},a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m}$. Output: Simplices $D_{1},D_{2},\ldots,D_{p}$(denoted by matrices) such that $[[a_{1},a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m}=\bigcup_{i=1}^{p}D_{i},\ {\rm int}(D_{i})\bigcap{\rm int}(D_{j})=\emptyset,\ {\rm for}\ i\neq j.$ V1: Let $F:=\\{[[a_{1},a_{2},\ldots,a_{s}],[b_{1},b_{2},\ldots,b_{t}]]_{m}\\}$, ${\rm temp}:=\emptyset$. V2: When $F\neq\emptyset$, repeat the following procedures V21: Choose a polytope $N\in F$. If $N$ is simplicial, then ${\rm temp}:={\rm temp}\cup\\{N\\}$, $F:=F\setminus\\{N\\}$. V22: If the polytope $N$ is not simplicial, then by Theorem 2.1 de- compose it into two convex polytope $B_{1},B_{2}$. $F:=F\setminus\\{N\\}\cup\\{B_{1},B_{2}\\}$. Go to step V2. V3: Return ${\rm temp}$. We have written a function in Maple [31] to implement the above algorithm. Lastly, we will present a formula for computing the number of simplices given by the polytope $L_{k}^{-}$ subdivision. ###### Lemma 2.7 According to algorithm Vmatrix, the convex polytope $[[1,\ldots,k]$, $[k+1,\ldots,m]]_{m}$ $(0\leq k\leq m-1,m\geq 2)$ can be subdivided just into $f(k,m)$ simplices, where $f(k,m)=\left(\begin{array}[]{c}m-1\\\ k\end{array}\right)=\frac{(m-1)!}{k!(m-1-k)!}.$ We know that $f(k,m)$ has the same recurrence formula as binomial coefficients by Theorem 2.1. Thus the proof of Lemma 2.7 is easy via an induction argument. This formula will be used to estimate the cost of Algorithm 2 in the next section. ## 3 Determining algorithm for copositive matrices In this section, we will present the complete determining algrorithm of a copositive matrix. Given an $n\times n$ symmetric matrix $A=[\alpha_{ij}]=\left[\begin{array}[]{cc}\alpha_{11}&\alpha^{T}\\\ \alpha&A_{2}\end{array}\right],$ compute $\widehat{A}$ (see (1)) $\widehat{A}=\left[\begin{array}[]{cc}\alpha_{11}&\widehat{\alpha}^{T}\\\ \widehat{\alpha}&DA_{2}D\end{array}\right].$ Let $B=\alpha_{11}DA_{2}D-\widehat{\alpha}\widehat{\alpha}^{T}$, and let $\widehat{\alpha}=({\rm sign}(\alpha_{12}),\ldots,{\rm sign}(\alpha_{1n}))^{T}=(\beta_{1},\ldots,\beta_{n-1})^{T}.$ Define the projection operator Proj of the matrix $A$ as follows, $\bullet$ If $\beta_{i}\geq 0,\ i=1,\ldots,n-1$, then ${\rm Proj(A)}=\\{DA_{2}D\\}.$ $\bullet$ If there is at least one -1 in $\beta_{i}$, then ${\rm Proj(A)}=\\{DA_{2}D,\ W_{1}^{T}BW_{1},\ \ldots,\ W^{T}_{p}BW_{P}\\}.$ Here the matrices $W_{1},\ldots,W_{p}$ is fixed by the simplicial subdivision of the convex polytope $\widehat{T}^{-}$ (see (2)). Algorithm 2 (COPOMATRIX) Input: Symmetric matrix $A\in{\mathbf{R}}^{n\times n}(n\geq 2)$. Output: $A$ is copositive, or $A$ is not copositive. C1: Let $F:=\\{A\\}$. C2: Repeat the following steps for the set $F$. C21: If the set $F$ is empty, then return “$A$ is copositive”. C22: Check the (1,1)th entry of every matrix $K$ in set $F$. If at least one of them is negative, then return “$A$ is not copositive”. C23: Compute the projective set $P:=\bigcup_{K\in F}{\rm Proj}(K)$ of set $F$. $F:=P\setminus\\{\hbox{the nonnegative matrices of $P$}\\}$. Go to step C21. Note that the above algorithm is also valid for $2\times 2$ matrices. Furthermore, for strictly copositive matrices we can also formulate a similar algorithm. The correctness of the algorithm COPOMATRIX is guaranteed by Lemma 1.1, and the algorithm obviously terminates. The cost of the algorithm mainly depends on the number of simplicial subdivisions of the polytope. According to Lemma 2.7, we can estimate that in the worst case it is at most: $\begin{array}[]{l}(\left(\begin{array}[]{c}n-2\\\ \left[\frac{n-2}{2}\right]\end{array}\right)+1)(\left(\begin{array}[]{c}n-3\\\ \left[\frac{n-3}{2}\right]\end{array}\right)+1)\cdots(\left(\begin{array}[]{c}2\\\ 1\end{array}\right)+1)\\\ \leq(2^{n-3})(2^{n-4})\cdots(2)(2)\\\ =2^{(n-2)(n-3)/2+1}.\end{array}$ The bound $2^{(n-2)(n-3)/2+1}$ is already much lower than doubly-exponential cost of CAD [2,9]. We have written a function in Maple to implement the algorithm COPOMATRIX. For non-commercial request, we will offer for free. Please sent e-mail to the address yaoyong@casit.ac.cn, or, j.jia.xu@gmail.com. ## 4 Acknowledgement The work of the authors were supported by the Chinese National Science Foundation under contracts 11001228 and 10901116. The authors also would like to thank the referees for their helpful suggestions. ## References * [1] L.E. Andersson, G. Chang, T. 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Li, Algorithms for determining the copositivity of a given symmetric matrix, Linear Algebra Appl. 430 (2009) 609-618.	arxiv-papers	2010-11-09T11:04:25	2024-09-04T02:49:14.649034	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jia Xu, Yong Yao", "submitter": "Yong Yao", "url": "https://arxiv.org/abs/1011.2039" }
1011.2274	# Relativistic corrections to the axial vector and vector currents in the $\bm{\overline{b}c}$ meson system at order $\bm{\alpha_{s}}$ Jungil Lee and WenLong Sang Department of Physics, Korea University, Seoul 136-701, Korea jungil@korea.ac.kr swlong@korea.ac.kr Seyong Kim Department of Physics, Sejong University, Seoul 143-747, Korea skim@sejong.ac.kr ###### Abstract: We compute the short distance coefficients for the NRQCD factorization formulas of the meson-to-vacuum matrix elements for the axial vector and vector parts of the charged weak current in the $S$-wave spin-singlet and -triplet $\bar{b}c$ mesons, respectively. The computation is carried out to order $\alpha_{s}$ including relativistic corrections of all orders in $\bm{q}^{2n}$, where $\bm{q}$ is the relative momentum of the $\bar{b}$ and $c$ in the meson rest frame. The relativistic corrections at order $\alpha_{s}$ are new. The results reveal that the relativistic corrections to the leptonic decay rate of the $B_{c}$ meson at order $\alpha_{s}$ or less converge rapidly, which shows a strong contrast to the uncomfortably large corrections of order $\alpha_{s}^{2}\|\bm{q}\|^{0}$. The short distance coefficients listed in this paper can be employed to compute the resummation of relativistic corrections to the phenomenological measurables that involve $B_{c}$ and $B_{c}^{}$ production and decay. $B_{c}$, NRQCD, decay, Relativistic Corrections ## 1 Introduction Among various quarkonia including charmonia and bottomonia, the bound state of a $\bar{b}c$ pair is a distinct heavy quarkonium system composed of two different heavy quark flavors. As the spin-singlet $S$-wave bound state of the $\bar{b}c$ pair, the $B_{c}$ meson was discovered by the CDF Collaboration at the Fermilab Tevatron through the decay mode $B_{c}\to J/\psi+\ell^{+}+\nu_{\ell}$ [1] and the $S$-wave spin-triplet state $B_{c}^{}$ has not been observed, yet111$B_{c}^{+}$ is the bound state of a $\bar{b}c$ pair and $B_{c}^{-}$ is that of $\bar{c}b$. Throughout this paper, $B_{c}$ denotes $B_{c}^{+}$. However, our analysis can be equally applied to the charge-conjugate state $B_{c}^{-}$.. The $\bar{b}c$ meson offers a unique laboratory for the nonrelativistic quantum chromodynamics (NRQCD) factorization framework [2] because the typical heavy quark ($Q$) velocity $v_{Q}$ in the $\bar{b}c$ meson lies between that ($v_{c}^{2}\sim 0.3$) in the charmonium system and that ($v_{b}^{2}\sim 0.1$) in the bottomonium system. In addition, unlike the spin-triplet $S$-wave $Q\bar{Q}$ mesons, $J/\psi$ and $\Upsilon$ that can decay into lepton pairs through the electromagnetic current, the decay of the $B_{c}$ meson can proceed only through the charged weak current and various dynamics play their roles in the decay of the $\bar{b}c$ meson [3, 4, 5, 6, 7, 8]. As of now, except for the mass, $m_{B_{c}}=6.277\pm 0.006$ GeV, and the life time, $\tau_{B_{c}}=(0.45\pm 0.04)\times 10^{-12}$ s, for the spin-singlet $S$-wave state $B_{c}$, little is known about various properties of the $\bar{b}c$ bound states experimentally [9]. In near future, one may probe many unknown properties of the bound states in detail as the CERN Large Hadron Collider accumulates orders of magnitude larger number of events than the currently available data [10] for the $\bar{b}c$ bound states. Therefore, it is desirable and necessary to achieve better accuracies in theoretical predictions. Earlier theoretical studies on the $\bar{b}c$ bound states cover the spectroscopy [11, 12, 13, 14, 15] and the production mechanism at colliders [16, 17, 18, 19, 20]. In order to achieve better accuracies in the predictions for the $\bar{b}c$-meson production and decay, it is necessary to know accurate values for the decay constants. The NRQCD factorization formula is useful in making a systematic series expansion of the decay constant for the $\bar{b}c$ meson system in powers of $v_{Q}$. The NRQCD factorization theorems have been proved for the electromagnetic and light hadronic decays of heavy quarkonia [2] and for a few exclusive production processes of heavy quarkonia [21, 22, 23]. The NRQCD factorization formulas for the hadronic part of the charged weak current that involves the decays of $B_{c}$ and $B_{c}^{}$ are similar to those for the electromagnetic decays of $Q\bar{Q}$ mesons. Braaten and Fleming computed the one-loop QCD corrections to the $B_{c}$ decay constant in the static limit $v_{Q}=0$ and the relativistic corrections of relative order $\alpha_{s}^{0}v_{Q}^{2}$ [24] by computing the short distance coefficients of the NRQCD factorization formula for the decay constant. Based on the same strategy, Hwang and Kim calculated the $B_{c}^{}$ counterparts [25]. Two-loop QCD correction to the short distance coefficient for axial vector current involving $B_{c}$ decay constant was calculated by Onishchenko and Veretin [26], which shows uncomfortably large correction like those for the leptonic decays of the spin-triplet $S$-wave quarkonia $J/\psi$ and $\Upsilon$ [27, 28]. In fact, according to the velocity-scaling rules of NRQCD [2], the corrections of relative orders $v_{Q}^{4}$ and $\alpha_{s}v_{Q}^{2}$ should be equally important as that of relative order $\alpha_{s}^{2}v_{Q}^{0}$. In addition, the large separation between $m_{b}$ and $m_{c}$, where $m_{Q}$ is the mass of the heavy quark $Q=b$ or $c$, in the $\bar{b}c$ system gives rise to factors of $\log(m_{b}/m_{c})$ in the one-loop corrections [24, 25, 26], which may potentially deteriorate the convergence of the power expansion in $v_{Q}$. Therefore, it is worthwhile to check if such large corrections indeed arise when one includes the relativistic effects. In this paper, we compute the relativistic corrections to the NRQCD factorization formulas of the meson-to-vacuum matrix elements for the axial vector and vector parts of the charged weak current in the $S$-wave spin- singlet and -triplet $\bar{b}c$ mesons, respectively. The calculation is carried out at first order of the strong coupling $\alpha_{s}$ including relativistic corrections to all orders in $v_{Q}$. The short distance coefficients for the NRQCD factorization formula are usually obtained after subtracting the infrared (IR)-sensitive contributions of the NRQCD correction from the full QCD corrections by perturbative matching. It requires laborious bookkeeping of the Feynman rules in the NRQCD perturbation theory which grows tremendously extensive as the order in $v_{Q}^{2n}$ increases [29, 30]. Instead, we use a new method introduced recently in [31] to compute the relativistic corrections to the short distance coefficients at order $\alpha_{s}$ covering all orders in $v_{Q}^{2n}$. The method integrates out the temporal component of the loop momentum by contour integration in the one- loop corrections to the full QCD amplitude and, then, expands the integrands in powers of the external momenta divided by $m_{Q}$ and the spatial components of the loop momentum divided by $m_{Q}$. In comparison with, so called, the method of region given in [32], this new method is particularly useful in computing relativistic corrections of higher orders in $v_{Q}$ and even makes it possible to find the closed form of the expression that includes the relativistic corrections resummed to all orders in $v_{Q}^{2n}$ at one loop. This paper is organized as follows. In section 2, we discuss the perturbative matching of NRQCD onto QCD at one loop. Kinematics of the problem and the definitions of the variables that are useful in computing the short distance coefficients are given in section 3. Section 4 contains the strategy and detailed formulas to compute the short distance coefficients. We compute the QCD one-loop corrections in section 5 followed by the NRQCD corrections in section 6. Our final results for the short distance coefficients are listed in section 7 and we summarize in section 8. In appendices, we provide the formulas for the tensor-integral reduction and list the values for the loop integrals that appear in the one-loop correction to the QCD and NRQCD amplitudes. ## 2 Perturbative matching to all orders in $\bm{v}$ We define the hadronic parts of the weak decay amplitudes $i{\cal A}^{\mu}_{5,{B_{c}}}$ and $i{\cal A}^{\mu}_{{B_{c}^{}}}$ for $B_{c}$ and $B_{c}^{}$ as the meson-to-vacuum matrix elements of the axial vector current ${\overline{b}}{\gamma}^{\mu}{\gamma}_{5}c$ and the vector current ${\overline{b}}{\gamma}^{\mu}c$, respectively as222 See, for example, [4].: $\displaystyle i{\cal A}^{\mu}_{5,{B_{c}}}$ $\displaystyle\equiv$ $\displaystyle\langle 0\|\,{\overline{b}}{\gamma}^{\mu}{\gamma}_{5}c\,\|B_{c}\rangle=if_{B_{c}}K^{\mu},$ (1a) $\displaystyle i{\cal A}^{\mu}_{{B_{c}^{}}}$ $\displaystyle\equiv$ $\displaystyle\langle 0\|\,{\overline{b}}{\gamma}^{\mu}c\,\|B_{c}^{}\rangle=if_{B^{}_{c}}m_{B^{}_{c}}{\varepsilon}^{\mu},$ (1b) where $b$ and $c$ are the Dirac field operators for the bottom quark and charm quark, respectively. The amplitudes (1) are scaled by the leptonic decay constants $f_{H}$ for $H=B_{c}$ and $B_{c}^{}$. Here, $K$ is the meson momentum, $\varepsilon$ is the polarization vector of the $B_{c}^{}$ meson, $m_{B_{c}}=$ 6.277 (6) GeV [9] and $m_{B_{c}^{}}=$ 6.330 (7)(2)(6) GeV [33] are the masses of the $B_{c}$ and $B_{c}^{}$ mesons. The quarkonium state $\|H\rangle$ in (1) for $H=B_{c}$ and $B_{c}^{}$ is normalized relativistically: $\langle H(K^{\prime})\|H(K)\rangle=2\,K^{0}(2\pi)^{3}\delta^{(3)}(\bm{K}^{\prime}-\bm{K})$. In the meson rest frame, the matrix elements in (1) become simple: Because $K=(m_{H},\bm{0})$ in this frame, only the $0$-th component survives in the matrix element (1a) for the $B_{c}$. Due to the transverse condition $\varepsilon\cdot K=0$ for the $B_{c}^{}$, $\varepsilon=(0,\bm{\varepsilon})$ in this frame and, therefore, only the spatial components are nonvanishing in the matrix element (1b) for the $B_{c}^{}$. According to the NRQCD factorization [2], we can write the nonvanishing components of $i{\cal A}^{\mu}_{5,{B_{c}}}$ and $i{\cal A}^{\mu}_{{B_{c}^{}}}$ in the rest frame of the $\bar{b}c$ bound states as $\displaystyle i{\cal A}^{0}_{5,{B_{c}}}$ $\displaystyle=$ $\displaystyle\sqrt{2m_{\\!B_{c}}}\sum_{n}P_{n}\langle 0\|\mathcal{O}_{n}\|B_{c}\rangle,$ (2a) $\displaystyle i{\cal A}^{i}_{{B_{c}^{}}}$ $\displaystyle=$ $\displaystyle\sqrt{2m_{\\!B_{c}^{}}}\sum_{n}V_{n}\langle 0\|\mathcal{O}^{i}_{n}\|B_{c}^{}\rangle,$ (2b) where $P_{n}$ and $V_{n}$ are the short distance coefficients and $\mathcal{O}_{n}$ and $\mathcal{O}_{n}^{i}$ are the NRQCD operators. The operator matrix elements in (2) are regularized dimensionally in $d=4-2\epsilon$ space-time dimensions. The overall factor $\sqrt{2m_{H}}$ for $H=B_{c}$ and $B_{c}^{}$ in (2) has been taken out because the state $\|H\rangle$ in (2) is normalized nonrelativistically: $\langle H(\bm{K}^{\prime})\|H(\bm{K})\rangle=(2\pi)^{3}\delta^{(3)}(\bm{K}^{\prime}-\bm{K})$, while the amplitudes on the left sides of (2) have the relativistic normalization for the quarkonium $H$ like those in (1). The main purpose of this paper is to compute the short distance coefficients $P_{n}$ and $V_{n}$ in (2) that correspond to $\bar{b}c$ color-singlet operators at order $\alpha_{s}$. These coefficients can be determined by the matching equations $\displaystyle i{\cal A}^{0}_{5,\bar{b}c_{1}}$ $\displaystyle=$ $\displaystyle\sum_{n}P_{n}\langle 0\|\mathcal{O}_{n}\|\bar{b}c_{1}\rangle,$ (3a) $\displaystyle i{\cal A}^{i}_{\bar{b}c_{1}}$ $\displaystyle=$ $\displaystyle\sum_{n}V_{n}\langle 0\|\mathcal{O}^{i}_{n}\|\bar{b}c_{1}\rangle,$ (3b) which is the statement of NRQCD factorization for the perturbative color- singlet $\bar{b}c$ state. Here, $\bar{b}c_{1}$ denotes the color-singlet $\bar{b}c$ pair whose invariant mass is the same as the meson mass. Throughout this paper, we suppress the factor $\sqrt{N_{c}}$ that comes from the implicit color trace in $i{\cal A}^{0}_{5,\bar{b}c_{1}}$ and $i{\cal A}^{i}_{\bar{b}c_{1}}$, where $N_{c}=3$ is the number of colors. Note that the coefficients $P_{n}$ and $V_{n}$ in (3) are identical to those in (2) because the short distance coefficients must not depend on the long distance nature of the heavy quarkonium state. While the amplitudes (2) contain nonperturbative quantities, the amplitudes (3) are calculable perturbatively. However, it is possible that the amplitudes (3) acquire singularities in the IR or ultraviolet (UV) regions at order $\alpha_{s}$ or higher. These divergences are to be regularized dimensionally. The matching equations that contain the terms upto order $\alpha_{s}$ are $\displaystyle i{\cal A}^{0(0)}_{5,\bar{b}c_{1}}+i{\cal A}^{0(1)}_{5,\bar{b}c_{1}}$ $\displaystyle=$ $\displaystyle\sum_{n}(P_{n}^{(0)}+P_{n}^{(1)})\langle 0\|\mathcal{O}_{n}\|\bar{b}c_{1}\rangle^{(0)}+\sum_{n}P_{n}^{(0)}\langle 0\|\mathcal{O}_{n}\|\bar{b}c_{1}\rangle^{(1)},$ (4a) $\displaystyle i{\cal A}^{i(0)}_{\bar{b}c_{1}}+i{\cal A}^{i(1)}_{\bar{b}c_{1}}$ $\displaystyle=$ $\displaystyle\sum_{n}(V_{n}^{(0)}+V_{n}^{(1)})\langle 0\|\mathcal{O}^{i}_{n}\|\bar{b}c_{1}\rangle^{(0)}+\sum_{n}V_{n}^{(0)}\langle 0\|\mathcal{O}^{i}_{n}\|\bar{b}c_{1}\rangle^{(1)},$ (4b) where the superscripts $(0)$ and $(1)$ indicate the order in $\alpha_{s}$. In the first sum of each line in (4), only color-singlet $\bar{b}c$ operators contribute, while in the next sum additional operators may enter once they mix with color-singlet $\bar{b}c$ operators under one-loop QCD corrections. Through order $\alpha_{s}$, the NRQCD amplitudes are defined by $\displaystyle{[}i{\cal A}^{0(j)}_{5,\bar{b}c_{1}}]_{\rm NRQCD}$ $\displaystyle=$ $\displaystyle\sum_{n}P_{n}^{(0)}\langle 0\|\mathcal{O}_{n}\|\bar{b}c_{1}\rangle^{(j)},$ (5a) $\displaystyle{[}i{\cal A}^{i(j)}_{\bar{b}c_{1}}]_{\rm NRQCD}$ $\displaystyle=$ $\displaystyle\sum_{n}V_{n}^{(0)}\langle 0\|\mathcal{O}^{i}_{n}\|\bar{b}c_{1}\rangle^{(j)},$ (5b) where $j=0$ or 1. At order $\alpha_{s}^{0}$, the NRQCD matrix elements $\langle 0\|\mathcal{O}_{n}\|\bar{b}c_{1}\rangle^{(0)}$ and $\langle 0\|\mathcal{O}^{i}_{n}\|\bar{b}c_{1}\rangle^{(0)}$ are finite and the short distance coefficients $P_{n}^{(0)}$ and $V_{n}^{(0)}$ can be determined from the identities $\displaystyle{[}i{\cal A}^{0(0)}_{5,\bar{b}c_{1}}]_{\rm NRQCD}$ $\displaystyle=$ $\displaystyle i{\cal A}^{0(0)}_{5,\bar{b}c_{1}}=\sum_{n}P_{n}^{(0)}\langle 0\|\mathcal{O}_{n}\|\bar{b}c_{1}\rangle^{(0)},$ (6a) $\displaystyle{[}i{\cal A}^{i(0)}_{\bar{b}c_{1}}]_{\rm NRQCD}$ $\displaystyle=$ $\displaystyle i{\cal A}^{i(0)}_{\bar{b}c_{1}}=\sum_{n}V_{n}^{(0)}\langle 0\|\mathcal{O}^{i}_{n}\|\bar{b}c_{1}\rangle^{(0)}.$ (6b) Each expansion of (6) is a power series in $\bm{q}/m_{Q}$, where $\bm{q}$ is half the relative three-momentum of the $\bar{b}$ and $c$ in the center-of- momentum (CM) frame of the $\bar{b}c$ pair. However, at order $\alpha_{s}$, the short distance coefficients $P_{n}^{(1)}$ and $V_{n}^{(1)}$ must be determined after subtracting the long distance contributions that are contained in the order-$\alpha_{s}$ matrix elements $\langle 0\|\mathcal{O}_{n}\|\bar{b}c_{1}\rangle^{(1)}$ and $\langle 0\|\mathcal{O}^{i}_{n}\|\bar{b}c_{1}\rangle^{(1)}$, which include the contributions from the potential, soft, and ultrasoft regions: $\displaystyle i{\cal A}^{0(1)}_{5,\bar{b}c_{1}}-{[}i{\cal A}^{0(1)}_{5,\bar{b}c_{1}}]_{\rm NRQCD}$ $\displaystyle=$ $\displaystyle\sum_{n}P_{n}^{(1)}\langle 0\|\mathcal{O}_{n}\|\bar{b}c_{1}\rangle^{(0)},$ (7a) $\displaystyle i{\cal A}^{i(1)}_{\bar{b}c_{1}}-{[}i{\cal A}^{i(1)}_{\bar{b}c_{1}}]_{\rm NRQCD}$ $\displaystyle=$ $\displaystyle\sum_{n}V_{n}^{(1)}\langle 0\|\mathcal{O}^{i}_{n}\|\bar{b}c_{1}\rangle^{(0)}.$ (7b) In this way, one can determine the order-$\alpha_{s}$ short distance coefficients $P_{n}^{(1)}$ and $V_{n}^{(1)}$, which are free of IR sensitivity. The computation of ${[}i{\cal A}^{0(1)}_{5,\bar{b}c_{1}}]_{\rm NRQCD}$ and ${[}i{\cal A}^{i(1)}_{\bar{b}c_{1}}]_{\rm NRQCD}$ is very complicated because it involves operators and interactions of all orders in $v_{Q}$. Fortunately, the authors of [31] recently introduced a way to compute the NRQCD amplitudes $[i{\cal A}^{0(1)}_{5,\bar{b}c_{1}}]_{\rm NRQCD}$ and $[i{\cal A}^{i(1)}_{\bar{b}c_{1}}]_{\rm NRQCD}$ directly from the full QCD counterparts $i{\cal A}^{0(1)}_{5,\bar{b}c_{1}}$ and $i{\cal A}^{i(1)}_{\bar{b}c_{1}}$ by expanding the integrands in powers of the momentum divided by $m_{Q}$. Before the expansion, temporal component of the loop momentum has been integrated out, using contour integration, in order to avoid the generation of ill defined pinch singularities that may develop if one expands the heavy quark propagators too early. The integrands for the integration over the remaining spatial components of the loop momentum are then expanded in powers of the external momenta divided by $m_{Q}$ and the spatial components of the loop momenta divided by $m_{Q}$. Divergent integrals over the spatial components of the loop momenta are regularized dimensionally while scaleless power-divergent integrals are dropped in accordance with the dimensional regularization scheme. As the last step, remaining UV divergences are renormalized according to the modified minimal subtraction ($\overline{\textrm{MS}}$) scheme. In this work, we apply this method to compute the NRQCD amplitudes ${[}i{\cal A}^{0(1)}_{5,\bar{b}c_{1}}]_{\rm NRQCD}$ and ${[}i{\cal A}^{i(1)}_{\bar{b}c_{1}}]_{\rm NRQCD}$ and determine the short distance coefficients $P_{n}^{(1)}$ and $V_{n}^{(1)}$ from (7). ## 3 Kinematics and notations In this section, we define notations for the kinematics of the problem. We take $p_{c}$ and $p_{\bar{b}}$ to be the momenta of the incoming heavy quark $c$ and heavy antiquark $\bar{b}$, respectively, which are on their mass shells: $p_{c}^{2}=m_{c}^{2}$ and $p_{\bar{b}}^{2}=m_{b}^{2}$. They are expressed as linear combinations of half the total momentum $p=\tfrac{1}{2}(p_{c}+p_{\bar{b}})=\tfrac{1}{2}K$ and half their relative momentum $q=\tfrac{1}{2}(p_{c}-p_{\bar{b}})$: $\displaystyle p_{c}$ $\displaystyle=$ $\displaystyle p+q,$ (8a) $\displaystyle p_{\bar{b}}$ $\displaystyle=$ $\displaystyle p-q.$ (8b) In the CM frame of the $\bar{b}c$ pair, the momenta are given by $\displaystyle p_{c}$ $\displaystyle=$ $\displaystyle(E_{c},\bm{q}),$ (9a) $\displaystyle p_{\bar{b}}$ $\displaystyle=$ $\displaystyle(E_{b},-\bm{q}),$ (9b) $\displaystyle p$ $\displaystyle=$ $\displaystyle[\tfrac{1}{2}(E_{c}+E_{b}),\bm{0}],$ (9c) $\displaystyle q$ $\displaystyle=$ $\displaystyle[\tfrac{1}{2}(E_{c}-E_{b}),\bm{q}],$ (9d) where $E_{c}=(m_{c}^{2}+\bm{q}^{2})^{1/2}$ and $E_{b}=(m_{b}^{2}+\bm{q}^{2})^{1/2}$. Note that $4p\cdot q=(p_{c}+p_{\bar{b}})\cdot(p_{c}-p_{\bar{b}})=m_{c}^{2}-m_{b}^{2}\neq 0,$ (10) unlike the case of the $Q\bar{Q}$ pair considered in [31]. For later use, it is convenient to define parameters $\delta$ which is the magnitude of the three-momentum of $\bar{b}$ or $c$ and $e_{c}$ which is the energy of the charm quark scaled by the invariant mass $\sqrt{4p^{2}}=E_{c}+E_{b}$ of the $\bar{b}c$ pair in the CM frame: $\displaystyle\delta$ $\displaystyle=$ $\displaystyle\frac{\|\bm{q}\|}{\sqrt{4p^{2}}}=\frac{1}{2}\sqrt{\left[1-\frac{(m_{b}+m_{c})^{2}}{4p^{2}}\right]\left[1-\frac{(m_{b}-m_{c})^{2}}{4p^{2}}\right]},$ (11a) $\displaystyle e_{c}$ $\displaystyle=$ $\displaystyle\frac{E_{c}}{\sqrt{4p^{2}}}=\frac{1}{2}\left(1-\frac{m_{b}^{2}-m_{c}^{2}}{4p^{2}}\right).$ (11b) Note that $\delta/e_{c}=\|\bm{q}\|/E_{c}$. The following relations are also useful: $\displaystyle p_{\bar{b}}\cdot p_{c}+m_{b}m_{c}$ $\displaystyle=$ $\displaystyle(E_{b}E_{c}+m_{b}m_{c})+\bm{q}^{2}=2m_{b}m_{c}+{O}(\bm{q}^{2}),$ (12a) $\displaystyle p_{\bar{b}}\cdot p_{c}-m_{b}m_{c}$ $\displaystyle=$ $\displaystyle(E_{b}E_{c}-m_{b}m_{c})+\bm{q}^{2}=\frac{(m_{b}+m_{c})^{2}}{2m_{b}m_{c}}\,\bm{q}^{2}+{O}(\bm{q}^{4}).$ (12b) In the derivation of the full QCD amplitudes, we use the Dirac spinors for the $c$ and $\bar{b}$ with the nonrelativistic normalization. In the CM frame of the $\bar{b}c$ pair, they are $\displaystyle u_{c}(p_{c})$ $\displaystyle=$ $\displaystyle\mathcal{N}_{c}\begin{pmatrix}(E_{c}+m_{c})\xi_{c}\\\ \bm{q}\cdot\bm{\sigma}\xi_{c}\end{pmatrix},$ (13a) $\displaystyle v_{c}(p_{\bar{b}})$ $\displaystyle=$ $\displaystyle\mathcal{N}_{b}\begin{pmatrix}-\bm{q}\cdot\bm{\sigma}\eta_{b}\\\ (E_{b}+m_{b})\eta_{b}\end{pmatrix},$ (13b) where $\mathcal{N}_{Q}=[2E_{Q}(E_{Q}+m_{Q})]^{-1/2}$ for $Q=c$ and $b$, $\xi_{c}$ and $\eta_{b}$ are Pauli spinors for the $c$ and $\bar{b}$, respectively. The spinors in (13) are convenient in making nonrelativistic expansions. The threshold expansion method in [34] and its dimensionally regularized version in [35] also use this form except that the relativistic normalization is used. To extract the spin-singlet and -triplet states from a full QCD amplitude for the $\bar{b}c$ pair, one can also make use of the spin- projection operators for those states. In [36, 23], for example, one can find the spin-projection operators for the spin-singlet and -triplet states of a heavy quark-antiquark pair with different flavors. We use the representation for the Dirac matrices introduced in [35]: $\gamma^{0}=\begin{pmatrix}\mathbbm{1}&\phantom{+}0\\\ 0&-\mathbbm{1}\end{pmatrix},\quad\gamma^{i}=\begin{pmatrix}\phantom{+}0&\phantom{+}\sigma^{i}\\\ -\sigma^{i}&\phantom{+}0\end{pmatrix},$ (14) where $\mathbbm{1}$ is the identity matrix. In (14) $\gamma^{i}$ and the Pauli matrix $\sigma^{i}$ are defined for $i=1,$ 2, $\cdots$, $d-1$. The requirement of the Clifford algebra for the Dirac matrices in $d$ space-time dimensions $\\{\gamma^{\mu},\gamma^{\nu}\\}=2g^{\mu\nu}\mathbbm{1},$ (15) for $\mu,\,\nu=0$, 1, 2, $\cdots$, $d-1$ forces the anticommutation relations for the Pauli matrices, $\\{\sigma^{i},\sigma^{j}\\}=2\delta^{ij}\mathbbm{1},$ (16) for $i,\,j=1$, 2, $\cdots$, $d-1$. In our computation of the spin-singlet case, we encounter the loop correction to the axial vector current. We carry out the Dirac algebra by making use of the naive dimensional regularization, in which $\gamma_{5}$ anticommutes with $\gamma^{\mu}$ for any indices $\mu$ in $d$ dimensions. This prescription is self consistent for the case considered in this paper [37]. As commented in section 3 of [38], the matrix representation of the Dirac $\gamma_{5}$ that is consistent with the choice (14) is then $\gamma_{5}=\begin{pmatrix}0&\phantom{+}\mathbbm{1}\\\ \mathbbm{1}&\phantom{+}0\end{pmatrix}.$ (17) which guarantees $\\{\gamma^{\mu},\gamma_{5}\\}=0$. With the matrix representations for the spinors in (13) and with the set of Dirac matrices (14) and (17), we can carry out the calculation consistent with naive dimensional regularization. For $d-1=3$, the Pauli matrices satisfy the commutation relations $[\sigma^{i},\sigma^{j}]=2i\epsilon^{ijk}\sigma^{k}.$ (18) However, for the spatial dimensions greater than 3 the totally antisymmetric combination of three Pauli matrices $\\{[\sigma^{i},\sigma^{j}],\sigma^{k}\\}$, which may arise in the threshold expansion of the products of three or more Dirac matrices of different spatial indices [34], is linearly independent of both $\mathbbm{1}$ and $\sigma^{\ell}$ for $\ell=1$, 2, $\cdots$, $d-1$. And the reduction $\\{[\sigma^{i},\sigma^{j}],\sigma^{k}\\}=4i\epsilon^{ijk}\mathbbm{1}$ (19) is allowed only at $3$ spatial dimensions [35]. Therefore, unless divergent contributions disappear, we do not use the reduction (19). ## 4 Formulas for short distance coefficients Let us first classify the operators that appear in the matching conditions (6) and (7). For the spin-singlet $S$-wave case, only a single type of operators $\mathcal{O}_{n}$ contributes and for the spin-triplet $S$-wave case, there are two kinds of operators $\mathcal{O}^{i}_{An}$ and $\mathcal{O}^{i}_{Bn}$: $\displaystyle\mathcal{O}_{n}$ $\displaystyle=$ $\displaystyle\chi_{b}^{\dagger}(-\tfrac{i}{2}\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{\nabla}}})^{2n}\psi_{c},$ (20a) $\displaystyle\mathcal{O}^{i}_{An}$ $\displaystyle=$ $\displaystyle\chi_{b}^{\dagger}(-\tfrac{i}{2}\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{\nabla}}})^{2n}\sigma^{i}\psi_{c},$ (20b) $\displaystyle\mathcal{O}^{i}_{Bn}$ $\displaystyle=$ $\displaystyle\chi_{b}^{\dagger}(-\tfrac{i}{2}\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{\nabla}}})^{2n-2}(-\tfrac{i}{2}\stackrel{{\scriptstyle\leftrightarrow}}{{{\nabla}}}^{i})(-\tfrac{i}{2}\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{\nabla}}})\cdot\bm{\sigma}\psi_{c},$ (20c) where $\psi_{c}$ is the Pauli spinor field that annihilates the charm quark and $\chi_{b}^{\dagger}$ is the Pauli spinor field that annihilates the antibottom quark. The operators in (20) contain ordinary derivatives, rather than covariant derivatives so that they are not gauge invariant. We evaluate their matrix elements in the Coulomb gauge, in which inclusion of the $\bar{b}c$ operators involving the gauge fields brings in corrections of relative order $v_{Q}^{4}$ [23]. While the operators $\mathcal{O}_{n}$ and $\mathcal{O}^{i}_{An}$ have only the $S$-wave contributions, the operator $\mathcal{O}^{i}_{Bn}$ also contains the $D$-wave contribution as well as the $S$-wave one. The operator ${\cal O}_{Bn}^{i}$ can be decomposed into a linear combination of ${\cal O}_{An}^{i}$ and the $D$-wave operator ${\cal O}_{Dn}^{i}$: ${\cal O}_{Bn}^{i}=\frac{1}{d-1}{\cal O}_{An}^{i}+{\cal O}_{Dn}^{i},$ (21) where ${\cal O}_{Dn}^{i}$ is defined by ${\cal O}_{Dn}^{i}=\chi_{b}^{\dagger}(-\tfrac{i}{2}\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{\nabla}}})^{2n-2}\left[(-\tfrac{i}{2}\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla}}^{i})(-\tfrac{i}{2}\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{\nabla}}})\cdot\bm{\sigma}-\frac{1}{d-1}(-\tfrac{i}{2}\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{\nabla}}})^{2}\sigma^{i}\right]\psi_{c}.$ (22) In the basis of operators ${\cal O}^{i}_{An}$ and ${\cal O}^{i}_{Bn}$ for the spin-triplet case, the matching conditions (6) and (7) become $\displaystyle i{\cal A}_{\bar{b}c_{1}}^{i(0)}$ $\displaystyle=$ $\displaystyle\sum_{n}a_{n}^{(0)}\langle 0\|{\cal O}^{i}_{An}\|\bar{b}c_{1}\rangle^{(0)}+\sum_{n}b_{n}^{(0)}\langle 0\|{\cal O}^{i}_{Bn}\|\bar{b}c_{1}\rangle^{(0)},$ (23a) $\displaystyle i{\cal A}_{\bar{b}c_{1}}^{i(1)}-\left[i{\cal A}_{\bar{b}c_{1}}^{i(1)}\right]_{\rm NRQCD}$ $\displaystyle=$ $\displaystyle\sum_{n}a_{n}^{(1)}\langle 0\|{\cal O}^{i}_{An}\|\bar{b}c_{1}\rangle^{(0)}+\sum_{n}b_{n}^{(1)}\langle 0\|{\cal O}^{i}_{Bn}\|\bar{b}c_{1}\rangle^{(0)},$ (23b) where $a_{n}$ and $b_{n}$ are the short distance coefficients corresponding to the operators ${\cal O}^{i}_{An}$ and ${\cal O}^{i}_{Bn}$, respectively. A similar equation holds in the basis ${\cal O}^{i}_{An}$ and ${\cal O}^{i}_{Dn}$, where the associated short distance coefficients are $\displaystyle S_{n}$ $\displaystyle=$ $\displaystyle a_{n}+\frac{1}{d-1}\,b_{n},$ (24a) $\displaystyle D_{n}$ $\displaystyle=$ $\displaystyle b_{n},$ (24b) where $S_{n}$ and $D_{n}$ are the $S$-wave and $D$-wave components of the short distance coefficient $V_{n}$, respectively. For more details, see [31]. The $\bar{b}c$ matrix elements for the spin-singlet case in (6a) and (7a) and those for the spin-triplet case in (6b) and (7b) are calculable perturbatively as $\displaystyle\langle 0\|{\cal O}_{n}\|\bar{b}c_{1}\rangle^{(0)}$ $\displaystyle=$ $\displaystyle\bm{q}^{2n}\eta_{b}^{\dagger}\xi_{c},$ (25a) $\displaystyle\langle 0\|{\cal O}_{An}^{i}\|\bar{b}c_{1}\rangle^{(0)}$ $\displaystyle=$ $\displaystyle\bm{q}^{2n}\eta_{b}^{\dagger}\sigma^{i}\xi_{c},$ (25b) $\displaystyle\langle 0\|{\cal O}_{Bn}^{i}\|\bar{b}c_{1}\rangle^{(0)}$ $\displaystyle=$ $\displaystyle\bm{q}^{2n-2}q^{i}\eta^{\dagger}_{b}\bm{q}\cdot\bm{\sigma}\xi_{c},$ (25c) where $\xi_{c}$ and $\eta_{b}$ are two-component spinors for the charm quark and the antibottom quark, respectively. In order to maintain consistency with our calculations in full QCD, we have taken the $\bar{b}c$ states to have the nonrelativistic normalization and we have suppressed the factor $\sqrt{N_{c}}$ that comes from the color trace. The most general Lorentz covariant forms of $i{\cal A}^{\mu}_{5,\bar{b}c_{1}}$ and $i{\cal A}^{\mu}_{\bar{b}c_{1}}$ are $\displaystyle i{\cal A}^{\mu}_{5,\bar{b}c_{1}}$ $\displaystyle=$ $\displaystyle\overline{v}_{b}(p_{\bar{b}})(G_{5}\gamma^{\mu}+H_{5}q^{\mu}+Q_{5}p^{\mu})\gamma_{5}u_{c}(p_{c}),$ (26a) $\displaystyle i{\cal A}^{\mu}_{\bar{b}c_{1}}$ $\displaystyle=$ $\displaystyle\overline{v}_{b}(p_{\bar{b}})(G\gamma^{\mu}+Hq^{\mu}+Qp^{\mu})u_{c}(p_{c}),$ (26b) for the axial vector and vector currents, respectively. Here, $\displaystyle G_{5}$ $\displaystyle=$ $\displaystyle\sqrt{Z_{b}Z_{c}}(1+\Lambda_{5}),$ (27a) $\displaystyle G$ $\displaystyle=$ $\displaystyle\sqrt{Z_{b}Z_{c}}(1+\Lambda),$ (27b) where $Z_{Q}$ for $Q=b$ or $c$ is the wavefunction renormalization factor of the heavy quark $Q$ at order $\alpha_{s}$ [24]: $\sqrt{Z_{Q}}=1-\frac{\alpha_{s}C_{F}}{2\pi}\left(\frac{1}{4\epsilon_{\rm UV}}+\frac{1}{2\epsilon_{\rm IR}}+\frac{3}{4}\log\frac{4\pi\mu^{2}e^{-\gamma_{\rm E}}}{m_{Q}^{2}}+1\right),$ (28) where $C_{F}=(N_{c}^{2}-1)/(2N_{c})=4/3$, $\mu$ is the renormalization constant, $\gamma_{\rm E}$ is the Euler-Mascheroni constant, and the subscripts on $1/\epsilon$ indicate the origins of the singularities. $\Lambda_{5}$ and $\Lambda$ in (27) are the multiplicative corrections to the axial vector and vector vertices, respectively. Note that the terms proportional to $p^{\mu}$ survive in (26) because the weak currents are not conserved while those terms vanish in the electromagnetic current which is conserved. At order $\alpha_{s}^{0}$, the only nonvanishing contributions in (26) are $G^{(0)}_{5}=G^{(0)}=1$ and all the other contributions are absent: $H^{(0)}_{5}=Q^{(0)}_{5}=H^{(0)}=Q^{(0)}=0$. The leading nonvanishing contributions to $H_{5}$, $Q_{5}$, $H$, and $Q$ appear from order $\alpha_{s}$. Similarly, the nonvanishing components of the NRQCD counterparts to $i{\cal A}^{\mu}_{5,\bar{b}c_{1}}$ and $i{\cal A}^{\mu}_{\bar{b}c_{1}}$ in the CM frame of the $\bar{b}c$ pair are $\displaystyle i\left[\mathcal{A}_{5,\bar{b}c_{1}}^{0}\right]_{\textrm{NRQCD}}$ $\displaystyle=$ $\displaystyle\bar{v}_{b}(p_{\bar{b}})(G_{5,\textrm{NRQCD}}\gamma^{0}+H_{5,\textrm{NRQCD}}q^{0}+Q_{5,\textrm{NRQCD}}p^{0})\gamma_{5}u_{c}(p_{c}),\phantom{xxx}$ (29a) $\displaystyle i\left[\mathcal{A}_{\bar{b}c_{1}}^{i}\right]_{\textrm{NRQCD}}$ $\displaystyle=$ $\displaystyle\bar{v}_{b}(p_{\bar{b}})(G_{\textrm{NRQCD}}\gamma^{i}+H_{\textrm{NRQCD}}q^{i})u_{c}(p_{c}),$ (29b) where we have used $p^{i}=0$ in the CM frame of the $\bar{b}c$ pair and $\displaystyle G_{5,\textrm{NRQCD}}$ $\displaystyle=$ $\displaystyle[\sqrt{Z_{b}}\,]_{\textrm{NRQCD}}[\sqrt{Z_{c}}\,]_{\textrm{NRQCD}}(1+\Lambda_{5,\textrm{NRQCD}}),$ (30a) $\displaystyle G_{\textrm{NRQCD}}$ $\displaystyle=$ $\displaystyle[\sqrt{Z_{b}}\,]_{\textrm{NRQCD}}[\sqrt{Z_{c}}\,]_{\textrm{NRQCD}}(1+\Lambda_{\textrm{NRQCD}}).$ (30b) The heavy quark wavefunction renormalization in NRQCD, $[\sqrt{Z_{Q}}\,]_{\textrm{NRQCD}}$ for $Q=b$ or $c$ at order $\alpha_{s}$, is given by [31] $\big{[}\sqrt{Z_{Q}}\,\big{]}_{\rm NRQCD}=1+\frac{\alpha_{s}C_{F}}{2\pi}\left(\frac{1}{\epsilon_{\rm UV}}-\frac{1}{\epsilon_{\rm IR}}\right).$ (31) By making use of the Dirac spinors in (13) and the Dirac matrices in (14) and (17), we find that the expressions in (26) and (29) are reduced into $\displaystyle\bar{v}_{b}(p_{\bar{b}})\gamma^{0}\gamma_{5}u_{c}(p_{c})$ $\displaystyle=$ $\displaystyle\mathcal{N}_{b}\mathcal{N}_{c}[(E_{b}+m_{b})(E_{c}+m_{c})-\bm{q}^{2}]\,\eta_{b}^{\dagger}\xi_{c},$ (32a) $\displaystyle\bar{v}_{b}(p_{\bar{b}})\gamma_{5}u_{c}(p_{c})$ $\displaystyle=$ $\displaystyle-\mathcal{N}_{b}\mathcal{N}_{c}[(E_{b}+m_{b})(E_{c}+m_{c})+\bm{q}^{2}]\,\eta_{b}^{\dagger}\xi_{c},$ (32b) $\displaystyle\bar{v}_{b}(p_{\bar{b}})\gamma^{i}u_{c}(p_{c})$ $\displaystyle=$ $\displaystyle\mathcal{N}_{b}\mathcal{N}_{c}\big{\\{}[(E_{b}+m_{b})(E_{c}+m_{c})+\bm{q}^{2}]\,\eta_{b}^{\dagger}\sigma^{i}\xi_{c}-2q^{i}\,\eta_{b}^{\dagger}\bm{q}\cdot\bm{\sigma}\xi_{c}\big{\\}},\phantom{xx}$ (32c) $\displaystyle\bar{v}_{b}(p_{\bar{b}})u_{c}(p_{c})$ $\displaystyle=$ $\displaystyle-\mathcal{N}_{b}\mathcal{N}_{c}(E_{b}+m_{b}+E_{c}+m_{c})\,\eta_{b}^{\dagger}\bm{q}\cdot\bm{\sigma}\xi_{c},$ (32d) where the expressions in (32) are valid to all orders in $\bm{q}$ and we have used the identities $\displaystyle(\bm{q}\cdot\bm{\sigma})^{2}$ $\displaystyle=$ $\displaystyle\bm{q}^{2},$ (33a) $\displaystyle(\bm{q}\cdot\bm{\sigma})\sigma^{i}(\bm{q}\cdot\bm{\sigma})$ $\displaystyle=$ $\displaystyle 2q^{i}(\bm{q}\cdot\bm{\sigma})-\sigma^{i}\bm{q}^{2},$ (33b) that derive from (16) in $d-1$ spatial dimensions. Note that we do not encounter the products of Dirac matrices which involve products of three Pauli matrices of different indices that bring in the contribution in (19). Therefore, the threshold expansion (32) is free of ambiguities in $d-1$ spatial dimensions. Substituting (32) into (26), we find that $\displaystyle i{\cal A}^{0}_{5,\bar{b}c_{1}}$ $\displaystyle=$ $\displaystyle\mathcal{N}_{b}\mathcal{N}_{c}\bigg{\\{}G_{5}\big{[}(E_{b}+m_{b})(E_{c}+m_{c})-\bm{q}^{2}\big{]}$ (34a) $\displaystyle\qquad\quad-(H_{5}q^{0}+Q_{5}p^{0})\big{[}(E_{b}+m_{b})(E_{c}+m_{c})+\bm{q}^{2}\big{]}\bigg{\\}}\eta_{b}^{\dagger}\xi_{c},$ $\displaystyle i{\cal A}^{i}_{\bar{b}c_{1}}$ $\displaystyle=$ $\displaystyle\mathcal{N}_{b}\mathcal{N}_{c}\bigg{\\{}G\big{[}(E_{b}+m_{b})(E_{c}+m_{c})+\bm{q}^{2}\big{]}\eta_{b}^{\dagger}\sigma^{i}\xi_{c}$ (34b) $\displaystyle\qquad\quad-\big{[}2G+(E_{b}+m_{b}+E_{c}+m_{c})H\big{]}q^{i}\eta_{b}^{\dagger}(\bm{\sigma}\cdot\bm{q})\xi_{c}\bigg{\\}}.$ Similarly, the NRQCD amplitudes $[i{\cal A}^{0}_{5,\bar{b}c_{1}}]_{\rm NRQCD}$ and $[i{\cal A}^{i}_{\bar{b}c_{1}}]_{\rm NRQCD}$ in (29) are the same as $i{\cal A}^{0}_{5,\bar{b}c_{1}}$ and $i{\cal A}^{i}_{\bar{b}c_{1}}$ in (34) except that the coefficients $F$ and $F_{5}$ are replaced with $F_{\rm NRQCD}$ and $F_{5,{\rm NRQCD}}$, respectively, for $F=G$, $H$, and $Q$. According to the matching conditions in (6), $[i{\cal A}^{0(0)}_{5,\bar{b}c_{1}}]_{\rm NRQCD}$ and $[i{\cal A}^{i(0)}_{\bar{b}c_{1}}]_{\rm NRQCD}$ are identical to the order-$\alpha_{s}^{0}$ full QCD counterparts in (34). In order to obtain the short distance coefficients at order $\alpha_{s}^{0}$, we need to expand $[i{\cal A}^{0(0)}_{5,\bar{b}c_{1}}]_{\rm NRQCD}=i{\cal A}^{0(0)}_{5,\bar{b}c_{1}}$ and $[i{\cal A}^{i(0)}_{\bar{b}c_{1}}]_{\rm NRQCD}$ $=i{\cal A}^{i(0)}_{\bar{b}c_{1}}$ as linear combinations of the perturbative NRQCD matrix elements in (25). By making use of these order-$\alpha_{s}^{0}$ perturbative matrix elements and the matching conditions in (6), (23a), and (24), we can obtain the short distance coefficients at order $\alpha_{s}^{0}$: $\displaystyle P^{(0)}_{n}$ $\displaystyle=$ $\displaystyle\left.\frac{1}{n!}\left(\frac{\partial}{\partial\bm{q}^{2}}\right)^{n}\mathcal{N}_{b}\mathcal{N}_{c}\big{[}(E_{b}+m_{b})(E_{c}+m_{c})-\bm{q}^{2}\big{]}\right\|_{\bm{q}^{2}=0},$ (35a) $\displaystyle S^{(0)}_{n}$ $\displaystyle=$ $\displaystyle a_{n}^{(0)}+\frac{1}{3}\,b_{n}^{(0)},$ (35b) $\displaystyle D^{(0)}_{n}$ $\displaystyle=$ $\displaystyle b_{n}^{(0)},$ (35c) where we have used the fact that $G^{(0)}_{5}=G^{(0)}=1$ and $H^{(0)}_{5}=Q_{5}^{(0)}=H^{(0)}=Q^{(0)}=0$ and the short distance coefficients $a_{n}^{(0)}$ and $b_{n}^{(0)}$ are given by $\displaystyle a_{n}^{(0)}$ $\displaystyle=$ $\displaystyle\left.\frac{1}{n!}\left(\frac{\partial}{\partial\bm{q}^{2}}\right)^{n}\mathcal{N}_{b}\mathcal{N}_{c}\big{[}(E_{b}+m_{b})(E_{c}+m_{c})+\bm{q}^{2}\big{]}\right\|_{\bm{q}^{2}=0},$ (36a) $\displaystyle b_{n}^{(0)}$ $\displaystyle=$ $\displaystyle-\left.\frac{2}{(n-1)!}\left(\frac{\partial}{\partial\bm{q}^{2}}\right)^{n-1}\mathcal{N}_{b}\mathcal{N}_{c}\right\|_{\bm{q}^{2}=0}.$ (36b) At order $\alpha_{s}^{1}$, the quantities $\Lambda_{5}$, $\Lambda$, and $Z_{Q}$ in (27) contain divergences. The multiplicative vertex correction factors $\Lambda_{5}$ and $\Lambda$ have logarithmic divergences in the UV and IR regions. They also contain Coulomb divergence, which is not analytic in the limit $\|\bm{q}\|\to 0$. The wavefunction renormalization constant $Z_{Q}$ has logarithmic divergences in the UV and IR regions. However, because of the usual cancellation between the vertex and fermion wavefunction renormalizations, $G_{5}^{(1)}$ and $G^{(1)}$ are free of UV divergences and contain only IR divergences. The quantities $H^{(1)}_{5}$, $Q^{(1)}_{5}$, $H^{(1)}$, and $Q^{(1)}$, which contribute only from this order, may have only Coulomb divergences in the limit $\|\bm{q}\|\to 0$. Therefore, in the full QCD amplitudes $i{\cal A}^{0(1)}_{5,\bar{b}c_{1}}$ and $i{\cal A}^{i(1)}_{\bar{b}c_{1}}$, UV divergences cancel and the amplitudes may have singularities only in the IR region, which are either Coulomb or logarithmic divergences. Because NRQCD reproduces full QCD in the IR region, the IR divergences in $G^{(1)}_{5,\textrm{NRQCD}}$, $H^{(1)}_{5,\textrm{NRQCD}}$, $Q^{(1)}_{5,\textrm{NRQCD}}$, $G^{(1)}_{\textrm{NRQCD}}$, and $H^{(1)}_{\textrm{NRQCD}}$, cancel those in $G_{5}^{(1)}$, $H_{5}^{(1)}$, $Q_{5}^{(1)}$, $G^{(1)}$, and $H^{(1)}$, respectively, to make the following quantities IR finite: $\displaystyle\Delta G_{5}^{(1)}$ $\displaystyle=$ $\displaystyle G_{5}^{(1)}-G_{5,\rm NRQCD}^{(1)},$ (37a) $\displaystyle\Delta H_{5}^{(1)}$ $\displaystyle=$ $\displaystyle H_{5}^{(1)}-H_{5,\rm NRQCD}^{(1)},$ (37b) $\displaystyle\Delta Q_{5}^{(1)}$ $\displaystyle=$ $\displaystyle Q_{5}^{(1)}-Q_{5,\rm NRQCD}^{(1)},$ (37c) $\displaystyle\Delta G^{(1)}$ $\displaystyle=$ $\displaystyle G^{(1)}-G_{\rm NRQCD}^{(1)},$ (37d) $\displaystyle\Delta H^{(1)}$ $\displaystyle=$ $\displaystyle H^{(1)}-H_{\rm NRQCD}^{(1)}.$ (37e) Therefore, the right sides of (7) which are determined by the expressions in (37) are free of IR sensitivities. Now we can obtain the short distance coefficients at order $\alpha_{s}^{1}$, by making use of the matching conditions in (7), (23b), and (24) as $\displaystyle P^{(1)}_{n}$ $\displaystyle=$ $\displaystyle\frac{1}{n!}\left(\frac{\partial}{\partial\bm{q}^{2}}\right)^{n}\mathcal{N}_{b}\mathcal{N}_{c}\bigg{\\{}\Delta G_{5}^{(1)}\big{[}(E_{b}+m_{b})(E_{c}+m_{c})-\bm{q}^{2}\big{]}$ (38a) $\displaystyle\qquad\quad-[\Delta H_{5}^{(1)}q^{0}+\Delta Q_{5}^{(1)}p^{0}]\big{[}(E_{b}+m_{b})(E_{c}+m_{c})+\bm{q}^{2}\big{]}\bigg{\\}}\Bigg{\|}_{\bm{q}^{2}=0},$ $\displaystyle S^{(1)}_{n}$ $\displaystyle=$ $\displaystyle a_{n}^{(1)}+\frac{1}{d-1}\,b_{n}^{(1)},$ (38b) $\displaystyle D^{(1)}_{n}$ $\displaystyle=$ $\displaystyle b_{n}^{(1)},$ (38c) where the short distance coefficients $a_{n}^{(1)}$ and $b_{n}^{(1)}$ are given by $\displaystyle a_{n}^{(1)}$ $\displaystyle=$ $\displaystyle\left.\frac{1}{n!}\left(\frac{\partial}{\partial\bm{q}^{2}}\right)^{n}\mathcal{N}_{b}\mathcal{N}_{c}\Delta G^{(1)}\big{[}(E_{b}+m_{b})(E_{c}+m_{c})+\bm{q}^{2}\big{]}\right\|_{\bm{q}^{2}=0},$ (39a) $\displaystyle b_{n}^{(1)}$ $\displaystyle=$ $\displaystyle-\left.\frac{1}{(n-1)!}\left(\frac{\partial}{\partial\bm{q}^{2}}\right)^{n-1}\\!\\!\\!\\!\\!\mathcal{N}_{b}\mathcal{N}_{c}\big{[}2\Delta G^{(1)}+(E_{b}+m_{b}+E_{c}+m_{c})\Delta H^{(1)}\big{]}\right\|_{\bm{q}^{2}=0}.$ Although we have completely removed the IR singularities in the short distance coefficients (38) and (39), they have logarithmic UV divergences that are originated from the one-loop NRQCD matrix elements in $G_{5,\rm NRQCD}^{(1)}$ and $G_{\rm NRQCD}^{(1)}$. As stated before, the quantities $\Delta H_{5}^{(1)}$, $\Delta Q_{5}^{(1)}$, and $\Delta H^{(1)}$ are free of UV divergences as well as IR divergences. We renormalize $\Delta G_{5}^{(1)}$, $\Delta G^{(1)}$, and the short distance coefficients in (38) and (39) according to the $\overline{\rm MS}$ scheme to find that $\big{[}f^{\,(1)}_{n}\big{]}_{\overline{\rm MS}}=f^{\,(1)}_{n}\big{\|}_{\Delta F^{(1)}\to\,\Delta F^{(1)}_{\overline{\rm MS}}},$ (40) where $F=G_{5}$ or $G$ and $f=P$, $a$, $b$, $S$, and $D$. In deriving the expression for $\big{[}f_{n}^{\,(1)}\big{]}_{\overline{\rm MS}}$ in (40), we have used the fact that, in minimal subtraction, one removes the $1/\epsilon$ pole times the order-$\alpha_{s}^{0}$ $d$-dimensional matrix element. Hence, a term proportional to $(d-1)^{-1}\epsilon^{-1}$ is subtracted in (38b) in carrying out the renormalization. ## 5 QCD corrections In this section, we compute the one-loop QCD corrections to the axial vector and vector parts of the charged weak current $i\mathcal{A}_{5,\bar{b}c_{1}}^{0(1)}$ and $i\mathcal{A}_{\bar{b}c_{1}}^{0(1)}$, respectively. The order-$\alpha_{s}$ QCD corrections are composed of the vertex corrections and the wavefunction renormalization contributions. In the Feynman gauge, the vertex correction contributions to $i\mathcal{A}_{5,\bar{b}c_{1}}^{0(1)}$ and $i\mathcal{A}_{\bar{b}c_{1}}^{0(1)}$ are given by $\displaystyle\Lambda^{\mu}_{5}$ $\displaystyle=$ $\displaystyle- ig_{s}^{2}C_{F}\int_{k}\frac{\overline{v}(p_{\bar{b}})\gamma_{\alpha}(-\\!\not{\\!p_{\bar{b}}}\,+\\!\not{\\!k}+m_{b})\gamma^{\mu}\gamma_{5}(\not{\\!p_{c}}+\\!\not{\\!k}+m_{c})\gamma^{\alpha}u(p_{c})}{D_{0}D_{1}D_{2}},$ (41a) $\displaystyle\Lambda^{\mu}$ $\displaystyle=$ $\displaystyle- ig_{s}^{2}C_{F}\int_{k}\frac{\overline{v}(p_{\bar{b}})\gamma_{\alpha}(-\\!\not{\\!p_{\bar{b}}}\,+\\!\not{\\!k}+m_{b})\gamma^{\mu}(\not{\\!p_{c}}+\\!\not{\\!k}+m_{c})\gamma^{\alpha}u(p_{c})}{D_{0}D_{1}D_{2}},$ (41b) where $g_{s}^{2}=4\pi\alpha_{s}$ is the strong coupling and the symbol $\int_{k}$ and the denominator factors $D_{i}$’s are defined by $\displaystyle\int_{k}$ $\displaystyle\equiv$ $\displaystyle\mu^{2\epsilon}\int\frac{d^{d}k}{(2\pi)^{d}},$ (42a) $\displaystyle D_{0}$ $\displaystyle=$ $\displaystyle k^{2}+i\varepsilon,$ (42b) $\displaystyle D_{1}$ $\displaystyle=$ $\displaystyle k^{2}-2k\cdot p_{\bar{b}}+i\varepsilon,$ (42c) $\displaystyle D_{2}$ $\displaystyle=$ $\displaystyle k^{2}+2k\cdot p_{c}+i\varepsilon.$ (42d) Here, $\mu$ is the renormalization scale and we have taken the gluon momentum as the loop momentum $k$. By making use of the anticommutation relation of $\gamma_{5}$ and $\gamma^{\mu}$ we rearrange each term of the numerator in (41) as a linear combination of terms $\gamma_{\alpha}\Gamma_{i}\gamma^{\alpha}\gamma_{5}$ or $\gamma_{\alpha}\Gamma_{i}\gamma^{\alpha}$, where $\Gamma_{i}$ is a product of three or less Dirac matrices. Applying the anticommutation relation (15) summed over the $d$-dimensional index $\alpha$ and making use of the on-shell conditions $\displaystyle/\\!\\!\\!p_{c}u(p_{c})$ $\displaystyle=$ $\displaystyle m_{c}u(p_{c}),$ (43a) $\displaystyle/\\!\\!\\!p_{\bar{b}}v(p_{\bar{b}})$ $\displaystyle=$ $\displaystyle-m_{b}v(p_{\bar{b}}),$ (43b) we can reduce the expressions in (41) as $\displaystyle\Lambda^{\mu}_{5}$ $\displaystyle=$ $\displaystyle- ig_{s}^{2}C_{F}\int_{k}\frac{\overline{v}(p_{\bar{b}})\Gamma^{\mu}(k,p,q,m_{b},-m_{c})\gamma_{5}u(p_{c})}{D_{0}D_{1}D_{2}},$ (44a) $\displaystyle\Lambda^{\mu}$ $\displaystyle=$ $\displaystyle- ig_{s}^{2}C_{F}\int_{k}\frac{\overline{v}(p_{\bar{b}})\Gamma^{\mu}(k,p,q,m_{b},m_{c})u(p_{c})}{D_{0}D_{1}D_{2}},$ (44b) where $\Gamma^{\mu}(k,p,q,m_{b},m_{c})$ is defined by $\displaystyle\Gamma^{\mu}(k,p,q,m_{b},m_{c})$ $\displaystyle=$ $\displaystyle\big{[}(d-2)k^{2}-2(4p^{2}-m_{b}^{2}-m_{c}^{2})+8k\cdot q\big{]}\gamma^{\mu}$ (45) $\displaystyle+2(m_{b}\gamma^{\mu}/\\!\\!\\!k+m_{c}/\\!\\!\\!k\gamma^{\mu})+2(2-d)k^{\mu}/\\!\\!\\!k-8q^{\mu}/\\!\\!\\!k.$ By making use of the standard reduction methods for the tensor loop integrals, we can express all of the loop-momentum dependence in terms of $p_{\bar{b}}$ and $p_{c}$, which are linear combinations of $p$ and $q$. Because $p\cdot q\neq 0$, the reduction formulas for the tensor integrals of the $S$-wave $\bar{b}c_{1}$ decay are slightly more complicated than those for the spin- triplet $S$-wave $Q\bar{Q}_{1}$ decay in [31] where $p\cdot q=0$. In appendix A we list the formulas for the tensor reduction. Once we apply the equations of motion in (43), $\Gamma^{\mu}$’s in (44) are reduced into a linear combination of $\gamma^{\mu}$, $p^{\mu}\mathbbm{1}$, and $q^{\mu}\mathbbm{1}$ as $\displaystyle\Lambda^{\mu}_{5}$ $\displaystyle=$ $\displaystyle- ig_{s}^{2}C_{F}\overline{v}(p_{\bar{b}})\big{[}A_{1}(-m_{c})\gamma^{\mu}+A_{2}(-m_{c})p^{\mu}+A_{3}(-m_{c})q^{\mu}\big{]}\gamma_{5}u(p_{c}),$ (46a) $\displaystyle\Lambda^{\mu}$ $\displaystyle=$ $\displaystyle- ig_{s}^{2}C_{F}\overline{v}(p_{\bar{b}})\big{[}A_{1}(m_{c})\gamma^{\mu}+A_{2}(m_{c})p^{\mu}+A_{3}(m_{c})q^{\mu}\big{]}u(p_{c}).$ (46b) As mentioned in [25], the Lorentz scalar coefficients $A_{i}$’s in (46a) and (46b) are the same except for the replacement $A_{i}(-m_{c})\leftrightarrow A_{i}(m_{c})$ and $A_{i}(m_{c})$’s are defined by $\displaystyle A_{1}(m_{c})$ $\displaystyle=$ $\displaystyle(d-2)(J_{1}-2J_{4})-4p_{\bar{b}}\cdot p_{c}J_{2}+4J_{3}+\frac{(m_{b}-m_{c})^{2}J_{3}+(m_{b}^{2}-m_{c}^{2})J_{5}}{p_{\bar{b}}\cdot p_{c}+m_{b}m_{c}},\phantom{xxxxx}$ (47a) $\displaystyle A_{2}(m_{c})$ $\displaystyle=$ $\displaystyle 2\bigg{[}\frac{(m_{b}-m_{c})J_{3}+(m_{b}+m_{c})J_{5}}{p_{\bar{b}}\cdot p_{c}+m_{b}m_{c}}+\frac{(2-d)J_{7}(m_{c})}{2p^{2}q^{2}}\bigg{]},$ (47b) $\displaystyle A_{3}(m_{c})$ $\displaystyle=$ $\displaystyle 2\bigg{[}\frac{(m_{b}+m_{c})J_{3}+(m_{b}-m_{c})J_{5}}{p_{\bar{b}}\cdot p_{c}-m_{b}m_{c}}+\frac{(2-d)J_{6}(m_{c})}{2q^{2}}\bigg{]}.$ (47c) Here, the $J_{i}$’s are scalar integrals which are defined and evaluated in appendix B. Except for the two scalar integrals $J_{6}$ and $J_{7}$, the other scalar integrals $J_{i}$ for $i\leq 5$ are even functions of $m_{c}$ and, therefore, we suppress the arguments of $J_{i}$ for $i\leq 5$. The expressions in (46) are new, in which the relativistic corrections to all orders in $\bm{q}^{2}$ are included. According to (26a) and (27), the vertex corrections can be parametrized as $\Lambda^{\mu}_{5}=\bar{v}(p_{\bar{b}})(\Lambda_{5}\gamma^{\mu}+H_{5}p^{\mu}+Q_{5}q^{\mu})\gamma_{5}u(p_{c})$ and $\Lambda^{\mu}=\bar{v}(p_{\bar{b}})(\Lambda\gamma^{\mu}+Hp^{\mu}+Qq^{\mu})u(p_{c})$. Therefore, the multiplicative factors $\Lambda_{5}$ and $\Lambda$ for the vertex corrections can be determined by $A_{1}(\mp m_{c})$, which are the coefficients of $\gamma^{\mu}$ in (46). In similar ways, $H_{5}$ and $H$ are determined by $A_{2}(\mp m_{c})$ and $Q_{5}$ and $Q$ are determined by $A_{3}(\mp m_{c})$, respectively. Substituting the values for the scalar integrals $J_{i}$’s evaluated in appendix B into $A_{1}(\mp m_{c})$ in (47) and then substituting $A_{1}(\mp m_{c})$ into (46), we obtain the multiplicative factors for the vertex corrections as $\Lambda_{5}=-ig_{s}^{2}C_{F}A_{1}(-m_{c})$ and $\Lambda=-ig_{s}^{2}C_{F}A_{1}(m_{c})$. The results are $\displaystyle\Lambda_{5}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}C_{F}}{4\pi}\Bigg{\\{}\frac{1}{\epsilon_{\rm UV}}+\frac{1}{2}\log\frac{(4\pi\mu^{2}e^{-\gamma_{\rm E}})^{2}}{m_{b}^{2}m_{c}^{2}}+\bigg{[}\frac{p_{\bar{b}}\cdot p_{c}}{2p^{2}}\bigg{(}\frac{1}{\epsilon_{\rm IR}}+\frac{1}{2}\log\frac{(4\pi\mu^{2}e^{-\gamma_{\rm E}})^{2}}{m_{b}^{2}m_{c}^{2}}\bigg{)}$ (48a) $\displaystyle+\,\delta^{2}\left(6+\frac{2(m_{b}+m_{c})^{2}}{p_{\bar{b}}\cdot p_{c}-m_{b}m_{c}}\right)\bigg{]}\left[L_{1}(\delta,e_{c})-\frac{i\pi}{\delta}\right]-\frac{p_{\bar{b}}\cdot p_{c}}{p^{2}}\,K(\delta,e_{c})-\frac{1}{8p^{2}}\big{[}m_{b}^{2}-m_{c}^{2}$ $\displaystyle+2p_{\bar{b}}\cdot p_{c}\,L_{2}(\delta,e_{c})\big{]}\log\frac{m_{c}^{2}}{m_{b}^{2}}+\frac{p_{\bar{b}}\cdot p_{c}}{2p^{2}}\bigg{[}\frac{\pi^{2}}{\delta}-\frac{i\pi}{2\delta}\log\frac{p^{4}m_{b}^{2}m_{c}^{2}}{[(p_{\bar{b}}\cdot p_{c})^{2}-m_{b}^{2}m_{c}^{2}]^{2}}\bigg{]}\Bigg{\\}},$ $\displaystyle\Lambda$ $\displaystyle=$ $\displaystyle\Lambda_{5}\|_{m_{c}\to-m_{c}},$ (48b) where $\Lambda_{5}\|_{m_{c}\to-m_{c}}$ represents the expression that can be obtained from $\Lambda_{5}$ after replacing $m_{c}$ with $-m_{c}$ and the variables $\delta$ and $e_{c}$ are defined in (11). The functions $L_{1}(\delta,e_{c})$, $L_{2}(\delta,e_{c})$, and $K(\delta,e_{c})$ are defined by $\displaystyle L_{1}(\delta,e_{c})$ $\displaystyle=$ $\displaystyle\frac{1}{2\delta}\log\frac{(e_{c}+\delta)}{(e_{c}-\delta)}\frac{(1-e_{c}+\delta)}{(1-e_{c}-\delta)}$ (49a) $\displaystyle=$ $\displaystyle\frac{(m_{b}+m_{c})^{2}}{m_{b}m_{c}}-\frac{(m_{b}+m_{c})^{2}\left(m_{b}^{2}-4m_{c}m_{b}+m_{c}^{2}\right)}{6m_{b}^{3}m_{c}^{3}}\,{\bm{q}}^{2}+{O}({\bm{q}}^{4}),$ $\displaystyle L_{2}(\delta,e_{c})$ $\displaystyle=$ $\displaystyle\frac{1}{2\delta}\log\frac{(e_{c}+\delta)}{(e_{c}-\delta)}\frac{(1-e_{c}-\delta)}{(1-e_{c}+\delta)}$ (49b) $\displaystyle=$ $\displaystyle\frac{m_{b}^{2}-m_{c}^{2}}{m_{b}m_{c}}-\frac{(m_{b}+m_{c})(m_{b}-m_{c})^{3}}{6m_{b}^{3}m_{c}^{3}}\,{\bm{q}}^{2}+{O}({\bm{q}}^{4}),$ $\displaystyle K(\delta,e_{c})$ $\displaystyle=$ $\displaystyle\frac{1}{4\delta}\left[{\rm Sp}\\!\left(\frac{2\delta}{e_{c}+\delta}\right)-{\rm Sp}\\!\left(\frac{-2\delta}{e_{c}-\delta}\right)+{\rm Sp}\\!\left(\frac{2\delta}{1-e_{c}+\delta}\right)-{\rm Sp}\\!\left(\frac{-2\delta}{1-e_{c}-\delta}\right)\right]$ (49c) $\displaystyle=$ $\displaystyle\frac{(m_{b}+m_{c})^{2}}{m_{b}m_{c}}-\frac{(m_{b}+m_{c})^{2}\left(m_{b}^{2}-10m_{c}m_{b}+m_{c}^{2}\right)}{18m_{b}^{3}m_{c}^{3}}\,{\bm{q}}^{2}+{O}({\bm{q}}^{4}).\phantom{xxxxx}$ We have listed the first two leading terms in the nonrelativistic expansions of these functions. According to (49), the functions $L_{1}(\delta,e_{c})$, $L_{2}(\delta,e_{c})$, and $K(\delta,e_{c})$ are of order 1 as $\|\bm{q}\|\to 0$. The function ${\rm Sp}(x)$ in (49c) is the Spence function, which is defined by ${\rm Sp}(x)=\int_{x}^{0}\frac{\log(1-t)}{t}dt.$ (50) Now we evaluate $G_{5}$ and $G$. Substituting $\Lambda_{5}$ and $\Lambda$ in (48) and the heavy quark wavefunction renormalization constant $Z_{Q}$ for $Q=b$ and $c$ in (28) into (27), we obtain $\displaystyle G_{5}$ $\displaystyle=$ $\displaystyle 1+\frac{\alpha_{s}C_{F}}{4\pi}\Bigg{\\{}\bigg{[}\frac{p_{\bar{b}}\cdot p_{c}}{2p^{2}}\left(L_{1}(\delta,e_{c})-\frac{i\pi}{\delta}\right)-2\bigg{]}\bigg{(}\frac{1}{\epsilon_{\rm IR}}+\frac{1}{2}\log\frac{(4\pi\mu^{2}e^{-\gamma_{\rm E}})^{2}}{m_{b}^{2}m_{c}^{2}}\bigg{)}$ (51a) $\displaystyle+\bigg{[}6+\frac{2(m_{b}+m_{c})^{2}}{p_{\bar{b}}\cdot p_{c}-m_{b}m_{c}}\bigg{]}\delta^{2}\bigg{[}L_{1}(\delta,e_{c})-\frac{i\pi}{\delta}\bigg{]}-\frac{p_{\bar{b}}\cdot p_{c}}{p^{2}}\,K(\delta,e_{c})-4-\frac{1}{8p^{2}}\big{[}m_{b}^{2}-m_{c}^{2}$ $\displaystyle+2p_{\bar{b}}\cdot p_{c}\,L_{2}(\delta,e_{c})\big{]}\log\frac{m_{c}^{2}}{m_{b}^{2}}+\frac{p_{\bar{b}}\cdot p_{c}}{2p^{2}}\bigg{[}\frac{\pi^{2}}{\delta}-\frac{i\pi}{2\delta}\log\frac{p^{4}m_{b}^{2}m_{c}^{2}}{[(p_{\bar{b}}\cdot p_{c})^{2}-m_{b}^{2}m_{c}^{2}]^{2}}\bigg{]}\Bigg{\\}},$ $\displaystyle G$ $\displaystyle=$ $\displaystyle G_{5}\|_{m_{c}\to-m_{c}}.$ (51b) As we have expected, $G_{5}$ and $G$ are free of UV divergences while they have Coulomb and logarithmic divergences in the IR region. Next we substitute the values for the scalar integrals $J_{i}$’s evaluated in appendix B into $A_{2}(\mp m_{c})$ and $A_{3}(\mp m_{c})$ in (47). Then we can determine $H_{5}=-ig_{s}^{2}C_{F}A_{2}(-m_{c})$, $Q_{5}=-ig_{s}^{2}C_{F}A_{3}(-m_{c})$, $H=-ig_{s}^{2}C_{F}A_{2}(m_{c})$, and $Q=-ig_{s}^{2}C_{F}A_{3}(m_{c})$. The results are $\displaystyle H_{5}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}C_{F}}{4\pi}\;\frac{2(m_{b}-m_{c})}{p_{\bar{b}}\cdot p_{c}+m_{b}m_{c}}\bigg{\\{}\delta^{2}\bigg{[}L_{1}(\delta,e_{c})-\frac{i\pi}{\delta}\bigg{]}+r_{-}\log\frac{m_{c}^{2}}{m_{b}^{2}}\bigg{\\}},$ (52a) $\displaystyle Q_{5}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}C_{F}}{4\pi}(m_{b}+m_{c})\Bigg{(}\frac{4}{p_{\bar{b}}\cdot p_{c}-m_{b}m_{c}}\bigg{\\{}\delta^{2}\bigg{[}L_{1}(\delta,e_{c})-\frac{i\pi}{\delta}\bigg{]}+r_{+}\log\frac{m_{c}^{2}}{m_{b}^{2}}\bigg{\\}}$ (52b) $\displaystyle-\frac{(m_{b}-m_{c})^{2}}{2p^{2}(p_{\bar{b}}\cdot p_{c}+m_{b}m_{c})}\bigg{\\{}\delta^{2}\bigg{[}L_{1}(\delta,e_{c})-\frac{i\pi}{\delta}\bigg{]}+r_{-}\log\frac{m_{c}^{2}}{m_{b}^{2}}\bigg{\\}}$ $\displaystyle+\frac{1}{p^{2}}\bigg{\\{}\delta^{2}\bigg{[}L_{1}(\delta,e_{c})-\frac{i\pi}{\delta}\bigg{]}-1+r_{+}\log\frac{m_{c}^{2}}{m_{b}^{2}}\bigg{\\}}\Bigg{)},\phantom{xxxxx}$ $\displaystyle H$ $\displaystyle=$ $\displaystyle H_{5}\|_{m_{c}\to-m_{c}},$ (52c) $\displaystyle Q$ $\displaystyle=$ $\displaystyle Q_{5}\|_{m_{c}\to- m_{c}},$ (52d) where $r_{\pm}$ is defined by $r_{\pm}=\frac{m_{b}\mp m_{c}}{4(m_{b}\pm m_{c})}\bigg{[}1-\frac{(m_{b}\pm m_{c})^{2}}{4p^{2}}\bigg{]}.$ (53) The variable $r_{-}$ is of order 1 and the variable $r_{+}$ is of order $\bm{q}^{2}$, which are both finite in the limit $\|\bm{q}\|\to 0$. The real parts of the quantities in (52) are finite. According to (12) and (11), $\textrm{Im}\,H_{5}$ is of order $\|\bm{q}\|$ and, therefore, finite and the leading contribution to $\textrm{Im}\,Q_{5,{\rm NRQCD}}$ is of order $1/\|\bm{q}\|$ and acquires Coulomb divergence. In the case of the vector counterparts, the leading contributions to $\textrm{Im}\,H$ and $\textrm{Im}\,Q$ are both Coulomb divergent. ## 6 NRQCD corrections In this section, we compute the NRQCD amplitudes of order $\alpha_{s}^{1}$. As shown in section 5, the one-loop QCD corrections $i\mathcal{A}^{0(1)}_{5,\bar{b}c_{1}}$ and $i\mathcal{A}^{i(1)}_{\bar{b}c_{1}}$ contain Coulomb and logarithmic divergences in the IR regions. In order to determine the short distance coefficients $P_{n}^{(1)}$ and $V_{n}^{(1)}$ in (7), which are insensitive to the long distance interactions, we remove those divergences based on the fact that NRQCD amplitudes must reproduce the corresponding full QCD amplitudes in the IR regions because NRQCD is a low energy effective field theory of QCD. We shall find that the divergences of $i\mathcal{A}^{0(1)}_{5,\bar{b}c_{1}}$ and $i\mathcal{A}^{i(1)}_{\bar{b}c_{1}}$ are identified as the one-loop corrections to the perturbative NRQCD matrix elements $\langle 0\|\mathcal{O}_{n}\|\bar{b}c_{1}\rangle^{(1)}$ and $\langle 0\|\mathcal{O}^{i}_{n}\|\bar{b}c_{1}\rangle^{(1)}$ in (5). Instead of following the direct NRQCD approach, we compute NRQCD quantities from the full QCD expressions $i\mathcal{A}^{0}_{5,\bar{b}c_{1}}$ and $i\mathcal{A}^{i}_{\bar{b}c_{1}}$ based on the method in [31]. First we carry out the integration over the temporal component $k^{0}$ of the loop integral and, then, expand the integrand in powers of $\bm{q}/m_{Q}$ and $\bm{k}/m_{Q}$, where $\bm{k}$ is the spatial component of the loop momentum. We regularize divergent integrals dimensionally and drop scaleless power- divergent integrals. The only nonvanishing divergent contributions are, then, either logarithmic or Coulomb divergent. As in [31], we use a special notation for this prescription for the loop integration as N$\int_{k}$. Once $k^{0}$ integral has been evaluated by contour integration, then the remaining integral is denoted as $\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{\bm{k}}\equiv\mu^{2\epsilon}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int\frac{d^{d-1}k}{(2\pi)^{d-1}},$ (54) where N$\int_{\bm{k}}$ indicates that the integrand of the spatial loop variable must be expanded in powers of $\bm{q}/m_{Q}$ and $\bm{k}/m_{Q}$ and then regulated dimensionally in $d-1$ spatial dimensions. To evaluate the vertex corrections $\Lambda^{\mu}_{5,{\rm NRQCD}}$ and $\Lambda^{\mu}_{\rm NRQCD}$ in NRQCD, we begin with the full QCD expressions in (44) by replacing the loop integrals $\int_{k}$ with N$\int_{k}$: $\displaystyle\Lambda^{\mu}_{5,{\rm NRQCD}}$ $\displaystyle=$ $\displaystyle- ig_{s}^{2}C_{F}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{k}\frac{\overline{v}(p_{\bar{b}})\Gamma^{\mu}(k,p,q,m_{b},-m_{c})\gamma_{5}u(p_{c})}{D_{0}D_{1}D_{2}},$ (55a) $\displaystyle\Lambda^{\mu}_{\rm NRQCD}$ $\displaystyle=$ $\displaystyle- ig_{s}^{2}C_{F}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{k}\frac{\overline{v}(p_{\bar{b}})\Gamma^{\mu}(k,p,q,m_{b},m_{c})u(p_{c})}{D_{0}D_{1}D_{2}},$ (55b) where $\Gamma^{\mu}(k,p,q,m_{b},m_{c})$ is defined in (45). In the NRQCD case, we omit the tensor reduction and directly evaluate the integrals N$\int_{k}$ because the tensor reduction does not simplify the intermediate steps of calculation considerably. Then the expressions in (55) becomes $\displaystyle\Lambda^{0}_{5,{\rm NRQCD}}$ $\displaystyle=$ $\displaystyle- ig_{s}^{2}C_{F}\overline{v}(p_{\bar{b}})\bigg{\\{}\big{[}(d-2)S_{1}-2(4p^{2}-m_{b}^{2}-m_{c}^{2})S_{2}+8q_{\nu}S_{3}^{\nu}\big{]}\gamma^{0}$ (56a) $\displaystyle+2(m_{b}\gamma^{0}\gamma_{\nu}-m_{c}\gamma_{\nu}\gamma^{0})S_{3}^{\nu}+2(2-d)S_{4}^{0\nu}\gamma_{\nu}-8q^{0}S_{3}^{\nu}\gamma_{\nu}\bigg{\\}}\gamma_{5}u(p_{c}),$ $\displaystyle\Lambda^{i}_{\rm NRQCD}$ $\displaystyle=$ $\displaystyle- ig_{s}^{2}C_{F}\overline{v}(p_{\bar{b}})\bigg{\\{}\big{[}(d-2)S_{1}-2(4p^{2}-m_{b}^{2}-m_{c}^{2})S_{2}+8q_{\nu}S_{3}^{\nu}\big{]}\gamma^{i}$ (56b) $\displaystyle+2(m_{b}\gamma^{i}\gamma_{\nu}+m_{c}\gamma_{\nu}\gamma^{i})S_{3}^{\nu}+2(2-d)S_{4}^{i\nu}\gamma_{\nu}-8q^{i}S_{3}^{\nu}\gamma_{\nu}\bigg{\\}}u(p_{c}),$ where the loop integrals $S_{1}$, $S_{2}$, $S_{3}^{\mu}$, and $S_{4}^{\mu\nu}$ are defined by $\displaystyle S_{1}$ $\displaystyle=$ $\displaystyle\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{k}\frac{1}{D_{1}D_{2}},$ (57a) $\displaystyle S_{2}$ $\displaystyle=$ $\displaystyle\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{k}\frac{1}{D_{0}D_{1}D_{2}},$ (57b) $\displaystyle S_{3}^{\mu}$ $\displaystyle=$ $\displaystyle\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{k}\frac{k^{\mu}}{D_{0}D_{1}D_{2}},$ (57c) $\displaystyle S_{4}^{\mu\nu}$ $\displaystyle=$ $\displaystyle\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{k}\frac{k^{\mu}k^{\nu}}{D_{0}D_{1}D_{2}}.$ (57d) Next we carry out the $k^{0}$ integrals in (56) by contour integration. In order to identify the residues of the $k^{0}$ integral, we express the denominator factors of the integrands in (57) in the following form: $\displaystyle D_{0}$ $\displaystyle=$ $\displaystyle(k^{0})^{2}-\bm{k}^{2}+i\varepsilon=(k^{0}-\|\bm{k}\|+i\varepsilon)(k^{0}+\|\bm{k}\|-i\varepsilon),$ (58a) $\displaystyle D_{1}$ $\displaystyle=$ $\displaystyle(k^{0}-E_{b})^{2}-\Delta_{b}^{2}+i\varepsilon=(k^{0}+\Delta_{b}-E_{b}-i\varepsilon)(k^{0}-\Delta_{b}-E_{b}+i\varepsilon),$ (58b) $\displaystyle D_{2}$ $\displaystyle=$ $\displaystyle(k^{0}+E_{c})^{2}-\Delta_{c}^{2}+i\varepsilon=(k^{0}+\Delta_{c}+E_{c}-i\varepsilon)(k^{0}-\Delta_{c}+E_{c}+i\varepsilon),$ (58c) where $\Delta_{Q}$ for $Q=c$ or $b$ are defined by $\Delta_{Q}=\sqrt{m_{Q}^{2}+(\bm{k}+\bm{q})^{2}}.$ (59) The resultant integrands of the $(d-1)$-dimensional integrals N$\int_{\bm{k}}$ are then expanded in powers of $\bm{q}/m_{Q}$ and $\bm{k}/m_{Q}$ for $Q=b$ or $c$. The following relations are useful in these expansions: $\displaystyle\Delta_{Q}-E_{Q}$ $\displaystyle=$ $\displaystyle\frac{{\bm{k}}^{2}+2{\bm{k}}\cdot{\bm{q}}}{\Delta_{Q}+E_{Q}},$ (60a) $\displaystyle\Delta_{b}^{2}-\Delta_{c}^{2}$ $\displaystyle=$ $\displaystyle m_{b}^{2}-m_{c}^{2},$ (60b) $\displaystyle\Delta_{Q}^{2}-(E_{Q}\pm\|{\bm{k}}\|)^{2}$ $\displaystyle=$ $\displaystyle\mp 2\|{\bm{k}}\|(E_{Q}\mp{\bm{q}}\cdot\hat{{\bm{k}}}).$ (60c) It is evident from (60) that the factors such as $1/(\Delta_{Q}-E_{Q})$ and $1/[\Delta_{Q}^{2}-(E_{Q}\pm\|{\bm{k}}\|)^{2}]$ may give rise to IR singularities. In this step, scaleless integrals that are power divergent in the UV regions are neglected under dimensional regularization. The nonvanishing elementary integrals $n_{0}$, $n_{1}$, $n_{2}$, and $n_{3}$ that survive in this step are evaluated in appendix C. Nonvanishing scaleless integrals are logarithmically divergent, which are proportional to $n_{0}$ defined in (102). The integrals $n_{1}$, $n_{2}$, and $n_{3}$ defined in (103) have scale dependencies on $\|\bm{q}\|$. Eventually, all of the loop integrals in (57) are decomposed into linear combinations of these elementary integrals. The resultant values for the integrals (57) are given in (110), (116), (120), and (123) of appendix C. Substituting these values to (56), we find the multiplicative vertex correction factors in NRQCD as $\displaystyle\Lambda_{5,{\rm NRQCD}}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}C_{F}}{4\pi}\frac{p_{\bar{b}}\cdot p_{c}}{2p^{2}}\bigg{\\{}\bigg{(}\frac{1}{\epsilon_{\rm IR}}-\frac{1}{\epsilon_{\rm UV}}\bigg{)}L_{1}(\delta,e_{c})+\frac{\pi^{2}}{\delta}-\frac{i\pi}{\delta}\bigg{[}\frac{1}{\epsilon_{\rm IR}}$ (61a) $\displaystyle+\log\bigg{(}\frac{\pi\mu^{2}e^{-\gamma_{\rm E}}}{\bm{q}^{2}}\bigg{)}+\bigg{(}\\!6+\frac{2(m_{b}+m_{c})^{2}}{p_{\bar{b}}\cdot p_{c}-m_{b}m_{c}}\bigg{)}\frac{\bm{q}^{2}}{2p_{\bar{b}}\cdot p_{c}}\bigg{]}\bigg{\\}},$ $\displaystyle\Lambda_{\rm NRQCD}$ $\displaystyle=$ $\displaystyle{[\Lambda_{5}]}_{\rm NRQCD}\big{\|}_{m_{c}\to-m_{c}}.$ (61b) Substituting the multiplicative vertex correction factors in (61) and the heavy quark wavefunction renormalization factor in (31) into (30), we obtain $G_{5,{\rm NRQCD}}$ and $G_{\rm NRQCD}$ as $\displaystyle G_{5,{\rm NRQCD}}$ $\displaystyle=$ $\displaystyle 1+\frac{\alpha_{s}C_{F}}{4\pi}\frac{p_{\bar{b}}\cdot p_{c}}{2p^{2}}\bigg{\\{}\bigg{(}\frac{1}{\epsilon_{\rm IR}}-\frac{1}{\epsilon_{\rm UV}}\bigg{)}\bigg{(}L_{1}(\delta,e_{c})-\frac{4p^{2}}{p_{\bar{b}}\cdot p_{c}}\bigg{)}+\frac{\pi^{2}}{\delta}$ (62a) $\displaystyle\\!\\!\\!\\!\\!\\!\\!-\frac{i\pi}{\delta}\bigg{[}\frac{1}{\epsilon_{\rm IR}}+\log\\!\bigg{(}\frac{\pi\mu^{2}e^{-\gamma_{\rm E}}}{\bm{q}^{2}}\bigg{)}\\!+\\!\bigg{(}\\!6+\frac{2(m_{b}+m_{c})^{2}}{p_{\bar{b}}\cdot p_{c}-m_{b}m_{c}}\bigg{)}\frac{\bm{q}^{2}}{2p_{\bar{b}}\cdot p_{c}}\bigg{]}\bigg{\\}},\phantom{xxxx}$ $\displaystyle G_{\rm NRQCD}$ $\displaystyle=$ $\displaystyle G_{5,{\rm NRQCD}}\|_{m_{c}\to-m_{c}}.$ (62b) As was expected, the logarithmic and Coulomb divergences in the IR regions of $G_{5}$ and $G$ in (51) are reproduced in $G_{5,{\rm NRQCD}}$ and $G_{\rm NRQCD}$ in (62), respectively. We notice that unlike $G_{5}$ and $G$, $G_{5,{\rm NRQCD}}$ and $G_{\rm NRQCD}$ contain logarithmic UV divergences. In a similar way, the remaining NRQCD correction factors are obtained as $\displaystyle H_{5,{\rm NRQCD}}$ $\displaystyle=$ $\displaystyle-\frac{\alpha_{s}C_{F}}{4\pi}i\pi\delta\;\bigg{[}\frac{2(m_{b}-m_{c})}{p_{\bar{b}}\cdot p_{c}+m_{b}m_{c}}\bigg{]},$ (63a) $\displaystyle Q_{5,{\rm NRQCD}}$ $\displaystyle=$ $\displaystyle-\frac{\alpha_{s}C_{F}}{4\pi}\frac{i\pi\delta(m_{b}+m_{c})}{p^{2}}\bigg{\\{}1-\bigg{[}\frac{(m_{b}-m_{c})^{2}}{2(p_{\bar{b}}\cdot p_{c}+m_{b}m_{c})}$ (63b) $\displaystyle-\frac{4p^{2}}{p_{\bar{b}}\cdot p_{c}-m_{b}m_{c}}\bigg{]}\bigg{\\}},$ $\displaystyle{H}_{\rm NRQCD}$ $\displaystyle=$ $\displaystyle H_{5,{\rm NRQCD}}\big{\|}_{m_{c}\to-m_{c}},$ (63c) $\displaystyle{Q}_{\rm NRQCD}$ $\displaystyle=$ $\displaystyle Q_{5,{\rm NRQCD}}\big{\|}_{m_{c}\to-m_{c}}.$ (63d) All of the quantities in (63) are pure imaginary. According to (11) and (12), $H_{5,{\rm NRQCD}}$ is of order $\|\bm{q}\|$ and, therefore, finite and the leading contribution to $Q_{5,{\rm NRQCD}}$ is of order $1/\|\bm{q}\|$ in the limit $\|\bm{q}\|\to 0$ and acquires Coulomb divergence. In the case of the vector counterparts, the leading contributions to $H_{\rm NRQCD}$ and $Q_{\rm NRQCD}$ are both Coulomb divergent, which show the behavior $\propto 1/\|\bm{q}\|$ in the limit $\|\bm{q}\|\to 0$. The expressions in (63) reproduce the IR behaviors of the full QCD counterparts in (52). ## 7 Results for the short distance coefficients In this section, we list our final results for the short distance coefficients $P_{n}^{(j)}$, $a_{n}^{(j)}$, and $b_{n}^{(j)}$ for $j=0$ and 1 and for $n=0$, 1, and 2. We have shown that the IR behaviors of $G_{5}$, $H_{5}$, $Q_{5}$, $G$, $H$, and $Q$ in (51) and (52) are exactly reproduced by the NRQCD counterparts $G_{5,{\rm NRQCD}}$, $H_{5,{\rm NRQCD}}$, $Q_{5,{\rm NRQCD}}$, $G_{\rm NRQCD}$, $H_{\rm NRQCD}$, and $Q_{\rm NRQCD}$ in (62) and (63), respectively. Therefore, all of the quantities $\Delta G_{5}$, $\Delta H_{5}$, $\Delta Q_{5}$, $\Delta G$, $\Delta H$, and $\Delta Q$ in (37) are free of IR singularities. Because the imaginary parts of the QCD amplitudes are the same as those of the NRQCD counterparts, all of the quantities $\Delta G_{5}$, $\Delta H_{5}$, $\Delta Q_{5}$, $\Delta G$, $\Delta H$, and $\Delta Q$ in (37) are real. Except for $\Delta G_{5}$ and $\Delta G$, which have logarithmic UV divergences originated from the NRQCD factors $G_{5,{\rm NRQCD}}$ and $G_{\rm NRQCD}$, all of the other quantities ($\Delta H_{5}$, $\Delta Q_{5}$, $\Delta H$, and $\Delta Q$) are finite in both UV and IR regions. We renormalize the UV divergences of $\Delta G_{5}$ and $\Delta G$ according to the $\overline{\rm MS}$ scheme. Our final results for $\Delta G^{(1)}_{5,\overline{\rm MS}}$, $\Delta H^{(1)}_{5}$, and $\Delta Q^{(1)}_{5}$ are $\displaystyle\Delta G^{(1)}_{5,\overline{\rm MS}}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}C_{F}}{4\pi}\bigg{\\{}-4+2\bigg{[}3+\frac{(m_{b}+m_{c})^{2}}{p_{\bar{b}}\cdot p_{c}-m_{b}m_{c}}\bigg{]}\delta^{2}L_{1}(\delta,e_{c})-\frac{p_{\bar{b}}\cdot p_{c}}{2p^{2}}\bigg{(}2K(\delta,e_{c})$ $\displaystyle+\frac{L_{2}(\delta,e_{c})}{2}\log\frac{m_{c}^{2}}{m_{b}^{2}}\bigg{)}+\bigg{[}\frac{p_{\bar{b}}\cdot p_{c}}{4p^{2}}L_{1}(\delta,e_{c})-1\bigg{]}\log\frac{\mu^{4}}{m_{b}^{2}m_{c}^{2}}-\frac{m_{b}^{2}-m_{c}^{2}}{8p^{2}}\log\frac{m_{c}^{2}}{m_{b}^{2}}\bigg{\\}},$ $\displaystyle\Delta H^{(1)}_{5}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}C_{F}}{2\pi}\bigg{[}\frac{m_{b}-m_{c}}{p_{\bar{b}}\cdot p_{c}+m_{b}m_{c}}\,\delta^{2}L_{1}(\delta,e_{c})+\frac{m_{b}+m_{c}}{8p^{2}}\log\frac{m_{c}^{2}}{m_{b}^{2}}\,\bigg{]},$ (64b) $\displaystyle\Delta Q^{(1)}_{5}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}C_{F}}{4\pi}\bigg{\\{}\bigg{[}\frac{4}{p_{\bar{b}}\cdot p_{c}-m_{b}m_{c}}-\frac{(m_{b}-m_{c})^{2}}{2p^{2}(p_{\bar{b}}\cdot p_{c}+m_{b}m_{c})}+\frac{1}{p^{2}}\bigg{]}(m_{b}+m_{c})\delta^{2}L_{1}(\delta,e_{c})$ (64c) $\displaystyle+\bigg{[}3-\frac{(m_{b}+m_{c})^{2}}{2p^{2}}\bigg{]}\frac{m_{b}-m_{c}}{4p^{2}}\log\frac{m_{c}^{2}}{m_{b}^{2}}-\frac{m_{b}+m_{c}}{p^{2}}\bigg{\\}}.$ The results for the vector part can be obtained by replacing $m_{c}$ in (64) with $-m_{c}$ as $\displaystyle\Delta G^{(1)}_{\overline{\rm MS}}$ $\displaystyle=$ $\displaystyle\Delta G^{(1)}_{5,\overline{\rm MS}}\big{\|}_{m_{c}\to-m_{c}},$ (65a) $\displaystyle\Delta H^{(1)}$ $\displaystyle=$ $\displaystyle\Delta H^{(1)}_{5}\big{\|}_{m_{c}\to-m_{c}},$ (65b) $\displaystyle\Delta Q^{(1)}$ $\displaystyle=$ $\displaystyle\Delta Q^{(1)}_{5}\big{\|}_{m_{c}\to-m_{c}}.$ (65c) Taking $m_{b}=m_{c}=m_{Q}$ in (65) for the vector current, we recover the corresponding results of the electromagnetic current for the spin-triplet $S$-wave $Q\bar{Q}$ pair of the same flavor in [31]. Now we are ready to obtain the short distance coefficients $P_{n}^{(j)}$, $a_{n}^{(j)}$, and $b_{n}^{(j)}$ for $n=0$, 1, and 2 and for $j=0$ and 1. The order-$\alpha_{s}^{0}$ short distance coefficients can be found from the expansion formulas in (35) and (36) as $\displaystyle P_{0}^{(0)}$ $\displaystyle=$ $\displaystyle 1,$ (66a) $\displaystyle P_{1}^{(0)}$ $\displaystyle=$ $\displaystyle-\frac{(m_{b}+m_{c})^{2}}{8m_{b}^{2}m_{c}^{2}},$ (66b) $\displaystyle P_{2}^{(0)}$ $\displaystyle=$ $\displaystyle\frac{(m_{b}+m_{c})^{2}}{128m_{b}^{4}m_{c}^{4}}(11m_{b}^{2}-10m_{b}m_{c}+11m_{c}^{2}),$ (66c) $\displaystyle a_{0}^{(0)}$ $\displaystyle=$ $\displaystyle 1,$ (66d) $\displaystyle a_{1}^{(0)}$ $\displaystyle=$ $\displaystyle-\frac{(m_{b}-m_{c})^{2}}{8m_{b}^{2}m_{c}^{2}},$ (66e) $\displaystyle a_{2}^{(0)}$ $\displaystyle=$ $\displaystyle\frac{(m_{b}-m_{c})^{2}}{128m_{b}^{4}m_{c}^{4}}(11m_{b}^{2}+10m_{b}m_{c}+11m_{c}^{2}),$ (66f) $\displaystyle b_{0}^{(0)}$ $\displaystyle=$ $\displaystyle-\frac{1}{2m_{b}m_{c}},$ (66g) $\displaystyle b_{1}^{(0)}$ $\displaystyle=$ $\displaystyle\frac{3(m_{b}^{2}+m_{c}^{2})}{16m_{b}^{3}m_{c}^{3}},$ (66h) $\displaystyle b_{2}^{(0)}$ $\displaystyle=$ $\displaystyle-\frac{31m_{b}^{4}+18m_{b}^{2}m_{c}^{2}+31m_{c}^{4}}{256m_{b}^{5}m_{c}^{5}}.$ (66i) It is straightforward to obtain $a_{n}^{(0)}$ and $b_{n}^{(0)}$ for $n\geq 3$ in the same way. The $S$\- and $D$-wave components of the short distance coefficient for the vector current can be obtained from (66) as $S_{n}^{(j)}=a_{n}^{(j)}+\tfrac{1}{3}b_{n}^{(j)}$ and $D_{n}^{(j)}=a_{n}^{(j)}$. The short distance coefficients of order $\alpha_{s}^{0}$ are free of scale dependence. The values for $P_{0}^{(0)}$ and $P_{1}^{(0)}$ agree with the previous results in [24, 25, 26]. $S_{0}^{(0)}$ and $S_{1}^{(0)}$ agree with the previous results in [25]. The order-$\alpha_{s}^{1}$ short distance coefficients $P_{n}^{(1)}$ for $n=0$, 1, and 2 can be obtained by substituting $[\Delta G_{5}^{(1)}]_{\rm\overline{MS}}$, $\Delta H_{5}^{(1)}$, and $\Delta Q_{5}^{(1)}$ into $\Delta G_{5}^{(1)}$, $\Delta H_{5}^{(1)}$, and $\Delta Q_{5}^{(1)}$ in (38), respectively. The results are $\displaystyle P_{0}^{(1)}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}C_{F}}{4\pi}\bigg{[}-6-\frac{3(m_{b}-m_{c})}{2(m_{b}+m_{c})}\log\frac{m_{c}^{2}}{m_{b}^{2}}\bigg{]},$ (67a) $\displaystyle\big{[}P_{1}^{(1)}\big{]}_{\overline{\rm MS}}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}C_{F}}{4\pi m_{b}^{2}m_{c}^{2}}\bigg{\\{}\frac{-1}{144}\bigg{[}4(m_{b}^{2}+98m_{b}m_{c}+m_{c}^{2})+3(7m_{b}^{2}+46m_{b}m_{c}+7m_{c}^{2})$ (67b) $\displaystyle\times\frac{m_{b}-m_{c}}{m_{b}+m_{c}}\log\frac{m_{c}^{2}}{m_{b}^{2}}\bigg{]}+\frac{2}{3}(m_{b}+m_{c})^{2}\log\frac{\mu^{2}}{m_{b}m_{c}}\bigg{\\}},$ $\displaystyle\big{[}P_{2}^{(1)}\big{]}_{\overline{\rm MS}}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}C_{F}}{4\pi m_{b}^{4}m_{c}^{4}}\bigg{\\{}\frac{-1}{57600}\bigg{[}4(4089m_{b}^{4}-10364m_{b}^{3}m_{c}-36586m_{b}^{2}m_{c}^{2}-10364m_{b}m_{c}^{3}$ (67c) $\displaystyle+4089m_{c}^{4})-15(177m_{b}^{4}+2212m_{b}^{3}m_{c}+4582m_{b}^{2}m_{c}^{2}+2212m_{b}m_{c}^{3}+177m_{c}^{4})$ $\displaystyle\times\frac{m_{b}-m_{c}}{m_{b}+m_{c}}\log\frac{m_{c}^{2}}{m_{b}^{2}}\bigg{]}-\frac{(m_{b}+m_{c})^{2}}{60}(21m_{b}^{2}+2m_{b}m_{c}+21m_{c}^{2})\log\frac{\mu^{2}}{m_{b}m_{c}}\bigg{\\}}.\phantom{xxxx}$ Except for $P_{0}^{(1)}$, $P_{n}^{(1)}$ for $n\geq 1$ are dependent on the NRQCD factorization scale $\mu$ that has been introduced in the process of renormalization. Our result for $P_{0}^{(1)}$ agrees with those in [24, 25, 26]. The results for $\big{[}P_{1}^{(1)}\big{]}_{\overline{\rm MS}}$ and $\big{[}P_{2}^{(1)}\big{]}_{\overline{\rm MS}}$ are new. The short distance coefficients $a_{n}^{(1)}$ and $b_{n}^{(1)}$ for $n=0$, 1, and 2 of the vector current at order $\alpha_{s}^{1}$ are obtained by substituting $[\Delta G^{(1)}]_{\rm\overline{MS}}$ and $\Delta H^{(1)}$ into $\Delta G^{(1)}$ and $\Delta H^{(1)}$ in (39), respectively. The results for $a_{n}^{(1)}$ are $\displaystyle a_{0}^{(1)}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}C_{F}}{4\pi}\bigg{[}-8-\frac{3(m_{b}-m_{c})}{2(m_{b}+m_{c})}\log\frac{m_{c}^{2}}{m_{b}^{2}}\bigg{]},$ (68a) $\displaystyle{[}a_{1}^{(1)}]_{\overline{\rm MS}}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}C_{F}}{4\pi m_{b}^{2}m_{c}^{2}}\bigg{\\{}\frac{1}{144}\bigg{[}32(m_{b}^{2}-m_{b}m_{c}+m_{c}^{2})-3(7m_{b}^{2}+10m_{b}m_{c}+7m_{c}^{2})$ (68b) $\displaystyle\times\frac{m_{b}-m_{c}}{m_{b}+m_{c}}\log\frac{m_{c}^{2}}{m_{b}^{2}}\bigg{]}+\frac{2}{3}(m_{b}+m_{c})^{2}\log\frac{\mu^{2}}{m_{b}m_{c}}\bigg{\\}},$ $\displaystyle{[}a_{2}^{(1)}]_{\overline{\rm MS}}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}C_{F}}{4\pi m_{b}^{4}m_{c}^{4}}\bigg{\\{}\frac{-1}{19200}\bigg{[}16(547m_{b}^{4}+128m_{b}^{3}m_{c}+122m_{b}^{2}m_{c}^{2}+128m_{b}m_{c}^{3}+547m_{c}^{4})$ (68c) $\displaystyle-15(59m_{b}^{4}+164m_{b}^{3}m_{c}+194m_{b}^{2}m_{c}^{2}+164m_{b}m_{c}^{3}+59m_{c}^{4})\frac{m_{b}-m_{c}}{m_{b}+m_{c}}\log\frac{m_{c}^{2}}{m_{b}^{2}}\bigg{]}$ $\displaystyle-\frac{1}{20}(m_{b}+m_{c})^{2}(7m_{b}^{2}-6m_{b}m_{c}+7m_{c}^{2})\log\frac{\mu^{2}}{m_{b}m_{c}}\bigg{\\}}.$ And the short distance coefficients $b_{n}^{(1)}$ are $\displaystyle b_{0}^{(1)}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}C_{F}}{4\pi m_{b}m_{c}}\bigg{[}2+\frac{m_{b}-m_{c}}{4(m_{b}+m_{c})}\log\frac{m_{c}^{2}}{m_{b}^{2}}\bigg{]},$ (69a) $\displaystyle{[}b_{1}^{(1)}]_{\overline{\rm MS}}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}C_{F}}{4\pi m_{b}^{3}m_{c}^{3}}\bigg{\\{}\frac{-1}{288}\bigg{[}8(19m_{b}^{2}-10m_{b}m_{c}+19m_{c}^{2})-21(m_{b}^{2}+4m_{b}m_{c}+m_{c}^{2})$ (69b) $\displaystyle\times\frac{m_{b}-m_{c}}{m_{b}+m_{c}}\log\frac{m_{c}^{2}}{m_{b}^{2}}\bigg{]}-\frac{1}{3}(m_{b}+m_{c})^{2}\log\frac{\mu^{2}}{m_{b}m_{c}}\bigg{\\}},$ $\displaystyle{[}b_{2}^{(1)}]_{\overline{\rm MS}}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}C_{F}}{4\pi m_{b}^{5}m_{c}^{5}}\bigg{\\{}\frac{1}{115200}\bigg{[}8(5647m_{b}^{4}-972m_{b}^{3}m_{c}+922m_{b}^{2}m_{c}^{2}-972m_{b}m_{c}^{3}+5647m_{c}^{4})$ (69c) $\displaystyle-15(527m_{b}^{4}+1912m_{b}^{3}m_{c}+2082m_{b}^{2}m_{c}^{2}+1912m_{b}m_{c}^{3}+527m_{c}^{4})\frac{m_{b}-m_{c}}{m_{b}+m_{c}}\log\frac{m_{c}^{2}}{m_{b}^{2}}\bigg{]}$ $\displaystyle+\frac{1}{120}(m_{b}+m_{c})^{2}(31m_{b}^{2}-8m_{b}m_{c}+31m_{c}^{2})\log\frac{\mu^{2}}{m_{b}m_{c}}\bigg{\\}}.$ As in the case of $a_{n}^{(0)}$ and $b_{n}^{(0)}$, it is straightforward to obtain $a_{n}^{(1)}$ and $b_{n}^{(1)}$ for $n\geq 3$. Except for $a_{0}^{(1)}$ and $b_{0}^{(1)}$, $a_{n}^{(1)}$ and $b_{n}^{(1)}$ for $n\geq 1$ are dependent on the NRQCD factorization scale $\mu$. Our result for $S_{0}^{(1)}=a_{0}^{(1)}+\tfrac{1}{3}b_{0}^{(1)}$ agrees with that in [25]. The results for $\big{[}a_{1}^{(1)}\big{]}_{\overline{\rm MS}}$, $\big{[}b_{1}^{(1)}\big{]}_{\overline{\rm MS}}$, $\big{[}a_{2}^{(1)}\big{]}_{\overline{\rm MS}}$, and $\big{[}b_{2}^{(1)}\big{]}_{\overline{\rm MS}}$ are new. Let us make a rough estimate of the effect of relativistic corrections at order $\alpha_{s}$. It is convenient to define the ratio of the NRQCD matrix element of relative order-$\bm{q}^{2n}$ to the leading-order matrix element: $\langle\bm{q}^{2n}\rangle_{B_{c}}\equiv\frac{\langle 0\|{\cal O}_{n}\|B_{c}\rangle}{\langle 0\|{\cal O}_{0}\|B_{c}\rangle},\,\,\,\,\,\,\,\langle\bm{q}^{2n}\rangle_{B_{c}^{}}\equiv\frac{\langle 0\|{\cal O}^{i}_{An}\|B_{c}^{}\rangle}{\langle 0\|{\cal O}^{i}_{A0}\|B_{c}^{}\rangle},$ (70) where ${\cal O}_{n}$ and ${\cal O}^{i}_{An}$ are defined in (20), and we have used the property that $\langle\bm{q}^{2n}\rangle_{B_{c}^{}}$ is independent of $i$. The ratios $\langle\bm{q}^{2n}\rangle_{B_{c},B_{c}^{}}$ are normalized to be consistent with that, $\langle\bm{q}^{2n}\rangle_{H}$, for the $Q\bar{Q}$ quarkonium $H$ considered in [39, 40, 31]. In [39], it was shown that the $\overline{\rm MS}$ value for $\langle\bm{q}^{2n}\rangle_{H}$ satisfies a generalized Gremm-Kapustin relation [41]: ${[}\langle\bm{q}^{2n}\rangle]_{\overline{\rm MS}}={[}\langle\bm{q}^{2}\rangle]_{\overline{\rm MS}}^{n}.$ (71) We assume that this relation is still valid in the case of the $\bar{b}c$ meson. Unfortunately, unlike the case of the $S$-wave bound states of the $Q\bar{Q}$ pairs, the leptonic decay rates for $B_{c}$ and $B_{c}^{}$ have not been measured accurately so that one cannot determine $\langle\bm{q}^{2n}\rangle_{B_{c},B_{c}^{}}$ with empirical data. Instead, by taking upper and lower bounds of the ratios $\langle\bm{q}^{2n}\rangle_{B_{c},B_{c}^{}}$ as $\langle\bm{q}^{2}\rangle_{J/\psi}=0.441\,\textrm{GeV}^{2}$ [40] and $\langle\bm{q}^{2}\rangle_{\Upsilon(1S)}=-0.193\,\textrm{GeV}^{2}$ [42], respectively, we make a rough estimate of the sums of products of $S$-wave short distance coefficients and operator matrix elements. Taking the central values for the variables $m_{b}=4.6\,{\rm GeV}$, $m_{c}=1.5\,{\rm GeV}$, $\mu=(m_{b}+m_{c})/2=3.05\,{\rm GeV}$, and $\langle{\bm{q}}^{2}\rangle_{B_{c},B_{c}^{}}=(\langle\bm{q}^{2}\rangle_{J/\psi}+\langle\bm{q}^{2}\rangle_{\Upsilon(1S)})/2=0.124\,{\rm GeV}^{2}$, we find that $\displaystyle\sum_{n=0}^{0}\left[P_{n}^{(1)}\right]_{\overline{\rm MS}}\,[\langle\bm{q}^{2}\rangle_{B_{c}}]^{n}_{\overline{\rm MS}}$ $\displaystyle=$ $\displaystyle-\frac{\alpha_{s}C_{F}}{4\pi}\times 4.292,$ (72a) $\displaystyle\sum_{n=0}^{1}\left[P_{n}^{(1)}\right]_{\overline{\rm MS}}\,[\langle\bm{q}^{2}\rangle_{B_{c}}]^{n}_{\overline{\rm MS}}$ $\displaystyle=$ $\displaystyle-\frac{\alpha_{s}C_{F}}{4\pi}\times 4.293^{+0.185}_{-0.330},$ (72b) $\displaystyle\sum_{n=0}^{2}\left[P_{n}^{(1)}\right]_{\overline{\rm MS}}\,[\langle\bm{q}^{2}\rangle_{B_{c}}]^{n}_{\overline{\rm MS}}$ $\displaystyle=$ $\displaystyle-\frac{\alpha_{s}C_{F}}{4\pi}\times 4.294^{+0.152}_{-0.306},$ (72c) $\displaystyle\sum_{n=0}^{\infty}\left[P_{n}^{(1)}\right]_{\overline{\rm MS}}\,[\langle\bm{q}^{2}\rangle_{B_{c}}]^{n}_{\overline{\rm MS}}$ $\displaystyle=$ $\displaystyle-\frac{\alpha_{s}C_{F}}{4\pi}\times 4.294^{+0.152}_{-0.308},$ (72d) $\displaystyle\sum_{n=0}^{0}\left[S_{n}^{(1)}\right]_{\overline{\rm MS}}\,[\langle\bm{q}^{2}\rangle_{B_{c}^{}}]^{n}_{\overline{\rm MS}}$ $\displaystyle=$ $\displaystyle-\frac{\alpha_{s}C_{F}}{4\pi}\times 6.209,$ (72e) $\displaystyle\sum_{n=0}^{1}\left[S_{n}^{(1)}\right]_{\overline{\rm MS}}\,[\langle\bm{q}^{2}\rangle_{B_{c}^{}}]^{n}_{\overline{\rm MS}}$ $\displaystyle=$ $\displaystyle-\frac{\alpha_{s}C_{F}}{4\pi}\times 6.168^{+0.214}_{-0.289},$ (72f) $\displaystyle\sum_{n=0}^{2}\left[S_{n}^{(1)}\right]_{\overline{\rm MS}}\,[\langle\bm{q}^{2}\rangle_{B_{c}^{}}]^{n}_{\overline{\rm MS}}$ $\displaystyle=$ $\displaystyle-\frac{\alpha_{s}C_{F}}{4\pi}\times 6.170^{+0.213}_{-0.247},$ (72g) $\displaystyle\sum_{n=0}^{\infty}\left[S_{n}^{(1)}\right]_{\overline{\rm MS}}\,[\langle\bm{q}^{2}\rangle_{B_{c}^{}}]^{n}_{\overline{\rm MS}}$ $\displaystyle=$ $\displaystyle-\frac{\alpha_{s}C_{F}}{4\pi}\times 6.170^{+0.213}_{-0.252},$ (72h) where the uncertainties are estimated by varying the values $m_{c}\leq\mu\leq m_{b}$ and $-0.193\,\textrm{GeV}^{2}$ $\leq\langle\bm{q}^{2}\rangle_{B_{c}}\leq 0.441\,\textrm{GeV}^{2}$. The estimates in (72) show that the series expansions for the relativistic corrections to the axial vector and vector currents at order $\alpha_{s}$ converge rapidly in spite of the large uncertainties in the ratio $\langle\bm{q}^{2}\rangle_{B_{c}}$. In addition, the order-one contributions of the form $(m_{b}-m_{c})/(m_{b}+m_{c})\log(m_{c}^{2}/m_{b}^{2})$ do not deteriorate the convergence of the short distance coefficients for the axial vector and vector currents. In [26], the authors obtained the short distance coefficient $P_{0}^{(2)}$ of order-$\alpha_{s}^{2}v^{0}$ as $-24.4\times[\alpha_{s}(m_{b})/\pi]^{2}$ with the input parameters $m_{b}=\mu=4.8\,$GeV and $m_{c}=1.65\,$GeV. If we use these values for $m_{b}$, $m_{c}$, $\mu$ and set $\langle{\bm{q}}^{2}\rangle_{B_{c}}=0.124\,{\rm GeV}^{2}$, then the coefficients (72) vary by about $\lesssim 3\,\%$. As an example, (72d) becomes $\sum_{n=0}^{\infty}\left[P_{n}^{(1)}\right]_{\overline{\rm MS}}\,[\langle\bm{q}^{2}\rangle_{B_{c}}]^{n}_{\overline{\rm MS}}=-\frac{\alpha_{s}C_{F}}{4\pi}\times 4.399.$ (73) ## 8 Summary We have computed the short distance coefficients for the NRQCD factorization formulas of the meson-to-vacuum matrix elements for the axial vector and vector parts of the charged weak current in the $S$-wave spin-singlet and -triplet $\bar{b}c$ mesons, respectively. The computation was carried out at order $\alpha_{s}^{0}$ and $\alpha_{s}^{1}$ including relativistic corrections of all of the $\bar{b}c$ NRQCD operators that contain any number of ordinary derivatives without gauge fields. In the Coulomb gauge, gauge field contributions first appear from relative order $v_{Q}^{4}$. The numerical value for the correction of order $\alpha_{s}v_{Q}^{4}$ is tiny ($<$0.1% of the leading order contribution). We have reproduced all available short distance coefficients of order $\alpha_{s}$ or less and our results of order $\alpha_{s}^{1}$ with relativistic corrections are new. By taking the limit $m_{b}=m_{c}=m_{Q}$, we have reproduced the results for the order-$\alpha_{s}$ corrections to the quarkonium electromagnetic current for the spin-triplet $S$-wave $Q\bar{Q}$ pair with the same flavor. Although we have listed explicitly the first few short distance coefficients of order-$\alpha_{s}$ for the relativistic corrections, it is straightforward to obtain the terms of higher orders in $\bm{q}^{2n}$. The results reveal that, in spite of the large uncertainties in the ratios $\langle\bm{q}^{2}\rangle_{B_{c},B_{c}^{}}$, the relativistic corrections to the leptonic decay rate of the $B_{c}(Bc^{})$ meson at order $\alpha_{s}^{1}$ or less converge rapidly, which shows a strong contrast to the uncomfortably large corrections of order $\alpha_{s}^{2}\|\bm{q}\|^{0}$. The short distance coefficients listed in this paper can be employed to compute the resummation of relativistic corrections to the phenomenological measurables that involve $B_{c}$ and $B_{c}^{}$ production and decay. ###### Acknowledgments. S.K. was supported by the National Research Foundation of Korea grant funded by the Korea government (MEST) No. 2010-0022219. The work of J.L. and W.S. was supported by Basic Science Research Program through the NRF of Korea funded by the MEST under contracts 2010-0015682 (J.L.) and 2010-0027811 (W.S.). J.L. and W.S. were also supported in part by a Korea University fund. ## Appendix A Tensor-integral reduction In this appendix, we describe the tensor-integral reduction that we use to simplify (44) to obtain (46). Tensor integrals of rank 1 and 2 that depend on $p$ or on both $p$ and $q$ can be expressed in terms of scalar integrals as follows: $\displaystyle\int_{k}k^{\mu}f(k,p)$ $\displaystyle=$ $\displaystyle\frac{p^{\mu}}{p^{2}}\int_{k}p\cdot kf(k,p),$ (74) $\displaystyle\int_{k}k^{\mu}k^{\nu}f(k,p)$ $\displaystyle=$ $\displaystyle g^{\mu\nu}\int_{k}d_{1}(k,p)f(k,p)+p^{\mu}p^{\nu}\int_{k}d_{2}(k,p)f(k,p),$ (75) $\displaystyle\int_{k}k^{\mu}f(k,p,q)$ $\displaystyle=$ $\displaystyle p^{\mu}\int_{k}d_{3}(k,p,q)f(k,p,q)+q^{\mu}\int_{k}d_{4}(k,p,q)f(k,p,q),$ (76) $\displaystyle\int_{k}k^{\mu}k^{\nu}f(k,p,q)$ $\displaystyle=$ $\displaystyle g^{\mu\nu}\int_{k}d_{5}(k,p,q)f(k,p,q)+p^{\mu}p^{\nu}\int_{k}d_{6}(k,p,q)f(k,p,q)$ (77) $\displaystyle+q^{\mu}q^{\nu}\int_{k}d_{7}(k,p,q)f(k,p,q)$ $\displaystyle+(p^{\mu}q^{\nu}+p^{\nu}q^{\mu})\int_{k}d_{8}(k,p,q)f(k,p,q),$ where $k$ is the loop momentum, the symbol $\int_{k}$ is defined in (42), and $f$ is an arbitrary scalar function of the argument four-vectors. The functions $d_{i}$’s are defined by $\displaystyle d_{1}(k,p)$ $\displaystyle=$ $\displaystyle\frac{1}{d-1}\left[k^{2}-\frac{(k\cdot p)^{2}}{p^{2}}\right],$ (78) $\displaystyle d_{2}(k,p)$ $\displaystyle=$ $\displaystyle\frac{1}{(d-1)p^{2}}\left[-k^{2}+d\frac{(k\cdot p)^{2}}{p^{2}}\right],$ (79) $\displaystyle d_{3}(k,p,q)$ $\displaystyle=$ $\displaystyle\frac{k\cdot q\,p\cdot q-q^{2}\,k\cdot p}{(p\cdot q)^{2}-p^{2}q^{2}},$ (80) $\displaystyle d_{4}(k,p,q)$ $\displaystyle=$ $\displaystyle d_{3}(k,q,p),$ (81) $\displaystyle d_{5}(k,p,q)$ $\displaystyle=$ $\displaystyle\frac{1}{d-2}\left[k^{2}+\frac{p^{2}(q\cdot k)^{2}+q^{2}(p\cdot k)^{2}-2\,p\cdot q\,p\cdot k\,q\cdot k}{(p\cdot q)^{2}-p^{2}q^{2}}\right],$ (82) $\displaystyle d_{6}(k,p,q)$ $\displaystyle=$ $\displaystyle\frac{1}{(d-2)\left[(p\cdot q)^{2}-p^{2}q^{2}\right]^{2}}\bigg{\\{}\big{[}(k\cdot p)^{2}(d-1)-p^{2}k^{2}\big{]}q^{4}+(d-2)(p\cdot q)^{2}(k\cdot q)^{2}$ (83) $\displaystyle+\big{[}k^{2}(p\cdot q)^{2}-2(d-1)(k\cdot p)(k\cdot q)(p\cdot q)+p^{2}(k\cdot q)^{2}\big{]}q^{2}\bigg{\\}},$ $\displaystyle d_{7}(k,p,q)$ $\displaystyle=$ $\displaystyle d_{6}(k,q,p),$ (84) $\displaystyle d_{8}(k,p,q)$ $\displaystyle=$ $\displaystyle\frac{1}{(d-2)\left[(p\cdot q)^{2}-p^{2}q^{2}\right]^{2}}\bigg{\\{}\left[q^{2}(k\cdot p)^{2}-k^{2}(p\cdot q)^{2}+p^{2}(k\cdot q)^{2}+p^{2}q^{2}k^{2}\right]p\cdot q$ (85) $\displaystyle-$ $\displaystyle 2p^{2}q^{2}\,k\cdot p\,k\cdot q+d\left[p\cdot q\,k\cdot p-p^{2}\,k\cdot q\right]\left[p\cdot q\,k\cdot q-q^{2}\,k\cdot p\right]\bigg{\\}}.$ ## Appendix B Scalar integrals for the vertex corrections In this appendix, we list the definitions and the values for the scalar integrals $J_{i}$ that appear in the vertex corrections in (46). The scalar integrals $J_{i}$’s are defined by $J_{i}=\int_{k}\frac{N_{i}}{D_{0}D_{1}D_{2}},$ (86) where $k$ is the loop momentum, the symbol $\int_{k}$ and the denominator factors $D_{i}$’s are defined in (42), and the numerators $N_{i}$’s of the integrand are defined by $\displaystyle N_{1}$ $\displaystyle=$ $\displaystyle k^{2},$ (87) $\displaystyle N_{2}$ $\displaystyle=$ $\displaystyle 1,$ (88) $\displaystyle N_{3}$ $\displaystyle=$ $\displaystyle 2k\cdot q,$ (89) $\displaystyle N_{4}$ $\displaystyle=$ $\displaystyle\frac{1}{d-2}\Bigg{[}k^{2}+\frac{p^{2}(q\cdot k)^{2}+q^{2}(p\cdot k)^{2}-2(p\cdot q)(p\cdot k)(q\cdot k)}{(p\cdot q)^{2}-p^{2}q^{2}}\Bigg{]},$ (90) $\displaystyle N_{5}$ $\displaystyle=$ $\displaystyle 2k\cdot p,$ (91) $\displaystyle N_{6}$ $\displaystyle=$ $\displaystyle\frac{m_{b}+m_{c}}{d-2}\;\frac{q^{2}}{(p\cdot q)^{2}-p^{2}q^{2}}\Bigg{\\{}k^{2}p^{2}-(p\cdot k)^{2}+\frac{(d-1)[p^{2}(q\cdot k)-(p\cdot q)(p\cdot k)]^{2}}{(p\cdot q)^{2}-p^{2}q^{2}}$ (92) $\displaystyle+$ $\displaystyle\frac{m_{c}-m_{b}}{m_{c}+m_{b}}\Bigg{[}-k^{2}(p\cdot q)+(p\cdot k)(q\cdot k)+\frac{(d-1)[q^{2}(p\cdot k)-(p\cdot q)(q\cdot k)]}{(p\cdot q)^{2}-p^{2}q^{2}}$ $\displaystyle\qquad\qquad\quad\times[p^{2}(q\cdot k)-(p\cdot q)(p\cdot k)]\Bigg{]}\Bigg{\\}},$ $\displaystyle N_{7}$ $\displaystyle=$ $\displaystyle\frac{m_{b}+m_{c}}{d-2}\;\frac{p^{2}q^{2}}{(p\cdot q)^{2}-p^{2}q^{2}}\Bigg{\\{}-k^{2}(p\cdot q)+(p\cdot k)(q\cdot k)$ (93) $\displaystyle+$ $\displaystyle\frac{(d-1)[q^{2}(p\cdot k)-(p\cdot q)(q\cdot k)][p^{2}(q\cdot k)-(p\cdot q)(p\cdot k)]}{(p\cdot q)^{2}-p^{2}q^{2}}$ $\displaystyle+$ $\displaystyle\frac{m_{c}-m_{b}}{m_{c}+m_{b}}\Bigg{[}k^{2}q^{2}-(q\cdot k)^{2}+\frac{(d-1)[q^{2}(p\cdot k)-(p\cdot q)(q\cdot k)]^{2}}{(p\cdot q)^{2}-p^{2}q^{2}}\Bigg{]}\Bigg{\\}},$ where $d=4-2\epsilon$ is the number of space-time dimensions. The external momenta $p$ and $q$ are defined by $p=\tfrac{1}{2}(p_{c}+p_{\bar{b}})$ and $q=\tfrac{1}{2}(p_{c}-p_{\bar{b}})$, where $p_{\bar{b}}$ and $p_{c}$ are the momenta for the $\bar{b}$ and $c$, respectively, which are on their mass shells: $p_{\bar{b}}^{2}=m_{b}^{2}$ and $p_{c}^{2}=m_{c}^{2}$. The values for the scalar integrals $J_{i}$’s are $\displaystyle J_{1}$ $\displaystyle=$ $\displaystyle\frac{i}{(4\pi)^{2}}\Bigg{[}\frac{1}{\epsilon_{\rm UV}}+2+2i\pi\delta-2\delta^{2}L_{1}(\delta,e_{c})+e_{c}\log\frac{4\pi\mu^{2}e^{-\gamma_{\rm E}}}{m_{c}^{2}}$ (94) $\displaystyle+(1-e_{c})\log\frac{4\pi\mu^{2}e^{-\gamma_{\rm E}}}{m_{b}^{2}}\Bigg{]},$ $\displaystyle J_{2}$ $\displaystyle=$ $\displaystyle\frac{i}{(4\pi)^{2}}\;\frac{1}{8p^{2}}\Bigg{\\{}\bigg{(}\frac{1}{\epsilon_{\rm IR}}+\frac{1}{2}\log\frac{(4\pi\mu^{2}e^{-\gamma_{\rm E}})^{2}}{m_{b}^{2}m_{c}^{2}}\bigg{)}\bigg{[}\frac{i\pi}{\delta}-L_{1}(\delta,e_{c})\bigg{]}+2K(\delta,e_{c})-\frac{\pi^{2}}{\delta}$ (95) $\displaystyle+\frac{i\pi}{2\delta}\log\bigg{[}\frac{p^{4}m_{b}^{2}m_{c}^{2}}{[(p_{\bar{b}}\cdot p_{c})^{2}-m_{b}^{2}m_{c}^{2}]^{2}}\bigg{]}+\frac{1}{2}L_{2}(\delta,e_{c})\log\frac{m_{c}^{2}}{m_{b}^{2}}\Bigg{\\}},$ $\displaystyle J_{3}$ $\displaystyle=$ $\displaystyle-\frac{i}{(4\pi)^{2}}\Bigg{[}2i\pi\delta-2\delta^{2}L_{1}(\delta,e_{c})+\frac{m_{b}^{2}-m_{c}^{2}}{8p^{2}}\log\frac{m_{c}^{2}}{m_{b}^{2}}\Bigg{]},$ (96) $\displaystyle J_{4}$ $\displaystyle=$ $\displaystyle\frac{1}{4}\Bigg{[}\frac{i}{(4\pi)^{2}}+J_{1}\Bigg{]},$ (97) $\displaystyle J_{5}$ $\displaystyle=$ $\displaystyle\frac{i}{2(4\pi)^{2}}\,\log\frac{m_{c}^{2}}{m_{b}^{2}},$ (98) $\displaystyle J_{6}$ $\displaystyle=$ $\displaystyle\frac{i}{(4\pi)^{2}}\frac{(m_{b}+m_{c})q^{2}}{p_{\bar{b}}\cdot p_{c}-m_{b}m_{c}}\\!\Bigg{[}\delta^{2}L_{1}(\delta,e_{c})-i\pi\delta+\frac{1}{4}\bigg{(}\frac{m_{b}-m_{c}}{m_{b}+m_{c}}-\frac{m_{b}^{2}-m_{c}^{2}}{4p^{2}}\bigg{)}\log\frac{m_{c}^{2}}{m_{b}^{2}}\Bigg{]},\phantom{xxxxx}$ (99) $\displaystyle J_{7}$ $\displaystyle=$ $\displaystyle\frac{i}{4(4\pi)^{2}}\frac{(m_{b}+m_{c})q^{2}}{p_{\bar{b}}\cdot p_{c}-m_{b}m_{c}}\Bigg{\\{}\bigg{[}m_{b}^{2}-m_{c}^{2}+2(p_{\bar{b}}\cdot p_{c}-m_{b}m_{c})\frac{m_{b}-m_{c}}{m_{b}+m_{c}}\bigg{]}\big{[}\delta^{2}L_{1}(\delta,e_{c})-i\pi\delta\big{]}$ (100) $\displaystyle+\bigg{[}\frac{m_{b}^{2}-m_{c}^{2}}{4}\bigg{(}\frac{m_{b}-m_{c}}{m_{b}+m_{c}}-\frac{m_{b}^{2}-m_{c}^{2}}{4p^{2}}\bigg{)}-\frac{p_{\bar{b}}\cdot p_{c}-m_{b}m_{c}}{2}\bigg{(}1-\frac{m_{b}^{2}-m_{c}^{2}}{4p^{2}}\bigg{)}\bigg{]}$ $\displaystyle\times\log\frac{m_{c}^{2}}{m_{b}^{2}}\Bigg{\\}},$ where the variables $\delta$ and $e_{c}$ are defined in (11) and the functions $L_{1}(\delta,e_{c})$, $L_{2}(\delta,e_{c})$, and $K(\delta,e_{c})$ are defined in (49). ## Appendix C Integrals for NRQCD vertex corrections In this appendix, we list elementary loop integrals that are useful in computing the NRQCD corrections considered in section 6. We also list the values for the integrals defined in (57). We follow the strategy of evaluating the integrals given in [31]. ### C.1 Elementary scalar integrals In dimensional regularization, scaleless power-divergent integrals vanish: $\int_{\bm{k}}\frac{1}{\|\bm{k}\|^{n}}=0$ (101) for $n\neq 3$. The only nonvanishing scaleless integral is $n_{0}\equiv\int_{\bm{k}}\frac{1}{\|\bm{k}\|^{3}}=\frac{1}{4\pi^{2}}\left(\frac{1}{\epsilon_{\textrm{UV}}}-\frac{1}{\epsilon_{\textrm{IR}}}\right),$ (102) which diverges logarithmically. Except for the integral (102), nonvanishing integrals are depending on $\|\bm{q}\|$, which are $\displaystyle n_{1}$ $\displaystyle\equiv$ $\displaystyle\int_{\bm{k}}\frac{1}{\bm{k}^{2}+2\bm{k}\cdot\bm{q}-i\varepsilon}=\frac{i}{4\pi}\|\bm{q}\|,$ (103a) $\displaystyle n_{2}$ $\displaystyle\equiv$ $\displaystyle\int_{\bm{k}}\frac{1}{\bm{k}^{2}(\bm{k}^{2}+2\bm{k}\cdot\bm{q}-i\varepsilon)}$ (103b) $\displaystyle=$ $\displaystyle-\frac{i}{16\pi\|\bm{q}\|}\left(\frac{1}{\epsilon_{\textrm{IR}}}+\log\frac{\pi\mu^{2}e^{-\gamma_{{}_{\\!\textrm{E}}}}}{\bm{q}^{2}}+i\pi\right),$ $\displaystyle n_{3}$ $\displaystyle\equiv$ $\displaystyle\int_{\bm{k}}\frac{\bm{k}^{2}}{\bm{k}^{2}+2\bm{k}\cdot\bm{q}-i\varepsilon}=\frac{i}{2\pi}\|\bm{q}\|^{3}.$ (103c) In projecting out the $S$-wave contribution from a scalar integral that depends on $\bm{q}$, we have to take the average over the angle of $\bm{q}$. The following formulas are useful in that step: $\displaystyle\int_{\bm{k}}\frac{f(\bm{k}^{2})}{E_{Q}\pm\bm{q}\cdot\hat{\bm{k}}}$ $\displaystyle=$ $\displaystyle\frac{1}{2\|\bm{q}\|}\log\left(\frac{E_{Q}+\|\bm{q}\|}{E_{Q}-\|\bm{q}\|}\right)\int_{\bm{k}}f(\bm{k}^{2}),$ (104) $\displaystyle\int_{\bm{k}}\frac{f(\bm{k}^{2})}{(E_{b}-\bm{q}\cdot\hat{\bm{k}})(E_{c}+\bm{q}\cdot\hat{\bm{k}})}$ $\displaystyle=$ $\displaystyle\frac{1}{E_{b}+E_{c}}\int_{\bm{k}}f(\bm{k}^{2})\left(\frac{1}{E_{b}+\bm{q}\cdot\hat{\bm{k}}}+\frac{1}{E_{c}-\bm{q}\cdot\hat{\bm{k}}}\right)$ (105) $\displaystyle=$ $\displaystyle\frac{1}{2(E_{b}+E_{c})\|\bm{q}\|}\log\left[\frac{(E_{b}+\|\bm{q}\|)(E_{c}+\|\bm{q}\|)}{(E_{b}-\|\bm{q}\|)(E_{c}-\|\bm{q}\|)}\right]\int_{\bm{k}}f(\bm{k}^{2}),\phantom{xxxx}$ where $Q=b$ or $c$, $\hat{\bm{k}}=\bm{k}/\|\bm{k}\|$, and $f(\bm{k}^{2})$ is any function of $\bm{k}^{2}$. In the following sections, we express the integrals $S_{1}$, $S_{2}$, $S_{3}^{\mu}$, and $S_{4}^{\mu\nu}$ defined in (57) in linear combinations of $n_{0}$, $n_{1}$, $n_{2}$, and $n_{3}$ in (102) and (103). We also find the covariant forms of the integrals $S_{1}$, $S_{2}$, $S_{3}^{\mu}$, and $S_{4}^{\mu\nu}$. ### C.2 $\bm{S_{1}}$ The $k^{0}$ integral of $S_{1}$ defined in (57) is the sum of two contributions: $S_{1}=S_{1c}+S_{1\bar{b}}$, where $S_{1c}$ and $S_{1\bar{b}}$ are the contributions from the poles of the charm quark and antibottom quark, respectively. The contribution from the $c$ pole is $S_{1c}=-\frac{i}{4(E_{b}+E_{c})}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{\bm{k}}\frac{1}{\Delta_{c}(\Delta_{c}+E_{c})}.$ (106) We find that all of the factors in the denominator of the integrands are of order $m_{Q}$ as $\bm{q}\to 0$ and $\bm{k}\to 0$. Therefore, the expansion of $\Delta_{Q}$ in powers of $({\bm{k}}+{\bm{q}})^{2}/m_{Q}^{2}$ gives only scaleless, power-divergent integrals, which vanish, so that $S_{1c}=0.$ (107) The contribution from the $\bar{b}$ pole is $S_{1\bar{b}}=\frac{i}{4(E_{b}+E_{c})}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{\bm{k}}\frac{1+(E_{b}/\Delta_{b})}{\bm{k}^{2}+2\bm{k}\cdot\bm{q}-i\varepsilon}.$ (108) Expanding $\Delta_{Q}$ in powers of $({\bm{k}}+{\bm{q}})^{2}/m_{Q}^{2}$, we obtain $S_{1\bar{b}}=\frac{i}{2(E_{b}+E_{c})}n_{1}=-\frac{\|{\bm{q}}\|}{8\pi(E_{b}+E_{c})},$ (109) By adding the two contributions (107) and (109), we obtain $S_{1}=\frac{i}{(4\pi)^{2}}2\pi i\delta,$ (110) where $\delta$ is defined in (11). ### C.3 $\bm{S_{2}}$ The $k^{0}$ integral of $S_{2}$ is the sum of three contributions: $S_{2}=S_{2g}+S_{2c}+S_{2\bar{b}}$, where $S_{2g}$, $S_{2c}$, and $S_{2\bar{b}}$ are the contributions from the poles of the gluon, charm quark, and antibottom quark, respectively. The gluon pole contribution is $\displaystyle S_{2g}$ $\displaystyle=$ $\displaystyle\frac{i}{8}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{\bm{k}}\frac{1}{\|\bm{k}\|^{3}(E_{b}-\bm{q}\cdot\hat{\bm{k}})(E_{c}+\bm{q}\cdot\hat{\bm{k}})}$ (111) $\displaystyle=$ $\displaystyle\frac{i}{16(E_{b}+E_{c})\|\bm{q}\|}\,n_{0}\log\left[\frac{(E_{b}+\|\bm{q}\|)(E_{c}+\|\bm{q}\|)}{(E_{b}-\|\bm{q}\|)(E_{c}-\|\bm{q}\|)}\right],$ which is proportional to the scaleless logarithmically divergent integral $n_{0}$. The contribution from the $c$ pole is $S_{2c}=\frac{i}{4(E_{b}+E_{c})}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{\bm{k}}\frac{1}{\Delta_{c}(\Delta_{c}+E_{c})[\bm{k}^{2}-(\Delta_{c}+E_{c})^{2}-i\varepsilon]}.$ (112) We find that the factors $\Delta_{c}$ and $\Delta_{c}+E_{c}$ are of order $m_{Q}$ as $\bm{q}\to 0$ and $\bm{k}\to 0$. Therefore, the expansion of the factors $1/\Delta_{c}$ and $1/(\Delta_{c}+E_{c})$ are trivial and the expansion gives only scaleless power-divergent integrals. The expansion of the last factor $1/[\bm{k}^{2}-(\Delta_{c}+E_{c})^{2}-i\varepsilon]$ can be done in powers of $\bm{k}^{2}/(\Delta_{c}+E_{c})^{2}$ and then the factor $1/(\Delta_{c}+E_{c})^{2n}$ is expanded in powers of $({\bm{k}}+{\bm{q}})^{2}/m_{Q}^{2}$. We find that all of the contributions are scaleless, power-divergent integrals so that $S_{2c}=0.$ (113) The contribution from the $\bar{b}$ is $S_{2\bar{b}}=-\frac{i}{4(E_{b}+E_{c})}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{\bm{k}}\frac{1+(E_{b}/\Delta_{b})}{[\bm{k}^{2}-(\Delta_{b}-E_{b})^{2}-i\varepsilon][\bm{k}^{2}+2\bm{k}\cdot\bm{q}-i\varepsilon]}.$ (114) The expansion of the integrand for $S_{2\bar{b}}$ is similar to that used in deriving (67) of [31]. Following that method, we find that $S_{2\bar{b}}=-\frac{i}{2(E_{b}+E_{c})}n_{2}=-\frac{1}{32\pi(E_{b}+E_{c})\|\bm{q}\|}\left(\frac{1}{\epsilon_{\textrm{IR}}}+\log\frac{\pi\mu^{2}e^{-\gamma_{{}_{\\!\textrm{E}}}}}{\bm{q}^{2}}+i\pi\right).$ (115) The sum of the three contributions in (111), (113), and (115) is $S_{2}=\frac{i}{(4\pi)^{2}}\frac{1}{8p^{2}}\bigg{[}\bigg{(}\frac{1}{\epsilon_{\rm UV}}-\frac{1}{\epsilon_{\rm IR}}\bigg{)}L_{1}(\delta,e_{c})-\frac{\pi^{2}}{\delta}+\frac{i\pi}{\delta}\bigg{(}\frac{1}{\epsilon_{\rm IR}}+\log\frac{\pi\mu^{2}e^{-\gamma_{\rm E}}}{{\bm{q}\;}^{2}}\bigg{)}\bigg{]},$ (116) where $\delta$ and $e_{c}$ are defined in (11) and the function $L_{1}(\delta,e_{c})$ is defined in (49). ### C.4 $\bm{S_{3}^{\mu}}$ The integral $S_{3}^{\mu}$ is the sum of three contributions: $S_{3}^{\mu}=S_{3g}^{\mu}+S_{3c}^{\mu}+S_{3\bar{b}}^{\mu}$, where $S_{3g}^{\mu}$, $S_{3c}^{\mu}$, and $S_{3\bar{b}}^{\mu}$ are the contributions from the poles of the gluon, charm quark, and antibottom quark, respectively. Following the same way that has been used to evaluate the $k^{0}$ integrals of $S_{1}$ and $S_{2}$, we carry out the $k^{0}$ integrals for $S_{3g}^{\mu}$, $S_{3c}^{\mu}$, and $S_{3\bar{b}}^{\mu}$ by contour integration. The sum of the three contributions is $\displaystyle S_{3}^{0}$ $\displaystyle=$ $\displaystyle-\frac{i}{8}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{\bm{k}}\frac{1}{\|\bm{k}\|^{2}(E_{b}-\bm{q}\cdot\hat{\bm{k}})(E_{c}+\bm{q}\cdot\hat{\bm{k}})}-\frac{i}{4(E_{b}+E_{c})}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{\bm{k}}\bigg{[}\frac{1}{\Delta_{c}[\bm{k}^{2}-(\Delta_{c}+E_{c})^{2}-i\varepsilon]}$ (117) $\displaystyle-\frac{1}{\Delta_{b}[\bm{k}^{2}-(\Delta_{b}-E_{b})^{2}-i\varepsilon]}\bigg{]},$ $\displaystyle S_{3}^{i}$ $\displaystyle=$ $\displaystyle\frac{i}{8}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{\bm{k}}\frac{k^{i}}{\|\bm{k}\|^{3}(E_{b}-\bm{q}\cdot\hat{\bm{k}})(E_{c}+\bm{q}\cdot\hat{\bm{k}})}+\frac{i}{4(E_{b}+E_{c})}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{\bm{k}}\bigg{[}\frac{k^{i}}{\Delta_{c}(\Delta_{c}+E_{c})}\;$ (118) $\displaystyle\times\frac{1}{\bm{k}^{2}-(\Delta_{c}+E_{c})^{2}-i\varepsilon}-\frac{[1+(E_{b}/\Delta_{b})]k^{i}}{[\bm{k}^{2}-(\Delta_{b}-E_{b})^{2}-i\varepsilon][\bm{k}^{2}+2\bm{k}\cdot\bm{q}-i\varepsilon]}\bigg{]}.$ $S_{3}^{0}$ and $S_{3}^{i}$ include three terms, which correspond to $S_{3g}^{\mu}$, $S_{3c}^{\mu}$, and $S_{3\bar{b}}^{\mu}$, respectively. The integrands of the gluon pole contributions $S_{3g}^{0}$ and $S_{3g}^{i}$, which are the first terms in $S_{3}^{0}$ and $S_{3}^{i}$, respectively, have factors $1/(E_{Q}\pm\bm{q}\cdot\hat{\bm{k}})$ that expand in powers of $\bm{q}\cdot\hat{\bm{k}}/E_{Q}$ producing scaleless factors and the factor $1/\|\bm{k}\|^{2}$ does not generate logarithmic divergence. Therefore, $S_{3g}^{\mu}=0$. The second terms of $S_{3}^{0}$ and $S_{3}^{i}$ are the charm quark pole contributions $S_{3c}^{0}$ and $S_{3c}^{i}$. We can follow the same procedure that was employed in expanding the integrand for $S_{2c}$ in (112) to find that $S_{3c}^{\mu}=0$. The last terms in $S_{3}^{0}$ and $S_{3}^{i}$ are the antibottom quark pole contributions $S_{3\bar{b}}^{0}$ and $S_{3\bar{b}}^{i}$, whose structure is similar to that of $S_{2\bar{b}}$ in (114). We find that $S_{3\bar{b}}^{0}$ contains only scaleless, power- divergent integrals, which vanish, and the only nonvanishing contribution is $S_{3\bar{b}}^{i}$: $S^{i}_{3\bar{b}}=\frac{i\,{q}^{i}}{4(E_{b}+E_{c})\bm{q}^{2}}\,n_{1}=-\frac{q^{i}}{16\pi(E_{b}+E_{c})\|\bm{q}\|}.$ (119) The Lorentz covariant expression for $S_{3}^{\mu}$ is, then, obtained as $S_{3}^{\mu}=\frac{i}{(4\pi)^{2}}\frac{i\pi}{4\delta p^{2}}\left[-\frac{p\cdot q}{p^{2}}p^{\mu}+q^{\mu}\right].$ (120) ### C.5 $\bm{S_{4}^{\mu\nu}}$ The integral $S_{4}^{\mu\nu}$ is the sum of three contributions: $S_{3}^{\mu}=S_{4g}^{\mu\nu}+S_{4c}^{\mu\nu}+S_{4\bar{b}}^{\mu\nu}$, where $S_{4g}^{\mu\nu}$, $S_{4c}^{\mu\nu}$, and $S_{4\bar{b}}^{\mu\nu}$ are the contributions from the poles of the gluon, charm quark, and antibottom quark, respectively. After evaluating the $k^{0}$ integral by contour integration, we find that $\displaystyle S_{4}^{00}$ $\displaystyle=$ $\displaystyle\frac{i}{8}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{\bm{k}}\frac{1}{\|\bm{k}\|(E_{b}-\bm{q}\cdot\hat{\bm{k}})(E_{c}+\bm{q}\cdot\hat{\bm{k}})}+\frac{i}{4(E_{b}+E_{c})}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{\bm{k}}\bigg{[}\frac{1+(E_{c}/\Delta_{c})}{\bm{k}^{2}-(\Delta_{c}+E_{c})^{2}-i\varepsilon}$ (121a) $\displaystyle-\frac{1-(E_{b}/\Delta_{b})}{\bm{k}^{2}-(\Delta_{b}-E_{b})^{2}-i\varepsilon}\bigg{]},$ $\displaystyle S_{4}^{0i}$ $\displaystyle=$ $\displaystyle-\frac{i}{8}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{\bm{k}}\frac{k^{i}}{\|\bm{k}\|^{2}(E_{b}-\bm{q}\cdot\hat{\bm{k}})(E_{c}+\bm{q}\cdot\hat{\bm{k}})}-\frac{i}{4(E_{b}+E_{c})}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{\bm{k}}\bigg{[}\frac{k^{i}}{\Delta_{c}[\bm{k}^{2}-(\Delta_{c}+E_{c})^{2}-i\varepsilon]}$ (121b) $\displaystyle-\frac{k^{i}}{\Delta_{b}[\bm{k}^{2}-(\Delta_{b}-E_{b})^{2}-i\varepsilon]}\bigg{]},$ $\displaystyle S_{4}^{ij}$ $\displaystyle=$ $\displaystyle\frac{i}{8}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{\bm{k}}\frac{k^{i}k^{j}}{\|\bm{k}\|^{3}(E_{b}-\bm{q}\cdot\hat{\bm{k}})(E_{c}+\bm{q}\cdot\hat{\bm{k}})}+\frac{i}{4(E_{b}+E_{c})}\mathcal{N}\\!\\!\\!\\!\\!\\!\\!\int_{\bm{k}}\bigg{[}\frac{k^{i}k^{j}/[\Delta_{c}(\Delta_{c}+E_{c})]}{[\bm{k}^{2}-(\Delta_{c}+E_{c})^{2}-i\varepsilon]}$ (121c) $\displaystyle-\frac{[1+(E_{b}/\Delta_{b})]k^{i}k^{j}}{[\bm{k}^{2}-(\Delta_{b}-E_{b})^{2}-i\varepsilon][\bm{k}^{2}+2\bm{k}\cdot\bm{q}-i\varepsilon]}\bigg{]}.$ Like $S_{3}^{\mu}$, the three terms in each of $S_{4}^{00}$, $S_{4}^{0i}$, and $S_{4}^{ij}$ correspond to $S_{4g}^{\mu\nu}$, $S_{4c}^{\mu\nu}$, and $S_{4\bar{b}}^{\mu\nu}$, respectively. 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Yu, NRQCD matrix elements for $S$-wave bottomonia and $\Gamma[\eta_{b}(nS)\to\gamma\gamma]$ with relativistic corrections, 1011.1554.	arxiv-papers	2010-11-10T05:09:47	2024-09-04T02:49:14.664474	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jungil Lee, WenLong Sang (Korea U.), Seyong Kim (Sejong U.)", "submitter": "Wen-long Sang", "url": "https://arxiv.org/abs/1011.2274" }
1011.2449	# Testing the CP-violating MSSM in stau decays at the LHC and ILC Herbi Dreiner1, Olaf Kittel2, Suchita Kulkarni1, Anja Marold1 1 Bethe Center for Theoretical Physics & Physikalisches Institut, Universität Bonn, D-53115 Bonn, Germany 2 Departamento de Física Teórica y del Cosmos and CAFPE, Universidad de Granada, E-18071 Granada, Spain & II. Institut für Theoretische Physik, Universität Hamburg, D-22761 Hamburg, Germany ###### Abstract We study CP violation in the two-body decay of a scalar tau into a neutralino and a tau, which should be probed at the LHC and ILC. From the normal tau polarization, a CP asymmetry is defined which is sensitive to the CP phases of the trilinear scalar coupling parameter $A_{\tau}$, the gaugino mass parameter $M_{1}$, and the higgsino mass parameter $\mu$ in the stau-neutralino sector of the Minimal Supersymmetric Standard Model. Asymmetries of more than $70\%$ are obtained in scenarios with strong stau mixing. As a result, detectable CP asymmetries in stau decays at the LHC are found, motivating further detailed experimental studies for probing the SUSY CP phases. ## I Introduction The surplus of matter over anti-matter within the universe can only be explained with a thorough understanding of CP violation. The CP phase in the quark mixing matrix of the Standard Model, which has been confirmed by B-meson experiments Belle , is not sufficient to understand the baryon asymmetry of the Universe Sakharov . However, the Minimal Supersymmetric Standard Model (MSSM) Haber:1984rc provides new physical phases that are manifestly CP- sensitive. After absorbing non-physical phases, we chose the complex parameters to be the higgsino mass parameter $\mu$, the U(1), and SU(3) gaugino mass parameters $M_{1}$, and $M_{3}$, and the trilinear scalar coupling parameters $A_{f}$ of the third generation sfermions $(f=b,t,\tau)$. The corresponding phases violate CP and are generally constrained by experimental bounds on electric dipole moments (EDMs) Li:2010ax . However, these restrictions are strongly model dependent cancellations3 ; cancellations1 ; cancellations2 , such that additional measurements outside the low energy EDM sector are required. Many CP observables have been proposed and studied in order to measure CP violation. Total cross sections totalsigma , masses masses , and branching ratios BRs , are CP-even quantities. For a direct evidence of CP violation, however, CP-odd (T-odd) observables are required. Examples are rate asymmetries of either branching ratios rateasymBRs , cross sections rateasymsigma , or angular distributions angulardistrib . Since these rate asymmetries require the presence of absorptive phases, they are typically small, of the order of $<10\%$, if they are not resonantly enhanced res . Larger CP-odd observables which already appear at tree-level are desirable. These are T-odd triple products of momenta and/or spins, from which CP-odd asymmetries can be constructed. Such triple product asymmetries are highly CP- sensitive, and have been intensively studied both at lepton and hadron colliders tripleproducts ; CPreview . Third generation sfermions have a rich phenomenology at high energy colliders like the LHC LHC or ILC ILC due to a sizable mixing of left and right states. In addition, the CP phases of the trilinear coupling parameters $A_{f}$ are rather unconstrained by the EDMs Semertzidis:2004uu ; cancellations3 ; Choi:2004rf . The phases of $A_{b}$ and $A_{t}$ have been studied in stop Bartl:2004jr ; Ellis:2008hq ; Deppisch:2009nj ; MoortgatPick:2009jy and sbottom Bartl:2006hh ; Deppisch:2010nc decays, respectively. Since these are decays of a scalar particle, the spin-spin correlations have to be taken into account. The triple product asymmetries can then be up to $40\%$, for sizable squark mixing. Similarly for probing the CP- violating phase of $A_{\tau}$ in the stau vertex, $\tilde{\tau}$-$\tilde{\chi}^{0}$-$\tau$, it is essential to include the tau spin. Only then is there a sensitivity to the phase of $A_{\tau}$ Bartl:2003gr ; Bartl:2003ck . If the spin of the tau is summed over, this crucial information is lost. Triple product asymmetries including the tau polarization have been studied in neutralino decays $\tilde{\chi}_{i}^{0}\to\tilde{\tau}\tau$ Bartl:2003gr , and also in chargino decays $\tilde{\chi}_{i}^{\pm}\to\tilde{\nu}_{\tau}\tau^{\pm}$ MKD . It was shown that the normal tau polarization itself is CP-sensitive, and that the asymmetries are large and of the order of $60\%$ to $70\%$. We are thus motivated to study CP violation, including the tau polarization, in the two-body decay of a stau $\tilde{\tau}_{m}\to\tau+\tilde{\chi}^{0}_{i},\quad m=1,2,\quad i=2,3,4,$ (1) followed by the subsequent chain of two-body decays $\displaystyle\tilde{\chi}^{0}_{i}$ $\displaystyle\to\ell_{1}+\tilde{\ell}_{n};$ (2a) $\displaystyle\tilde{\ell}_{n}$ $\displaystyle\to\tilde{\chi}^{0}_{1}+\ell_{2};\quad n=L,R,\quad\ell=e,\mu.$ (2b) See Fig. 1 for a schematic picture of the entire stau decay. This process is kinematically open for a mass hierarchy $\displaystyle m_{\tilde{\tau}}$ $\displaystyle>$ $\displaystyle m_{\tilde{\chi}_{i}^{0}}\;>\;m_{\tilde{e}}\,=\,m_{\tilde{\mu}},$ (3) where the staus are heavier than the smuons and selectrons. We thus work in MSSM scenarios with heavier stau soft SUSY breaking parameters $\displaystyle M_{\tilde{E}_{\tau}}$ $\displaystyle>$ $\displaystyle M_{\tilde{E}_{e}}=M_{\tilde{E}_{\mu}}$ (4) $\displaystyle M_{\tilde{L}_{\tau}}$ $\displaystyle>$ $\displaystyle M_{\tilde{L}_{e}}=M_{\tilde{L}_{\mu}}.$ (5) We show that the normal tau polarization, with respect to the plane spanned by the $\tau$ and $\ell_{1}$ momentum, is a triple product asymmetry which is sensitive to the phases of $A_{\tau}$, $M_{1}$, and $\mu$ in the stau- neutralino sector. For nearly degenerate stau masses, $M_{\tilde{E}_{\tau}}\approx M_{\tilde{L}_{\tau}},$ a strong stau mixing is obtained which results in tau polarization asymmetries of more than $70\%$. This should be measurable at colliders111Note that we do not include the tau decay in our calculations. However, some of the decay products of the tau have to be reconstructed in order to measure the tau spin. The main goal of our work is to motivate such an experimental study, to address the feasibility of measuring the CP phases at the LHC or ILC. . Since the stau is a scalar particle, its particular production does not contribute to CP-sensitive spin- spin correlations, and can thus be considered separately. This allows a collider-independent study, where we only discuss the boost dependence of the CP asymmetries. Figure 1: Schematic picture of stau decay. The paper is organized as follows. In Section II we review stau mixing and the stau-neutralino Lagrangian with complex couplings. We calculate the amplitude squared for the entire stau decay in the spin-density matrix formalism Haber:1994pe . We construct the CP asymmetry from the normal tau polarization, and discuss its MSSM parameter dependence, as well its boost dependence for colliders like the ILC and LHC. In Section III, we numerically study the phase and parameter dependence of the asymmetry, and the stau and neutralino branching ratios. We comment on the impact of the ${\tilde{\tau}}_{2}$ decay in scenarios with nearly degenerate stau masses. We summarize and conclude in Section IV. The Appendices contain the definitions of momenta and spin vectors, the analytical expressions for the stau decay amplitudes in the spin- density matrix formalism, and formulae for the stau decay widths. ## II Formalism ### II.1 Stau mixing In the complex MSSM, the stau mixing matrix in the $(\tilde{\tau}_{L},\tilde{\tau}_{R})$-basis is Haber:1984rc ; Bartl:2002uy $\mathcal{M}_{\tilde{\tau}}=\left(\begin{array}[]{cc}m^{2}_{\tilde{\tau}_{L}}&e^{-i\phi_{\tilde{\tau}}}m_{\tau}\|\Lambda_{\tilde{\tau}}\|\\\ \\\ e^{i\phi_{\tilde{\tau}}}m_{\tau}\|\Lambda_{\tilde{\tau}}\|&m^{2}_{\tilde{\tau}_{R}}\\\ \end{array}\right).$ (6) CP violation is parameterized by the physical phase $\displaystyle\phi_{\tilde{\tau}}$ $\displaystyle=$ $\displaystyle{\rm arg}[\Lambda_{\tilde{\tau}}],$ (7) $\displaystyle\Lambda_{\tilde{\tau}}$ $\displaystyle=$ $\displaystyle A_{\tau}-\mu^{}\cot{\beta},$ (8) with the complex trilinear scalar coupling parameter $A_{\tau}$, the complex higgisino mass parameter $\mu$, and $\tan\beta=v_{1}/v_{2}$, the ratio of the vacuum expectation values of the two neutral Higgs fields. The left and right stau masses are $\displaystyle m^{2}_{\tilde{\tau}_{L}}$ $\displaystyle=$ $\displaystyle M^{2}_{{\tilde{L}}_{\tau}}+(-\frac{1}{2}+\sin^{2}\theta_{w})m^{2}_{Z}\cos(2\beta)+m^{2}_{\tau},$ (9) $\displaystyle m^{2}_{\tilde{\tau}_{R}}$ $\displaystyle=$ $\displaystyle M^{2}_{{\tilde{E}}_{\tau}}-\sin^{2}\theta_{w}m^{2}_{Z}\cos(2\beta)+m^{2}_{\tau},$ (10) with the real soft SUSY breaking parameters $M^{2}_{{\tilde{L}_{\tau}},{\tilde{E}}_{\tau}}$, the electroweak mixing angle $\theta_{w}$, and the masses of the $Z$ boson $m_{Z}$, and of the tau lepton, $m_{\tau}$. In the mass basis, the stop eigenstates are Haber:1984rc ; Bartl:2002uy $\left(\begin{array}[]{c}\tilde{\tau}_{1}\\\ \tilde{\tau}_{2}\end{array}\right)=\mathcal{R}^{\tilde{\tau}}\left(\begin{array}[]{c}\tilde{\tau}_{L}\\\ \tilde{\tau}_{R}\end{array}\right),$ (11) with the diagonalization matrix $\mathcal{R}^{\tilde{\tau}}=\left(\begin{array}[]{cc}e^{i\phi_{\tilde{\tau}}}\cos\theta_{\tilde{\tau}}&\sin\theta_{\tilde{\tau}}\\\ -\sin\theta_{\tilde{\tau}}&e^{-i\phi_{\tilde{\tau}}}\cos\theta_{\tilde{\tau}}\end{array}\right),$ (12) and the stau mixing angle $\displaystyle\cos\theta_{\tilde{\tau}}$ $\displaystyle=$ $\displaystyle\frac{\displaystyle- m_{\tau}\|\Lambda_{\tilde{\tau}}\|}{\displaystyle\sqrt{m_{\tau}^{2}\|\Lambda_{\tilde{\tau}}^{2}\|+\left(m_{\tilde{\tau}_{1}}^{2}-m_{\tilde{\tau}_{2}}^{2}\right)^{2}}},$ (13) $\displaystyle\sin\theta_{\tilde{\tau}}$ $\displaystyle=$ $\displaystyle\frac{\displaystyle m_{\tilde{\tau}_{L}}^{2}-m_{\tilde{\tau}_{1}}^{2}}{\displaystyle\sqrt{m_{\tau}^{2}\|\Lambda_{\tilde{\tau}}^{2}\|+\left(m_{\tilde{\tau}_{1}}^{2}-m_{\tilde{\tau}_{2}}^{2}\right)^{2}}}.$ (14) The stau mass eigenvalues are $\displaystyle m^{2}_{\tilde{\tau}_{1,2}}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\bigg{[}\left(m_{\tilde{\tau}_{L}}^{2}+m_{\tilde{\tau}_{R}}^{2}\right)\mp$ (15) $\displaystyle\phantom{\frac{1}{2}}\sqrt{\left(m_{\tilde{\tau}_{L}}^{2}-m_{\tilde{\tau}_{R}}^{2}\right)^{2}+4m_{\tilde{\tau}}^{2}\|\Lambda_{\tilde{\tau}}\|^{2}}~{}\bigg{]}.$ ### II.2 Lagrangian and complex couplings The relevant Lagrangian terms for the stau decay $\tilde{\tau}_{m}\to\tau\tilde{\chi}_{i}^{0}$ are Haber:1984rc ; Bartl:2002uy $\displaystyle{\mathscr{L}}_{\tau\tilde{\tau}\tilde{\chi}^{0}}=g\,\bar{\tau}\,(a_{mi}^{\tilde{\tau}}\,P_{R}+b_{mi}^{\tilde{\tau}}\,P_{L})\,\tilde{\chi}_{i}^{0}\,\tilde{\tau}_{m}+{\rm h.c.},$ (16) with $P_{L,R}=(1\mp\gamma_{5})/2$, and the weak coupling constant $g=e/\sin\theta_{w}$, $e>0$. The couplings are defined as Bartl:2002uy $a_{mi}^{\tilde{\tau}}\equiv\sum^{2}_{n=1}\,(\mathcal{R}^{\tilde{\tau}}_{mn})^{\ast}\,{\cal A}_{in}^{\tau},\quad b_{mi}^{\tilde{\tau}}\equiv\sum^{2}_{n=1}\,(\mathcal{R}^{\tilde{\tau}}_{mn})^{\ast}\,{\cal B}_{in}^{\tau}.$ (17) The stau diagonalization matrix $\mathcal{R}^{\tilde{t}}$ is given in Eq. (2), and ${\cal A}_{i}^{\tau}\equiv\left(\begin{array}[]{ccc}f_{\tau i}^{L}\\\\[5.69054pt] h_{\tau i}^{R}\end{array}\right),\qquad{\cal B}_{i}^{\tau}\equiv\left(\begin{array}[]{ccc}h_{\tau i}^{L}\\\\[5.69054pt] f_{\tau i}^{R}\end{array}\right).$ (18) In the photino, zino, higgsino basis ($\tilde{\gamma},\tilde{Z},\tilde{H}^{0}_{a},\tilde{H}^{0}_{b}$), we have $\displaystyle f_{\tau i}^{L}$ $\displaystyle=$ $\displaystyle\sqrt{2}\bigg{[}\frac{1}{\cos\theta_{w}}\left(\frac{1}{2}-\sin^{2}\theta_{w}\right)N_{i2}+\sin\theta_{w}N_{i1}\bigg{]},\qquad$ (19) $\displaystyle f_{\tau i}^{R}$ $\displaystyle=$ $\displaystyle\sqrt{2}\sin\theta_{w}\left(\tan\theta_{w}N_{i2}^{}-N_{i1}^{}\right),$ (20) $\displaystyle h_{\tau i}^{L}$ $\displaystyle=$ $\displaystyle(h_{\tau i}^{R})^{\ast}=-Y_{\tau}(N_{i3}^{\ast}\cos\beta+N_{i4}^{\ast}\sin\beta),$ (21) $\displaystyle Y_{\tau}$ $\displaystyle=$ $\displaystyle\frac{m_{\tau}}{\sqrt{2}\,m_{W}\cos\beta},$ (22) with $m_{W}$ the mass of the $W$ boson, and $N$ the complex, unitary $4\times 4$ matrix that diagonalizes the neutralino mass matrix Haber:1984rc $N^{}\cdot{\mathcal{M}}_{\tilde{\chi}^{0}}\cdot N^{\dagger}={\rm diag}(m_{\tilde{\chi}^{0}_{1}},\dots,m_{\tilde{\chi}^{0}_{4}}).$ (23) The interaction Lagrangian relevant for the neutralino decay $\tilde{\chi}_{i}^{0}\to\tilde{\ell}_{R,L}^{\pm}\ell^{\mp}$, for $\ell=e,\mu$ is Haber:1984rc $\displaystyle{\mathscr{L}}_{\ell\tilde{\ell}\tilde{\chi}^{0}}$ $\displaystyle=$ $\displaystyle g\bar{\ell}f_{\ell i}^{L}P_{R}\tilde{\chi}_{i}^{0}\tilde{\ell}_{L}+g\bar{\ell}f_{\ell i}^{R}P_{L}\tilde{\chi}_{i}^{0}\tilde{\ell}_{R}+\mbox{h.c.},$ (24) with the couplings $f_{\ell i}^{L,R}$ given in Eqs. (19) and (20). ### II.3 Tau spin density matrix The unnormalized, $2\times 2$, hermitian, $\tau$ spin density matrix for stau decay, Eqs. (1) and (2), reads $\rho^{\lambda_{\tau}\lambda^{\prime}_{\tau}}\equiv\int\left(\|\mathcal{M}\|^{2}\right)^{\lambda_{\tau}\lambda^{\prime}_{\tau}}{{\rm d}\mathscr{L}\\!\textsl{ips}},$ (25) with the amplitude $\mathcal{M}$, and the Lorentz invariant phase space element ${{\rm d}\mathscr{L}\\!\textsl{ips}}$, for details see Appendix B. The $\tau$ helicities are denoted by $\lambda_{\tau}$ and $\lambda^{\prime}_{\tau}$. In the spin density matrix formalism Haber:1994pe , the amplitude squared is given by $\displaystyle\left(\|\mathcal{M}\|^{2}\right)^{\lambda_{\tau}\lambda^{\prime}_{\tau}}=\|\Delta(\tilde{\chi}^{0}_{i})\|^{2}\|\Delta(\tilde{\ell})\|^{2}\times$ (26) $\displaystyle\sum_{\lambda_{i}\lambda_{i}^{\prime}}\rho_{D}(\tilde{\tau})^{\lambda_{\tau}\lambda_{\tau}^{\prime}}_{\lambda_{i}\lambda_{i}^{\prime}}\;\rho_{D_{1}}(\tilde{\chi}^{0}_{i})^{\lambda_{i}^{\prime}\lambda_{i}}\;D_{2}(\tilde{\ell}),$ with the neutralino helicities $\lambda_{i},\,\lambda_{i}^{\prime}$. The amplitude squared decomposes into the remnants of the propagators $\Delta(j)=\dfrac{i}{s_{j}-m_{j}^{2}+im_{j}\Gamma_{j}},$ (27) with mass $m_{j}$, and width $\Gamma_{j}$ of particle $j=\tilde{\chi}^{0}_{i}$ or $\tilde{\ell}$, and the unnormalized spin density matrices for stau decay $\rho_{D}(\tilde{\tau})$, and neutralino decay $\rho_{D_{1}}(\tilde{\chi}^{0}_{i})$. The decay matrix of the spinless slepton is a factor since the polarizations of the final lepton and LSP are not accessible. The corresponding amplitude is denoted by $D_{2}(\tilde{\ell})$. Defining a set of spin basis vectors $s_{\tau}^{a}$ for the tau, see Eqs. (60) in Appendix A, and $s_{\tilde{\chi}^{0}_{i}}^{b}$ for the neutralino Kittel:2004rp , the spin density matrices can be expanded in terms of the Pauli matrices $\sigma$ $\displaystyle\rho_{\rm D}(\tilde{\tau})^{\lambda_{\tau}\lambda_{\tau}^{\prime}}_{\lambda_{i}\lambda_{i}^{\prime}}={\rm D}\,\delta^{\lambda_{\tau}\lambda_{\tau}^{\prime}}\delta_{\lambda_{i}\lambda_{i}^{\prime}}+\Sigma_{\rm D}^{a}\,(\sigma^{a})^{\lambda_{\tau}\lambda_{\tau}^{\prime}}\delta_{\lambda_{i}\lambda_{i}^{\prime}}+$ $\displaystyle\Sigma_{\rm D}^{b}\,\delta^{\lambda_{\tau}\lambda_{\tau}^{\prime}}(\sigma^{b})_{\lambda_{i}\lambda_{i}^{\prime}}+\Sigma_{\rm D}^{ab}\,(\sigma^{a})^{\lambda_{\tau}\lambda_{\tau}^{\prime}}(\sigma^{b})_{\lambda_{i}\lambda_{i}^{\prime}},$ (28) $\displaystyle\rho_{{\rm D}_{1}}({\tilde{\chi}}_{i}^{0})^{\lambda_{i}^{\prime}\lambda_{i}}={\rm D}_{1}\,\delta^{\lambda_{i}^{\prime}\lambda_{i}}+\Sigma_{{\rm D}_{1}}^{b}\,(\sigma^{b})^{\lambda_{i}^{\prime}\lambda_{i}},\phantom{spacespace}$ (29) with an implicit sum over $a,b=1,2,3$, respectively. The real expansion coefficients ${\rm D}$, ${\rm D_{1}}$, $\Sigma_{\rm D}^{a}$, $\Sigma_{\rm D}^{b}$, $\Sigma_{{\rm D}_{1}}^{b}$ and $\Sigma_{\rm D}^{ab}$ contain the physical information of the process. ${\rm D}$ denotes the unpolarized part of the amplitude for stau decay ${\tilde{\tau}}_{m}\to\chi^{0}_{i}\tau$ , ${\rm D_{1}}$ denotes the unpolarized part for neutralino decay $\chi^{0}_{i}\to{\tilde{\ell}}_{R}\ell_{1}$ , respectively. $\Sigma_{\rm D}^{a}$ gives the tau polarization, $\Sigma_{\rm D}^{b}$, and $\Sigma_{{\rm D}_{1}}^{b}$ describe the contributions from the neutralino polarization, and $\Sigma_{\rm D}^{ab}$ is the spin-spin correlation term, which contains the CP-sensitive parts. We give the expansion coefficients explicitly in Appendix C. Inserting the density matrices, Eqs. (28) and (29), into Eq. (26), we get for the amplitude squared $\displaystyle(\|\mathcal{M}\|^{2})^{\lambda_{\tau}\lambda_{\tau}^{\prime}}$ $\displaystyle=$ $\displaystyle 2\|\Delta({\tilde{\chi}^{0}}_{i})\|^{2}\|\Delta(\tilde{\ell})\|^{2}\times$ (30) $\displaystyle\Big{[}~{}({\rm D}{\rm D}_{1}+\Sigma_{\rm D}^{b}\Sigma_{{\rm D}_{1}}^{b})\delta^{\lambda_{\tau}\lambda_{\tau}^{\prime}}$ $\displaystyle+(\Sigma_{\rm D}^{a}{\rm D}_{1}+\Sigma_{\rm D}^{ab}\Sigma_{{\rm D}_{1}}^{b})(\sigma^{a})^{\lambda_{\tau}\lambda_{\tau}^{\prime}}\Big{]}{\rm D}_{2},\quad$ with an implicit sum over $a,b=1,2,3$. The amplitude squared $(\|\mathcal{M}\|^{2})^{\lambda_{\tau}\lambda_{\tau}^{\prime}}$ is now decomposed into an unpolarized part (first summand), and into the part for the tau polarization (second summand), in Eq. (30). By using the completeness relations for the neutralino spin vectors, Eq. (62), the products in Eq. (30) can be written222The formulas are given for the decay of a negatively charged stau $\tilde{\tau}_{m}\to\tau^{-}\tilde{\chi}_{i}^{0}$, followed by $\tilde{\chi}^{0}_{i}\to\ell_{1}^{+}\tilde{\ell}_{R}^{-}$. The signs in parentheses in Eqs. (31) and (32) hold for the charge conjugated stau decay $\tilde{\tau}_{m}^{\ast}\to\tau^{+}\tilde{\chi}_{i}^{0}$; $\tilde{\chi}^{0}_{i}\to\ell_{1}^{+}\tilde{\ell}_{R}^{-}$. In order to obtain the terms for the decay $\tilde{\tau}_{m}^{(\ast)}\to\tau^{\mp}\tilde{\chi}_{i}^{0}$, however, followed by the neutralino decay into a positively charged slepton, $\tilde{\chi}^{0}_{i}\to\ell_{1}^{-}\tilde{\ell}_{R}^{+}$, one has to reverse the signs of Eqs. (31) and (32). This is due to the sign change of $\Sigma^{b}_{{\rm D}_{1}}$, see Eqs. (74). In Appendix C, we also give the terms for the neutralino decay into a left slepton, $\tilde{\chi}^{0}_{i}\to\ell_{1}^{\pm}\tilde{\ell}_{L}^{\mp}$. Note that the term proportional to $m_{\tau}$ in Eq. (32) is negligible at high particle energies $E\gg m_{\tau}$. , $\displaystyle\Sigma^{b}_{\rm D}~{}\Sigma^{b}_{{\rm D}_{1}}$ $\displaystyle=$ $\,{}^{\;\,+}_{(-)}\frac{g^{4}}{2}\left(\|a^{\tilde{\tau}}_{mi}\|^{2}-\|b^{\tilde{\tau}}_{mi}\|^{2}\right)\|f^{R}_{\ell i}\|^{2}\times$ (31) $\displaystyle\left[m_{\tilde{\chi}_{i}^{0}}^{2}(p_{\tau}\cdot p_{\ell_{1}})-(p_{\tilde{\chi}_{i}^{0}}\cdot p_{\tau})(p_{\ell_{1}}\cdot p_{\tilde{\chi}_{i}^{0}})\right],\quad$ $\displaystyle\Sigma^{ab}_{\rm D}~{}\Sigma^{b}_{{\rm D}_{1}}$ $\displaystyle=$ $\,{}^{\;\,+}_{(-)}\frac{g^{4}}{2}\left(\|a^{\tilde{\tau}}_{mi}\|^{2}+\|b^{\tilde{\tau}}_{mi}\|^{2}\right)\|f^{R}_{\ell i}\|^{2}m_{\tau}\times$ (32) $\displaystyle\left[(s^{a}_{\tau}\cdot p_{\tilde{\chi}_{i}^{0}})(p_{\tilde{\chi}_{i}^{0}}\cdot p_{\ell_{1}})-m_{\tilde{\chi}_{i}^{0}}^{2}(s^{a}_{\tau}\cdot p_{\ell_{1}})\right]$ $\,{}^{\;\,+}_{(-)}g^{4}\mathfrak{Re}\\{a^{\tilde{\tau}}_{mi}(b^{\tilde{\tau}}_{mi})^{\ast}\\}\|f^{R}_{\ell i}\|^{2}m_{\tilde{\chi}_{i}^{0}}\times$ $\displaystyle\left[(p_{\tau}\cdot p_{\tilde{\chi}_{i}^{0}})(s^{a}_{\tau}\cdot p_{\ell_{1}})-(p_{\tau}\cdot p_{\ell_{1}})(s^{a}_{\tau}\cdot p_{\tilde{\chi}_{i}^{0}})\right]$ $\,{}^{\;\,+}_{(-)}g^{4}\|f^{R}_{\ell i}\|^{2}m_{\tilde{\chi}_{i}^{0}}\times$ $\displaystyle\mathfrak{Im}\\{a^{\tilde{\tau}}_{mi}(b^{\tilde{\tau}}_{mi})^{\ast}\\}[p_{\tilde{\tau}},~{}p_{\ell_{1}},~{}p_{\tau},~{}s^{a}_{\tau}].$ The spin-spin correlation term $\Sigma^{ab}_{\rm D}~{}\Sigma^{b}_{{\rm D}_{1}}$, Eq. (32), explicitly depends on the imaginary part $\mathfrak{Im}\\{a^{\tilde{\tau}}_{mi}(b^{\tilde{\tau}}_{mi})^{\ast}\\}$ of the stau-tau-neutralino couplings, Eq. (16). Thus this term is manifestly CP- sensitive, i.e., it depends on the phases $\phi_{A_{\tau}}$, $\phi_{1}$, $\phi_{\mu}$ of the stau-tau-neutralino sector. The imaginary part is multiplied by the totally anti-symmetric (epsilon) product, $\mathcal{E}^{a}\equiv[p_{\tilde{\tau}},~{}p_{\ell_{1}},~{}p_{\tau},~{}s^{a}_{\tau}]\equiv\epsilon_{\mu\nu\rho\sigma}\,p_{\tilde{\tau}}^{\mu}\,p_{\ell_{1}}^{\nu}\,p_{\tau}^{\rho}\,s^{a,\sigma}_{\tau},$ (33) with the convention $\epsilon_{0123}=1$. Since each of the spatial components of the four-momenta $p$, or the spin vectors $s^{a}_{\tau}$, changes sign under a time transformation, $t\to-t$, the epsilon product $\mathcal{E}^{a}$ is T-odd. In the stau rest frame, $p_{\tilde{\tau}}^{\mu}=(m_{\tilde{\tau}},\mathbf{0})$, the epsilon product reduces to the T-odd triple product ${\mathcal{T}}^{a}$ $[p_{\tilde{\tau}},~{}p_{\ell_{1}},~{}p_{\tau},~{}s^{a}_{\tau}]=m_{\tilde{\tau}}\;\,(\mathbf{p}_{\ell_{1}}\times\mathbf{p}_{\tau})\cdot\mathbf{s}^{a}_{\tau}\equiv m_{\tilde{\tau}}\,{\mathcal{T}}^{a}.$ (34) The task in the next section is to define an observable that projects out from the amplitude squared the part proportional to $\mathcal{E}^{a}$ (or ${\mathcal{T}}^{a}$), in order to probe the CP-sensitive coupling combination $\mathfrak{Im}\\{a^{\tilde{\tau}}_{mi}(b^{\tilde{\tau}}_{mi})^{\ast}\\}$. ### II.4 Normal tau polarization and CP asymmetry The $\tau$ polarization is given by the expectation value of the Pauli matrices $\sigma$$=(\sigma_{1},\sigma_{2},\sigma_{3})$ Renard:1981de $\mbox{\boldmath$\mathcal{P}$ }=\frac{{\rm Tr}\\{\rho\mbox{\boldmath$\sigma$}\\}}{{\rm Tr}\\{\rho\\}},$ (35) with the $\tau$ spin density matrix $\rho$, as given in Eq. (25). In our convention for the polarization vector $\mathcal{P}$ $=(\mathcal{P}_{1},\mathcal{P}_{2},\mathcal{P}_{3})$, the components $\mathcal{P}_{1}$ and $\mathcal{P}_{3}$ are the transverse and longitudinal polarizations in the plane spanned by ${\bf p}_{\ell_{1}}$ and ${\bf p}_{\tau}$, respectively, and $\mathcal{P}_{2}$ is the polarization normal to that plane. See our definition of the tau spin basis vectors $s_{\tau}^{a}$ in Appendix A. The normal $\tau$ polarization is equivalently defined as $\displaystyle\mathcal{P}_{2}$ $\displaystyle\equiv$ $\displaystyle\frac{N(\uparrow)-N(\downarrow)}{N(\uparrow)+N(\downarrow)},$ (36) with the number of events $N$ with the $\tau$ spin up $(\uparrow)$ or down $(\downarrow)$, with respect to the quantization axis ${\bf p}_{\ell_{1}}\times{\bf p}_{\tau}$, see Eq. (60). The normal $\tau$ polarization can thus also be regarded as an asymmetry $\mathcal{P}_{2}=\frac{\sigma(\mathcal{T}>0)-\sigma(\mathcal{T}<0)}{\sigma(\mathcal{T}>0)+\sigma(\mathcal{T}<0)},$ (37) of the triple product $\mathcal{T}=({\bf p}_{\ell_{1}}\times{\bf p}_{\tau})\cdot{\mbox{\boldmath$\xi$}}_{\tau},$ (38) where $\xi$τ is the direction of the $\tau$ spin vector for each event. The triple product $\mathcal{T}$ is included in the spin-spin correlation term $\Sigma^{ab}_{\rm D}~{}\Sigma^{b}_{{\rm D}_{1}}$, Eq. (32), cf. Eq. (34), and the asymmetry thus probes the term which contains the CP-sensitive coupling combination $\mathfrak{Im}\\{a^{\tilde{\tau}}_{mi}(b^{\tilde{\tau}}_{mi})^{\ast}\\}$. Since under naive time reversal, $t\to-t$, the triple product $\mathcal{T}$ changes sign, the tau polarization $\mathcal{P}_{2}$, Eq. (37), is T-odd. Due to CPT invariance Luders:1954zz , $\mathcal{P}_{2}$ would thus be CP-odd at tree level. In general, $\mathcal{P}_{2}$ also has contributions from absorptive phases, e.g. from intermediate $s$-state resonances or final-state interactions, which do not signal CP violation. Although such absorptive contributions are a higher order effect, and thus expected to be small, they can be eliminated in the true CP asymmetry Bartl:2003gr ${\mathcal{A}_{\tau}^{\rm CP}}=\frac{1}{2}(\mathcal{P}_{2}-\bar{\mathcal{P}}_{2}),$ (39) where $\bar{\mathcal{P}}_{2}$ is the normal tau polarization for the charged conjugated process $\tilde{\tau}_{m}^{\ast}\to\tau^{+}\tilde{\chi}_{i}^{0}$. For our analysis at tree level, where no absorptive phases are present, we find $\bar{\mathcal{P}}_{2}=-\mathcal{P}_{2}$, see the sign change in Eqs. (31) and (32), and thus ${\mathcal{A}_{\tau}^{\rm CP}}=\mathcal{P}_{2}$. We study ${\mathcal{A}_{\tau}^{\rm CP}}$ in the following, which is, however, equivalent to $\mathcal{P}_{2}$ at tree level. Inserting now the explicit form of the density matrix $\rho$, Eq. (25), into Eq. (35), together with Eq. (30), we obtain the CP asymmetry ${\mathcal{A}_{\tau}^{\rm CP}}=\mathcal{P}_{2}=\frac{\int\Sigma^{a=2,b}_{\rm D}~{}\Sigma^{b}_{{\rm D}_{1}}\,{{\rm d}\mathscr{L}\\!\textsl{ips}}}{\int{\rm D}{\rm D}_{1}\,{{\rm d}\mathscr{L}\\!\textsl{ips}}},$ (40) where we have used the narrow width approximation for the propagators in the phase space element ${{\rm d}\mathscr{L}\\!\textsl{ips}}$, see Eq. (87). Note that in the denominator of ${\mathcal{A}_{\tau}^{\rm CP}}$, Eq. (40), the spin correlation terms vanish, $\int\Sigma^{b}_{\rm D}~{}\Sigma^{b}_{{\rm D}_{1}}\,{{\rm d}\mathscr{L}\\!\textsl{ips}}=0$, see Eq. (31), when integrated over phase space. In the numerator only the spin-spin correlation term $\Sigma^{ab}_{\rm D}~{}\Sigma^{b}_{{\rm D}_{1}}$ for $a=2$ contributes, which contains the T-odd epsilon product $\mathcal{E}^{a}$, see Eq. (33). ### II.5 Parameter dependence of the CP asymmetry To qualitatively understand the dependence of the asymmetry ${\mathcal{A}_{\tau}^{\rm CP}}$, Eq. (40), on the MSSM parameters, we study in some detail its dependence on the $\tilde{\tau}_{m}$-$\tau$-$\tilde{\chi}_{i}^{0}$ couplings, $a^{\tilde{\tau}}_{mi}$ and $b^{\tilde{\tau}}_{mi}$, see Eq. (81). From the explicit form of the decay terms $\Sigma^{b}_{\rm D}~{}\Sigma^{b}_{{\rm D}_{1}}$ Eq. (31), and ${\rm D}$, ${\rm D}_{1}$, Eqs. (69), (73), respectively, we find that the asymmetry $\displaystyle{\mathcal{A}_{\tau}^{\rm CP}}$ $\displaystyle=$ $\displaystyle\eta_{mi}\;\frac{m_{{\tilde{\chi}^{0}}_{i}}\int[p_{\tilde{\tau}},~{}p_{\ell_{1}},~{}p_{\tau},~{}s^{a=2}_{\tau}]\,{\rm d}\mathscr{L}\\!\textsl{ips}}{(p_{{\tilde{\chi}}_{i}^{0}}\cdot p_{\tau})(p_{{\tilde{\chi}}_{i}^{0}}\cdot p_{\ell_{1}})\,\int{\rm d}\mathscr{L}\\!\textsl{ips}},$ (41) with $(p_{{\tilde{\chi}}_{i}^{0}}\cdot p_{\tau})=(m_{\tilde{\tau}}^{2}-m_{\tilde{\chi}_{i}^{0}}^{2})/2$, and $(p_{\tilde{\chi}^{0}_{i}}\cdot p_{\ell_{1}})=(m_{\tilde{\chi}_{i}^{0}}^{2}-m_{\tilde{\ell}}^{2})/2$, is proportional to the decay coupling factor $\eta_{mi}=\frac{\mathfrak{Im}\\{a^{\tilde{\tau}}_{mi}(b^{\tilde{\tau}}_{mi})^{\ast}\\}}{\frac{1}{2}(\|a^{\tilde{\tau}}_{mi}\|^{2}+\|b^{\tilde{\tau}}_{mi}\|^{2})},$ (42) with $\eta_{mi}\in[-1,1]$. We thus expect maximal asymmetries for equal moduli of left and right couplings, $\|a^{\tilde{\tau}}_{mi}\|\approx\|b^{\tilde{\tau}}_{mi}\|$, which have a phase difference of about $\pi/2$, where the coupling factor can be maximal $\eta_{mi}=\pm 1$, see Eq. (42). To study the dependence of $\eta$ on the CP phase $\phi_{\tilde{\tau}}$ of the stau sector, and the stau mixing angle $\theta_{\tilde{\tau}}$, we expand the imaginary part of the product of $\tilde{\tau}_{m}$-$\tau$-$\tilde{\chi}_{i}^{0}$ couplings $\displaystyle\mathfrak{Im}\\{a^{\tilde{\tau}}_{mi}(b^{\tilde{\tau}}_{mi})^{\ast}\\}$ $\displaystyle=$ $\displaystyle\mathfrak{Im}\Big{\\{}\|\mathcal{R}_{m1}\|^{2}f^{L}_{\tau i}h^{R}_{\tau i}+\|\mathcal{R}_{m2}\|^{2}f^{\ast R}_{\tau i}h^{R}_{\tau i}$ (43) $\displaystyle+$ $\displaystyle\mathcal{R}_{m1}\mathcal{R}_{m2}^{\ast}\big{[}(h^{R}_{\tau i})^{2}-f^{R}_{\tau i}f^{\ast L}_{\tau i}\big{]}\Big{\\}},$ in terms of the stau mixing matrix $\mathcal{R}$, the gauge couplings $f_{\tau i}^{L,R}$ and the higgs couplings $h_{\tau i}^{L,R}$. In particular, for a CP- conserving neutralino sector, $\phi_{1}=\phi_{\mu}=0$, we have $\displaystyle\mathfrak{Im}\\{a^{\tilde{\tau}}_{mi}(b^{\tilde{\tau}}_{mi})^{\ast}\\}=\,^{\;\,+}_{(-)}\sin\phi_{\tilde{\tau}}\sin(2\theta_{\tilde{\tau}})\frac{1}{2}\left[(h^{R}_{\tau i})^{2}-f^{R}_{\tau i}f^{L}_{\tau i}\right],$ (44) for $m=1$, and the sign in parentheses holds for $m=2$. Thus we expect a maximal $\eta$ and thus maximal asymmetries for maximal stau mixing333 Note that a maximal mixing is naturally achieved for nearly degenerate staus. However then the asymmetries for $\tilde{\tau}_{1}$ and $\tilde{\tau}_{2}$ decay typically have similar magnitude but opposite sign, and thus might cancel. See the discussion at the end of the numerics in Section III.4. , $\theta_{\tilde{\tau}}\approx\pm\pi/4$, and a maximal CP phase in the stau mixing matrix, $\phi_{\tilde{\tau}}\approx\pm\pi/2$. Note that, in particular, the dependence of $\phi_{\tilde{\tau}}$ on $\phi_{A_{\tau}}$ is strong for $\|A_{\tau}\|>\|\mu\|\tan\beta$. We will study numerically the phase and parameter dependence on ${\mathcal{A}_{\tau}^{\rm CP}}$ and $\eta$ further in Section III. ### II.6 Boost dependence Figure 2: Boost distributions of the $\tau$ polarization asymmetry ${\mathcal{A}_{\tau}^{\rm CP}}$, Eq. (39), normalized by ${\mathcal{A}_{\tau}^{\rm CP}}(\beta_{\tilde{\tau}}=0)$, for three different sets of stau masses, $m_{\tilde{\tau}_{1,2}}\approx 200$ GeV (solid, red), $400$ GeV (dashed, green), and $1000$ TeV (dotted, blue), see text, for stau decay $\tilde{\tau}_{1}\to\tau\tilde{\chi}_{2}^{0}$, followed by $\tilde{\chi}_{2}^{0}\to\ell_{1}\tilde{\ell}_{R}$, and $\tilde{\ell}_{R}\to\tilde{\chi}^{0}_{1}\ell_{2}$ ($\ell=e$ or $\mu$), see Fig. 1, The SUSY parameters are given in Table 1. The triple product asymmetry ${\mathcal{A}_{\tau}^{\rm CP}}$, Eq. (40), is not Lorentz invariant but depends on the boost of the decaying stau, $\beta_{\tilde{\tau}}=\dfrac{\|{\bf p}_{\tilde{\tau}}\|}{E_{\tilde{\tau}}}.$ (45) In Fig. 2, we show the boost dependence of the asymmetry ${\mathcal{A}_{\tau}^{\rm CP}}(\beta_{\tilde{\tau}})$, normalized by ${\mathcal{A}_{\tau}^{\rm CP}}(\beta_{\tilde{\tau}}=0)$. The SUSY parameters are given in Table 1, and we have chosen three sets of different $\tilde{\tau}$ soft-breaking parameters $\\{M_{{\tilde{E}}_{\tau}},M_{{\tilde{L}}_{\tau}}\\}=\\{195,200\\}$ GeV (solid, red); $\\{395,400\\}$ GeV (dashed, green); and $\\{998,1000\\}$ GeV (dotted, blue). The corresponding stau masses are $\\{m_{\tilde{\tau}_{1}},m_{\tilde{\tau}_{2}}\\}=\\{194,209\\}$; $\\{395,404\\}$; $\\{998,1002\\}$ GeV, respectively. The corresponding asymmetries in the stau rest frame are ${\mathcal{A}_{\tau}^{\rm CP}}(\beta_{\tilde{\tau}})=-66\%$; $-72\%$, $-71\%$. Note that we have chosen nearly degenerate stau masses which lead to an enhanced stau mixing and thus to maximal asymmetries; see also the discussion in Section III. For the stau masses $\\{m_{\tilde{\tau}_{1}},m_{\tilde{\tau}_{2}}\\}=\\{194,209\\}$ GeV, the staus can be produced at the ILC with $\sqrt{s}=500$ GeV, and have a fixed boost of $\beta_{\tilde{\tau}}=0.63$. The corresponding asymmetry is then reduced to ${\mathcal{A}_{\tau}^{\rm CP}}=-53\%$ if the stau rest frame cannot be reconstructed. Typical ILC cross section for these masses are of the order of some $20$ fb Alwall:2007st . If the staus are produced at the LHC, they will have a distinct boost distribution depending on their mass, which typically peaks at high values $\beta_{\tilde{\tau}}\approx 0.9$ for stau masses of the order of a few $100$ GeV up to a $1$ TeV, see e.g. Refs Deppisch:2009nj ; Deppisch:2010nc . Then the normal tau polarization in the laboratory frame is obtained by folding the boost dependent polarization ${\mathcal{A}_{\tau}^{\rm CP}}$ with the normalized stau boost distribution Deppisch:2009nj , ${\mathcal{A}_{\tau}^{\rm CP}}_{\rm lab}=\dfrac{1}{\sigma_{P}}\int_{0}^{1}\dfrac{{\rm d}\sigma_{P}}{{\rm d}\beta_{\tilde{\tau}}}{\mathcal{A}_{\tau}^{\rm CP}}(\beta_{\tilde{\tau}})\,{\rm d}\beta_{\tilde{\tau}},$ (46) with the production cross section $\sigma_{P}=\sigma(pp\to\tilde{\tau}^{+}\tilde{\tau}^{-})$. The typical reduction of the normal tau polarization ${\mathcal{A}_{\tau}^{\rm CP}}_{\rm lab}$ is of the order of two thirds of the asymmetry compared to that in the stau rest frame ${\mathcal{A}_{\tau}^{\rm CP}}(0)$. However, it has been recently shown (for similar asymmetries in stop decays at the LHC), that the rest frame can be partly reconstructed event by event using on-shell mass conditions, see Refs. MoortgatPick:2009jy . The LHC cross section for stau pair production, $\sigma(pp\to\tilde{\tau}_{1}^{+}\tilde{\tau}_{1}^{-})$, also sensitively depends on the stau masses, e.g., for our benchmark scenario in Table 1, we find cross sections up to $10$ fb at $\sqrt{s}=14$ TeV Alwall:2007st . Table 1: Benchmark scenario. The mass parameters $M_{2}$, $\|\mu\|$, $A_{\tau}$, $M_{\tilde{E}}$, $M_{\tilde{L}}$ $M_{{\tilde{E}}_{\tau}}$, and $M_{{\tilde{L}}_{\tau}}$ are given in GeV. ${{\phi_{1}}_{\phantom{I}}}_{\phantom{I}}$ \| \| ${{\phi_{\mu}}_{\phantom{I}}}_{\phantom{I}}$ \| \| ${{\phi_{A_{\tau}}}_{\phantom{I}}}_{\phantom{I}}$ \| \| ${M_{2}}_{\phantom{I}}$ \| \| ${\|\mu\|}_{\phantom{I}}$ \| \| ${A_{\tau}}_{\phantom{I}}$ \| \| ${\tan\beta}_{\phantom{I}}$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- 0 \| \| 0 \| \| $\pi/2$ \| \| 250 \| \| 250 \| \| 2000 \| \| 3 ${M_{{\tilde{E}}_{\tau}}}_{\phantom{I}}$ \| \| \| \| ${M_{{\tilde{L}}_{\tau}}}_{\phantom{I}}$ \| \| \| \| ${M_{\tilde{E}}}_{\phantom{I}}$ \| \| \| \| ${M_{\tilde{L}}}_{\phantom{I}}$ 495 \| \| \| \| 500 \| \| \| \| 150 \| \| \| \| 200 Table 2: Mass spectrum for the scenario in Table 1. ${\tilde{\ell}}$ \| \| $m$ [GeV] \| \| \| \| \| \| \| \| $\tilde{\chi}$ \| \| $m$ [GeV] ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- ${\tilde{e}}_{R},{\tilde{\mu}}_{R}$ \| \| 155 \| \| \| \| \| \| \| \| ${\tilde{\chi}}_{1}^{0}$ \| \| 112 ${\tilde{e}}_{L},{\tilde{\mu}}_{L}$ \| \| 204 \| \| \| \| \| \| \| \| ${\tilde{\chi}}_{2}^{0}$ \| \| 190 ${\tilde{\nu}}_{e},{\tilde{\nu}}_{\mu}$ \| \| 192 \| \| \| \| \| \| \| \| ${\tilde{\chi}}_{3}^{0}$ \| \| 254 ${\tilde{\nu}}_{\tau}$ \| \| 497 \| \| \| \| \| \| \| \| ${\tilde{\chi}}_{4}^{0}$ \| \| 327 ${\tilde{\tau}}_{1}$ \| \| 495 \| \| \| \| \| \| \| \| ${\tilde{\chi}}_{1}^{\pm}$ \| \| 181 ${\tilde{\tau}}_{2}$ \| \| 504 \| \| \| \| \| \| \| \| ${\tilde{\chi}}_{2}^{\pm}$ \| \| 325 ## III Numerical results We quantitatively study the tau polarization asymmetry, and the branching ratios for the two-body decay chain $\tilde{\tau}_{1}\to\tau+\tilde{\chi}_{2}^{0};\quad\tilde{\chi}_{2}^{0}\to\ell_{1}^{+}+\tilde{\ell}_{R}^{-};\quad\tilde{\ell}_{R}^{-}\to\tilde{\chi}^{0}_{1}+\ell_{2}^{-},$ (47) for $\ell=e,\mu$. The asymmetry probes the MSSM phases $\phi_{1}$, $\phi_{\mu}$ and $\phi_{A_{\tau}}$, of the neutralino and stau sector. We center our numerical discussion around a general MSSM benchmark scenario, see Table 1. We choose heavier soft breaking parameters in the stau sector than in the $\tilde{e},\tilde{\mu}$ sector, to enable the mass hierarchy $m_{\tilde{\tau}_{m}}>m_{\tilde{\chi}_{i}^{0}}>m_{\tilde{\ell}_{R}}>m_{\tilde{\chi}_{1}^{0}}.$ (48) Further we choose almost degenerate staus which enhances their mixing, leading to maximal asymmetries. We choose a large value of the trilinear scalar coupling parameter, $\|A_{\tau}\|>\|\mu\|\tan\beta$444The value of $\|A_{\tau}\|$ is restricted bz the vacuum stability contidtion as $\|A_{\tau}^{2}<3(m_{\tilde{\tau}}^{2}+m_{{\tilde{\nu}}_{\tau}}^{2}+M_{H}^{2}+\mu^{2}\|$ Frere:1983 ., to enhance the impact of $\phi_{A_{\tau}}$ in the stau sector. Finally, to reduce the number of MSSM parameters, we use the (GUT inspired) relation $\|M_{1}\|=5/3\,M_{2}\tan^{2}\theta_{w}$ Haber:1984rc for the gaugino mass parameters. The resulting masses of the staus, neutralinos and charginos are summarized in Table 2. ### III.1 Phase dependence For the benchmark scenario given in Table 1, we study the phase dependence of the asymmetry ${\mathcal{A}_{\tau}^{\rm CP}}$ in the stau rest frame. In Fig. 3, we show the dependence on the CP phases in the neutralino sector, $\phi_{1}$ and $\phi_{\mu}$. In Fig. 3, we show the dependence on the phases in the stau secotor $\phi_{A_{\tau}}$ and $\phi_{\mu}$. The asymmetry strongly depends on $\phi_{A_{\tau}}\approx\phi_{\tilde{\tau}}$, which we expect for $\|A_{\tau}\|\gg\|\mu\|\tan\beta$ as in our benchmark scenario, see Table 1. In particular for $\phi_{\mu}=0$ in Fig. 3, the asymmetry follows the approximation formula Eq. (44), and attains its maximal values at $\phi_{\tilde{\tau}}\approx\phi_{A_{\tau}}\approx\pm\pi/2$. Figure 3: Phase dependence of (a) the $\tau$ polarization asymmetry ${\mathcal{A}_{\tau}^{\rm CP}}$, Eq. (39), in percent, in the $\phi_{1}$–$\phi_{\mu}$ plane (for $\phi_{A_{\tau}}=0$), and (b) in the $\phi_{A_{\tau}}$–$\phi_{\mu}$ plane (for $\phi_{1}=0$), in the stau rest frame. We consider the decay $\tilde{\tau}_{1}\to\tau\tilde{\chi}_{2}^{0}$, followed by $\tilde{\chi}_{2}^{0}\to\ell_{1}^{+}\tilde{\ell}_{R}^{-}$, and $\tilde{\ell}_{R}^{-}\to\tilde{\chi}^{0}_{1}\ell_{2}^{-}$ where $\ell=e$ or $\mu$, cf. Fig. 1. The other MSSM parameters are defined in Table 1. ### III.2 $\|A_{\tau}\|$–$\tan\beta$ dependence and stau mixing Figure 4: $\|A_{\tau}\|$–$\tan\beta$ dependence of (a) the $\tau$ polarization asymmetry ${\mathcal{A}_{\tau}^{\rm CP}}$, Eq. (39), in percent, in the stau rest frame (for the decay $\tilde{\tau}_{1}\to\tau\tilde{\chi}_{2}^{0}$, followed by $\tilde{\chi}_{2}^{0}\to\ell_{1}^{+}\tilde{\ell}_{R}^{-}$, and $\tilde{\ell}_{R}^{-}\to\tilde{\chi}^{0}_{1}\ell_{2}^{-}$ for $\ell=e$ or $\mu$, cf. Fig. 1), (b) the coupling factor $\eta$, Eq. (42), (c) the phase $\phi_{\tilde{\tau}}$ in the stau sector, Eq. (7), and (d) $\sin(2\theta_{\tilde{\tau}})$, with $\theta_{\tilde{\tau}}$ the stau mixing angle, Eqs. (13), (14). The plots are for $\phi_{A_{\tau}}=\pi/4,$ the other MSSM parameters are given in Table 1. In Fig. 4, we show the $\|A_{\tau}\|$ and $\tan\beta$ dependence of the asymmetry ${\mathcal{A}_{\tau}^{\rm CP}}$ in the stau rest frame. We can observe that the asymmetry obtains its maximum, ${\mathcal{A}_{\tau}^{\rm CP}}\approx-77\%$, where also the coupling factor is maximal, $\eta\approx 0.95$, see Fig. 4. As discussed in Subsection II.5, the imaginary part of the product of the stau couplings $\mathfrak{Im}\\{a^{\tilde{\tau}}_{mi}(b^{\tilde{\tau}}_{mi})^{\ast}\\}$ is maximal for a maximal CP phase $\phi_{\tilde{\tau}}=\pi/2$ in the stau sector, which we show in Fig. 4. Note that the location of the maximum of ${\mathcal{A}_{\tau}^{\rm CP}}$ is not at maximal stau mixing, $\sin(\theta_{\tilde{\tau}})=1/\sqrt{2}\approx 0.7$, since $\eta\propto\sin(2\theta_{\tilde{\tau}})/(\|a^{\tilde{\tau}}\|^{2}+\|b^{\tilde{\tau}}\|^{2})$ starts to decrease for increasing $A_{\tau}$ and $\tan\beta$. To study the stau mixing, we show the $M_{{\tilde{E}}_{\tau}}$–$M_{{\tilde{L}}_{\tau}}$ dependence of the asymmetry ${\mathcal{A}_{\tau}^{\rm CP}}$ in Fig. 5. In the entire $M_{{\tilde{E}}_{\tau}}$–$M_{{\tilde{L}}_{\tau}}$ plane, the CP phase in the stau sector is almost maximal, $\phi_{\tilde{\tau}}=0.61\pi$. However, the asymmetry obtains its maxima in the small corridor $M_{{\tilde{E}}_{\tau}}\approx M_{{\tilde{L}}_{\tau}}$, where the stau mixing is maximal, $\theta_{\tilde{\tau}}=\pi/4$. ### III.3 $\|\mu\|$–$M_{2}$ dependence and branching ratios We show the $\|\mu\|$–$M_{2}$ dependence of the asymmetry ${\mathcal{A}_{\tau}^{\rm CP}}$ in Fig. 5. The maxima of ${\mathcal{A}_{\tau}^{\rm CP}}$ are obtained where the coupling factor $\eta$ is also maximal, see Eq. (42). In Fig. 6, we show the corresponding stau branching ratio, ${\rm BR}(\tilde{\tau}_{1}\to\tau\tilde{\chi}_{2}^{0})$, which can be as large as $40$%. Other competing channels can reach ${\rm BR}(\tilde{\tau}_{1}\to\tau\tilde{\chi}_{1}^{0})\approx 65\%$, and ${\rm BR}(\tilde{\tau}_{1}\to\nu_{\tau}\tilde{\chi}_{1(2)}^{\pm})\approx 20(10)\%$. The stau decay into the chargino $\tilde{\chi}_{1}^{\pm}$ is always open since typically the second lightest neutralino and the lightest chargino are almost degenerate, $m_{\tilde{\chi}_{2}^{0}}\approx m_{\tilde{\chi}_{1}^{\pm}}$. The neutralino branching ratio ${\rm BR}(\tilde{\chi}_{2}^{0}\to\ell\tilde{\ell}_{R})$, summed over $\ell=e,\mu$, is shown in Fig. 6, which reaches up to $100$%. The other important competing decay channels are ${\rm BR}(\tilde{\chi}_{2}^{0}\to\nu_{\ell}\tilde{\nu}_{\ell})$, and ${\rm BR}(\tilde{\chi}_{2}^{0}\to\ell\tilde{\ell}_{L})$, which open around $\mu\approx 250$ GeV and $\mu\approx 300$ GeV, respectively, for $M_{2}=250$ GeV. Note that in our benchmark scenario, see Table 1, we have ${\rm BR}(\tilde{\ell}_{R}\to\tilde{\chi}^{0}_{1}\ell)=1$. ### III.4 Impact of $\tilde{\tau}_{2}$ decay As we discussed in Section III.2, we find large asymmetries for nearly degenerate staus, where we naturally obtain a maximal stau mixing. However, then typically the asymmetries for $\tilde{\tau}_{1}$ and $\tilde{\tau}_{2}$ decay are similar in magnitude, but opposite in sign. For example in our benchmark scenario we find ${\mathcal{A}_{\tau}^{\rm CP}}=-71\%$ for $\tilde{\tau}_{1}$ decay, but ${\mathcal{A}_{\tau}^{\rm CP}}=+32\%$ for the decay of $\tilde{\tau}_{2}$. If the production and decay process of $\tilde{\tau}_{1}$ cannot be experimentally disentangled from that of $\tilde{\tau}_{2}$ properly, the two asymmetries might cancel. We show their sum in Fig. 7 in the $M_{{\tilde{E}}_{\tau}}$–$M_{{\tilde{L}}_{\tau}}$ plane. In Fig. 7, we show the corresponding stau mass splitting. Note that also the stau branching ratios are similar in size; for example in our benchmark scenario we have ${\rm BR}(\tilde{\tau}_{1}\to\tau\tilde{\chi}_{2}^{0})=18\%$, and ${\rm BR}(\tilde{\tau}_{2}\to\tau\tilde{\chi}_{2}^{0})=30\%$. For the $M_{{\tilde{E}}_{\tau}}$–$M_{{\tilde{L}}_{\tau}}$ plane shown in Fig. 5, the decay branching ratio ${\rm BR}(\tilde{\tau}_{1}\to\tau\tilde{\chi}_{2}^{0})$ is at least $10\%$, and that of $\tilde{\tau}_{2}$ is larger by roughly a factor of $2$ to $4$. Figure 5: Dependence of the $\tau$ polarization asymmetry ${\mathcal{A}_{\tau}^{\rm CP}}$, Eq. (39), in percent, in the stau rest frame (for the decay $\tilde{\tau}_{1}\to\tau\tilde{\chi}_{2}^{0}$, followed by $\tilde{\chi}_{2}^{0}\to\ell_{1}^{+}\tilde{\ell}_{R}^{-}$, and $\tilde{\ell}_{R}^{-}\to\tilde{\chi}^{0}_{1}\ell_{2}^{-}$ for $\ell=e$ or $\mu$, see Fig. 1), on (a) the soft breaking parameters in the stau sector $M_{{\tilde{E}}_{\tau}}$, $M_{{\tilde{L}}_{\tau}}$, Eqs. (9), Eqs. (10). In (b) the dependence of ${\mathcal{A}_{\tau}^{\rm CP}}$ on the gaugino and higgsino parameters $\|\mu\|$, $M_{2}$. Below the contour $m_{\tilde{e}_{R}}=m_{\tilde{\chi}_{2}^{0}}$ the two-body decay $\tilde{\chi}_{2}^{0}\to\ell\tilde{\ell}_{R}$ is kinematically forbidden, above the contour $m_{\tilde{e}_{R}}=m_{\tilde{\chi}_{1}^{0}}$ the lightest neutralino is no longer the LSP since $m_{\tilde{e}_{R}}<m_{\tilde{\chi}_{1}^{0}}$. Below the contour $m_{\tilde{\chi}_{1}^{\pm}}=100$ GeV the lightest chargino is lighter than $100$ GeV. The MSSM parameters are given in Table 1. Figure 6: Contour lines in the $\|\mu\|$–$M_{2}$ plane of (a) the stau branching ratio ${\rm BR}(\tilde{\tau}_{1}\to\tau\tilde{\chi}_{2}^{0})$ in percent, and (b) the neutralino branching ratio ${\rm BR}(\tilde{\chi}_{2}^{0}\to\ell\tilde{\ell}_{R})$, in percent, summed over both lepton flavors $\ell=e$, $\mu$ and charges, for the MSSM parameters as given in Table 1. Below the contours $m_{\tilde{e}_{R}}=m_{\tilde{\chi}_{2}^{0}}$ in Figs. 6, 6, the two-body decay $\tilde{\chi}_{2}^{0}\to\ell\tilde{\ell}_{R}$ is kinematically forbidden, above the contours $m_{\tilde{e}_{R}}=m_{\tilde{\chi}_{1}^{0}}$ the lightest neutralino is no longer the LSP since $m_{\tilde{e}_{R}}<m_{\tilde{\chi}_{1}^{0}}$. Below the contours $m_{\tilde{\chi}_{1}^{\pm}}=100$ GeV the lightest chargino is lighter than $100$ GeV. Figure 7: Contour lines of (a) the sum of the $\tau$ polarization asymmetries ${\mathcal{A}_{\tau}^{\rm CP}}$, Eq. (39), in percent, for the decays $\tilde{\tau}_{1}\to\tau\tilde{\chi}_{2}^{0}$ and $\tilde{\tau}_{2}\to\tau\tilde{\chi}_{2}^{0}$, each in the stau rest frame and followed by $\tilde{\chi}_{2}^{0}\to\ell_{1}^{+}\tilde{\ell}_{R}^{-}$, $\tilde{\ell}_{R}^{-}\to\tilde{\chi}^{0}_{1}\ell_{2}^{-}$, for $\ell=e$ or $\mu$, see Fig. 1, and (b) the stau mass splitting $m_{\tilde{\tau}_{2}}-m_{\tilde{\tau}_{1}}$ in GeV. Both plots are shown in the plane of the soft breaking parameters of the stau sector, $M_{{\tilde{E}}_{\tau}}$–$M_{{\tilde{L}}_{\tau}}$, see Eqs. (9), (10). The other MSSM parameters are given in Table 1. ## IV Summary and conclusions We have analyzed the normal tau polarization and the corresponding CP asymmetry in the two-body decay chain of a stau $\tilde{\tau}_{1}\to\tau+{\tilde{\chi}}_{2}^{0}.$ (49) The CP-sensitive parts appear only in the spin-spin correlations, which can be probed by the subsequent neutralino decay ${\tilde{\chi}}_{2}^{0}\to\ell_{1}+{\tilde{\ell}}_{R};\quad{\tilde{\ell}}_{R}\to{\tilde{\chi}}_{1}^{0}+\ell_{2},$ (50) for $\ell=e,\mu$. The T-odd tau polarization normal to the plane spanned by the $\tau$ and $\ell_{1}$ momenta, can then be used to define a CP-odd tau polarization asymmetry. It is based on a triple product, which probes the CP phases of the trilinear scalar coupling parameter $A_{\tau}$, the higgsino mass parameter $\mu$, and the U(1) gaugino mass parameter $M_{1}$. We have analyzed the analytical and numerical dependence of the asymmetry on these parameters in detail. In particular, for nearly degenerate staus where the stau mixing is strong, the asymmetry obtains its maxima and can be larger than $70\%$. The normal tau polarization can thus be considered as an ideal CP observable to probe the CP phases in the stau and neutralino sector of the MSSM. Since the CP-sensitive parts appear only in the subsequent stau decay products the stau production process can be separated. Thus both, ILC, and LHC collider studies are possible. Concerning the kinematical dependence, the asymmetry is not Lorentz invariant, since it is based on a triple product. At the LHC, staus are produced with a distinct boost distribution. Evaluated in the laboratory frame, the resulting tau polarization asymmetries get typically reduced by a factor of two thirds, compared to the stau rest frame. We want to stress that a thorough experimental analysis, addressing background processes, detector properties, and event rate reconstruction efficiencies, will be needed in order to explore the measurability of CP phases in the stau sector at the LHC or ILC. We hope that our work motivates such a study. ## Acknowledgements We thank M. Drees and F. von der Pahlen for enlightening discussions and helpful comments. This work has been supported by MICINN project FPA.2006-05294. AM was supported by the Konrad Adenauer Stiftung, BCGS, and a fellowship of Bonn University. HD was supported by the Hemholtz Alliance “Physics at the Terascale” and BMBF “Verbundprojekt HEP-Theorie” under the contract 0509PDE. SK was supported by BCGS. OK acknowledges support from CPAN. ## Appendix A Momenta and spin vectors For the stau decay $\tilde{\tau}_{m}\to\tau\tilde{\chi}_{i}^{0}$, we choose the coordinate frame in the laboratory (lab) system, such that the momentum of decaying $\tilde{\tau}$ points in the $z$-direction. $\displaystyle p^{\mu}_{\tilde{\tau}}$ $\displaystyle=$ $\displaystyle(E_{\tilde{\tau}},0,0,\|{\mathbf{p}}_{\tilde{\tau}}\|),$ (51) $\displaystyle p^{\mu}_{\tau}$ $\displaystyle=$ $\displaystyle E_{\tau}(1,\sin\theta_{\tau},0,\cos\theta_{\tau}),$ (52) with the decay angle $\theta_{\tau}=\varangle({\mathbf{p}}_{\tilde{\tau}},{\mathbf{p}}_{\tau})$, and $\displaystyle E_{\tau}$ $\displaystyle\approx$ $\displaystyle\|{\mathbf{p}}_{\tau}\|\approx\frac{(m_{\tilde{\tau}}^{2}-m_{\tilde{\chi}^{0}_{i}}^{2})}{2(E_{\tilde{\tau}}-\|{\mathbf{p}}_{\tilde{\tau}}\|\cos\theta_{\tau})},$ (53) in the limit $m_{\tau}\to 0$. The momenta of the leptons from the subsequent neutralino decay $\tilde{\chi}_{i}^{0}\to\ell_{1}\tilde{\ell}$; $\tilde{\ell}\to\tilde{\chi}_{1}^{0}\ell_{2}$ (1), can be parameterized by $\displaystyle p^{\mu}_{\ell_{1}}$ $\displaystyle=$ $\displaystyle E_{\ell_{1}}(1,\sin\theta_{1}\cos\phi_{1},\sin\theta_{1}\sin\phi_{1},\cos\theta_{1}),$ (54) $\displaystyle p^{\mu}_{\ell_{2}}$ $\displaystyle=$ $\displaystyle E_{\ell_{2}}(1,\sin\theta_{2}\cos\phi_{2},\sin\theta_{2}\sin\phi_{2},\cos\theta_{2}),$ (55) with the energies $\displaystyle E_{\ell_{1}}$ $\displaystyle=$ $\displaystyle\frac{m_{\tilde{\chi}^{0}_{i}}^{2}-m_{\tilde{\ell}}^{2}}{2(E_{\tilde{\chi}^{0}_{i}}-\|{\mathbf{p}}_{\tilde{\chi}^{0}_{i}}\|\cos\theta_{D_{1}})},$ (56) $\displaystyle E_{\ell_{2}}$ $\displaystyle=$ $\displaystyle\frac{m_{\tilde{\ell}}^{2}-m_{\tilde{\chi}^{0}_{i}}^{2}}{2(E_{\tilde{\ell}}-\|{\mathbf{p}}_{\tilde{\ell}}\|\cos\theta_{D_{2}})},$ (57) and the decay angles $\theta_{D_{1}}=\varangle({\mathbf{p}}_{\tilde{\chi}^{0}_{i}},{\mathbf{p}}_{\ell_{1}})$, $\theta_{D_{2}}=\varangle({\mathbf{p}}_{\tilde{\ell}},{\mathbf{p}}_{\ell_{2}})$, that is, $\displaystyle\cos\theta_{D_{1}}$ $\displaystyle=$ $\displaystyle\frac{({\mathbf{p}}_{\tilde{\tau}}-{\mathbf{p}}_{\tau})\cdot\hat{\mathbf{p}}_{\ell_{1}}}{\|{\mathbf{p}}_{\tilde{\tau}}-{\mathbf{p}}_{\tau}\|},$ (58) $\displaystyle\cos\theta_{D_{2}}$ $\displaystyle=$ $\displaystyle\frac{({\mathbf{p}}_{\tilde{\tau}}-{\mathbf{p}}_{\tau}-{\mathbf{p}}_{\ell_{1}})\cdot\hat{\mathbf{p}}_{\ell_{2}}}{\|{\mathbf{p}}_{\tilde{\tau}}-{\mathbf{p}}_{\tau}-{\mathbf{p}}_{\ell_{1}}\|},$ (59) with the unit momentum vector $\hat{\mathbf{p}}={\mathbf{p}}/\|{\mathbf{p}}\|$. We define the tau spin vectors by $\displaystyle s_{\tau}^{1,\mu}$ $\displaystyle=$ $\displaystyle\left(0,\frac{{\mathbf{s}}_{\tau}^{2}\times{\mathbf{s}}_{\tau}^{3}}{\|{\mathbf{s}}_{\tau}^{2}\times{\mathbf{s}}_{\tau}^{3}\|}\right),\qquad s_{\tau}^{2,\mu}=\left(0,\frac{{\mathbf{p}}_{\ell_{1}}\times{\mathbf{p}}_{\tau}}{\|{\mathbf{p}}_{\ell_{1}}\times{\mathbf{p}}_{\tau}\|}\right),$ $\displaystyle s_{\tau}^{3,\mu}$ $\displaystyle=$ $\displaystyle\frac{1}{m_{\tau}}\left(\|{\bf{p}}_{\tau}\|,\frac{E_{\tau}}{\|{\bf{p}}_{\tau}\|}{\bf{p}}_{\tau}\right).$ (60) The spin vectors $s^{a}_{\tau},$ $a=1,2,3,$ for the tau, and $s^{b}_{\tilde{\chi}_{i}^{0}},$ $b=1,2,3,$ for the neutralino $\tilde{\chi}^{0}_{i}$, fulfil completeness relations $\displaystyle\sum_{a}s_{\tau}^{a,\,\mu}s_{\tau}^{a,\,\nu}$ $\displaystyle=$ $\displaystyle-g^{\mu\nu}+\frac{p_{\tau}^{\mu}p_{\tau}^{\nu}}{m_{\tau}^{2}},$ (61) $\displaystyle\sum_{b}s_{\tilde{\chi}_{i}^{0}}^{b,\,\mu}s_{\tilde{\chi}_{i}^{0}}^{b,\,\nu}$ $\displaystyle=$ $\displaystyle-g^{\mu\nu}+\frac{p_{\tilde{\chi}_{i}^{0}}^{\mu}p_{\tilde{\chi}_{i}^{0}}^{\nu}}{m_{\tilde{\chi}_{i}^{0}}^{2}},$ (62) and they form orthonormal sets $\displaystyle s_{\tau}^{a}\cdot s_{\tau}^{c}$ $\displaystyle=$ $\displaystyle-\delta^{ac},\qquad s_{\tau}^{a}\cdot\hat{p}_{\tau}=0,$ (63) $\displaystyle s_{\tilde{\chi}^{0}_{i}}^{b}\cdot s_{\tilde{\chi}^{0}_{i}}^{c}$ $\displaystyle=$ $\displaystyle-\delta^{bc},\qquad s_{\tilde{\chi}^{0}_{i}}^{b}\cdot\hat{p}_{\tilde{\chi}^{0}_{i}}=0,$ (64) with $\hat{p}^{\mu}=p^{\mu}/m$. Note that the asymmetry ${\mathcal{A}_{\tau}^{\rm CP}}$, Eq. (40), does not depend on the explicit form of the neutralino spin vectors, since they are summed in the amplitude squared, see Eq. (31), using the completeness relation. ## Appendix B Phase space The Lorentz invariant phase-space element for the stau decay chain, see Eqs. (1) - (2), can be decomposed into two-body phase-space elements Bycki $\displaystyle{\rm d}\mathscr{L}\\!\textsl{ips}(s_{\tilde{\tau}};\,p_{\ell_{1}},p_{\ell_{2}},p_{\tilde{\chi}_{1}^{0}})=\dfrac{1}{(2\pi)^{2}}\displaystyle{\rm d}\mathscr{L}\\!\textsl{ips}(s_{\tilde{\tau}};\,p_{\tau},p_{{\tilde{\chi}^{0}_{i}}})$ $\displaystyle\times{\rm d}s_{\tilde{\chi}^{0}_{i}}\,{\rm d}\mathscr{L}\\!\textsl{ips}(s_{\tilde{\chi}^{0}_{i}};\,p_{\ell_{1}},p_{\tilde{\ell}})\,{\rm d}s_{\tilde{\ell}}\,{\rm d}\mathscr{L}\\!\textsl{ips}(s_{\tilde{\ell}};\,p_{\ell_{2}},p_{{\tilde{\chi}}_{1}^{0}}).\quad$ (65) The different contributions are $\displaystyle{\rm d}\mathscr{L}\\!\textsl{ips}(s_{\tilde{\tau}};\,p_{\tau},p_{\tilde{\chi}^{0}_{i}})$ $\displaystyle=$ $\displaystyle\dfrac{1}{4\pi}\dfrac{\|{\bf p}_{\tau}\|^{2}}{m_{\tilde{\tau}}^{2}-m_{\tilde{\chi}^{0}_{i}}^{2}}\sin\theta_{\tau}\,{\rm d}\theta_{\tau},\quad$ (66) $\displaystyle{\rm d}\mathscr{L}\\!\textsl{ips}(s_{\tilde{\chi}^{0}_{i}};\,p_{\ell_{1}},p_{\tilde{\ell}})$ $\displaystyle=$ $\displaystyle\dfrac{1}{2(2\pi)^{2}}\dfrac{\|{\bf p}_{\ell_{1}}\|^{2}}{m_{\tilde{\chi}^{0}_{i}}^{2}-m_{\tilde{\ell}}^{2}}{\rm d}\Omega_{1},$ (67) $\displaystyle{\rm d}\mathscr{L}\\!\textsl{ips}(s_{\tilde{\ell}};\,p_{\ell_{2}},p_{{\tilde{\chi}}_{1}^{0}})$ $\displaystyle=$ $\displaystyle\dfrac{1}{2(2\pi)^{2}}\dfrac{\|{\bf p}_{\ell_{2}}\|^{2}}{m_{\tilde{\ell}}^{2}-m_{{\tilde{\chi}}_{1}^{0}}^{2}}{\rm d}\Omega_{2},$ (68) with $s_{j}=p_{j}^{2}$ and ${\rm d}\Omega_{j}=\sin\theta_{j}\,{\rm d}\theta_{j}\,{\rm d}\phi_{j}$. ## Appendix C Density matrix formalism The coefficients of the stau decay matrix, Eq. (28), are $\displaystyle{\rm D}$ $\displaystyle=$ $\displaystyle\frac{g^{2}}{2}\left(\|a_{mi}^{\tilde{\tau}}\|^{2}+\|b_{mi}^{\tilde{\tau}}\|^{2}\right)(p_{\tilde{\chi}^{0}_{i}}\cdot p_{\tau})$ (69) $\displaystyle-g^{2}\mathfrak{Re}\\{a_{mi}^{\tilde{\tau}}\,(b_{mi}^{\tilde{\tau}})^{\ast}\\}m_{\tilde{\chi}^{0}_{i}}m_{\tau},$ $\displaystyle\Sigma_{\rm D}^{a}$ $\displaystyle=$ $\,{}^{\;\,-}_{(+)}\frac{g^{2}}{2}\left(\|a^{\tilde{\tau}}_{mi}\|^{2}-\|b^{\tilde{\tau}}_{mi}\|^{2}\right)m_{\tau}(p_{\tilde{\chi}_{i}^{0}}\cdot s^{a}_{\tau}),$ (70) $\displaystyle\Sigma_{\rm D}^{b}$ $\displaystyle=$ $\,{}^{\;\,-}_{(+)}\frac{g^{2}}{2}\left(\|a^{\tilde{\tau}}_{mi}\|^{2}-\|b^{\tilde{\tau}}_{mi}\|^{2}\right)m_{\tilde{\chi}^{0}_{i}}(p_{\tau}\cdot s_{\tilde{\chi}^{0}_{i}}^{b}),$ (71) $\displaystyle\Sigma_{\rm D}^{ab}$ $\displaystyle=$ $\displaystyle\frac{g^{2}}{2}\left(\|a^{\tilde{\tau}}_{mi}\|^{2}+\|b^{\tilde{\tau}}_{mi}\|^{2}\right)(s^{a}_{\tau}\cdot s^{b}_{{\tilde{\chi}_{i}^{0}}})m_{\tau}m_{\tilde{\chi}_{i}^{0}}$ (72) $\displaystyle+g^{2}\mathfrak{Re}\\{a^{\tilde{\tau}}_{mi}(b^{\tilde{\tau}}_{mi})^{\ast}\\}\times$ $\displaystyle\left[(s^{a}_{\tau}\cdot p_{\tilde{\chi}_{i}^{0}})(s^{b}_{\tilde{\chi}_{i}^{0}}\cdot p_{\tau})-(s^{a}_{\tau}\cdot s^{b}_{\tilde{\chi}_{i}^{0}})(p_{\tilde{\chi}_{i}^{0}}\cdot p_{\tau})\right]$ $\displaystyle-g^{2}\mathfrak{Im}\\{a^{\tilde{\tau}}_{mi}(b^{\tilde{\tau}}_{mi})^{\ast}\\}[s^{a}_{\tau},~{}p_{\tau},~{}s^{b}_{\tilde{\chi}_{i}^{0}},~{}p_{\tilde{\chi}_{i}^{0}}].$ The formulas are given for the decay of a negatively charged stau, $\tilde{\tau}_{m}\to\tau^{-}\tilde{\chi}_{i}^{0}$. The signs in parentheses hold for the charge conjugated decay $\tilde{\tau}_{m}^{\ast}\to\tau^{+}\tilde{\chi}_{i}^{0}$. Note that the terms proportional to $m_{\tau}$ in Eqs. (69), (70), and (72), are negligible at high particle energies $E\gg m_{\tau}$, in particular $\Sigma_{\rm D}^{a}$ can be neglected. The coefficients of the $\tilde{\chi}_{1}^{0}$ decay matrix, Eq. (29), are Kittel:2004rp $\displaystyle{\rm D}_{1}$ $\displaystyle=$ $\displaystyle\frac{g^{2}}{2}\|f_{\ell i}^{R}\|^{2}(m_{\tilde{\chi}_{i}^{0}}^{2}-m_{\tilde{\ell}}^{2}),$ (73) $\displaystyle\Sigma_{{\rm D}_{1}}^{b}$ $\displaystyle=$ $\,{}^{\;\,+}_{(-)}g^{2}\|f_{\ell i}^{R}\|^{2}m_{\tilde{\chi}^{0}_{i}}(s_{\tilde{\chi}^{0}_{i}}^{b}\cdot p_{\ell_{1}}),$ (74) and the selectron decay factor is $\displaystyle{\rm D}_{2}$ $\displaystyle=$ $\displaystyle g^{2}\|f_{\ell_{1}}^{R}\|^{2}(m_{\tilde{\ell}}^{2}-m_{\chi_{1}^{0}}^{2}).$ (75) The signs in parentheses hold for the charge conjugated processes, that is $\tilde{\chi}^{0}_{i}\to\ell_{1}^{-}\tilde{\ell}_{R}^{+}$ in Eq. (74). For the decay into a left slepton $\tilde{\chi}^{0}_{i}\to\ell_{1}^{+}\tilde{\ell}_{L}^{-}$, Eqs. (73), (74), and (75) read Kittel:2004rp $\displaystyle{\rm D}_{1}$ $\displaystyle=$ $\displaystyle\frac{g^{2}}{2}\|f^{L}_{\ell i}\|^{2}(m_{\tilde{\chi}_{i}^{0}}^{2}-m_{\tilde{\ell}}^{2}),$ (76) $\displaystyle\Sigma^{b}_{{\rm D}_{1}}$ $\displaystyle=$ $\,{}^{\;\,-}_{(+)}g^{2}\|f^{L}_{\ell i}\|^{2}m_{\tilde{\chi}_{i}^{0}}(s^{b}_{\tilde{\chi}_{i}^{0}}\cdot p_{\ell_{1}}),$ (77) $\displaystyle{\rm D}_{2}$ $\displaystyle=$ $\displaystyle g^{2}\|f^{L}_{\ell 1}\|^{2}(m_{\tilde{\ell}}^{2}-m_{\tilde{\chi}_{1}^{0}}^{2}),$ (78) respectively. The expressions for Eqs. (31) and (32) have to be changed accordingly. The sign in parenthesis in Eq. (77) holds for the charge conjugated process $\tilde{\chi}^{0}_{i}\to\ell_{1}^{-}\tilde{\ell}_{L}^{+}$. ## Appendix D Stau decay widths The partial decay width for the decay $\tilde{\tau}_{m}\to\tau\tilde{\chi}_{i}^{0}$ in the stau rest frame is Bartl:2002uy $\Gamma(\tilde{\tau}_{m}\to\tau\tilde{\chi}_{i}^{0})=\frac{m^{2}_{\tilde{\tau}}-m^{2}_{\tilde{\chi}^{0}_{i}}}{4\pi m^{3}_{\tilde{\tau}}}\,D,$ (79) with the decay function $D$ given in Eqs. (69), and the approximation $m_{\tau}=0$. For the decay $\tilde{\tau}_{m}\to\nu_{\tau}\tilde{\chi}_{j}^{\pm}$ the width is Bartl:2002uy $\Gamma(\tilde{\tau}_{m}\to\nu_{\tau}\tilde{\chi}_{j}^{\pm})=\frac{(m^{2}_{\tilde{\tau}}-m^{2}_{\tilde{\chi}^{\pm}_{j}})^{2}}{16\pi m^{3}_{\tilde{\tau}}}g^{2}\|l^{\tilde{\tau}}_{mj}\|^{2},$ (80) with the stau-chargino-neutrino coupling Haber:1984rc ; Bartl:2002uy $l_{mj}^{\tilde{\tau}}=-({\mathcal{R}}^{\tilde{\tau}}_{m1})^{\ast}\,U_{j1}+Y_{\tau}\,({\mathcal{R}}^{\tilde{\tau}}_{m2})^{\ast}\,U_{j2},$ (81) and the stau diagonalization matrix ${\mathcal{R}}^{\tilde{\tau}}$, Eq. (12), the Yukawa coupling $Y_{\tau}$, Eq. (22), and the matrix $U$, that diagonalizes the chargino matrix Haber:1984rc , $U^{\ast}\cdot{\mathcal{M}}_{\tilde{\chi}^{\pm}}\cdot V^{\dagger}={\rm diag}(m_{\tilde{\chi}^{\pm}_{1}},m_{\tilde{\chi}^{\pm}_{2}}).$ (82) The stau decay width for the entire decay chain, Eqs. (1) - (2), is then given by $\displaystyle\Gamma(\tilde{\tau}\to\tau\ell_{1}\ell_{2}{\tilde{\chi}}_{1}^{0})=$ (84) $\displaystyle=$ $\displaystyle\frac{1}{2m_{\tilde{\tau}}}\int\,\|\mathcal{M}\|^{2}\,{\rm d}\mathscr{L}\\!\textsl{ips}(s_{\tilde{\tau}};\,p_{\tau},p_{\ell_{1}},p_{\ell_{2}},p_{\tilde{\chi}_{1}^{0}})$ $\displaystyle=$ $\displaystyle\Gamma(\tilde{\tau})\times{\rm BR}(\tilde{\tau}\to\tau\tilde{\chi}_{i}^{0})\times{\rm BR}(\tilde{\chi}_{i}^{0}\to\ell_{1}\tilde{\ell})$ $\displaystyle\times{\rm BR}(\tilde{\ell}\to\ell_{2}\tilde{\chi}_{1}^{0}),$ with the phase-space element ${\rm d}\mathscr{L}\\!\textsl{ips}$, as given in the Appendix A, the amplitude squared $\displaystyle\|\mathcal{M}\|^{2}$ $\displaystyle=$ $\displaystyle 4\|\Delta(\tilde{\chi}^{0}_{i})\|^{2}\|\Delta(\tilde{\ell})\|^{2}\,D\,D_{1}\,D_{2},$ (86) obtained from Eqs. (30) by summing the tau helicities $\lambda_{\tau}$, $\lambda_{\tau}^{\prime}$. The neutralino branching ratios are given, for example, in Ref. 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Richardson, arXiv:0805.3037 [hep-ph].	arxiv-papers	2010-11-10T18:20:13	2024-09-04T02:49:14.685984	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Herbi Dreiner, Olaf Kittel, Suchita Kulkarni, Anja Marold", "submitter": "Anja Marold", "url": "https://arxiv.org/abs/1011.2449" }
1011.2484	# Nondistillability of distillable qutrit-qutrit states under depolarizing noise Salman Khan, M. K. Khan Department of Physics, Quaid-i-Azam University, Islamabad 45320, Pakistan sksafi@phys.qau.edu.pk ###### Abstract We study the effects of decoherence on some particular bipartite qutrit states under the influence of global, collective, local and multilocal depolarizing noise. We show that certain free entangled distillable qutrit density matrices become bound entangled or separable and hence convert into nondistillable density matrices in global noise. The collective noise increases the degree of entanglement of the qutrit bipartite states. Furthermore, we show that some particular local operation cannot avoid the Nondistillability of the distillable states. PACS: 03.65.Ud; 03.65.Yz; 03.67.Mn;03.67.Pp Keywords: Entanglement; Decoherence; qutrits. ## 1 Introduction Entanglement is one of the potential sources of quantum theory. It is the key concept and a major resource for quantum communication and computation [1]. In the last two decades, enormous efforts have been made to investigate various aspects of quantum entanglement and its benefits in a number of setups, such as teleportation of unknown states [2], quantum key distribution [3], quantum cryptography [4] and quantum computation [5, 6]. However, a realistic quantum system cannot be completely isolated from its environment. Generally, the coupling between an environment and a quantum system leads to the phenomenon of decoherence and it gives rise to an irreversible transfer of information from the system to the environment [7, 8, 9]. Nevertheless, It has also been shown [10, 11, 12] that the environment can induced entanglement between two otherwise totally decoupled systems. It is, thus, necessary to investigate the behavior of initial state entanglement in the presence of decoherence. Yu and Eberly [13, 14] showed that entanglement loss occurs in a finite time under the action of pure vacuum noise in a bipartite state of qubits. They found that, even though it takes infinite time to complete decoherence locally, the entanglement may be lost in finite time. The name ”entanglement sudden death” (ESD) was tossed for the phenomenon of sudden loss of entanglement. The finite time loss of entanglement definitely limits the application of entangled states in quantum information processing. The phenomenon of ESD is not limited only to two qubit entangled state, it is investigated in systems of larger spaces such as qutrits and qudits [15, 16, 17, 18, 19, 20, 21, 22]. A geometric interpretation of the effect of ESD is given in Ref. [23]. The experimental evidences of the phenomenon of ESD have been reported for optical setups [24] and atomic ensembles [25]. The extraction of a pure entangled singlet state from an ensemble of mixed state through local quantum operation and classical communication (LOCC) is called distillation [26]. An inseparable state is said to possess some degree of entanglement. Quantum states are grouped into separable and entangled states for qubit-qubit and qubit-qutrit system by using Peres-Horodecki criterion [27, 28]. According to this criterion, the partial transpose of a separable density matrix must has non-negative eigenvalues. The partial transpose of a bipartite density matrix $\rho_{m\nu,n\mu}$ over the second qutrit B is given by $\rho_{m\mu,n\nu}^{T_{B}}=\rho_{m\nu,n\mu}$ and for the first qutrit, it can similarly be defined. Nevertheless, such a characterization for higher dimensional bipartite states is difficult [29]. A bipartite entangled state can either be free or bound entangled state. A free entangled state can be distilled to a singlet state through LOCC whereas a bound entangled state cannot be, no matter how many copies are available. A bound entangled state cannot be used for reliable quantum information processing [30]. A nondistillable bound entangled state have positive partial transpose (PPT), whereas a negative partial transpose (NPT) states are regarded distillable [30]. However, no single criterion is sufficient to detect bound entangled states. For example, for a qutrit-qutrit system there are many bound entangled states and no single criterion can fully describe all of them [31]. However, it is shown in Ref. [32] that realignment criterion can be used to detect certain bound entangled states. For a bipartite density matrix $\rho_{m\nu,n\mu}$, the realignment criterion is given by $\left(\rho^{R}\right)_{mn,\nu\mu}=\rho_{m\nu,n\mu}$. A state is separable under realignment criterion if $\left\\|\rho^{R}\right\\|\leq 1$ and a PPT state is bound entangled if the quantity $\left\\|\rho^{R}\right\\|-1$ is positive. This is important to point out here that realignment criterion is not capable to detect all bound entangled states. The phenomenon in which a free entangled state becomes nondistillable is called distillability sudden death (DSD). It is shown in Refs. [22, 33, 34] that in the presence of dephasing noise some free entangled states of qutrit-qutrit system become nondistillable in a finite time. The effect of amplitude damping channel on such a system is studied in Ref. [35], where the authors found that a simple local unitary transformation can completely avoid DSD. In this paper we study the behavior of entanglement by revisiting a particular family of density matrices that are considered in Refs. [22, 33, 34, 35] of qutrit-qutrit system in the presence of depolarizing noise. We use partial transpose criterion and realignment criterion for our investigation. We consider various coupling of the system and environment in which the system is influenced by global, collective, local or multilocal depolarizing noise. It is shown that under certain particular conditions, the depolarizing noise can increase the degree of entanglement in the qutrit-qutrit system. For example, under the action of multilocal depolarizing noise, the negativity increases linearly with the increase of depolarizing noise of one qutrit’s local environment by controlling the other qutrit’s local environment. On the other hand, the free entangled density matrices can become bound entangled or separable and hence nondistillable under another particular condition in the depolarizing noise. For example, all the free entangled density matrices become bound entangled under the action of global environment. Furthermore, it is shown that DSD cannot be avoided by performing simple local unitary operation, considered in this paper, for the density matrices. Since no definitive criterion for separability or entanglement of density matrices of dimensions greater than six is available, we believe that our results are valid for the family of density matrices considered in this paper. ### 1.1 Qutrit-Qutrit System in a Depolarizing Noise We consider a composite system of two qutrits A and B that are coupled to a noisy environment both collectively and individually. The qutrits are spatially separated and has no direct interaction with each other. The collective coupling refers to the situation when both the qutrits are influenced by the same environment and the multilocal coupling describes the situation when each qutrit is independently influenced by its own environment. The system is said to be coupled to a global environment when it is influenced by both collective and multilocal noises at the same time. Let the bases of Hilbert space of each qutrit be denoted by $\|0\rangle$, $\|1\rangle$ and $\|2\rangle$. Then the bases of the composite system are given in the order $\|00\rangle$, $\|01\rangle$, $\|02\rangle$, $\|10\rangle$, $\|11\rangle$, $\|12\rangle$, $\|20\rangle$, $\|21\rangle$, $\|22\rangle$. The dynamics of the composite system in the presence of depolarizing noise can best be described in the Kraus operators formalism. The Kraus operators for a single qutrit depolarizing noise, which satisfy the completeness relation $\sum_{i}E_{i}^{{\dagger}}E_{i}=I$, are given as [36] $\displaystyle E_{0}$ $\displaystyle=$ $\displaystyle\sqrt{1-p}I_{3},\quad E_{1}=\sqrt{\frac{p}{8}}Y,\quad E_{2}=\sqrt{\frac{p}{8}}Z,$ $\displaystyle E_{3}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{p}{8}}Y^{2},\quad E_{4}=\sqrt{\frac{p}{8}}YZ,\quad E_{5}=\sqrt{\frac{p}{8}}Y^{2}Z,$ $\displaystyle E_{6}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{p}{8}}YZ^{2},\quad E_{7}=\sqrt{\frac{p}{8}}Y^{2}Z^{2},\quad E_{8}=\sqrt{\frac{p}{8}}Z^{2},$ (1) with $\displaystyle Y$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\ 1&0&0\end{array}\right),\quad Z=\left(\begin{array}[]{ccc}1&0&0\\\ 0&\omega&0\\\ 0&0&\omega^{2}\end{array}\right),$ (8) $\displaystyle I_{3}$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&1\end{array}\right)$ (12) where $\omega=e^{2i\pi/3}$ and $p=1-e^{-\Gamma t/2}\in\left[0,1\right]$ is the decoherence parameter. The lower and upper limits of $p$ stand, respectively, for undecohered and fully decohered cases of the noisy environment. The evolution of the density matrix when it is influenced by the global depolarizing noise is given by $\rho(t)=\sum_{i=0}^{80}\sum_{j=0}^{8}\left(E_{i}^{AB}E_{j}^{B}E_{j}^{A}\right)\rho\left(E_{j}^{A{\dagger}}E_{j}^{B{\dagger}}E_{i}^{AB{\dagger}}\right),$ (13) where $E_{j}^{A}=E_{m}\otimes I_{3}$, $E_{j}^{B}=I_{3}\otimes E_{m}$ are the Kraus operators of the multilocal coupling of each individual qutrit and $E_{i}^{AB}$ are the Kraus operators of the collective coupling that are formed from all the possible combinations of the tensor product of the Kraus operators of a single qutrit depolarizing noise in the form $E_{m}\otimes E_{m}$. The subscripts $m=0,1,2,...8$ stand for a single qutrit Kraus operators of a depolarizing noise given in Eq. (1). The system is said to be under the action of global depolarizing noise when both multilocal and collective couplings are switched on simultaneously (global = multilocal + collective). We consider the following initial density matrix of the bipartite qutrit system $\rho(0)=\frac{2}{7}\|\Psi_{+}\rangle\langle\Psi_{+}\|+\frac{\alpha}{7}\sigma_{+}+\frac{5-\alpha}{7}\sigma_{-},$ (14) where $2\leq\alpha\leq 5$. In Eq. (14) $\|\Psi_{+}\rangle$ is a maximally entangled bipartite qutrit state given by $\|\Psi_{+}\rangle=1/\sqrt{3}\left(\|00\rangle+\|11\rangle+\|22\rangle\right)$, $\sigma_{+}$ and $\sigma_{-}$ are separable states, which are given by $\sigma_{+}=1/3\left(\|01\rangle\langle 01\|+\|12\rangle\langle 12\|+\|20\rangle\langle 20\|\right)$, $\sigma_{-}=1/3\left(\|10\rangle\langle 10\|+\|21\rangle\langle 21\|+\|02\rangle\langle 02\|\right)$. The density matrix of Eq. (14) is separable for $2\leq\alpha\leq 3$, bound entangled for $3\leq\alpha\leq 4$ and free entangled for $4<\alpha\leq 5$ [37]. For a further detailed study of the properties of these density matrices we refer the readers to Refs. [31, 32, 38]. We will concentrate on the entanglement behavior of free entangled density matrix in the range $4<\alpha\leq 5$. Using initial density matrix of Eq. (14) in Eq. (13) and taking the partial transpose over the second qutrit, it is easy and straightforward to find the eigenvalues. Let the decoherence parameters for multilocal noise of the two qutrits and collective noise of the composite system be $p_{1}$, $p_{2}$ and $p$ respectively. Then, the eigenvalues of the partial transpose of the final density matrix of Eq. (13) when only the first qutrit is coupled to the noisy environment are given by $\displaystyle\lambda_{1,3,5}$ $\displaystyle=$ $\displaystyle\frac{1}{336}\left(40-3p_{1}+\sqrt{\left(41-20\alpha+41\alpha^{2}\right)\left(9p_{1}-8\right)^{2}}\right)$ $\displaystyle\lambda_{2,4,6}$ $\displaystyle=$ $\displaystyle\frac{1}{336}\left(40-3p_{1}-\sqrt{\left(41-20\alpha+41\alpha^{2}\right)\left(9p_{1}-8\right)^{2}}\right)$ $\displaystyle\lambda_{7,8,9}$ $\displaystyle=$ $\displaystyle\frac{2}{21}+\frac{1}{56}p_{1}$ (15) The eigenvalues of the partial transpose of the final density matrix when both the qutrits are coupled to multilocal depolarizing noise are given by $\displaystyle\lambda_{1,3,5}$ $\displaystyle=$ $\displaystyle\frac{1}{2688}(320-24p_{2}+3p_{1}(-8+9p_{2})$ $\displaystyle+\sqrt{(41-20\alpha+4\alpha^{2})(-8+9p1)^{2}(-8+9p2)^{2}})$ $\displaystyle\lambda_{2,4,6}$ $\displaystyle=$ $\displaystyle\frac{1}{2688}(320-24p_{2}+3p_{1}(-8+9p_{2})$ $\displaystyle-\sqrt{(41-20\alpha+4\alpha^{2})(-8+9p_{1})^{2}(-8+9p_{2})^{2}})$ $\displaystyle\lambda_{7,8,9}$ $\displaystyle=$ $\displaystyle\frac{1}{1344}(128+24p_{1}+24p_{2}-27p_{1}p_{2})$ (16) It is easy to see that Eq. (16) reduces to Eq. (15), if we switch off the coupling of second qutrit with its local environment. The eigenvalues when both qutrits are coupled to the same environment, that is, for collective coupling of the environment and system are obtained by replacing $p_{1}=p_{2}=p$ in Eq. (16). The only eigenvalues that possibly become negative in Eqs. (15), (16) and hence for the collective noise are those which are given by $\lambda_{2,4,6}$. To observe the behavior of entanglement and see whether the phenomena of DSD and ESD happen in these cases, we use the partial transpose criterion and realignment criterion in the following. The degree of entanglement in NPT states is quantified by using negativity [27, 28]. It is given by the sum of the absolute value of the negative eigenvalues of the partial transpose of a density matrix. The negativity for local, multilocal and collective noise becomes $\mathcal{N}(\rho)=\max\left\\{0,\left\|\lambda_{2,4,6}\right\|\right\\}=\max\left\\{0,\left\|3\lambda_{2}\right\|\right\\}.$ (17) \put(-320.0,220.0){} \| \| ---\|---\|--- Figure 1: (color online) The negativity for various values of parameter $\alpha$ is plotted against the decoherence parameter $p_{1}$, for the case when only one qutrit undergoes decoherence. The realignment criterion $\|\|\rho^{R}\|\|-1$ for single qutrit decoherence is also plotted for $\alpha=5$. The negativity for different values of the parameter $\alpha$, when only one qutrit is coupled to the environment, is plotted against $p_{1}$ in Fig. $1$. It can be seen that the negativity becomes zero at a finite value of $p_{1}$ (around $p_{1}=0.2$) only for density matrices that correspond to the upper limit of the parameter $\alpha$ and hence becomes PPT that might possess some degree of entanglement. For density matrices in the lower limit of $\alpha$, the negativity first decreases to a minimum value, however, still positive and then increases with $p_{1}$, thereby increasing the degree of NPT (free entanglement). The partial transpose criterion fails to detect the behavior of entanglement in density matrices of large $\alpha$ beyond certain values of $p_{1}$. Further investigation of the entanglement in these states can be carried out by using realignment criterion. The quantity $\left\\|\rho^{R}\right\\|-1$ for density matrices of large $\alpha$ is also plotted in Fig. $1$. The positive value of this quantity in the range $0.21\leq p_{1}\leq 0.288$ shows that these density matrices are bound entangled. However for $p_{1}>0.288$, the realignment criterion also fails to detect the possible entanglement for these states. \put(-320.0,220.0){} \| \| ---\|---\|--- Figure 2: (color online) The negativity for various values of parameter $\alpha$ is plotted against the decoherence parameter $p_{2}$ when both the qutrits are coupled to their local environments. The value of decoherence parameter $p_{1}=0.1$. The negativity for the case when both qutrits are individually coupled to its own environment, that is, for multilocal coupling is plotted in Fig. $2$ against the decoherence parameter $p_{2}$ for decoherence parameter $p_{1}=0.1$. It can be seen that neither DSD nor ESD occurs in any density matrix ($4<\alpha\leq 5$) as the negativity is positive for all of them. For density matrices of lower values of $\alpha$ in the range $4<\alpha\leq 5$, the negativity increases linearly with $p_{2}$ and hence the degree of entanglement. On the other hand, for density matrices that correspond to intermediate and upper values of $\alpha$, it first decreases, however remain positive, and then increases linearly. It can be shown that the negativity increases linearly for all density matrices, irrespective of the value of $\alpha$ ($4<\alpha\leq 5$), for larger values of the decoherence parameter $p_{1}$. Thus, both DSD and ESD can be completely avoided in the case of multilocal coupling of the system and environment by controlling the environment of one qutrit at least. Nevertheless, for $p_{1}<0.1$, some density matrices that correspond to large values of $\alpha$ become PPT. For example, for $p_{1}=0.05$, the negativity of density matrix that corresponds to $\alpha=5$ becomes zero at $p_{2}=0.165$. The negativity and the realignment criterion for such density matrix is plotted in Fig. $3$. It shows that the density matrix becomes bound entangled for $0.165<p_{2}\leq 1$. \put(-320.0,220.0){} \| \| ---\|---\|--- Figure 3: (color online) The negativity and realignment criterion $\|\|\rho^{R}\|\|-1$ for the multilocal coupling are plotted against the decoherence parameter $p_{2}$. The values of other parameters are set to $p_{1}=0.05$, $\alpha=5$. \put(-320.0,220.0){} \| \| ---\|---\|--- Figure 4: (color online) The negativity for various values of $\alpha$ is plotted against decoherence parameter $p$ when the system evolves under collective depolarizing noise. In Fig. $4$, we have plotted the negativity for various density matrices against the decoherence parameter $p$ for the case of collective noise. As can be seen from the figure, non of the density matrices undergoes DSD or ESD. According to partial transpose criterion, all the density matrices in the range $4<\alpha\leq 5$ are free entangled under the action of collective depolarizing noise. Though the negativity for each density matrix drops to a minimum, but positive value, in the range of lower values of decoherence parameter. The degree of NPT increases for all density matrices with the values of decoherence parameter $p$ in the large values limit. \put(-320.0,220.0){} \| \| ---\|---\|--- Figure 5: (color online) The eigenvalues for the case of global noise are plotted against the decoherence parameter $p$. The values of other parameters are set to $p_{1}=p_{2}=0.5$, $\alpha=4.3$. The general form of eigenvalues for the global noise have quite lengthy expressions, instead of writing their expression, we prefer to see their behavior by plotting them against decoherence parameter $p$. Such a plot for $p_{1}=p_{2}=0.5$ and $\alpha=4.3$ is shown in Fig. $5$. Since all the eigenvalues for the whole range of decoherence parameter are positive, the density matrices under global noise are PPT. The realignment criterion also fails to detect any possible entanglement as the quantity $\left\\|\rho^{R}\right\\|-1$ for the range of interest of the parameter $\alpha$ against the decoherence parameter $p$ is negative. We conclude that the distillable free entangled qutrit-qutrit states becomes completely nondistillable in the presence of global depolarizing noise. Whereas under the action of global dephasing noise, it is shown [34] that these density matrices are free entangled in certain limited time and undergo DSD at a later time. To see the effect of local unitary operation on the dynamics of the state of Eq. 14, we consider the unitary operator $U=I_{3}\otimes\theta$, with $\theta=\|0\rangle\langle 1\|+\|1\rangle\langle 0\|+\|2\rangle\langle 2\|$. When this operator is applied locally to the state of Eq. 14, it converts the maximally entangled state $\|\Psi_{+}\rangle$ into another maximally entangled state given by $\|\tilde{\Psi}_{+}\rangle=1/\sqrt{3}\left(\|00\rangle+\|11\rangle+\|22\rangle\right)$ and the two separable states $\sigma_{+}$, $\sigma_{-}$ into other two separable states, respectively, given by $\tilde{\sigma}_{+}=1/3\left(\|00\rangle\langle 00\|+\|12\rangle\langle 12\|+\|21\rangle\langle 21\|\right)$, $\tilde{\sigma}_{-}=1/3\left(\|11\rangle\langle 11\|+\|20\rangle\langle 20\|+\|02\rangle\langle 02\|\right)$. The density matrix after $U$ acts on $\rho(0)$ becomes $\sigma_{\theta}=U\rho(0)U^{\dagger}=\frac{2}{7}\|\tilde{\Psi}_{+}\rangle\langle\tilde{\Psi}_{+}\|+\frac{\alpha}{7}\tilde{\sigma}_{+}+\frac{5-\alpha}{7}\tilde{\sigma}_{-}$ (18) The evolution of $\sigma_{\theta}$ in the noisy environment and its partial transpose can be found straightforwardly in the same manner as done for the state $\rho(t)$. It is found that all the eigenvalues for each individual case, considered for the coupling of $\rho(t)$, remains unchanged. In conclusion, the local operation that we consider here does not change the behavior of entanglement in the presence of any of the aforementioned coupling under the action of depolarizing noise. It is important to state that under the action of amplitude damping noise, a local unitary operation changes the behavior of entanglement and completely avoids DSD [35]. ## 2 Summary We study the dynamics of entanglement for particular bipartite qutrit density matrices under global, collective, multilocal and local depolarizing noise. We show that unlike the cases of dephasing [34] and amplitude damping noises [35], the influence of depolarizing noise is completely different. Using partial transpose criterion and realignment criterion, it is shown that only those density matrices that correspond to large values of $\alpha$ becomes PPT and thus nondistillable in the local depolarizing noise. In the case of multilocal depolarizing noise, both ESD and DSD can be avoided by controlling one or the other local environment. We also show that instead of ESD and DSD to occur, the degree of entanglement for free entangled density matrices under collective depolarizing noise increases with the increasing value of decoherence parameter. All the free entangled density matrices in the specified range of parameter $\alpha$ becomes PPT under the action of global depolarizing noise. In conclusion, we show that the free entangled distillable density matrices convert into bound entangled or separable density matrices and thus become completely nondistillable under global noise. Also, the depolarizing noise can be used to increase the degree of entanglement in the specific family of density matrices that correspond to a particular values of the parameter $\alpha$ under certain couplings with the environment. Furthermore, It is shown that the local operation, we used in this paper, does not change the dynamics of entanglement under each coupling of the system and environment for the case of the specific family of the density matrices. Since there is no single known definitive criterion for separability and entanglement in states of dimension greater than six, therefore, we emphasis that the validity of our results is true for the family of density matrices that we consider here and may not be generalized to all bipartite qutrit states. ## References * [1] Bouwmeester D, Ekert A and Zeilinger A 2000 The Physics of Quantum Information (Springer-Verlag Berlin) * [2] Bennett C H, Brassard G, Crepeau C, Jozsa R, Peres A and W K Wootters 1993 Phys. Rev. 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B 43 045504 * [36] Salimi S, Soltanzadeh M M, 2009 International Journal of Quantum Information 7 615-626 * [37] Horodecki P, Horodecki M and Horodecki R 1999 Phys. Rev. Lett. 82 1056–1059 * [38] Figure Captions Figure $1$. The negativity for various values of parameter $\alpha$ is plotted against the decoherence parameter $p_{1}$, for the case when only one qutrit undergoes decoherence. The realignment criterion $\|\|\rho^{R}\|\|-1$ for single qutrit decoherence is also plotted for $\alpha=5$. Figure $2$. The negativity for various values of parameter $\alpha$ is plotted against the decoherence parameter $p_{2}$ when both the qutrits are coupled to their local environments. In the inset of the figure the realignment criterion is plotted against decoherence parameter $p_{2}$, which shows that the density matrices become bound entangled for $p_{N}=$ $0.165<p_{2}\leq 1$. Figure $3$. The negativity and realignment criterion $\|\|\rho^{R}\|\|-1$ for the multilocal coupling are plotted against the decoherence parameter $p_{2}$. The values of other parameters are set to $p_{1}=0.05$, $\alpha=5$. Figure $4$. The negativity for various values of $\alpha$ is plotted against decoherence parameter $p$ when the system evolves under collective depolarizing noise. Figure $5$. The eigenvalues for the case global noise are plotted against the decoherence parameter $p$. The values of other parameters are set to $p_{1}=p_{2}=0.5$, $\alpha=4.3$.	arxiv-papers	2010-11-10T20:31:08	2024-09-04T02:49:14.697967	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Salman Khan, M.K. Khan", "submitter": "Salman Khan", "url": "https://arxiv.org/abs/1011.2484" }
1011.2625	# First-principles investigation of dynamical properties of molecular devices under a steplike pulse Yanxia Xing1,2, Bin Wang1 and Jian Wang1,∗ 1Department of Physics and the Center of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China. 2Department of Physics, Beijing Institute of Technology, Beijing 100081, China ###### Abstract We report a computationally tractable approach to first principles investigation of time-dependent current of molecular devices under a step-like pulse. For molecular devices, all the resonant states below Fermi level contribute to the time-dependent current. Hence calculation beyond wideband limit must be carried out for a quantitative analysis of transient dynamics of molecules devices. Based on the exact non-equilibrium Green’s function (NEGF) formalism of calculating the transient current in Ref.Maciejko, , we develop two approximate schemes going beyond the wideband limit, they are all suitable for first principles calculation using the NEGF combined with density functional theory. Benchmark test has been done by comparing with the exact solution of a single level quantum dot system. Good agreement has been reached for two approximate schemes. As an application, we calculate the transient current using the first approximated formula with opposite voltage $V_{L}(t)=-V_{R}(t)$ in two molecular structures: Al-${\rm C}_{5}$-Al and Al-${\rm C}_{60}$-Al. As illustrated in these examples, our formalism can be easily implemented for real molecular devices. Importantly, our new formula has captured the essential physics of dynamical properties of molecular devices and gives the correct steady state current at $t=0$ and $t\rightarrow\infty$. ###### pacs: 71.15.Mb, 72.30.+q, 85.35.-p 73.23.-b ## I introduction With the rapid progress in molecular electronics,Book quantum transport in molecular device has received increasing attention. In particular, the dynamic response of molecular devices to external parametersgross3 ; Zhu ; yip ; Maciejko ; diventra ; chen1 ; gross2 , in which the external time-dependent fields or internal parametric pump potentials drive the electrons to tunnel through the molecular device, is one of the most important issues in molecular electronics. The simplest molecular device structure is the two-probe lead- device-lead (LDL) configuration, where “device” is the molecular device connected to the external probes by the “leads”. In such a device, all the atomic details of the device material can be treated using density functional theory (DFT) and the non-equilibrium physics can be taken into account using non-equilibrium Green’s function (NEGF). Up to now, from an atom point of view, one of the most popular theoretical approaches used to study the quantum transport properties of molecular device is Keldysh nonequilibrium Green’s functions coupled with density-functional theory (NEGF-DFT).mcdcal Using this approach, the steady state quantum transport properties in molecular devices have been widely studied. For time dependent response of molecular devices, there have been many different theoretical approaches, such as evolution of time-dependent Schrodinger equation,Schro , time development operator approach,evolution and the NEGF technique.NEGF These approaches are convenient to deal with dynamic response of time-dependent external field that is sinusoidal (e.g., microwave radiation). Under such an external field an electron can tunnel through the system by emitting or absorbing photons, giving rise to the photon-assisted tunneling (PAT). Concerning the steady state ac response to harmonic external field, the Floquet approach is convenient.Floquet For the transient transport, however, the pulse like ac signal is the optimal driven force since they can provide a less ambiguous measure of time scales.time In this case, besides PAT, one of the most interesting questions to ask is how fast a device can turn on or turn off a current. With the development of molecular electronics, providing a particular viable switching device has become a key technical issue. Concerning the transient dynamics, different approaches such as path-integral techniques,pathintegral the solution of Wigner distribution function,Wigner the time-dependent numerical renormalization group,NRG time- dependent DFT (TDDFT),TDDFT ; gross3 and Keldish Green’s funcitonZhu ; Maciejko ; pulse have also been developed and applied to different systems. Up to now, most of these approaches can only be implemented in simple systems such as quantum dotsMaciejko ; pulse or one-dimension tight-binding chains.gross3 Numerical calculation of transient current for molecular devices is very difficult at present stage due to the huge computational cost. This is because if we calculate the current as a function of time $t$, the amount of calculation scales as $t^{3}$ if the time-evolution method is used. This scaling can be reduced to a linear scaling in $t$ if the wideband limit is used.zheng As we have demonstrated,WangBin the wideband limit is not a good approximation for molecular devices. If one uses the exact solution from NEGF,Maciejko one can calculate the transient current at a particular time. However, the calculation involves a triple integration over energy which is extremely time consuming. Clearly an approximate scheme that is suitable for numerical calculation of transient properties for real molecular devices while still captured essential physics is needed. It is the purpose of this paper to provide such a practical scheme. To study transient dynamics, in this paper, we consider a system that consists of a scattering region coupled to two leads with the external time dependent pulse bias potential $V_{\alpha}(t)=\theta(\pm t)V_{\alpha}$. For this case, the time-dependent current for a step-like pulse has been derived exactly without using the wide-band limit by Maciejko et alMaciejko . Since the general expression for the current involves triple integrations, it is extremely difficult to perform them in a real systems like molecules devices. So, approximation has to be made. The simplest approximation is the so called wide-band approximation where self-energies $\Sigma^{r,a}$ are assumed to be constants independent of energy.Jauho1 Unfortunately, this approximation can not give the correct result since in general there are several resonant levels that significantly contribute to the transient current in molecules devices. To go beyond the wideband limit, we propose an approximate scheme of calculating the transient current that is suitable for numerical calculation of real molecules devices.foot1 Our scheme is an approximation of the exact solution of Maciejko et alMaciejko . It is very fast computationally and gives the correct limits at $t=0$ and $t=\infty$. Since the exact solution of transient current is available for a single level quantum dot system, we have compared our result with the exact solution on the quantum dot system to test our approximate schemes. Good agreement is obtained. Therefore, our approximated scheme maintains essential physics of transient dynamics. Using our scheme, we calculate the transient current for the upward pulse (turn-on) and downward pulse (turn-off) in two molecular structures: Al-${\rm C}_{5}$-Al and Al-${\rm C}_{60}$-Al. We find that different from the single level quantum dot system, upon switching on the current oscillates rapidly in the first a few or tens fs with several characteristic time scales. Furthermore, due to the resonant states in molecular devices, transient currents have a much longer decay time $\tau$, especially for the molecule device having a complex electronic structure such as Al-${\rm C}_{60}$-Al. The rest of paper is organized as follows: In Sec.II, starting from the typical molecular device Hamiltonian which is expressed in an non-orthogonal basis, we shall derive a general DC and AC current expressions for a non- orthogonal basis set. It is found that for DC bias, the expressions of current for orthogonal and non-orthogonal basis sets are the same. For ac current, however, the expressions are different as will be demonstrated in Sec.II. The reason that we study the difference between orthogonal and non-orthogonal basis sets is the following. For the NEGF formalism, it is assumed that the basis set is orthogonal. It turns out that for ac transport, the current expression becomes extremely complicated if non-orthogonal basis is used. For DFT calculation, however, most people work in molecular orbitals that are non- orthogonal. Our results show that we must orthogonalizing the nonorthogonal molecular Hamiltonian, so that the present approach in Ref.Maciejko, can be used. In Sec.III, based on the exact solution of Maciejko et al, we derive two approximate expressions for transient current with different levels of approximation. They are all suitable for numerical calculation for real molecular devices. In addition, the initial current and its asymptotic long time limit are shown to be correct. In Sec.IV, in order to appreciate our approximate formulas, we compare our result with the exact result obtained in Ref.Maciejko, for a single-level quantum dot connected to external leads with a Lorentzian linewidth. In Sec.V, we apply our formalism to several molecular devices. Finally, a conclusion is presented in Sec.VI. Two appendices are given at the end of the paper. In Appendix A, we give a detailed derivation of orthogonalization relation for an non-orthogonal basis. This relation is used to derive the effective Green’s function which is the key to approximate exact current expression of Maciejko et al. In Appendix B, we show how to orthogonalize an nonorthogonal Hamiltonian so that the general AC current for real molecules device can be derived. ## II general AC current ### II.1 Hamiltonian The transport properties of a molecular device can be described by the following general Hamiltonian: $\displaystyle H=H_{c}+H_{T}+\sum_{\alpha=L,R}H_{\alpha}$ (1) where $H_{L}$ and $H_{R}$ describe the left and right macroscopic reservoir, respectively; $H_{c}$ is Hamiltonian of the central molecular device; $H_{T}$ couples the reservoirs to the molecular device. For a particular basis set, the above Hamiltonian can be written in the following matrix form: $\displaystyle H_{\alpha}$ $\displaystyle=$ $\displaystyle\sum\limits_{\mu_{\alpha}\nu_{\alpha}}c^{\dagger}_{\mu_{\alpha}}\left[{\mathbf{H}}^{0}_{\mu_{\alpha}\nu_{\alpha}}+eV_{\alpha}(t)\delta_{\mu_{\alpha}\nu_{\alpha}}\right]c_{\nu_{\alpha}}$ $\displaystyle H_{c}$ $\displaystyle=$ $\displaystyle\sum\limits_{\mu_{c}\nu_{c}}d^{\dagger}_{\mu_{c}}\left[{\mathbf{H}}^{0}_{\mu_{c}\nu_{c}}+{\mathbf{U}}_{\mu_{c}\nu_{c}}(t)\right]d_{\nu_{c}}$ $\displaystyle H_{T}$ $\displaystyle=$ $\displaystyle\sum\limits_{\nu_{\alpha},\nu_{c}}c^{\dagger}_{\nu_{\alpha}}{\mathbf{T}}^{0}_{\nu_{\alpha}\nu_{c}}d_{\nu_{c}}+h.c.$ (2) where $e$ is the electron charge, $c_{\nu_{\alpha}}$ ($c_{\nu^{\dagger}_{\alpha}}$) and $d_{\nu_{c}}$ ($d^{\dagger}_{\nu_{c}}$) are Fermionic annihilation (creation) operators at the state $\nu$ in the lead-$\alpha$ and the state $\nu$ in central molecular device. $\nu_{\alpha}$, $\nu_{c}$ are the indices of the given basis set. The Hamiltonian of lead-$\alpha$ are divided into two parts: the time independent part ${\bf H}_{\alpha}^{0}$ and time dependent part due to external bias $V_{\alpha}(t)$ on the lead-$\alpha$. Here we consider two kinds of step-like bias: upwards pulse (turn-on case) $V_{\alpha}^{U}(t)$ and downwards pulse (turn-off case) $V_{\alpha}^{D}(t)$, where $\displaystyle V^{D}_{\alpha}(t)=\left\\{\begin{array}[]{cc}V_{\alpha},~{}~{}~{}t<0\\\ 0,~{}~{}~{}~{}~{}t>0\end{array}\right.,~{}~{}~{}V^{U}_{\alpha}(t)=\left\\{\begin{array}[]{cc}0,~{}~{}~{}~{}~{}t<0\\\ V_{\alpha},~{}~{}~{}t>0\end{array}\right.$ (7) In the adiabatic approximation it is assumed that the single particle energies acquire a rigid time-dependent shift as ${\bf H}_{\alpha}^{0}+{\bf I}V_{\alpha}(t)$. The energy shift in the leads is assumed to be uniform throughout. This assumption is reasonable since the pulse rising time is slower than the usual metallic plasma oscillation time, which ensures that the external electric field is effectively screened.Jauho2 Since Green’s function ${\bf G}^{r}(t,t^{\prime})$ is obtained by solving Dyson equation from the known history, it is better to set time dependent external bias $V_{\alpha}(t>0)=0$ so that the uncertainty of future can be eliminated.Maciejko From Eq.(7), this is satisfied only in the downward case. In the following, we will discuss how to eliminate this uncertainty for the upward pulse. To use the Dyson equation, we will separate the Hamiltonian into two pieces: the unperturbed Hamiltonian that can be exactly resolved and the interacting term which contributes to the self energy in Dyson equations. For the downward pulse, we define the non-biased open system as the unperturbed system. It is described by the Hamiltonian ${\bf H}^{0}={\bf H}^{0}_{\alpha}+{\bf H}^{0}_{c}+{\bf H}^{0}_{T}$. For the upward pulse, however, the situation is different, in which we will set the DC biased open system ${\bf H}^{V}=[{\bf H}^{0}_{\alpha}+V_{\alpha}{\bf I}]+[{\bf H}^{0}_{c}+{\bf U}^{V}]+{\bf H}^{V}_{T}$ as the unperturbed Hamiltonian and set ${\tilde{V}}^{U}_{\alpha}(t)=V^{U}_{\alpha}(t)-V_{\alpha}$ as the new time dependent part. Here ${\bf H}^{V}_{T}$ denotes the coupling between scattering region and biased leads and ${\bf U}^{V}$ is the induced coulomb potential due to the external bias. Now, the time dependent bias ${\tilde{V}}^{U}$ satisfies ${\tilde{V}}^{U}(t>0)=0$, and the uncertainty of the future in the upward case is eliminated. Then, for the downward case, we have ${\tilde{V}}^{D}_{\alpha}(t)=V^{D}_{\alpha}(t)$ and ${\bf H}^{er}={\bf H}^{0}$ while for the upward case we have ${\tilde{V}}^{U}_{\alpha}(t)=V^{U}_{\alpha}(t)-V_{\alpha}$ and ${\bf H}^{ex}={\bf H}^{V}$. From now on we will use superscript $``ex"$ to denote the unperturbed system that is exactly resolvable. When the system is biased, the incoming electron will polarize the system. The induced Coulomb potential in the central scattering region consists of two parts: DC and AC parts. The DC part can be put into the exactly resolvable Hamiltonian ${\bf H}^{ex}$. The induced time dependent coulomb potential ${\bf U}(t)$ due to the external bias ${\tilde{V}}_{\alpha}(t)$ is included as part of the non-equilibrium Hamiltonian. Because the electric field is not screened in the small scattering region where the potential drop occurs, the coulomb potential landscape ${\bf U(t)}$ in the central region is not uniform, which is different from the semi-infinite leads. Note that it is rather difficult to treat the time-dependent coulomb potential and no close formed solution exists if one does not assume wide band limit. In the small bias limit, we can expand the time-dependent coulomb potential to linear order in bias ${\bf U}(t)=e\sum_{\alpha}{\bf u}_{\alpha}{\tilde{V}}_{\alpha}(t)$ so that the analytic expression for current can be obtained. Here $u_{\alpha}$ is the characteristic potential.buttiker9 From the gauge invariance, c2gauge $\sum_{\alpha}{\bf u}_{\alpha}={\bf I}$, and ${\bf u}_{\alpha}$ is determined from a poisson like equation.yadong In this paper, we consider the symmetric coupling so that for the external bias ${\tilde{V}}_{L}(t)=-{\tilde{V}}_{R}(t)$ it is a good approximation to assume that the time dependent coulomb potential $U(t)$ is roughly zero in the the molecular device regime. In the following, we will derive an exact solution of transient current using a non-orthogonal basis set.foot2 To facilitate the derivation, we take a unitary transformation $\hat{O}(t)$ to the Hamiltonian (2) with $\displaystyle\hat{O}(t)$ $\displaystyle=$ $\displaystyle{\rm exp}\left\\{ie\sum_{\nu_{\alpha}}\int_{0}^{t}d\tau~{}\left[{\tilde{V}}_{\alpha}(\tau)c^{\dagger}_{\nu_{\alpha}}c_{\nu_{\alpha}}\right]\right\\}$ where ${\tilde{V}}_{\alpha}(\tau)=\theta(-\tau)V_{\alpha}$ for the downward pulse and ${\tilde{V}}_{\alpha}(\tau)=-\theta(-\tau)V_{\alpha}$ for the upward pulse. Note that the time $t$ in $\hat{O}(t)$ can be negative or positive, and $\hat{O}(t)=1$ only when $t>0$. The new Hamiltonian ${\mathcal{H}}=\hat{O}H\hat{O}^{\dagger}(t)+i(\frac{\partial}{\partial t}\hat{O}(t))\hat{O}^{\dagger}(t)$, in which ${\mathcal{H}}_{\alpha}$ and ${\mathcal{H}}_{T}$ are different from original ones: $\displaystyle{\mathcal{H}}_{\alpha}$ $\displaystyle=$ $\displaystyle\sum\limits_{\mu_{\alpha}\nu_{\alpha}}{\bar{c}}^{\dagger}_{\mu_{\alpha}}{\mathbf{H}}^{0}_{\mu_{\alpha}\nu_{\alpha}}{\bar{c}}_{\nu_{\alpha}}$ $\displaystyle{\mathcal{H}}_{T}$ $\displaystyle=$ $\displaystyle\sum\limits_{\nu_{\alpha},\nu_{c}}{\bar{c}}^{\dagger}_{\nu_{\alpha}}{\mathbf{T}}_{\nu_{\alpha}\nu_{c}}(t){d}_{\nu_{c}}+h.c.$ (8) where $\displaystyle{\bar{c}}_{\nu_{\alpha}}=c_{\nu_{\alpha}}\exp[{ie\sum_{\mu_{\alpha}}\int_{0}^{t}d\tau~{}{\tilde{V}}_{\alpha}(\tau){c}^{\dagger}_{\mu_{\alpha}}{c}_{\mu_{\alpha}}}],$ $\displaystyle{\mathbf{T}}_{\nu_{\alpha}\nu_{c}}(t)={\mathbf{T}}^{0}_{\nu_{\alpha}\nu_{c}}{\mathfrak{W}}_{\alpha}(t)$ $\displaystyle{\mathfrak{W}}_{\alpha}(t)=\exp[{ie\int_{0}^{t}{\tilde{V}}_{\alpha}(\tau)d\tau}]$ (9) For the original Hamiltonian with nonorthogonal basis, the overlap between nonorthogonal basis is expressed as the matrix form ${\bf S}^{0}_{\mu\nu}=\langle\mu\|\nu\rangle$. After the unitary transform, annihilation (creation) operators $c_{\alpha}$ ($c^{\dagger}_{\alpha}$) and consequently the orbital basis $\mu_{\alpha}$ in the leads are changed, then overlap matrices between the leads and the scattering region become $\displaystyle{\bf S}_{\nu_{\alpha}\nu_{c}}(t)={\bf S}^{0}_{\nu_{\alpha}\nu_{c}}{\mathfrak{W}}_{\alpha}(t)$ $\displaystyle{\bf S}_{\nu_{c}\nu_{\alpha}}(t)={\mathfrak{W}}^{\dagger}_{\alpha}(t){\bf S}^{0}_{\nu_{c}\nu_{\alpha}}.$ (10) In the following, we will use the transformed Hamiltonian [Eq.(8,9), in which ${\bar{c}}_{\nu_{\alpha}}$, ${d}_{\nu_{c}}$ are used] to derive the time dependent current expression. ### II.2 The current The current operator from a particular lead-$\alpha$ to the molecular junction can be calculated from the evolution of the number operator of the electron in the semi-infinite lead-$\alpha$. Assuming there is no direct coupling between the left and right leads, the current operator can be expressed as:acCur $\displaystyle\hat{J}_{\alpha}(t)$ $\displaystyle=$ $\displaystyle-e\sum_{\nu_{\alpha}}\frac{d}{dt}\hat{N}_{\nu_{\alpha}}(t)$ (11) $\displaystyle=$ $\displaystyle-e\sum_{\nu_{\alpha}}\left[{\bar{c}}^{\dagger}_{\nu_{\alpha}}(t)\frac{d}{dt}{\bar{c}}_{\nu_{\alpha}}(t)+\left(\frac{d}{dt}{\bar{c}}^{\dagger}_{\nu_{\alpha}}(t)\right){\bar{c}}_{\nu_{\alpha}}(t)\right]$ $\displaystyle=$ $\displaystyle e\sum_{\nu_{\alpha},\nu_{c}}{\bar{c}}^{\dagger}_{\nu_{\alpha}}(t)\left(i{\mathbf{T}}_{\nu_{\alpha}\nu_{c}}(t)+{\mathbf{S}}_{\nu_{\alpha}\nu_{c}}(t)\frac{d}{dt}\right)d_{\nu_{c}}(t)+H.c.$ where ‘H.c.’ denotes the Hermitian conjugate. The current is obtained by taking average over the nonequilibrium quantum state ‘$<...>$’, $\displaystyle J_{\alpha}(t)=e~{}\sum_{\nu_{\alpha},\nu_{c}}$ $\displaystyle\left[\mathbf{G}^{<}_{\nu_{c}\nu_{\alpha}}(t,t^{\prime})\left(\mathbf{T}_{\nu_{\alpha},\nu_{c}}(t^{\prime})-{\bf S}_{\nu_{\alpha},\nu_{c}}(t^{\prime})i\grave{\frac{\partial}{\partial t}}\right)\right.$ $\displaystyle-$ $\displaystyle\left.\left(\mathbf{T}_{\nu_{c},\nu_{\alpha}}(t^{\prime})-{\bf S}_{\nu_{c},\nu_{\alpha}}(t^{\prime})i\acute{\frac{\partial}{\partial t}}\right)\mathbf{G}^{<}_{\nu_{\alpha}\nu_{c}}(t^{\prime},t)\right]_{t=t^{\prime}},$ (12) where “$\grave{\frac{\partial}{\partial t}}$” and “$\acute{\frac{\partial}{\partial t}}$” denotes the left and right derivation respectively, and ${\bf G}^{<}_{\nu_{c},\nu_{\alpha}}(t,t^{\prime})=i\left\langle{\bar{c}}^{\dagger}_{\nu_{\alpha}}(t^{\prime})d_{\nu_{c}}(t)\right\rangle,~{}~{}{\bf G}^{<}_{\nu_{\alpha},\nu_{c}}(t^{\prime},t)=i\left\langle d^{\dagger}_{\nu_{c}}(t){\bar{c}}_{\nu_{\alpha}}(t^{\prime})\right\rangle.$ Using the Keldysh equation and the theorem of analytic continuation, we have $\displaystyle{\bf G}^{<}_{c\alpha}(t,t^{\prime})=\int dt_{1}$ $\displaystyle\left[{\bf G}^{r}_{cc}(t,t_{1}){\bf B}_{c\alpha}(t_{1}){\bf g}^{<}_{\alpha\alpha}(t_{1},t^{\prime})+\right.$ (13) $\displaystyle\left.{\bf G}^{<}_{cc}(t,t_{1}){\bf B}_{c\alpha}(t_{1}){\bf g}^{a}_{\alpha\alpha}(t_{1},t^{\prime})\right]$ where $\displaystyle{\bf B}_{c\alpha}(t_{1})$ $\displaystyle=$ $\displaystyle{\bf T}_{c\alpha}(t_{1})-{\bf S}_{c\alpha}(t_{1})i\grave{\frac{\partial}{\partial t}}$ (14) For simplicity, we have dropped the subscript $\mu$, and keep only the symbol $c$ and $\alpha$ to indicate the central scattering region and lead-$\alpha$, respectively. In the above expression and in the following, the summation convention on repeated sub-indices is assumed. Substituting Eq.(13) into Eq.(12), we have the general expression for the current: $\displaystyle J_{\alpha}(t)$ $\displaystyle=$ $\displaystyle-2e{\rm Re}\int dt_{1}~{}{\rm Tr}$ (15) $\displaystyle\left[{\bf G}^{r}_{cc}(t,t_{1}){\bf B}_{c\alpha}(t_{1}){\bf g}^{<}_{\alpha\alpha}(t_{1},t^{\prime}){\bf B}_{\alpha c}(t^{\prime})-\right.$ $\displaystyle\left.{\bf G}^{<}_{cc}(t,t_{1}){\bf B}_{c\alpha}(t_{1}){\bf g}^{a}_{\alpha\alpha}(t_{1},t^{\prime}){\bf B}_{\alpha c}(t^{\prime})\right]_{t=t^{\prime}}$ When the system reaches a stationary state, $V_{\alpha}(t)=V_{\alpha}$ becomes time independent, from definition Eq.(9), (10) and (14), we can find ${\bf B}_{c\alpha}(t_{1})X{\bf B}_{\alpha c}(t)=e^{-ieV_{\alpha}(t_{1}-t)}{\bf B}^{0}_{c\alpha}X{\bf B}^{0}_{\alpha c},$ with ${\bf B}^{0}_{c\alpha/\alpha c}={\bf T}^{0}_{c\alpha/\alpha c}-i\grave{\frac{\partial}{\partial t}}{\bf S}^{0}_{c\alpha/\alpha c}$, where “0” denotes the zero bias system.In addition, all the propagators ${\bf G}$ and ${\bf g}$ depend only on the time difference $t_{1}-t$. Taking the Fourier transformation, from Eq.(12) or Eq.(15), we can easily obtain DC current expressed in the energy representation: $\displaystyle J_{\alpha}$ $\displaystyle=$ $\displaystyle\int d\epsilon~{}\mathcal{J}_{\alpha}(\epsilon)$ (16) $\displaystyle=$ $\displaystyle\rm{Re}~{}2e\int d\epsilon~{}{\rm Tr}\left[\mathbf{G}^{r}(\epsilon)\mathbf{\Sigma}^{<}_{\alpha}(\epsilon)+\mathbf{G}^{<}(\epsilon)\mathbf{\Sigma}^{a}_{\alpha}(\epsilon)\right]$ where ${\mathbf{G}}$ and ${\mathbf{\Sigma}}$ are the Green’s function and the self-energy. They have the same matrix dimension as that of the Hamiltonian ${\bf H}_{c}$. The Green’s function ${\bf G}^{r/a}$ and self-energy ${\bf\Sigma}^{r/a}$ is defined as $\displaystyle{\bf G}^{r/a}(\epsilon)=\left[\epsilon{\bf I}-{\bf H}_{c}-{\bf\Sigma}^{r/a}(\epsilon)\right]^{-1}$ $\displaystyle\mathbf{\Sigma}^{\gamma}_{\alpha}(\epsilon)=\left[\mathbf{T}^{0}_{c\alpha}-\epsilon^{\alpha}{\bf S}^{0}_{c\alpha}\right]\mathbf{g}^{\gamma}_{\alpha\alpha}(\epsilon^{\alpha})\left[\mathbf{T}^{0}_{\alpha c}-\epsilon^{\alpha}{\bf S}^{0}_{\alpha c}\right]$ (17) where $\epsilon^{\alpha}=\epsilon-eV_{\alpha}$, ${\bf I}$ is the unitary matrix with same dimension as ${\bf H}_{c}$, $\gamma=r,a,<$, and $\displaystyle{\bf g}^{r/a}_{\alpha\alpha}(\epsilon)$ $\displaystyle=$ $\displaystyle\left[\left((\epsilon\pm i0^{+}){\bf S}^{0}_{\alpha\alpha}-{\bf H}^{0}_{\alpha\alpha}\right)^{-1}\right]_{\nu_{\alpha}\in{\rm sur},\mu_{\alpha}\in{\rm sur}}$ $\displaystyle{\bf g}^{<}_{\alpha\alpha}(\epsilon)$ $\displaystyle=$ $\displaystyle f(\epsilon)\left[{\bf g}^{a}_{\alpha\alpha}(\epsilon)-{\bf g}^{r}_{\alpha\alpha}(\epsilon)\right]$ (18) is the surface Green’s function of the semi-infinite periodic lead which can be calculated numerically using a transfer matrix method.transfer Here, $f(\epsilon)$ is the Fermi distribution. Eq.(16) shows that the dc current expressions are the same for both orthogonal and non-orthogonal basis sets. When the time dependent field $V_{\alpha}(t)$ is present, however, the current expressed in energy representation will be very complicated for nonorthogonal basis due to the term ${\bf S}(t^{\prime})i\frac{\partial}{\partial t}$ in Eq.(12), since ${\bf B}(t_{1})X{\bf B}(t)$ can’t be expressed as a function of time difference $t_{1}-t$. One thing is clear, the transient current expressions are different for orthogonal and non-orthogonal basis sets. Instead of deriving a complicated transient current expression using a non- orthogonal basis set, we will eliminate ${\bf S}_{c\alpha/\alpha c}(t^{\prime})i\frac{\partial}{\partial t}$ in Eq.(12) and work on an orthogonal basis set. In Appendix B, from the overlap matrix ${\bf S}$, we derive the orthogonal basis set and new Hamiltonian ${\tilde{H}}$ expressed in this orthogonal basis. With the new orthogonal Hamiltonian, the overlap matrix ${\bf S}_{c\alpha/\alpha c}(t^{\prime})$ will be eliminated since the overlap matrix of orthogonal basis ${\bf S}^{orth}={\bf I}$. Then, replacing Hamiltonian ${\bf H}$ in Eq.(2) with $\tilde{\bf H}$ and go through the derivation leading to Eqs.(2-15) again, we arrive at a new AC current expression: $\displaystyle J_{\alpha}(t)=2e{\rm Re}\int dt_{1}{\rm Tr}\left\\{\mathbf{G}_{cc}^{r}(t,t_{1})\left[{\bf T}_{c\alpha}(t_{1}){\bf g}^{<,ex}_{\alpha\alpha}(t_{1}-t){\bf T}_{\alpha c}(t)\right]\right\\}$ $\displaystyle+2e{\rm Re}\int dt_{1}{\rm Tr}\left\\{\mathbf{G}^{<}_{cc}(t,t_{1})\left[{\bf T}_{c\alpha}(t_{1}){\bf g}^{a,ex}_{\alpha\alpha}(t_{1}-t){\bf T}_{\alpha c}(t)\right]\right\\}$ (19) Defining the self-energy on the orthogonal basis $\displaystyle{\bf\Sigma}^{\gamma=r,a,<}_{\alpha}(t,t^{\prime})={\bf T}_{c\alpha}(t){\bf g}^{\gamma,ex}_{\alpha\alpha}(t-t^{\prime}){\bf T}_{\alpha c}(t^{\prime})$ (20) where ${\bf g}^{\gamma,ex}_{\alpha\alpha}(t-t^{\prime})=\int\frac{d\epsilon}{2\pi}~{}e^{-i\epsilon(t-t^{\prime})}{\bf g}^{\gamma,ex}_{\alpha\alpha}(\epsilon)$ is the surface Green’s function of semi-infinite lead-$\alpha$ in the unperturbed state as defined in the Sec.II.1. For the downward pulse we have set the unperturbed system as the open system at zero bias, in which ${\bf g}^{\gamma,ex}_{\alpha\alpha}(\epsilon)=\left[\epsilon-H^{0}_{\alpha}+i0^{+}\right]^{-1}_{\alpha\in{\rm sur}}$. For the upward pulse, the unperturbed system means $V_{\alpha}$ biased open system, in which ${\bf g}^{\gamma,eq}_{\alpha\alpha}(\epsilon)=\left[\epsilon- eV_{\alpha}-H^{0}_{\alpha}+i0^{+}\right]^{-1}_{\alpha\in{\rm sur}}$. From Eq.(19),(20), we have the general current formula $\displaystyle J_{\alpha}(t)$ $\displaystyle=$ $\displaystyle 2e{\rm Re}\int dt_{1}{\rm Tr}\left[\mathbf{G}^{r}(t,t_{1})\Sigma^{<}_{\alpha}(t_{1},t)+\mathbf{G}^{<}(t,t_{1})\Sigma^{a}_{\alpha}(t_{1},t)\right]$ At $t<0$, AC external bias $V_{\alpha}(t)$ or time dependent part in Hamiltonian ${\tilde{V}}_{\alpha}(t)$ is a constant and the system is in a steady state. Consequently, the total current is known from DC transport theory that is expressed in the form of Eq.(16) but with the Green’s function and self-energy obtained from the orthogonal Hamiltonian defined above. Hence in the following we shall derive only the Ac current when $t>0$. First, we shall look at the self-energy. From Eq.(9) and (20), $\displaystyle\mathbf{\Sigma}^{\gamma}_{\alpha}(t,t^{\prime})$ $\displaystyle=$ $\displaystyle\mathfrak{W}^{\dagger}_{\alpha}(t)\left[\mathbf{T}^{0}_{c\alpha}\mathbf{g}_{\alpha\alpha}^{\gamma}(t,t^{\prime})T^{0}_{\alpha c}\right]\mathfrak{W}_{\alpha}(t^{\prime})$ $\displaystyle=$ $\displaystyle\mathfrak{W}^{\dagger}_{\alpha}(t)\left[\int\frac{d\epsilon}{2\pi}~{}e^{i\epsilon(t-t^{\prime})}\mathbf{\Sigma}_{\alpha}^{\gamma,ex}(\epsilon)\right]\mathfrak{W}_{\alpha}(t^{\prime})$ $\displaystyle=$ $\displaystyle\mathfrak{W}^{\dagger}_{\alpha}(t)\mathfrak{V}^{\dagger}_{\alpha}(t)\left[\int\frac{d\epsilon}{2\pi}~{}e^{i\epsilon(t-t^{\prime})}\mathbf{\Sigma}_{\alpha}^{\gamma,0}(\epsilon)\right]\mathfrak{V}_{\alpha}(t^{\prime})\mathfrak{W}_{\alpha}(t^{\prime})$ where $\mathfrak{V}_{\alpha}(t)=1$ for the downward pulse and $\mathfrak{V}_{\alpha}(t)=e^{ieV_{\alpha}t}$ for the upward pulse. Here $\mathbf{\Sigma}_{\alpha}^{\gamma,0}(\epsilon)$ is the self-energy at zero bias, $\mathbf{\Sigma}_{\alpha}^{\gamma,ex}(\epsilon)=\mathbf{T}^{0}_{c\alpha}\mathbf{g}_{\alpha\alpha}^{\gamma,ex}(\epsilon)\mathbf{T}^{0}_{\alpha c}$ is the self-energy at the unperturbed state defined above. In the downward case ${\bf\Sigma}^{\gamma,ex}_{\alpha}={\bf\Sigma}^{\gamma,0}_{\alpha}$; In the upward case ${\bf\Sigma}^{\gamma,ex}_{\alpha}={\bf\Sigma}^{\gamma,V}_{\alpha}$. Setting ${\bf S}^{0}_{\alpha c}={\bf S}^{0}_{c\alpha}=0$, ${\bf\Sigma}^{\gamma,0}_{\alpha}$ and ${\bf\Sigma}^{\gamma,V}_{\alpha}$ are defined in Eq.(17) with zero and nonzero $V_{\alpha}$, respectively. We have ${\bf\Sigma}^{r/a,V}_{\alpha}(\epsilon)={\bf\Sigma}^{r/a,0}_{\alpha}(\epsilon- eV_{\alpha})$. From Eq.(LABEL:generalCur) and (LABEL:self2), we find $\displaystyle J_{\alpha}(t)$ $\displaystyle=$ $\displaystyle 2e\rm{Re}\int\frac{d\epsilon}{2\pi}\int^{t}_{-\infty}dt_{1}~{}~{}e^{i\epsilon(t-t_{1})}$ (23) $\displaystyle\left[\mathbf{G}^{r}(t,t_{1})\tilde{\mathbf{\Sigma}}^{<}_{\alpha}(\epsilon,t_{1},t)+\mathbf{G}^{<}(t,t_{1})\tilde{\mathbf{\Sigma}}^{a}_{\alpha}(\epsilon,t_{1},t)\right]$ where the first term is the current flowing into the molecular device while the second one is the current flowing from the molecular device, and $\displaystyle\tilde{\mathbf{\Sigma}}^{\gamma}_{\alpha}(\epsilon,t_{1},t)$ $\displaystyle=$ $\displaystyle\mathcal{W}_{\alpha}^{\dagger}(t_{1})\mathbf{\Sigma}^{\gamma,0}_{\alpha}(\epsilon)\mathcal{W}_{\alpha}(t)$ (24) where $\mathcal{W}_{\alpha}(t)=\mathfrak{V}_{\alpha}(t)\mathfrak{W}_{\alpha}(t)$. Here ${\bf\Sigma}^{\gamma,0}_{\alpha\alpha}$ is the self-energy of lead-$\alpha$ at zero bias. The lesser Green’s function is given by $\displaystyle\mathbf{G}^{<}(t,t^{\prime})=\int dt_{1}\int dt_{2}~{}\mathbf{G}^{r}(t,t_{1})\left[\sum_{\beta}\mathbf{\bf{\Sigma}}^{<}_{\beta}(t_{1},t_{2})\right]\mathbf{G}^{a}(t_{2},t^{\prime})$ (25) $\displaystyle=$ $\displaystyle\sum_{\beta}\int\frac{d\epsilon}{2\pi}~{}e^{-i\epsilon(t-t^{\prime})}\left[\int^{t}_{-\infty}dt_{1}~{}e^{i\epsilon(t-t_{1})}\mathcal{W}_{\beta}(t)\mathbf{G}^{r}(t,t_{1})\mathcal{W}^{\dagger}_{\beta}(t_{1})\right]$ $\displaystyle\mathbf{\Sigma}^{<,0}_{\beta}(\epsilon)\left[\int^{t^{\prime}}_{-\infty}dt_{2}~{}e^{-i\epsilon(t^{\prime}-t_{2})}\mathcal{W}_{\beta}(t_{2})\mathbf{G}^{a}(t^{\prime},t_{2})\mathcal{W}^{\dagger}_{\beta}(t)\right]$ Substitute Eq.(24) and (25) into Eq.(23) and introducing a spectrum function $\displaystyle{\bf A}_{\alpha}(t,\epsilon)=\int_{-\infty}^{t}dt_{1}~{}e^{i\epsilon(t-t_{1})}\mathcal{W}_{\alpha}(t){\bf G}^{r}(t,t_{1})\mathcal{W}^{\dagger}_{\alpha}(t_{1})$ (26) we have $\displaystyle J^{in}_{\alpha}(t)$ $\displaystyle=$ $\displaystyle 2e\rm{Re}\int\frac{d\epsilon}{2\pi}~{}{\bf A}_{\alpha}(t,\epsilon){\bf\Sigma}_{\alpha}^{<,0}(\epsilon)$ (27) $\displaystyle J^{out}_{\alpha}(t)$ $\displaystyle=$ $\displaystyle 2e\rm{Re}\int\frac{d\epsilon}{2\pi}~{}\sum_{\beta}{\bf A}_{\beta}(t,\epsilon){\bf\Sigma}^{<,0}_{\beta}(\epsilon)\tilde{\bf F}_{\beta\alpha}(t,\epsilon)$ (28) where $\displaystyle\tilde{\bf F}_{\beta\alpha}(t,\epsilon)$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{t}dt^{\prime}~{}e^{-i\epsilon(t-t^{\prime})}\int\frac{dE}{2\pi}~{}e^{iE(t-t^{\prime})}$ (29) $\displaystyle{\bf A}_{\beta}^{\dagger}(t^{\prime},\epsilon){\mathcal{W}}_{\alpha}^{\dagger}(t^{\prime}){\bf\Sigma}_{\alpha}^{a,0}(E){\mathcal{W}}_{\alpha}(t)$ Very often, ${\bf\Sigma}^{r/a}(t-t^{\prime})$ is singular at $t=t^{\prime}$, such as the quantum dot system with the wide-band limit ${\bf\Sigma}^{r/a}(0)=\int\frac{dE}{2\pi}~{}{\bf\Sigma}^{r/a}(E)=\delta(0)(\mp\Gamma/2)$, or the superconducting-quantum dot-normal metal system, and so on. In these cases, we should be careful with Eq.(29), $\displaystyle\tilde{\bf F}_{\beta\alpha}(t,\epsilon)$ $\displaystyle=$ $\displaystyle{\bf F}_{\beta\alpha}(t,\epsilon)+\bar{\bf F}_{\beta\alpha}(t,\epsilon)$ (30) $\displaystyle=$ $\displaystyle\left(\int_{-\infty}^{t^{-}}+\frac{1}{2}\int_{t^{-}}^{t^{+}}\right)dt^{\prime}~{}e^{-i\epsilon(t-t^{\prime})}\int\frac{dE}{2\pi}~{}e^{iE(t-t^{\prime})}$ $\displaystyle{\bf A}_{\beta}^{\dagger}(t^{\prime},\epsilon){\mathcal{W}}_{\alpha}^{\dagger}(t^{\prime}){\bf\Sigma}_{\alpha}^{a,0}(E){\mathcal{W}}_{\alpha}(t)$ The first integral $\int_{-\infty}^{t^{-}}$ is the same as Eq.(29), the second integral $\frac{1}{2}\int_{t^{-}}^{t^{+}}$ now becomes $\bar{\bf F}_{\beta\alpha}(t,\epsilon)={\bf A}_{\beta}^{\dagger}(t,\epsilon){\bf\Delta}^{a}_{\alpha}$, where we have defined $\displaystyle{\bf\Delta}^{r/a}_{\alpha}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int_{t^{-}}^{t^{+}}dt^{\prime}~{}\left[\int\frac{dE}{2\pi}~{}{\bf\Sigma}_{\alpha}^{r/a,0}(E)\right]$ (31) $\displaystyle=$ $\displaystyle\frac{1}{2}\int_{t^{-}}^{t^{+}}dt^{\prime}~{}{\bf\Sigma}_{\alpha}^{r/a,0}(0)$ Then, Eq.(28) becomes $\displaystyle J^{out}_{\alpha}(t)$ $\displaystyle=$ $\displaystyle 2e\rm{Re}\int\frac{d\epsilon}{2\pi}~{}\sum_{\beta}{\bf A}_{\beta}(t,\epsilon){\bf\Sigma}^{<,0}_{\beta}(\epsilon){\bf F}_{\beta\alpha}(t,\epsilon)$ (32) $\displaystyle+$ $\displaystyle 2e\rm{Re}\int\frac{d\epsilon}{2\pi}~{}\sum_{\beta}{\bf A}_{\beta}(t,\epsilon){\bf\Sigma}^{<,0}_{\beta}(\epsilon){\bf A}^{\dagger}_{\beta}(t,\epsilon){\bf\Delta}^{a}_{\alpha}$ We note that Eq.(32) is the same as that derived in Ref.Maciejko, . Different from Ref.Maciejko, , we have split the expression into two terms. The first term corresponds to the non-wideband limit, i.e., when the linewidth function $\bf\Gamma$ goes to zero at large energy. The second term of Eq.(32) is related to the wideband limit. Hence, for a quantum dot with a Lorentzian linewidth functionMaciejko , only the first term is nonzero while for the system in contact with a superconducting lead both terms are nonzero. So far, we have discussed the ac conduction current $J_{\alpha}(t)$ under the time dependent bias derived from the evolution of the number operator of the electron in the semi-infinite lead-$\alpha$. Now we wish to address the issue of charge accumulation in the scattering region. In principle, this can be done by including the self-consistent Coulomb potential due to ac bias.yadong However, at finite voltages, there is no close form expression for ac current if Coulomb potential is included. Alternatively, one can treat Coulomb potential phenomenologically as follows. From the continuity equation, $\sum_{\alpha}J_{\alpha}(t)+dQ(t)/dt=0$, we see that the conduction current is not a conserved quantity. In the presence of ac bias, the displacement current $J_{\alpha}^{d}$ due to the charge pileup $dQ/dt$ inside the scattering region becomes important and must be considered. Since we have neglected the Coulomb interaction in our calculation, we can use the method of current partitionbuttiker4 ; wbg to include the displacement current. This can be done by partitioning the total displacement current $\sum_{\alpha}J^{d}_{\alpha}=dQ/dt$ into each leads giving rise to a conserving total current $I_{\alpha}=J_{\alpha}+J^{d}_{\alpha}$. For symmetric systems like what we shall study below, it is reasonable to assume that $J^{d}_{L}=J^{d}_{R}$ from which we find $J^{d}_{\alpha}=-(J_{L}+J_{R})/2$. Hence the total current is given by $I_{L}=(J_{L}-J_{R})/2$Jauho2 which satisfies the current conservation $I_{L}+I_{R}=0$. ## III transient AC current Up to now, we have derived the general expression for time dependent current, Eq.(26,27,29,32) which can be used for orthogonal as well as nonorthogonal basis set. To calculate the transient current we have to solve the retarded Green’s function ${\bf G}^{r}(t,t^{\prime})$ and integrate it over time to find ${\bf A}_{\beta}(t,\epsilon)$ and $\tilde{\bf F}_{\beta\alpha}(t,\epsilon)$. For the pulse-like voltage ${\tilde{V}}_{\alpha}(t)=\pm\theta(-t)$, we can obtain the Green’ function ${\bf G}^{r}(t,t^{\prime})$ by solving Dyson equation ${\bf G}^{r}={\bf G}^{r,eq}+{\bf G}^{r,eq}{\bf\Xi}{\bf G}^{r}$ from the known history in the time domain. Depending on what is the chosen unperturbed system that can be solved exactly, the Dyson equation can be written in a different but equivalent form. In the study of time-dependent transport, it is better to treat the time-independent, open steady state system as the unperturbed system as described in Sec.II.1, and treat the time dependent part $\tilde{V}_{\alpha}(t)$ and ${\bf U}(t)$ as a perturbation. As a result, the effective self-energy ${\bf\Xi}$, which is due to the ac bias, would have two sources: the perturbation in leads $\bar{\bf\Sigma}^{r}_{\alpha}$ and the induced Coulomb interaction in molecular device ${\bf U}(t)$. Then, $\displaystyle\mathbf{G}^{r}(t,t^{\prime})$ $\displaystyle=$ $\displaystyle\mathbf{G}^{r,ex}(t,t^{\prime})+\int_{-\infty}^{0}dt_{1}~{}\mathbf{G}^{r,ex}(t,t_{1})\mathbf{U}(t_{1})\mathbf{G}^{r}(t_{1},t^{\prime})$ $\displaystyle+$ $\displaystyle~{}\int dt_{1}~{}dt_{2}~{}\mathbf{G}^{r,ex}(t,t_{1})\left[\sum_{\alpha}\mathbf{\bar{\Sigma}}^{r}_{\alpha}(t_{1},t_{2})\right]\mathbf{G}^{r}(t_{2},t^{\prime})$ where ${\bf U}(t)$ is the response of the molecular device that is due to the Coulomb interaction when the time-dependent voltage is turned on. Here we have assumed an adiabatic response since most of time the variance of the applied electric field is much slower than the particles’ intrinsic lifetime inside the scattering region. Then we have ${\bf U}(t)=\pm{\bf U}\theta(-t)$ for downward case and upward case with ${\bf U}={\bf H}^{V}_{c}-{\bf H}^{0}_{c}$. $\int dt_{1}~{}dt_{2}=\left(\int^{0}_{-\infty}dt_{1}\int^{t_{1}}_{-\infty}dt_{2}+\int^{t}_{0}dt_{1}\int^{0}_{-\infty}dt_{2}\right)$ $\displaystyle\mathfrak{\bar{{\bf\Sigma}}}^{r}_{\alpha}(t,t^{\prime})={\bf\Sigma}^{r}_{\alpha}(t,t^{\prime})-{\bf\Sigma}_{\alpha}^{r,ex}(t-t^{\prime})$ $\displaystyle{\bf\Sigma}_{\alpha}^{r,ex}(t-t^{\prime})={\mathfrak{V}}^{\dagger}_{\alpha}(t){\bf\Sigma}_{\alpha}^{r,0}(t-t^{\prime}){\mathfrak{V}}_{\alpha}(t^{\prime})$ ### III.1 Exact expression of ${\bf A}_{\beta}(t,\epsilon)$ and ${\bf F}_{\beta\alpha}(t,\epsilon)$ Following the derivations in Ref.Maciejko, , we can get the exact expression for ${\bf A}_{\beta}(t,\epsilon)$ and ${\bf F}_{\beta\alpha}(t,\epsilon)$ with the aid of the expressions $\epsilon_{\beta}=\epsilon+eV_{\beta}$ and $\epsilon_{\beta\alpha}=\epsilon+eV_{\beta}-eV_{\alpha}$: $\displaystyle{\bf A}^{D}_{\beta}(t,\epsilon)$ $\displaystyle=$ $\displaystyle{\bf G}^{r,0}(\epsilon)+\int\frac{dE}{2\pi}~{}e^{i(\epsilon-E)t}$ (33) $\displaystyle\times$ $\displaystyle{\bf G}^{r,0}(E)\left[Z(\epsilon_{\beta})-Z(\epsilon)+{\bf P}_{D}{\bf G}^{r,V}(\epsilon_{\beta})\right]$ $\displaystyle{\bf F}^{D}_{\beta\alpha}(t,\epsilon)$ $\displaystyle=$ $\displaystyle\int\frac{dE}{2\pi}~{}Z^{}(\epsilon){\bf G}^{a,0}(\epsilon){\bf\Sigma}^{a,0}_{\alpha}(E)+\int\frac{dE}{2\pi}~{}e^{-i(\epsilon-E)t}$ (34) $\displaystyle\times$ $\displaystyle\left\\{\left[Z^{}(\epsilon_{\beta})-Z^{}(\epsilon)+{\bf G}^{a,V}(\epsilon_{\beta}){\bf P}^{\dagger}_{D}\right]{\bf G}^{a,0}(E){\bf Q}_{D}(E)\right.$ $\displaystyle+$ $\displaystyle\left.\left[Z^{}(\epsilon_{\beta\alpha}){\bf G}^{a,V}(\epsilon_{\beta})-Z^{}(\epsilon){\bf G}^{a,0}(\epsilon)\right]{\bf\Sigma}^{a,0}_{\alpha}(E)\right\\}$ $\displaystyle{\bf A}^{U}_{\beta}(t,\epsilon)$ $\displaystyle=$ $\displaystyle{\bf G}^{r,V}(\epsilon_{\beta})+\int\frac{dE}{2\pi}~{}e^{i(\epsilon_{\beta}-E)t}$ (35) $\displaystyle\times$ $\displaystyle{\bf G}^{r,V}(E)\left[Z(\epsilon)-Z(\epsilon_{\beta})+{\bf P}_{U}{\bf G}^{r,0}(\epsilon)\right]$ $\displaystyle{\bf F}^{U}_{\beta\alpha}(t,\epsilon)$ $\displaystyle=$ $\displaystyle\int\frac{dE}{2\pi}~{}Z^{}(\epsilon_{\beta\alpha}){\bf G}^{a,V}(\epsilon_{\beta}){\bf\Sigma}^{a,0}_{\alpha}(E)+\int\frac{dE}{2\pi}~{}e^{-i(\epsilon_{\beta}-E)t}$ $\displaystyle\times$ $\displaystyle\left\\{\left[Z^{}(\epsilon)-Z^{}(\epsilon_{\beta})+{\bf G}^{a,0}(\epsilon){\bf P}^{\dagger}_{U}\right]{\bf G}^{a,V}(E){\bf Q}_{U}(E)\right.$ $\displaystyle+$ $\displaystyle\left.e^{ieV_{\alpha}t}\left[Z^{}(\epsilon){\bf G}^{a,0}(\epsilon)-Z^{}(\epsilon_{\beta\alpha}){\bf G}^{a,V}(\epsilon_{\beta})\right]{\bf\Sigma}^{a,0}_{\alpha}(E)\right\\}$ where $\displaystyle{\bf P}_{D}$ $\displaystyle=$ $\displaystyle Z(\epsilon_{\beta}){\bf U}+\sum_{\delta}\left[Z(\epsilon_{\beta})-Z(\epsilon_{\beta\delta})\right]\left[{\bf\Sigma}^{r,0}_{\delta}(\epsilon_{\beta\delta})-{\bf\Sigma}^{r,0}_{\delta}(E)\right]$ $\displaystyle{\bf P}_{U}$ $\displaystyle=$ $\displaystyle-Z(\epsilon){\bf U}+\sum_{\delta}\left[Z(\epsilon)-Z(\epsilon_{\delta})\right]\left[{\bf\Sigma}^{r,0}_{\delta}(\epsilon)-{\bf\Sigma}^{r,0}_{\delta}(E-V_{\delta})\right]$ (37) $\displaystyle{\bf Q}_{D}(E)=\int\frac{d\epsilon^{\prime}}{2\pi}~{}\left[1-e^{i(\epsilon^{\prime}-E)t}\right]Z(\epsilon^{\prime}){\bf\Sigma}^{a,0}_{\alpha}(\epsilon^{\prime})$ $\displaystyle{\bf Q}_{U}(E)=\int\frac{d\epsilon^{\prime}}{2\pi}~{}\left[1-e^{i(\epsilon^{\prime}_{\alpha}-E)t}\right]Z(\epsilon^{\prime}_{\alpha}){\bf\Sigma}^{a,0}_{\alpha}(\epsilon^{\prime})$ with $\displaystyle Z(\epsilon)=[i(E-\epsilon-i0^{+})]^{-1}$ (38) In the absence of the ac bias, the quantity $A_{\alpha}$ is the Fourier transform of the retarded Green’s function while the quantity $F_{\beta\alpha}$ is related to the Fourier transform of the advanced Green’s function. They are all expressed in terms of the unperturbed Green’s functions ${\bf G}^{r/a,0/V}$ and self energy ${\bf\Sigma}^{0/V}$ which have been widely studied in molecular device using the NEGF-DFT formalism. ${\bf G}^{r/a,0/V}$ and self energy ${\bf\Sigma}^{0/V}$ can be expressed as $\displaystyle{\bf G}^{r/a,0/V}(\epsilon)=\left[\epsilon{\bf I}-{\bf H}^{0/V}_{c}-{\bf\Sigma}^{r/a,0/V}(\epsilon)\right]^{-1}$ $\displaystyle\mathbf{\Sigma}^{\gamma,0}_{\alpha}(\epsilon)=\left[\mathbf{T}^{0}_{c\alpha}-\epsilon{\bf S}^{0}_{c\alpha}\right]\mathbf{g}^{\gamma}_{\alpha\alpha}(\epsilon)\left[\mathbf{T}^{0}_{\alpha c}-\epsilon{\bf S}^{0}_{\alpha c}\right]$ $\displaystyle\mathbf{\Sigma}^{\gamma,V}_{\alpha}(\epsilon)=\left[\mathbf{T}^{0}_{c\alpha}-\epsilon^{\alpha}{\bf S}^{0}_{c\alpha}\right]\mathbf{g}^{\gamma}_{\alpha\alpha}(\epsilon^{\alpha})\left[\mathbf{T}^{0}_{\alpha c}-\epsilon^{\alpha}{\bf S}^{0}_{\alpha c}\right]$ where $\gamma=r,a,<$, $\epsilon^{\alpha}=\epsilon-eV_{\alpha}$. Obviously, ${\bf\Sigma}^{\gamma,V}_{\alpha}(\epsilon)={\bf\Sigma}^{\gamma,0}_{\alpha}(\epsilon- eV_{\alpha})$. In the wideband limit, Eq.(33-LABEL:FU1) will reduce to the formula first derived by Jauho et al.Jauho2 With ${\bf A}$ and ${\bf F}$ obtained we can, in principle, solve the AC current biased by downwards or upwards pulse exactly. In practice, however, its computational cost is expensive for a realistic molecular device. For example, to calculate $J_{\alpha}^{out}(t)$, we have to do triple integrals over energy and repeat this procedure to collect data for all time sequence. In the numerical calculation especially in ab-initio modeling, it is practically very difficult if not impossible to calculate the transient current for the complex structure in molecular devices. So approximation must be made so that Eq.(33-LABEL:FU1) can be simplified. ### III.2 Approximate scheme of ${\bf A}_{\beta}(t,\epsilon)$ and ${\bf F}_{\beta\alpha}(t,\epsilon)$ The approximate solution of ${\bf A}_{\beta}(t,\epsilon)$ and ${\bf F}_{\beta\alpha}(t,\epsilon)$ in Eq.(33-LABEL:FU1) have to satisfy the following requirements. First, it has to greatly reduce the calculational cost. Second, it has to keep essential physics of transient dynamics. Third, it must have the correct initial current at $t=0$ and approach the correct asymptotic limit at $t\rightarrow\infty$. The first goal is realized by eliminating double energy integral using a reasonable ansatz, with which the dynamical properties of molecular device is maintained. To find such an ansatz, we first assume that ${\bf\Sigma}^{a,0}(E)$ changes smoothly and slightly with $E$ and is analytic in the upper half plane, so that the typical integral like $\int d\epsilon dE~{}\frac{e^{i(\epsilon-E)t}}{-i(E-\epsilon+i0^{+})}{\bf\Sigma}^{a,0}(E)$ is roughly zero due to the different phase in $e^{\i(\epsilon-E)t}$. Then the last term of ${\bf F}^{U/D}$ and the second term of ${\bf Q}^{U/D}$ disappear. Considering the following identity, $\displaystyle\int\frac{dE}{2\pi}~{}\frac{{\bf\Sigma}^{a}_{\alpha}(E)}{-i(E-\epsilon+i0^{+})}$ $\displaystyle=$ $\displaystyle\left[\int_{-\infty}^{0^{-}}+\frac{1}{2}\int^{0^{+}}_{0^{-}}\right]d\tau~{}{\bf\Sigma}^{a}_{\alpha}(\tau)\int\frac{dE}{2\pi}~{}\frac{e^{iE\tau}}{-i(E-\epsilon+i0^{+})}$ $\displaystyle=$ $\displaystyle\left[\int_{-\infty}^{0^{+}}-\frac{1}{2}\int^{0^{+}}_{0^{-}}\right]d\tau~{}e^{i\epsilon\tau}{\bf\Sigma}^{a}_{\alpha}(\tau)={\bf\Sigma}^{a}_{\alpha}(\epsilon)-{\bf\Delta}^{a}_{\alpha}$ and defining ${\bf\Sigma}^{a}_{\alpha}(E,\Delta)={\bf\Sigma}^{a}_{\alpha}(E)-{\bf\Delta}^{a}_{\alpha}$, the first term of ${\bf F}_{U/D}$ and ${\bf Q}_{U/D}$ in Eqs.(37) can be simplified, ${\bf F}_{U/D}$ now becomes $\displaystyle{\bf F}^{D}_{\beta\alpha}$ $\displaystyle\simeq$ $\displaystyle{\bf G}^{a,0}(\epsilon){\bf\Sigma}^{a,0}_{\alpha}(\epsilon,\Delta)+\int\frac{dE}{2\pi}~{}e^{-i(\epsilon-E)t}$ (39) $\displaystyle\times$ $\displaystyle\left[Z^{}(\epsilon_{\beta})-Z^{}(\epsilon)+{\bf G}^{a,V}(\epsilon_{\beta}){\bf P}^{\dagger}_{D}\right]{\bf G}^{a,0}(E){\bf\Sigma}^{a,0}_{\alpha}(E,\Delta)$ $\displaystyle{\bf F}^{U}_{\beta\alpha}$ $\displaystyle\simeq$ $\displaystyle{\bf G}^{a,V}(\epsilon_{\beta}){\bf\Sigma}^{a,0}_{\alpha}(\epsilon_{\beta\alpha},\Delta)+\int\frac{dE}{2\pi}~{}e^{-i(\epsilon_{\beta}-E)t}$ $\displaystyle\times$ $\displaystyle\left[Z^{}(\epsilon)-Z^{}(\epsilon_{\beta})+{\bf G}^{a,0}(\epsilon){\bf P}^{\dagger}_{U}\right]{\bf G}^{a,V}(E){\bf\Sigma}^{a,0}_{\alpha}(E-eV_{\alpha},\Delta)$ We note that, in the wide-band limit, Eq.(39,LABEL:newFU) is exact. With our approximation we have eliminated one of the energy integrals in $J^{out}$, and ${\bf A}$ and ${\bf F}$ now have similar structures since $\tilde{\bf F}\sim{\bf A}^{\dagger}{\bf\Sigma}^{a}$. With the approximation defined in Eq.(39,LABEL:newFU), the current can be written in a compact form (see section C) if we introduce the effective Green’s function $\displaystyle\tilde{\bf G}^{r/a,0}(E,\epsilon)=\left[E{\bf S}-{\bf H}^{0}_{c}-\sum_{\alpha}{\bf\Sigma}^{r/a,0}_{\alpha}(\epsilon)\right]^{-1}$ (41) $\displaystyle\tilde{\bf G}^{r/a,V}(E,\epsilon)=\left[E{\bf S}-{\bf H}^{V}_{c}-\sum_{\alpha}{\bf\Sigma}^{r/a,V}_{\alpha}(\epsilon)\right]^{-1}$ (42) In general we have to consider the overlap matrix ${\bf S}$. However, we should keep in mind that in the deriving of the time dependent current, we have to orthogonalize the basis set, which would lead to ${\bf S}={\bf I}$. Here $\tilde{\bf G}^{r/a}(E,\epsilon)$ can be regarded as the Green’s functions at energy $E$ and constant parameter $\epsilon$ for open system with the effective Hamiltonian ${\bf H}^{r/a}_{eff}={\bf H}_{c}+{\bf\Sigma}^{r}_{\alpha}(\epsilon)$. For a given ${\bf H}_{eff}$, Eqs.(41,42) are equivalent to $\displaystyle(E{\bf S}-{\bf H}^{r}_{eff})\tilde{\bf G}^{r}={\bf I}$ (43) On the other hand, Green’s function can be expanded in terms of the eigenfunctions of the corresponding Hamiltonian, $\tilde{\bf G}^{r}=\sum_{n}{\bf\Psi}^{n}C_{n}.$ (44) where ${\bf H}_{eff}{\bf\Psi}^{n}=E_{n}(\epsilon){\bf\Psi}_{n}$. Substituting Eq.(44) into Eq.(43), and using the general orthogonality relation ${\bf\Phi}^{n,\dagger}{\bf S}{\bf\Psi}^{m}=C_{m}\delta_{nm}$ [see Appendix A] and the eigenvalue equation ${\bf H}_{eff}{\bf\Psi}^{n}=E_{n}(\epsilon){\bf\Psi}^{n}$, we have $\displaystyle\tilde{\bf G}^{r}(E,\epsilon)=\sum_{n}\frac{{\bf\Psi}^{n}{\bf\Phi}^{n,\dagger}}{[E-E_{n}(\epsilon)]{\bf\Phi}^{n,\dagger}{\bf S}{\bf\Psi}^{n}}$ (45) Obviously, this Green’s function can be calculated by finding the residues ${\rm Res}_{n}={\bf\Psi}^{n}{\bf\Phi}^{n,\dagger}/{{\bf\Phi}^{n,\dagger}{\bf S}{\bf\Psi}^{n}}$ at various poles $E=E_{n}(\epsilon)$. Then, we replace $Z(\epsilon){\bf G}^{r/a}(E)$ in Eqs.(33,35,39,LABEL:newFU) by $Z(\epsilon)\tilde{\bf G}^{r/a}(E,\epsilon)$. Although $\tilde{\bf G}^{r/a}(E,\epsilon)$ is different from initial Green’s function ${\bf G}^{r/a}(E)=\left[E-{\bf H}_{c}-{\bf\Sigma}^{r/a}(E)\right]^{-1}$, this substitution is reasonable since the major contribution of the integration in Eqs.(33-LABEL:FU1) comes from the pole $\epsilon$ in $Z(\epsilon)$ (see Eq.(38)). Similarly, considering the major contribution of the pole of $Z(\epsilon)$, we replace $Z(\epsilon){\bf\Sigma}^{a,0}(E)$ in Eqs.(33,35,39,LABEL:newFU) by $Z(\epsilon){\bf\Sigma}^{a,0}(\epsilon)$. Since ${\bf\Sigma}(\epsilon)$ in $\tilde{\bf G}^{r}(E,\epsilon)$ is independent of energy $E$, we can perform contour integration over energy $E$ in Eqs.(33) and (35) by closing a contour on lower half plane and perform the integration over energy $E$ in Eqs.(34) and (LABEL:FU1) by closing a contour on upper half plane. Thus, energy integration over $E$ can be analytically performed. It should be noted that the self energy ${\bf\Sigma}^{r/a}$ is not independent of energy in contrast to the wide-band limit, this energy dependence is on $\epsilon$ but not on $E$. In this way, we can reduce the computational cost and keep the essential physics of the dynamics as we will show later. ### III.3 Approximate expression of ${\bf A}_{\beta}(t,\epsilon)$ and ${\bf F}_{\beta\alpha}(t,\epsilon)$ Now, considering the initial current and the asymptotic long time limit, we can write the approximate expression of ${\bf A}_{\beta}(t,\epsilon)$ and ${\bf F}_{\beta\alpha}(t,\epsilon)$ from Eqs.(33,35,39,LABEL:newFU): $\displaystyle{\bf A}^{D/U}_{\beta}(t,\epsilon)={\bf A}^{D/U}_{\beta,1}+{\bf A}^{D/U}_{\beta,2}$ (46) $\displaystyle{\bf F}^{D}_{\beta\alpha}(t,\epsilon)={\bf A}^{D,\dagger}_{\beta,1}{\bf\Sigma}^{a,0}_{\alpha}(\epsilon_{\beta\alpha},\Delta)+{\bf A}^{D,\dagger}_{\beta,2}{\bf\Sigma}^{a,0}_{\alpha}(\epsilon,\Delta)$ (47) $\displaystyle{\bf F}^{U}_{\beta\alpha}(t,\epsilon)={\bf A}^{U,\dagger}_{\beta,1}{\bf\Sigma}^{a,0}_{\alpha}(\epsilon,\Delta)+{\bf A}^{U,\dagger}_{\beta,2}{\bf\Sigma}^{a,0}_{\alpha}(\epsilon_{\beta\alpha},\Delta)$ (48) with $\displaystyle{\bf A}^{D}_{\beta,1}=\int\frac{dE}{2\pi}~{}e^{i(\epsilon-E)t}\left[Z(\epsilon_{\beta})\tilde{\bf G}^{r,0}(E,\epsilon_{\beta})\left({\bf I}+{\bf\Xi}^{D}{\bf G}^{r,V}(\epsilon_{\beta})\right)\right]$ (49) $\displaystyle{\bf A}^{D}_{\beta,2}={\bf G}^{r,0}(\epsilon)-\int\frac{dE}{2\pi}~{}e^{i(\epsilon-E)t}\left[Z(\epsilon)\tilde{\bf G}^{r,0}(E,\epsilon)\right]$ (50) $\displaystyle{\bf A}^{U}_{\beta,1}=\int\frac{dE}{2\pi}~{}e^{i(\epsilon_{\beta}-E)t}\left[Z(\epsilon)\tilde{\bf G}^{r,V}(E,\epsilon)\left({\bf I}+{\bf\Xi}^{U}{\bf G}^{r,0}(\epsilon)\right)\right]$ (51) $\displaystyle{\bf A}^{U}_{\beta,2}={\bf G}^{r,V}(\epsilon_{\beta})-\int\frac{dE}{2\pi}~{}e^{i(\epsilon_{\beta}-E)t}\left[Z(\epsilon_{\beta})\tilde{\bf G}^{r,V}(E,\epsilon_{\beta})\right]$ (52) where $\displaystyle{\bf\Xi}^{D}$ $\displaystyle=$ $\displaystyle{\bf U}+\sum_{\delta}\left[{\bf\Sigma}^{r,0}_{\delta}(\epsilon_{\beta\delta})-{\bf\Sigma}^{r,0}_{\delta}(\epsilon_{\beta})\right]$ $\displaystyle=$ $\displaystyle{\bf U}+\sum_{\delta}\left[{\bf\Sigma}^{r,V}_{\delta}(\epsilon_{\beta})-{\bf\Sigma}^{r,0}_{\delta}(\epsilon_{\beta})\right]$ $\displaystyle{\bf\Xi}^{U}$ $\displaystyle=$ $\displaystyle-{\bf U}+\sum_{\delta}\left[{\bf\Sigma}^{r,0}_{\delta}(\epsilon)-{\bf\Sigma}^{r,0}_{\delta}(\epsilon- eV_{\delta})\right]$ (53) $\displaystyle=$ $\displaystyle-{\bf U}+\sum_{\delta}\left[{\bf\Sigma}^{r,0}_{\delta}(\epsilon)-{\bf\Sigma}^{r,V}_{\delta}(\epsilon)\right]$ This is the second level of approximation. As we will see later that it is better than the first level approximation described below. Now we can make further approximation (the first level). To do this, we note that the Green’s function ${\bf G}^{r}$ can be obtained using the Dyson equation, ${\bf G}^{r,tot}={\bf G}^{r,ex}+{\bf G}^{r,ex}{\bf\Xi}{\bf G}^{r,tot}$ (54) where ${\bf G}^{r,tot}$ is the Green’s function of system denoted by ${\bf H}^{tot}={\bf H}^{ex}+{\bf H}^{\prime}$, ${\bf G}^{r,ex}$ is the unperturbed Green’s function corresponding to ${\bf H}^{ex}$ that can be exactly solved, ${\bf\Xi}$ is the effective self energy describing ${\bf H}^{\prime}$. If we set ${\bf H}^{ex}$ and ${\bf H}^{tot}$ as zero biased open system and $V_{\alpha}$ biased open system respectively, we have ${\bf G}^{r,tot}={\bf G}^{r,V}(\epsilon)={\bf G}^{r,0}(\epsilon)+{\bf G}^{r,0}(\epsilon){\bf\Xi}^{D}{\bf G}^{r,V}(\epsilon)$ (55) Similarly, if we treat ${\bf H}^{ex}$ and ${\bf H}^{tot}$ as $V_{\alpha}$ biased open system and zero biased open system, respectively, we obtain another Dyson equation ${\bf G}^{r,tot}={\bf G}^{r,0}(\epsilon)={\bf G}^{r,V}(\epsilon)+{\bf G}^{r,V}(\epsilon){\bf\Xi}^{U}{\bf G}^{r,0}(\epsilon)$ (56) Similar to the derivation of the second level of approximation, we can also replace ${\bf G}^{ex}(\epsilon)$ by $\tilde{\bf G}^{ex}(E,\epsilon)$ in Eq.(55,56) which leads to $\displaystyle\tilde{\bf G}^{r,V}(E,\epsilon)\simeq\tilde{\bf G}^{r,0}(E,\epsilon)\left[{\bf I}+{\bf\Xi}^{D}{\bf G}^{r,V}(\epsilon)\right]$ $\displaystyle\tilde{\bf G}^{r,0}(E,\epsilon)\simeq\tilde{\bf G}^{r,V}(E,\epsilon)\left[{\bf I}+{\bf\Xi}^{U}{\bf G}^{r,0}(\epsilon)\right]$ (57) Then, Eqs.(49) and (51) can be further approximated as $\displaystyle{\bf A}^{D}_{\beta,1}=\int\frac{dE}{2\pi}~{}e^{i(\epsilon-E)t}\left[Z(\epsilon_{\beta})\tilde{\bf G}^{r,V}(E,\epsilon_{\beta})\right]$ (58) $\displaystyle{\bf A}^{U}_{\beta,1}=\int\frac{dE}{2\pi}~{}e^{i(\epsilon_{\beta}-E)t}\left[Z(\epsilon)\tilde{\bf G}^{r,0}(E,\epsilon)\right]$ (59) This is the first level of approximation. It is easy to confirm that when the self-energy is energy independent these two approximations lead to exactly the same expression of transient current in the wide-band limit. In the next section we will numerically compare these two approximations with the exact solution. ### III.4 initial and asymptotic currents We now show that the currents calculated from Eqs.(27,32,46-52) and from Eqs.(27,32,46-48,50,52,58,59) satisfy the correct current limit at initial $t=0$ and asymptotic limit $t\rightarrow\infty$ times. Note that the initial current and asymptotic currents can be calculated from a standard DC transport nonequilibrium Green’s function analysis. It is expected that the asymptotic current for the downward pulse $J^{D}_{\alpha}(t\rightarrow\infty)$ and initial current for the upward pulse $J^{U}_{\alpha}(t=0)$ are zero since there is no bias in the system. Now we discuss the limiting cases for two versions of approximations developed in section IIIC. When $t=0$, $e^{i(\epsilon-E)t}=1$, we can perform integration over energy $E$ in Eqs.(49-52) by closing a contour at upper half plane, where only a single residual exists at an energy pole of $Z$. At $t=0$, $\tilde{\bf G}^{r/a,0/V}(E,\epsilon)={\bf G}^{r/a,0/V}(\epsilon)$, therefore Eqs.(49,51) and Eqs.(58,59) are equivalent. Now we focus on the current obtained from Eqs.(27,32,46-48,50,52,58,59). After integrating over $\epsilon$, the two terms in Eqs.(50,52) cancels to each other, then from Eq.(58, 59), ${\bf A}^{D/U}_{\beta}(t=0)$ becomes $\displaystyle{\bf A}^{D}_{\beta}(t=0)$ $\displaystyle=$ $\displaystyle\tilde{\bf G}^{r,V}(\epsilon_{\beta},\epsilon_{\beta})={\bf G}^{r,V}(\epsilon_{\beta})$ (60) $\displaystyle{\bf A}^{U}_{\beta}(t=0)$ $\displaystyle=$ $\displaystyle\tilde{\bf G}^{r,0}(\epsilon,\epsilon)={\bf G}^{r,0}(\epsilon)$ (61) For ${\bf F}_{\beta\alpha}$, we can perform integration over energy $E$ by closing a contour at lower half plane. Similarly, there also exists only a single residual on energy pole $E_{Z}$ of $Z^{}$ in the lower half plane, and $\displaystyle{\bf F}^{D}_{\beta\alpha}(t=0)$ $\displaystyle=$ $\displaystyle\tilde{\bf G}^{a,V}(\epsilon_{\beta},\epsilon_{\beta}){\bf\Sigma}^{a,0}_{\alpha}(\epsilon_{\beta\alpha},\Delta)={\bf G}^{a,V}(\epsilon_{\beta}){\bf\Sigma}^{a,V}_{\alpha}(\epsilon_{\beta},\Delta)$ $\displaystyle{\bf F}^{U}_{\beta\alpha}(t=0)$ $\displaystyle=$ $\displaystyle\tilde{\bf G}^{a,V}(\epsilon,\epsilon){\bf\Sigma}^{a,0}_{\alpha}(\epsilon,\Delta)={\bf G}^{a,0}(\epsilon){\bf\Sigma}^{a,0}_{\alpha}(\epsilon,\Delta)$ (63) Substituting Eq.(60-63) into Eq.(27,32), and considering $\displaystyle{\bf\Sigma}^{\gamma,0}_{\beta}(\epsilon)={\bf\Sigma}^{\gamma,V}_{\beta}(\epsilon_{\beta})$ $\displaystyle{\bf G}^{<,0/V}(\epsilon)={\bf G}^{r,0/V}(\epsilon)\left[\sum_{\beta}{\bf\Sigma}^{<,0/V}_{\beta}(\epsilon)\right]{\bf G}^{a,0/V}(\epsilon)$ $\displaystyle{\bf\Sigma}^{<,0}_{\beta}(\epsilon)=f(\epsilon)\left[{\bf\Sigma}^{a,0}_{\beta}(\epsilon)-{\bf\Sigma}^{r,0}_{\beta}(\epsilon)\right]$ $\displaystyle{\bf\Sigma}^{<,V}_{\beta}(\epsilon)=f(\epsilon- eV_{\beta})\left[{\bf\Sigma}^{a,V}_{\beta}(\epsilon)-{\bf\Sigma}^{r,V}_{\beta}(\epsilon)\right]$ (64) where $f(\epsilon)$ is Fermi distribution function, we have initial current at $t=0$ $\displaystyle J^{D}_{\alpha}=2e\rm{Re}\int\frac{d\epsilon}{2\pi}~{}{\bf G}^{r,V}(\epsilon){\bf\Sigma}_{\alpha}^{<,V}(\epsilon)+{\bf G}^{<,V}(\epsilon){\bf\Sigma}^{a,V}_{\alpha}(\epsilon)$ (65) $\displaystyle J^{U}_{\alpha}=2e\rm{Re}\int\frac{d\epsilon}{2\pi}~{}{\bf G}^{r,0}(\epsilon){\bf\Sigma}_{\alpha}^{<,0}(\epsilon)+{\bf G}^{<,0}(\epsilon){\bf\Sigma}^{a,0}_{\alpha}(\epsilon)$ (66) Eqs.(65) and (66) are the same as the formal DC current expression in the case of nonzero bias and zero bias, respectively. $J^{U}_{\alpha}(t=0)$ in Eq.(66) is exactly zero since the Fermi distribution in ${\bf\Sigma}^{<}_{\alpha}$ and ${\bf G}^{<}$ are equal for $\alpha=L$ and $\alpha=R$. When $t\rightarrow\infty$, by virtue of the Riemann-Lebesgue lemma,lemma the Fourier integral over $\epsilon$ vanishes, i.e., $\int\frac{d\epsilon}{2\pi}~{}e^{-i\epsilon t}{\bf G}^{r}{\bf\Sigma}^{r}...$ equal to zero at $t\rightarrow\infty$ since there always exist poles in lower half plane. With this in mind, we have $\displaystyle{\bf A}^{D}_{\beta}(t\rightarrow\infty,\epsilon)$ $\displaystyle=$ $\displaystyle{\bf G}^{r,0}(\epsilon)$ (67) $\displaystyle{\bf F}^{D}_{\beta\alpha}(t\rightarrow\infty,\epsilon)$ $\displaystyle=$ $\displaystyle{\bf G}^{a,0}(\epsilon){\bf\Sigma}^{a,0}_{\alpha}(\epsilon,\Delta)$ (68) $\displaystyle{\bf A}^{U}_{\beta}(t\rightarrow\infty,\epsilon)$ $\displaystyle=$ $\displaystyle{\bf G}^{r,V}(\epsilon_{\beta})$ (69) $\displaystyle{\bf F}^{U}_{\beta\alpha}(t\rightarrow\infty,\epsilon)$ $\displaystyle=$ $\displaystyle{\bf G}^{a,V}(\epsilon_{\beta}){\bf\Sigma}^{a,0}_{\alpha}(\epsilon_{\beta\alpha},\Delta)={\bf G}^{a,V}(\epsilon_{\beta}){\bf\Sigma}^{a,V}_{\alpha}(\epsilon_{\beta},\Delta)$ From Eq.(67-LABEL:FU4) and Eq.(27,32), we have the asymptotic current $\displaystyle J^{D}_{\alpha}=2e\rm{Re}\int\frac{d\epsilon}{2\pi}~{}{\bf G}^{r,0}(\epsilon){\bf\Sigma}_{\alpha}^{<,0}(\epsilon)+{\bf G}^{<,0}(\epsilon){\bf\Sigma}^{a,0}_{\alpha}(\epsilon)$ (71) $\displaystyle J^{U}_{\alpha}=2e\rm{Re}\int\frac{d\epsilon}{2\pi}~{}{\bf G}^{r,V}(\epsilon){\bf\Sigma}_{\alpha}^{<,V}(\epsilon)+{\bf G}^{<,V}(\epsilon){\bf\Sigma}^{a,V}_{\alpha}(\epsilon)$ (72) It is easy to see, Eqs.(71) and (72) are the formal DC current expression in the case of zero bias and nonzero bias, respectively, and $J^{D}_{\alpha}(t\rightarrow\infty)$ in Eq.(71) is exactly zero. ## IV comparison with the exact result in quantum dot system Now we consider a system composed of a single-level quantum dot connected to external leads with a Lorentzian linewidth. This system can be solved exactly to give a transient current for pulse-like biasMaciejko . We can obtain transient current using three methods: (i) the exact current expressed by Eqs.(27,32, 33-LABEL:FU1), (ii) the first level of approximation from Eqs.(27,32,46-48,50,52,58,59) and (iii) the second level of approximation from Eqs.(27,32,46-52). We will compare the current obtained from these three methods. The system is described by the following simple Hamiltonian $\displaystyle H=\sum_{k_{\alpha}}\epsilon_{k_{\alpha}}(t)c^{\dagger}_{k_{\alpha}}c_{k_{\alpha}}+\epsilon_{d}(t)d^{\dagger}d+\sum_{k_{\alpha}}(t_{k_{\alpha}}c^{\dagger}_{k_{\alpha}}d+h.c.)$ (73) where $\epsilon_{d}(t)=\epsilon_{d}^{0}+U(t)$ and $\epsilon_{k_{\alpha}}(t)=\epsilon^{0}_{k_{\alpha}}+V_{\alpha}(t)$. Because the scattering region has only one state with energy level $\epsilon^{0}_{d}$, the Green’s functions $G(\epsilon)$ and self energy $\Sigma(\epsilon)$ thus become scalars instead of matrices. If we choose linewidth function $\Gamma_{\alpha}(\omega)\equiv 2\pi\rho_{\alpha}(\omega)\|t_{k_{\alpha}}\|^{2}$ to be Lorentzian with the linewidth amplitude $\Gamma^{0}_{\alpha}$, $\Gamma_{\alpha}(\omega)=\frac{W^{2}}{\omega^{2}+W^{2}}\Gamma^{0}_{\alpha}$ then $G^{\gamma}(\epsilon)$ and $\Sigma^{\gamma}(\epsilon)$ can be expressed as $\displaystyle G^{r/a,0}(\epsilon)=\left[\epsilon-\epsilon_{d}^{0}-\sum_{\alpha}\Sigma^{r/a,0}(\epsilon)\right]^{-1}$ $\displaystyle G^{r/a,V}(\epsilon)=\left[\epsilon-\epsilon_{d}^{0}-U^{V}-\sum_{\alpha}\Sigma^{r/a,V}(\epsilon)\right]^{-1}$ $\displaystyle G^{<,0/V}(\epsilon)=G^{<,0/V}(\epsilon)\left[\sum_{\alpha}\Sigma^{<,0/V}(\epsilon)\right]G^{<,0/V}(\epsilon)$ $\displaystyle\Sigma^{r/a,0}_{\alpha}(\epsilon)=\int\frac{d\omega}{2\pi}~{}\Gamma_{\alpha}(\omega)/(\epsilon-\omega\pm i0^{+})$ $\displaystyle\Sigma^{r/a,V}_{\alpha}(\epsilon)=\int\frac{d\omega}{2\pi}~{}\Gamma_{\alpha}(\omega)/(\epsilon- eV_{\alpha}-\omega\pm i0^{+})$ $\displaystyle\Sigma^{<,0}_{\alpha}(\epsilon)=f(\epsilon)\left[\Sigma^{a,0}_{\alpha}(\epsilon)-\Sigma^{r,0}_{\alpha}(\epsilon)\right]$ $\displaystyle\Sigma^{<,V}_{\alpha}(\epsilon)=f(\epsilon- eV_{\alpha})\left[\Sigma^{a,V}_{\alpha}(\epsilon)-\Sigma^{r,V}_{\alpha}(\epsilon)\right]$ Using the theorem of residual, we can analytically perform integral in $A_{\beta}$ and $F_{\beta\alpha}$ for either exact formula or two approximate formulas. In the calculation, we set $\Gamma=\Gamma^{0}_{L}+\Gamma^{0}_{R}$ as the energy unit, and set $\Gamma^{0}_{L}=\Gamma^{0}_{R}=0.5$. We first consider the transient current induced by opposite voltage $V_{L}(t)=-V_{R}(t)$. In this case, the equilibrium coulomb potential in quantum dot $U^{0,V}=0$, and the time dependent perturbation coming from coulomb response $U(t)$ is assumed to be zero. It is a reasonable assumption since the coulomb potential in scattering region is canceled by the opposite voltage in left and right lead. In Fig.1, we plot two approximated transient currents and exact transient current in downward [panel (a), (b), (c)] and upward [panel (d), (e), (f)] case vs time for different bandwidth $W$. We find that for all bandwidth $W$, the approximated current and exact current have the same dynamical behaviors. Fig.2 gives direct comparison where we merge panels (a), (b) and (c) in Fig.1 as panel (a) in Fig.2, and merge panels (d), (e) and (f) in Fig.1 as panel (b) in Fig.2. We can see that for the downward pulse [panel (a)], transient current using three formulas are almost indistinguishable. This means that in the opposite voltage, our approximation, the first approximation [Eqs.(50,52,58,59)] and the second approximation [Eqs.(49-52)] are all very good for studying transient dynamics. For the upward pulse, although the approximations are not as good as in downward case, the currents calculated from approximate scheme are still in good agreement with the exact solution especially for the second approximation. Hence we may conclude that the two approximations are all reasonable in the opposite voltage $V_{L}(t)=-V_{R}(t)$. They can be used to study transient dynamics in the real molecular device to speed up the calculation. Figure 1: (Color online) Time dependent current $J(t)$ corresponding to an opposite downward pulse or upward pulse in three versions: the exact solution and two approximations. The different black lines are for different bandwidth $W$. The red line is for $W=\infty$, i.e., the wide-band limit. The current is in the unit of $e\Gamma$, and the time is in the unit of $2\pi/\Gamma$. $eV_{L}=-eV_{R}=5.$ Figure 2: (Color online) Merged version of Fig.1 for $W=1$, $2$, $5$ and $20$. Panel (a) corresponding to the downward pulse current comes from panel (a), (b) and (c) in Fig.1, panel (b) corresponding to upward pulse current comes from panel (d), (e) and (f). Along the black arrow, the bandwidth are $W=1$, $2$, $5$ and $20$, respectively. Next, we focus on the asymmetric voltage, i.e., $V_{L}(t)\neq V_{R}(t)$. In this case, the equilibrium coulomb potential in quantum dot $U^{0/V}$, and the time dependent perturbation coming from coulomb response $U(t)$ can’t be canceled by the voltage in left and right lead. In principle, perturbation $U(t)$ should be calculated by solving time dependent Schrödinger equation, it will be very difficult and computational demanding therefore can’t be implemented in real molecular device. As an alternative scheme, we have set $U(t)=[eV_{L}(t)\Gamma^{0}_{L}+eV_{L}(t)\Gamma^{0}_{L}]/\Gamma$. For the single level quantum dot system, this is exact because the central scattering region now is expressed in a scalar instead of matrices, which leads to the same transient current for the opposite voltage $V_{L}(t)=-V_{R}(t)$ and asymmetric voltage $V_{L}(t)=V(t)$, $V_{R}(t)=0$ or $V_{L}(t)=0$, $V_{R}(t)=-V(t)$ in the exact solution. For the first approximation the poles in time dependent term $e^{i(\epsilon-E)t}$ are different from that in the second level approximation, i.e., the poles of $\tilde{\bf G}^{r,0}$ in Eq.49 and $\tilde{\bf G}^{r,V}$ in Eq.51 are replaced by the poles of $\tilde{\bf G}^{r,V}$ in Eq.58 and $\tilde{\bf G}^{r,0}$ in Eq.59, respectively. Because of this, the time evolution process are not as accurate in the first approximation, especially for the large $V_{\alpha}$. So, for the asymmetric voltage, the second approximation is better. In Fig.3 and Fig.4, we compare the transient current obtained from the second approximation [panel (b-d)] for opposite or asymmetric voltage with the exact transient current [panel (a)] in response to the downward pulse and upward pulse, respectively. We find that all transient currents from the second approximation in Fig.3 and Fig.4 [panel (b)] are very close to the exact result [panel (a)]. Moreover, in Fig.3 and Fig.4, the approximate transient current in panel (b), (c), (d) have almost the same behavior. It is safe to say that our approximations have kept essential physics of dynamical transport properties. Figure 3: (Color online) Panel (a): exact time dependent current $J(t)$ corresponding to downward pulse for $dV=V_{L}-V_{R}=5$. Panel (b-d) are corresponding to the second approximate transient current corresponding to downward pulse for opposite voltage $V_{L}=-V_{R}=2.5$, asymmetric voltage $V_{L}=5$, $V_{R}=0$ and $V_{L}=0$, $V_{R}=-5$, respectively. The different black lines are for different bandwidths $W$. The red line is wide-band limit for $W=\infty$. Figure 4: (Color online) Same to Fig.3, transient current corresponding to upward pulse vs time are plotted. ## V several examples for real molecular devices In this section, we implement our approximate formula in two representative molecular devices including a short carbon chain coupled to aluminum leads and a $C_{60}$ molecule coupled to aluminum leads. These systems were chosen because they are typical in first-principles calculation and their practical importance to nano-electronics. In Fig.5(a) and Fig.5(b), we show the structure of Al-${\rm C}_{5}$-Al and Al-${\rm C}_{60}$-Al, respectively, where Al leads are along (100) direction, one unit cell of Al lead consists of 9 Al atoms and total 40 atoms were included in the simulation box. For the Al-C5-Al device, the nearest distance between Al leads and the carbon chain is 3.781 a.u. and the distance of C-C bond is 2.5 a.u.(1 a.u.=0.529Å). In Al-C60-Al device, the distance between the Al atom and the nearest C atom equal to 3.625 a.u.. To calculate the dynamic response of molecular devices, we have used the first-principles quantum transport package MATDCAL.matdcal Considering the complicated coulomb response in scattering region, we set $V_{L}(t)=-V_{R}(t)$. In this case, the first approximation is simple but as good as the second one. So, in the following, the first approximate formula [Eqs.(27,32,46-48,50,52,58,59)] is used. In principle, the calculation involves the following steps: (1) calculate the device Hamiltonian including central scattering Hamiltonian and lead Hamiltonian using NEGF-DFT package to get two potential landscapes $U^{0}$ at zero bias and $U^{V}$ at $V_{\alpha}$ bias, respectively. They are originally expressed in a nonorthogonal fireball basis. (2) orthogonalize the nonorthogonal device Hamiltonian using the approachthesis introduced in Appendix B so that they are finally expressed in an orthogonal basis. (3) with the orthogonal lead Hamiltonian $H_{\alpha}$, one calculates zero biased self energy ${\bf\Sigma}^{r/a,0}_{\alpha}$ and $V_{\alpha}$ biased self energy ${\bf\Sigma}^{r/a,V}_{\alpha}$ from Eqs.(17,18) using the transfer matrix method.transfer (4) with orthogonalized central scattering Hamiltonian ${\bf H}^{0}_{c}$ and ${\bf H}^{V}_{c}$ and self energy ${\bf\Sigma}^{r/a,0}_{\alpha}$ and ${\bf\Sigma}^{r/a,V}_{\alpha}$ obtained from two potential landscapes $U^{0}$ and $U^{V}$, one solves the effective Green’s function ${\bf G}^{r/a,0/V}$ using Eqs.(41,42) by calculating its poles and residuals from Eq.(45). Step (1)-(4) are time independent processes and easy to perform. (5) calculate time dependent quantities ${\bf A}^{D/U}_{\beta,1}$ and ${\bf A}^{D/U}_{\beta,2}$ from Eqs.(58,59) and Eqs.(50,52). Then ${\bf A}_{\beta}$ and ${\bf F}_{\beta\alpha}$ can be calculated from Eqs.(46-48). (6) integrate over $\epsilon$ and obtain the final AC current $J^{D/U}(t)=[J^{D/U}_{L}(t)-J^{D/U}_{R}(t)]/2$ from Eqs.(27,32). Figure 5: (Color online) Panel (a): Structure of Al-C5-Al. Panel(b): structure of Al-${\rm C}_{60}$-Al. First we study the Al-${\rm C}_{5}$-Al structure. In Fig.6, we plot the transient current $J(t)$ corresponding to the upward pulse [panel (a) and (b)] and the downward pulse [panel (c) and (d)] for different external voltages $V_{R}=-V_{L}=0.001a.u.$ [panel (a) and (c)] and $V_{R}=-V_{L}=0.01a.u.$ [panel (b) and (d)] in Al-${\rm C}_{5}$-Al structure. Following observations are in order: (1) as we have discussed in Sec.III.4, for all bias voltages $V_{\alpha}$ the transient currents indeed reach the correct limits at $t=0$ and $t\rightarrow\infty$. For the upward pulse, $J(t=0)=0$ and $J(t\rightarrow\infty)=J_{dc}$ while for the downward pulse we have $J(t=0)=J_{dc}$ and $J(t\rightarrow\infty)=0$. (2) for both upward pulse (turn-on voltage) and downward pulse (turn-off voltage), once the bias voltage is switched, currents oscillate rapidly in the first a few or tens fs and then gradually approach to the steady-state values, i.e., $J_{dc}$ for turn-on voltage and zero for turn-off voltage. The larger the voltage $V_{\alpha}$, the more rapid the current oscillates. (3) concerning the long time behavior, the time dependent current oscillates with a frequency proportional to $\|V_{\alpha}\|$.WangBin This is because the time dependent term $e^{i(\epsilon-E)t}$ in Eqs.(50,52,58,59) are $V_{\alpha}$ dependent. For the upward pulse, $e^{i(\epsilon_{\alpha}-E)t}\propto e^{iV_{\alpha}t}$, which directly leads to the oscillating frequency proportional to $\|V_{\alpha}\|$. For the downward pulse, although $e^{i(\epsilon-E)t}$ is $V_{\alpha}$ independent, in the energy integral on $E$, the pole $E_{n}$ of $\tilde{\bf G}^{r}(E,\epsilon)$ are determined by the self energy ${\bf\Sigma}^{r,V}_{\alpha}$. Since ${\bf\Sigma}^{r,V}_{\alpha}$ depends on $V_{\alpha}$, this leads to $V_{\alpha}$ dependent oscillating frequency. In addition, we notice that although the properties of dc conductance of short carbon chains are different for the chains with odd and even number atomsWangBin2 due to the completely different electronic structure near Fermi level, the ac signals are similar (see Ref.WangBin, where Al-${\rm C}_{4}$-Al structure was analyzed). This indicates that in AC transport, all states with energy from $-\infty$ to the Fermi energy are contributing, which is very different from dc case where only the states near Fermi level contribute to transport processes. Figure 6: (Color online) Time dependent current $J(t)$ corresponding to the upward pulse [panel (a) and (b)] and the downward pulse [panel (c) and (d)] for different external voltages $V_{\alpha}$ in Al-${\rm C}_{5}$-Al device. The inset of panel (a) shows the long time behavior of the time-dependent current. The red (gray in print) dashed lines in panels indicate asymptotic current $J(t\rightarrow\infty)$ which the DC current biased by $V_{L/R}$ labeled in corresponding panels for the upward pulse, and arrive at zero for the downward case. Next, we study the second sample: the Al-${\rm C}_{60}$-Al structure. In Fig.7, the transient current $J(t)$ of the structure corresponding to an upward pulse [panel (a) and (b)] and a downward pulse [panel (c) and (d)] for different external voltages $V_{R}=-V_{L}=0.001a.u.$ [panel (a) and (c)] and $V_{R}=-V_{L}=0.01a.u.$ [panel (b) and (d)] are plotted. Similar to the Al-${\rm C}_{5}$-Al structure, correct initial current $J(t=0)$ and asymptotic current $J(t\rightarrow\infty)$ are also obtained in Al-${\rm C}_{60}$-Al structure. In addition, there are also rapidly oscillations at short times after the switch although the oscillation is not as rapid as that in the Al-${\rm C}_{5}$-Al structure. Furthermore, similar to Al-${\rm C}_{5}$-Al structure, in gradually reaching the steady-state values, the current oscillates with a frequency proportional to $\|V_{\alpha}\|$ but its decay rate is much slower than that in Al-${\rm C}_{5}$-Al structure. It indicates that there are much more quasi-resonant state that contribute to the transient current in Al-${\rm C}_{60}$-Al structure which is reasonable considering the complex electronic structure of isolated ${\rm C}_{60}$. In the following, we will analyze in detail how the current decays for the Al-${\rm C}_{60}$-Al structure. Figure 7: (Color online) Time dependent current $J(t)$ corresponding to the upward pulse [panel (a) and (b)] and the downward pulse [panel (c) and (d)] in Al-${\rm C}_{60}$-Al device for different $V_{\alpha}$. In panel (a) and (c), $V_{R}=-V_{L}=0.001a.u.$. In panel (b) and (d), $V_{R}=-V_{L}=0.01a.u.$. Same to Fig.6, the red (gray in print) dashed lines in panels indicate asymptotic current $J(t\rightarrow\infty)$. The long time AC current or detailed short time AC current are shown in inset of panels. Physically, decay time of current corresponds to the width of the quasi-bound state. In molecular devices, because the linewidth function ${\bf\Gamma}(\epsilon)$ are complex and energy dependent matrix, we can’t extract characteristic time scale directly from $1/{\bf\Gamma}$. As such, the transmission coefficient $T(\epsilon)$ is needed to understand the resonant state and corresponding characteristic time scale. In Fig.8(a), we plot transmission coefficient $T(\epsilon)$ in the energy range from the energy band bottom to the Fermi energy for Al-${\rm C}_{60}$-Al structure at zero bias. Here, the sharp peaks [some of them, see red crossed signed peaks in Fig.8(a)] correspond to resonant states with large lifetimes. At a particular resonant state, the incoming electron can dwell for a long time, which contributes to a much more slowly decaying current than other non-resonant states. In Fig.8(b), (c) and (d), we amplify the first, second and forth labeled quasi-resonant transmission, respectively, where the peaks’ width $\Gamma_{peak}\sim 10^{-5}a.u.$ are indicated, corresponding to a decay time $\tau\sim 2400fs$ from the expression $\Gamma_{peak}t=1$. In Fig.8(e)-(g), corresponding to different $\epsilon$ where the resonant peaks in Fig.8(b)-(d) are located, we plot long time behavior of current element $J_{L}(\epsilon)$. Here $J_{L}(\epsilon)$ is the time dependent current for each energy $\epsilon$, the integration over which gives the final current $J_{\alpha}(t)$. We can see that for each resonant state the current $J_{L}(\epsilon)$ keeps oscillating in a long time comparable to the decay time $\tau\sim 2400fs$. Furthermore the intensity of the oscillation $\Delta J\sim 0.2\mu A$ is not very small comparing to the DC signal $J_{dc}=5.1\mu A$. Figure 8: (Color online) Panel (a): transmission coefficient $T(\epsilon)$ in the energy range from the energy band bottom to the Fermi energy. In the whole energy range, there are some resonant states corresponding to the very sharp transmission coefficient $T(\epsilon)$, as we have indicated (see red cross) and labeled (by 1, 2, 3 and 4) in panel (a), some of them contribute to the current at long time. We amplify the first, second and forth labeled resonant transmission in panel (b), (c) and (d), respectively. In panel (e)-(g), we plot the long time behavior of current $J_{L}(\epsilon)$ at a fixed $\epsilon$ for the first, second and forth resonant states. The external voltage $V_{\alpha}=0.001a.u.$. After integration over energy, these slowly decaying currents $J_{L}(\epsilon)$ due to the resonant states may cancel to each other partially due to the difference in their phases. However, we should keep in mind that it is these resonant peaks that may give rise to convergence problem. Hence in the calculation, we should first scan the equilibrium and non-equilibrium transmission coefficient (100,000 energy points for example) to resolve sharp resonant peaks in the whole energy range from minimum energy to Fermi energy. Then, for each sharp resonant peak, enough (100 for example) energy points should be chosen to converge the integration of the current $J_{L}(\epsilon)$ over $\epsilon$, i.e., $\int d\epsilon J(\epsilon)$. For the non-resonant state, i.e., the smoothly changed region in $T(\epsilon)$, the current $J(\epsilon)$ are integrated using less energy points. As we have discussed that the resonant states are important for the transient current and they must be carefully treated in calculation. However, in the calculation of the effective Green’s function $\tilde{\bf G}^{r/a,0/V}$, a small imaginary part that is usually added to the real energy $\epsilon\rightarrow\epsilon+i\eta$ to help resolving the retarded or advanced self-energies. This in turn introduces pseudo resonant states. In order to eliminate the pseudo resonant state in effective Green’s function $\tilde{\bf G}^{r/a,0/V}$ [Eqs.(41,42)], one has to calculate the self-energy by setting $\eta=0$ and resolve the retarded or advanced self-energies with the aid of the group velocity $v_{k}=(\partial E(k)/\partial k)$.Sanvito ## VI conclusion By orthogonalizing the Hamiltonian expressed in the nonorthogonal basis and considering the singularity of self-energy ${\Sigma}^{r/a}(t,t^{\prime})$ at $t=t^{\prime}$, we have generalized the solution [ developed in Ref.Maciejko, ] of the transient current driven either by a downward step voltage pulse or by a upward step pulse. This generalized result can be applied to both the quantum dot model and real molecular device. Based on the exact solution given in Ref.Maciejko, , we derived two approximate formulas that are suitable for numerical calculation of the transient current for molecular devices. We have tested our approximate formula in a quantum dot system where exact numerical solution exists. For the quantum dot system, we chose a Lorentzian linewidth (beyond wideband limit) and compared the time-dependent current calculated using both exact formula and our approximate formula. We found that for the opposite voltage $V_{L}(t)=-V_{R}(t)$, the results obtained from the exact formalism and two approximate scheme agree very well with each other especially in the downward pulse case. For the nonsymmetric voltage $V_{L}(t)=V(t)$, $V_{R}(t)=0$ or $V_{L}(t)=0$, $V_{R}(t)=-V(t)$, the second approximation is better. This shows that our approximate formulas captured the essential physics of the transient current. In addition, it gives the correct initial current at $t=0$ and correct asymptotic current at $t\rightarrow\infty$. Since we have reduced the calculation from triple integral to single integral over the energy, the approximated approach reduces the computational cost drastically and it can be easily implemented in first principles calculation for molecular devices. To demonstrate this, we calculated the transient current using the first approximated scheme with an opposite voltage $V_{L}(t)=-V_{R}(t)$ for two molecular structures: Al-${\rm C}_{5}$-Al and Al-${\rm C}_{60}$-Al. Different from the quantum dot system, because of the complex electronic structure in molecular devices, transient currents oscillate rapidly in the first a few or tens fs as the bias voltage is switched, then gradually approach to the steady-state values. Furthermore, due to the resonant state in molecular devices, transient currents have a very long decay time $\tau$. ## Appendix A orthogonality relation for the nonorthogonal basis For a system described by $H$, the time independent eigenvalue equation is written as: $\displaystyle H\|n\rangle$ $\displaystyle=$ $\displaystyle E_{n}\|n\rangle$ (74) the eigenvectors $\|n\rangle$ form an orthogonal complete basis set. However, in many systems such as a molecular device connected to external leads, the basis set constructed by eigen vectors is not convenient. We usually expand the eigen vector $\|n\rangle$ in other basis $\|\mu\rangle$, which is non- orthogonal complete set (or nearly complete). $\displaystyle\|n\rangle\simeq\sum_{\mu}\|\mu\rangle\langle\mu\|n\rangle$ (75) the eigenvalue equation now becomes $\displaystyle\sum_{\mu}H\|\mu\rangle\langle\mu\|n\rangle$ $\displaystyle=$ $\displaystyle E_{n}\sum_{\mu}\|\mu\rangle\langle\mu\|n\rangle$ $\displaystyle\sum_{\mu}\langle\nu\|H\|\mu\rangle\langle\mu\|n\rangle$ $\displaystyle=$ $\displaystyle E_{n}\sum_{\mu}\langle\nu\|\mu\rangle\langle\mu\|n\rangle$ $\displaystyle\sum_{\mu}{\bf H}_{\nu\mu}{\bf\Psi}_{\mu}^{n}$ $\displaystyle=$ $\displaystyle E_{n}\sum_{\mu}{\bf S}_{\nu\mu}{\bf\Psi}_{\mu}^{n}$ (76) where ${\bf S}_{\nu\mu}=\langle\nu\|\mu\rangle$. In the matrix form, we have ${\bf H}{\bf\Psi}^{n}=E_{n}{\bf S}{\bf\Psi}^{n}$. If we use the self-energy to replace the effect of leads the effective Hamiltonian for the open system becomes ${\bf H}={\bf H}_{0}+{\bf\Sigma}^{r}$. Since the effective Hamiltonian is not Hermitian, we can define the adjoint operator ${\bf H}^{\dagger}={\bf H}={\bf H}_{0}+{\bf\Sigma}^{a}$ and corresponding eigen-equation becomes ${\bf H}^{\dagger}\|\phi_{n}\rangle=E^{}_{n}{\bf S}\|\phi_{n}\rangle$. Then $\displaystyle{\bf\Phi}^{m,\dagger}{\bf H}{\bf\Psi}^{n}$ $\displaystyle=$ $\displaystyle E_{n}{\bf\Phi}^{m,\dagger}{\bf S}{\bf\Psi}^{n},$ (77) $\displaystyle{\bf\Psi}^{n,\dagger}{\bf H}^{\dagger}{\bf\Phi}^{m}$ $\displaystyle=$ $\displaystyle E^{}_{m}{\bf\Psi}^{n,\dagger}{\bf S}^{\dagger}{\bf\Phi}^{m}$ (78) Taking hermitian conjugate of Eq.(78), $\displaystyle{\bf\Phi}^{m,\dagger}{\bf H}{\bf\Psi}^{n}$ $\displaystyle=$ $\displaystyle E_{m}{\bf\Phi}^{m,\dagger}{\bf S}{\bf\Psi}^{n}$ (79) From (77) and (79), we have $\displaystyle{\bf\Phi}^{n,\dagger}{\bf S}{\bf\Psi}^{m}=C_{m}\delta_{nm}$ (80) For the normalized wave function $\|\psi_{n}\rangle$ and $\|\phi_{n}\rangle$, $\displaystyle{\bf\Phi}^{\dagger}{\bf S\Psi}={\bf I}$ (81) It is the usual orthogonality relation for eigenvectors expressed in a nonorthogonal basis set. For an hermitian Hamiltonian ${\bf H}={\bf H}^{\dagger}$, $\|\psi_{n}\rangle=\|\phi_{n}\rangle$, we have ${\bf\Psi}^{\dagger}{\bf S\Psi}={\bf I}.$ ## Appendix B Orthogonalize Hamiltonian expressed in nonorthogonal basis In this appendix, we will show how to construct a new orthogonal basis from the atomic real-space nonorthogonal basis. We will transform the original Hamiltonian ${\bf H}$ which is expressed in the nonorthogonal basis into Hamiltonian $\tilde{\bf H}$ expressed in the new orthogonal basis. Of course, instead of ${\bf S}$, the overlap matrix in the new basis will be ${\bf I}$. Denoting nonorthogonal basis $\|\mu\rangle$ and orthogonal basis $\|j\rangle$, they are related by unitary transform operator ${\bf\mathcal{U}}$ $\displaystyle\|{\mu}\rangle=\sum_{j}\|j\rangle\langle j\|{\mu}\rangle=\sum_{j}\|j\rangle{\bf\mathcal{U}}_{j\mu}$ $\displaystyle{\bf\mathcal{U}}_{j\mu}=\langle j\|{\mu}\rangle$ (82) where we have used the completeness of orthogonal basis $\|j\rangle$. Using the orthogonality $\langle i\|j\rangle=\delta_{ij}$ $\displaystyle\sum_{\mu\nu}\langle i\|\mu\rangle\langle\mu\|\nu\rangle\langle\nu\|j\rangle$ $\displaystyle=$ $\displaystyle\sum_{\mu\nu}{\bf{\mathcal{U}}}_{i\mu}{\bf S}_{\mu\nu}{\bf{\mathcal{U}}}^{\dagger}_{\nu j}=\delta_{ij}$ where we have used the completeness of nonorthogonal basis. In the matrix form, ${\bf{\mathcal{U}}}{\bf S}{\bf{\mathcal{U}}}^{\dagger}={\bf I}$. We can formally define ${\bf{\mathcal{U}}}={\bf S}^{-\frac{1}{2}},~{}~{}{\bf{\mathcal{U}}}^{\dagger}=\left[{\bf S}^{-\frac{1}{2}}\right]^{\dagger}.$ Then new Hamiltonian ${\tilde{\bf H}}$ expressed in basis $\|i\rangle$ can be expressed as: $\displaystyle{\tilde{\bf H}}_{ij}$ $\displaystyle=$ $\displaystyle\langle i\|H\|j\rangle$ (83) $\displaystyle=$ $\displaystyle\sum_{\mu\nu}\langle i\|\mu\rangle\langle\mu\|H\|\nu\rangle\langle\nu\|j\rangle$ $\displaystyle=$ $\displaystyle\sum_{\mu\nu}{\bf{\mathcal{U}}}_{i\mu}{\bf H}_{\mu\nu}{\bf{\mathcal{U}}}^{\dagger}_{\nu j}$ In the matrix form, ${\tilde{\bf H}}={\bf S}^{-\frac{1}{2}}{\bf H}\left[{\bf S}^{-\frac{1}{2}}\right]^{\dagger}$. We now discuss how to find the matrix ${\bf S}^{-\frac{1}{2}}$. Without loss generality, we assume the real overlap matrix ${\bf S}$ satisfies eigen function ${\bf SV}={\bf V}{\rm diag}(\lambda_{1},...,\lambda_{n})$ with the eigenvalues $\lambda_{1},...,\lambda_{n}$ and eigenvectors ${\bf V}=[v_{1},...,v_{n}]$. Since ${\bf S}$ is real and symmetric, the eigenvectors are real and orthogonal, and it thus holds that ${\bf V}^{\dagger}{\bf V}={\bf V}{\bf V}^{\dagger}={\bf I}$. Then $\displaystyle{\bf S}$ $\displaystyle=$ $\displaystyle{\bf V}{\rm diag}(\lambda_{1},...,\lambda_{n}){\bf V}^{\dagger}$ $\displaystyle=$ $\displaystyle{\bf V}{\rm diag}(\sqrt{\lambda_{1}},...,\sqrt{\lambda_{n}}){\bf V}^{\dagger}{\bf V}{\rm diag}(\sqrt{\lambda_{1}},...,\sqrt{\lambda_{n}}){\bf V}^{\dagger}$ It follows that $\displaystyle{\bf S}^{\frac{1}{2}}={\bf V}{\rm diag}(\sqrt{\lambda_{1}},...,\sqrt{\lambda_{n}}){\bf V}^{\dagger}$ (84) From ${\bf S}^{-\frac{1}{2}}{\bf S}^{\frac{1}{2}}={\bf I}$ and Eq.(84), we have $\displaystyle{\bf S}^{-\frac{1}{2}}{\bf V}{\rm diag}(\sqrt{\lambda_{1}},...,\sqrt{\lambda_{n}}){\bf V}^{\dagger}={\bf I}$ $\displaystyle{\bf S}^{-\frac{1}{2}}{\bf V}{\rm diag}(\sqrt{\lambda_{1}},...,\sqrt{\lambda_{n}}){\bf V}^{\dagger}{\bf V}{\rm diag}(\frac{1}{\sqrt{\lambda_{1}}},...,\frac{1}{\sqrt{\lambda_{n}}}){\bf V}^{\dagger}$ $\displaystyle=$ $\displaystyle{\bf S}^{-\frac{1}{2}}={\bf V}{\rm diag}(\frac{1}{\sqrt{\lambda_{1}}},...,\frac{1}{\sqrt{\lambda_{n}}}){\bf V}^{\dagger}$ (85) In general, the dimension of matrix ${\bf S}$ is infinity, we can’t calculate its eigenvalue $\lambda_{i}$ and eigenvector $v_{i}$ by diagonalizing ${\bf S}$. However, in the tight-binding representation, the state $\mu$ and $\nu$ hardly overlap when their separation is large enough in real space, i.e., ${\bf S}_{\mu\nu}\approx 0$ for most of off-diagonal elements. Considering the periodic properties in semi-infinite leads, we can select a block matrix which is large enough to include all the overlap between leads and central molecular regions. For the non-orthogonal basis including several unit cell of atomic leads as a buffer layer into the central scattering region is enough to get a good screening for dc transport calculation. In transforming the Hamiltonian to the orthogonal basis needed for ac transport calculation, however, it turns out that we have to include at least 10 unit cells of atomic leads into the central scattering region. Partly because the overlap of orthogonal basis has longer range than that of non-orthogonal basis. 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1011.2662	# Universal Conductance Fluctuations in Mesoscopic Systems with Superconducting Leads: Beyond the Andreev Approximation Yanxia Xing1,2 and Jian Wang1,∗ 1Department of Physics and the center of theoretical and computational physics, The University of Hong Kong, Hong Kong, China 2Department of Physics, Beijing Institute of Technology, Beijing 100081, China ###### Abstract We report our investigation of the sample to sample fluctuation in transport properties of phase coherent normal metal-superconductor hybrid systems. Extensive numerical simulations were carried out for quasi-one dimensional and two dimensional systems in both square lattice (Fermi electron) as well as honeycomb lattice (Dirac electron). Our results show that when the Fermi energy is within the superconducting energy gap $\Delta$, the Andreev conductance fluctuation exhibits a universal value (UCF) which is approximately two times larger than that in the normal systems. According to the random matrix theory, the electron-hole degeneracy (ehD) in the Andreev reflections (AR) plays an important role in classifying UCF. Our results confirm this. We found that in the diffusive regime there are two UCF plateaus, one corresponds to the complete electron-hole symmetry (with ehD) class and the other to conventional electron-hole conversion (ehD broken). In addition, we have studied the Andreev conductance distribution and found that for the fixed average conductance $\langle G\rangle$ the Andreev conductance distribution is a universal function that depends only on the ehD. In the localized regime, our results show that ehD continues to serve as an indicator for different universal classes. Finally, if normal transport is present, i.e., Fermi energy is beyond energy gap $\Delta$, the AR is suppressed drastically in the localized regime by the disorder and the ehD becomes irrelevant. As a result, the conductance distribution is that same as that of normal systems. ###### pacs: 72.80.Vp, 74.45.+c, 73.23.-b, 68.65.Pq ## I introduction It is well known that quantum interference leads to significant sample-to- sample fluctuations in the conductance at low temperatures. These fluctuations can be observed in a single sample as a function of external parameters such as the magnetic field since the variation of magnetic field has a similar effect on the interference pattern as the variation in impurity configuration. One of the fundamental problems of mesoscopic physics is to understand the statistical distribution of the conductance in disordered systemsbook ; Beenakker ; add . It has been established that in the diffusive regime, the conductance of any metallic sample fluctuates as a function of chemical potential, impurity configuration (or magnetic field) with a universal conductance fluctuation (UCF) that depends only on the dimensionality and the symmetry of the system.lee85 The UCF is given by ${\rm Var}(G/G_{0})=2/(16\beta)$, $2/(15\beta)$, $3/(16\beta)$, $5/(17\beta)$ for quantum dot (QD), quasi-one dimension (1D), two dimensions (2D) square and three dimensions cubic sample with $G_{0}=2e^{2}/h$. Here the index $\beta$ corresponds to circular orthogonal ensemble (COE) when the time-reversal and spin-rotation symmetries are present ($\beta=1$), circular unitary ensemble (CUE) if time-reversal symmetry is broken ($\beta=2$) and circular symplectic ensemble (CSE) if the spin-rotation symmetry is broken while time-reversal symmetry is maintained ($\beta=4$), respectively.lee85 In the crossover regime from diffusive to localized regimes, the conductance distribution was found to be a universal function that depends only on the average conductance for quasi-1D, 2D, and QD mesoscopic systems and for $\beta=1,2,4$.saenz2 ; ucf4 In the localized regime, the conductance distribution seems to be independent of dimensionality and ensemble symmetry.ucf4 In the presence of a superconducting lead, using random matrix theory (RMT), the conductance fluctuations in the mesoscopic normal and superconductor hybrid systems have been studied in the diffusive regime for quasi-1D systemsBeenakker ; Beenakker1 ; Beenakker2 ; Takane and QD system.randmatri It was found that the UCF in a COE system shows approximately a twofold increase over the normal systems, i.e., ${\rm rms}(G_{NS})\simeq 2~{}{\rm rms}(G_{N})$.Takane ; randmatri Different from the normal conductor, in the presence of superconducting lead, electron-hole degeneracy (ehD) plays a similar role of ”symmetry”. UCF assumes different value depending on whether ehD is broken or not. According to RMT,Beenakker the Andreev conductance fluctuation ${\rm rms}(G_{NS})=\sqrt{4.3}~{}{\rm rms}(G_{N})$ (with ehD) and ${\rm rms}(G_{NS})=\sqrt{4}~{}{\rm rms}(G_{N})$ (with ehD broken) were predicted. Up to now, however, most of the investigations on Andreev conductance fluctuation have been done for systems with ehD ($\epsilon=0$) and low energy regime ($\Delta\ll E_{c}$ where $E_{c}$ is Thouless energy). There is not yet a numerical study on the NS hybrid system where ehD is broken. In fact, for the existing studies on the NS hybrid system with ehD, there is no consensus on the theoretical predicted value of ${\rm rms}(G_{NS})$. Specifically, concerning the increase factor $\alpha_{0}$ in “${\rm rms}(G_{NS})=\alpha_{0}~{}{\rm rms}(G_{N})$”, a diagrammatic theory predicted $\sqrt{6}$,Takane1 and a numerical calculation using tight binding model gave $\alpha_{0}=\sqrt{4}$,Takane and the random matrix theory indicated $\alpha_{0}=\sqrt{4.3}$Beenakker and $\sqrt{4.5}$.Beenakker1 Recently, graphene based normal-metal-superconductor (GNS) systems were intensively studied because good contacts between the superconductor electrodes and graphene have been realized experimentally.ref16 ; ref17 In a conventional (quadratic energy dispersion relation) normal-metal- superconductor (CNS) system, the usual Andreev reflection (AR) occurs.ref15 For GNS systems, AR can be either intravalley or intervalley, which are called Andreev retroreflection (ARR) and specular Andreev reflection (SAR), respectively.ref5 When the excitation energy $2\epsilon$ is smaller than that of incident energy relative to Dirac Point $E_{F}-E_{0}$, ARR happens, otherwise SAR occurs. At the transition point $2\epsilon=E_{F}-E_{0}$ between ARR and SAR, the reflection angle $\theta$ (measured relative to the NS junction normal) jumps from $+90^{\circ}$ to $-90^{\circ}$,Beenakker the shot noise vanishes and the Fano factor has a universal value.Wangbg In general, SAR differs from ARR or conventional AR (CAR) where an extra phase $\pi$ which can be observed in the quantum interference of the two SAR reflections.Xing So far most of investigations on UCF focus on the Fermi electrons (quadratic dispersion relation) with the zero or low energy and less attention is paid on the Dirac electrons. In addition there is no numerical work reporting Andreev conductance fluctuation when ehD is broken. It would be interesting to ask the following questions. What happens to UCF for GNS systems? Is it the same as that in CNS systems? Is there any difference between ARR and SAR? Which theoretically predicted value of UCF for the quasi-1D CNS system (with ehD) is favored? What happened when ehD is broken? What about the conductance distributions in these systems? It is the purpose of this paper to address these questions. In this paper, using the tight-binding model, we carry out a theoretical study on the sample to sample fluctuation in transport properties of phase coherent systems with normal metal-superconductor heterojunction. In view of the possible difference among CAR, ARR and SAR, we consider both the CNS systems using the square lattice and GNS system using the honeycomb lattice. Extensive numerical simulations on quasi-1D and 2D systems in the presence of a superconducting lead show that when the Fermi energy is within the superconducting gap $E_{F}<\Delta$, UCF roughly doubles the value in the absence of the superconducting lead. This is the case for both CAR in CNS system and ARR and SAR in GNS system. So there is no distinct difference between ARR and SAR. Besides, concerning ehD in the NS hybrid system, new universal classes are present in agreement with the prediction of RMT.Beenakker Two plateaus of UCF were found in our numerical results, one corresponds to the complete electron-hole symmetryehsymm1 class (with ehD) and the other to conventional electron-hole conversion (with ehD broken). It was found that the case of “ehD broken” decreases the value of UCF, again in agreement with the theoretical analysis.Beenakker Specifically, in the quasi-1D systems, ${\rm rms}(G_{NS})/{\rm rms}(G_{N})$ for both Fermi and Dirac electrons is $2.07\pm 0.04$ that is close to $\sqrt{4.3}$ when ehD is present while when ehD is broken it is $1.99\pm 0.08$ that is close to $\sqrt{4}$. For 2D systems, when ehD is present, ${\rm rms}(G_{NS})/{\rm rms}(G_{N})$ is $1.91\pm 0.07$ for Fermi electrons and $1.96\pm 0.07$ for Dirac electrons while it is $1.82\pm 0.08$ when ehD is broken for both Fermi electrons and Dirac electrons. Furthermore, the different conductance distributions $P(G)$ for the fixed average conductance $\langle G\rangle$ also indicate this new symmetry class in localized regime. We also point out that the new universality class due to the ehD is quite different from the conventional ensemble symmetries. It was shown numerically that the conductance distribution $P(G)$ in the deep localized regime for normal systems is a universal function which depends only on the average conductance $\langle G\rangle$ but not on the Fermi energies as well as other parameters.ucf4 In addition, it does not seem to depend on the ensemble symmetry and dimensionality of the system. In the presence of the superconducting lead, our numerical results for 2D systems with $\beta=1$ show that the conductance distribution is still an universal function that depends only the average conductance $\langle G\rangle$. Different from normal system, however, it depends on whether the system has the ehD. Finally, when $E_{F}$ is above $\Delta$, normal transport is present. We found that the AR is suppressed by the disorder especially in the localized regime where normal transmission dominates transport processes. In this case, the ehD is irrelevant and the same universal conductance distribution is found as that in the normal systems in the localized regime. The rest of the paper is organized as follows. In Sec. II, with the tight- binding representation, the model system including central disordered region and attached ideal normal lead and superconducting lead is introduced. The formalisms for calculating the conductance and fluctuation of conductance are then derived. Sec. III gives numerical results along with detailed discussions. Finally, a brief summary is presented in Sec. IV. ## II model and Hamiltonian Figure 1: (Color online) Schetch of CNS [panel (a)] and GNS [panel (b)] system, in which ideal superconducting lead (left, orange), normal lead (right, blue) and disordered normal scattering region (shadowed blue region) are concluded. The scattering theory of electronic conduction is developed by Landauer,Landauer Imry,Imry and B$\ddot{u}$ttiker.Buttiker It provides a complete description of quantum transport in the system without electron- electron interactions. A mesoscopic conductor can be modeled by a phase- coherent disordered region connected by ideal leads (without disorder) to two electron reservoirs (normal metal or superconductor), which are in equilibrium at zero temperature with fixed electrochemical potential (or Fermi energy) $E_{F}$. Here we assume that the central scattering region is normal region, the same as the right normal lead. Then the total system Hamiltonian $H=H_{S}+H_{N}+H_{T}$ (1) where $H_{S}$, $H_{N}$ and $H_{T}$ are the Hamiltonian of superconducting lead (orange region in Fig.1), semi-infinite normal ribbon (blue region in Fig.1) and tunneling between the normal region and superconducting terminal, respectively. Two kinds of structure were considered in this paper: the structure with quadratic energy dispersion [square lattice, Fig.1(a)] and structure with conical energy spectrum [honeycomb lattice, Fig.1(b)]. In the absence of the superconductor, the whole system $H_{0}$ including $H_{S}$, $H_{N}$ and $H_{C}$ can be written in the tight-binding representation:ref19 ; ref20 $\displaystyle H_{0}=\sum_{\bf i}(E_{0}+\delta\epsilon_{\bf i})a^{\dagger}_{\bf i}a_{\bf i}-\sum_{<{\bf ij}>}ta_{\bf i}^{\dagger}a_{\bf j}$ (2) where ${\bf i}=(i_{x},i_{y})$ is the index of the discrete square lattice or honeycomb lattice site which is arranged as in inset of Fig.1. Here $a_{\bf i}$ and $a_{\bf i}^{\dagger}$ are the annihilation and creation operators at the discrete site ${\bf i}$. $E_{0}$ is the constant on-site energy. In the square lattice, $E_{0}$ is center of energy band, and in honeycomb lattice, $E_{0}$ is the energy reference point (the Dirac Point). $\delta\epsilon_{\bf i}$ is random on-site potential which is nonzero only in the center region to simulate the disordered scattering region. Here $\delta\epsilon_{\bf i}$ is uniformly distributed with $\delta\epsilon_{\bf i}=[-w/2,w/2]$ where $w$ is disorder strength. The data for fluctuations are obtained by averaging over up to 10,000 disorder configurations and the data for distribution are obtained over 1,000,000 disorder configurations. The second term in Eq.(2) is the nearest neighbor hopping with hopping elements $t$ and “$<>$” denotes the sum over the nearest sites. Due to the superconductor, it is convenient to write the Hamiltonian $H$ in the Nambu representation,Nambu . In this representation the Fermi energy of the right normal lead in equilibrium (at zero bias) is set to be the superconductor condensate. It is conventionally set to zero. As a result, the spin up electrons and the spin down holes have the positive and negative energy, respectively. Taking this into account the Hamiltonian (2) is cross multiplied by spin representation. $H_{N}$ and $H_{T}$ in Eq.(1) can be rewritten as $H_{0,N/T}\otimes\sigma_{z}$ and $H_{S}=\left(\begin{array}[]{cc}H_{0,S}&\tilde{\Delta}\\\ \tilde{\Delta}^{}&-H_{0,S}\end{array}\right)$ (3) where $\tilde{\Delta}=\Delta e^{i\varphi}$ is the energy gap or the pair potential of the semi-infinite superconducting lead. Here we can assume $\tilde{\Delta}=\Delta$ to be a real parameter by selecting a special phase of the superconductor lead in our calculation.realGap In the calculation, for simplicity we set external voltage in the normal and superconducting terminal as $V_{N}=V$, $V_{S}=0$. The current flowing from the normal lead can be calculated from the Landauer-B$\ddot{u}$ttiker formula:footnote $\displaystyle J_{N}=J^{e}_{N}-J^{h}_{N}$ $\displaystyle J^{e/h}_{N}=\pm\frac{e}{\hbar}\int\frac{dE}{2\pi}~{}\left\\{T_{e/h}(E)[f_{\pm}(E)-f_{0}(E)]\right.$ $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.T_{A}(E)[f_{\pm}(E)-f_{\mp}(E)]\right\\}$ (4) where $e$ is the electron charge, $f_{0}(E)=\left[e^{E/k_{B}\mathcal{T}}+1\right]^{-1}$ is the Fermi distribution in the superconducting lead, $f_{\pm}(E)=\left[e^{(E\mp eV_{N})/k_{B}\mathcal{T}}+1\right]^{-1}$ are the Fermi distribution functions in the normal terminal for the electrons and holes, respectively. $T_{e/h}$ is the transmission coefficient that the particles incident from superconducting lead traverse to the normal terminal as electrons/holes and $T_{A}$ is AR coefficient representing the reflection probability that the incident electrons from the normal terminal are reflected as holes or vice versa. Note that the two processes are symmetric and have the same AR coefficient $T_{A}$. $T_{e/h}$ and $T_{A}$ are calculated from $\displaystyle T_{e}={\rm Tr}\\{\Gamma^{N}_{\uparrow\uparrow}[G^{r}\Gamma^{S}G^{a}]_{\uparrow\uparrow}\\},~{}~{}T_{h}={\rm Tr}\\{\Gamma^{N}_{\downarrow\downarrow}[G^{r}\Gamma^{S}G^{a}]_{\downarrow\downarrow}]\\}$ $\displaystyle T_{A}={\rm Tr}[\Gamma^{N}_{\uparrow\uparrow}G^{r}_{\uparrow\downarrow}\Gamma^{N}_{\downarrow\downarrow}G^{a}_{\downarrow\uparrow}]={\rm Tr}[\Gamma^{N}_{\downarrow\downarrow}G^{r}_{\downarrow\uparrow}\Gamma^{N}_{\uparrow\uparrow}G^{a}_{\uparrow\downarrow}]$ (5) the line-width function $\Gamma^{N/S}(E)=i[\Sigma_{N/S}^{r}(E)-\Sigma_{N/S}^{r\dagger}(E)]$. The Green’s function $G^{r}(E)=[G^{a}(E)]^{\dagger}=[EI- H_{C}-\Sigma^{r}_{N}(E)-\Sigma^{r}_{S}(E)]^{-1}$ where $H_{C}$ is Hamiltonian matrix of the central scattering region and $I$ is the unit matrix with the same dimension as that of $H_{C}$, $\Sigma_{l=N,S}^{r}$ is the matrix of retarded self-energy from the normal/superconducting lead with the only nonzero elements in the sub-block that are neighbor of normal or superconducting lead. The self-energy is calculating according to $\Sigma^{r}_{l}=H_{Cl}g^{r}H_{lC}$ where $H_{Cl}$ ($H_{lC}$) is the coupling from central region (leads) to leads (central region) and $g^{r}$ is the surface retarded Green’s function of semi-infinite lead which can be calculated using a transfer matrix method.transfer Due to electron-hole symmetry, $T_{e}(E)=T_{h}(-E)$ and $T_{A}(E)=T_{A}(-E)$, which leads to $J_{N}=2J^{e}_{N}=-2J^{h}_{N}$. At zero temperature limit, the energy dependent conductance can be expressed as: $\displaystyle G_{NS}(E_{F})$ $\displaystyle=$ $\displaystyle d(J^{e}_{N}-J^{h}_{N})/dV$ (6) $\displaystyle=$ $\displaystyle\frac{e^{2}}{h}\left\\{[T_{e}(E_{F})+T_{h}(-E_{F})]+4T_{A}(E_{F})\right\\}$ $\displaystyle=$ $\displaystyle\frac{2e^{2}}{h}\left[T_{e/h}(\pm E_{F})+2T_{A}(E_{F})\right].$ When the incident energy $E_{F}<\Delta$, there is no normal quasi-particle transport $T_{e/h}=0$ and only AR contributes to conductance $G$. We will focus mainly on this quantity in this paper. In this case, the conductance fluctuation defined as ${\rm rms}(G)=\sqrt{\langle[G-\langle G\rangle]^{2}\rangle}$ becomes $\displaystyle{\rm rms}(G)=\frac{4e^{2}}{h}\sqrt{\langle T_{A}^{2}\rangle-\langle T_{A}\rangle^{2}}$ (7) where $\langle...\rangle$ denotes averaging over an ensemble of samples with different disorder configurations of the same strength $w$. When $E_{F}$ is beyond superconducting energy gap $\Delta$, normal transmission $T_{e/h}$ is present, conductance variance now consists of three components: (1). the Andreev related fluctuation ${\rm Var}(G)_{\rm Andr}$ from AR coefficient $T_{A}$. (2). the normal fluctuation ${\rm Var}(G)_{\rm Norm}$ from normal transmission coefficient $T_{e/h}$. (3). the cross term ${\rm Var}(G)_{\rm cross}$. They are expressed as $\displaystyle{\rm Var}(G)={\rm Var}(G)_{\rm Andr}+{\rm Var}(G)_{\rm Norm}+{\rm Var}(G)_{\rm cross}$ $\displaystyle=[\frac{e^{2}}{h}]^{2}\left[\left\langle(4\delta T_{A})^{2}\right\rangle+\left\langle(\delta T_{N})^{2}\right\rangle+8\left\langle\delta T_{A}\delta T_{N}\right\rangle\right]$ (8) where $\delta T_{A}=T_{A}-\langle T_{A}\rangle$, $\delta T_{N}=(T_{e}+T_{h})-\langle T_{e}+T_{h}\rangle$. ## III results and discussion In the numerical calculations, the energy is measured in the unit of the nearest coupling elements $t$. For the square lattice, $t=\frac{\hbar^{2}}{2m^{}a^{2}}$ with $m^{}$ the effective electron mass and $a$ the lattice constant. For the honeycomb lattice, $t=\frac{2}{3b}\hbar v_{F}$ with the carbon-carbon distance $b=0.142nm$ and the Fermi velocity $v_{F}=0.89\times 10^{6}ms^{-1}$. The size of the scattering region $N\times M$ is described by integer $N$ and $M$ corresponding to the width and length, respectively. For example in Fig.1, the width $W=Na$ with $N=3$, the length $L=Ma$ with $M=5$ in the panel (a), and the width $W=N\times 3b$ with $N=3$, the length $L=M\times\sqrt{3}b$ with $M=7$ in the panel (b). Table 1: The parameter $E_{F}$, $E_{0}$, $\Delta$ used in the square lattice model and honeycomb model. The different columns are corresponding to the different transport processes denoted by ‘AR’, ‘ARR’, ‘SAR’ ,‘NT’ and so on. Here, ‘AR’ is for the pure conventional AR assisted tunneling processes (only conventional AR exists) in square lattice. ‘ARR‘ and ‘SAR’ denotes the pure ARR assisted process and pure SAR assisted process in honeycomb lattice, respectively. ‘NT’ is for the transport beyond the superconducting Gap where NT can also contribute to the transport processes. sq \| \| \| \| \| \| \| \| \| \| \| ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- AR \| $E_{F}$ \| $E_{0}$ \| $\Delta$ \| AR \| $E_{F}$ \| $E_{0}$ \| $\Delta$ \| NT \| $E_{F}$ \| $E_{0}$ \| $\Delta$ 1 \| 0 \| 2.1 \| 0.1 \| 4 \| 0.2 \| 2.2 \| 0.3 \| 1 \| 0.2 \| 2.2 \| 0.3 2 \| 0 \| 2.3 \| 0.1 \| 5 \| 0.3 \| 2.3 \| 0.4 \| 2 \| 0.3 \| 2.3 \| 0.4 3 \| 0 \| 2.4 \| 0.1 \| 6 \| 0.4 \| 2.4 \| 0.5 \| 3 \| 0.4 \| 2.4 \| 0.5 hc \| \| \| \| \| \| \| \| \| \| \| ARR \| $E_{F}$ \| $E_{0}$ \| $\Delta$ \| SAR \| $E_{F}$ \| $E_{0}$ \| $\Delta$ \| NT \| $E_{F}$ \| $E_{0}$ \| $\Delta$ 1 \| 0 \| 0.6 \| 0.1 \| 1 \| 0.6 \| 0.0 \| 0.7 \| 1 \| 0.7 \| 0 \| 0.5 2 \| 0 \| 0.7 \| 0.1 \| 2 \| 0.6 \| 0.1 \| 0.7 \| 2 \| 0.7 \| 0.1 \| 0.5 3 \| 0 \| 0.8 \| 0.1 \| 3 \| 0.7 \| 0.0 \| 0.8 \| 3 \| 0.7 \| 0 \| 0.3 4 \| 0 \| 0.9 \| 0.1 \| 4 \| 0.7 \| 0.1 \| 0.8 \| 4 \| 0.7 \| 0.1 \| 0.3 5 \| 0.1 \| 0.7 \| 0.2 \| \| \| \| \| 5 \| 0.7 \| 0 \| 0.1 6 \| 0.1 \| 0.8 \| 0.2 \| \| \| \| \| 6 \| 0.7 \| 0.1 \| 0.1 As documented in the literature, in order to get the saturated UCF plateaus,saenz2 ; ucf4 ; LiDafang the number of transmission channels for incoming electron should be large enough in the numerical calculation. We denote $N_{c}$ as the chain number which determines directly the number of channels. $N_{c}$ is defined in the following way: take Fig.1 as an example, in panel (a), $N_{c}=3$ and $N_{c}=6$ in panel (b). In 2D systems we set $N_{c}=40$ and $60$. For quasi 1D systems we use only $N_{c}=40$ because it is more computational demanding than 2D systems. To get a larger channel number, the incident energy $E_{F}$ should be set away from the bottom of energy band $E_{b}$. In the square lattice, to mimic the parabolic energy spectrum for Fermi electrons, the constraints for incident energy $E_{F}<E_{b}+2t$ and Andreev reflected energy $-E_{F}<E_{b}+2t$ are needed. While for Dirac electrons in the honeycomb lattice, the absolute value of relative incident energy (to the Dirac point $E_{0}$) $\|E_{F}-E_{0}\|<t$ and relative Andreev reflected energy $\|-E_{F}-E_{0}\|<t$ are set. In Table.1, we list all the parameters used in the following calculations including the incident energy $E_{F}$, the superconducting gap $\Delta$ and the on-site energy $E_{0}$ which is the center of energy band for the square lattice and the Dirac Point for the honeycomb lattice. From these parameters in the clean system with NS heterojunction, we can easily calculate the channel number for electron or hole, AR coefficient $T_{A}$ and the normal transmission coefficient of electron or hole $T_{se/sh}$ for Fermi energy beyond superconducting gap. At the same time, we can also get the normal transmission coefficient $T_{ne/nh}$ in normal system without the superconducting lead. ### III.1 Conductance fluctuation and conductance distribution in the diffusive regime Figure 2: (Color online) panel(a) and panel(b): in the presence of superconducting lead, conductance fluctuation ${\rm rms}(G)$ vs average conductance $\langle G\rangle$ in the square lattice for $N_{c}=40$ and $N_{c}=60$, respectively. The symbol is labeled in first column in Tab.1 for the square lattice case denoted by ‘sq’. The red dotted lines indicate two plateaus in the values (with the unit of $0.866e^{2}/h$) of $1.94\pm 0.03$ and $1.85\pm 0.05$ in panel (a) and $1.98\pm 0.02$ and $1.89\pm 0.03$ in panel (b). For comparison, corresponding to panel (a) and panel (b), in panel (c) and (d), we plot ${\rm rms}(G)$ vs $\langle G\rangle$ in the absence of superconducting lead, i.e., $\Delta=0$, respectively. The plateaus in the values of $1.02\pm 0.02$ and $1.04\pm 0.02$ in the unit of $0.866e^{2}/h$ are indicated in panel (c) and panel (d). The system size: $N_{c}=40$ corresponds to width $W=40a$, considering the square shape sample, we set $L=40a$. For $N_{c}=60$, we have $W=60a$, $L=60a$. Figure 3: (Color online) Same as Fig.2 except the model is quasi-1D square lattice with chain number $N_{c}=40$. The system size: width $W=40a$, length $L=1000a$. In the unit of $0.73e^{2}/h$, two plateaus with the values of $2.01\pm 0.02$ and $1.93\pm 0.03$ in panel (a) and a single plateau in the value of $0.97\pm 0.01$ is indicated in panel (b). Figure 4: (Color online) ${\rm rms}(G)$ contributed by ARR [panel(a)] and SAR [panel(b)] vs $\langle G\rangle$ in 2D honeycomb lattice for $N_{c}=40$ [open symbols] and $N_{c}=60$ [symbols with ‘-’]. The symbols are labeled in second column in Tab.1 for the honeycomb lattice case denoted by ‘hc’. The red dotted lines indicate two plateaus with the values of $1.96\pm 0.03$ and $1.82\pm 0.02$ in the unit of $0.866e^{2}/h$ in panel (a) and a single plateau with the value of $1.82\pm 0.02$ in panel (b). Panel (c) and (d): ${\rm rms}(G)$ contributed by normal quasi-particle transmission $T_{ne}$ vs $\langle G\rangle$ in case of $\Delta=0$, corresponding to panel (a) and (b), respectively. The plateaus in the values of $1.00\pm 0.02$ in the unit of $0.866e^{2}/h$ are indicated in panel (c) and panel (d). The system size: for $N_{c}=40$ its width is equal to $W=60b$; considering the square shape sample, the length $L=35\sqrt{3}b$. Similarly, for $N_{c}=40$ its width is $W=90b$, $L=52\sqrt{3}b$. Figure 5: (Color online) Same as Fig.2 except the model is quasi-1D honeycomb lattice with chain number $N_{c}=40$. The system size: width $W=60b$, length $L=500\sqrt{3}b$. The red dotted lines indicate two plateaus with the values of $2.03\pm 0.03$ and $1.93\pm 0.03$ in the unit of $0.73e^{2}/h$ in panel (a), a single plateau with the value of $1.94\pm 0.04$ in panel (b), the value of $0.98\pm 0.02$ in panel (c) and panel (d). We first examine conductance fluctuations in the diffusive regime. In our calculation the size of 2D square lattice is set to be $40\times 40$ for $N_{c}=40$ and $60\times 60$ for $N_{c}=60$. The size of 2D honeycomb lattice is chosen to be $20\times 35$ for $N_{c}=40$ and $30\times 52$ for $N_{c}=60$. For quasi-1D systems, the size is chosen to be $40\times 1000$ in square lattice and $20\times 500$ in honeycomb lattice with $N_{c}=40$. In Fig.2 and 3, 4 and 5, we plot conductance fluctuations ${\rm rms}(G)$ vs the average conductance $\langle G\rangle$ in 2D square lattice, quasi-1D square lattice, 2D honeycomb lattice and quasi-1D honeycomb lattice, respectively. Each point in the figure is obtained by averaging over 10,000 configurations. Different parameters used in all figures are tabulated in Table.1. From Fig.2-5, we see following general behaviors. (1). in the localized regime where $\langle G\rangle<1$, all the curves collapse into a single curve indicating the universal behavior of the conductance distribution function.ucf4 (2). in the diffusive regime where $\langle G\rangle>1$, there is a plateau region for ${\rm rms}(G)$ where the fluctuation is nearly independent of average conductance $\langle G\rangle$ and other system parameters. This is the regime for the universal conductance fluctuation. The plateau value [labeled by red dotted line in top panels] is approximately twice the value of the known UCF values ${\rm rms}(G)=0.866e^{2}/h$ for 2D system and $0.73e^{2}/h$ for quasi-1D system [labeled by red dotted line in bottom panels]. This doubling seems to be true for both Fermi electrons (square lattice) [Fig.2 and Fig.3] and Dirac electrons (graphene system) [Fig.4 and Fig.5]. (3). There are two separate UCF plateaus for the AR assisted transport processes in the CNS system [Fig.2(a), Fig.3(a)] and the ARR assisted transport processes in GNS system [Fig.4(a),Fig.5(a)], while for the SAR assisted transport processes [panel (b)] there is only one UCF plateau. It appears that this difference in UCF can be used to distinguish ARR and SAR. However, it turns out to be incorrect when considering the ehD “symmetry”. In Fig.4(a) and Fig.5(a), Andreev conductance fluctuations corresponding to $E_{F}=0$ (with ehD) and $E_{F}\neq 0$ (ehD broken) from diffusive regime all the way to localized regime are plotted and two UCF plateaus associated to ehD “symmetry” are then indicated. For SAR in graphene systems, we have $\|E_{F}\|>\|E_{0}\|\neq 0$ (ehD broken). Fig.4(b) and Fig.5(b) then show only one UCF plateau. (4). Denoting the increase factor $\alpha_{0}$ through the relation ${\rm rms}(G_{NS})=\alpha_{0}~{}{\rm rms}(G_{N})$ [$G_{N}$ is shown in bottom panels in Fig.2-5] in the plateau region in diffusive regime, it is very different for square 2D system and quasi-1D system and slightly different for Fermion electrons and Dirac electrons. Specifically, our results for the quasi-1D systems for both Fermi electrons and Dirac electron is as follows: (a). when ehD is present ${\rm rms}(G_{NS})/{\rm rms}(G_{N})$ is $2.07\pm 0.04$ that is very close to $\sqrt{4.3}$. (b). when ehD is broken it is $1.99\pm 0.08$ that is close to $\sqrt{4}$. For 2D systems, when ehD is present, ${\rm rms}(G_{NS})/{\rm rms}(G_{N})$ is $1.91\pm 0.07$ for Fermi electrons and $1.96\pm 0.07$ for Dirac electrons. When ehD is broken it is $1.82\pm 0.08$ for both Fermi electrons and Dirac electrons. (5). For larger $\langle G\rangle$ (in the ballistic regime) the conductance fluctuation falls down quickly to zero. This is because the number of conducting channels $N_{c}$ is finitesaenz2 ; LiDafang ; ucf4 . The width of plateau region is longer with a larger $N_{c}$. In the limit of the infinite $N_{c}$, the plateaus of conductance fluctuation will extend to infinite. We now take a closer look at each figure discussed above. In the top panels of Fig.2-5, we can see that all curves of ${\rm rms}(G)$ vs $\langle G\rangle$ collapse into universal curves that are slightly separated in the region of $1<\langle G\rangle<10$. To make the discussion of separate UCF plateaus quantitative, we plot ${\rm rms}(G)$ vs small $\langle G\rangle$ ($\langle G\rangle<10$) in Fig.6(a), (b), (c) and (d) corresponding to Fig.2, Fig.3, Fig.4 and Fig.5. In Fig.6, we clearly see two separate UCF in the regime where $1<\langle G\rangle<10$. For each UCF plateau, the conductance fluctuation ${\rm rms}(G)$ vs average conductance $\langle G\rangle$ is a universal function, i.e., it is independent of system parameters such as $E_{F}$, $E_{0}$, $\Delta$, system size and so on and depends only on $\langle G\rangle$. In fact, not only the ${\rm rms}(G)$ (the second moment), the third, forth, …, and higher moments are universal function of $\langle G\rangle$. This means that the conductance distribution $P(G)$ is a universal function that depends only on the average conductance $\langle G\rangle$ in addition to the symmetry and dimensionality of the system. Figure 6: (Color online) Corresponding to Fig.2, Fig.3, Fig.4 and Fig.5, ${\rm rms}(G)$ vs small $\langle G\rangle$ ($<10$) are plotted in panel(a), (b), (c) and (d), respectively. Figure 7: (Color online) In the diffusive regime, corresponding to eight selected parameters with $\|E_{F}\|<\Delta$ from Tab.1, the conductance distribution $P(G)$ obtained from 1,000,000 configurations is plotted for the fixed $\langle G\rangle\simeq 3$ in square [panel (a)] and honeycomb lattices [panel (b)]. Table 2: In square lattice or honeycomb model, corresponding to eight selected parameter labeled in Tab.1 with $\|E_{F}\|>\Delta$, the average conductance $\langle G\rangle$ and the second, third, …, ninth moments are listed for the first (with ehD, 1 and 3 column) and the second (ehD broken, 2 and 4 column) class in the diffusive regime with $\langle G\rangle\simeq 3$. sq \| $\langle G\rangle$ \| $\sqrt{\mu_{2}}$ \| $\sqrt[3]{\mu_{3}}$ \| $\sqrt[4]{\mu_{4}}$ \| $\sqrt[5]{\mu_{5}}$ \| $\sqrt[6]{\mu_{6}}$ \| $\sqrt[7]{\mu_{7}}$ \| $\sqrt[8]{\mu_{8}}$ \| $\sqrt[9]{\mu_{9}}$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $1_{\rm AR}$ \| 3.001 \| 1.445 \| 0.929 \| 1.840 \| 1.750 \| 2.196 \| 2.266 \| 2.512 \| 2.645 $2_{\rm AR}$ \| 3.013 \| 1.443 \| 0.923 \| 1.838 \| 1.745 \| 2.184 \| 2.266 \| 2.518 \| 2.660 $4_{\rm AR}$ \| 2.986 \| 1.366 \| 0.862 \| 1.737 \| 1.639 \| 2.061 \| 2.129 \| 2.367 \| 2.486 $6_{\rm AR}$ \| 2.976 \| 1.365 \| 0.864 \| 1.736 \| 1.639 \| 2.061 \| 2.129 \| 2.367 \| 2.486 hc \| $\langle G\rangle$ \| $\sqrt{\mu_{2}}$ \| $\sqrt[3]{\mu_{3}}$ \| $\sqrt[4]{\mu_{4}}$ \| $\sqrt[5]{\mu_{5}}$ \| $\sqrt[6]{\mu_{6}}$ \| $\sqrt[7]{\mu_{7}}$ \| $\sqrt[8]{\mu_{8}}$ \| $\sqrt[9]{\mu_{9}}$ $1_{\rm ARR}$ \| 2.998 \| 1.417 \| 0.899 \| 1.805 \| 1.708 \| 2.143 \| 2.218 \| 2.463 \| 2.591 $2_{\rm ARR}$ \| 2.988 \| 1.421 \| 0.901 \| 1.809 \| 1.713 \| 2.148 \| 2.224 \| 2.470 \| 2.602 $5_{\rm ARR}$ \| 3.005 \| 1.347 \| 0.836 \| 1.713 \| 1.606 \| 2.031 \| 2.091 \| 2.328 \| 2.441 $4_{\rm SAR}$ \| 3.009 \| 1.344 \| 0.833 \| 1.710 \| 1.602 \| 2.027 \| 2.087 \| 2.324 \| 2.438 To demonstrate the conductance distribution has two different universalities, we plot in Fig.7 the conductance distribution $P(G)$ obtained from 1,000,000 configurations for a fixed average conductance $\langle G\rangle\simeq 3$ in the square lattice [panel (a)] and the honeycomb lattice [panel (b)]. In this figure, we choose eight parameters from Tab.1 with $\|E_{F}\|<\Delta$. We see that for both square and honeycomb lattices, the conductance distributions corresponding to $E_{F}=0$ and $E_{F}\neq 0$ are clearly different. In addition, for each case, $E_{F}=0$ or $E_{F}\neq 0$, conductance distributions for square and honeycomb lattices are almost the same, as can be seen from Tab.2 in which the second, third, …, ninth moments are listed for the parameters labeled in Tab.1 corresponding to the first ($E_{F}=0$ with ehD) and the second ($E_{F}\neq 0$ where ehD is broken) classes with fixed $\langle G\rangle\simeq 3$. Here, the n-th moment is defined as $\mu_{n}=\left\langle[G-\langle G\rangle]^{n}\right\rangle$. In Tab.2, the n-th moments labeled by “$1_{AR}$” and “$2_{AR}$” correspond to the first class in square lattice, they are close to the n-th moments labeled by ‘$1_{ARR}$’ and ‘$2_{ARR}$’ in honeycomb lattice. Since the universal behavior is determined only by the symmetry and dimensionality, why there are two universal curves for AR? This can be qualitatively understood as follows. When the energy of incoming electron is within the superconducting energy gap, only AR exists. The AR amplitude of total NS system $T_{A}$ is contributed by multiple Andreev reflections and can be expressed in terms of transmission amplitude $t$ and $r$ in the absence of superconducting leads and the pure AR matrix $r_{A}$ of the only NS interface [not consider the clean or disordered normal scattering region] in the following formBeenakker $\displaystyle T_{A}(\epsilon)={\rm Tr}[m(\epsilon)m^{\dagger}(\epsilon)]$ (9) with $\displaystyle m(\epsilon)=t^{e}_{12}(\epsilon)Mt^{e,\dagger}_{12}(-\epsilon)$ $\displaystyle M=[I-r^{eh}_{A}(\epsilon)r^{e,}_{22}(-\epsilon)r^{eh,T}_{A}(\epsilon)r^{e}_{22}(\epsilon)]^{-1}r^{eh}_{A}(\epsilon)$ (10) where we have used the electron-hole symmetry relation $t^{h}_{21}(\epsilon)=t^{e,}_{21}(-\epsilon)$, $r^{he}_{A}(\epsilon)=r^{eh,T}_{A}(\epsilon)$ and the symmetry relation of normal transmission matrix $t^{e}_{21}(\epsilon)=t^{e,T}_{12}(\epsilon)$ in the absence of magnetic filed, where ‘T’ denotes transpose. Eq.(10) can be expanded in power series which gives multiple Andreev reflections. For qualitative understanding, we can focus on the first term in the series, i.e., $m(\epsilon)=t_{12}(\epsilon)t_{12}^{\dagger}(-\epsilon)$ and $T^{(1)}_{A}=T_{12}(\epsilon)T_{12}(-\epsilon)$. It is similar for the higher order of $T_{A}$. Now it is clear why we obtain two universal conductance distributions for Andreev conductance. For $\epsilon=0$ (with ehD) the total Andreev reflection coefficient $T_{A}$ is expressed in terms of only one type of normal transmission coefficient $T(0)$. For $\epsilon\neq 0$ (ehD broken), however, $T_{A}$ consists of two kinds of transmission coefficient $T(\epsilon)$ and $T(-\epsilon)$ that have the completely different statistics. It is the statistical interference of $T(\epsilon)$ and $T(-\epsilon)$ that leads to the new universal conductance distribution. It should be noted in order to get the uniform statistical interference, $T(\epsilon)$ and $T(-\epsilon)$ must be separated far enough from each other, i.e., $\epsilon$ is larger than Thouless energy. The incident energy $\epsilon$ (related to condensed energy, equal to $E_{F}$ in our calculation) is so large that it is comparable to energy gap $\Delta$, so we must go beyond Andreev approximation (AA). While in the present works, AA are widely used, it is why the present works can’t present this new symmetry class. We will show [Fig.9] in the AA, the conductance distribution is smoothly changed with $E_{F}$, in stead of the two universal functions corresponding to $E_{F}=0$ and $E_{F}\neq 0$ in the case with non-Andreev approximation (NAA). ### III.2 Statistical properties in the localized regime As we have shown, different universal conductance distributions corresponding to $E_{F}=0$ and $E_{F}\neq 0$ are found in the diffusive regime. It has been demonstrated numericallyucf4 that the conductance distribution for a fixed $\langle G\rangle$ in the localized regime seems to be a universal function which does not depend on dimensionality (quasi-1D, 2D and quantum dot systems) and ensemble symmetry (COE, CUE or CSE). For normal-superconductor hybrid systems, it is interesting to know whether this conclusion is still valid. Figure 8: (Color online) the skewness $\gamma_{1}$ and the kurtosis $\gamma_{2}$ vs $\langle G\rangle$ for the 1D square or honeycomb lattice with $N_{c}=40$ and 2D square or honeycomb lattice with $N_{c}=40$ and $N_{c}=60$. Different symbols (1)-(9) are described as in panel (b) and labeled in Tab.1. Table 3: Same to Tab.2 except we consider localized regime with fixed average conductance $\langle G\rangle\simeq 0.3$. sq \| $\langle G\rangle$ \| $\sqrt{\mu_{2}}$ \| $\sqrt[3]{\mu_{3}}$ \| $\sqrt[4]{\mu_{4}}$ \| $\sqrt[5]{\mu_{5}}$ \| $\sqrt[6]{\mu_{6}}$ \| $\sqrt[7]{\mu_{7}}$ \| $\sqrt[8]{\mu_{8}}$ \| $\sqrt[9]{\mu_{9}}$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $1_{\rm AR}$ \| .3005 \| 0.669 \| 0.986 \| 1.287 \| 1.527 \| 1.728 \| 1.900 \| 2.052 \| 2.192 $2_{\rm AR}$ \| .2997 \| 0.669 \| 0.986 \| 1.287 \| 1.527 \| 1.727 \| 1.898 \| 2.049 \| 2.186 $4_{\rm AR}$ \| .3002 \| 0.632 \| 0.936 \| 1.229 \| 1.464 \| 1.658 \| 1.823 \| 1.964 \| 2.087 $6_{\rm AR}$ \| .2990 \| 0.631 \| 0.936 \| 1.229 \| 1.464 \| 1.660 \| 1.826 \| 1.970 \| 2.099 hc \| $\langle G\rangle$ \| $\sqrt{\mu_{2}}$ \| $\sqrt[3]{\mu_{3}}$ \| $\sqrt[4]{\mu_{4}}$ \| $\sqrt[5]{\mu_{5}}$ \| $\sqrt[6]{\mu_{6}}$ \| $\sqrt[7]{\mu_{7}}$ \| $\sqrt[8]{\mu_{8}}$ \| $\sqrt[9]{\mu_{9}}$ $1_{\rm ARR}$ \| .2998 \| 0.667 \| 0.982 \| 1.281 \| 1.520 \| 1.718 \| 1.887 \| 2.036 \| 2.170 $2_{\rm ARR}$ \| .3005 \| 0.669 \| 0.985 \| 1.286 \| 1.526 \| 1.725 \| 1.895 \| 2.044 \| 2.179 $5_{\rm ARR}$ \| .3001 \| 0.631 \| 0.934 \| 1.226 \| 1.460 \| 1.654 \| 1.817 \| 1.958 \| 2.081 $4_{\rm SAR}$ \| .3002 \| 0.631 \| 0.936 \| 1.229 \| 1.463 \| 1.658 \| 1.823 \| 1.966 \| 2.094 Figure 9: (Color online) In the localized regime, corresponding to selected parameters labeled in Tab.1 with $\|E_{F}\|=0$ and $0<\|E_{F}\|<\Delta$, the conductance distribution $P[\log_{10}(G)]$ obtained from 1,000,000 configurations are plotted in panel (a) and panel(b) respectively for the fixed $\langle G\rangle\simeq 0.3$ in square lattice [marked with sq] and honeycomb lattice [marked with hc]. In addition, we also plot $P[\log_{10}(G)]$ within AA for $E_{F}=0$ and $E_{F}\neq 0$ in panel(a) and panel(b), respectively. There are two ways to examine the universal conductance distribution $P(G)$: (1). plot $P(G)$ at each $\langle G\rangle$ for different system parameters to see whether all $P(G)$ collapse into a single curve. One can only plot $P(G)$ at a few selected $\langle G\rangle$. (2). plot the moments of $P(G)$ as a function of $\langle G\rangle$ to see the universal behavior. However one can only plot several moments of conductance. Here we focus on the higher order moments $\mu_{3}$ and $\mu_{4}$. In Fig.8, we plot $\sqrt[3]{\mu_{3}}$ [panel(a)] and $\sqrt[4]{\mu_{4}}$ [panel(4)] vs $\langle G\rangle$ for 2D and quasi-1D systems on square and honeycomb lattices. Symbols (1)-(9) are described as in panel (b) and labeled in Tab.1. From the figure, it is clear that the data do not collapse into a single curve. In this calculation, we have used only 10,000 configurations per data point which is not enough to resolve the universality class if any. To improve this, we fix the average conductance $\langle G\rangle$ and calculate higher moments by averaging over 1,000,000 configurations. In Tab.3, we choose the same set of parameters as used in the diffusive regime [Tab.2], and tabulate the average conductance $\langle G\rangle$ and the second, third, …, ninth moments for the fixed $\langle G\rangle\simeq 0.3$. Similar to Tab.2, two universality classes can be identified. The first universality class has ehD and consists of data points from four different set of parameters labeled by “$1_{AR}$” and “$2_{AR}$”(square lattice) and labeled by “$1_{ARR}$” and “$2_{ARR}$”(honeycomb lattice). The rest of data form the second universality class where ehD is broken. Hence it is expected that the conductance distributions for $E_{F}=0$ (with ehD) and $E_{F}\neq 0$ (without ehD) belong to different universality class in the localized regime. This indeed can be seen from Fig.9(a) and (b) where we have plotted the conductance distributions of $\log_{10}(G)$ for $E_{F}=0$ and $E_{F}\neq 0$. Fig.9(a) shows the conductance distribution with ehD for six different sets of parameters where two of them are for AA and the other four are NAA. Obviously, they fall into the same universality class. In Fig.9(b), we show the data for the case with broken ehD. We see that four set of data with NAA collapse into a single curve indicating the universal conductance distribution that is clearly different from Fig.9(a). When AA is made, however, the conductance distribution depends on $E_{F}$ which is non-universal. The results from Fig.9 show that even in the localized regime, the Andreev conductance distributions for $E_{F}=0$ (with ehD) and $E_{F}\neq 0$ (ehD broken) belong to different universality class. ### III.3 statistics beyond superconducting gap Figure 10: (Color online) ${\rm Var}(G)_{\rm Norm}$, ${\rm Var}(G)_{\rm Andr}$ and ${\rm Var}(G)_{\rm cross}$, the three compositions of variance of $G$ vs $\langle G\rangle$ for the 2D square lattice [the left column] and 2D honeycomb lattice [the right column] with $N_{c}=40$ [open symbols] and $N_{c}=60$ [symbols with ‘-’]. Figure 11: (Color online) Panel (a): ${\rm Var}(G)_{\rm Norm}$ vs $\langle G\rangle$ for the 1D [with $N_{c}=40$, corresponding to the crossed symbols] or 2D [with $N_{c}=40$, corresponding to the open symbols and $N_{c}=60$, corresponding to the symbols with ‘-’] square lattice [symbols (1), (2) and (3)] or honeycomb lattice [symbols (4), (5) and (6)]. Panel (b): in a 2D square or honeycomb lattice system with $N_{c}=40$, corresponding to four selected parameter with $E_{F}$ beyond $\Delta$ and labeled in Tab.1, conductance distribution $P[\log_{10}(G)]$ exported from 1,000,000 configurations is plotted. In comparation, we also plot $P[\log_{10}(G)]$ for different parameters in the normal system with $\Delta=0$ from 1,000,000 configurations. Table 4: Beyond the superconducting Gap $\Delta$, the average conductance $\langle G\rangle$ and the second, third, …, ninth moments of normal conductance $G_{N}$ are listed in the localized regime with $\langle G\rangle\simeq 0.3$. NT \| $\langle G\rangle$ \| $\sqrt{\mu_{2}}$ \| $\sqrt[3]{\mu_{3}}$ \| $\sqrt[4]{\mu_{4}}$ \| $\sqrt[5]{\mu_{5}}$ \| $\sqrt[6]{\mu_{6}}$ \| $\sqrt[7]{\mu_{7}}$ \| $\sqrt[8]{\mu_{8}}$ \| $\sqrt[9]{\mu_{9}}$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $1_{\rm sq}$ \| .2997 \| 0.311 \| 0.356 \| 0.470 \| 0.546 \| 0.622 \| 0.691 \| 0.755 \| 0.814 $3_{\rm sq}$ \| .2999 \| 0.310 \| 0.355 \| 0.467 \| 0.544 \| 0.620 \| 0.689 \| 0.752 \| 0.812 $1_{\rm hc}$ \| .2998 \| 0.310 \| 0.357 \| 0.472 \| 0.551 \| 0.631 \| 0.703 \| 0.771 \| 0.835 $6_{\rm hc}$ \| .3004 \| 0.310 \| 0.351 \| 0.462 \| 0.537 \| 0.612 \| 0.679 \| 0.742 \| 0.801 In previous sub-sections, we have studied the statistical properties of pure AR assisted conductance with incident energy $\|E_{F}\|<\Delta$. In this sub- section, we will focus on the case in which the incident energy $E_{F}$ is above $\Delta$. In this case, conductance is contributed by both normal transmission and Andreev reflection. The conductance variance ${\rm Var}(G)$ consists of three terms, the Andreev conductance fluctuation ${\rm Var}(G)_{\rm Andr}$, the normal conductance fluctuation ${\rm Var}(G)_{\rm Norm}$ and the cross term between them ${\rm Var}(G)_{\rm cross}$ [see Eq.(6) and Eq.(8)]. In Fig.10, we plot ${\rm Var}(G)_{\rm Norm}$, ${\rm Var}(G)_{\rm Andr}$ and ${\rm Var}(G)_{\rm cross}$ vs $\langle G\rangle$ for the 2D square lattice [left panels] and 2D honeycomb lattice [right panels] with $N_{c}=40$ [open symbols] and $N_{c}=60$ [symbols with ‘-’]. Our results can be summarized as follows. (1) The Andreev related variance ${\rm Var}(G)_{\rm Andr}$ is drastically suppressed by the disorder. In localized regime [$\langle G\rangle<1$], due to strong disorder, it is completely suppressed to almost zero. As a result only ${\rm Var}(G)_{\rm Norm}$ plays a dominant pole in the localized regime. (2) in the localized regime, the dominant ${\rm Var}(G)_{\rm Norm}$ exhibits a universal behavior, i.e., it is independent of system parameters (such as $E_{F}$, $E_{0}$, $N_{c}$, $\Delta$ and so on). In Fig.11(a), we plot ${\rm Var}(G)_{\rm Norm}$ of 2D system [Fig.10(b1),(b2)] and quasi-1D system for square lattice and honeycomb lattice. We find that ${\rm Var}(G)_{\rm Norm}$ in the localized regime is also independent of dimensionality and type of lattice. It is not surprising since in localized regime, all AR related process are suppressed by the strong disorder. In absence of electron-hole conversion, statistics of NS system are same as that of normal system. In order to improve the accuracy in the calculation, we also calculate the higher order moments and conductance distribution by averaging over 1,000,000 configurations and tabulate average conductance $\langle G\rangle$ and the second, third, …, ninth moments for the fixed $\langle G\rangle\simeq 0.3$ in Tab.4. It is found that the n-th moment for the square lattice and the honeycomb lattice are the same. Correspondingly, in Fig.11(b), we plot the conductance distribution of $\log_{10}(G)$ in a 2D square and honeycomb lattices with $\Delta=0$ and $\Delta\neq 0$. The symbols for $\Delta\neq 0$ are labeled as in Tab.1 and the symbols for $\Delta=0$ is described in Fig.11(b). We see that those data labeled by “$1_{NT}$” belong to the first class ($E_{F}=0$), and the other data belong to the second class ($E_{F}\neq 0$). We can see that when the incident energy is above the superconducting gap $\Delta$, the conductance distributions of NS system are almost indistinguishable from that of normal system with $\Delta=0$. This again confirms that the normal transmission is dominant, electron-hole conversion and consequently the ehD is irrelevant in the localized regime. On experimental side, conductance fluctuationstaley ; aris and magnetoconductance fluctuationbra has been measured for mono and multi-layer graphene normal systems. The conductance fluctuations of normal- superconducting hybrid systems (non-graphene) has also been studied.hartog Hence, our results can be checked experimentally. ## IV conclusion Using the tight-binding model, we have carried out a theoretical study on the sample to sample fluctuation in transport properties of phase coherent systems with conventional NS hybrid systems or graphene based NS hybrid systems. Extensive numerical simulations on quasi-1D or 2D systems show that (1). When $E_{F}<\Delta$, the UCF due to AR is found to be roughly doubled comparing to the system in the absence of the superconducting lead. Denoting the increase factor $\alpha_{0}$ through the relation ${\rm rms}(G_{NS})=\alpha_{0}~{}{\rm rms}(G_{N})$, we found that the difference between $\alpha_{0}$ in 2D system and quasi-1D system is quite large while the difference is small between Fermi electrons and Dirac electrons. (2). Our results show that ehD in the NS hybrid system can lead to a new universality class. In the diffusive regime we found two slightly separated UCF plateaus, one corresponds to the complete electron- hole symmetry class (with ehD) and the other to conventional electron-hole conversion (with ehD broken). In addition, the AR conductance distribution for the fixed average conductance $\langle G\rangle$ in diffusive regime also confirms that the new universality class can be classified using ehD. (3). In the localized regime, we found that the conductance distribution is a universal function that depends only on the average conductance and the ehD. We emphasize that one has to go beyond AA to make sure that the AR conductance distribution is universal in the localized regime. (4). Finally, when $E_{F}$ is beyond $\Delta$, normal transport is present. In general, the conductance distributions of NS systems and normal systems are different. In the localized regime, however, the AR is suppressed significantly by the disorder. Hence in the localized regime normal transmission dominates the transport processes. In this case, the ehD is irrelevant and the conductance distribution is a universal function that depends only on the average conductance in the localized regime. ${\bf ACKNOWLEDGMENTS}$ We gratefully acknowledge the financial support by a RGC grant (HKU705409P) from the Government of HKSAR. ## References (1) lectronic address: jianwang@hkusua.hku.hk * (2) B. L. Altshuler, P. A. Lee, and R. A. Webb, Mesoscopic Phenomena in Solids (North-Holland, Amsterdam, 1991). * (3) C. W. J. Beenakker, Rev. Mod. 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1011.2666	# Nernst and Seebeck effect in a graphene nanoribbon Yanxia Xing1, Qing-feng Sun2, and Jian Wang1 1Department of Physics and the Center of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China 2Beijing National Lab for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China ###### Abstract The thermoelectric power, including the Nernst and Seebeck effects, in graphene nanoribbon is studied. By using the non-equilibrium Green function combining with the tight-binding Hamiltonian, the Nernst and Seebeck coefficients are obtained. Due to the electron-hole symmetry, the Nernst coefficient is an even function of the Fermi energy while the Seebeck coefficient is an odd function regardless of the magnetic field. In the presence of a strong magnetic field, the Nernst and Seebeck coefficients are almost independent of the chirality and width of the nanoribbon, and they show peaks when the Fermi energy crosses the Landau levels. The height of $n$-th (excluding $n=0$) peak is $[\ln 2/\|n\|]$ for the Nernst effect and is $\ln 2/n$ for the Seebeck effect. For the zeroth peak, it is abnormal with height $[2\ln 2]$ for the Nernst effect and the peak disappears for the Seebeck effect. When the magnetic field is turned off, however, the Nernst effect is absent and only Seebeck effect exists. In this case, the Seebeck coefficient strongly depends on the chirality of the nanoribbon. The peaks are equidistant for the nanoribbons with zigzag edge but are irregularly distributed for the armchair edge. In particular, for the insulating armchair ribbon, the Seebeck coefficient can be very large near the Dirac point. When the magnetic field varies from zero to large values, the differences among the Seebeck coefficients for different chiral ribbons gradually vanish and the nonzero value of Nernst coefficient appears first near the Dirac point then gradually extents to the whole energy region. ###### pacs: 72.15.Jf, 73.23.-b, 73.43.-f, 81.05.Uw ## I introduction As a single atomic layer extracted from graphite, graphene has been successfully fabricated experimentally.ref1 ; ref01 Due to its peculiar topological structure, the graphene exhibits peculiar properties.ref2 For the graphene sheet, the conduction and valence band in graphene intersect at Dirac points, the corners of the hexagonal first Brillouin zone. Around the Dirac points graphene has a unique band structure and its quasi particles satisfy the massless Dirac equation where the speed of light is replaced by the Fermi velocity of graphene ($v_{F}\approx 10^{6}m/sec$). Experimentally, by varying the gate voltage, the charge carriers of graphene can be easily tuned, globallyref1 or locallyref3 . As a result the Fermi level can be above or below the Dirac points, which is viewed as electron-like or hole-like system. Along the different crystal direction in honeycomb lattice, the band structureref4 and the transport properties are different. For the graphene ribbon with the zigzag edge, a special edge state exists.ref5 While for the graphene ribbon with armchair edge, it is metallic when the transverse layer $N=3M-1$ with integer M and insulator otherwise.ref5 When the perpendicular magnetic field is strong enough to form Landau levels (LLs), these differences due to different chirality at the zero magnetic field disappear. In addition, both theoreticallytheory2 and experimentally,ref2 the Hall conductance was found to be the half-integer in the values $g(n+1/2)e^{2}/h$ with degeneracy $g=4$, indicating that the quantization condition is shifted by a half-integer compared with the usual integer quantum Hall effect. It is a direct manifestation of the unique electronic structure of graphene. The thermoelectric power (TEP), or the thermal gradient induced current (or bias with an open boundary), results from a balance of electric and thermal forces acting on the charge carriers. In general, we consider two thermoelectric powers, the Nernst effect which is the transverse TEP induced by a longitudinal thermal gradient in a perpendicular magnetic field and the Seebeck effect which is the thermal gradient induced bias in a two probe system. TEP is of great importance in understanding electronic transport because it is more sensitive to the details of the density of states ref6 and the particle-hole asymmetryref7 than the conductance. In the early days, because of the experimental difficulty (particularly in low-dimensional systems or nano-devices), the Nernst effect and Seebeck effect are often neglected. Instead, one usually measures the Hall effect and the resistivity. Now, with the development of the micro-fabrication technology and the low- temperature measurement technology, the thermoelectric measurement in low- dimensional samples has been feasible.ref8 Recently, the Nernst effect and Seebeck effect have been widely observed and experimentally investigated in many systems, including the high-Tc superconductivity,ref9 ferromagnets,ref10 semimetallic,ref11 Pyrochlore Molybdates,ref12 Bismuch,ref13 single walled carbon nanotube,ref14 etc. For the graphene, the study of thermoelectric properties can elucidate details of the electronic structure of the ambipolar nature that cannot be realized by probing conductance alone. Very recently, using a microfabricated heater and thermometer electrodes, the conductance and the diffusive TEP of graphene are simultaneously measured by Zuev et.al.Gtherm1 and Wei et.al..Gtherm2 Zuev et.al. found electrons and holes contribute to Seebeck effect in opposite ways. At high temperatures direct measurement of Seebeck coefficient $S_{C}$ can be compared with that calculated from the Mott relation.Mott Furthermore, divergence of $S_{C}$ and the large Nernst signal were found near the charge neutral point (i.e. the Dirac point).Gtherm2 Also, at low temperatures, depending on $E_{F}$, TEP is oscillating. The temperature suppresses the oscillation and enhances the magnitude of TEP. Up to now, some theoretical investigations have been carried out on the thermal response in the graphene. The electronic transport coefficients including thermopower was semiclassically treated and only classical Hall effect (low field) in graphene was studied.theory1 In addition, the Nernst coefficient was studied only in weak magnetic field. It was found to be strong and positive near Dirac point.theory2 For a strong magnetic field in the quantum Hall regime, the Seebeck coefficient was studied and was focused on its dependence of the field orientation.theory3 In all these works, the quantum Nernst effect is absent because of the calculational subtleties in the presence of the strong magnetic field. For the normal two dimensional electron gas characterized by a parabolic dispersion, the Nernst effect has been studied.NernstTheory1 ; NernstTheory2 ; KuboNernst Of these works, two alternative boundary conditions were considered in calculating the thermal response functions. One is the adiabatic boundary condition that the temperatures in the upper and lower edge are fixed. In this case the Nernst coefficient is similar to the Seebeck coefficient.NernstTheory1 The other one is the non-adiabatic boundary condition on the upper and lower edges, in which the edge currents are in contact with two heat baths with different temperatures.KuboNernst The Nernst coefficient is different from the Seebeck coefficient. It is the purpose of our work to focus on the quantum Nernst effect in the graphene nanoribbon with the adiabatic boundary condition. In this paper, we carry out a theoretical study of the Nernst effect in a crossed graphene nanoribbon and the Seebeck effect in a single graphene nanoribbon in the strong perpendicular magnetic field, zero magnetic field, and weak magnetic field. By using the tight binding model and the nonequilibrium Green function method, the transmission coefficient and consequently the Nernst and Seebeck coefficients are obtained. In a strong perpendicular magnetic field $B$, high degenerated LLs are formed, and the edge states dominate the transport processes, so the Nernst (Seebeck) coefficients are almost the same along the different chiral directions. We find that the Nernst coefficient $N_{C}$ and the Seebeck coefficient $S_{C}$ show peaks when the Fermi energy $E_{F}$ passes the LLs. At $E_{F}=0$, because the zeroth LL is shared by electron-like and hole-like Landau states, $N_{C}$ which is an even function of $E_{F}$ has the highest peak while $S_{C}$ which is an odd function of $E_{F}$ vanishes. On the other hand, at zero $B$, there is no Lorentz force to bend the trajectories of the thermally diffusing carriers, so Nernst effect is absent. In this case, the Seebeck coefficient $S_{C}$ is strongly dependent on the chirality of graphene ribbon. In particular, for the insulating armchair ribbon, $S_{C}$ can be very large near the Dirac point. At last, the crossover behavior of the thermoelectric power from the zero magnetic field to the strong magnetic field is also studied. The rest of the paper is organized as follows. In Section II, the models for crossed graphene ribbon or single graphene ribbon are introduced. The formalisms for calculating the Nernst and Seebeck coefficient are then derived. Section III gives numerical results along with discussions. Finally, a brief summary is presented in Section IV. ## II model and formalism We consider two graphene systems: a four terminal crossed graphene nanoribbon and a two terminal graphene nanoribbon as shown in the left and right insets of Fig.1(b). Here we consider ballistic two dimensional electron gas in which the mean free path and the phase coherent length are greater than the device size. In the experiment, we can use the smaller sample to reduce the device size, and use the lower temperature or the higher magnetic field to enhance the phase coherent length. In the tight-binding representation, the Hamiltonian operator can be written in the following form:ref2 ; ref19 ; ref20 $\displaystyle H_{G}=\sum_{\bf i}\epsilon_{\bf i}a^{\dagger}_{\bf i}a_{\bf i}-\sum_{<{\bf ij}>}te^{i\phi_{ij}}a_{\bf i}^{\dagger}a_{\bf j},$ (1) where ${\bf i}=(i_{x},i_{y})$ is the index of the discrete honeycomb lattice site which is arranged as in inset of Fig.1b, and $a_{\bf i}$ and $a_{\bf i}^{\dagger}$ are the annihilation and creation operators at the site ${\bf i}$. $\epsilon_{\bf i}$ is the on-site energy (i.e. the energy of the Dirac point) which can be controlled experimentally by the gate voltage, here we set $\epsilon_{i}=0$ as an energy zero point. The second term in Eq.(2) is the hopping term with the hopping energy $t$. When the graphene ribbon is under a uniform perpendicular magnetic field $B_{z}=B$, a phase $\phi_{ij}$ is added in the hopping term, and $\phi_{ij}=\int_{i}^{j}\vec{A}\cdot d\vec{l}/\phi_{0}$ with the vector potential $\vec{A}=(-By,0,0)$ and the flux quanta $\phi_{0}=\hbar/e$. With this ballistic system, the current flowing to the p-th graphene lead can be calculated from the Landauer-B$\ddot{u}$ttiker formula:addnote1 $\displaystyle J_{p}$ $\displaystyle=$ $\displaystyle\frac{2e}{\hbar}\sum_{q}\int\frac{dE}{2\pi}[T_{pq}(E)(f_{p}(E)-f_{q}(E))].$ (2) where $p,q=1,2,3,4$ for the four terminal system or $p,q=1,2$ for the two terminal system, and $T_{pq}$ is the transmission coefficient from terminal-q to terminal-p. In Eq.(2), the transmission coefficient $T_{pq}$ can be calculated from $T_{pq}(E)=Tr[\Gamma_{p}G^{r}\Gamma_{q}G^{a}]$, where the line-width function $\Gamma_{p}(E)=i(\Sigma_{p}^{r}-\Sigma_{p}^{r\dagger})$. The Green’s function $G^{r}(E)=[G^{a}(E)]^{\dagger}=\\{EI-H_{0}-\sum_{p}\Sigma^{r}_{p}(E)\\}^{-1}$ where $H_{0}$ is Hamiltonian matrix of the central region and $I$ is the unit matrix with the same dimension as that of $H_{0}$, and $\Sigma_{p}^{r}$ is the retarded self-energy function from the lead-p. The self-energy function can be obtained from $\Sigma^{r}_{p}(E)=H_{c,p}g^{r}_{p}(E)H_{p,c}$, where $H_{c,p}$ ($H_{p,c}$) is the coupling from central region (lead-p) to lead-p (central region) and $g^{r}_{p}(E)$ is the surface retarded Green’s function of semi- infinite lead-p which can be calculated using transfer matrix method.transfer $f_{p}(E)$ in Eq.(2) is the Fermi distribution function, it is also a function of the Fermi energy $E_{F}$ and temperature $\mathcal{T}$, and can be written as $f_{p}(E,E_{F}^{p},\mathcal{T}_{p})=\frac{1}{e^{(E-E^{p}_{F})/k_{B}\mathcal{T}_{p}}+1}$ (3) where $E^{p}_{F}=E_{F}+eV_{p}$ with $e$ the electron charge and $V_{p}$ is the external bias. In the four terminal system, the thermal gradient $\Delta\mathcal{T}$ is added between the longitudinal terminal-1 and terminal-3, and $\mathcal{T}_{1}=\mathcal{T}+0.5\Delta\mathcal{T}$, $\mathcal{T}_{3}=\mathcal{T}-0.5\Delta\mathcal{T}$, $V_{1}=V_{3}=0$. Due to the Lorentz force, the longitudinal thermal gradient induces a transverse current $J_{2,4}$ in the closed boundary condition or a transverse bias $V_{2,4}$ in the open boundary condition in the terminal-2 and terminal-4. Here we consider the open boundary ($J_{2}=J_{4}=0$) and calculate the balanced bias $V_{2,4}$. While in the two terminal system, both original thermal gradient $\Delta\mathcal{T}$ and induced balanced bias are considered in the longitudinal terminal-1 and terminal-2, and we have $\mathcal{T}_{1}=\mathcal{T}+0.5\Delta\mathcal{T}$ and $\mathcal{T}_{2}=\mathcal{T}-0.5\Delta\mathcal{T}$. Assuming small thermal gradient and consequently the small induced external bias, the Fermi distribution function in Eq.(3) can be expanded linearly around the Fermi energy $E_{F}$ and the temperature $\mathcal{T}$, $\displaystyle f_{p}(E,E_{F}^{p},\mathcal{T}_{p})$ $\displaystyle=$ $\displaystyle f_{0}+eV_{p}\left.\frac{\partial f}{\partial E^{p}_{F}}\right\|_{V_{p}=0,\mathcal{T}_{p}=\mathcal{T}}+\Delta\mathcal{T}_{p}\left.\frac{\partial f}{\partial\mathcal{T}_{p}}\right\|_{V_{p}=0,\mathcal{T}_{p}=\mathcal{T}}$ (4) $\displaystyle=$ $\displaystyle f_{0}+f_{0}(f_{0}-1)\left[\frac{eV_{p}}{k_{B}\mathcal{T}}+(E-E_{F})\frac{\Delta\mathcal{T}_{p}}{k_{B}\mathcal{T}^{2}}\right]$ where $f_{0}=\left[e^{(E-E_{F})/k_{B}\mathcal{T}}+1\right]^{-1}$ is the Fermi distribution in the zero bias and zero thermal gradient. Then for the four terminal system, the current $J_{2}$ of the terminal-2 can be rewritten as: $\displaystyle J_{2}$ $\displaystyle=$ $\displaystyle\frac{2e}{h}\int dE~{}f_{0}(f_{0}-1)T_{21}(E)\left[(E-E_{F})\frac{\Delta\mathcal{T}}{2k_{B}\mathcal{T}^{2}}+\frac{qV_{2}}{k_{B}\mathcal{T}}\right]$ (5) $\displaystyle+$ $\displaystyle\frac{2e}{h}\int dE~{}f_{0}(f_{0}-1)T_{23}(E)\left[(E-E_{F})\frac{-\Delta\mathcal{T}}{2k_{B}\mathcal{T}^{2}}+\frac{qV_{2}}{k_{B}\mathcal{T}}\right]$ $\displaystyle+$ $\displaystyle\frac{2e}{h}\int dE~{}f_{0}(f_{0}-1)T_{24}(E)\left[e\frac{V_{2}-V_{4}}{k_{B}\mathcal{T}}\right]$ Similarly, the expression for the current $J_{4}$ of the terminal-4 can also be obtained. Using the open boundary condition with $J_{2}=J_{4}=0$ and considering the system symmetry ($T_{21}=T_{43}$, $T_{23}=T_{41}$ and $T_{24}=T_{42}$), the Nernst coefficient $N_{C}$ in the four terminal system is: $\displaystyle N_{C}$ $\displaystyle=$ $\displaystyle-\frac{V_{2}-V_{4}}{\Delta\mathcal{T}}$ (6) $\displaystyle=$ $\displaystyle\frac{1}{e\mathcal{T}}\frac{\int dE~{}(E-E_{F})(T_{21}-T_{23})f_{0}(f_{0}-1)}{\int dE~{}(T_{21}+T_{23}+2T_{24})f_{0}(f_{0}-1)}$ In the two terminal system, the current $J_{1}=-J_{2}$ is $\displaystyle J_{1}$ $\displaystyle=$ $\displaystyle\int dEf_{0}(f_{0}-1)T_{21}(E)\left[(E-E_{F})\frac{\Delta\mathcal{T}}{2k_{B}\mathcal{T}^{2}}+e\frac{(V_{1}-V_{2})}{k_{B}\mathcal{T}}\right]$ Let $J_{1}=0$, we have Seebeck coefficient $S_{c}$ $\displaystyle S_{C}$ $\displaystyle=$ $\displaystyle-\frac{V_{1}-V_{2}}{\Delta\mathcal{T}}$ (7) $\displaystyle=$ $\displaystyle\frac{1}{e\mathcal{T}}\frac{\int dE~{}(E-E_{F})T_{21}(E)f_{0}(1-f_{0})}{\int dE~{}T_{21}(E)f_{0}(1-f_{0})}$ ## III numerical results and discussion In the numerical calculations, we set the carbon-carbon distance $a=0.142nm$ and the hopping energy $t=2.75eV$ as in a real graphene sample.ref3 ; ref4 Throughout this paper the energy is measured in the unit of $t$. The magnetic field $B$ is expressed in terms of magnetic flux $BS_{0}$ in the unit of $\phi_{0}/\pi$ where $S_{0}=\frac{3}{2}\sqrt{3}a^{2}$ is the area of a honeycomb unit cell and $\phi_{0}=\hbar/e$ is the flux quanta. If we set $BS_{0}=0.001\phi_{0}/\pi$, the real magnetic field is around $4T$. The width of the graphene ribbon is described by an integer $N$, and the corresponding real width is $3Na$ for zigzag edge nanoribbon and $\sqrt{3}Na$ for the armchair edge nanoribbon. In the schematic setup-1I and setup-1II in the inset of Fig.1b, $N=2$. In the presence of the strong perpendicular magnetic field, since transport properties are independent of the chirality, we choose the setup-1I shown in the left inset of Fig.1(b) to study the Nernst effect and the setup-1II shown in right inset of Fig.1(b) to study the Seebeck effect. On the other hand, when the magnetic field is zero, the Seebeck effect strongly depends on the edge chirality, so we will study both zigzag and armchair edge nanoribbons, respectively. Figure 1: (Color online) Nernst coefficient $N_{C}$ (a) in the four terminal system and Seebeck coefficient $S_{C}$ (b) in two terminal system vs Fermi Energy $E_{F}$ with the strong magnetic field $BS_{0}=0.008\phi_{0}/\pi$ and ribbon width $N=80$. Different curves are for different temperatures $k_{B}\mathcal{T}$. The four terminal system and the two terminal system are shown in left and right inset in panel (b), respectively. ### III.1 the strong perpendicular magnetic field case Firstly, we study the system with strong perpendicular magnetic field. Fig.1 shows the Nernst coefficient $N_{C}$ and Seebeck coefficient $S_{C}$ versus Fermi energy $E_{F}$ for different temperatures $\mathcal{T}=0.001t$, $0.003t$, $0.006t$ and $0.01t$. Considering the ambipolar nature of the graphene and the electron-hole symmetry, the Nernst coefficient $N_{C}$ is an even function of $E_{F}$ [$N_{C}(E_{F})=N_{C}(-E_{F})$], because both the energy $E-E_{F}$ and the direction of the particle movement (or $T_{21}-T_{23}$) reverse their signs under the electron-hole transformation. From Fig.1(a), we see that the Nernst coefficient $N_{C}$ show peaks when $E_{F}$ passes the LLs $E_{n}=sign(n)\sqrt{2e\hbar v_{F}^{2}\|n\|B}$ and show valleys between adjacent LLs. With the increase of the temperature, the peak heights roughly remain unchanged, but the valleys rise. For convenience, the peaks are numbered and the peak at $E_{F}=0$ is denoted as the zeroth peak. In the low temperature limits, for the $n$-th peak with $n\not=0$, the height is $[\ln 2/\|n\|]$, and the zeroth peak height is $[2\ln 2]$. In Fig.2(a) we plot inverse of the peak heights versus the peak number $n$ (see the crossed circle symbols) at the low temperature $\mathcal{T}=0.001t$. It satisfies the relation $[\|n\|/\ln 2]$. For comparison, the inverse of the peak’s height for the conventional metal is also plotted (see dotted pentagram symbol), which is $[(n+1/2)/\ln 2]$. In Fig.1(b), we plot the Seebeck coefficient $S_{C}$ versus $E_{F}$ at different temperatures $\mathcal{T}$. Similarly, the Seebeck coefficient $S_{C}$ display peaks when $E_{F}$ passes the LLs and show valleys between adjacent LLs. However, $S_{C}$ shows two essential differences from the Nernst effect: First, $S_{C}$ is an odd function of $E_{F}$, which means that contributions to $S_{C}$ from electrons and holes differ by a sign due to the electron-hole symmetry. So the Seebeck coefficient $S_{C}$ is negative for $E_{F}<0$. Second, when $E_{F}$ is on the zero-th LL, $S_{C}$ is zero instead of the highest peak in the curve of $N_{C}$-$E_{F}$. This is because the zero- th LL with the fourfold degeneracy is shared equally by electrons and holes and the electrons and holes give the opposite contributions to $S_{C}$. The inverse of the peak height of Seebeck coefficient at the low temperature ($k_{B}\mathcal{T}=0.001t$) is plotted in Fig.2(b). It is found that in graphene, the pseudospin related Berry phase ref01 introduces an additional phase shift in the magneto-oscillation of TEP. As a result of this phase shift, the inverse of peak height is $\propto n$ (see the crossed circle symbol in Fig.2(b)). While in the conventional metal or semiconductor with massive carriers, there is no pseudospin related berry phase, the inverse of peak height is $\propto n+\frac{1}{2}$ (see dotted pentacle symbol in Fig.2(b)). Figure 2: (Color online) The panel (a) and (b) are respectively the inverse of peak height of Nernst and Seebeck coefficients vs. the peak number $n$. The crossed circle symbols are for the graphene and dotted pentagram symbols are for the conventional metal. The temperature $k_{B}\mathcal{T}=0.001t$ and other parameters are the same as Fig.1. In panel (a), the two lines are $\|n\|/\ln 2$ and $(n+1/2)/\ln 2$ and in panel (b) the two lines are $n/\ln 2$ and $(n+1/2)/\ln 2$. Figure 3: (Color online) Panel (a): the magnifications of the zeroth and first Nernst peaks in Fig.1(a). Along the arrow direction, temperature $k_{B}\mathcal{T}$ increases from $0.001t$ to $0.029t$ with increment of $0.002t$. Panel (b): the disorder effect of panel (a) at a fixed temperature $k_{B}\mathcal{T}=0.01t$. Next, we study the temperature effect. Since TEP (Nernst effect or Seebeck effect) represents the entropy transported per unit charge, both Nernst coefficient and Seebeck coefficient increase with the increasing temperature which are exhibited in Fig.1(a) and (b). To take a closer look in Fig.3, we plot the zeroth and first peak for the temperature range $\in[0.001t,0.029t]$ in the step of $0.002t$. The temperature effect of $S_{C}$ is similar to that of $N_{C}$, so we only show the Nernst coefficient $S_{C}$ in Fig.3(a). At low temperatures, with the increase of the temperature $k_{B}\mathcal{T}$, the peak height and position do not vary much, but the peak half-width is broadened proportional to $k_{B}\mathcal{T}$, so the valley between the LLs rises. When the temperature $k_{B}\mathcal{T}$ exceeds the spacing of nearest LLs, the Nernst and Seebeck coefficients $N_{C}$ and $S_{C}$ are enhanced in the whole range of energies including both the peak and valley because of the overlap of the neighboring peaks. In addition, except for the zeroth peak, the peak positions for all other peaks shift towards the zeroth peak. Now we study the disorder effect on the Nernst and Seeback effect. To consider the effect of disorder, random on-site potentials $\delta\epsilon_{i}$ in the center region are added with a uniform distribution $[-W/2,W/2]$ with disorder strength W. The data is obtained by averaging over up to 1200 disorder configurations. It is known that when the magnetic field is absent, the Seebeck effect is strongly affected by the disorder, and the peaks are suppressed even in the small disorder. On the other hand, in the presence of the strong magnetic field, the Seebeck effect and Nernst effect are robust to the disorder, because of the existence of the quantized Landau level. The bigger the sample is (or the stronger the magnetic field is), the more robust the Nernst effect and Seebeck effect. Similar to Fig.3(a), in Fig.3(b) we plot the zeroth and first peak at fixed temperature $k_{B}\mathcal{T}=0.01t$ with different disorder strengths. Here sample size ($N=40$) is smaller than that in Fig.1(a) (in which $N=80$). With the smaller sample size, the zeroth universal values of peak height $2\ln 2$ can still remain until disorder $W$ is larger than $1t$. For the first peak, the universal values of height $\ln 2/\|n\|$ remains at $W=0.3t$ and washes out at stronger disorder. It means that the Nernst peak corresponding to the lower Landauer level can resist stronger disorders. In fact, this effect of disorder has been studied for the thermal response to the charge currenttheory1 or to the spin current.Cheng So, in the following, we will focus only on the clean system. Figure 4: (Color online) Seebeck coefficient $S_{C}$ vs. Fermi energy $E_{F}$ for the different temperatures $k_{B}\mathcal{T}$ at zero magnetic field. Panel (a) is for the zigzag ribbon as sketched in the inset of panel (a), with the width $N=40$. Panel (b) and (c) are for the armchair ribbon sketched in inset of panel (b) with the width $N=41$ (b) and $N=40$ (c). The gray solid curves in panels (a), (b), and (c) are the corresponding transmission coefficients $T$. Figure 5: (Color online) The Seebeck coefficient of Fig.4(c) with the Fermi Energy interval $[-0.03,0.03]$. ### III.2 the case of zero magnetic field In this subsection, we study the TEP at zero magnetic field. Because there is no Lorentz force to bend the trajectories of the thermally diffusing carriers, the Nernst effect is absent and $N_{C}=0$. At $B=0$, the Seebeck coefficient $S_{C}$ is strongly dependent on the chirality of graphene ribbon. In addition, for the armchair edge ribbon, it is metallic when $N=3M-1$ ($M$ is an integer) and insulator otherwise.ref5 The Seebeck coefficient $S_{C}$ has essential difference for the metallic and insulator armchair ribbons. In the following we consider three different systems: (1) zigzag edge ribbon with width $N=40$ (sketched in inset of Fig.4(a)), (2) metallic armchair edge ribbon with width $N=41$ (sketched in inset of Fig.4(b)), and (3) insulating armchair edge ribbon with width $N=40$ (sketched in inset of Fig.4(b)). Fig.4(a), (b), and (c) show the Seebeck coefficient $S_{C}$ versus $E_{F}$ for the above three systems, respectively. For the convenience of discussion, we also plot corresponding transmission coefficient $T=T_{21}=T_{12}$ versus $E_{F}$ in each panel. We can see that $S_{C}$ is an odd function of $E_{F}$ and $S_{C}$ increases when the temperature increases. In addition, $S_{C}$ peaks when Fermi energy crosses the discrete transverse channels where quantized transmission coefficient jumps from one step to another. These properties are similar for the above three cases. But there are also many essential different behaviors. (1). For the zigzag edge ribbon, the transverse channels are equidistance with the energy interval $\Delta=\|t\|\pi/(2N)$ in the conduction band or the valence band (except that the interval from the first transmission channel in the conduction band to the first transmission channel in the valence band is $3\Delta$). So peaks of $S_{C}$ are uniformly distributed over energies and the peak height of $S_{C}$ satisfies $[\ln 2/2n]$ where $n$ is the peak number (see Fig.4(a)). (2). In metallic armchair edge ribbon, however, the transverse channel and consequently the peaks of $S_{C}$ are irregularly distributed. The peak height of $S_{C}$ is closely related to the transmission coefficient $T=T_{21}=T_{12}$ and it can be expressed as $2\Delta Tln2/(2T+\Delta T)$ at low temperatures, where $\Delta T$ is the change of $T$ when $E_{F}$ scans over the certain transverse channel. With increasing of the temperature, some of peaks that are very close to each other merge together so that both peak height and position are irregular (see Fig.4(b)). (3). Finally, for the insulating armchair edge ribbon, except for the irregularly distributed peaks for $\|E_{F}\|>\Delta$, the Seebeck coefficient $S_{C}$ is very large for $E_{F}$ near the Dirac point (0) at low temperatures. Fig.5 magnifies the curves of $S_{C}$-$E_{F}$ near the Dirac point. At low temperatures, $S_{C}$ can be very large when $E_{F}$ approaches the Dirac point. For example $S_{C}$ can reach about $10$ at $\mathcal{T}=0.0022t$. At the Dirac point the sign of $S_{C}$ changes abruptly. This is because near the Dirac point the transmission coefficient $T_{12}$ is zero and the carriers can’t be transmitted. In order to balance the thermal forces acting on the charge carriers, we have to add a very large bias leading to a very large Seebeck coefficient near the Dirac point at low temperatures. When temperature increase such that $k_{B}\mathcal{T}$ is greater than the gap of the insulating armchair edge ribbon $S_{C}$ decreases gradually. We emphasize that if the armchair edge ribbon is narrow enough (such as $W\approx 10nm$ as in our calculation), $S_{C}\approx 10$ at the temperature $\mathcal{T}=0.0022t/k_{B}\approx 60K$. This very large $S_{C}$ can be observed in the present technology. Figure 6: (Color online) Panel (a)-(c) plot the Nernst coefficient $N_{C}$ vs Fermi Energy $E_{F}$ at different temperatures $k_{B}\mathcal{T}=0.001t$, $0.005t$ and $0.01t$ in the magnetic field $BS_{0}=0.0005\phi_{0}/\pi,0.002\phi_{0}/\pi$ and $0.005\phi_{0}/\pi$, respectively. In left panels the thermal gradient is added along the zigzag edge ribbon as shown in the left top sketch. While in the right panels, the thermal gradient is added along the armchair edge ribbon as shown in the right top sketch. The ribbon width $N=80$. ### III.3 the crossover from zero magnetic field to high magnetic field In this subsection, we study the Nernst and Seebeck effect when the magnetic field varies from zero to finite values (strong magnetic field). At zero magnetic field, the Nernst coefficient $N_{C}$ is zero and the Seebeck effect $S_{C}$ is dependent on the chirality of graphene ribbon. At high magnetic fields, however, both $N_{C}$ and $S_{C}$ are independent of the ribbon chirality. What happens with the magnetic field in the intermediate range? First, we study the Nernst effect, in which two different setups (the setup-6I and setup-6II) sketched in the top of Fig.6 are considered. In Fig.6 we plot the Nernst coefficient $N_{C}$ versus $E_{F}$ at different temperatures and magnetic fields. From Fig.6(a) to (c), the magnetic field increases from weak to strong enough to form edge state. At the weak magnetic field (such as $BS_{0}/\phi_{0}=0.0005/\pi$), the Nernst coefficient $N_{C}$ peaks sharply near the Dirac point at low temperatures. Because on two sides of the Dirac point, the carriers are electron-like and hole-like and they are shifted to the opposite direction under the weak magnetic field, the Nernst effect is largest at the Dirac point. In particular, in the setup-6I, the Nernst coefficient $N_{C}$ is very large at the Dirac point, which is much larger than that in setup-6II and in the case of high magnetic field. Because for the setup-6I, the longitudinal leads (lead-1 and lead-3) are metallic with a large transmission coefficient but the transverse leads (lead-2 and lead-4) are almost insulator near the Dirac point. As a result, we have to add a much larger bias to balance the thermal current so that the Nernst coefficient $N_{C}$ is very large in the setup-6I at the low magnetic field (see Fig.6(a1) and (b1)). With increasing of $B$, LLs are formed one by one. The zeroth LL located at the Dirac point is formed first (at about $BS_{0}/\phi_{0}=0.0015/\pi$, no shown), then is the first LL, the second and so on. For example, In Fig.6(a), no LL is formed while in Fig.6(b), the zeroth, first and second LL are formed. As soon as LLs are formed, the Nernst coefficient $N_{C}$ will satisfy the relation that its peak heights are equal to $\ln 2/\|n\|$ (or $2\ln 2$ for $n=0$). From Fig.6(c), we can see that as $BS_{0}/\phi_{0}=0.005/\pi$, electrons (or holes) with Fermi energy $\|E_{F}\|\leq 0.3t$ all belong to robust edge states. In this case, the Nernst coefficient $N_{C}$ are almost the same for the setup-6I and setup-6II. For the Seebeck effect, armchair edge ribbon can either be metal or insulator, we also consider three different systems as in the case of zero magnetic field. In Fig.7 we plot the Seebeck coefficient $S_{C}$ versus $E_{F}$ at different temperatures and magnetic fields for three different systems. The first column is for the zigzag edge ribbon with width $N=80$, the second column is for metallic armchair edge ribbon with $N=80$, and the third column is for insulating armchair edge ribbon with $N=81$. From Fig.7(a) to Fig.7(c), the magnetic field increases gradually. We can see that in the weak magnetic field, the peaks of $S_{C}$ are still regularly distributed for the zigzag ribbon and are irregular for the armchair ribbon due to the different band structure for the zigzag edge and armchair edge ribbon. Moreover, for the insulating armchair edge ribbon, the energy gap near Dirac point is diminished because of the magnetic field $B$, the very high and sharp $S_{C}$ at $B=0$ (see Fig.5) is gradually dropped with the increasing of $B$. But at the weak magnetic field $BS_{0}/\phi_{0}=0.0005/\pi$, the $N_{C}$ can still reach 3 (see Fig.7(a3)), which is much larger than all peaks of $S_{C}$ in the high magnetic field case. Similar to Fig.6, with the increasing of $B$ further, the LLs is gradually formed from Dirac point to the high $E_{F}$, the the properties of $S_{C}$ for three systems gradually tend to the same. At the high magnetic field $BS_{0}/\phi_{0}=0.005/\pi$, LLs are completely formed for $\|E_{F}\|<0.3$, then Seebeck coefficient $S_{C}$ for three different systems are all the same to that in the Hall region. ## IV conclusion In summary, by using the Landauer-B$\ddot{u}$ttiker formula combining with the non-equilibrium Green’s function method, the Nernst effect in the crossed graphene ribbon and the Seebeck effect in the single graphene ribbon are investigated. It is found that due to the electron-hole symmetry, the Nernst coefficient $N_{C}$ is an even function while the Seebeck coefficient $S_{C}$ is an odd function of the Fermi energy $E_{F}$. $N_{C}$ and $S_{C}$ show peaks when $E_{F}$ crosses the Landau levels at high magnetic fields or crosses the transverse sub-bands at the zero magnetic field. In the strong magnetic field, due to the fact that high degenerated Landau levels dominate transport processes the Nernst and Seebeck coefficients are similar for different chirality ribbons. The peak height of $N_{C}$ and $S_{C}$, respectively, are $[\ln 2/\|n\|]$ and $[\ln 2/n]$ with the peak number $n$, except for $n=0$. For zeroth peak, it is abnormal. Its peak height is $[2\ln 2]$ for the Nernst effect and it disappears for the Seebeck effect. While in zero magnetic field, Nernst effect is absent and the Seebeck effect is strongly dependent on the chirality of the ribbon. For the zigzag edge ribbon, the peaks of $S_{C}$ are equidistance, but they are irregularly distributed for armchair edge ribbon. Surprisingly, for the insulating armchair edge ribbon, the Seebeck coefficient $S_{C}$ can be very large near the Dirac point due to the energy gap. When the magnetic field increases from zero to high values, the irregularly or regularly distributed peaks of $S_{C}$ in different chiral ribbons gradually tends to be the same. In addition, the nonzero values of the Nernst coefficient $N_{C}$ appear first near the Dirac point and then gradually in the whole energy region. 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However, the Landauer-Buttiker formula is to calculate the current flowing from a terminal to the center region, in which the both interior and surface currents have been included. So it is validity regardless of the magnetic field. * (28) D. H. Lee and J. D. Joannopoulos, Phys. Rev. B 23, 4997 (1981); ibid, 23, 4988 (1981). * (29) S.-G Cheng, Y. Xing, Q.-F Sun and X. C. Xie, Phys. Rev. B 78, 045302 (2008). Figure 7: (Color online) Panel (a)-(c) plot the Nernst coefficient $N_{C}$ vs Fermi Energy $E_{F}$ at different temperatures $k_{B}\mathcal{T}$ and different magnetic fields $BS_{0}/\phi_{0}$. The other parameters and the chirality of ribbon for the first, second, and third column panels are the same as Fig.4(a), (b), and (c), respectively.	arxiv-papers	2010-11-11T14:22:29	2024-09-04T02:49:14.730305	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Yanxia Xing, Qing-feng Sun and Jian Wang", "submitter": "Xing Yanxia", "url": "https://arxiv.org/abs/1011.2666" }
1011.2785	# Optimal detection of losses by thermal probes Carmen Invernizzi Carmen.Invernizzi@unimi.it Dipartimento di Fisica dell’Università degli Studi di Milano, I-20133 Milano, Italy Matteo G. A. Paris matteo.paris@fisica.unimi.it Dipartimento di Fisica dell’Università degli Studi di Milano, I-20133 Milano, Italy CNISM, Udr Milano, I-20133 Milano, Italy Stefano Pirandola pirs@cs.york.ac.uk Department of Computer Science, University of York, York YO10 5GH, United Kingdom ###### Abstract We consider the discrimination of lossy bosonic channels and focus to the case when one of the values for the loss parameter is zero, i.e., we address the detection of a possible loss against the alternative hypothesis of an ideal lossless channel. This discrimination is performed by inputting one-mode or two-mode squeezed thermal states with fixed total energy. By optimizing over this class of states, we find that the optimal inputs are pure, thus corresponding to single- and two-mode squeezed vacuum states. In particular, we show that for any value of the damping rate smaller than a critical value there is a threshold on the energy that makes the two-mode squeezed vacuum state more convenient than the corresponding single-mode state, whereas for damping larger than this critical value two-mode squeezed vacua are always better. We then consider the discrimination in realistic conditions, where it is unlikely to have pure squeezing. Thus by fixing both input energy and squeezing, we show that two-mode squeezed thermal states are always better than their single-mode counterpart when all the thermal photons are directed into the dissipative channel. Besides, this result also holds approximately for unbalanced distribution of the thermal photons. Finally, we also investigate the role of correlations in the improvement of detection. For fixed input squeezing (single-mode or two-mode), we find that the reduction of the quantum Chernoff bound is a monotone function of the two-mode entanglement as well as the quantum mutual information and the quantum discord. We thus verify that employing squeezing in the form of correlations (quantum or classical) is always a resource for loss detection whenever squeezed thermal states are taken as input. ###### pacs: 03.67.-a, 42.50.Dv, 42.50.Ex ## I Introduction One of the main obstacles to the development of quantum technologies is the decoherence associated to losses and absorption processes occurring during the propagation of a quantum signal. The description of the dynamics of systems subject to noisy environments Breuer , as well the detection, quantification and estimation of losses and, more generally, the characterization of lossy channels at the quantum level, received much attention in the recent years Breuer ; DecoRev ; Dau06 ; Zurek . An efficient characterization of decoherence is relevant for quantum repeaters Brask , quantum memories Jensen , cavity QED systems Haroche08 , superconducting quantum circuits Wang , quantum teleportation Qtele , quantum cryptography QKD1 ; QKD2 and secret-key capacities Devetak ; PirSKcapacity . In this paper we address the discrimination of lossy channels, i.e., we consider a situation where the loss (damping) rate of a channel may assume only two possible values and we want to discriminate between them by probing the channel with a given class of states. In particular, we address the discrimination of lossy channels for bosonic systems using squeezed thermal states as probing states, and focus attention to the case when one of the values for the loss parameter is zero, i.e., we address the detection of a possible loss against the alternative hypothesis of an ideal lossless channel. Such kind of discrimination is crucial since a recent analysis Mon11 has revealed the importance of assessing the deviation from ideal conditions, i.e., the identity channel, in implementing large-scale quantum communication. Despite the discrimination of lossy channels has been already considered in the literature, the approach of this paper is novel. Previous works have in fact considered this kind of discrimination by constraining the energy irradiated over the unknown lossy channel but not the total energy employed in preparing the input states. For instance, in the quantum reading of digital memories QreadingPRL , the discrimination of lossy channels has been analyzed by fixing the mean total number of photons irradiated over the channel, independently on the number of probing modes (this approach has been also followed by the recent Ref. NairLAST ). Previously, in the quantum illumination of targets Qillumination ; QilluminationOTHERS , the channel discrimination was performed by fixing the mean number of photons in each of the modes probing the channel. In both these models there was no restriction for the energy involved in the use of ancillary modes. Our approach considers the discrimination of lossy channels by constraining the total energy of the input state, thus including both the probing mode (irradiated over the unknown lossy channel) and a possible ancillary mode (bypassing the lossy channel and detected by the output measurement). Thus, while previous models were more focussed on restricting the energy irradiated over the channel, we address the problem from the point of view of the input source, i.e., considering the global effort in preparing this source. By fixing the input energy, we then optimize over an important class of Gaussian states, i.e., single- and two- mode squeezed thermal states. The choice of these states relies in their experimental accessibility, being routinely generated in today’s quantum optics laboratories where they can be reliably controlled PRLtwo-mode . Furthermore, because of the squeezing, they represent important examples of non-classical states, i.e., states with non-positive P representation Glauber . This is another feature which diversifies our study from previous works QreadingPRL ; Qillumination , where the main goal was the comparison between non-classical states and classical states (i.e., with positive P representation). Note also that our problem, involving a discrete channel discrimination, is completely different from problems of parameter estimation, where one has to infer a parameter taking continuous values. The estimation of the damping constant of a bosonic channel has been recently addressed by Refs. Freyberger ; Mon07 ; AdessoR ; Mon10 . It is clear that, for a given input state, any problem of channel discrimination collapses into a problem of state discrimination Helstrom ; Chefles ; QSE ; QSE2 ; Kargin , where we have to compute the minimum error probability in identifying one of two possible output states. By assuming $M$ identical copies of the input state and a memoryless quantum channel, we have $M$ output states which are exact replicas of two equiprobable states. In this case, the minimum error probability is well-approximated by the quantum Chernoff bound (QCB) PRAQCB ; PRLQCB ; Nuss ; Aud ; Pirandola . Now the crucial step is to vary the input state, trying to optimize the value of the output QCB. In the case of two lossy bosonic channels, this kind of optimization must be constrained, meaning that we have to fix some crucial parameters of the input states, in particular, their energy. As we have already mentioned before, in our investigation we optimize the QCB on the class of single- and two-mode squeezed thermal states by fixing their total energy. Single-mode thermal states are sent through the lossy channel, while two-mode thermal states probe the channel with the one of the modes (probing mode) while bypassing the channel and assisting the output measurement with the other mode (reference mode). In this scenario, we find that the pure version of these states, i.e., single- and two-mode squeezed vacuum states, are optimal for detecting losses, i.e., discriminating a lossy from a lossless channel. Furthermore, we are able to show that, for any value of the damping rate smaller than a certain critical value, there is a threshold on the energy that makes the two-mode squeezed vacuum state more convenient than the corresponding single-mode state. More interestingly, for damping rates larger than the critical value, the two-mode squeezed vacuum state performs always better than the single-mode squeezed vacuum state with exactly the same energy. In order to stay close to schemes which are feasible with current technology, we also analyze the effect of the mixedness in the probe states. In this case, we study the channel discrimination by fixing not only the total energy of the input state but also its total amount of squeezing. Then, we are able to show that two-mode squeezed thermal states are always better than their single-mode counterpart when all the thermal photons are directed into the dissipative channel. We have numerically verified that this result also holds, approximately, for unbalanced distribution of the thermal photons. Finally, we also investigate the role of correlations in the improvement of loss detection. In order to quantify correlations, besides entanglement and mutual information, we exploit the recent results on the quantum discord Disc1 ; Disc2 ; Ferraro10 ; GQDSCa ; GQDSCb ; Vedral , which has been defined with the aim of capturing quantum correlations in mixed separable states that are not quantified by entanglement. Thus, at fixed input squeezing, we study the reduction of the QCB as a function of various correlation quantifiers, i.e., quantum mutual information, entanglement and quantum discord. This analysis allows us to conclude that employing squeezing in the form of correlations, either quantum or classical, is beneficial for the task of loss detection whenever squeezed thermal states are considered as input probes. The paper is structured as follows. In Sec. II we review the discrimination of quantum states together with the general definition of the QCB. In Sec. III we give the basic notation for Gaussian states and in Sec. IV we review the formula of the QCB for Gaussian states. Then, in Sec. V we discuss the discrimination of lossy channels by considering single-mode and two-mode squeezed thermal states. In particular, we provide the details on how to compute the QCB for distinguishing an ideal lossless channel from a lossy channel. Section VI reports the main results regarding the single- and two- mode states as a function of the total energy and squeezing. Finally, in Section VII we analyze the role of correlations in enhancing the discrimination. Section VIII closes the paper with some concluding remarks. ## II Quantum State Discrimination and Quantum Chernoff Bound As we have mentioned before, the problem of quantum channel discrimination collapses to the problem of quantum state discrimination when we fix the input state. As a result, mathematical tools such as the quantum fidelity and the QCB are fundamental in our analysis. Despite it has been introduced only very recently, the QCB has been already a crucial tool in several areas of quantum information: it has been exploited as a distinguishability measure between qubits and single-mode Gaussian states PRAQCB ; PRLQCB , to evaluate the degree of nonclassicality for single-mode Gaussian states Boca or the polarization of a two-mode state Ghiu . It has also been applied in the theory of quantum phase transitions to distinguish between different phases of the $XY$ model at finite temperature AbastoQCM , and to the discrimination of two ground states or two thermal states in the quantum Ising model JMOQCB . For continuous variables systems the quantum discrimination of Gaussian states is a central point in view of their experimental accessibility and their relatively simple mathematical description GSinQI ; QITCV . In fact, in the case of Gaussian states, the QCB can be computed from their first and the second statistical moments. A first formula, valid for single-mode Gaussian states, was derived in PRAQCB . Later, Ref. Pirandola provided a general closed formula for multimode Gaussian states, relating the QCB bound to their symplectic spectra. Furthermore, from these spectra one can derive larger upper bounds which are easier to compute than the QCB Pirandola . In this section, we start by establishing notation and reviewing the problem of quantum state discrimination, together with the general definition of QCB. Then, from the next section, we will specialize our attention to the case of Gaussian states and we will review the formula for Gaussian states in Sec. IV. In its simplest formulation, the problem of quantum state discrimination consists in distinguishing between two possible states, $\rho_{A}$ and $\rho_{B}$, which are equiprobable for a quantum system. We suppose that $M$ identical copies of the quantum system are available. Then, we have the following equiprobable hypotheses, $H_{A}$ and $H_{B}$, about the global state $\displaystyle H_{A}$ $\displaystyle:\rho_{A}^{M}=\underbrace{\rho_{A}\otimes\ldots\otimes\rho_{A}}_{M}$ $\displaystyle H_{B}$ $\displaystyle:\rho_{B}^{M}=\underbrace{\rho_{B}\otimes\ldots\otimes\rho_{B}}_{M}.$ In order to discriminate between these two hypotheses, one can measure the global system by using a two-outcome positive operator valued measure (POVM) $\\{E_{A},E_{B}\\}$, with $E_{A}+E_{B}={\mathbb{I}}$ and $E_{A},E_{B}\geq 0$. After observing the outcome $j=A$ or $B$, the observer infers that the state of the system was $\rho_{j}^{M}$. The error probability of inferring the state $\rho_{j}^{M}$ when the actual state is $\rho_{k}^{M}$ is thus given by the Born rule $P_{jk}=\hbox{Tr}\left[\rho_{k}^{M}E_{j}\right]$. As a result, the optimal POVM for this discrimination problem is the one minimizing the overall probability of misidentification, i.e., $P_{e}=\frac{1}{2}(P_{BA}+P_{AB})$. Since $E_{A}={\mathbb{I}}-E_{B}$, we have $\displaystyle P_{e}$ $\displaystyle=\frac{1}{2}\hbox{Tr}[\rho_{A}^{M}E_{B}]+\frac{1}{2}\hbox{Tr}[\rho_{B}^{M}E_{A}]$ $\displaystyle=\frac{1}{2}\left(1-\hbox{Tr}\left[E_{B}\Lambda\right]\right)~{},$ (1) where $\Lambda=\rho_{B}^{M}-\rho_{A}^{M}\,,$ is known as the Helstrom matrix Helstrom . Now, the error probability $P_{e}$ has to be minimized over $E_{B}$. Since $\hbox{Tr}[\Lambda]=0$, the Helstrom matrix has both positive and negative eigenvalues and the minimum $P_{e}$ is attained if $E_{B}$ is chosen as the projector over $\Lambda_{+}$, i.e., the positive subspace of $\Lambda$. Assuming this optimal operator we have $\hbox{Tr}[E_{B}\Lambda]=\mathop{\text{Tr}}\nolimits[\Lambda_{+}]=\frac{1}{2}\mathop{\text{Tr}}\nolimits\|\Lambda\|$ with $\|\Lambda\|=\sqrt{\Lambda^{\dagger}\Lambda}$. Thus the minimal error probability is given by $P_{e}=\frac{1}{2}\left[1-T(\rho_{A}^{M},\rho_{B}^{M})\right]~{},$ where $T(\rho,\sigma)=\frac{1}{2}\mathop{\text{Tr}}\nolimits\|\rho-\sigma\|$ is the so-called trace distance. The computation of the trace distance may be rather difficult. For this reason, one can resort to the QCB that gives an upper bound to the probability of error $P_{e}$ PRAQCB ; PRLQCB ; Nuss ; Aud ; Pirandola $P_{e}\leq\frac{Q^{M}}{2}~{},$ (2) where $Q=\inf_{0\leq s\leq 1}\mathop{\text{Tr}}\nolimits\left[\rho_{A}^{s}\rho_{B}^{1-s}\right]~{}.$ (3) The bound of Eq. (2) is attainable asymptotically in the limit $M\rightarrow\infty$ as follows from the results in PRLQCB ; Nuss . One may think that the trace distance has a more natural operational meaning than the QCB. In spite of this, it does not adapt to the case of many copies; indeed, one can find states $\rho,\sigma,\rho^{\prime},\sigma^{\prime}$ such that $T(\rho,\sigma)<T(\rho^{\prime},\sigma^{\prime})\quad\hbox{but}\quad T(\rho^{\prime}{}^{M},\sigma^{\prime}{}^{M})<T(\rho^{M},\sigma^{M})\,.$ By contrast, the QCB does resolve this problem since $Q(\rho,\sigma)<Q(\rho^{\prime},\sigma^{\prime})\;\Longrightarrow\;Q(\rho^{M},\sigma^{M})<Q(\rho^{\prime M},\sigma^{\prime M})\,.$ Because of this property, the minimization of the QCB over single-copy states ($\rho$ and $\sigma$) implies the minimization over multi-copy states ($\rho^{M}$ and $\sigma^{M}$). This is true as long as the minimization is unconstrained or if the constraints regard single-copy observables (e.g., the mean energy per copy). Finally, note that there is a close relation between the QCB and the Uhlmann fidelity $F(\rho_{A},\rho_{B})=\mathop{\text{Tr}}\nolimits\left(\sqrt{\sqrt{\rho_{A}}\rho_{B}\sqrt{\rho_{A}}}\right)^{2}$ which is one of the most popular measures of distinguishability for quantum states. In fact, for the single-copy state discrimination ($M=1$) we have PRAQCB ; Fuchs ; Nielsen ; Pirandola $\frac{1-\sqrt{1-F(\rho_{A},\rho_{B})}}{2}\leq P_{e}\leq\frac{Q}{2}\leq\frac{\sqrt{F(\rho_{A},\rho_{B})}}{2}.$ (4) More generally, by exploiting the multiplicativity of the fidelity under tensor products of density operators, i.e., $F(\rho_{A}\otimes\sigma_{A},\rho_{B}\otimes\sigma_{B})=F(\rho_{A},\rho_{B})F(\sigma_{A},\sigma_{B})~{},$ we can write $F(\rho_{A}^{M},\rho_{B}^{M})=F(\rho_{A},\rho_{B})^{M}=F^{M}~{}.$ This leads to the general multi-copy version of Eq. (4) which is given by QreadingSUPP $\frac{1-\sqrt{1-F^{M}}}{2}\leq P_{e}\leq\frac{Q^{M}}{2}\leq\frac{F^{M/2}}{2}~{}.$ (5) From the previous inequalities, it is clear that the QCB gives a tighter bound than the quantum fidelity. However, if one of the two states is pure, then the QCB just equals the fidelity, i.e., we have $Q(\rho_{A},\rho_{B})=F(\rho_{A},\rho_{B})=\mathop{\text{Tr}}\nolimits[\rho_{A}\,\rho_{B}]~{}.$ ## III Gaussian states In this section we give the basic notions on bosonic systems and Gaussian states, ending with the definition of squeezed thermal states. An $n$-mode bosonic system is described by a tensor-product Hilbert space $\mathcal{H}^{\otimes n}$ and a vector of canonical operators $\boldsymbol{R}=(q_{1},p_{1},\ldots,q_{n},p_{n})^{T}$ satisfying the commutation relations $[R_{l},R_{m}]=i\Omega_{lm}~{},$ where $l,m=1,\cdots,2n$ and $\Omega_{lm}$ are the elements of the symplectic form $\boldsymbol{\Omega}=\bigoplus_{k=1}^{n}\left(\begin{array}[c]{cc}0&1\\\ -1&0\end{array}\right)~{}.$ (6) Alternatively, we can use the mode operators $a_{k}$ which are given by the cartesian decomposition of the canonical operators, i.e., $q_{k}=\frac{1}{\sqrt{2}}(a_{k}+a_{k}^{\dagger})~{},~{}p_{k}=\frac{1}{i\sqrt{2}}(a_{k}-a_{k}^{\dagger})~{}.$ These operators satisfy the commutation relations $[a_{k},a_{k^{\prime}}^{\dagger}]=\delta_{kk^{\prime}}$ with $k,k^{\prime}=1,\cdots,n$. An arbitrary quantum state $\rho$ of the system is equivalently described by the characteristic function $\chi[\rho](\boldsymbol{\lambda})=\mathop{\text{Tr}}\nolimits[\rho D(\boldsymbol{\lambda})]$ where $D(\boldsymbol{\lambda})=\otimes_{k=1}^{n}D_{k}(\lambda_{k})$ is the $n$-mode displacement operator, with $\boldsymbol{\lambda}=(\lambda_{1},\ldots,\lambda_{n})^{T}$, $\lambda_{k}\in\mathbb{C}$, and $D_{k}(\lambda_{k})=\exp\\{\lambda_{k}a_{k}^{\dagger}-\lambda_{k}^{\ast}a_{k}\\}$ is the single mode displacement operator. A state $\rho$ is called Gaussian if the corresponding characteristic function is Gaussian $\chi[\rho](\boldsymbol{\Lambda})=\exp\left\\{-\frac{1}{2}\boldsymbol{\Lambda^{T}\sigma\Lambda}+\boldsymbol{X^{T}\Omega\Lambda}\right\\}$ (7) where $\boldsymbol{\Lambda}$ is the real vector $\boldsymbol{\Lambda}=(\text{{Re}}\lambda_{1},\text{{{Im}}}\lambda_{1}\mathrm{,\ldots,{Re}}\lambda_{n}\mathrm{,{Im}}\lambda_{n})^{T}~{}.$ In this case, the state is described by its first two statistical moments, i.e., the vector of mean values $\boldsymbol{X}$ and the covariance matrix (CM) $\boldsymbol{\sigma}$, whose elements are defined as $\displaystyle X_{l}=$ $\displaystyle\langle R_{l}\rangle$ $\displaystyle\sigma_{lm}=$ $\displaystyle\frac{1}{2}\langle\\{R_{l},R_{m}\\}\rangle-\langle R_{l}\rangle\langle R_{m}\rangle$ (8) where $\\{A,B\\}=AB+BA$ denotes the anti-commutator, and $\langle O\rangle=\mathop{\text{Tr}}\nolimits[\rho O]$ is the mean value of the operator $O$. In the remainder of this section, we consider only zero-mean Gaussian states, i.e., Gaussian states with $\boldsymbol{X}=0$, which are therefore fully specified by their CM. The properties of these states may be expressed in very simple terms by introducing the symplectic transformations. A matrix $\boldsymbol{S}$ is called symplectic when preserves the symplectic form of Eq. (6), i.e., $\boldsymbol{S\Omega S^{T}}=\boldsymbol{\Omega}.$ Then, according to the Williamson’s theorem, for every CM $\boldsymbol{\sigma}$, there exists a symplectic matrix $\boldsymbol{S}$ such that $\boldsymbol{\sigma}=\boldsymbol{SWS^{T}}$ (9) where $\boldsymbol{W}=\bigoplus_{k=1}^{n}d_{k}\left(\begin{array}[c]{cc}1&0\\\ 0&1\end{array}\right)~{},$ and the $d_{k}$’s are called the symplectic eigenvalues of $\boldsymbol{\sigma}$. The physical statement implied by the decomposition of Eq. (9) is that every zero-mean Gaussian state $\rho$ can be obtained from a thermal state by performing the unitary transformation $U_{\boldsymbol{S}}$ associated with the symplectic matrix $\boldsymbol{S}$, i.e., we have $\rho=U_{\boldsymbol{S}}~{}\boldsymbol{\nu~{}}U_{\boldsymbol{S}}^{{\dagger}}$ where $\boldsymbol{\nu}=\nu_{1}\otimes\ldots\otimes\nu_{n}$ is a tensor product of single-mode thermal states $\nu_{k}=\frac{1}{\bar{n}_{k}+1}\sum_{m}\left(\frac{\bar{n}_{k}}{\bar{n}_{k}+1}\right)^{m}\|m\rangle_{k}\langle m\|$ with average number of photons given by $\bar{n}_{k}=d_{k}-1/2$. For a single- mode system the most general zero-mean Gaussian state may be written as $\rho=S(\zeta)\nu S^{\dagger}(\zeta)$ where $S(\zeta)=\exp\\{\frac{1}{2}({\zeta a^{\dagger}}^{2}-\zeta^{\ast}{a}^{2})\\}$ is the single-mode squeezing operator and $\zeta=re^{i\phi}\in\mathbb{C}$. The corresponding covariance matrix is given by $\boldsymbol{\sigma}=\left(\begin{array}[c]{cc}a&c\\\ c&b\end{array}\right)$ (10) where $\displaystyle a$ $\displaystyle=(\bar{n}+\frac{1}{2})\left[\cosh(2r)-\sinh(2r)\cos\phi\right]$ $\displaystyle b$ $\displaystyle=(\bar{n}+\frac{1}{2})\left[\cosh(2r)+\sinh(2r)\cos\phi\right]$ $\displaystyle c$ $\displaystyle=(\bar{n}+\frac{1}{2})\sinh(2r)\sin\phi~{}.$ (11) In particular, we can consider the case of a real squeezing parameter, e.g., by fixing $\zeta=-r$ Segno . In this case, the previous expressions of Eq. (11) simplify into the following $\displaystyle a$ $\displaystyle=\frac{1}{2}(2\bar{n}+1)\exp(2r)$ $\displaystyle b$ $\displaystyle=\frac{1}{2}(2\bar{n}+1)\exp(-2r)$ $\displaystyle c$ $\displaystyle=0~{}.$ (12) This state defines the single-mode squeezed thermal state. It depends on two real parameters only, i.e., we have $\rho=S(r)\nu S^{\dagger}(r)=\rho(r,\bar{n})~{}.$ In particular, for $\bar{n}=0$ the state is pure and corresponds to a single- mode squeezed vacuum state $\rho(r,0)=S(r)\left\|0\right\rangle\left\langle 0\right\|S^{\dagger}(r)$. Now let us consider two-mode (zero-mean) Gaussian states. They are completely characterized by their $4\times 4$ CM $\boldsymbol{\sigma}=\left(\begin{array}[c]{cc}\mathbf{A}&\mathbf{C}\\\ \mathbf{C}^{T}&\mathbf{B}\end{array}\right)$ (13) where $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$ are $2\times 2$ blocks. It is useful to introduce the symplectic invariants $\displaystyle I_{1}$ $\displaystyle=\det\mathbf{A}~{},~{}I_{2}=\det\mathbf{B}\,,$ $\displaystyle I_{3}$ $\displaystyle=\det\mathbf{C}~{},~{}I_{4}=\det\boldsymbol{\sigma}~{}.$ (14) By means of these invariants, we can simply write the two symplectic eigenvalues as $d_{\pm}=\sqrt{\frac{\Delta\pm\sqrt{\Delta^{2}-4I_{4}}}{2}}~{},$ where $\Delta=I_{1}+I_{2}+2I_{3}$ DiagSympl ; PirCMs . By means of local symplectic operations, the CM of Eq. (13) can be recast in the standard form, where the three blocks $\mathbf{A}$ and $\mathbf{B}$ are proportional to the identity and $\mathbf{C}$ is diagonal DiagSympl . In the particular case of a two-mode squeezed thermal state, we can write $\boldsymbol{\sigma}=\frac{1}{2}\left(\begin{array}[c]{cc}A{\mathbb{I}}_{2}&C\sigma_{z}\\\ C\sigma_{z}&B{\mathbb{I}}_{2}\end{array}\right)$ (15) where $\displaystyle A$ $\displaystyle=\cosh(2r)+2\bar{n}_{1}\cosh^{2}r+2\bar{n}_{2}\sinh^{2}r$ $\displaystyle B$ $\displaystyle=\cosh(2r)+2\bar{n}_{1}\sinh^{2}r+2\bar{n}_{2}\cosh^{2}r$ $\displaystyle C$ $\displaystyle=(1+\bar{n}_{1}+\bar{n}_{2})\sinh 2r,$ (16) with ${\mathbb{I}}_{2}$ the $2\times 2$ identity matrix and $\sigma_{z}=\text{diag}(1,-1)$ the z-Pauli matrix. This corresponds to considering a density operator of the form $\rho=S_{2}(r)\left(\nu_{1}\otimes\nu_{2}\right)S_{2}(r)^{\dagger}\,,$ where $S_{2}(r)=\exp\\{r(a^{\dagger}b^{\dagger}-ab)\\}$ is the two-mode squeezing operator. This state depends on three real parameters: the squeezing parameter and the two thermal numbers, i.e., we have $\rho=\rho(r,\bar{n}_{1},\bar{n}_{2})~{}.$ In particular, for $\bar{n}_{1}=\bar{n}_{2}=0$ the state is pure and corresponds to a two-mode squeezed vacuum state $\rho(r,0,0)=S_{2}(r)(\left\|0\right\rangle_{1}\left\langle 0\right\|\otimes\left\|0\right\rangle_{2}\left\langle 0\right\|)S_{2}^{\dagger}(r)$. ## IV Quantum Chernoff bound for Gaussian states Here we review the formula of the QCB for multimode Gaussian states Pirandola . In particular, we adapt this formula to our notation and physical units (here the vacuum noise is $1/2$, while in Ref. Pirandola it was equal to $1$). Let us consider two Gaussian states $\rho$ (with statistical moments $\boldsymbol{X}$ and $\boldsymbol{\sigma}$) and $\rho^{\prime}$ (with statistical moments $\boldsymbol{X}^{\prime}$ and $\boldsymbol{\sigma}^{\prime}$). The CMs of these two states can be decomposed as $\displaystyle\boldsymbol{\sigma}$ $\displaystyle=\boldsymbol{S~{}\boldsymbol{W}}(\bar{n}_{1},\cdots,\bar{n}_{n})~{}\boldsymbol{S^{T}}$ (17) $\displaystyle\boldsymbol{\sigma}^{\prime}$ $\displaystyle=\boldsymbol{S}^{\prime}~{}\boldsymbol{\boldsymbol{W}}(\bar{n}_{1}^{\prime},\cdots,\bar{n}_{n}^{\prime})~{}\boldsymbol{S}^{\prime}\boldsymbol{{}^{T}},$ (18) where $\\{\bar{n}_{k}\\}$ and $\\{\bar{n}_{k}^{\prime}\\}$ are their thermal numbers, and $\boldsymbol{W}(x_{1},\cdots,x_{n})=\bigoplus_{k=1}^{n}(2x_{k}+1){\mathbb{I}}_{2}~{}.$ Now let us define the functions $G_{s}(x)=\frac{1}{(x+1)^{s}-x^{s}}~{},$ and $\Lambda_{s}(x)=\frac{x^{s}}{(x+1)^{s}-x^{s}}~{}.$ Then, the QCB is given by $Q=\inf_{0\leq s\leq 1}Q_{s}~{},$ where $Q_{s}=\frac{\Pi_{s}}{\sqrt{\det\boldsymbol{\Sigma}_{s}}}\exp\left(-\frac{1}{2}\mathbf{d}^{T}\boldsymbol{\Sigma}_{s}^{-1}\mathbf{d}\right)~{}.$ (19) In the formula of Eq. (19), we have $\mathbf{d}=\boldsymbol{X}-\boldsymbol{X}^{\prime}$, $\Pi_{s}=\prod_{k=1}^{n}G_{s}(\bar{n}_{k})G_{1-s}(\bar{n}_{k}^{\prime})~{},$ and $\displaystyle\boldsymbol{\Sigma}_{s}=$ $\displaystyle\boldsymbol{S~{}W}[\Lambda_{s}(\bar{n}_{1}),\cdots,\Lambda_{s}(\bar{n}_{n})]~{}\boldsymbol{S^{T}}$ $\displaystyle+\boldsymbol{S}^{\prime}~{}\boldsymbol{W}[\Lambda_{1-s}(\bar{n}_{1}^{\prime}),\cdots,\Lambda_{1-s}(\bar{n}_{n}^{\prime})]~{}\boldsymbol{S}^{\prime T}~{}.$ ### IV.1 Discrimination of squeezed thermal states For the discrimination of squeezed thermal states, the previous formula simplifies a lot. First of all, since they are zero-mean Gaussian states, we have $\mathbf{d}=0$ and, therefore, the exponential factor in Eq. (19) disappears. Then, the symplectic decompositions in Eqs. (17) and (18) are achieved using symplectic matrices $\boldsymbol{S}$ and $\boldsymbol{S}^{\prime}$ which are just one-parameter squeezing matrices, i.e., $\boldsymbol{S}=\boldsymbol{S}(r)$ and $\boldsymbol{S}^{\prime}=\boldsymbol{S}^{\prime}(r^{\prime})$. Thus, let us consider the discrimination of single-mode squeezed thermal states $\rho=\rho(r,\bar{n})$ and $\rho^{\prime}=\rho^{\prime}(r^{\prime},\bar{n}^{\prime})$. In this case, the QCB can be computed using $Q_{s}=\frac{\Pi_{s}(\bar{n},\bar{n}^{\prime})}{\sqrt{\det\boldsymbol{\Sigma}_{s}(r,\bar{n},r^{\prime},\bar{n}^{\prime})}}~{},$ (20) where $\Pi_{s}(\bar{n},\bar{n}^{\prime})=G_{s}(\bar{n})G_{1-s}(\bar{n}^{\prime})~{},$ and $\displaystyle\boldsymbol{\Sigma}_{s}(\bar{n},\bar{n}^{\prime},r,r^{\prime})$ $\displaystyle=\boldsymbol{S}(r)\boldsymbol{W}[\Lambda_{s}(\bar{n})]\boldsymbol{S}(r)\boldsymbol{{}^{T}}$ $\displaystyle+\boldsymbol{S}(r^{\prime})\boldsymbol{W}[\Lambda_{1-s}(\bar{n}^{\prime})]\boldsymbol{S}(r^{\prime})\boldsymbol{{}^{T}}.$ For the discrimination of two-mode squeezed thermal states $\rho=\rho(r,\bar{n}_{1},\bar{n}_{2})$ and $\rho^{\prime}=\rho^{\prime}(r^{\prime},\bar{n}_{1}^{\prime},\bar{n}_{2}^{\prime})$, we can use $Q_{s}=\frac{\Pi_{s}(\bar{n}_{1},\bar{n}_{2},\bar{n}_{1}^{\prime},\bar{n}_{2}^{\prime})}{\sqrt{\det\boldsymbol{\Sigma}_{s}(r,\bar{n}_{1},\bar{n}_{2},r^{\prime},\bar{n}_{1}^{\prime},\bar{n}_{2}^{\prime})}}~{},$ (21) where $\Pi_{s}(\bar{n}_{1},\bar{n}_{2},\bar{n}_{1}^{\prime},\bar{n}_{2}^{\prime})=G_{s}(\bar{n}_{1})G_{s}(\bar{n}_{2})G_{1-s}(\bar{n}_{1}^{\prime})G_{1-s}(\bar{n}_{2}^{\prime})~{},$ and $\displaystyle\boldsymbol{\Sigma}_{s}(r,\bar{n}_{1},\bar{n}_{2},r^{\prime},\bar{n}_{1}^{\prime},\bar{n}_{2}^{\prime})=\boldsymbol{S}(r)\boldsymbol{W}[\Lambda_{s}(\bar{n}_{1}),\Lambda_{s}(\bar{n}_{2})]\boldsymbol{S}(r)\boldsymbol{{}^{T}}$ $\displaystyle+\boldsymbol{S}(r^{\prime})\boldsymbol{W}[\Lambda_{1-s}(\bar{n}_{1}^{\prime}),\Lambda_{1-s}(\bar{n}_{2}^{\prime})]\boldsymbol{S}(r^{\prime})\boldsymbol{{}^{T}}.$ ## V Detection of losses by thermal probes In what follows, we study the evolution of a Gaussian state in a dissipative channel $\mathcal{E}_{\Gamma}$ characterized by a damping rate $\Gamma$, which may result from the interaction of the system with an external environment, as for example a bath of oscillators, or from an absorption process. We consider the problem of detecting whether or not the dissipation dynamics occurred. Given an input state $\rho$, this corresponds to discriminating between an output state identical to the input $\rho$, and another output state storing the presence of loss $\mathcal{E}_{\Gamma}(\rho)$. Lossy channels are Gaussian channels, meaning that they tansform Gaussian states into Gaussian states. Furthermore, if the input is a squeezed thermal state, then the output state is still squeezed thermal (this is discussed in detail afterwards). In general, we consider the schematic diagram depicted in Fig.1. In order to detect loss, we consider either a single-mode squeezed thermal state evolving in the lossy channel with parameter $\Gamma$ followed by a measurement at the output, or a two-mode squeezed thermal state with the damping process occurring in only one of the two modes (the probing mode), followed by a measurement on both of the modes. Our aim is to minimize the error probability in discriminating between the ideal case $\Gamma=0$ and the lossy case $\Gamma>0$. In the next section, this will be done by fixing some important parameters of the input state, such as total energy and squeezing ContQCB . Figure 1: Single- and two-mode schemes for the detection of losses. Top: a single-mode squeezed thermal state $\rho$ enters the lossy channel with damping rate $\Gamma$. A measurement apparatus detects the output state $\rho^{\prime}$. Bottom: the lossy channel acts on the probing mode (1) of a two-mode squeezed thermal state $\rho$, while the reference mode (2) bypasses the channel. The output state $\rho^{\prime}$ of both the modes is then measured. The propagation of a mode of radiation in a lossy channel corresponds to the coupling of the mode $a$ with a zero temperature reservoir made of large number of external modes. By assuming a Markovian reservoir and weak coupling between the system and the reservoir the dynamics of the system is described by the Lindblad Master equation Walls $\dot{\rho}=\frac{\Gamma}{2}\mathcal{L}[a]\rho$ (22) where $\mathcal{L}[a]\rho=2a\rho a^{\dagger}-a^{\dagger}a\rho-\rho a^{\dagger}a$. The general solution may be expressed by using the operator-sum representation of the associated completely-positive map, i.e., upon writing $\eta=e^{-\Gamma t}$, we have $\varrho(\eta)=\sum_{m}V_{m}\varrho V_{m}^{{\dagger}}$ where $V_{m}=\sqrt{\frac{(1-\eta)^{m}}{m!}}a^{m}\eta^{\frac{1}{2}(a^{{\dagger}}a-m)}\,,$ and $\varrho$ is the initial state. ### V.1 Single-mode case Let us now start with single-mode states. Eq. (22) can be recast into a Fokker-Planck equation for the Wigner function in terms of the quadrature variables $q$ and $p$, $\dot{W}=\frac{\Gamma}{2}\left[\partial_{\boldsymbol{X}}^{T}\boldsymbol{X}+\partial_{\boldsymbol{X}}^{T}\boldsymbol{\sigma}_{\infty}\partial_{\boldsymbol{X}}^{T}\right]W$ where $\boldsymbol{X}=(q,p)^{T}$, $\partial_{\boldsymbol{X}}=(\partial_{q},\partial_{p})^{T}$ and we introduced the diffusion matrix $\boldsymbol{\sigma}_{\infty}=\text{diag}(1/2,1/2)$. Solving the equation for the Wigner function of a single-mode Gaussian state one can obtain the evolution equation for the CM $\boldsymbol{\sigma}$. This is given by AS1 $\dot{\boldsymbol{\sigma}}=-\Gamma(\boldsymbol{\sigma}-\boldsymbol{\sigma}_{\infty})~{},$ which yields to $\boldsymbol{\sigma}(t)=e^{-\Gamma t}\boldsymbol{\sigma}_{0}+(1-e^{-\Gamma t}){\boldsymbol{\sigma}}_{\infty}~{}.$ The latter equation describes the evolution of an initial Gaussian state with CM $\boldsymbol{\sigma}_{0}$ towards the stationary state given by the Gaussian state of the environment with CM $\boldsymbol{\sigma}_{\infty}$. For simplicity, from now on we omit the index of $\boldsymbol{\sigma}_{0}$, we replace $\boldsymbol{\sigma}(t)\rightarrow\boldsymbol{\sigma}^{\prime}$, and we insert the time $t$ into the damping parameter $\Gamma$. Thus, the evolved CM of the single mode case simply reads $\boldsymbol{\sigma}^{\prime}=e^{-\Gamma}\boldsymbol{\sigma}+(1-e^{-\Gamma})\,\boldsymbol{\sigma}_{\infty}~{}.$ Now let us consider the specific case of an input squeezed thermal state $\rho=\rho(r,n_{T})$ with squeezing $r$ and thermal number $n_{T}$. According to Eqs. (10) and (12), its CM is given by $\boldsymbol{\sigma}=\left(\begin{array}[c]{cc}a&0\\\ 0&b\end{array}\right)~{},$ with $a=\frac{1}{2}\left(1+2n_{\scriptstyle T}\right)e^{2r}~{},~{}b=\frac{1}{2}\left(1+2n_{\scriptstyle T}\right)e^{-2r}\,~{}.$ (23) At the output of the channel, the state $\rho^{\prime}$ has CM $\boldsymbol{\sigma}^{\prime}=\left(\begin{array}[c]{cc}a_{\Gamma}&0\\\ 0&b_{\Gamma}\end{array}\right)~{},$ where $a_{\Gamma}=\frac{1}{2}\left(1+2n_{\Gamma}\right)e^{2r_{\Gamma}}~{},~{}b_{\Gamma}=\frac{1}{2}\left(1+2n_{\Gamma}\right)e^{-2r_{\Gamma}}~{},$ (24) and $\displaystyle n_{\Gamma}$ $\displaystyle=\sqrt{\det[\boldsymbol{\sigma}^{\prime}]}-1/2$ (25) $\displaystyle r_{\Gamma}$ $\displaystyle=\frac{1}{4}\log\left[\frac{e^{-\Gamma}a+(1-e^{-\Gamma})/2}{e^{-\Gamma}b+(1-e^{-\Gamma})/2}\right].$ (26) Thus, we still have a squeezed thermal state $\rho^{\prime}=\rho^{\prime}(r_{\Gamma},n_{\Gamma})$ with squeezing $r_{\Gamma}$ and thermal number $n_{\Gamma}$. Now, the discrimination between a lossless ($\Gamma=0$) and a lossy channel ($\Gamma>0$) corresponds to the discrimination between the input state $\rho=\rho(r,n_{T})$ and the output one $\rho^{\prime}=\rho^{\prime}(r_{\Gamma},n_{\Gamma})$. In order to estimate the error probability affecting this discrimination, we can compute the quantum Chernoff bound. This is achieved by replacing $(r,\bar{n})\rightarrow(r,n_{T})\;\;\hbox{and}\;\;(r^{\prime},\bar{n}^{\prime})\rightarrow(r_{\Gamma},n_{\Gamma})\,$ in Eq. (20). ### V.2 Two-mode case According to the scheme of Fig. 1, the map describing the evolution of a two- mode state is $\mathcal{E}_{\Gamma}\otimes\mathcal{I}$, where the lossy channel $\mathcal{E}_{\Gamma}$ acts on the probing mode while the identity channel $\mathcal{I}$ acts on the reference mode. At the level of the CM it corresponds to the following transformation $\displaystyle\boldsymbol{\sigma}^{\prime}=$ $\displaystyle\left(e^{-\Gamma/2}{\mathbb{I}}_{2}\oplus{\mathbb{I}}_{2}\right)\boldsymbol{\sigma}\left(e^{-\Gamma/2}{\mathbb{I}}_{2}\oplus{\mathbb{I}}_{2}\right)$ $\displaystyle+({\mathbb{I}}_{4}-e^{-\Gamma}{\mathbb{I}}_{2}\oplus{\mathbb{I}}_{2})\boldsymbol{\sigma}_{\infty}~{}.$ (27) As input state, let us consider a two-mode squeezed thermal state $\rho=\rho(r,n_{\scriptstyle T_{1}},n_{\scriptstyle T_{2}})$. Its CM is provided in Eq. (15) with the elements given in Eq. (16) by replacing $(r,\bar{n}_{1},\bar{n}_{2})\rightarrow(r,n_{\scriptstyle T_{1}},n_{\scriptstyle T_{2}})$. The CM of the output state can be derived using the Eq. (27). This CM can be put in the normal form of Eq. (15) with elements given by Eq. (16) by replacing $(r,\bar{n}_{1},\bar{n}_{2})\rightarrow(r_{\Gamma},n_{\scriptstyle\Gamma_{1}},n_{\scriptstyle\Gamma_{2}})$. Here the squeezing parameter $r_{\Gamma}$ and the thermal numbers $n_{\scriptstyle\Gamma_{1}}$ and $n_{\scriptstyle\Gamma_{2}}$ are function of the input parameters $r$, $n_{\scriptstyle T_{1}}$ and $n_{\scriptstyle T_{2}}$ (the explicit expression is too long to be shown here). Thus, the output state is still a two-mode squeezed thermal state $\rho^{\prime}=\rho^{\prime}(r_{\Gamma},n_{\scriptstyle\Gamma_{1}},n_{\scriptstyle\Gamma_{2}})$. As before, the discrimination between a lossless ($\Gamma=0$) and a lossy channel ($\Gamma>0$) corresponds to the discrimination between the input state $\rho=\rho(r,n_{\scriptstyle T_{1}},n_{\scriptstyle T_{2}})$ and the output one $\rho^{\prime}=\rho^{\prime}(r_{\Gamma},n_{\scriptstyle\Gamma_{1}},n_{\scriptstyle\Gamma_{2}})$. The error probability affecting this discrimination is estimated by the QCB which is computed by replacing $(r,\bar{n}_{1},\bar{n}_{2})\rightarrow(r,n_{\scriptstyle T_{1}},n_{\scriptstyle T_{2}})$ and $(r^{\prime},\bar{n}_{1}^{\prime},\bar{n}_{2}^{\prime})\rightarrow(r_{\Gamma},n_{\scriptstyle\Gamma_{1}},n_{\scriptstyle\Gamma_{2}})$ in Eq. (21). ## VI Optimization of the thermal probes In this section, we optimize the discrimination of a lossless ($\Gamma=0$) from a lossy channel ($\Gamma>0$) by maximizing over thermal probes, i.e., single- and two-mode squeezed thermal states. For this sake, we evaluate the QCB as a function of the most important parameters of the input state, i.e., its total energy and squeezing. In our first analysis, we show that for fixed total energy, single- and two-mode squeezed vacuum states are optimal. In particular, we show the conditions where the two-mode state outperforms the single-mode counterpart. Then, by fixing both the total energy and squeezing, we will find the optimal squeezed thermal state. According to Sec. II the minimization of the QCB over single-copy states implies the minimization over multi-copy states (when the minimization is unconstrained or subject to single-copy constraints). This implies that finding the optimal input state $\rho$ at fixed energy automatically assures that $\rho\otimes\rho\otimes\cdots$ is the optimal multi-copy state at fixed energy per copy when we consider a multiple access to the unknown (memoryless) channel. In order to perform our investigation we introduce a suitable parametrization of the input energy. Given a single-mode squeezed thermal state $\rho=\rho(r,n_{T})$, its energy (mean total number of photons) can be written as $N_{1}=n_{\scriptstyle T}+n_{\scriptstyle S}+2n_{\scriptstyle S}n_{\scriptstyle T}~{},$ (28) where $n_{\scriptstyle T}$ accounts for the mean number of thermal photons, $n_{\scriptstyle S}=\sinh^{2}r$ quantifies the squeezing, and $n_{\scriptstyle S}n_{\scriptstyle T}$ is a cross term. Alternatively, we can introduce a squeezing factor $\beta_{1}\in[0,1]$ such that $\displaystyle n_{\scriptstyle S}$ $\displaystyle=\beta_{1}N_{1}$ (29) $\displaystyle n_{\scriptstyle T}$ $\displaystyle=\frac{(1-\beta_{1})N_{1}}{1+2\beta_{1}N_{1}}~{}.$ (30) Thus the single-mode squeezed thermal state can be parametrized as $\rho=\rho(N_{1},\beta_{1})$, i.e., in terms of its total energy $N_{1}$ and the squeezing factor $\beta_{1}$. Note that for $\beta_{1}=0$ the state is completely thermal with energy $N_{1}=n_{\scriptstyle T}$, while for $\beta_{1}=1$ the state is a squeezed vacuum with energy $N_{1}=n_{\scriptstyle S}$. In our problem of loss detection ($\Gamma=0$ versus $\Gamma>0$), we denote by $Q_{1}(N_{1},\beta_{1})$ the output QCB which is computed by using the input state $\rho=\rho(N_{1},\beta_{1})$. Now, given a two-mode squeezed thermal state $\rho=\rho(r,n_{\scriptstyle T_{1}},n_{\scriptstyle T_{2}})$, its total energy can be written as $N_{2}=n_{\scriptstyle T_{1}}+n_{\scriptstyle T_{2}}+2n_{\scriptstyle S}+2n_{\scriptstyle S}(n_{\scriptstyle T_{1}}+n_{\scriptstyle T_{2}})~{},$ (31) where $n_{\scriptstyle T_{1}}$ ($n_{\scriptstyle T_{2}}$) quantifies the thermal photons in the probing (reference) mode, $n_{\scriptstyle S}=\sinh^{2}r$ quantifies the two-mode squeezing energy, and the last energetic term is a cross term. In this case, besides the squeezing factor $\beta_{2}$, we can also introduce an asymmetry parameter $\gamma\in[0,1]$ which quantifies the fraction of thermal energy used for the probing mode. In other words, we can write $\displaystyle n_{\scriptstyle S}$ $\displaystyle=\frac{1}{2}\beta_{2}N_{2}$ (32) $\displaystyle n_{\scriptstyle T_{1}}$ $\displaystyle=\gamma\,\frac{(1-\beta_{2})N_{2}}{1+\beta_{2}N_{2}}$ (33) $\displaystyle n_{\scriptstyle T_{2}}$ $\displaystyle=(1-\gamma)\frac{(1-\beta_{2})N_{2}}{1+\beta_{2}N_{2}}.$ (34) Thus the two-mode squeezed thermal state can be parametrized as $\rho=\rho(N_{2},\beta_{2},\gamma)$, i.e., in terms of the total energy $N_{2}$, the squeezing factor $\beta_{2}$, and the asymmetry parameter $\gamma$. Note that for $\beta_{2}=0$ we have two thermal states, one describing the probing mode with thermal energy $n_{\scriptstyle T_{1}}=\gamma\,N_{2}$, and the other one describing the reference mode with thermal energy $n_{\scriptstyle T_{2}}=(1-\gamma)N_{2}$. For $\beta_{2}=1$ we have instead a two-mode squeezed vacuum state with total energy $N_{2}=2n_{\scriptstyle S}$. In this case, the thermal energy is zero and $\gamma$ can be therefore arbitrary. In our problem of loss detection ($\Gamma=0$ versus $\Gamma>0$), we denote by $Q_{2}(N_{2},\beta_{2},\gamma)$ the output QCB which is computed by using the input state $\rho=\rho(N_{2},\beta_{2},\gamma)$. ### VI.1 Optimal input at fixed total energy In our first investigation we fix the mean total number of photons of the input state. In other words we fix $N_{1}=N_{2}=N~{}.$ (35) Then we minimize the output QCB among single-mode and two-mode squeezed thermal states. As a first step we compute the optimal quantities $\displaystyle Q_{1}(N)$ $\displaystyle:=\inf_{\beta_{1}}Q_{1}(N,\beta_{1})$ (36) $\displaystyle Q_{2}(N)$ $\displaystyle:=\inf_{\beta_{2},\gamma}Q_{2}(N,\beta_{2},\gamma)~{}.$ (37) Then, we compare $Q_{1}(N)$ with $Q_{2}(N)$. Figure 2: Output QCB$\ Q_{1}(N,\beta_{1})$ optimized over input single-mode squeezed thermal states $\rho=\rho(N,\beta_{1})$. From left to right we consider different values of the transmissivity: $\eta=0.1$ (left panel), $\eta=0.5$ (middle panel) and $\eta=0.9$ (right panel). In each panel, we plot $Q_{1}(N,\beta_{1})$ as function of the energy $N$ for different values of $\beta_{1}$. From top to bottom: $\beta_{1}=0.1$ (dashed line), $\beta_{1}=0.5$ (dotted line) and $\beta_{1}=1$ (solid line). The minimum curve is always achieved for $\beta_{1}=1$, i.e., for an input single-mode squeezed vacuum state. Figure 3: Output QCB$\ Q_{2}(N,\beta_{2},\gamma)$ optimized over input two-mode squeezed thermal states $\rho=\rho(N,\beta_{2},\gamma)$. From left to right we consider different values of the transmissivity: $\eta=0.1,$ $0.5$ and $0.9$. From top to bottom, we consider different values of the asymmetry parameter $\gamma=0,$ $0.5$ and $1$. In each panel, we then plot $Q_{2}$ as function of the energy $N$ for different values of $\beta_{2}$. From top to bottom: $\beta_{2}=0.1$ (dashed line), $\beta_{2}=0.5$ (dotted line) and $\beta_{2}=1$ (solid line). The minimum curve is always achieved for $\beta_{2}=1$ corresponding to an input two-mode squeezed vacuum state. According to our findings, in the Eqs. (36) and (37) the infima are achieved for $\beta_{1}=\beta_{2}=1$. This is numerically shown in Fig. 2 for the single-mode case and in Fig. 3 for the two-mode case. Thus, we have found that, at fixed input energy $N$, the optimal thermal probes are given by single- and two-mode squeezed vacuum states. In this case, the input state is pure and the QCB corresponds to the fidelity (which is the case when the s-overlap in Eq. (3) is minimized for $s$ approching the border). Let us adopt the transmissivity $\eta=e^{-\Gamma}$ to quantify the damping of the channel, so that $\Gamma=0$ (ideal channel) corresponds to $\eta=1$, and $\Gamma>0$ (lossy channel) corresponds to $0\leq\eta<1$. Then, for single-mode we can write $Q_{1}(N)=\frac{1}{\sqrt{1+N(1-\eta^{2})}}~{},$ (38) and for two-modes we derive $Q_{2}(N)=\frac{4}{\left[2+N(1-\sqrt{\eta})\right]^{2}}~{}.$ (39) In Fig. 4, we show the behaviors of the single-mode QCB $Q_{1}(N)$ and two- mode QCB $Q_{2}(N)$ as function of the input energy $N$ for several values of transmissivity $\eta$ (or, equivalently, the damping rate $\Gamma$). As expected the discrimination improves by increasing the input energy $N$ and decreasing the transmissivity $\eta$. Figure 4: (Color online). Single-mode QCB $Q_{1}(N)$ (solid lines) and two- mode QCB $Q_{2}(N)$ (dashed lines) as a function of the input energy $N$ for different damping rates. From top to bottom $\Gamma=0.1,0.3,1$ (red, green and blue, respectively) corresponding to $\eta\simeq 0.9,0.74,0.37$. By comparing curves with the same color (fixed damping $\Gamma$), we can see that $Q_{2}(N)$ outperforms $Q_{1}(N)$ only after a certain value of the input energy $N$. Figure 5: Threshold energy as a function of the transmissivity $N_{th}=N_{th}(\eta)$ (solid curve dividing the dark and the white areas). The dark area indicates the values of the energy $N$ for which the two-mode squeezed vacuum state is optimal. The white region indicates where the single- mode squeezed vacuum state is optimal. The dashed line denotes the behavior of the threshold energy $N_{th}$ close to the critical transmissivity $\eta_{c}\simeq 0.296$. As we can see from Fig. 4, for a given value of the transmissivity $\eta$, the two-mode QCB $Q_{2}(N)$ outperforms the single-mode QCB $Q_{1}(N)$ only after a threshold energy. In fact, for any value of the transmissivity $\eta$ larger than a critical value $\eta_{c}$ there is a threshold energy $N_{th}=N_{th}(\eta)$ that makes the two-mode squeezed vacuum state more convenient than the single-mode counterpart. This threshold energy decreases for decreasing values of $\eta$. In particular, for transmissivities less than the critical value $\eta_{c}$, the threshold energy becomes zero, i.e., the two-mode state is always better than single-mode state. We have numerically evaluated the critical value $\eta_{c}\simeq 0.296$ (corresponding to $\Gamma_{c}\simeq 1.22$). This phenomenon is fully illustrated in Fig. 5, where we have plotted the threshold energy as function of the transmissivity $N_{th}=N_{th}(\eta)$. For $N>N_{th}$ (dark area), the optimal state is the two-mode squeezed vacuum state, while for $N<N_{th}$ (white area) it is the single-mode squeezed vacuum state. In particular, note that $N_{th}=0$ at $\eta=\eta_{c}$. Close to the critical transmissivity we have OtherCurve $N_{th}\simeq 4(\eta-\eta_{c})+5.5(\eta-\eta_{c})^{2}~{}.$ (40) ### VI.2 Optimal input at fixed energy and squeezing It should be said that, in realistic conditions, it is unlikely to have pure squeezing. For this reason, it is important to investigate the performances of the squeezed thermal states by fixing this physical parameter together with the total energy. Thus, in this section, we fix both the input energy and squeezing, i.e., we set $\displaystyle N_{1}$ $\displaystyle=N_{2}=N,$ $\displaystyle\beta_{1}$ $\displaystyle=\beta_{2}=\beta~{}~{}(0\leq\beta\leq 1)~{}.$ (41) Then, we compare the single-mode squeezed thermal state $\rho=\rho(N,\beta)$ with the two-mode squeezed thermal states $\rho=\rho(N,\beta,\gamma)$ for various values of $\gamma$. In other words, we compare $Q_{1}(N,\beta)$ and $Q_{2}(N,\beta,\gamma)$. For fixed $N$ and $\beta$, we find that the minimum of $Q_{2}(N,\beta,\gamma)$ is achieved for $\gamma=1$ (easy to check numerically). This means that two- mode discrimination is easier when all the thermal photons are sent through the lossy channel. In this case we find numerically that $Q_{2}(N,\beta,1)<Q_{1}(N,\beta)\,,$ for every values of the input parameters $N$ and $\beta$, and every value of damping rate $\Gamma$ in the channel. In other words, at fixed energy and squeezing, there is a two-mode squeezed thermal state (the asymmetric one with $\gamma=1$) able to outperform the single-mode squeezed thermal state in the detection of any loss. In order to quantify the improvement we introduce the QCB reduction $\Delta Q=Q_{1}(N,\beta)-Q_{2}(N,\beta,1).$ The more positive this quantity is, the more convenient is the use of the two- mode state instead of the single-mode one. In Fig. 6 we show the behavior of $\Delta Q$ as function of the input energy and squeezing for two different values of the damping. As one can see from the plot, the QCB reduction is always positive. Its value increases with the energy while reaching a maximum for intermediate values of the squeezing. By comparing the two panels of Fig. 6, we can also note that the QCB reduction increases for increasing damping $\Gamma$ (i.e., decreasing transmissivity). Figure 6: (Color online) Density plot of the QCB reduction $\Delta Q$ as function of the input energy $N$ and the squeezing $\beta$. The left plot is for $\Gamma=0.1$ and the right one for $\Gamma=0.9$. Thus, we have just shown that, for fixed values of $N$ and $\beta$, the asymmetric two-mode squeezed thermal state ($\gamma=1$) is the optimal thermal probe in the detection of any loss $\Gamma$. Here we also show that this is approximately true for $\gamma\lesssim 1$. In other words, we show that the inequality $Q_{2}(N,\beta,\gamma)<Q_{1}(N,\beta)$ is robust against fluctuations of $\gamma$ below the optimal value $\gamma=1$. This property is clearly important for practical implementations. To study this situation, let us consider the $\gamma$-dependent QCB reduction $\Delta Q_{\gamma}=Q_{1}(N,\beta)-Q_{2}(N,\beta,\gamma)~{}.$ (42) In Fig. 7 we have specified this quantity for different values of the asymmetry parameter $\gamma$ (each panel refers to a different value of $\gamma$). Then, for every chosen $\gamma$, we have computed $\Delta Q_{\gamma}$ over a sample of $10^{3}$ random values of $N$, $\beta$, and $\Gamma$ (in each panel). As one can see from the figure, the quantity $\Delta Q_{\gamma}$ is approximately positive also when $\gamma$ is quite different from the unity. Figure 7: (Color online) QCB reduction $\Delta Q_{\gamma}$ for different values of $\gamma$ (top left $\gamma=0.99$, top right $\gamma=0.9$, bottom left $\gamma=0.8$, bottom right $\gamma=0.7$). In each panel, $\Delta Q_{\gamma}$ is computed over a sample of $10^{3}$ random values of $N$, $\beta$, and $\Gamma$. ## VII Analysis of the correlations Since two-mode squeezed thermal states are able to outperform the single-mode counterpart under several physical conditions, it is natural to investigate this improvement directly in terms of the correlations of the input state. The quantification of the correlations is realized by using the entanglement, the quantum discord and the quantum mutual information. In order to quantify the degree of entanglement of a two-mode Gaussian state, we can use the logarithmic negativity. Let us consider a bipartite Gaussian state with CM given in Eq. (13). It is easy to derive the symplectic eigenvalues of the partially transposed state. These are given by $\tilde{d}_{\pm}=\sqrt{\frac{\tilde{\Delta}-\sqrt{\tilde{\Delta}^{2}-4I_{4}}}{2}}~{},$ where $\tilde{\Delta}=I_{1}+I_{2}-2I_{3}$, and the symplectic invariants $I_{1}$, $I_{2}$, $I_{3}$ and $I_{4}$ are defined in Eq. (14). From the smallest of these symplectic eigenvalues, we can compute the logarithmic negativity, which is equal to $E=\max\\{0,-\log 2\tilde{d}_{-}\\}~{}.$ A bipartite Gaussian state is entangled iff $\tilde{d}_{-}<1/2$, so that the logarithmic negativity gives positive values for all the entangled states and $0$ otherwise. The quantum discord is defined as the mismatch of two different quantum analogues of classically equivalent expressions of the mutual information and may be used to quantify quantum correlations in mixed separable states. For a two-mode squeezed thermal state with CM as in Eq. (15), the quantum discord may be written as GQDSCa $\displaystyle D=$ $\displaystyle h(\sqrt{I_{2}})-h(d_{-})-h(d_{+})$ $\displaystyle+h\left(\frac{\sqrt{I_{1}}+2\sqrt{I_{1}I_{2}}+2I_{3}}{1+2\sqrt{I_{2}}}\right)$ (43) where $h(x)=\left(x+\frac{1}{2}\right)\log\left(x+\frac{1}{2}\right)-\left(x-\frac{1}{2}\right)\log\left(x-\frac{1}{2}\right)$ is the binary Shannon entropy. We have that, for $0\leq D\leq 1$, the state may be either entangled or separable, whereas all the states with $D>1$ are entangled GQDSCa ; GQDSCb . Finally, the quantum mutual information, which quantifies the amount of total, classical plus quantum, correlations, is given by $I=S(\rho_{A})+S(\rho_{B})-S(\rho_{AB})$, where $S(\rho)=-\mathop{\text{Tr}}\nolimits[\rho\log\rho]$ is the von Neumann entropy of the state $\rho$ and $\rho_{A(B)}=\mathop{\text{Tr}}\nolimits_{B(A)}[\rho_{AB}]$ are the partial traces over the two subsystems. For a two-mode squeezed thermal state with CM as in Eq. (15) the quantum mutual information can be computed using the formula $I=\frac{1}{2}\left[h(\sqrt{I_{1}})+h(\sqrt{I_{2}})-h(d_{+})-h(d_{-})\right]\,.$ Figure 8: (Color online) Upper panels. QCB reduction $\Delta Q_{\bar{\gamma}}$ (with $\bar{\gamma}=0.999$) as a function of the three correlation quantifiers X=I,D,E where I is the quantum mutual information (dotted red), D is the quantum discord (dashed blue) and E is the entanglement (solid black). The plots are for fixed squeezing: $\beta=0.1$ for the left panel and $\beta=0.9$ for the right one. For each quantifier we plot three different curves corresponding to different values of the damping (from top to bottom $\Gamma=0.9,0.5$ and $0.1$). Each curve is generated by varying the input energy $N$ between $0$ and $5$ photons. Middle panels. Density plots of the QCB reduction $\Delta Q_{\bar{\gamma}}$ as a function of the input discord and entanglement. The plots are for fixed damping: $\Gamma=0.2$ in the left panel and $\Gamma=0.8$ in the right one. In each panel, the density plot is generated by varying the squeezing $0\leq\beta\leq 1$ and the energy $0\leq N\leq 5$. Lower panels. Entanglement (left) and discord (right) as a function of the quantum mutual information. Plots are generated by taking a random sample of $10^{4}$ two-mode squeezed thermal states, i.e., random values of $N$ and $\beta$ with $\gamma=\bar{\gamma}$. For pure states the previous three measures are equivalent, whereas for mixed states, as in the case under investigation in this section, they generally quantify different kind of correlations. Here we consider the QCB reduction $\Delta Q_{\bar{\gamma}}=Q_{1}(N,\beta)-Q_{2}(N,\beta,\bar{\gamma})$ between a single-mode squeezed thermal state $\rho=\rho(N,\beta)$ and a two-mode squeezed thermal state $\rho=\rho(N,\beta,\bar{\gamma})$ with $\bar{\gamma}=0.999$. By fixing the input squeezing $\beta$ and varying the input energy $N$, we study the behaviour of $\Delta Q_{\bar{\gamma}}$ as function of the three correlation quantifiers, i.e., quantum mutual information, quantum discord and entanglement (computed over the input two- mode state). As shown in the upper panels of Fig. 8, the QCB reduction $\Delta Q_{\bar{\gamma}}$ is an increasing function of all the three correlation quantifiers for fixed input squeezing ($\beta=0.1$ for the left panel and $\beta=0.9$ for the right one). Note that, in each panel and for each quantifier, we plot three different curves corresponding to different values of the damping $\Gamma=0.9,$ $0.5$ and $0.1$. The monotonicity of the QCB reduction in all the correlation quantifiers suggests that the presence of correlations should definitely be considered as a resource for loss detection, whether these correlations are classical or genuinely quantum, i.e., those quantified by entanglement. In other words, employing the input squeezing in the form of correlations is always beneficial for loss detection when we consider squeezed thermal states as input sources. The importance of correlations is confirmed by the plots in the middle panels. Here we consider again the QCB reduction $\Delta Q_{\bar{\gamma}}=Q_{1}(N,\beta)-Q_{2}(N,\beta,\bar{\gamma})$ for $\bar{\gamma}=0.999$. Then, by varying input squeezing $\beta$ and energy $N$, we study $\Delta Q_{\bar{\gamma}}$ as function of both discord and entanglement (damping is $\Gamma=0.2$ in the left panel, and $\Gamma=0.8$ in the right one). These plots show how the QCB reduction is approximately an increasing function of both discord and entanglement. Finally, in the lower panels of Fig. 8, we also show how entanglement (left) and discord (right) are increasing functions of the quantum mutual information with good approximation (these plots are generated by choosing a random sample of $10^{4}$ two-mode squeezed thermal states). ## VIII Conclusions In this paper we have considered the quantum discrimination of lossy channels. In particular, we have focused to the case when one of the two channels is the identity, i.e., the problem of discriminating the presence of a damping process from its absence (loss detection). For this kind of discriminination we have considered thermal probes as input, i.e., single- and two-mode squeezed thermal states. The performance of the channel discrimination has been quantified using the QCB, computed over the two possible states at the output of the unknown channel for a given input state. Finding the optimal input state $\rho$ which minimizes this bound gives automatically the optimal multi-copy state $\rho\otimes\rho\otimes\cdots$ when we consider many accesses to the unknown channel (under the assumption of single-copy constraints). In this scenario, we have fixed the total energy of the input state and optimized the discrimination (detection of loss) over the class of single- and two-mode squeezed thermal states. We have found numerically that the optimal states are pure, thus corresponding to single- and two-mode squeezed vacuum states. Furthermore, we have determined the conditions where the two-mode state outperforms the single-mode counterpart. This happens when the energy exceeds a certain threshold, which becomes zero for suitably low values of the transmissivity (i.e., high values of damping). It is worth noticing that our approach (where we fix the total energy of probing and reference modes) also gives a sufficient condition for the problem where only the probing energy is fixed. In fact, if a two-mode state outperforms a single-mode state above a certain threshold value $N_{th}$ of the total energy, this also happens when just the energy of the probing mode is above that value $N_{th}$. This is a trivial consequence of the fact that the total energy is bigger than the probing energy for two-mode states ($N_{2}>N_{2}^{\mathrm{probe}}$) while the two quantities are the same for single-mode states ($N_{1}=N_{1}^{\mathrm{probe}}$). Thus, $N_{2}^{\mathrm{probe}}=N_{1}^{\mathrm{probe}}>N_{th}$ can be written as $N_{2}>N_{1}>N_{th}$ which is a stronger condition than $N_{2}=N_{1}>N_{th}$, since the QCB is decreasing in the total energy, as one can see from Eqs. (38) and (39). In our investigation we have then considered the problem of loss detection in more realistic conditions, where it is unlikely to have pure squeezing. In this case, we have studied the optimal state for fixed total energy and squeezing, i.e., by fixing all the relevant resources needed to create the input state. Under these constraints, we have shown that a two-mode squeezed thermal state which conveys all the thermal photons in the dissipative channel is the optimal thermal probe. In addition, this result is robust against fluctuations, i.e., it holds approximately also when the thermal photons are distributed in a more balanced way between the probe mode (sent through the dissipative channel) and the reference mode (bypassing the channel). 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Rev A 69, 022318 (2004). * (56) Clearly we can invert the curve and introduce a threshold transmissivity as function of the energy $\eta_{th}=\eta_{th}(N)$. For values $\eta<\eta_{th}$ the two-mode state is better than the single-mode state, while the opposite happens for $\eta>\eta_{th}$. We have $\eta_{th}\simeq\eta_{c}+0.18\,N^{0.7}$ for small $N$ and $\eta_{th}\simeq 1-2/N$ for large $N$.	arxiv-papers	2010-11-11T22:21:59	2024-09-04T02:49:14.741999	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Carmen Invernizzi, Matteo G. A. Paris, Stefano Pirandola", "submitter": "Matteo G. A. Paris", "url": "https://arxiv.org/abs/1011.2785" }
1011.2813	††thanks: Electronic mail: xjzuo@yahoo.com.cn # Striped morphologies induced by magnetic impurities in d-wave superconductors Xian-Jun Zuo Department of Physics, Zhejiang University of Technology, Hangzhou 310023, People s Republic of China ###### Abstract We study striped morphologies induced by magnetic impurities in d-wave superconductors (DSCs) near optimal doping by self-consistently solving the Bogoliubov-de Gennes equations based on the $t-t^{\prime}-U-V$ model. For the single impurity case, it is found that the stable ground state is a modulated checkerboard pattern. For the two-impurity case, the stripe-like structures in order parameters are induced due to the impurity-pinning effect. The modulations of DSC and charge orders share the same period of four lattice constants (4$a$), which is half the period of modulations in the coexisting spin order. Interestingly, when three or more impurities are inserted, the impurities could induce more complex striped morphologies due to quantum interference. Further experiments of magnetic impurity substitution in DSCs are expected to check these results. stripe, impurity, d-wave superconductors ## I introduction Recent studies of spatial inhomogeneous phases with stripe or checkerboard modulations in copper oxide-based compounds has received much attention. Neutron scattering (NS)experiments reveal the existence of incommensurate magnetic peaks in cuprates, which has led to discussions of a stripe phase lake1 ; tranq2 ; yamada3 ; mook4 ; fujita5 ; hoff6 ; howa7 ; hana9 ; chen6 ; franz7 ; chenhy8 . Meanwhile, scanning tunneling microscopy (STM) experiments observed checkerboard-like charge-density wave (CDW) modulations Mc9 ; vers10 ; Kohs11 , which break the square symmetry of the underlying lattice. It was proposed that the effect can be understood in terms of quasiparticle interference (QPI) due to scattering on impurities and other inhomogeneities zhujx12 ; wangqh13 ; zhangd14 ; Podolsky15 within the so-called ’octet’ model, or in terms of static or fluctuating stripes kivelson16 . Kohsaka et al. suggest that both dispersive and non-dispersive modulated patterns originate from different regions in momentum and energy space Kohs11 . The well-defined states in the nodal region are responsible for the low energy QPI structure, whereas the ill-defined quasiparticle states in the antinodal regions are responsible for the non-dispersive charge order above some energy scale. On the other hand, zero-field experiments on the well ordered YBCO samples show no evidence of static broken translational symmetry in nuclear magnetic resonance or NS experiments, which leads to the proposal that the checkerboard patterns observed by STM is caused by the local disorders yang17 . In brief, the physics that determines the ultimate patterns of competing orders is subtle. For instance, there are striped structures interpreted in terms of actual or incipient orders kivelson16 , or attributed to be induced by lattice distortion or impurity-pinning effect normand ; yang17 . To date, the issue of the nature of spatial inhomogeneous phases in cuprates is still controversial. Impurity effects have been proven to be a valuable probe to explore the fundamental properties of cuprates since it often provides important insights into the underlying physics. Experimentally, STM detection of the modulation of the local density of states by impurities was used to great advantage to probe the nature of quasi-particle states of DSCs both in the superconducting and pseudo-gap phases pan ; huds21 ; chatt . Another interesting issue is interference patterns and pinning stripes by impurities in DSCs. In general, impurity substitution causes additional slowing down of spin fluctuations and pinning of the stripes, leading to the formation of a static charge or spin order. Zhu et al. have proposed an explanation to the checkerboard pattern around a impurity or vortex cores based on an effective mean-field $t-U-V$ model zhujx12 . For the case without an applied magnetic field, the disorder can produce a similar pinning effect of the fluctuating stripes. Andersen et al. investigate disorder-induced freezing of incommensurate spin fluctuations, which agrees qualitatively with experimental observations andersen18 . In the present work, we study the striped morphologies induced by magnetic impurities in d-wave superconductors (DSCs) near optimal doping. Including the competitions and coexistence among the DSC, spin-density wave (SDW), and CDW orders, the system is explored by self-consistently solving the Bogoliubov - de Gennes (BdG) equations. It is found that the local disorders can induce interesting phenomena, including checkerboard, stripe modulations, and other complex striped morphologies. We expect that these phenomena could be observed in the STM and NS experiments. ## II Theory and calculation details We start from the two-dimensional $t-t^{\prime}-U-V$ model, which consists of two parts, $H=H_{0}+H_{imp}$. The Hamiltonian $H_{0}$ and $H_{imp}$ describe the superconductor and magnetic impurities, respectively, which can be written as $\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle-\sum_{ij\sigma}t_{ij}(c_{i\sigma}^{\dagger}c_{j\sigma}+H.c.)$ $\displaystyle+$ $\displaystyle\sum_{ij}(\Delta_{ij}c_{i\uparrow}^{\dagger}c_{j\downarrow}^{\dagger}+H.c.)+\sum_{i\sigma}(Un_{i,\bar{\sigma}}-\mu)c_{i\sigma}^{\dagger}c_{i\sigma},$ $\displaystyle H_{imp}$ $\displaystyle=$ $\displaystyle\sum_{i}h_{eff}(i)(c_{i\uparrow}^{\dagger}c_{i\uparrow}-c_{i\downarrow}^{\dagger}c_{i\downarrow}).$ (1) Here $c_{i\sigma}$ annihilates an electron of spin $\sigma$ at the $i$th site. The hopping integral $t_{ij}$ takes $t$ between nearest neighbor (NN) pairs, and $t^{\prime}$ between next-nearest neighbor (NNN) pairs. $U$ is the on-site Coulomb repulsion interaction. $\mu$ is the chemical potential, which is determined self-consistently in the calculation. The local effective field $h_{eff}(i)$ is introduced to model the exchange coupling between conducting electrons and the impurity spin, where we have treated the Kondo impurity spin as a Ising-like one. Some similar model had been employed to study the effects of magnetic impurities on cuprate superconductors bala19 ; zuo20 , which can qualitatively explain the observed impurity states well huds21 . Therefore, we employed the above model in this work. The self-consistent mean-field parameters are given by $n_{i}=\sum_{\sigma}<c_{i\sigma}^{\dagger}c_{i\sigma}>$, the magnetization $m_{i}=(1/2)(<c_{i\uparrow}^{\dagger}c_{i\uparrow}>-<c_{i\downarrow}^{\dagger}c_{i\downarrow}>)$, and the DSC order parameter $\Delta_{ij}=(V/2)<c_{i\uparrow}c_{j\downarrow}-c_{i\downarrow}c_{j\uparrow}>$ with V the phenomenological pairing interaction. The Hamiltonian $H$ can be diagonalized by solving the following BdG equations, $\displaystyle\left(\begin{array}[]{lr}H_{ij,\uparrow}&\Delta_{ij}\\\ \Delta_{ij}^{\ast}&-H_{ij,\downarrow}^{\ast}\end{array}\right)\Psi_{j}=E\Psi_{i},$ (4) where the quasiparticle wave function $\Psi_{i}=(u_{i\uparrow},v_{i\downarrow})^{T}$. The spin-dependent single- particle Hamiltonian reads $H_{ij\sigma}=-t\delta_{i+\tau,j}-t^{\prime}\delta_{i+\tau^{\prime},j}+[\sum_{i_{m}}\sigma h_{eff}(i)\delta_{i,i_{m}}+Un_{i,\bar{\sigma}}-\mu]\delta_{ij}$. Here the subscripts $\tau$ and $\tau^{\prime}$ denote the unit vector directing along four NN and NNN bonds respectively, and $i_{m}$ is the position of the impurity site. The self-consistent parameters are given by $n_{i\uparrow}=\sum_{n}\|u^{n}_{i\uparrow}\|^{2}f(E_{n})$, $n_{i\downarrow}=\sum_{n}\|v^{n}_{i\downarrow}\|^{2}[1-f(E_{n})]$, and $\Delta_{ij}=\frac{V}{4}\sum_{n}[{u^{n}_{i\uparrow}v_{j\downarrow}^{n\ast}+v_{i\downarrow}^{n\ast}u^{n}_{j\uparrow}}]tanh(\frac{\beta E_{n}}{2})$, where $f(E)=1/(1+e^{\beta E})$ is the Fermi-Dirac distribution function. Hereafter, the length is measured in units of the lattice constant $a$, and the energy in units of $t$. We set $U=2.5$ and $t^{\prime}=-0.2$ in this paper. The pairing interaction is chosen as $V=1.0$ to guarantee that the superconducting order $\Delta_{0}\simeq 0.08t$, comparable with the observed $T_{c}$ in cuprate superconductors. We study the impurity effects on the electronic states of DSCs near optimal doping with the filling factor $n_{f}=\sum_{i\sigma}c_{i\sigma}^{\dagger}c_{i\sigma}/(N_{x}N_{y})=0.83$ (i.e., the hole doping $x=0.17$), where $N_{x}$, $N_{y}$ are the linear dimension of the unit cell. The BdG equations are solved self-consistently for a square lattice of $32\times 32$ sites, and the periodic boundary conditions are adopted. The numerical calculation is performed at a very low temperature, $T=10^{-5}$K, to extract the low-energy physics. The local effective field is taken to be $h_{eff}$ at the impurity site and zero otherwise. The DSC order parameter at the $i$th site is defined as $\Delta_{i}=(\Delta_{i,i+e_{x}}+\Delta_{i,i-e_{x}}-\Delta_{i,i+e_{y}}-\Delta_{i,i-e_{y}})/4$, and the spin order parameter is $M_{i}=(-1)^{i}m_{i}$. ## III Numerical results and discussion In FIG. 1, we plot the spatial distributions of the DSC, spin, and charge orders around the impurity site. One can see that all the three orders display checkerboard modulations around the magnetic impurity. Similar to the nonmagnetic impurity case, a SDW checkerboard pattern is observed around the magnetic impurity zhujx12 ; chen22 , which coincides with the NS data lake1 . However, it is noteworthy that the checkerboard pattern of the DSC order can also be induced by the magnetic impurity, and a weak associated CDW pattern is observed, which is different from the nonmagnetic impurity case. Moreover, the modulated DSC and CDW orders share the same periodicity. Overall, the DSC order is strongly suppressed at the impurity site while the amplitudes of the CDW and SDW orders reach global maxima. This is the common feature of orders around the magnetic impurity despite various parameters. Therefore, in view of these phenomena, we clearly see the relationship of competition and coexistence between antiferromagnetic and DSC orderings. In comparison to the nonmagnetic impurity case zhujx12 ; chen22 , one finds that in both cases the DSC orders are suppressed around the impurity site. However, for the CDW order, the opposite tendency is observed due to the fact that the nonmagnetic impurity is repulsive. Next we consider the many-impurity case to show the striped morphologies which could be induced by magnetic impurities in DSCs. As shown in FIG. 2, upon inserting magnetic impurities in the same direction, a local stripe-like structure can be induced, which is analogous to the nonmagnetic impurity case chen22 . A second important feature is that the average separation between neighboring stripes is not expected to change upon impurity-doping. The impurity-pinned SDW stripe also show a modulation with periodicity 8$a$, which coexists with the DSC order. In addition, similar to the single impurity case, the modulated DSC and CDW stripes also share the same periodicity. For the three-impurity case, one notes that actually there are local ferromagnetic moments formed around impurities. If the three impurities are placed so as to form a right-angled shape, one finds that the striped structures also show a right-angled shape due to the impurity-pinning and quantum interference effects (see Fig.3). The impurity-pinned stripes along x or y direction are balanced by quantum interference effects so that structures with a right- angled shape are favorable on energy. Meanwhile, the alternately dark and bright features due to quantum interference effect are still obvious. Away from the impurities, the stripes eventually evolve to checkerboard structures due to quasiparticle interference, and tend to disappear on farther sites. This feature can be seen more clearly from the DSC order parameter [see Fig.3 (a)]. Moreover, we also calculate the cases with four or five impurities. As shown in Fig. 4, when inserting four or five magnetic impurities symmetrically around the center site, there are symmetrical impurity-pinned stripe structures formed around impurities. Interestingly, these stripe structures are somewhat analogous to ”quantum corrals”, structures built from adatoms using atomic manipulation processes, which have been observed on a metal surface crommie23 ; manoha24 , Here, these quantum-corral-like structures in order parameters are produced due to impurity-pinning and quantum interference effects in DSCs. This feature is especially clear in the spin order, in which the alternately dark and bright rectangular corrals surround the impurities near the center. Similar to the three-impurity case, the stripes eventually evolve to checkerboard structures in the sites away from the impurities and disappear on farther sites. ## IV conclusion To conclude, we have investigated the modulations around magnetic impurities of the DSC, spin, and charge orders in DSCs. It is found that magnetic impurities could induce rich and interesting striped morphologies. For the single impurity case, the checkerboard modulations are observed in all three orders. The weak associated CDW pattern is different from the nonmagnetic impurity case. When more magnetic impurities are inserted, more complex modulated structures could be induced, including rectilinear and right-angled stripes and quantum-corral-like structures due to impurity-pinning and quantum interference effects. The latter structures were only reported on metal surfaces previously. We expect that these phenomena could be observed in the high resolution STM and NS measurements on DSCs. ## Acknowledgements This work is supported by the National Nature Science Foundation of China (Grant No. 10904063). Inspiring discussions with Prof. Chang-De Gong and Dr. Yuan Zhou are acknowledged. ## References * (1) B. Lake, G. Aeppli, K. N. Clausen, D. F. McMorrow, K. Lefmann, N. E. Hussey, N. Mangkorntong, M. Nohara, H. Takagi, T. E. Mason, and A. Schroder, Science 291, 1759 (2001); B. Lake, H. M. Rønnow, N. B. Christensen, G. Aeppli, K. Lefmann, D. F. McMorrow, P. Vorderwisch, P. Smeibidl, N. Mangkorntong, T. Sasagawa, M. Nohara, H. Takagi and T. E. Mason, Nature (London) 415, 299 (2002). * (2) J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature (London) 375, 561 (1995); J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita, and K. Yamada, Nature (London) 429, 534 (2004). * (3) K. Yamada, C. H. Lee, K. Kurahashi, J. Wada, S. 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Lett. 92, 077203 (2004). * (29) M. F. Crommie, C.P. Lutz and D.M. Eigler, Science 262, 218 (1993). * (30) H.C. Manoharan, C.P. Lutz and D.M. Eigler, Nature 403, 512 (2000). Figure 1: (Color online) The surface plots of orders around the magnetic impurity on a unit cell of size $32\times 32$ sites. (a), (c), and (e) are the spatial distributions of the DSC, spin and charge orders. (b), (d), and (f) are their contour plots. Figure 2: (Color online) The spatial contour plots of orders by inserting two or three magnetic impurities in the same direction. (a), (c), and (e) are the contour plots of the DSC, spin and charge orders with two impurities, while (b), (d), and (f) are the three-impurity case. Figure 3: (Color online) The same plot as Fig. 1 but with three impurities forming a right-angled shape. Figure 4: (Color online) The spatial contour plots of orders by inserting four or five magnetic impurities symmetrically around the center site. (a), (c), and (e) are the contour plots of the DSC, spin and charge orders with four impurities, while (b), (d), and (f) are the five-impurity case.	arxiv-papers	2010-11-12T03:05:42	2024-09-04T02:49:14.753447	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Xian-Jun Zuo", "submitter": "Xian-Jun Zuo", "url": "https://arxiv.org/abs/1011.2813" }
1011.2821	# The focusing of electron flow in a bipolar Graphene ribbon with different chiralities Yanxia Xing1, Jian Wang1,∗, and Qing-feng Sun2 1Department of Physics and the Center of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China 2Beijing National Lab for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China ###### Abstract The focusing of electron flow in a symmetric p-n junction (PNJ) of graphene ribbon with different chiralities is studied. Considering the PNJ with the sharp interface, in a armchair ribbon, the electron flow emitting from $(-L,0)$ in n-region can always be focused perfectly at $(L,0)$ in p-region in the whole Dirac fermion regime, i.e. in whole regime $E_{0}<t$ where $E_{0}$ is the distance between Dirac-point energy and Fermi energy and $t$ is the nearest hopping energy. For the bipolar ribbon with zigzag edge, however, the incoming electron flow in n-region is perfectly converged in p-region only in a very low energy regime with $E_{0}<0.05t$. Moreover, for a smooth PNJ, electrons are backscattered near PNJ, which weakens the focusing effect. But the focusing pattern still remains the same as that of the sharp PNJ. In addition, quantum oscillation in charge density occurs due to the interference between forward and backward scattering. Finally, in the presence of weak perpendicular magnetic field, charge carriers are deflected in opposite directions in the p-region and n-region. As a result, the focusing effect is smeared. The lower energy $E_{0}$, the easier the focusing effect is destroyed. For the high energy $E_{0}$ (e.g. $E_{0}=0.9t$), however, the focusing effect can still survive in a moderate magnetic field on order of one Tesla. ###### pacs: 73.63.-b, 73.23.Ad, 73.40.Gk, ## I introduction Graphene is a single layer carbon atoms packed into honeycomb lattice. From the point of view of its electronic properties in the low energy regime, a graphene sheet is a two-dimensional (2D) zero-gap semiconductor with the conical energy spectrum around Dirac points, the corners of the hexagonal first Brillouin zone, and its quasi-particles are formally described by the massless Dirac equation where the speed of light is replaced by the Fermi velocity of graphene.Dirac The detailed electronic properties of graphene has been reviewed in Ref.[mod1, ]. Different from the usual zero-gap semiconductor in which the electrons and holes are normally described by separate Schrödinger equations with generally different effective masses, the electrons and holes in graphene are conjugately linked and described by different components of the same spinor wavefunction,Dirac which means they are interconnected as Dirac fermions in QED. So graphene is a relativistic counterpart in the condensed-matter system. So far, 2D graphene has been successfully fabricated experimentally.ref1 ; ref01 By varying the gate voltageref3 or doping the underlying substrate,ref4 the charge carriers of graphene can be easily tuned, the controllable ballistic PNJ or PNPJ are also realized experimentally.pnj Therefore intriguing phenomena exhibited in the bipolar graphene,microwave ; Andreev ; aref1 ; Klein ; Kleinback ; lens ; Kleininter ; aref2 such as microwave-induced reflection,microwave specular Andreev reflection,Andreev ; aref1 Klein tunneling,Klein Klein backscatteringKleinback and negative refraction index effectlens , are possible to be verified experimentally. In fact, a direct experimental observation of Klein tunneling has been realized through a extremely sharp graphene PNJ.Kleininter It was shown that due to the Berry phase $\pi$, which was derived from the intersection of the energy bands at Dirac points,Berry the backscattering is absent.Ando1 This naturally leads to the so-called Klein tunnelingKlein or interband tunneling that an incident electron tunnel from the conduction into the valence band without backscattering. Because of the conservation of momentum and energy, interband tunneling through the p-n interface may resemble the optical refraction at the surface of meta-materials with negative refractive index.negrefconcept In another word, the Klein paradox gives rise to the negative refraction.negtolens This means that an interface of the symmetric PNJ perpendicular to the current flow is able to focus the electric current whereas a ballistic strip of p-type graphene separated by two n-type regions acts as a lens. These intriguing phenomena have been described in Ref.[lens, ], in which the Kubo formula was applied to the single-particle Dirac-like Hamiltonian of graphene. It means that for an infinite 2D graphene system with ideal conical energy spectrum, i.e., in the very low energy regime, the electrons emitting from source are perfectly focused at the mirror symmetric point of the symmetric PNJ. For the realistic graphene system, however, one can not separate electrons and holes close to Dirac point due to the electron-hole puddlespuddle (about tens $meV$puddle1 ). Of course, we can experimentally increase the density of electrons and holes by the gate voltage to evade from the puddle region. However, in the high density case, the energy spectrum deviates from the linear relation. Hence, the effect due to the non- linear dispersion should be examined. For this purpose, we will use the tight binding Hamiltonian to study the graphene based PNJ. In addition, considering the chirality of graphene ribbon, it is better to use the tight-binding Hamiltonian to describe the transport processes along different chirality directions. In this paper, using the tight-binding model, we carry out a theoretical study on the focusing effect of electron flow in the graphene ribbon with a symmetric PNJ. Due to the chirality of graphene, the focusing effects may be different for the zigzag ribbon and the armchair ribbon. Indeed, it is found that for the armchair ribbon with a sharp p-n interface the electron flow emitting from $(-L,0)$ in n-region can always be focused perfectly at $(L,0)$ in p-region for all energy $E_{0}=\|E_{F}-E_{p/n}\|<t$. For the zigzag ribbon, however, the electron flow is perfectly focused only in the very low regime ($E_{0}<0.05t$). Furthermore, the perfect focusing in the bipolar ribbon is robust against disorders induced by the random potential. But the edge disorders drastically affect the perfect focusing. For a smooth PNJ, electrons are backscattered close to PNJ at a distance proportional to $k_{y}$ and interface width $d$.mod In this case a quantum interference between forward and backward scattering is present and the intensity of the focused spot is weakened, but the focusing pattern keeps almost. Finally, in the presence of a weak perpendicular magnetic field, the momentum $k_{y}$ is no longer conserved. Consequently particles are deflected in opposite directions in the p-region and n-region, which destroys the perfect focusing especially in the low energy regime. The rest of the paper is organized as follows. In Sec. II, the model system including bipolar graphene ribbon in the tight-binding representation with attached source or detector terminal is introduced. The formalisms for calculating the local particle density, the local current density vector and the local conductance are then derived. Sec. III gives numerical results along with some discussions. Finally, a brief summary is presented in Sec. IV. ## II model and formalism In order to study the scattering due to PNJ, we consider two kinds of open bipolar graphene systems (armchair and zigzag ribbons) as shown in Fig.1. The bipolar graphene ribbon consists of semi-infinite electronlike ribbon [orange region] and semi-infinite holelike ribbon [green region] along $x$-direction with a sharp p-n interface located at $x=0$. Electron flow is injected into graphene system from a source lead located at $(-L,0)$ in the n-region. Here we assume that the source lead and the bipolar ribbon are in contact with six lattices [see the blue area in Fig.1]. The injected electrons in the n-region can spread in all directions. Because of the open boundary condition, left- going electrons can finally escape into infinite graphene electrode while right-going electrons [shown in Fig.2] can then be scattered only by p-n interface [thick black line]. Consequently, the response signals are converged around the symmetric site $(L,0)$ [red area in Fig.1]. In order to investigate the focusing current, we couple a detecting electrode locally in the p-region and study the local current (conductance) flowing from that electrode. Clearly, the local current depends on the coupling position of the detecting electrode. The total Hamiltonian including the infinite graphene ribbon in the tight- binding representationham and the source/drain electrode that is expressed in $k$ space with the free electron model can be written as: $\displaystyle H$ $\displaystyle=$ $\displaystyle\sum_{\bf i}\epsilon_{\bf i}a^{\dagger}_{\bf i}a_{\bf i}-\sum_{<{\bf ij}>}te^{i\phi_{\bf ij}}a_{\bf i}^{\dagger}a_{\bf j}$ (1) $\displaystyle+$ $\displaystyle\sum\limits_{\alpha,k}\left[\epsilon_{\alpha,k}d_{\alpha,k}^{\dagger}d_{{\alpha},k}+(\gamma a_{{\bf i}_{\alpha}}^{\dagger}d_{{\alpha},k}+h.c)\right]$ where ${\bf i}=({\bf i}_{x},{\bf i}_{y})$ is the index of the discrete site on the honeycomb lattice which is sketched in the Fig.1, and $a_{\bf i}$ and $a_{\bf i}^{\dagger}$ are the annihilation and creation operators at the site ${\bf i}$. Here $\epsilon_{\bf i}$ in the first term of Eq.(1) is the on-site energy (i.e., the energy of the Dirac point) which can be controlled experimentally by the gate voltage. In the n-region or p-region far away from PNJ all the on-site energy are the same with $\epsilon_{{\bf i}}=E_{n}$ or $\epsilon_{{\bf i}}=E_{p}$. Near PNJ, $\epsilon_{\bf i}$ changes from $E_{n}$ to $E_{p}$ abruptly or smoothly for the sharp PNJ or smooth PNJ, respectively. The second term in Eq.(1) is the nearest neighbor hopping term with the hopping energy $t$, ‘$<{\bf ij}>$’ denotes the nearest neighbor lattice sites. When the graphene ribbon is under a uniform perpendicular magnetic field $B_{z}=B$, a phase $\phi_{\bf ij}$ is added in the hopping term, and $\phi_{\bf ij}=\int_{\bf i}^{\bf j}\vec{A}\cdot d\vec{l}/\phi_{0}$ with the vector potential $\vec{A}=(-By,0,0)$ and the flux quanta $\phi_{0}=\hbar/e$. Finally, the last term in Eq.(1) represents the Hamiltonian of the source and detector leads described in the $k$ space and their coupling to the graphene lattices ${\bf i}_{\alpha}$. Here $\alpha=s,d$ represent source and detecting electrodes and $d_{{\alpha},k}$ ($d^{\dagger}_{{\alpha},k}$) is the annihilation (creation) operator of the electrons in the electrode $\alpha$. When the electron flow is injected from the source electrode into the graphene in the n-region, the response signal is induced everywhere in the p-region. To make a thorough study on the focusing effect, we consider three physical quantities in the p-region: (1) the local current density vector, (2) the local particle density, and (3) the local current (conductance) of the detecting electrode. For quantities (1) and (2), we consider the system without the coupling of the detecting electrode so that the influence of the detecting electrode can be eliminated. ### II.1 Local current density vector The general current density vector $\mathbf{J}_{\mathbf{i}\mathbf{j}}$ from the site ${\bf i}$ to its nearest neighbor site ${\bf j}$ can be expressed as:DOC $\displaystyle J_{\mathbf{i}\mathbf{j}}$ $\displaystyle=$ $\displaystyle\frac{e}{h}\int dE~{}~{}\left[{\bf G}^{<}_{\mathbf{i}\mathbf{j}}(E){\bf H}_{\mathbf{j}~{}\mathbf{i}}-{\bf H}_{\mathbf{i}\mathbf{j}}{\bf G}^{<}_{\mathbf{j}~{}\mathbf{i}}(E)\right]$ (2) $\displaystyle=$ $\displaystyle 2\frac{e}{h}{\rm Re}\int dE~{}~{}\left[te^{i\phi_{\bf ji}}{\bf G}^{<}_{\mathbf{i}\mathbf{j}}(E)\right]$ where $e$ is the electron charge, ${\bf G}^{<}_{\bf ij}$ is the matrix element of the lesser Green’s function of the scattering region. Because the graphene ribbon is translation invariant in the p/n-region, the central scattering region can be chosen arbitrarily as long as the source sites ${\bf i}_{s}$ and the detector sites ${\bf i}_{d}$ are included. From the Keldysh equation, the lesser Green’s function is related to the retarded and advanced Green’s functions, $\displaystyle{\bf G}^{<}(E)={\bf G}^{r}(E)\left[\sum_{\alpha}{\bf\Sigma}^{<}_{\alpha}(E)\right]{\bf G}^{a}(E)$ (3) Here the sum index $\alpha=L,R,s$ denote the left/right graphene lead and source lead with $\alpha\neq d$ because of the decoupling of the detector lead. The retarded Green’s function ${\bf G}^{r}(E)=[{\bf G}^{a}(E)]^{\dagger}=\\{E{\bf I}-{\bf H}_{0}-\sum_{\alpha}{\bf\Sigma}^{r}_{\alpha}(E)\\}^{-1}$, where ${\bf H}_{0}$ is Hamiltonian matrix of the central scattering region and ${\bf I}$ is the unit matrix with the same dimension as that of ${\bf H}_{0}$, ${\bf\Sigma}_{\alpha}^{r}$ is the retarded self-energy function from the lead-$\alpha$. ${\bf\Sigma}^{r}_{\alpha}$ can be obtained from ${\bf\Sigma}^{r}_{L/R}(E)={\bf H}_{c,L/R}{\bf g}^{r}_{L/R}(E){\bf H}_{L/R,c}$, where ${\bf H}_{c,L/R}$ (${\bf H}_{L/R,c}$) is the coupling from central region (lead-L/R) to lead-L/R (central region) and ${\bf g}^{r}_{L/R}(E)$ is the surface retarded Green’s function of the semi-infinite lead which can be calculated using a transfer matrix method.transfer Concerning the source lead, we take the wide band approximation, then the non-zero elements of self energy matrix ${\bf\Sigma}^{r}_{s}(E)=-i{\bf\Gamma}_{s}/2$ is energy independent, where line-width function ${\bf\Gamma}_{s}=2\pi\gamma^{2}\rho_{s}(E_{F})$. ${\bf\Sigma}^{<}_{\alpha}(E)$ in Eq.(3) is the lesser self energy of the lead-$\alpha$. Because the isolated lead is in the equilibrium, ${\bf\Sigma}^{<}_{\alpha}$ can be obtained from the fluctuation-dissipation theorem: $\displaystyle{\bf\Sigma}^{<}_{\alpha}(E)$ $\displaystyle=$ $\displaystyle\left[{\bf\Sigma}^{a}_{\alpha}(E)-{\bf\Sigma}^{r}_{\alpha}(E)\right]f_{\alpha}(E)$ (4) $\displaystyle=$ $\displaystyle i{\bf\Gamma}_{\alpha}(E)f_{\alpha}(E)$ with ${\bf\Sigma}^{a}_{\alpha}={\bf\Sigma}^{r,\dagger}_{\alpha}$ and $f_{\alpha}(E)=f_{0}(E-eV_{\alpha})$ where $f_{0}(E)=1/[{\rm exp}(E/k_{B}\mathcal{T})+1]$ is the Fermi distribution function. $V_{\alpha}$ is the external bias in the terminal-$\alpha$. Since we are interested only in the local response due to the source flow, the external biases are set as $V_{s}=\delta V$ and $V_{L/R}=0$. In calculating transport properties, we divide $G^{<}(E)$ into equilibrium and non-equilibrium parts as $\displaystyle{\bf G}^{<}(E)$ $\displaystyle=$ $\displaystyle{\bf G}^{r}(E)\left[if_{0}(E)\sum_{\alpha}{\bf\Gamma}_{\alpha}(E)\right]{\bf G}^{a}(E)$ (5) $\displaystyle+$ $\displaystyle{\bf G}^{r}(E)\left[i\sum_{\alpha}\left\\{f_{\alpha}(E)-f_{0}(E)\right\\}{\bf\Gamma}_{\alpha}(E)\right]{\bf G}^{a}(E)$ where the equilibrium term does not contribute to the transport and can be dropped out from now on. It is the non-equilibrium term that gives rise to the system response to the electron injection from the source lead. Because of $V_{s}=\delta V$ and $V_{L/R}=0$, we have $\displaystyle{\bf G}^{<}(E)=i{\bf G}^{r}(E)\left[f_{s}(E)-f_{0}(E)\right]{\bf\Gamma}_{s}{\bf G}^{a}(E)$ (6) Substituting Eq.(6) into Eq.(2) and considering the limit of small source bias, the local current density vector ${\bf J}_{\bf ij}$ (or the local conductance density vector ${\bf J}_{\bf ij}/\delta V$) can be expressed in the following form at zero temperature: ${\bf J}_{\mathbf{i}\mathbf{j}}/\delta V=\frac{2e}{h}{\rm Im}\left\\{te^{i\phi_{\bf ji}}\left[{\bf G}^{r}(E_{F})i{\bf\Gamma}_{s}{\bf G}^{a}(E_{F})\right]_{\bf ij}\right\\}$ (7) It should be noted that the current density ${\bf J}_{\bf ij}$ in Eq.(7) is defined between the lattice sites ${\bf i}$ and ${\bf j}$ with the direction from site ${\bf i}$ to ${\bf j}$. In order to obtain the local current density vector ${\bf J_{i}}$ at the site ${\bf i}$, we take the weighted average on ${\bf J}_{\bf ij}$ over all the nearest neighbors ${\bf j}$. ### II.2 Local particle density The local particle density (i.e. the electron occupation number) is defined as $\displaystyle\rho_{\bf i}$ $\displaystyle=$ $\displaystyle- ie\int\frac{dE}{2\pi}~{}~{}{\bf G}^{<}_{\bf ii}(E)$ (8) where ${\bf G}^{<}_{\bf ii}$ is the diagonal element of the lesser Green’s function ${\bf G}^{<}$ in Eq.(3). Similar to the derivation of local current density vector ${\bf J}_{\bf ij}/\delta V$, here we consider only the variation of the local particle density caused by the electron injection from the source lead. At zero temperature and the small bias $\delta V$ limit, substituting Eq.(6) into Eq.(8), the variation of the local particle density is expressed as: $\displaystyle\delta\rho_{\bf i}/\delta V$ $\displaystyle\equiv$ $\displaystyle[\rho_{\bf i}(V_{s}=\delta V)-\rho_{\bf i}(V_{s}=0)]/\delta V$ (9) $\displaystyle=$ $\displaystyle\frac{e^{2}}{2\pi}\left[{\bf G}^{r}(E_{F}){\bf\Gamma}_{s}{\bf G}^{a}(E_{F})\right]_{\bf ii}$ Since the Hamiltonian is defined at discrete lattice sites, the local quantities can also be defined at each lattice site. Such a local quantity is feasible but not necessary. In fact, for graphene, we can define the “local” quantity by averaging over six discrete sites in a unit cell of honeycomb lattice. This average can eliminate the strong variation of local quantities in the A and B sublattices. Now every local site can be determined from the coordinates $(x,y)$ shown in Fig.1. For example, the local injection area displayed in a blue area in Fig.1 is located at $(-3,0)$ in Fig.1(a) and $(-5,0)$ in Fig.1(b). In the whole lattice region in Fig.1, there are $7\times 3$ and $11\times 3$ units in Fig.1(a) and Fig.1(b), respectively. ### II.3 Local conductance Concerning the local conductance, the detecting lead-d is coupled to the graphene ribbon in the p-region. Similar to the local particle density, here the detecting lead also couples to six sites ${\bf i}_{d}$ in a unit cell of honeycomb lattice. The current flowing to the detecting lead-$d$ can be expressed as $\displaystyle J_{d}$ $\displaystyle=$ $\displaystyle\frac{e}{\hbar}\sum_{k_{d}}\left[{\bf G}^{<}_{{\bf i}_{d},k_{d}}(t_{1},t_{2}){\bf H}_{k_{d},{\bf i}_{d}}-{\bf H}_{{\bf i}_{d},k_{d}}{\bf G}^{<}_{k_{d},{\bf i}_{d}}(t_{1},t_{2})\right]_{t_{1}=t_{2}}$ (10) $\displaystyle=$ $\displaystyle\frac{e}{\hbar}\sum_{k_{d}}\left]{\bf G}^{<}_{{\bf i}_{d},k_{d}}(t_{1},t_{2})\gamma-\gamma{\bf G}^{<}_{k_{d},{\bf i}_{d}}(t_{1},t_{2})\right]_{t_{1}=t_{2}}$ Using the Dyson equation in the time contour, we can get the Landauer-B$\ddot{u}$ttiker formulaLanduer which is expressed in terms of non- equilibrium Green’s functions: $\displaystyle J_{d}$ $\displaystyle=$ $\displaystyle\frac{e}{\hbar}\sum_{\alpha}\int\frac{dE}{2\pi}{\bf T}_{d,\alpha}(E)[f_{d}(E)-f_{\alpha}(E)].$ (11) with $\alpha=L,R,s$ representing the left/right graphene leads and source leads. Since we shall concentrate only on the response current induced by the current injected from the source lead, we use the following boundary conditions $V_{L,R}=V_{d}=0$ and $V_{s}=\delta V$. The current now becomes, $\displaystyle J_{d}$ $\displaystyle=$ $\displaystyle\frac{e}{\hbar}\int\frac{dE}{2\pi}[T_{d,s}(E)(f_{d}(E)-f_{s}(E))].$ (12) where $T_{d,s}$ is the transmission coefficient from the source lead located at the site ${\bf i}_{s}$ to the detecting lead located at the site ${\bf i}_{d}$ which can be calculated from $T_{d,s}(E)={\rm Tr}\left[{\bf\Gamma}_{d}{\bf G}^{r}(E){\bf\Gamma}_{s}{\bf G}^{a}(E)\right]$, where ${\bf G}^{a}={\bf G}^{r\dagger}$ is the advanced Green’s function in the scattering region. In the wide band limit, the line-width function ${\bf\Gamma}_{s/d}(E)=i({\bf\Sigma}_{s/d}^{r}-{\bf\Sigma}_{s/d}^{r\dagger})=2\pi\gamma^{2}\rho_{s/d}(E_{F})$. Here $f_{s/d}(E)$ in Eq.(12) is the Fermi distribution function of the source and detecting lead, and $f_{s}(E)=f_{0}(E-eV_{s})$ and $f_{d}(E)=f_{0}(E)$. Considering the zero temperature and small bias $V_{s}$ limits, local conductance contributed by the source electron flow can be expressed as: $\displaystyle G_{{\bf i}_{d}}=J_{d}/\delta V=\frac{e^{2}}{h}T_{d,s}(E_{F}).$ (13) ## III numerical results and discussion In the numerical calculations, we set the nearest-neighbor carbon-carbon distance $a=0.142nm$ and the second nearest-neighbor distance $b=\sqrt{3}a\simeq 0.25nm$, the hopping energy $t=2.75eV$ as in a real graphene sample.ref3 In this paper, we consider only the focusing effect of the symmetric PNJ, in which the electron density of n region is the same as the hole density in p region, i.e., $\rho_{e}=\rho_{h}$. For simplicity, we set $E_{F}=0$. Hence in the n-region far away from PNJ the on site energy $\epsilon_{{\bf i}}=E_{n}=-E_{0}$ while $\epsilon_{{\bf i}}=E_{p}=E_{0}$ in the p-region far away from PNJ. Near PNJ, $\epsilon_{\bf i}$ changes from $-E_{0}$ to $E_{0}$ abruptly (smoothly) for the sharp (smooth) PNJ. Experimentally, it is more convenient to measure the electric conductance. So, in order to detect the focusing effect by a single PNJ in graphene, one can use a small electric contact (such as a STM probe) as a source of electron flow in the n-region and another local probe located in the p-region as a detector. Electric conductance between the two contacts measures the transmission probability for a charged carrier from the source to the detector. Numerically, We have calculated the local conductance $G_{{\bf i}_{d}}$ and confirmed that the distribution of local conductance is similar to that of the local particle density $\delta\rho_{\bf i}/\delta V$. For this reason, only the numerical results on local particle density are shown in this paper. In addition, in order to visualize the focusing process, we also show the distribution of local current density vector in the p-region. ### III.1 Focusing effect in very low energy regime Now we study the focusing effect in the graphene ribbon with a sharp PNJ. For a zigzag ribbon or a armchair ribbon with sharp and symmetric PNJ, the spacial distribution of the local particle density $\delta\rho(x,y)/\delta V$ in p-region due to electrons coming from the source lead is shown in Fig.3(a) and Fig.3(b), respectively. Following observations are in order. First of all, electrons injected at $(-L,0)$ in n-region can be focused around $(L,0)$ shown as red spot in Fig.3 which is similar to Ref.[lens, ]. This is because the Fermi energy $E_{F}$ is close to Dirac energy $E_{0}=0.05t$ so that the energy dispersion is nearly linear, i.e., $E_{0}\simeq kb\frac{\sqrt{3}}{2}t$ where $k$ is module of momentum vector $\mathbf{k}$. The charged carriers scattering through PNJ can mimic the refraction of light by left-handed metamaterials with refraction index equal to -1. Secondly, besides the focusing spot (red and green region), there is also a weak interference pattern (blue wave pattern) shown in Fig.3, which is different from Ref.[lens, ] in which the wave pattern is absent. In fact, the wave pattern is solely due to the boundary of the nanoribbon. When an electron is injected from the source area, it can propagate in all directions and the right-going electrons can be scattered by either the boundary of nanoribbon (thin black lines) or sharp PNJ (thick black line) as shown in Fig.2. In the p-region, interference pattern is due to the interference between the state scattered by both boundary and PNJ (thick blue lines) and the state scattered only by PNJ (thick red lines). The spacial period of the interference is proportional to the momentum ${\mathbf{k}}$ or $E_{0}$. Finally, the focusing phenomena in zigzag ribbon is slightly different from that of armchair ribbon: the electron flow is perfectly focused in armchair ribbon [panel (b)], but can’t be fully focused in the zigzag ribbon [panel (a)]. This is because the energy band structures are different for the armchair ribbon and zigzag ribbon. In the following, we will examine the different focusing effects in detail. ### III.2 Focusing in zigzag ribbon When Fermi energy is gradually moved away from Dirac point, the energy spectrum is not ideal conical anymore. In Fig.4 we plot the contour lines of dispersion relation $E(k_{x},k_{y})$ of graphene sheetfootnote with energy interval between nearest contour lines $\delta E=0.1t$. The panel (a) is for the graphene sheet with the carbon-carbon bond along the $x$-direction which corresponds to the armchair graphene ribbon, and the panel (b) is for the graphene sheet with the carbon-carbon bond along the $y$-direction corresponding to the zigzag ribbon. The deviation of ideal conical energy spectrum is clearly exhibited even at small energy $E=0.2t$ (the second small contour lines around the Dirac points $K$ and $K^{\prime}$ show anisotropy behaviors). Since the p-n interface at $x=0$ is along $y$ direction, the $y$ component of momentum, $k_{y}$, is conserved during the scattering. As a result, the incident wave vector $k_{x,in}$, the reflecting wave vector $k_{x,r}$, and the transmitting wave vector $k_{x,t}$ must lie on the black dotted lines in Fig.4. When an electron with energy $E=E_{F}$, velocity $(v_{x},v_{y})$ and corresponding momentum $(k_{x,in},k_{y,in})$ with respect to Dirac point $K$, injects from n-region and is scattered at the p-n interface, according to the identical direction of $V_{x}$, we can solve the reflecting and scattering momentum $k_{x,r}$ and $k_{x,t}$ using the energy conservation and $k_{y}$ conservation. For the zigzag ribbon [corresponding to Fig.4(b)], $k_{x,in}$ can be intra-scattered to $k_{x,r/t}$ around $K$ [$K=(\frac{2}{3}\frac{2\pi}{b},0)$] valley or inter-scattered to $k_{x^{\prime},r/t}$ in $K^{\prime}$ [$K^{\prime}=-(\frac{2}{3}\frac{2\pi}{b},0)$] valley. The interband scattering states is symmetric which satisfies $k^{\prime}_{x,r/t}=-k_{x,in}$. The intraband scattering, however, exhibits asymmetric properties that $\delta k_{x,r/t}\equiv k_{x,in}+k_{x,r/t}\neq 0$ or $v_{x,in}\neq-v_{x,r}$, $v_{x,in}\neq v_{x,t}$ for any fixed $k_{y}$. In Fig.5, we plot the asymmetric relation between $k_{x,in}$ and $k^{intra}_{x,r/t}$ in the intraband scattering and the symmetric relation between $k_{x,in}$ and $k^{inter}_{x,r/t}$ in the interband scattering. We see that in the intraband scattering, the larger $k_{y}$, the larger derivation $\delta k^{intra}_{x,r/t}$ is, while in the interband scattering, $k^{inter}_{x,r/t}$ is always equal to $-k_{x,in}$ for all $k_{y}$. It is known that interband scattering is weakvalleysuppress in pure samples due to the large momentum shift, so the asymmetric intraband scattering is dominant in zigzag ribbon PNJ. As a result, the refraction index can not be strictly equal to $-1$ and the charge flow can’t be fully converged at the symmetric spot. In Fig.6 focusing effect for $E_{0}=0.1t$ and $E_{0}=0.2t$ are plotted, respectively. Comparing Fig.6(a) and Fig.6(b) we see that it is more difficult to focus the electron beam for larger momentum ${\mathbf{k}}$ (energy $E_{0}$). ### III.3 Focusing in armchair ribbon For the armchair ribbon [corresponding to Fig.4(a)] only intraband scattering occurs. Now $k_{x,in}$ can be symmetrically intra-scattered to $k_{x,r/t}$ in $K$ valley with $-k_{x,in}=k_{x,r}=k_{x,t}$ since $k_{x}$ in Fig.4(a) is symmetric about $k_{x}=0$. So, although the energy dispersion of armchair ribbon is also not strictly linear at high energies as in zigzag ribbon, due to the symmetric scattering, the focusing effect is always perfect in armchair ribbon for all $E_{0}<t$ (i.e. in Dirac fermion regime). Furthermore, with increasing of $E_{0}$, the electron flow coming from n-region shows better convergence in the p-region with smaller focusing spot and stronger intensity. Furthermore, The spacial period of the interference pattern is proportional to the momentum ${\mathbf{k}}$ or $E_{0}$, which can be clearly seen by comparing Fig.3(b), Fig.7(a) and Fig.7(b). . Roughly speaking, two energy regimes are considered for the Dirac Fermion according to band structure of graphene: (1). ’Near linear dispersion’ regime $0<E_{0}<0.5t$ where $E_{0}\approx kb\frac{\sqrt{3}}{2}t$. (2) ’Beyond linear dispersion’ regime $0.5t<E_{0}<t$ where the energy spectrum is non-conical. The focusing effect corresponding to these two regimes are plotted in Fig.7 and Fig.8, respectively. In the first regime, with the near linear dispersion relation, velocity $v_{x}$ or $v_{y}$ is roughly a constant, and ${k_{y,in}}/{k_{x,in}}\approx{v_{y,in}}/{v_{x,in}}$, ${k_{y,t}}/{k_{x,t}}\approx{v_{y,t}}/{v_{x,t}}$. For the symmetric scattering ($k_{x,in}=k_{x,t}$) in the armchair ribbon, refraction index $n\approx-1$ giving rise to the convergent spot shown in Fig.7. When $E_{0}$ is large enough (larger than 0.5t), energy spectrum is non-conical and velocity now depends on momentum. This leads to a different focusing effect shown in Fig.8 in which crossed focusing zone is present. In order to show the focusing of the electron flow vividly, instead of the contour of local particle density in Fig.7(b) and Fig.8(b), the quiver of local current density vector around the convergence spot in p-region are plotted in Fig.9 and Fig.10, respectively. For demonstration purpose, the local current density is plotted at every other site. The arrow on each site denotes the local current density vector whose module and direction are described by the size/color and orientation of the arrow respectively. In the low energy regime (Fig.9), the vectors of local current density converge conically to the focusing spot [red spot in Fig.7(b)]. On the other hand, current density is converged mainly from four crossed corners in the high energy regime (Fig.10). Furthermore, comparing Fig.9 and Fig.10, it is clear that electron flow with larger $E_{0}$ gives better convergence. ### III.4 Effect of disorders in armchair nanoribbon As discussed in the previous sections, the clean graphene PNJ is investigated. In a real device, the disorder is always present. In this subsection, we study the disorder effect on the perfect focusing in the armchair nanoribbon. We consider two kinds of disorders, one is induced by random on-site potential $\delta\epsilon_{\bf i}$ and the other is due to the edge defect.edgedefect The random on-site potentials $\delta\epsilon_{\bf i}$ with a uniform distribution $[-w/2,w/2]$ are added near PNJ within the width of $18a$ where $w$ is disorder strength. The edge defect is modeled through missing atoms on the graphene edge. We model the missing atom by setting the corresponding hopping matrix elements to zero. The edge roughness is controlled by $p$, the probability of a missing atom on the outermost row [the red line in Fig.11]. For both on-site potential disorder and edge defect, all data are obtained by averaging over $500$ configurations. In Fig.12 we plot the contour of local particle density in armchair ribbon with a sharp PNJ for $E_{0}=0.5t$ [same as in Fig.8(a)] in the presence of random on-site potential disorder. In Fig.12, the width of armchair ribbon is set to $105b$, the source flow is injected from the honeycomb unit cell at $(-210a,0)$ and focused around the spot located at $(210a,0)$. Panel (a), (b), (c) and (d) correspond to different disorder strengths $w=0$, $0.2$, $0.5$ and $1.0$. For the small random potential strength $w$ (e.g. $w=0.2$), the interference pattern and the focusing spot can be well kept. On the other hand, for the large $w$ (e.g. $w=1.0$), we can see that the random potential disturbs the interference between forward and backward scattering, so the interference pattern is smeared, which increases the density of state outside the focusing spot. Consequently, the intensity of focusing spot decreases. However, we emphasize that although random potential disturbs the interference pattern and reduces the intensity of the focusing spot, the focused spot is clearly visible and its size still remains unchanged. It means that the focusing effect is robust against random potential, especially in the weak disorder case. In Fig.13 we plot the contour of local particle density in the presence of edge defect for $E_{0}=0.5t$. Panel (a), (b), (c) and (d) are corresponding to the different probability $p$ of a missing atom on the outermost row with $p=0$, $0.1$, $0.2$ and $0.5$. We find that in the presence of edge disorder, the size of focusing spot increases clearly and the focusing intensity is greatly reduced. So the effect of the edge defect on the focusing effect is more significant than that of the random potential. But the focusing spot and interference pattern still survive and are clearly visible (see Fig.13d), even in the strong edge defect case with $p=0.5$. To estimate the disorder strength needed to reduce the intensity of focusing spot, in Fig.14(a) and (b), we plot the maximum value (the value at the focusing spot central $[L,0]$) of the focused spot ${\rm LDOS}_{max}$ vs strength of random potential $w$ and the probability of a missing atom $p$. Considering the computational cost, here we take 200 configurations and label the error bar. From Fig.14(a), we find that for weak random potential (when $w<0.5$), ${\rm LDOS}_{max}$ hardly changes with $w$, and focusing effect remains unchanged [see Fig.12(a), (b) and (c)]. Beyond the weak disorder regime ($w>0.5$), ${\rm LDOS}_{max}$ declines abruptly, and focusing effect can’t be kept as good as in the weak disorder regime [see Fig.12(d)]. On the other hand for the edge defect (see Fig.14(b)), we can see that the electron beam can be focused perfectly at $p=0$ and $p=1$ because the graphene ribbon edges are intact at both $p=0$ and $1$. When $p$ increases from $0$ to $1$, more and more atoms in edge are missing until two edges are completely peeled. Correspondingly ${\rm LDOS}_{max}$ decreases first and then increases since the edges are the most random when $p$ is around 0.5. We notice in Fig.14(b), comparing to ${\rm LDOS}_{max}$ near $p=1$, ${\rm LDOS}_{max}$ is reduced faster near $p=0$. It means that the vacancy defect (a few atoms are missing on edges) destroys focusing effect more significantly than the adsorption defect (a few atoms are attached to edges). ### III.5 Focusing of armchair ribbon with smooth PNJ Up to now, we have studied focusing effect by the sharp PNJ. But in realistic graphene based PNJ or PNPJ, the potential changes smoothly from $E_{n}$ to $E_{p}$ within a width $d$. The width $d$ is of the order of the separation between the graphene layer and the top gate and $d\sim~{}$tens nm.pnj In such a smooth PNJ, backscattering is present near PNJ in the distance proportional to $k_{y}$ and interface width $d$, which reduces the possibility of Klein tunneling. For a linear electrostatic potential $U(x)=(vk_{F}/d)x$, the angular dependent transport probabilityKlein $T(\theta)=e^{-\pi(k_{F}d)\sin^{2}(\theta)}$ where $\theta$ is incident angle. It is obvious that the smooth PNJ will reduce the intensity of the focused electron beam due to the decreased transport probability $T(\theta)$. It appears that it also increases the size of the focused spot. Actually, it is not the case due to the following reason. For a single n-p junction (whether smooth or sharp), the electrons (holes) with an energy equal to the chemical potential $\mu=0$ and momentum $k_{x}=k_{F}\cos(\theta)$ transport from conduction (valence) band to the the valence (conduction) band with the conserved $k_{y}=k_{F}\cos(\theta)$ but $k^{\prime}_{x}=-k_{x}$, leading to the almost unchanged focusing pattern, as shown in Fig.15. The smooth PNJ can be modeled by smoothly varied Dirac energy $U(x)$ across the PNJ. In the numerical calculation, we use the following $U(x)$, $\displaystyle U(x)=\left\\{\begin{array}[]{rr}-E_{0}\left[1+{{\rm sinh}(\frac{x_{0}}{L})}/{{\rm sinh}(\frac{x-x_{0}}{L})}\right],&x\leq 0\\\ &\\\ E_{0}\left[1-{{\rm sinh}(\frac{x_{0}}{L})}/{{\rm sinh}(\frac{x+x_{0}}{L})}\right],&x\geq 0\end{array}\right.$ (17) where $L$ is the distance between source probe located at $(-L,0)$ and PNJ at $x=0$. In Fig.15(a), with $L=320\times 0.43nm$, $U(x)$ for different $x_{0}$ have been plotted. In Fig.15(b, c, d) with the same parameters used in Fig.7(a) in which the sharp PNJ is used, the contour of local particle density is re-plotted for the smooth PNJ shown in Fig.15(a). We can see that the quantum interference between the states scattered by forward and backward scattering is present. The density of state outside focused spot is increased. Moreover, intensity of the convergent spot is reduced comparing to that of sharp PNJ due to the reduced transmission probability. The wider the PNJ interface, the more the focusing effect is reduced because of the smaller $T(\theta)$. For example, when the PNJ width $d=0$, the maximum of local particle density of state ${\rm LDOS}_{max}=0.0095$ [see Fig.7(a)], increasing $d$ gradually, ${\rm LDOS}_{max}=0.0068$, $0.0035$ [see Fig.15(b) and (c)]. When the PNJ width $d$ is reaching to Fermi wavelength [such that $k_{F}d\sim 1$, in Fig.15, $k_{F}\sim 1/(15a)$], the intensity of focused electron beam decrease very slowly [see Fig.15(c) and (d)]. For the very big $d>>1/k_{F}$, the intensity of focused spot decreases and its size increases continually. In addition, due to the Klein tunneling, the focusing effect can still occur and the convergent contour is almost the same as that of the sharp PNJ, although the intensity of focusing spot is reduced. ### III.6 Focusing of armchair ribbon in the presence of small perpendicular magnetic field In the presence of small perpendicular magnetic field, the momentum $k_{y}$ is not a conserved quantity. In this case, electrons and holes are deflected in opposite directions due to the opposite Lorentz force. So the injecting electron flow in the n-region now can’t be effectively converged in the p-region and the focusing spot is smeared. Considering the opposite deflection for the electrons and holes, the smeared convergent spot will move away from symmetric point $(L,0)$. The larger the size of scattering region, the focusing effect is less significant because of the more deflection in the larger size. On the other hand, the transmission probability $T$ of a single PNJ becomes magnetic field dependent on the field scale $B^{}=(\hbar/e)k_{F}/d$ with which the cyclotron radius $l_{cycl}=\hbar k_{F}/eB$ becomes comparable to the width $d$ of PNJ. The maximum angle rotating away from normal incidence $\theta_{max}=\pm\arcsin(B/B^{})$. The transmission probability of a bipolar ribbon is suppressed as $T(B<B^{})\propto\frac{W}{d}(1-(B/B^{})^{2})^{3/4}$ where $d$ is width of PNJ and $W$ is width of ribbon.Magnet The influence of transmission probability on focusing effect is mainly to reduce the intensity of focused spot. So in the presence of the weak magnetic field, not only the intensity of focused spot is reduced but the focusing pattern is destroyed by the deflection of electron beam as well. In Fig.16, we plot the local particle density in the armchair ribbon with small magnetic field for sharp and symmetric PNJ. For the sharp PNJ, intensity of focused spot is reduced not so severely as in the smooth PNJ. The magnetic field $B$ is expressed in terms of magnetic flux $BS_{0}$ in the unit of $\phi_{0}/\pi$ where $S_{0}=\frac{3}{2}\sqrt{3}a^{2}$ is the area of a honeycomb unit cell and $\phi_{0}=\hbar/e$ is the flux quanta. Here $BS_{0}=0.0001\phi_{0}/\pi$ corresponds to the magnetic field $B=0.4T$. From Fig.16, we can see that in the low energy regime [panel (a), $E_{0}=0.2t$] the focusing effect is destroyed severely, and focusing effect is hardly destroyed in the higher energy regime [panel (b), $E_{0}=0.9t$] due to the less influence of magnetic field on electron flow with higher energy. ## IV conclusion In conclusion, using the tight-binding Hamiltonian, we report the focusing of electron flow in zigzag or armchair graphene ribbon with a single symmetric PNJ. For a sharp PNJ, in the very low energy regime ($\|E_{F}-E_{n/p}\|=E_{0}<0.05$), graphene ribbon exhibits almost conical energy spectrum and the electron flow coming from n-region can be converged in the p-region perfectly. When energy $E_{0}$ increases, however, energy spectrum gradually deviates from linear behavior. And the band structures are different for zigzag ribbon and armchair ribbon. For the zigzag ribbon, although the interband scattering is symmetric with $-k^{inter}_{x,in}=k^{inter}_{x,r/t}$ but the dominant scattering, intraband scattering, is asymmetric with $k^{intra}_{x,in}+k^{intra}_{x,r/t}\neq 0$. As a result, the electron flow coming from source in n-region can’t be converged in the p-region. As for the armchair ribbon, only the intraband scattering exists which is always symmetric with $-k_{x,in}=k_{x,r/t}$ for all $k_{y}$. This leads to a perfect focusing effect for all energy $E_{0}<t$ regardless of linear or nonlinear dispersion relationship of Dirac Fermion. Specifically, the electron flow converges conically in the low energy regime ($E_{0}<0.5t$), and converge mainly from four crossed corners in the high energy regime ($0.5t<E_{0}<t$). When disorder is present, the perfect focusing in the bipolar ribbon can be robust against disorder induced by the random potential. The perfect focusing is however drastically affected by the edge defect, not only the intensity of focused spot is reduced, the size of spot is also increased. Furthermore, when the real smooth PNJ is considered, Klein tunneling is reduced significantly due to the backscattering. In this case the intensity of convergent spot is reduced, but the convergent contour still remains the same. 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Heinzel, Phys. Rev. B 78, 161407(R) (2008); Hengyi Xu and T. Heinzel, I. V. Zozoulenko, Phys. Rev. B 80, 045308 (2009). * (31) A. V. Shytov, Nan Gu, L. S. Levitov, cond-mat/0708.3081 (2007). Figure 1: (Color online) A schematic diagram of the graphene PNJ in a armchair ribbon [panel (a)] and in a zigzag ribbon [panel (b)]. The graphene ribbon is along $x$-direction and sharp p-n interface is located at $x=0$. The electron flow injected at $(-L,0)$ [the blue area] in the n-region [orange lattice region] is focused around the symmetric site $(L,0)$ (the red area) in the p-region [green lattice region]. Figure 2: (Color online) After injecting from the source lead in the n-region, the right-going electrons are scattered by the PNJ and boundary of the bipolar graphene ribbon. Figure 3: (Color online) Distribution of local particle density $\delta\rho(x,y)/\delta V$ in a graphene ribbon with a single sharp PNJ at $x=0$. Panel (a): zigzag ribbon with ribbon width $W=500\times 3a$. The source flow is injected from the honeycomb unit cell at $(-800b,0)$ and focused around the spot located at $(800b,0)$. Panel (b): the armchair ribbon with ribbon width $W=501b$. The source flow is injected from the honeycomb unit cell at $(-320\times 3a,0)$. The other parameter used: $E_{0}=0.05t$. Figure 4: (Color online) The contour of dispersion relation $E(k_{x},k_{y})$ of graphene sheet. The energy interval between nearest contour lines is $\delta E=0.1t$. Panel (a): $E(k_{x},k_{y})$ of the graphene sheet with the carbon-carbon bond is along the $x$-direction corresponding to armchair ribbon or Fig.LABEL:structure(a). Panel (b): $E(k_{x},k_{y})$ of the graphene sheet with the carbon-carbon bond is along the $y$-direction corresponding to zigzag ribbon or Fig.1(b). Figure 5: (Color online) Scattering momentum $k_{x,r/t}^{intra}$ and $k_{x,r/t}^{inter}$ vs injecting momentum $k_{x,in}$ for the zigzag ribbon with sharp PNJ. In the interband case, $k_{x,r/t}^{inter}=-k_{x,in}$ for all conserved $k_{y}$ (red line). While in the intraband case (black lines), $k_{x,r/t}^{inter}$ is not equal to $-k_{x,in}$. Different black lines along the black arrow are corresponding to $k_{y}=0$, $0.1/b$, $0.15/b$, $0.2/b$, respectively. Figure 6: (Color online) Contour of local particle density in zigzag ribbon with a sharp PNJ for $E_{0}=0.1t$ [panel (a)] and $E_{0}=0.2t$ [panel (b)]. Figure 7: (Color online) Contour of local particle density in armchair ribbon with a sharp PNJ for $E_{0}=0.1t$ [panel (a)] and $E_{0}=0.2t$ [panel (b)]. Figure 8: (Color online) Contour of local particle density in armchair ribbon with a sharp PNJ for $E_{0}=0.5t$ [panel (a)] and $E_{0}=0.9t$ [panel (b)]. Figure 9: (Color online) Instead of contour of local particle density in Fig.7(b), the quiver of local current density vector around convergence spot is plotted. Figure 10: (Color online) Instead of contour of local particle density in Fig.8(b), the quiver of local current density vector around convergence spot is plotted. Figure 11: (Color Online) panel (a): sketch of edge in the scattering region with zigzag edge. panel (b) sketch of edge in the scattering region with armchair edge. Figure 12: (Color Online) Contour of local particle density in the armchair ribbon with a sharp PNJ for $E_{0}=0.5t$. Panel (a), panel (b), panel (c) and panel (d) are corresponding to random on-site potential strengths $w=0$, $0.2$, $0.5$ and $1.0$, respectively. Figure 13: (Color Online) Contour of local particle density in armchair ribbon with a sharp PNJ for $E_{0}=0.5t$. Panel (a), panel (b), panel (c) and panel (d) are corresponding to $p=0$, $0.1$, $0.2$ and $0.5$, respectively. Figure 14: According to Fig.12 and Fig.13, maximum value (the value at the focusing spot central $[L,0]$) of the focusing spot ${\rm LDOS}_{max}$ vs the strength of random potential $w$ [in panel (a)] and probability of one missing atom $p$ [in panel (b)] are plotted, respectively. Figure 15: (Color Online) panel (a): smoothly changed $U(x)$ forming smooth PNJ used in panel (b, c, d), in which $x_{0}=0.253a$, $0.753a$ and $3*3a$, respectively. Panel (b, c, d): Contour of local particle density in armchair ribbon for $E_{0}=0.1$ [same to Fig.7(a) where sharp PNJ is used]. For different panels, smooth PNJ shown in panel (a) are used respectively. Figure 16: (Color online) Contour of local particle density in the p-region for a armchair ribbon in the perpendicular weak magnetic filed. The sharp PNJ is located at $x=0$. The magnetic field $BS_{0}=0.0001\phi_{0}/\pi$, $E_{0}=0.2t$ in panel (a) and $E_{0}=0.9t$ in panel (b).	arxiv-papers	2010-11-12T04:05:43	2024-09-04T02:49:14.759665	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Yanxia Xing, Jian Wang and Qing-feng Sun", "submitter": "Xing Yanxia", "url": "https://arxiv.org/abs/1011.2821" }
1011.2946	A traditional paper-based passport contains a Machine-Readable Zone (MRZ) and a Visual Inspection Zone (VIZ). The MRZ has two lines of the holder’s personal data, some document data, and verification characters encoded using the Optical Character Recognition font B (OCR-B). The encoded data includes the holder’s name, date of birth, and other identifying information for the holder or the document. The VIZ contains the holder’s photo and signature, usually on the data page. However, the MRZ and VIZ can be easily duplicated with normal document reproduction technology to produce a fake passport which can pass traditional verification. Neither of these features actively verify the holder’s identity; nor do they bind the holder’s identity to the document. A passport also contains pages for stamps of visas and of country entry and exit dates, which can be easily altered to produce fake permissions and travel records. The electronic passport, supporting authentication using secure credentials on a tamper-resistant chip, is an attempt to improve on the security of the paper-based passport at minimum cost. This paper surveys the security mechanisms built into the first-generation of authentication mechanisms and compares them with second-generation passports. It analyzes and describes the cryptographic protocols used in Basic Access Control (BAC) and Extended Access Control (EAC). # A Survey of System Security in Contactless Electronic Passports Anshuman Sinha ###### Index Terms: ePassport, Electronic Passport, RFID, Security, Contactless, EAC, BAC, MAC, PKI ## I Introduction In an effort to secure the borders of the United States of America, Congress has legislated requirements for Electronic Passports (ePassports) [16] for all visitors from countries participating in the Visa Waiver Program (VWP) [4]. Any passport issued by a participating state after October 2010 must be machine-readable with an electronic facial image encoded on a secure chip. As a requirement of the US-VISIT program [5], all visitors to the U.S. must have their photo, and a fingerprint of their index finger, taken for electronic comparison. In a reciprocal move, the U.S. has started issuing electronic passports to its citizens from all domestic issuance agencies since August 2007. European Union countries have advanced to a second-generation electronic passport. Countries like Germany, France, and the Czech Republic have passports at the next level, whereby biometric information is stored in a secure passport chip. Similarly, Asian countries like Malaysia and European Union (EU) countries have already advanced to biometrics and Extended Access Control (EAC) like access control. Any effort to make passports electronic and secure requires adding hardware, firmware, and software at different levels to the existing verification infrastructure. The centerpiece of the next generation of passport technology lies in Java programmable secure controllers with advanced cryptographic capabilities. An ePassport has an embedded Radio Frequency Identification chip (RFID) with processing capability for cryptographic computations. Unfortunately, the wireless link between this passport tag, which is called chip in the remainder of this paper, and a passport verification reader can lead to security and privacy threats. Threat modeling is based on first principles which analyzes compromise to common security goals of any system. ## II Threat Model for ePassports Electronic passports [11] must prevent the known attacks common to secure radio frequency-based identification and access control systems. Comprehensive threat modeling and designing countermeasures are key to the robustness of any security system; ePassports are no different in this regard. Each threat must have one or more mechanisms (or countermeasures) built into the system. A security system can be designed only to secure against known threats and attacks. A threat is defined as a weak link in the system which can be broken to compromise system’s security. A threat model should be comprehensive enough to include all known attacks. Threat and attack model will have overlaps, however this may not be the case for every noted threat. TABLE I: ePassport Threat Model S.No. \| Security Threats \| Compromises ---\|---\|--- 1. \| Forging \| Integrity 2. \| Repudiation \| Availability 3. \| Skimming \| Confidentiality 4. \| Cryptographic Analysis \| Confidentiality \| \| & Integrity 5. \| Unauthorized System \| Authorization 6. \| Unauthorized User \| Verification 7. \| Privacy Threats \| Consent, verification \| \| & authentication 8. \| Platform Integrity \| Integrity However, system may fail to secure against unknown threats or those discovered after the design has been committed and the product taken to field, side channel attacks [20] is example of one of the threat which was not modeled and accounted for in the design of early hardware security systems. Cryptographer these days are aiming to build systems which would be safe to unknown attacks. A threat model for ePassports is given in Table I. The left column shows known threats that must have“antidotes” in the design to ensure that fundamental security requirements are not violated. The right column shows the requirement(s) at risk. System security, including passport system security, comprises the following fundamental security requirements: Secrecy - preventing system information from flowing in unauthorized paths; Integrity - preventing unauthorized modification of system state; Legitimate Use - preventing unauthorized use of system resources; Availability - preventing unauthorized interference in the use of the system. Notice that all of these issues involve the notion of authorization. ##### Forging A passport can be forged by replacing a complete chip with a different one. There are two cases of such an attack. The first is replacement of a chip with a cloned or duplicate LDS, the unprotected contents of an ePassport’s Logical Data Structure. In this case, the duplicated passport LDS matches that of an original electronic passport. This is a cloning of a chip and its complete data set. The second case is replacement of a chip with a tampered LDS. (A secure chip generally has hardware mechanisms built-in to protect against data alteration.) Forging clearly violates integrity. ##### Non-Repudiation A non-repudiation attack concerns an ePassport that has been tampered with to withhold information embedded in its secure chip. The chip or its antenna could be set to a state where it can not be read by a valid reader. In such a scenario, the passport holder can not be verified electronically. Due to the non-availability of the secure chip, two situations emerge. The first is the failure to electronically authenticate the passport. If the secure chip is not available, the passport can not be authenticated and verified electronically. The authentication must be performed by relying on MRZ data. The second case that emerges is failure to verify the passport holder. Due to the non- availability of the passport chip, the holder’s biometrics and other data stored on the chip may not be verified. If the passport holder’s biometrics are stored only on the secure chip and not in a backend system, the biometrics cannot be verified. ##### Skimming The unprotected contents of an ePassport’s LDS could be read by an unauthorized reader close enough to the passport to use the wireless link. A skimmer could utilize a reader which has been modified to read data from distances greater than the passport designers anticipated. ##### Cryptographic Analysis The cryptographic keys stored in an ePassport chip could be exposed by deploying mass computing power after gathering skimmed or eavesdropped data from different electronic passports. The keys could be used to illicitly communicate with other passports. ##### Unauthorized System A system may be authentic but may not have the correct privileges to read information from the passport. ##### Unauthorized User Unauthorized user is anyone who is not the person to whom the identity credentials had been issued. Such a person misuses the identity documents by using someone else’s information or appearance. ##### Privacy Threats Privacy threats [17] are unapproved use of personal information or tracking of the passport holder. ##### Platform Integrity The platform on which applications run must be free from any malicious code which can act as a Trojan Horse for the information stored in the ePassport. Usually, such methods are added to the system for the ease of testing during development and must be removed to completely secure the system. ## III Known Attacks for ePassports Security mechanisms built into ePassports must counter the specific known attacks shown in Table II. The attack profile is as important as the threat profile. TABLE II: ePassport Attack Model S.No. \| Known Attacks ---\|--- 1. \| Forging 2. \| Skimming 3. \| Eavesdropping 4. \| Illicit Verification 5. \| Data/Noise Injection 6. \| Impostor/False Biometrics 7. \| Rogue Reader/Hardware 8. \| Duplication/Cloning 9. \| Tracking ##### Forging A passport can be forged by replacing its complete chip with a different one. There are two such cases. First, the chip is replaced with a cloned or duplicated LDS. In this case, the chip on the duplicated passport matches the data contents of the original ePassport. This is cloning of a chip and its complete data set. Second, the chip is replaced with a tampered LDS. A new chip, with a modified copy of credentials is put in place of the original chip. Such a modified secure chip is embedded into a passport from which the original chip was removed. A secure chip has hardware mechanisms built to protect against data alteration. ##### Skimming An ePassport can be skimmed to read the unprotected contents of its LDS. The skimmer may gather sensitive details like the passport holder’s name, age, address, and travel information. The skimmer could utilize readers which are modified, extended, or rogue to read data from distances greater than designed for. ##### Eavesdropping Eavesdropping is an attack to intercept communication between card and reader. The stolen information could either be document or personal information or system related cryptographic information which can be replayed to illicitly communicate with other passports. ##### Illicit Verification Illicit verifiers are like fake ATM(s) installed at locations to falsely retrieve the PIN of a banking card. The personal information could be skimmed if such a system with copied keys is installed. ##### Data/Noise Injection A data injector is a device which can manipulate or alter the data being sent by a secure chip to a reader or vice versa. It can interject data with its own data frames which may not only interfere with communication but also alter add or subtract data exchanged between secure reader and chip. ##### Imposter/False Biometrics An impostor can fake the biometrics of an authentic passport holder with methods like wax fingers or face masks to fool the system. Impostors break the security of a system by faking the identity of the actual passport holder. ##### Rogue Reader/Hardware A modified reader can be used to read the contactless data from a card beyond it normal operating range. Other modified hardware can store communications while legitimately communicating with a reader. ##### Duplication/Cloning Cloning of a chip and duplication of its complete or partial data is an attack requiring sophisticated machinery and means. Such attacks are conducted only by advanced attackers who have the finances to invest in such systems. ##### Tracking A tracker is a person who is only interested in knowing where the passport holder is traveling. A tracker may not have access to all the information which is stored in the secure chip. The tracker is only interested to know the whereabouts of a person. The tracker skims the data which could be used to trace a passport holder’s location. ## IV Authentication in ePassport Systems International Civil Aviation Organization (ICAO) [6] has defined two different mechanisms to authenticate secure chips embedded in ePassports: active and passive authentication. In active authentication, the secure controller processes cryptographic information in the chip; in passive authentication, no computation is involved and the contents of a tamper-proof chip are read only by a verification device. Consequently, passive authentication is implemented on secure memory devices whereas active authentication requires a processor. Lately, a new type of authentication for EAC, called Chip Authentication has been proposed by European Union (EU) [2]. Similar authentication was also proposed by some of the far east countries, but this paper details only the EU proposal. ### IV-A Passive Authentication An ePassport’s document security object ${SO}_{D}$ is digitally signed by issuing country at the time of personalization and the certificate is stored in secure chip. Hash of each data group (DG) is computed and stored in secure passport chip. In Passive Authentication (PA), the inspection system first verifies issuing country’s signed security data object [12] using public keys stored in the inspection system. If the signature matches, the hash of each data group is verified. By verifying hash of data groups inspection system infers if the data has been tampered. Certificate of document signer may be distributed to visited country in lieu of storing it on the chip. Typically, a visited country enters into an agreement with the issuing country to obtain the certificate and distribute it at different entry check points. Before checking the signature, validity of signed certificate is verified by checking Certificate Revocation List (CRL) for any updates. The revocation lists are regularly updated in a secure but mutually agreed storage area known as the Public Key Directory (PKD). The ePassport Public Key Infrastructure (PKI) symbols for passive authentication are defined in Section E of the appendix. #### IV-A1 Weaknesses Passive authentication does not prevent copying of chip data onto another chip, skimming, or unauthorized access to contents stored in the chip. The certificate of a document signer from an issuing country is stored on the secure chip. Reading a certificate from a secure chip to authenticate and verify a signature is not a good security practice, since it’s quite possible that the certificate was revoked or it’s invalid. The verification device must check the CRL while reading certificate from secure chip. For electronic passports the Certificate Revocation List (CRL) is stored in PKD. #### IV-A2 Strengths An ePassport’s security object has provisions to select hashing and signature algorithms. In case an algorithm in use turns obsolete, either because of weakness or otherwise, it can be switched to an alternate one. However, passports which have already been issued can not be substituted with the latest algorithm. The ICAO specifications for passive authentication have provisions for using larger key lengths which improve cryptographic security. The choice of a strong cryptographic algorithm for computing a hash of data structures improves on possible collisions of signature values. Passive authentication does not necessarily require the verification device to be online except to get the updated CRL from the PKD. This requires fewer infrastructure and can be performed in a remote location. ### IV-B Active Authentication Active Authentication (AA) of electronic passports is performed using a unique cryptographic key pair $KPu_{AA}$ and $KPr_{AA}$. The AA public key $KPu_{AA}$ is stored in Data Group 15 (DG15), one of 16 Data Groups (DGs), in the secure chip. The private key is stored in secure chip and never leaves the chip. Typically, an active authentication key pair is generated inside the secure chip; however, many system designers prefer creation of keys outside of the chip to improve personalization speed. The correctness of an AA key is verified by checking the signature of DG15, which is signed by passport signer’s private key. To check signature, signer’s certificate should be retrieved from PKD or from the chip, and the CRL checked for updates. Visual inspection, electronic verification of the security object, and challenge response authentication using asymmetric key pair, determine if keys are read from a passport which has not been copied or cloned. The chip also stores signed MRZ information in the logical data structure (LDS). Active authentication is designed to detect if a passport chip has been replaced by a fake one or its contents have been copied to another chip. #### IV-B1 Challenge Response The challenge response protocol for AA is outlined below. (a) The authentication device checks for validity of the security object and retrieves the $KPu_{AA}$ either from the chip or PKD. (b) Before using the key in this cryptographic protocol, it checks for its validity by verifying the signature of the key. (c) The passport’s chip contains the secret authentication private key $KPr_{AA}$ which is not accessible. The certificate and public key $KPu_{AA}$ are stored on the chip as well. (d) The terminal generates a nonce and sends it to the secure chip as a challenge. The verification device may choose to send a date and time in addition to the nonce, which can optionally be stored in the chip upon signature. $Challenge=R{VD}\rightarrow ePassport$ (1) (e) The challenge, and at times an additional counter, is signed by the authentication private key. $Response=Sign(RSA){KPr_{AA}(R{VD}+C)}$ (2) (f) Verification device checks digital signature sent to it. (g) If the signature is verified, the secure chip is actively authenticated. #### IV-B2 Key Lifetime The authentication key pair is normally valid for the lifetime of an ePassport. Typically, ePassports are issued at least for five years or more. Therefore, the key lifetime of an active authentication key pair is five years. The strength of active authentication lies with the strength of the secure chip to securely hold the private key. Secure controllers with all the tamper-proofing mechanisms are strong key vaults. #### IV-B3 Weaknesses AA is designed to detect passports with cloned chips. It requires a secure chip with cryptographic capabilities, which may increase authentication time compared to PA. However, with faster processing and better algorithms this difference in time may not be noticeable. AA is usually performed in combination with BAC, which uses diversified keys that do not have high entropy. Use of AA with BAC overcomes the strongest weakness of BAC. In AA, the same key pair is used for every authentication session. There are no temporal keys for every new session of authentication. AA does not perform any type of external or terminal authentication. It assumes all terminals are trustworthy. This may not be an issue since, unlike second generation passports, first generation passports do not hold any private biometric data. Chip Authentication $\displaystyle\mathbf{Terminal}$ $\displaystyle\mathbf{Passport}$ Request $\displaystyle\underrightarrow{Pu_{Passport}}$ Read DG15 Verify Signature $\displaystyle\underleftarrow{Signed[Pu_{Passport}]}$ Send Send $\displaystyle\underrightarrow{Pu_{Terminal}}$ Receive Compute $\displaystyle K_{Session}$ Compute Figure 1: Messaging in Chip Authentication #### IV-B4 Strengths Introducing an asymmetric key pair allows signed trace and track information of visitors. An authenticated time stamp using the private key, which is stored only in the secure chip, allows system to be updated with trace information regarding entry or exit of a visitor. The private key is bound to the identity of person holding the ePassport. AA could improve privacy since each access to the passport can be logged in secure memory of the chip. The audit trail could be helpful in tracing and tracking an illicit request to access the passport. The AA public keys are signed and often stored on the chip; therefore, it does not require verification devices to be online. ### IV-C Chip Authentication Chip authentication is used in second generation ePassports with BAC to improve security by introducing encryption of all messages exchanged between the inspection system and ePassport. In BAC with key diversification, the keys must be generated using holder-specific data which may not have high entropy. The active authentication and diversified keys are not unique to each and every session. Chip authentication improves these two authentication mechanisms by introducing keys which are unique to every chip and every session. Additionally, a message authentication code (MAC) is added to every message from the chip. Like AA, it introduces asymmetric key pair, signed, with public key of the issuing country. In chip authentication, every chip has its own key pair assigned and stored in it. The public key is signed and stored in one of the public data groups and the private key is stored in secure memory of the chip [21]. See Figure 1. #### IV-C1 Ephemeral Static Diffie-Hellman Key Exchange The terminal chooses an ephemeral key pair which is used to encrypt a single session of communication between the secure chip and the interface device. The Elliptic Curve Diffie-Hellman (ECDH) key agreement scheme is chosen to agree upon keys to encrypt data exchanged between chip and verification device [21]. Subsequently, a symmetric key, 3DES is used for encryption of all messages between the secure chip and terminal. This improves weaknesses of key diversification using static keys and PA challenge-response mechanisms. It also improves asymmetric key challenge response used in AA; the symmetric keys are used to encrypt messages. The following is a conceptual outline of chip authentication. (a) An elliptic curve and corresponding public curve point P is chosen for a given field and shared between the passport chip and verification device. (b) The verification device generates a random number which is its private key. The passport’s key, and private keys $k_{p}$ and $k_{v}$, are stored in the chip. (c) Both generate public keys $Q_{p}$ and $Q_{v}$ and send them to each other. $Q=\begin{cases}Q_{p}=k_{p}P\\\ Q_{v}=k_{v}P\end{cases}$ (d) The chip and the verification device generate common shared data $Q_{s}$ using each other’s public key. $Q_{s}=\begin{cases}k_{p}Q_{v}=k_{p}k_{v}P\\\ k_{v}Q_{p}=k_{v}k_{p}P\end{cases}$ (e) After the above step, $Q_{s}$ becomes the shared symmetric key for any encryption of any subsequent communication between the chip and the verification device. There are many different forms of the ECDH protocol, each with its merits. Menezes Qu Vanstone (MQV), another form of ECDH, has additional security mechanisms built in. An ePassport uses one of the forms of ECDH to perform the key agreement. #### IV-C2 Weaknesses Chip authentication requires high-end processors which can perform Diffie- Hellman key exchange. Standard interfaces for Diffie-Hellman key exchange are available only in Java Card 2.2.x. #### IV-C3 Strengths Chip authentication covers all the weaknesses with the different schemes mentioned above. Along with message authentication as defined for BAC, chip authentication is the strongest known authentication mechanism. Terminal Authentication $\displaystyle\mathbf{Terminal}$ $\displaystyle\mathbf{Passport}$ Send $\displaystyle\underrightarrow{Pr_{CVCA}Signed[{Pu_{DVCA}}]}$ Verify DVCA Certificate Send $\displaystyle\underrightarrow{Pr_{DVCA}Signed[{Pu_{IS}}]}$ Verify IS Certificate Sign $\displaystyle\underleftarrow{Challenge}$ Send Send $\displaystyle\underrightarrow{Pr_{IS}Signed[Challenge]}$ Verify Receive $\displaystyle\underleftarrow{GrantAccess}$ Verify Access Figure 2: Messaging in Terminal Authentication ### IV-D Terminal Authentication The second generation ePassport introduces concept of terminal rights and their authentication using asymmetric cryptography. In EAC [2], holder’s biometrics are stored on the chip. The holder’s biometrics should not be released to terminals which do not have rights to read the information. A secure chip has no power source, so can not maintain time. Due to this limitation, the chip can not verify standard X.509 certificates. The secure chip in an electronic passport has limited access to the network, and therefore can not reliably update itself from the CRL. To overcome these limitations, a different type of certificate, known as a ’card/chip verifiable’ certificate is used for external authentication. The terminal is authenticated if an asymmetric key pair is verified. The terminal stores private key of key pair and ePassport stores the public key, which is embedded with other information in a certificate, known as Card Verifiable Certificate (CVC). The CVC is securely stored to the ePassport at the time of personalization. The knowledge of private key in the terminal is verified by an asymmetric challenge response protocol. See Figure 2. To authenticate the terminal, the ePassport sends a challenge to the verification device, which has access to the corresponding private key. The verification device attaches document number, challenge, and hash of the session-unique data and signs it with its private key. Either the RSA or ECDSA signature algorithm is used to sign and verify. The terminal contains its private key either in tamper-resistant memory or on the connected network from where it can be securely fetched. The chip contains ’trust anchor’ or ’root certificate’ which is used to verify signature received from the terminal. Built-in with authentication are access rights to read or update biometrics of the passport holder. Terminal authentication is preceded by internal chip authentication. As described above, chip authentication concludes with a mutually agreed symmetric key which envelops all the subsequent communication between ePassport and verification terminal. All messages are encrypted and attached with MAC computed as per specifications of Chip Authentication. Figure 3: PKI for Terminal Authentication #### IV-D1 PKI for Terminal Authentication Figure 3 illustrates the two-level PKI necessary for terminal authentication. Country Verifying CA ($CVCA_{Issuer}$) issues certificate to document verifying CA ($DVCA_{Issuer}$). The document verifying CA issues terminal certificates to each and every verification terminal deployed in the location. Figure 4: PKI for Terminal Authentication with Cross Certification Terminal verification extends beyond the boundaries of an issuing country. Cross certification allows foreign countries to verify the identity of the passport holder using biometrics. The $CVCA_{Issuer}$ certifies visited country’s document verifying CA ($DVCA_{Visit}$). The terminals at visited country now have signatures of the issuing country CA which can be verified by electronic passports from the issuing country. Figure 4 represents the PKI for cross certification. #### IV-D2 Access Rights EAC introduces access rights to verification terminals allowing only authorized terminals to read or modify certain data. The terminal’s application has controlled access to different biometrics stored on the chip. Role-based access right mechanisms are implemented by encoded tables in card- verifiable terminal certificates. Such access right control mechanisms are used to limit or grant access to perform read or update operations. The card- verifiable certificates have been defined to contain privileges for different roles. Different roles include certificate authority, foreign country, and domestic inspection systems. Germany has chosen to implement such a system [21]. #### IV-D3 Weaknesses Terminal authentication can not revoke a certificate once it has been issued to a country or an organization. Once issued, there is no known way to revoke a certificate using mechanisms like CRL and time. #### IV-D4 Strengths Terminal authentication adds terminal authentication and access rights to the facility of authentication, which is useful in controlling access to the biometrics stored inside the chip. ## V Access Control in ePassport Systems Access control using data stored in a secure chip is defined for various levels of classification by the issuing country. The U.S. had chosen Plain Access Control222Plain Access Control (PAC) is a term coined only in this paper by the author and may be viewed as special case of BAC. Per ICAO, BAC has mandatory document signing but the key diversification to read the data stored in chip is optional. (PAC) as the primary method to control access. However, this was later changed to BAC after skimming and eavesdropping concerns were raised by different privacy groups. ICAO mandates storage of only a facial image in the chip; other biometrics, like fingerprints, are not stored in the chip, which limits the use of biometrics for access control. Mutual authentication of a secure chip and a verification device is optional. These limitations have been removed in EAC, which is better suited for biometrics-based access control at the verification point. TABLE III: Optional and Mandatory Security Mechanisms Mechanism \| Mandatory/Optional ---\|--- Passive Authentication \| ICAO Mandatory BAC \| ICAO Optional, EU Mandatory Active Authentication \| Issuing Country EAC \| EU Mandatory for biometrics ### V-A Plain Access Control PAC is a special case of BAC, in which key diversification is not required to read the data. This scheme of access control allows any reader to read data from a chip. The secure data object is hashed, signed, and stored in the LDS. Authenticity of a passport is verified by checking the authenticity of data and its signature via a signed object. According to ISO 14443, read range is limited to 10 cm; however, the data can be skimmed using modified readers that achieve a range greater than 10 cm. Therefore, PAC does not counter skimming and eavesdropping types of attacks. PAC may be secure for contact only communication; however, contactless communication introduces risks and mere signature verification is inadequate security. See Table III. #### V-A1 Security Weaknesses ##### Confidentiality The data is unencrypted and can be read by any reader. There is no confidentiality of data and this scheme allows any reader to read the data [15]. ##### Authentication Any reader can read the data encoded on the chip. There is no security mechanism to authenticate the reader, passport holder, or the secure chip. ##### Data Cloning The data can be easily read and written to another chip without any modification. ### V-B Basic Access Control Per ICAO, BAC is an optional but recommended way to achieve interoperability of ePassport-based border control between countries since the European Union mandated diversified key authentication. The diversified key is generated, using MRZ data, and used to mutually authenticate the passport and inspection devices. Since common knowledge is required to generate the diversified keys, this scheme implicitly authenticates the inspection system; however, mutual authentication is not strong. In BAC, data between inspection or verification reader is not encrypted using the session keys. An ePassport could be authenticated using either active or passive authentication. The mandatory and optional stages of BAC are listed below. 1. 1. Key Diversification [Optional]: BAC diversified key authentication and opening of an unencrypted communication channel is an optional but suggested method to authenticate the ePassport and inspection device. 2. 2. Passive Authentication [Mandatory]: Passive authentication and checking of the signature is a mandatory step in BAC. Passive authentication verifies authenticity of the data only. 3. 3. Active Authentication [Optional]: Active Authentication, as described above, is an optional step in BAC. #### V-B1 Key Diversification The fixed seed key is diversified using the passport number, date of birth, and expiration date of the document. Key diversification creates a unique key for each passport; however, keys are diversified using known data which is not random and therefore has low entropy. A true random number can not be known to both the inspection system and the secure chip without doing one of the following. * • Use a back-end system to store a known secret random number. The same number can be stored in the card. * • Exchange the random number as part of a setup message. However, this must be passed in the clear, which weakens security. * • Use a known but secret coding algorithm to generate the same codes. A known algorithm could be used to generate a pseudo random number. However, this is not a good approach since a secret black box algorithm could be broken. #### V-B2 Session Key Derivation BAC uses two keys, one for encryption and another for calculating the message authentication code being exchanged between reader and secure chip. Sessions keys are generated using a seed key derived using the data read from the MRZ. 1. 1. The passport contains $K_{seed},MRZData,C_{Enc}$ and $C_{MAC}$. The session keys are derived from seed key and personal data stored in MRZ. $\begin{array}[]{\|cc\|}K_{seed}&C_{MAC}\\\ MRZData&C_{ENC}\end{array}$ 2. 2. The seed key $K_{seed}$ is diversified using data stored in the MRZ. $MRZData\otimes K_{seed}\rightarrow K_{div}$ 3. 3. The $K_{div}$ is concatenated with 32-bit counters. One for encryption and the other for MAC. $D=\begin{cases}K_{Enc}\Leftarrow K_{div}\|\|C_{Enc},\\\ K_{MAC}\Leftarrow K_{div}\|\|C_{MAC}\end{cases}$ 4. 4. Next, a message digest is generated on key set D. SHA-1 is calculated on both the keys defined in key set D, now called H. This is a 160-bit or a 20-byte long number. $H=\begin{cases}H_{Enc}\Leftarrow SHA(K_{Enc}),\\\ H_{MAC}\Leftarrow SHA(K_{MAC})\end{cases}$ 5. 5. The first 56 bits of H constitutes K(a) and checksum is calculated per the DES algorithm. The computed checksum is appended to the key to make it 64 bits long. $H\rightarrow K_{a}$ 6. 6. The next 56 bits from the 64th bit of H constitutes K(b) and a checksum is calculated per the DES algorithm. The computed checksum is appended to the key to make it 64 bits long. $H\rightarrow K_{b}$ 7. 7. The last 20 bits or four bytes of H are discarded with no affect. 8. 8. The challenge response messaging occurs to verify the passport as per ISO 11770-2 using 3DES in block-cipher mode. 9. 9. Random numbers (nonces) generated by the passport and the reader are of eight bytes long. The keys generated are valid from the start to the end of communication between the verification device and ePassport. The lifetime of the document signing key is equal to the longest time for which the passport is valid plus time for which the key has been used to sign other passports. The document- signing key is erased once the key has expired. Countries choose the frequency at which a new document-signing certificate is issued. #### V-B3 Challenge Response The message sequence for challenge response is summarized below. The nonce is eight bytes or 64 bits long. 1. 1. The verification device generates a random number $R_{v}$ and encrypts with the triple DES keys generated. $M_{vp}=3DES(K_{ab}(R_{v}))$ 2. 2. The passport decrypts it and verifies if the random number matches. $R_{v}$encrypts with the triple DES keys generated. $R_{v}=3DES^{-1}(K_{ab}(R_{v}))$ 3. 3. Passport generates a random number $R_{p}$ encrypts with the triple DES keys. $M_{pv}=3DES(K_{ab}(R_{p}))$ 4. 4. The verification device decrypts the challenge and verifies if the random number matches. $R_{p}$encrypts with the triple DES keys generated. $R_{p}=3DES^{-1}(K_{ab}(R_{p}))$ #### V-B4 Message Authentication Every message between passport and reader can be appended with a Message Authentication Code (MAC). The MAC is calculated per the ISO 9797-1 MAC algorithm. The MAC is calculated using the DES algorithm with Cipher Block Chaining (CBC). For all blocks except $B_{N-1}$, $K_{a}$ is used for encryption. For every block, an incremental counter is appended to the message to make its MAC unique. This technique works well against message replay attacks, which are common in wireless communication. Note that the next to last block is encrypted using $K_{b}$ instead of $K_{a}$. #### V-B5 Security Weaknesses ##### Key Generation Both the encryption and the MAC key are generated from the same seed key. Although there are additional counters attached to generate the encryption and MAC keys, the compromise of one single key $K_{seed}$ could be sufficient to break the encryption as well as the message authentication code. Note that the counter is only an additional number incremented by one. To overcome this threat, the counters can be easily replaced by a random number which can be either encrypted or attached to the message. ##### Key Diversification The seed key $K_{seed}$ is diversified using the passport holder’s name and date of birth, or the keys are diversified using the passport’s information which can be easily read from the data page of the passport. The entropy of the diversification information is limited and the diversification information is available in the data page of the passport. This data in most cases is encoded in the machine-readable zone. Despite the use of the SHA 200 hashing algorithm, the key generated may not be truly random. ##### Cryptographic Algorithm The choice of cryptographic algorithm has been DES, a well known and proven standard for symmetric encryption/decryption of data. However, with Rijndael’s algorithm already announced as the AES, the choice of DES does not seem appropriate. AES has already been designed into some networking and banking protocols. AES’s stronger substitution permutation network structure and the fact that it can be implemented well on a medium- or small-sized chip, like the one used for passports, makes it a better candidate than DES. The lack of adoption of AES in the ePassport standards should probably be revisited sometime. ##### Chip and Data Cloning The secure chip can not be easily protected against a clone attack. A cloned chip is assumed to have copied all data bit-wise and replicated to similar silicon. Detection of such clones is a challenge which industry is presently facing. The anti-cloning solution may lie beyond the secure chip is some special type of printing or material that is added to passports. ### V-C Extended Access Control Following rollout of U.S. passports, the EU came up with its set of passport identification standards and security protocols [2] [26]. EAC is designed for a second generation of passports which involves storing biometrics of the holder. The common biometrics used are either finger prints or iris images. Terminal Authentication is being implemented by EU countries with some country-specific extensions. EAC keys are exchanged bilaterally between the issuing and visited state. The signed certificates must be available to the verification device for reading biometric information stored on the secure chip. EAC replaces active authentication by chip authentication. As per [28], an EAC mutual authentication session consists of the following stages with mandatory and optional parts. 1. 1. BAC [Mandatory for all Passports]: Establishes a secure channel with diversified keys. 2. 2. Chip Authentication [Mandatory for Second Generation Passports]: EAC replaces active authentication by mandatory chip authentication. 3. 3. Passive Authentication [First Generation Passports]: Passive authentication is performed as per ICAO 9303 specifications. 4. 4. Terminal Authentication [Mandatory for Second Generation Passports]: EAC adds terminal authentication to mutual authentication. All terminals are not trusted by the passport, as is the case for first generation passports. EAC message exchange consists of two stages: chip authentication followed by terminal authentication. The chip is “deduced” to be verified at the beginning of the second stage since it is able to encrypt and decrypt messages using session keys negotiated in the first stage. The session keys are established in the chip authentication stage. The first stage consists of an Elliptic Curve Diffie-Hellman or Diffie-Hellman key exchange. The terminal authentication stage consists of signature verification using either RSA or ECDSA. #### V-C1 Security Weaknesses ##### Card Verifiable Certificates EAC involves use of card-verifiable certificates which are not standard. Unlike X.509 and PGP certificates, such non-standard certificates are not widely deployed. There have been limited implementations [25] of verification of X.509 certificates on smart card chips. A card-verifiable certificate can not be revoked or its validity checked with time since cards do not have a clock. This weakness limits the effectiveness of security mechanisms based on card-verifiable certificates. ##### Loss of Verification Terminal In extended authentication schemes, a private key is stored in the terminal; therefore any theft or loss of a verification terminal could jeopardize the security of all passports using the corresponding public key. This could possibly mean a breach of security; however, to overcome this weakness, tamper-resistance mechanisms should be deployed to secure the storage of keys and sensitive information. ## VI Biometrics Facial images and fingerprints are the most useful biometrics for ePassports [10]. Biometrics add another dimension to the authentication of ePassports by allowing checks for “what you have” and “what you are” at the same time. Biometrics can help to move from manned border crossing stations to unmanned border verification stations. The automated verification of credentials was first implemented by Malaysia and followed by Australia. Brazil has decided to roll out its ePassports with all ten fingerprints and a facial image encoded on the chip. Many countries use biometrics for law enforcement, using a one-to-many search- and-match to identify criminals. Since criminals often try to cross country borders, having similar mechanisms such as FIPS 201 or Personal Identity Verification (PIV) is useful for ePassports. A one-to-many comparison requires additional storage and computing for quick results. This can be achieved only by letting the minutiae out of the secure chip and matching them to the database of foul fingerprints. EAC grants or denies access based on biometric information stored in the secure chip. Just like electronic signatures, a biometrics template is unique to the passport holder, and therefore raises different privacy concerns about the distribution of template information. Storing biometrics securely in the secure chip with access control is the best way to maximize security and privacy concerns. As per ICAO, the facial image of a passport holder is not sensitive or secret biometric data. A facial image is used in BAC for personal identification and therefore it is not encrypted. Table IV shows provisions available in electronic passports for additional biometrics like an iris or retinal image and visual marks. TABLE IV: ePassport Biometrics Name \| Classification ---\|--- Facial Image \| Public Encoded Finger \| Private Encoded Iris/Retina \| Private Visual Mark \| Private Signature \| Private ### VI-A Electronic Verification Electronic verification is possible only if the passport holder is present and the verification station has the necessary devices to capture the biometric data of the passport holder. If the biometrics data is encrypted, it must be decrypted after generating the diversified keys which are calculated only after reading the MRZ. The biometrics of the passport holder are captured using a capturing station and biometrics sensor and matched against those stored on the secure chip. ICAO specifications do not mandate a comparison of biometrics on the secure chip, commonly know as “match on chip” Since the match is typically done on the verification device, to secure the verification the biometrics should be encrypted since they will travel wirelessly. Besides the risk of losing minutiae to an illegitimate terminal, the verification of fingerprints at an unmanned station introduces the ’gelatin finger’ or ’prosthetic finger’ threat. The threat of introducing a prosthetic or gelatin finger with ridges as per stolen images of an individual can not be easily mitigated. The author can propose reading more than one finger to reduce the chances of such a threat succeeding. Enrolling all ten fingers on the chip and randomly selecting the finger to verify helps improve security. The security against this threat also improves if more than one fingerprint of the individual is verified. Extending this logic, if both the finger and facial biometrics are stored on the chip and electronically verified at the unmanned station, the gelatin finger threat is mitigated. Experts have argued that the facial images could also be easily found and replicated to fool the facial recognition systems; however, security with “dual checks” should be an acceptable level of security. ### VI-B Terminal Verification All terminals are usually not granted access to read the biometric templates stored on the card. Some countries like Germany have designed their systems to validate the access rights of terminals to read the template before exposing templates to them. Terminal authentication is performed and the access bits decoded to check for the access rights. ## VII Protection Profiles Common criteria profiles [19] [18] have been defined for each type of ePassport access control. Most of the hardware, embedded software, manufacturing process and issuance, and handling systems are evaluated to meet high degrees of assurances. The two different protection profiles cover threats as perceived by the developers of such profiles. The protection profiles may be comprehensive but may not be complete. An example of such a threat is a side-channel attack. Protection against such an attack has been built in most of the secure controllers. ## VIII Conclusions Since ePassports are being issued in place of traditional paper passports, which do not have secure RF-enabled contactless chips, the security of ePassports must be better than paper-based passports, which are not vulnerable to skimming, eavesdropping, or tracking attacks. This is an unwritten and probably unrealized expectation from the issuing countries, passport holders, and society. The world is seeing the first few generations of RFID passports. The convenience and security of contactless access control transactions are here to stay. The second generation passport with biometric access control will be more prevalent in the coming years. The future of passports may shift from single-chip electronic RF to multi-chip modules with a combined data storage capacity from multiple chips. Future ePassports may look like a secured solid disk with onboard sensors and limited or no printed information. This would be a true generation shift from paper electronic to fully electronic passports. Such a passport may have its own sensors on board to validate its holder. Some other interesting security features like RF-DNA [30] [29] can be used as certificates of authenticity in lieu of electronic certificates. These methods rely on physical creation of fingerprints which would identify each passport. See Table V. TABLE V: Generations of Passport Generation \| Type ---\|--- Generation 0 \| MRZ Generation 1 \| BAC + PA, EA Generation 2 \| EAC + TA, CA Generation 3 \| eVisas Generation 4 \| Advanced Electronics ### -A Construction #### -A1 Inlay The tamper-proof chip embedded in a passport is connected to a coil and made into an inlay which is inserted on either the cover or first page of a machine-readable travel document (MRTD). The term MRTD refers to ePassports as well as other travel authorization documents including visas and permits for entry. The inlay is usually composed of different layers of synthetic polymers which protect the coil, chip, and more importantly the interconnect. The most important of all components is a secure RF-enabled tamper-proof chip which allows reading of data while securely storing it. An ePassport’s physical construction is required to last for at least 10 years. The physical construction should be strong enough to survive the pressure and temperature that it could be subjected to in those years [28]. The construction of a passport consists of layers of laminated poly-carbonate or polyester (PET/PETG) or some other material which is a combination of similar materials [3]. The inlay is usually 300-480 $\mu$m thick, depending on the thickness of the silicon. The dimensions of the inlay are about 600 x 400 mm in length and width. The chip is made into a module before bonding the antenna to the module. Often, layers of different materials are used to strengthen the composition of the inlay. An inlay goes through numerous test cycles, including the ISO 10373 physical, temperature, magnetic, electric, and chemical tests. Besides this, an inlay must go through another important test known as the high-pressure impact or “stamping” test. Since passports are often stamped, they need to survive the stamping impact stress. #### -A2 Antenna The antenna is typically made of copper or some other alloy of copper to construct a coil with the right attenuation and RF characteristics. The antenna dimensions of a passport are specified in ISO 7810 ID-1. The size of such an antenna is between 85.60 and 53.98 mm (3.370 to 2.125 in). The number of turns in the coil is not specified but left to the antenna designer. The same goes for material used to manufacture the antenna. The electrical and magnetic characteristics of the magnetic field are specified in ISO 14443. Figure 5: Directions for Measuring Attenuation #### -A3 RF Shield An RF shield is a mesh of electro-magnetically opaque material, usually put on the cover of a passport to avoid skimming and eavesdropping [1]. The shield, also known as a “Faraday Cage”, is often in the form of a pouch or cover on the passport [22]. The shield offers -40 to -60 dB attenuation in the frequency band around 13.56 MHz to limit skimming or eavesdropping. This level of attenuation is perceived to block the communicating signals between a passport and interrogating readers. An RF shield is not fully effective in blocking communication between a reader and a passport. A reader with strong signal can be designed that could nullify attenuation provided by an RF shield when the ePassport is open and the shield is not effective. Shield is less effective if it is placed in the cover page and the cover page is opened. The shield provides some safety but not the surety of blocking communication between a passport and a rogue-interrogating reader. It should not be relied upon as the sole mechanism to avoid such attacks. The exact methods of measuring the attenuation of electro magnetic shielding is defined in [22]. For a geometry of the ID1 format, attenuation should be calculated at all six sides and preferably at regular angular intervals of inclination. The different directions of measurement are illustrated in Figure 5. #### -A4 The Secure Chip The secure chip module is usually smaller than 20 mm2. Of six pins of a secure chip, only two are connected to the antenna. Other pads that are not used must be secured against probing attacks. The contactless air interface could be either Type A or B. Per ICAO, the minimum size of available persistent non- volatile memory on the chip should not be less than 30 kB. Chips with 30 kB of available data space, are sufficient only for storing a minimal set of mandatory biometrics and can be used only for first-generation ePassports. Higher capacity chips are required for countries intending to store visas and travel information on the passport. The writable information on a chip includes data groups DG17, DG18, and DG19. The significance of each data group is explained in Section -B below. In such cases, the minimum size of persistent non-volatile memory must be greater than 256 kB. The chip should be capable of generating random numbers and performing at least DES/3DES cryptographic operations like encryption, hashing, and signing for BAC. The chip must be capable of performing asymmetric cryptographic operations like Elliptic Curve Diffie-Hellman key exchange for EAC. #### -A5 Radio Frequency All ePassports are specified to work only at 13.56 MHz for the advantages of high-frequency communication over low frequency. The 13.56 MHz communication band is more immune to processor noise, noise from the earth’s field, and the surrounding environment, which is necessary for stable and consistent RF communication. The RF communication is based on ISO 14443 standards developed for near-field inductively coupled systems. ICAO defines communication to be either Type A or Type B. ICAO has additional test standards for stronger adherence to field strengths and modulation indices. Also defined are strong test criteria for its qualifications. ### -B Logical Data Structure To achieve interoperability, the LDS of electronic passports is defined by an ICAO standard. The data structure has various mandatory or optional components. Data groups DG 1-16 are written by the issuing country; data groups DG 17-19 are written by a receiving country. #### -B1 Mandatory Issuing State or Organization Data The mandatory encoded LDS is shown in Table VI. This includes two data groups DG1 and DG2. TABLE VI: Mandatory Logical Data Location \| Group \| Type ---\|---\|--- MRZ + Chip \| DG1 \| Holder and Passport Info. Data Page + Chip \| DG2 \| Encoded Face of Holder TABLE VII: Issuing State or Organization Logical Data Name \| Data ---\|--- DG3-DG7 \| Holder’s Biometrics DG8 \| Data Feature(s) DG9-DG14 \| Structure Feature(s) DG10 \| Substance Feature(s) DG11-DG14 \| Additional Info / RFU DG15 \| Active Authentication Key Info DG16 \| Person to Notify #### -B2 Optional Logical Data The optional LDS which can be encoded is shown in Table VII. ### -C Electronic Passport Public Key Infrastructure An ePassport requires a PKI for every issuing country and a cross state (central) repository to exchange any updates. A PKI is required even for BAC and passive authentication of ePassports. Every issuing country has its own Country Signing Certificate Authority (CSCA) and Country Verifying Certificate Authority (CVCA). A CSCA’s public signature key is denoted as $KPu_{CSCA}$ and its private signature key is $KPr_{CSCA}$. A CVCA’s public signature key is denoted as $KPu_{CVCA}$ and its private signature key is $KPr_{CVCA}$. A CSCA issues certificates to the document signer which are used to sign the LDS. The document signer’s public key is denoted as $KPu_{DS}$ and the private key is $KPr_{DS}$. A CVCA issues certificates to a Document Verifier (DV), which may or may not be the same entity as the document signer. For active authentication, an additional key pair is added. This key pair is only used in a challenge response to actively authenticate the ePassport chip. The public active authentication key is denoted as $KPu_{AA}$ and the private key is denoted as $KPr_{AA}$. The authenticity of an ePassports is verified by the document verifier. The issuing country can revoke certificates and have a need to update the CRL and set of document verification certificates. These certificates are normally stored in a common shared and secure repository, which is regularly scanned for updates. ## References * [1] International Civil Aviation Organization, _Doc 9303 Part 1, Vol. 2 Specifications for electronically enabled passports with biometric identification capability_ 2006 * [2] BSI, _Technical Guide TR 03110, Ver. 1.11 Advanced Security Mechanism for Machine Readable Travel Documents - Enhanced Access Control_ 2008 * [3] Smartrac Website, _http://www.smartrac-group.com/en/personalausweise.php_ , 2008 * [4] Department of Homeland Security Website, _http://www.dhs.gov/files/programs/content_multi_image_002.shtm\\#1_ , 2010 * [5] Department of Homeland Security US-VISIT Program, _http://www.dhs.gov/files/programs/usv.shtm_ , 2010 * [6] International Civil Aviation Organization, _Machine Readable Travel Documents (MRTDs) : History, interoperability and implementation. Release 1, Draft 1.4_ March 23, 2007 * [7] International Civil Association Organization, _RF protocol and application test standards for ePassport Part 2, Tests for air interface initialization, anti-collision and transport protocol. Version 1.02_ Feb 20, 2007 * [8] International Civil Association Organization, _RF protocol and application test standards for ePassport Part 3, Tests for air interface initialization, anti-collision and transport protocol, Tests for application protocol and logical data structure. Version 1.01_ Feb 20, 2007 * [9] International Civil Association Organization, _RF protocol and application test standards for ePassport Part 4, ePassport reader tests for air interface, initialization, anti-collision and transport protocol. Version 1.01_ Feb 20, 2007 * [10] M. Nüsken, _Lecture Notes Electronic Passport & Biometrics, b-it Bonn-Aachen International Center for Information Technology_ Winter 2006/2007 * [11] A. Juels, D. Molnar & D. Wagner, _Security and Privacy Issues in ePassport_ April 17, 1995 * [12] Federal Information Processing Standards (FIPS) _Secure Hash Standard, Publication 180-1_ April 17, 1995 * [13] Z. Cheng & R. Comley, _Attacks on an ISO/IEC 11770-2 Key Establishment Protocol_ Sept 23, 2004 * [14] E. Bjelk sen & L. W. Olsen, _Security Issues in ePassports - ICAO Standard and National Implementations as Part of the US Visa-Waiver Program_ May, 2006 * [15] B. Schneier, _Schneier on Security, RFID Passport Security_ http://www.schneier.com/blog/archives/2005/04/rfid_passport_s.html, April 28, 2005 * [16] United States of America, Department of State, _Electronic Passport, 22 CFR Part 51 [Public Notice 5208] RIN 1400-AB93._ Final rule, October 25, 2005 * [17] United States Department of Transportation, _FAA Privacy Impact Assessment of IDMS and PIV Cards_ , http://www.dot.gov/pia/faa_idms.htm, Feb. 2008 * [18] Common Criteria, _Common Criteria Protection Profile, MRTD Extended Access Control_ , Bundesamt für Sicherheit in der Informationstechnik, Version 1.1, September 2006 * [19] Common Criteria, _Common Criteria Protection Profile, MRTD Basic Access Control_ , Bundesamt für Sicherheit in der Informationstechnik, Version 1.0, August 2005 * [20] P. Kocher, J. Jaffe & B. Jun, _Differential Power Analysis_ , Cryptography Research Inc., 1999 * [21] Dennis K. Kugler, _Extended Access Control - Infrastructure and Protocol, Interop-Test_ , Federal Office for Information Security, June 2006 * [22] IEEE, _Standard Method for Measuring the Effectiveness of Electromagnetic Shielding Enclosures_ , IEEE 299-2 1997 * [23] Z. Riha, _JRC Ispra European Commission_ , Masaryk University, Sept 2008 * [24] C. Mitchell, _Limitations of Challenge Response Entity Authentication_ , HP Laboratories, May 1989 * [25] O. Henniger, K. Lafou, D. Scheuermann & B. Struif, _Verifying X.509 Certificates on Smart Cards_ , HP Laboratories, November 2006 * [26] Dr. T. Nguyen, _Contactless Authentication Protocol for MRTDs: BAC & EAC_, Bundesdruckerei GmbH, RFID Security, 2006 Graz * [27] M. Schl ter, Z. Riha, B. Hofbauer, _ePassports EAC Conformity and Interoperability Test Results_ , Prague 2008 * [28] V. Krishnan & H. Wang, _Security Analysis of Australian and E.U. E-Passport Implementation_ , Journal of Research and Practice in Information Technology, Vol. 40, No. 3, August 2008 * [29] Y. Chen, M.K. Mihcak & D. Kirovski, _Certifying Authenticity via fiber infused paper_ , ACM SIGecom Exchanges, Volume 5, Issue 3, April 2005 * [30] G. DeJean, M.K. Mihcak & D. Kirovski, _RF-DNA: Radio-Frequency Certificates of Authenticity, Cryptographic Hardware and Embedded Systems - CHES 2007_ , Lecture Notes in Computer Science, Volume 4727/2007, 2007	arxiv-papers	2010-11-12T15:28:26	2024-09-04T02:49:14.773270	{ "license": "Public Domain", "authors": "Anshuman Sinha", "submitter": "Anshuman Sinha", "url": "https://arxiv.org/abs/1011.2946" }
1011.2989	# A Decoding Approach to Fault Tolerant Control of Linear Systems with Quantized Disturbance Input Sophie M. Fosson Abstract. The aim of this paper is to propose an alternative method to solve a Fault Tolerant Control problem. The model is a linear system affected by a disturbance term: this represents a large class of technological faulty processes. The goal is to make the system able to tolerate the undesired perturbation, i.e., to remove or at least reduce its negative effects; such a task is performed in three steps: the detection of the fault, its identification and the consequent process recovery. When the disturbance function is known to be _quantized_ over a finite number of levels, the detection can be successfully executed by a recursive _decoding_ algorithm, arising from Information and Coding Theory and suitably adapted to the control framework. This technique is analyzed and tested in a flight control issue; both theoretical considerations and simulations are reported. ## 1 Introduction Fault Tolerant Control (FTC for short, [5],[11],[6]) aims to cancel or contain the consequences of faults in an automation system. Such an operation is fundamental in modern technological processes, which are required to assure robust performance, stability and safety even in case of partial malfunctions or degradations. Often, robustness is achieved by redundancy, say by the introduction of many control components like sensors; nevertheless, this sophistication naturally increases the probability of breakdown and then continues to motivate the research on reliable control systems. The problem of upholding the functionality of an apparatus affected a disturbance is ubiquitous in the industrial and transport fields. In particular, FTC systems are widely applied in those contexts where human health and environment are concerned, for example, in the design of mechanical and chemical plants; nuclear power reactors; medical systems; aircrafts, helicopters and spacecrafts; automotive engines, railway and marine vehicles. Another interesting application is in the communication networks (for instance, wireless sensor networks), where the aim of FTC is to avoid unexpected interruptions of data flow in case of troubled connectivity or impaired nodes. In all these contexts, a satisfying FTC design can prevent non-reversible failures and stops, with the ultimate objective of reducing health, environmental and economic damages. The literature about FTC is definetely widespread and contributions arise from diverse applied mathematical domains. In order to get into the argument, there are many survey works that introduce the main theoretical concepts and provide classifications of the outstanding FTC approaches, with detailed references. For example, we refer the reader to the recent review [28], which supplies a comprehensive bibliography, and to [12], [21], [17], [25]. As far as the applications are concerned, aircraft flight control has been motivating FTC research since 1970s, given the evident danger that aircraft faults may cause to human safety. Therefore, a significant amount of papers has been produced on the argument, taking account of the wide variety of issues and models introduced in the study of flight dynamics. For a general overview see [20], [8] and the up-to-date book [6] that in Chapter II provides the list of the most common flight control systems, with the relative references. In this work, a linear model with a multiplicative disturbance factor is considered, which is very common in flight framework ([26]); in particular, we will adopt a system presented in [2],[1] and studied also in [27], [10] as an application test. Even if FTC systems can be designed in many different ways according to the specific aim they are conceived for, in general they all have to perform the following main tasks: 1. 1. the Fault Detection, i.e., the controller makes a binary decision on the presence of a malfunction; 2. 2. the Fault Identification, i.e., the controller determines or estimates the size of the disturbance; if necessary, Identification is preceded by Fault Isolation, that is, the location of the impaired component; 3. 3. the eventual active compensation to the fault, i.e., the reconfiguration of the system inputs and/or parameters in order to maintain, as much as possible, the integrity of the process. Fault Detection and Identification (FDI) can be undertaken in diverse ways. In the cited works, in particular [6] a comprehensive discussion about the most popular FDI schemes is presented: among them, we remind the unknown input observers (UIO, [18], [24]) and residual generation, Kalman filtering, the statistical methods and the more recent techniques based on neural networks ([14]). This paper is devoted to the case when a quantized disturbance input is introduced in a continuous linear system. Such an _hybrid_ model, which combines discrete and continuous dynamics, is motivated by the upcoming digitalization of modern devices: a quantized disturbance may represent the switches of actuators or sensors and the malfunctions in digital components; moreover, it may describe the behavior of any mechanical device that is known to occupy only certain positions and also the approximation of a continuous disturbance. Results about FTC for hybrid systems are not very common. In part, they can be retrieved in the extensive discussion about the detection of _abrupt_ changes in dynamical systems, whose leading work is [4] (while some further contributions are given by [13] and [15]). The problem of estimating brusque alterations is always actual (as an example, see [23] and [22], which respectively concern medical imaging and ground-penetrating radar issues) and in general is approched by classical estimation techniques, such as Kalman Filtering. Recently, input quantization in linear systems has been studied in particular with the aim of reducing the effects of a coarse quantization ([16], [7]). In this work, instead, our purpose is exploiting the information that the disturbance input is quantized to detect the fault occurrene: it follows that quantization is supposed to be already performed in a satisfactory way. In order to evaluate the quantized input disturbance, an original Information theoretic approach is proposed in this paper: given the discrete nature of the disturbance, FDI is performed by a _decoding technique_ derived from the framework of digital transmissions and Coding Theory ([19]). The algorithm we will introduce has already been tested in Deconvolution issues ([9]). The problem we address here still is a Deconvolution problem, given that we assume a linear system as model, but in addition a compensation task is introduced to minimize the consequence of faults: our FTC is conceived with a feedback loop that supplies a compensation input in real-time and then continuously reconfigures the system (which naturally does not happen in classical Deconvolution issues). The structure of the paper is the following: in Section II, we describe the problem we aim to study; in Section III, we introduce the decoding algorithm furtherly used for the Fault Detection; in Section IV, we provide a theoretical analysis of the algorithm in terms of minimization of a suitably defined _Error Function_ that represents the distance between the optimal behavior (i.e., without disturbance) and the output of the FTC itself; sensitivity to the false alarm (_false positive_) and to miss fault detection (_false negative_); promptness of detection and reconfiguration. In Section V, wi give the design criteria to obtain the best performance from our algorithm, while in Section VI we show a few significant simulations about a specific numerical example, arisen from Flight Control literature; finally, Section VII is devoted to some conclusive observations. ### 1.1 Notation In this paper, the following notation will be used: * • given a subset $A$ of a set $X$, $\mathds{1}_{A}:X\to\\{0,1\\}$ will denote the indicator function, defined by $\mathds{1}_{A}(x)=1$ if $x$ belongs to $A$ and $\mathds{1}_{A}(x)=0$ otherwise; * • the function erfc is defined by $\text{erfc}(x)=\int_{x}^{+\infty}e^{-s}ds$ for any $x\in\mathds{R}$; * • random variables will be indicated by capital letters; * • given any variable $X$, $\hat{x}$ will denote its estimation. ## 2 Problem Statement In this paper, we consider processes that can be modeled by the following linear, finite-dimensional system: $\left\\{\begin{array}[]{l}\dot{x}(t)=\mathrm{A}x(t)+\mathrm{B}z(t)f(t)\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ t\in[0,T]\\\ x(0)=0\\\ y(t)=\mathrm{C}x(t)\end{array}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \right.$ (1) where $x(t)\in\mathds{R}^{n}$, $y(t)\in\mathds{R}^{m}$, $f(t)$ and $z(t)$ are scalar functions and $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are constant matrices with consistent dimensions. $f(t)$ is a known input signal, while $z(t)$ is a disturbance modelling some fault in the system. Typically, $z(t)\in(0,1]$; if $z(t)=1$, the system operates in its nominal regime and is totally driven by $f(t)$: this is the condition that one aims to reproduce even when $z(t)\in(0,1)$, i.e., when some unexpected breakdown, interruption or loss of effectiveness affects the dynamics. In order to achieve that, a control input $u$ is introduced, which adjusts the dynamics as follows: $\dot{x}(t)=\mathrm{A}x(t)+\mathrm{B}z(t)\left(f(t)+u(t)\right)$ (2) Notice that to maintain the error-free behavior, say $\mathrm{B}z(t)\left(f(t)+u(t)\right)=\mathrm{B}f(t)$, in principle it is sufficient to fix $u(t)=f(t)\left(\frac{1}{z(t)}-1\right)$, but, in the real applications, this is often impossible for the following motivations. Generally, the disturbance $z$ is not known and the the controller can access it only through the observation of the output $y$. In order to determine $z$ one has to perform a _deconvolution_ , that is, to invert the solution of equation (2) with initial condition $x(0)=0$: $y(t)=\mathrm{C}x(t)=\mathrm{C}\int_{0}^{t}e^{(t-s)\mathrm{A}}\mathrm{B}z(s)(f(s)+u(s))ds$ (3) Furthermore, the acquisition of the data usually is not exact. This inaccuracy can be modeled by an additive noise $n(t)$ in the output (in this work, $n(t)$ will be defined as a white gaussian noise): the available function now is $r(t)=y(t)+n(t)$. Under this condition, the inversion of expression (3) becomes tricky: deconvolution is in fact known to be an ill-posed and ill-conditioned problem, that is, the uniqueness of solution is not guaranteed and also small errors in the data may raise large errors in the solution. In conclusion, the reconstruction of $z(t)$ by inversion may produce outcomes very far from the correct ones; for this reason, an estimation approach to the problem is the most suitable one. In addition to that, in this work we make the following The controller can access $y$ only at certain time instants, say each $\tau$ time instants. Hence, the available data are the samples $r_{k}=r(k\tau)$ where $K\in\mathds{N}$, $k\in\\{0,\dots,K-1\\}$ (for simplicity, let us suppose that $K\tau=T$). Moreover, in this work, two further main assumptions are made. ###### Assumption 1 The controller can access $r(t)$ only at each $\tau$ time instants. The available data are the samples $r_{k}=r(k\tau)$ where $k\in\\{1,\dots,K\\}$ and $K\in\mathds{N}$ is supposed to be such that $K\tau=T$. ###### Assumption 2 The disturbance function $z(t)$ is known to be quantized over two levels, say $z(t)$ can assume only two values $\zeta_{0}$ and $\zeta_{1}$. $\zeta_{0}$ and $\zeta_{1}$ may respectively represent the nominal and the faulty conditions ($\zeta_{0}=1$, $\zeta_{1}\in(0,1)$). Such a binary situation naturally occurs in many engineering applications: it can model, for instance, the abrupt blocking of an actuator, the sharp loss of efficiency of a device, the sudden disconnection of some component, the functioning of alarm sensors. In the next, we will generally refer to the jumps from $\zeta_{0}$ and $\zeta_{1}$ and vice-versa as _switch points_. Notice that Fault Detection and Identification are coincident under this assumption: the decision on the fault presence automatically determines also its size. In this work, being aware of all these conditions, we aim to estimate $z(t)$ as well as possible in order to provide the best control input to the system. Clearly, the estimation has to be performed on-line, that is, each time a sample is acquired (notice that the sampling inevitably undertakes some delay): each $\tau$ instants the controller tries to detect eventual faults and consequently updates the system design. For mathematical simplicity, the eventual switch points of $z(t)$ are suppoed to occur at the time instants $k\tau$, in order to have synchronization with the output sampling. Hence, we can write: $z(t)=\sum_{k=0}^{K-1}z_{k}\mathds{1}_{[k\tau,(k+1)\tau[}(t)\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ z_{k}\in\left\\{\zeta_{0},\zeta_{1}\right\\}$ (4) Now, $z(t)$ is equivalent to the binary sequence $(z_{0},\dots,z_{K-1})\in\\{\zeta_{0},\zeta_{1}\\}^{K}$: the estimation problem is actually discrete. Let $\hat{z}_{k}$ be an estimate of $z_{k}$: since the operation must be performed on-line, we expect $\hat{z}_{k-1}=\mathcal{D}(r_{1},\dots,r_{k})$, where $\mathcal{D}$ indicates a detection/estimation function. Taking account of the conditions mentioned before, the natural definition of the control input is: $u(t)=f(t)\left(\frac{1}{\hat{z}_{k-1}}-1\right)\mathds{1}_{[k\tau,(k+1)\tau)}(t)\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ k=0,\dots,K-1$ (5) $u(t)$ is computed and introduced in the system each $\tau$ time instants. Consider now a generic interval $[k\tau,(k+1)\tau)$. Being based on the estimate $\hat{z}_{k-1}$ relative to the previous interval, $u(t)$ is deceptive when a switch occurs at $k\tau$: the delay $\tau$ underlies a temporary, unavoidable deviation (even in case of correct detection) from the right trajectory. This issue will be widely discussed in the next; for the moment, let us just observe that switch points cause the most of the problems in our FTC model. For this reason, _permanent_ interruptions, i.e., _failures_ (which involve just one switch point) are definitely preferable than _transient_ faults for our purpose, though this should appear as a paradox in the practice. ### 2.1 Illustrative Example: a Flight Control Problem A typical example of FTC problem arises from the literature of Flight Control. Systems of kind (1) are often used to model different aspects of the aerospace dynamics. For instance, if we consider the matrices $\mathrm{A}=\left[\begin{array}[]{ccc}-0.5162&26.96&178.9\\\ -0.6896&-1.225&-30.38\\\ 0&0&-14\end{array}\right]$ (6) $\mathrm{B}=\left[\begin{array}[]{c}-175.6\\\ 0\\\ 14\end{array}\right],\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{C}=\left[1\leavevmode\nobreak\ \leavevmode\nobreak\ 12.43\leavevmode\nobreak\ \leavevmode\nobreak\ 0\right]$ (7) the system (1) represents the longitudinal short-period mode of an F4-E jet with additional horizontal canards, in supersonic conditions. The vector $x$ determines the longitudinal trajectory: its three entries respectively represent the normal acceleration, the pitch rate and the deviation of elevator deflection from the trim position. The output $y(t)$ is the $C^{}$ response, a usual parameter in flight mechanics that synthesizes the aircraft response to the pilot inputs; typically, the $C^{}$ response must lie in a given admissible envelope. This application example is illustrated in the Appendix D.1 of [2] and studied also in [1],[27],[10]. In this context, $f(t)$ can be interpreted as the elevator deflection command and $z(t)$ as the indicator of the status of the elevators: $z=\zeta_{0}$ may attest a good status, while the switch to $z=\zeta_{1}$ may denote an abrupt loss of effectiveness. In such a case, the controller is required to detect the accident and add the suitable control input $u(t)$ in order to recover the optimal trajectory, say the one imposed by the flight plan. In terms of the output $y(t)$, one aims to maintain or to bringit it back into the prescribed envelope. Notice that in this case, it makes sense to suppose the fault to be definitive, that is, the elevator cannot recover its efficiency during the flight. We then talk about a failure. This situation often occurs in the applications, which motivates us to focus on it in our following analysis. This Flight Problem will be retrieved later and used as test application for the implementation of our detection algorithm, which is introduced in the next section. ## 3 Fault Detection: The One State Algorithm Given the quantization of $z_{k}\in\\{\zeta_{0},\zeta_{1}\\}$, it makes sense to settle the same set for the estimation: $\hat{z}_{k}\in\\{\zeta_{0},\zeta_{1}\\}$. This consideration arises from coding/decoding techniques in digital transmissions, where unknown input messages, that are combinations of symbols from a known finite alphabet, must be recovered within the same alphabet. In other terms, the _decoder_ is an estimator that exploits the prior information about the input source. The detection method that we introduce in this section is derived from an optimal decoding algorithm named BCJR after its authors Bahl, Cocke, Jelinek and Raviv (see [3]). Given the noisy output of a digital transmission, the BCJR computes the probabilities of all the possible codewords, implementing a maximum a posteriori (MAP, [19]) estimation through a recursive procedure. In particular, given codes defined on trellises, it evaluates the a posteriori probabilities of each state. The classical version of the algorithm is constituted by two recursions (one forward, one backward) and requires the transmission of the whole message before decoding. Moreover, it also requires the system to have a finite number of states. Nevertheless, it is possibile to modify the proceudre to avoid these bonds: in spite of reliability, one can make it causal (hence to work on line) by removing the backward recursion and also it can be simplified by considering not all the possible states, but just a fixed number of states. In [9], these variations are widely discussed. The algorithm we introduce here is exactly a causal BCJR considering just one state at each step (for this reason we refer to it as the One State Algorithm). The compuations of the probabilities is in this case straightforward and reduces to the comparison between two Euclidean distances at each step. This makes the algorithm definetely low-complexity, which encourages its implementation. Its performance actually depends on the specific application case and will be analysed in the next sections. Now, let us describe the operative structur of the One State Agorithm in detail. ### 3.1 One State Algorithm’s pattern Before showing the algorithm, notice that the solution of the equation (3) can be written recursively as $\begin{split}x_{k}&=e^{\tau\mathrm{A}}x_{k-1}+z_{k-1}(1-u_{k-1})\int_{0}^{\tau}e^{s\mathrm{A}}\mathrm{B}f(k\tau-s)ds\\\ &=e^{\tau\mathrm{A}}x_{k-1}+\frac{z_{k-1}}{\hat{z}_{k-2}}\int_{0}^{\tau}e^{s\mathrm{A}}\mathrm{B}f(k\tau-s)ds\\\ x_{0}&=0\end{split}$ (8) where $x_{k}=x(k\tau)$, $k=0,\dots,K$. Now, the key idea of the One State procedure is to provide a recursive estimation of the state $x_{k}$ and of $z_{k-1}$ given the current lecture $r_{k}$ and the estimate of the previous state $x_{k-1}$. In the next, let us use the following notation: $n_{k}=n(k\tau)$, $d_{E}$ indicates the Euclidean distance and finally: $\mathrm{M}_{\tau,k}=\int_{0}^{\tau}e^{s\mathrm{A}}\mathrm{B}f(k\tau-s)ds$ (9) The One State Algorithm’s pattern is then the following: 1. 1. $k=0$. Initialization: $\hat{x}_{0}=0$; 2. 2. $k=1$. System evolution (with no compensation): $x_{1}=z_{0}\mathrm{M}_{\tau,1}$. Lecture: $r_{1}=y_{1}+n_{1}=\mathrm{C}x_{1}+n_{1}$. Disturbance Estimation: $\hat{z}_{0}=\left\\{\begin{array}[]{ll}\zeta_{0}&\text{if }d_{E}(r_{1},\zeta_{0}\mathrm{C}\mathrm{M}_{\tau,1})\leq d_{E}(r_{1},\zeta_{1}\mathrm{C}\mathrm{M}_{\tau,1})\\\ \zeta_{1}&\text{otherwise }\\\ \end{array}\right.$ State Estimation : $\hat{x}_{1}=\hat{z}_{0}\mathrm{M}_{\tau,1}$. 3. 3. $k=2,\dots,K$. System evolution (with compensation): $x_{k}=e^{\tau\mathrm{A}}x_{k-1}+\frac{z_{k-1}}{\hat{z}_{k-2}}\mathrm{M}_{\tau,k}$. Lecture: $r_{k}=y_{k}+n_{k}=\mathrm{C}x_{k}+n_{k}$. Disturbance Estimation: $\hat{z}_{k-1}=\left\\{\begin{array}[]{ll}\zeta_{0}&\text{if }d_{E}(r_{k},\mathrm{C}e^{\tau\mathrm{A}}\hat{x}_{k-1}+\frac{\zeta_{0}}{\hat{z}_{k-2}}\mathrm{C}\mathrm{M}_{\tau,k})\\\ &\leq d_{E}(r_{k},\mathrm{C}e^{\tau\mathrm{A}}\hat{x}_{k-1}+\frac{\zeta_{1}}{\hat{z}_{k-2}}\mathrm{C}\mathrm{M}_{\tau,k})\\\ z_{1}&\text{otherwise }\\\ \end{array}\right.$ State Estimation: $\hat{x}_{k}=e^{\tau\mathrm{A}}\hat{x}_{k-1}+\frac{\hat{z}_{k-1}}{{\hat{z}_{k-2}}}\mathrm{M}_{\tau,k}$. Notice that the system does not have compensation in the first interval $[0,\tau)$, as the first useful lecture is performed at time $t=\tau$. For the binary nature of each $z_{k}$, the process of estimation/detection reduces here to the comparison of two distances. Moreover, the storage required is of two locations (one float for the current state and one boolean for the current disturbance): the algorithm is definitely low-complexity. ## 4 Theoretical Analysis of the One State Algorithm This section is devoted to the theoretical description of the behavior and performance of the One State Algorithm applied to the system (1)-(4) with a failure, that is, there exists a time instant $T_{F}=k_{F}\tau\in[0,T]$, $k_{F}\in\mathds{N}$ such that $z(t)=\left\\{\begin{array}[]{ll}\zeta_{0}=1&t\in[0,T_{F})\\\ \zeta_{1}\in(0,1)&t\in[T_{F},T]\\\ \end{array}\right.$ (10) or equivalently, $z_{k}=\zeta_{0}$ for $k=0,1,\dots,k_{F}-1$ and $z_{k}=\zeta_{1}$ for $k=k_{F},1,\dots,K-1$. Switch points are particulary tricky and the choice to focus on a system with just one switch point allows to isolate the problem and to understand completely the consequences of a switch. On the other hand, this case is crucial for the applications, where the problem of failures is dramatically serious. Our model can be naturally described in probabilistic terms: the fact that lecture noise is supposed to be white gaussian, (that is, a sequence of independent gaussian random variables $N_{k}\sim\mathcal{N}(0,\sigma^{2})$) introduces some amount of uncertainty in the system. In particular, also $\hat{z}$, $x$, $y$, $r$, $\hat{x}$ are random variables, as they are directly or indirectly functions of the noise. To emphasize that stochastic nature, from now onwards, we will indicate random variables by capital letters. Let us resume the complete recursive system in probabilistic terms: $\begin{split}&X_{0}=0,\leavevmode\nobreak\ \hat{X}_{0}=0,\leavevmode\nobreak\ \hat{Z}_{-1}=\zeta_{0}=1\\\ &X_{k}=e^{\tau\mathrm{A}}X_{k-1}+\frac{z_{k-1}}{\hat{Z}_{k-2}}\mathrm{M}_{\tau,k}\\\ &Y_{k}=\mathrm{C}X_{k}\\\ &R_{k}=Y_{k}+N_{K}\\\ &\hat{Z}_{k-1}=\mathcal{D}_{1}(R_{k},\hat{X}_{k-1},\hat{Z}_{k-2})\\\ &\hat{X}_{k}=e^{\tau\mathrm{A}}\hat{X}_{k-1}+\frac{\hat{Z}_{k-1}}{\hat{Z}_{k-2}}\mathrm{M}_{\tau,k},\leavevmode\nobreak\ \leavevmode\nobreak\ k=1,\dots,K\\\ \end{split}$ (11) where $\mathcal{D}_{1}$ indicates the One State detection function. Notice that $X_{0}$, $\hat{Z}_{-1}$, $X_{1}$, $Y_{1}$ are actually deterministic, in particular, fixing $\hat{Z}_{-1}=\zeta_{0}=1$ is just an other way to state that there is no compensation for the system in the first interval $[0,\tau)$. Finally, we remark that $z(t)$ is not supposed to be driven by some probabilistic law. Such an information on the input might be useful to improve the detection and has been studied in other deconvolutio contexts (see, for instance, [9]). Nevertheless, in this work we rather prefer to focus on a specific disturbance. ### 4.1 The Error Function The performace of the algorithm must be determined through the evaluation of a suitable _error function_ , say a distance between the desired and the real trajectories. In this work, we adopt as error function the discrete stochastic process $(E_{k})_{k=0,1,\dots}$ that describes the signed distance between the trajectory of the system with control and compensation $X_{k}$ and the nominal trajectory $x^{N}(t)$, at time instants $k\tau$, $k=0,1,\dots$: $\left\\{\begin{array}[]{l}E_{k}=X_{k}-x^{N}(k\tau)\\\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ =e^{\tau\mathrm{A}}E_{k-1}+\left(\frac{z_{k-1}}{\hat{Z}_{k-2}}-1\right)\mathrm{M}_{\tau,k}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ k=1,\dots,K\\\ E_{0}=0.\\\ \end{array}\right.$ (12) The so-defined error function is characterized by the following fact:. ###### Proposition 1 For any $k_{0},n\in\mathds{N}$, the event $\\{E_{k_{0}+n}=e^{n\tau A}E_{k_{0}}\\}$ corresponds to the event $\\{\hat{Z}_{k-1}=z_{k}$ for all $k=k_{0},k_{0}+1,\dots k_{0}+n]\\}$. Proof It immediately follows from the definition of $E_{k}$: for any $n\in\mathds{N}$, the event $\\{E_{k+1}=e^{\tau\mathrm{A}}E_{k}\\}$ is equivalent to $\\{\hat{Z}_{k-1}=z_{k}\\}$ and then $\\{E_{k_{0}+n}=e^{n\tau\mathrm{A}}E_{k_{0}}\\}$ corresponds to the event $\\{\hat{Z}_{k_{0}-1}=z_{k_{0}},\hat{Z}_{k_{0}}=z_{k_{0}+1},\dots,\hat{Z}_{k_{0}+n-1}=z_{k_{0}+n}\\}$. Notice that under the hypothesis of the proposition and if $\mathrm{A}$ is asymptotically stable, $E_{k}$ exponentially decades to zero, regardless of the initial value $E_{k_{0}}$. Moreover, observe that the condition $\hat{Z}_{k-1}=z_{k}$ is not the event of correct detection $\hat{Z}_{k}=z_{k}$, since the feedback in the system implies a delay $\tau$; however, if $z_{k}$ is constant over the considered interval, the two events are the same. In the next, we will focus on this context of constant disturbance, which models the state of the system before and after an irreversible failure. In particular, we will study the conditions to obtain correct detection, which leads to the exponential decay of the error; we will show that even if we cannot achieve the certainty of decodig exaclty in the presence of noise, however we can approximate this condition satisfactorily, that is, with a probability close to one, at least in some common situations. More precisely, our goal is to study the probability of the event $E_{k_{0}+n}=e^{n\tau\mathrm{A}}E_{k_{0}}$ conditioned to the fact that $z_{k}\text{ constant for any}k\in[k_{0},k_{0}+n]$ and given some initial conditions at $k_{0}$ concerning the state of the algorithm, which will be defined later. In particular, we will find out the conditions that make this probability sufficiently close to one, for a sufficiently large $n$. This includes the probability to obtain a very small $E_{K}$, starting from any initial error $E_{k_{0}}$, and to preserve it from further perturbations. In the next, we will give the formal definition of the probability described now and we will refer to it as the probability of _n_ -_step error decay_. Before that, we need to evaluate the detection error probability, which is defined and computed in the next paragraph. ### 4.2 Computation of the Detection Error Probability Let us define the stochastic process $(D_{k})_{k=0,1,\dots}$ that represents the distance between the states estimated by the One State procedure and the ones corresponding to the system with compensation: $\left\\{\begin{array}[]{l}D_{k}=\hat{X}_{k}-X_{k}=e^{\tau\mathrm{A}}D_{k-1}+\frac{\hat{Z}_{k-1}-z_{k-1}}{\hat{Z}_{k-2}}\leavevmode\nobreak\ \mathrm{M}_{\tau,k}\\\ D_{0}=0.\\\ \end{array}\right.$ Then, ###### Definition 2 Given $k\in\mathds{N}$, $\mathrm{d}\in\mathds{R}^{n}$ and $\zeta\in\\{\zeta_{0},\zeta_{1}\\}$, we define the Detection Error Probability (_DEP_ for short) as $\text{\emph{DEP}}(k,\mathrm{d},\zeta)=P\left(\hat{Z_{k}}\neq z_{k}\|D_{k}=\mathrm{d},\hat{Z}_{k-1}=\zeta\right).$ By the definition of $D_{k}$, the DEP is equal to $P(\hat{Z_{k}}\neq z_{k},D_{k+1}=e^{\tau\mathrm{A}}d+\frac{z_{k}^{c}-z_{k}}{z_{k-1}}\mathrm{M}_{\tau,k+1}\|D_{k}=\mathrm{d},\hat{Z}_{k-1}=z_{k-1})$ (13) where $z_{k}^{c}$ indicates the complementary of $z_{k}$ in $\\{\zeta_{0},\zeta_{1}\\}$. This probability may be interpreted as the transition probability of the Markov Process $(D_{k},\hat{Z}_{k-1})_{k=0,1,\dots}$ in the state space $\mathbf{D}\times\\{\zeta_{0},\zeta_{1}\\}$, $\mathbf{D}\subset\mathds{R}^{n}$, with starting state $(D_{0},\hat{Z}_{-1})=(0,\zeta_{0})$. The DEP, which is fundamental to calculate the probability of the event $\\{E_{k_{0}+n}=e^{n\tau A}E_{k_{0}}\\}$ as shown in the next paragraph, can be analytically evaluated in the case of scalar output ($m=1$ in the system (1)) and extended to the case $m>1$ with no particular difficulty, through some numerical techniques. In this paper, we discuss in the case $m=1$, which turns out to be interesting for the possibility of analytically describing the behavior of the DEP with respect to the parameters and to analytically derive design criteria for the fault detection. In the sequel, we then assume $Y_{k},R_{k}\in\mathds{R}$, $k=1,\dots,K$. Let $S_{k}^{w}=\mathrm{C}e^{\tau\mathrm{A}}\hat{X}_{k-1}+\frac{w}{\hat{Z}_{k-2}}\mathrm{C}\mathrm{M}_{\tau,k}\in\mathds{R}$ with $w\in\\{\zeta_{0},\zeta_{1}\\}$ be the two possible received signals estimated by the One State Algorithm at the generic step $k$. The DEP is then computed in the following ###### Proposition 3 For any $k=1,2,\dots,K$, $\begin{split}&\emph{\text{DEP}}(k-1,\mathrm{d},\zeta)=\\\ &\leavevmode\nobreak\ =\frac{1}{2}\emph{\text{erfc}}\left(\frac{\Big{\|}\frac{\zeta_{0}-\zeta_{1}}{2\zeta}\mathrm{C}\mathrm{M}_{\tau,k}\Big{\|}+\mathrm{C}e^{\tau\mathrm{A}}\mathrm{d}\left[\left(1-2\mathds{1}_{\\{\zeta_{0}\\}}(z_{k-1})\right)\left(1-2\mathds{1}_{(S_{k}^{\zeta_{1}},+\infty)}(S_{k}^{\zeta_{0}})\right)\right]}{\sigma\sqrt{2}}\right)\end{split}$ (14) Proof Under the hypothesis that $z_{k-1}=\zeta_{1}$ the DEP is given by: $\begin{split}&\text{DEP}(k-1,\mathrm{d},\zeta)\|_{(z_{k-1}=\zeta_{1})}=P\left(\hat{Z}_{k-1}=\zeta_{0}\Big{\|}D_{k-1}=\mathrm{d},\hat{Z}_{k-2}=\zeta,z_{k-1}=\zeta_{1}\right)\\\ &=P\left(\|R_{k}-S_{k}^{\zeta_{0}}\|<\|R_{k}-S_{k}^{\zeta_{1}}\|\leavevmode\nobreak\ \leavevmode\nobreak\ \Big{\|}D_{k-1}=\mathrm{d},\hat{Z}_{k-2}=\zeta,z_{k-1}=\zeta_{1}\right)\\\ &=\left\\{\begin{array}[]{ll}P\left(R_{k}<\frac{S_{k}^{\zeta_{1}}+S_{k}^{\zeta_{0}}}{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \Big{\|}D_{k-1}=\mathrm{d},\hat{Z}_{k-2}=\zeta,z_{k-1}=\zeta_{1}\right)&\text{ if }S_{k}^{\zeta_{1}}>S_{k}^{\zeta_{0}}\\\ P\left(R_{k}\geq\frac{S_{k}^{\zeta_{1}}+S_{k}^{\zeta_{0}}}{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \Big{\|}D_{k-1}=\mathrm{d},\hat{Z}_{k-2}=\zeta,z_{k-1}=\zeta_{1}\right)&\text{ otherwise.}\\\ \end{array}\right.\end{split}$ If $S_{k}^{\zeta_{1}}>S_{k}^{\zeta_{0}}$: $\begin{split}&P\left(R_{k}<\frac{S_{k}^{\zeta_{1}}+S_{k}^{\zeta_{0}}}{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \Big{\|}D_{k-1}=\mathrm{d},\hat{Z}_{k-2}=\zeta,z_{k-1}=\zeta_{1}\right)=\\\ &=P\left(R_{k}<\mathrm{C}e^{\tau\mathrm{A}}\hat{X}_{k-1}+\frac{\zeta_{0}+\zeta_{1}}{2\zeta}\mathrm{C}\mathrm{M}_{\tau,k}\leavevmode\nobreak\ \leavevmode\nobreak\ \|D_{k-1}=\mathrm{d}\right)\\\ &=P\left(\mathrm{C}X_{k}+N_{k}<\mathrm{C}e^{\tau\mathrm{A}}\hat{X}_{k-1}+\frac{\zeta_{0}+\zeta_{1}}{2\zeta}\mathrm{C}\mathrm{M}_{\tau,k}\leavevmode\nobreak\ \leavevmode\nobreak\ \|D_{k-1}=\mathrm{d}\right)\\\ &=P\left(\mathrm{C}e^{\tau\mathrm{A}}X_{k-1}+\frac{\zeta_{1}}{\zeta}\mathrm{C}\mathrm{M}_{\tau,k}+N_{k}<\mathrm{C}e^{\tau\mathrm{A}}\hat{X}_{k-1}+\frac{\zeta_{1}+\zeta_{0}}{2\zeta}\mathrm{C}\mathrm{M}_{\tau,k}\leavevmode\nobreak\ \leavevmode\nobreak\ \|D_{k-1}=\mathrm{d}\right)\\\ &=P\left(N_{k}<\mathrm{C}e^{\tau\mathrm{A}}\mathrm{d}+\frac{\zeta_{0}-\zeta_{1}}{2\zeta}\mathrm{C}\mathrm{M}_{\tau,k}\right)\\\ &=\frac{1}{2}\text{erfc}\left(\frac{-\mathrm{C}e^{\tau\mathrm{A}}\mathrm{d}+\frac{\zeta_{1}-\zeta_{0}}{2\zeta}\mathrm{C}\mathrm{M}_{\tau,k}}{\sigma\sqrt{2}}\right).\end{split}$ The last step depends on the gaussian distribution of $N_{k}$; notice also that $\frac{\zeta_{1}-\zeta_{0}}{\zeta}\mathrm{C}\mathrm{M}_{\tau,k}=S_{k}^{\zeta_{1}}-S_{k}^{\zeta_{0}}>0$. It follows also that for $S_{k}^{\zeta_{1}}\leq S_{k}^{\zeta_{0}}$: $P\left(R_{k}\geq\frac{S_{k}^{\zeta_{1}}+S_{k}^{\zeta_{0}}}{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \Big{\|}D_{k-1}=\mathrm{d},\hat{Z}_{k-2}=\zeta,z_{k-1}=\zeta_{1}\right)=1-\frac{1}{2}\text{erfc}\left(\frac{-\mathrm{C}e^{\tau\mathrm{A}}\mathrm{d}+\frac{\zeta_{1}-\zeta_{0}}{2\mathrm{z}}\mathrm{C}\mathrm{M}_{\tau,k}}{\sigma\sqrt{2}}\right).$ where $\frac{\zeta_{1}-\zeta_{0}}{\zeta}\mathrm{C}\mathrm{M}_{\tau,k}=S_{k}^{\zeta_{1}}-S_{k}^{\zeta_{0}}\leq 0$. Summing up, $\begin{split}&\text{DEP}(k-1,\mathrm{d},\zeta)\|_{(z_{k-1}=\zeta_{1})}=\\\ &=P\left(\|R_{k}-S_{k}^{\zeta_{0}}\|<\|R_{k}-S_{k}^{\zeta_{1}}\|\leavevmode\nobreak\ \leavevmode\nobreak\ \Big{\|}D_{k-1}=\mathrm{d},\hat{Z}_{k-2}=\zeta,z_{k-1}=\zeta_{1}\right)\\\ &=\left\\{\begin{array}[]{ll}\frac{1}{2}\text{erfc}\left(\frac{-\mathrm{C}e^{\tau\mathrm{A}}\mathrm{d}+\frac{\zeta_{1}-\zeta_{0}}{2\zeta}\mathrm{C}\mathrm{M}_{\tau,k}}{\sigma\sqrt{2}}\right)&\text{ if }S_{k}^{\zeta_{1}}>S_{k}^{\zeta_{0}}\\\ 1-\frac{1}{2}\text{erfc}\left(\frac{-\mathrm{C}e^{\tau\mathrm{A}}\mathrm{d}+\frac{\zeta_{1}-\zeta_{0}}{2\zeta}\mathrm{C}\mathrm{M}_{\tau,k}}{\sigma\sqrt{2}}\right)&\text{ otherwise.}\\\ \end{array}\right.\end{split}$ This actually corresponds to the false negative probability. The false positive probability $\text{DEP}(k-1,\mathrm{d},\zeta)\|_{(z_{k-1}=\zeta_{0})}$ can be computed in the same way and the result is: $\begin{split}&\text{DEP}(k-1,\mathrm{d},\zeta)\|_{(z_{k-1}=\zeta_{0})}=P\left(\hat{Z}_{k-1}=\zeta_{1}\Big{\|}D_{k-1}=\mathrm{d},\hat{Z}_{k-2}=\zeta,z_{k-1}=\zeta_{0}\right)\\\ &=P\left(\|R_{k}-S_{k}^{\zeta_{1}}\|<\|R_{k}-S_{k}^{\zeta_{0}}\|\leavevmode\nobreak\ \leavevmode\nobreak\ \Big{\|}D_{k-1}=\mathrm{d},\hat{Z}_{k-2}=\zeta,z_{k-1}=\zeta_{0}\right)\\\ &=\left\\{\begin{array}[]{ll}1-\frac{1}{2}\text{erfc}\left(\frac{-\mathrm{C}e^{\tau\mathrm{A}}\mathrm{d}-\frac{\zeta_{1}-\zeta_{0}}{2\zeta}\mathrm{C}\mathrm{M}_{\tau,k}}{\sigma\sqrt{2}}\right)&\text{ if }S_{k}^{\zeta_{1}}>S_{k}^{\zeta_{0}}\\\ \frac{1}{2}\text{erfc}\left(\frac{-\mathrm{C}e^{\tau\mathrm{A}}\mathrm{d}-\frac{\zeta_{1}-\zeta_{0}}{2\zeta}\mathrm{C}\mathrm{M}_{\tau,k}}{\sigma\sqrt{2}}\right)&\text{ otherwise.}\\\ \end{array}\right.\end{split}$ The thesis is then proved. ###### Remark 1 If $\mathrm{d}=0\in\mathds{R}^{n}$, $\begin{split}\emph{\text{DEP}}(k-1,0,\zeta)&=\frac{1}{2}\emph{\text{erfc}}\left(\frac{\Big{\|}\frac{\zeta_{0}-\zeta_{1}}{2\zeta}\mathrm{C}\mathrm{M}_{\tau,k}\Big{\|}}{\sigma\sqrt{2}}\right)\\\ &=\frac{1}{2}\text{\emph{erfc}}\left(\frac{\|S_{k}^{\zeta_{0}}-S_{k}^{\zeta_{1}}\|/2}{\sigma\sqrt{2}}\right).\end{split}$ (15) This expression suggests an Information theoretic intepretation of our problem. In fact, the presence of the gaussian noise in the data lecture can be thought as if signal $y_{k}$ was transmitted on an Additive White Gaussian Noise (AWGN) channel. If $D_{k-1}=0$, $y_{k}$ can be $S_{k}^{\zeta_{0}}$ or $S_{k}^{\zeta_{1}}$. Moreover, if we shift the signals by their average, so that they become antipodal $\pm\frac{S_{k}^{\zeta_{0}}-S_{k}^{\zeta_{1}}}{2}$, the average energy per channel use at step $k$ is $\mathcal{E}_{k}=\left(\frac{S_{k}^{\zeta_{0}}-S_{k}^{\zeta_{1}}}{2}\right)^{2}$. Given that the spectral density of the gaussian noise is $N_{0}=2\sigma^{2}$, the argument of the erfc function in (15) turns out to be the square root of the so called Signal-to-Noise Ratio (SNR), defined as $\text{SNR}_{k}=\mathcal{E}_{k}/N_{0}$, of our ideal channel. Generally, the SNR compares the magnitudes of the transmitted signal and of the channel noise and it is widely used in Informatiom Theory to describe channel performance. In our framework, the SNR determines the reliability of the detection, say the reliability of the channel where $y_{k}$ is ideally transmitted. This remark emphasizes that our problem is analogous to a common digital-transmission paradigm and bears out the idea of using decoding techniques to the detection task. In the next, we will use the common dB notation for the SNR, that is, we express it as $10\log_{10}$ of its value. ###### Remark 2 Since typically $\zeta_{1}<\zeta_{0}$, by expression (15) we have $\emph{\text{DEP}}(k-1,0,\zeta_{1})<\emph{\text{DEP}}(k-1,0,\zeta_{0}).$ Given that $\hat{Z}_{k-2}=\zeta_{1}$ is generally more likely when $z_{k-2}=\zeta_{1}$ (otherwise our detection method would be improper), we can conclude that our detection algorithm is more reliable after the failure, or, in other terms, it is more sensitive to false positives. ### 4.3 Computation of the Probability of _n_ -step Error Decay Given a time interval $[k_{0},k_{0}+n)$, $k_{0},n\in\mathds{N}$, $k_{0}\geq 1$, we can formally define the probability of _n_ -step error decay (EDPn for short) as $\begin{split}&\text{EDP}^{n}(k_{0},\mathrm{d},\zeta,\eta)=\\\ &P\left(E_{k_{0}+n}=e^{n\tau\mathrm{A}}E_{k_{0}}\big{\|}D_{k_{0}-1}=\mathrm{d},\hat{Z}_{k_{0}-2}=\zeta,z_{k}=\eta\text{ for any }k=k_{0}-1,\dots,k_{0}+n-1)\right)\\\ \end{split}$ where $\mathrm{d}\in\mathds{R}^{n}$, $\zeta,\eta\in\\{\zeta_{0},\zeta_{1}\\}$. Notice that $z_{k}$ is assumed to be constant in $[k_{0}-1,k_{0}+n-1]$, that is, we consider the system before or after a failure event. Recalling the Proposition 1, the EDP is connected to the DEP by the following expression: $\begin{split}&\text{EDP}^{1}(k_{0},\mathrm{d},\zeta,\eta)=P\left(E_{k_{0}+1}=e^{\tau\mathrm{A}}E_{k_{0}}\big{\|}D_{k_{0}-1}=\mathrm{d},\hat{Z}_{k_{0}-2}=\zeta,z_{k_{0}-1}=z_{k_{0}}=\eta\right)\\\ &=P\left(\hat{Z}_{k_{0}-1}=z_{k_{0}}\big{\|}D_{k_{0}-1}=\mathrm{d},\hat{Z}_{k_{0}-2}=\zeta,z_{k_{0}-1}=z_{k_{0}}=\eta\right)\\\ &=1-\text{DEP}(k_{0}-1,\mathrm{d},\zeta)_{\big{\|}z_{k_{0}-1=\eta}}\\\ \end{split}$ that is, the Error decays when the detection is correct. Notice that this relation between EDP and DEP subsists in virtue of the condition $z_{k_{0}-1}=z_{k_{0}}$: if $k_{0}$ were a switch point, the feedback delay would produce a deviation in the Error Function in case of correct detection. Generalizing to $n$ steps, $\begin{split}&\text{EDP}^{n}(k_{0},\mathrm{d},\zeta,\eta)=\\\ &=P(\hat{Z}_{k_{0}-1}=\hat{Z}_{k_{0}}=\dots=\hat{Z}_{k_{0}+n-2}=\eta\big{\|}D_{k_{0}-1}=\mathrm{d},\hat{Z}_{k_{0}-2}=\zeta)\\\ &=P\left((D_{k_{0}},\hat{Z}_{k_{0}-1})=(e^{\tau\mathrm{A}}\mathrm{d},\eta)\|(D_{k_{0}-1},\hat{Z}_{k_{0}-2})=(\mathrm{d},\zeta)\right)\cdot\\\ &\leavevmode\nobreak\ \leavevmode\nobreak\ \cdot\prod_{m=1}^{n-1}P\left((D_{k_{0}+m},\hat{Z}_{k_{0}+m-1})=(e^{(m+1)\tau\mathrm{A}}\mathrm{d},\eta)\big{\|}(D_{k_{0}+m-1},\hat{Z}_{k_{0}+m-2})=(e^{m\tau\mathrm{A}}\mathrm{d},\eta)\right)\\\ &=\text{EDP}^{1}(k_{0},\mathrm{d},\zeta,\eta)\prod_{m=1}^{n-1}\text{EDP}^{1}(k_{0}+m,e^{m\tau\mathrm{A}}\mathrm{d},\eta,\eta)\\\ &=\big{(}1-\text{DEP}(k_{0}-1,\mathrm{d},\zeta)\big{)}_{\big{\|}z_{k_{0}-1}=\eta}\prod_{m=1}^{n-1}\big{(}1-\text{DEP}(k_{0}+m-1,e^{m\tau A}\mathrm{d},\eta)\big{)}_{\big{\|}z_{k_{0}+m-1}=\eta}\\\ \end{split}$ By Proposition 3, this is equal to $\begin{split}&\text{EDP}^{n}(k_{0},\mathrm{d},\zeta,\eta)=\\\ &=\frac{1}{2}\text{erfc}\left(-\frac{\Big{\|}\frac{\zeta_{0}-\zeta_{1}}{2\zeta}\mathrm{C}\mathrm{M}_{\tau,k_{0}}\Big{\|}+\mathrm{C}e^{\tau\mathrm{A}}\mathrm{d}\left[\left(1-2\mathds{1}_{\\{\zeta_{0}\\}}(\eta)\right)\left(1-2\mathds{1}_{(S_{k}^{\zeta_{1}},+\infty)}(S_{k}^{\zeta_{0}})\right)\right]}{\sigma\sqrt{2}}\right)\\\ &\cdot\prod_{m=1}^{n-1}\frac{1}{2}\text{erfc}\left(-\frac{\Big{\|}\frac{\zeta_{0}-\zeta_{1}}{2\eta}\mathrm{C}\mathrm{M}_{\tau,k_{0}+m}\Big{\|}+\mathrm{C}e^{(m+1)\tau\mathrm{A}}\mathrm{d}\left[\left(1-2\mathds{1}_{\\{\zeta_{0}\\}}(\eta)\right)\left(1-2\mathds{1}_{(S_{k+m}^{\zeta_{1}},+\infty)}(S_{k+m}^{\zeta_{0}})\right)\right]}{\sigma\sqrt{2}}\right).\end{split}$ (16) Our next goal is to evaluate the $\text{EDP}^{n}$ in different instances of system (1,10). First of all, let us distinguish what happens before and after the failure. ### 4.4 False positive evaluation Let suppose the system to be affected by a failure according to the model (10) with $k_{F}\geq 1$, that is, the system is not faulty from the beginning. In particular, since there is no compensation at the first time step (or equivalently $\hat{Z}_{-1}=\zeta_{0}$), no false positive is produced at $k=0$. Then, studying the EDP in $[1,k_{F})$ actually corresponds to evaluate the probability that no false postives occur during the whole pre-failure transient regime. Given that $D_{0}=0$, we have $\begin{split}&\text{EDP}^{k_{F}-1}(1,0,\zeta_{0},\zeta_{0})=\prod_{m=1}^{k_{F}-1}\frac{1}{2}\text{erfc}\left(-\frac{\Big{\|}\frac{\zeta_{0}-\zeta_{1}}{2\zeta_{0}}\mathrm{C}\mathrm{M}_{\tau,m}\Big{\|}}{\sigma\sqrt{2}}\right).\end{split}$ (17) Since $E_{1}=0$ and $D_{0}=0$, then $\text{EDP}^{k_{F}-1}(1,0,\zeta_{0},\zeta_{0})=P(E_{k_{F}}=0)=P(D_{k_{F}}=0)$. ### 4.5 Switch Point Suppose that $D_{k_{F}}=0$, then in particular, $\hat{Z}_{k_{F}-1}=z_{k_{F}-1}$ and $\hat{Z}_{k_{F}-1}\neq z_{k_{F}}$. In other terms, the detection is correct, but the compensation, based on the detection at the previous step, is not efficient in correspondance of a switch point. Our detection method cannot control what happens at step at step $k_{F}$, that is, in the time interval $[T_{F},T_{F}+\tau)$. ### 4.6 False negative evaluation Given that we cannot control the system immediately after the switch point, it is likely that $E_{k_{F}+1}\neq 0$. We now want to study the probability of decay of the Error Function towards zero, which actually corresponds to the evaluation of the false negatives. In fact, under the hypothesis $D_{k_{F}}=0$ (i.e., no false positives and in particular $\hat{Z}_{k_{F}-1}=\zeta_{0}$), for any $n\in\mathds{N}$, $\begin{split}&\text{EDP}^{n}(k_{F}+1,0,\zeta_{0},\zeta_{1})=\text{EDP}^{1}(k_{F}+1,0,\zeta_{0},\zeta_{1})\prod_{m=1}^{n-1}\text{EDP}^{1}(k_{F}+1+m,0,\zeta_{1},\zeta_{1})\\\ &\frac{1}{2}\text{erfc}\left(-\frac{\Big{\|}\frac{\zeta_{0}-\zeta_{1}}{2\zeta_{1}}\mathrm{C}\mathrm{M}_{\tau,k_{F}+1}\Big{\|}}{\sigma\sqrt{2}}\right)\prod_{m=1}^{n-1}\frac{1}{2}\text{erfc}\left(-\frac{\Big{\|}\frac{\zeta_{0}-\zeta_{1}}{2\zeta_{1}}\mathrm{C}\mathrm{M}_{\tau,k_{F}+1+m}\Big{\|}}{\sigma\sqrt{2}}\right).\end{split}$ (18) Notice that $n$ can be any positive integer, since the failure state is not reversible. Moreover, it is clear that if $n\to\infty$, then $EDP^{n}\to 0$, that is, it is not likely that the Error decays to zero and remains null forever. However, we can approximate this ideal situation, as we will see in the next. The considerations about the EDP made in this section are now applied to the case of constant input $f(t)$. More precisely we will exploit them to establish suitable design criteria, that is, which is the best choice of parameters to obtain the maximum performance from the One State Algorithm. ### 4.7 Constant input $f(t)$ If the input $f(t)$ is constant, say $f\equiv 1$, the system evolution does not depend on time step $k$. In fact, $\mathrm{M}_{\tau,k}=\mathrm{M}_{\tau}=(e^{\tau\mathrm{A}}-\mathds{I})\mathrm{A}^{-1}\mathrm{B}$ for any $k=1,\dots,K$. Hence, $\begin{split}&\text{EDP}^{n}(1,0,\zeta_{0},\zeta_{0})=\left[\frac{1}{2}\text{erfc}\left(-\frac{\Big{\|}\frac{\zeta_{0}-\zeta_{1}}{2\zeta_{0}}\mathrm{C}\mathrm{M}_{\tau}\Big{\|}}{\sigma\sqrt{2}}\right)\right]^{n}\end{split}$ (19) for any $n\in\mathds{N}$ such that $n+1\leq k_{F}$ and $\begin{split}&\text{EDP}^{n}(k_{F}+1,0,\zeta_{0},\zeta_{1})=\frac{1}{2}\text{erfc}\left(-\frac{\Big{\|}\frac{\zeta_{0}-\zeta_{1}}{2\zeta_{0}}\mathrm{C}\mathrm{M}_{\tau}\Big{\|}}{\sigma\sqrt{2}}\right)\left[\frac{1}{2}\text{erfc}\left(-\frac{\Big{\|}\frac{\zeta_{0}-\zeta_{1}}{2\zeta_{1}}\mathrm{C}\mathrm{M}_{\tau}\Big{\|}}{\sigma\sqrt{2}}\right)\right]^{n-1}.\end{split}$ (20) In terms of signal-to-noise ratio, we can write $\sqrt{\text{SNR}(\eta)}=\frac{\left\|\frac{\zeta_{1}-\zeta_{0}}{2\eta}\mathrm{C}\mathrm{M}_{\tau}\right\|}{\sigma\sqrt{2}}$ so that $\begin{split}&\text{EDP}^{n}(1,0,\zeta_{0},\zeta_{0})=\left[\frac{1}{2}\text{erfc}\left(-\sqrt{\text{SNR}(\zeta_{0})}\right)\right]^{n}\\\ &\text{EDP}^{n}(k_{F}+1,0,\zeta_{0},\zeta_{1})=\frac{1}{2}\text{erfc}\left(-\sqrt{\text{SNR}(\zeta_{0})}\right)\left[\frac{1}{2}\text{erfc}\left(\sqrt{\text{SNR}(\zeta_{1})}\right)\right]^{n-1}.\\\ \end{split}$ Under the hypothesis $0<\zeta_{1}<\zeta_{0}=1$, $\text{SNR}(\zeta_{0})<\text{SNR}(\zeta_{1})$, that is $\text{EDP}^{m}(k_{0},0,\zeta_{0},\zeta_{0})<\text{EDP}^{m}(k_{1},0,\zeta_{1},\zeta_{1})$; in other terms, our detection algorithm is more sensitive to false positives, then our fault tolerant control method is more efficient _after_ the failure. Hence, the suitable design criteria for the pre-failure state will automatically be appropriate also for the post-failure state. This is why in the next we will generically name $\text{SNR}=\text{SNR}(\zeta_{0})\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ and }\leavevmode\nobreak\ \leavevmode\nobreak\ \text{EDP}^{n}=\text{EDP}^{n}(k_{0},0,\zeta_{0},\zeta_{0})=\left[\frac{1}{2}\text{erfc}\left(-\sqrt{\text{SNR}}\right)\right]^{n}.$ (21) The next section is devoted to the study of design criteria for our FTC system, on the basis of the theoretical analysis developed in the last pages. Particular attention will be paid to the case of constant $f(t)$, for which optimal criteria can be formulated. ## 5 Design Criteria In this section, our aim is to provide the design criteria to obtain the best performance from our FTC scheme, based on the One State Algorithm. The key point of this issue is that the controller is supposed to be free to choose the sampling time step $\tau$, hence our goal is to give the criteria to determine the _otpimal_ $\tau$, which, in our framework, can be defined as the one that _minimizes_ the Error Function, in the sense that we now explain. Given the failure system (1,10) and a time window $W=n\tau$ not containing the switch point, our first purpose is to maximize the probability that $E_{k}$ remains null (if we set before the failure) or decays to zero (if we set after the failure) along the interval $W$. Furthermore, given that in $(T_{F},T_{F}+\tau]$ a correct detection causes a failed compensation and a consequent abrupt deviation in the output $y$ (as we will show in the numerical simulations), our second purpose is to minimize the peak of this unavoidable deviation. This qualitative discussion is now quantified in two different input instances: $f(t)$ constant and $f(t)$ sinusoidal. As far as the first case in concerned, we will show that the theoretic analysis of Section 4 provides the instrument to determine the sampling time that minimizes the Error Function in an analytic way. On the other hand, when the input is not constant some difficulties arise in the definition of the optimal $\tau$; however, we will explain how to obtain suitable values of $\tau$ by a numerical numerical computation, still based on the analysis of Section 4. ### 5.1 Design Criteria in the case of constant input $f(t)$ Recalling the Paragraph 4.7 and in particular the simplified notation (21), let us explain how to define the optimal $\tau$ when $f(t)\equiv 1$. As just said, we aim to maximize the EDP in a given time window $W$ not containing the failure instant and to minimize the peak of the deviation immediately after the failure. In particular, if $E_{k_{F}}=0$, by definition 12, the extent of the peak in the output is given by $\max_{t\in(0,\tau]}\|\frac{\zeta_{1}-\zeta_{0}}{\zeta_{0}}\mathrm{C}\mathrm{M}_{t}\|$. In brief, we intend to provide $\begin{split}&\tau_{1}=\underset{\tau>0}{\operatorname{argmax}}\leavevmode\nobreak\ \text{EDP}^{W/\tau}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \tau_{2}=\underset{\tau>0}{\operatorname{argmin}}\leavevmode\nobreak\ \left(\max_{t\in(0,\tau]}\|\mathrm{C}\mathrm{M}_{t}\|\right)\end{split}$ (22) The optimum will be $\tau_{1}=\tau_{2}$, but in general this is not the case. Then, we define the optimal $\tau$ as follows: we do not look for the maximum EDP, but we just require $\text{EDP}^{W/\tau}>1-\varepsilon$ where $\varepsilon<<1$ is a fixed tolerance. In other terms, we demand that the EDP be very close to 1. Then, the optimal $\tau$, indicated by $\tau_{\text{opt}}=\tau_{\text{opt}}(\varepsilon)$, is : $\tau_{\text{opt}}=\underset{\tau:\leavevmode\nobreak\ \text{EDP}^{W/\tau}>1-\varepsilon}{\operatorname{argmin}}\left(\max_{t\in(0,\tau]}\|\mathrm{C}\mathrm{M}_{t}\|\right).$ (23) #### 5.1.1 Application to the Flight Control Problem Let us now compute $\tau_{\text{opt}}$ for the Flight Control Problem introduced in the Paragraph 2.1, in the case of constant input $f(t)$. Figure 1: $m_{\tau}$ In the Figure 1, the graph of $\mathrm{C}\mathrm{M}_{\tau}$ in function of $\tau$ is shown. In particular, we notice that $\mathrm{C}\mathrm{M}_{\tau}$ is negative for any $\tau>0$, achieves a global minimun at $\tau_{0}=0.55$ and converges to a constant value for a sufficienlty large $\tau$. Then, if $\tau>\tau_{0}$, $\max_{t\in(0,\tau]}\|\mathrm{C}\mathrm{M}_{t}\|=\|\mathrm{C}\mathrm{M}_{\tau_{0}}\|$, that is, the peak is fixed and we cannot control it. This undesired occurrence can be prevented by imposing $\tau\in(0,\tau_{0}].$ In this interval, $\mathrm{C}\mathrm{M}_{\tau}$ is monotone decreasing and $\max_{t\in(0,\tau]}\|\mathrm{C}\mathrm{M}_{t}\|=\|\mathrm{C}\mathrm{M}_{\tau}\|$. Then, fixed the tolerance $\varepsilon$, our aim is the computation of $\tau_{\text{opt}}=\underset{\tau\in(0,\tau_{0}]:\text{EDP}^{W/\tau}>1-\varepsilon}{\operatorname{argmin}}\|\mathrm{C}\mathrm{M}_{\tau}\|.$ (24) Notice that $\text{EDP}^{W/\tau}=\left[\frac{1}{2}\text{erfc}\left(-\sqrt{\text{SNR}}\right)\right]^{W/\tau}=\left[\frac{1}{2}\text{erfc}\left(-\frac{\|\frac{\zeta_{1}-\zeta_{0}}{2\zeta_{0}}\mathrm{C}\mathrm{M}_{\tau}\|}{\sigma\sqrt{2}}\right)\right]^{W/\tau}$ is monotone increasing as a function of $\tau$. Then, let $\tau_{m}=\tau_{m}(\varepsilon)$ be the minimum $\tau$ in $(0,\tau_{0}]$ such that $\text{EDP}^{W/\tau}>1-\varepsilon$ (if it exists). Then $\tau_{\text{opt}}=\underset{\tau\geq\tau_{m}}{\operatorname{argmin}}\|\mathrm{C}\mathrm{M}_{\tau}\|=\tau_{m}.$ (25) Now, let assign numerical values to the parameter and solve the corresponding instance. Suppose that: $\begin{split}&\zeta_{0}=1\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \zeta_{1}=\frac{1}{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \sigma^{2}=2\\\ &\varepsilon=10^{-}3\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ W=20\\\ \end{split}$ (26) Figure 2: $EDP^{W/\tau}$ in function of $\tau$ in the instance (26). The second graph is a zoom that allows to see that $\tau_{\text{opt}}=0.112$ In this case, $\tau_{\text{opt}}=0.112$ as shown in Figure 2. Figure 3: The optimal $\tau$’s as the noise variance $\sigma^{2}$ changes ($\zeta_{0}=1,\zeta_{1}=\frac{1}{2},\varepsilon=10^{-}3,W=20$) The value of $\tau_{\text{opt}}$ clearly depends on the noise and in particular there can exist noise values for which there is no $\tau$ making $EDP^{W/\tau}>1-\varepsilon$: for instance, this occurs if we consider $\sigma^{2}>34.72$ in the example (26) (the range of admittible $\sigma^{2}$’s with the corresponding $\tau_{\text{opt}}$’s is shown in Figure 3). In such situation, one should allow a lower threshold $1-\varepsilon$. In Section 6 we will show a few simulations about the Flight Example. ### 5.2 Design Criteria in the case of input $f(t)=\sin t$ When $f(t)$ is not constant, it is more difficult to study analytical design criteria as the quality of the detection depends on time. In particular, at each time step $k\tau$ the detection is affected by the values of $f(t)$, $t\in((k-1)\tau,k\tau)$, then any detection step is different from the others and an analogous of (23) cannot be provided: roughly speaking, the optimum would be to change $\tau$ according to the shape of $f(t)$ in each considered interval. When $f(t)$ is periodic, we can suggest some numerical computation in order to fix a suitable $\tau$. In fact, if we compute $\text{EDP}^{W/\tau}(1,0,\zeta_{0},\zeta_{0})$ for a sufficiently large $W$, we get an idea about the sampling times that are more suitable. On the other hand, there is no way to control the amplitude of the deviation in case of failure, since this again depends on time. The idea is then to choose as samling time that maximises $\text{EDP}^{W/\tau}(1,0,\zeta_{0},\zeta_{0})$ or that makes it larger than a given threshold, being conscious that this does not arrange the anavoidable deviation. Figure 4: $EDP^{W/\tau}(1,0,\zeta_{0},\zeta_{0})$ in function of $\tau$ in the instance (26) ($\zeta_{0}=1,\zeta_{1}=\frac{1}{2},\sigma^{2}=2,W=20$). Let us illustrate these observations in the Flight Control Problem with $f(t)=\sin t$ and parameters given by (26). First, let us numerically compute $\text{EDP}^{W/\tau}(1,0,\zeta_{0},\zeta_{0})$ in function of $\tau$, the result being presented in Figure 4: the graph shows a clear unsettled behavior which cannot be described analytically. However, it also suggests the values of $\tau$ that give an high $\text{EDP}^{W/\tau}(1,0,\zeta_{0},\zeta_{0})$ and which can then considered suitable. No general consideration can be derived, except that a very small $\tau$ is in general not preferable. More details about this instance can be retrieved in the simulations presented in the next Section. ## 6 Flight Control Problem: a few simulations In this section, we show some simulations concerning the application of the One State Algorithm to the Flight FTC example presented in the Paragraph 2.1 and studied in the previous paragraphs. In a time interval $[0,T]=[0,40]$, we suppose that a failure occurs at $T_{F}=20$ and causes the switch of the disturbance function $z(t)$ from $\zeta_{0}=1$ to $\zeta_{1}=1/2$ ($\zeta_{1}=1/2$ might represent a loss of effectiveness of $50\%$ of the elevator of the aircraft). The lecture noise is a gaussian random variable $\mathcal{N}(0,2)$. We consider boht the cases of input $f\equiv 1$ and $f(t)=\sin t$ and we show the behavior of the One State procedure for different values of $\tau$. The graphs represent the output $y(t)$ of the system. Figure 5: Output y(t): Nominal System vs System with a failure at $T_{F}=20$, with lecture noise of variance $\sigma^{2}=2$ and $f\equiv 1$. Six different cases are shown: the first graph represents the system with no control and compensation; the other ones are with compensation, respectively with time step $\tau$ equal to $0.4$, $0.12$, $0.09$, $0.07$, $0.01$ Figure 5 reproduces the case $f\equiv 1$. The first graph compares the nominal system, that is, the desirable trajectory, to the faulty system with no compensation: after the failure, the trajectory of the latter is sensibily uncorrect. In the other graphs, we introduce the compensation using the One State Algorithm: as proved in the Paragraph 5.1.1 , $\tau_{\text{opt}}=0.112$. In the second graph, we fix $\tau=0.4$, which is larger than $\tau_{\text{opt}}$: we obtain a correct detection at each step, but the unavoidable deviation is not optimized: in fact, considering $\tau_{\text{opt}}$ (third graph), we have a smaller peak after the failure. Furthermore, we see that also $\tau=0.09$ is suitable, even if, the corresponding $\text{EDP}^{W/\tau}>1-\varepsilon$. On the other hand, $\tau=0.07$ assures a good detection only after the failure (this is consistent with our observation about the different sensitivity ot false positives and false negatives), while a too small sampling time ($\tau=0.001)$ causes instabililty: the detection is not reliable and the Error is always nonnull. Figure 6: Output y(t): Nominal System vs System with a failure at $T_{F}=20$, with lecture noise of variance $\sigma^{2}=2$ and $f(t)=\sin t$. Six different cases are shown: the first graph represents the system with no control and compensation; the other ones are with compensation, respectively with time step $\tau$ equal to $0.525$, $0.35$, $0.3$, $0.01$, $0.001$ Figure 6 concerns the case $f(t)=\sin t$. Again, the output of the system with no compensation in the first graph undergoes an evident change after the failure at $T_{F}=20$. Instead, applying the One State Algorithm with time step $\tau=0.525$ (this value being suggested by the numerical computation of the EDP) allows to recover the nominal condition. The same occurs with $\tau=0.35$, which is preferable for the smaller amplitude of the unavoidable deviation in correspondence to the switch point. When $\tau=0.3$, some detections fail (the error percentage is about $4\%$), but the output $y$ is not dramatically affected by them. Furthermore, when $\tau=0.01$ the error percentage is about $9\%$: many deviations occur, but they are not very large. In particular, they are quite null when the slope of $y(t)$ is steeper. In correspondence to the switch point a plain oscillation is present, but it is less remarkable than in the cases of larger $\tau$. Decreasing $\tau$ again, the percentage of wrong detections does not overpass $10\%$, but for very small values of $\tau$, the system is unstable (see for instance, the last graph corresponding to $\tau=0.001$) and many oscillations occur. ## 7 Conclusions In this paper, an original Fault Tolerant Control method, based on Information and Coding Theory, has been introduced. Given a linear system with a disturbance and supposing that the disturbance function is quantized over two levels, the detection task can be tackled by decoding techniques. In particular, we have introduced the One State Algorithm which is a low- complexity, recursive decoding algorithm, derived from the BCJR. Its application to a Flight FTC problem has generated satisfactory outcomes even in case of relative high noise in the data acquisition. The low-complexity encourages the implementation of this method; moreover, adjusting the sampling time step $\tau$, one can improve its performance, according to the different values of noise and of input $f$. In some cases, for instance when $f$ is constant, an optimal value of $\tau$ can be analytically computed with sufficient precision, where the optimality is intended in terms of trade-off between convergence conditions and amplitude of the deviations. Other arrangements might be obtained changing the values and the number of levels of quantization. ## References * [1] J. Ackermann. Robustness against sensor failures. Automatica, 20(2):211–215, 1984. * [2] J. Ackermann. Sampled-data control systems: analysis and synthesis, robust system design. Springer, Verlag New York, USA, 1985. * [3] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv. 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In Proceedings of the Institution of Mechanical Engineers – Part G – Journal of Aerospace Engineering, volume 219, pages 263–276. Professional Engineering Publishing, 2005. * [21] R.F. Stengel. Intelligent failure-tolerant control. IEEE Control Systems Magazine, 11(4):14–23, June 1991. * [22] U. Sumbul, J.M. Santos, and J.M. Pauly. A practical acceleration algorithm for real-time imaging. Medical Imaging, IEEE Transactions on, 28(12):2042–2051, dec. 2009\. * [23] V. Venkatasubramanian, H. Leung, and B. Moorman. An interacting multiple-model-based abrupt change detector for ground-penetrating radar. Geoscience and Remote Sensing Letters, IEEE, 4(4):634–638, oct. 2007. * [24] N. Viswanadham and R. Srichander. Fault detection using unknown input observers. Control Theory Adav. Technol., 3:91–101, 1987. * [25] A.S. Willsky. A survey of design methods for failure detection in dynamic systems. Automatica–J. IFAC, 12(6):601–611, 1976. * [26] Dan Ye and Guang-Hong Yang. Adaptive fault-tolerant tracking control against actuator faults with application to flight control. Control Systems Technology, IEEE Transactions on, 14(6):1088–1096, 2006. * [27] Jiong-Sang Yee, Jian Liang Wang, and Bin Jiang. Actuator fault estimation scheme for flight applications. Journal of Dynamic Systems, Measurement, and Control, 124(4):701–704, 2002. * [28] Y. Zhang and J. Jiang. Bibliographical review on reconfigurable fault-tolerant control systems. Annual Reviews in Control, 32(2):229–252, December 2008.	arxiv-papers	2010-11-12T17:16:18	2024-09-04T02:49:14.785766	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Sophie M. Fosson", "submitter": "Sophie Fosson", "url": "https://arxiv.org/abs/1011.2989" }
1011.3291	# Shift of Dirac points and strain induced pseudo-magnetic field in graphene Hua Tong Yang yanght653@nenu.edu.cn Center for Advanced Optoelectronic Functional Materials Research, Key Laboratory for UV-Emitting Materials and Technology of Ministry of Education, and School of Physics, Northeast Normal University, Changchun 130024, China ###### Abstract We propose that the strain induced effective pseudo-magnetic field in graphene can also be explained by a curl movement of the Dirac points, if the Dirac points can be regarded as a slowly varying function of position. We also prove that the Dirac points must be confined within two triangles, each one has 1/8 the area of the Brillouin zone. ###### pacs: 73.22.Pr, 73.22.Dj, 73.22.Gk, 73.20.At The discovery of graphene, a monolayer carbon atom sheet Novoselov-sci306 , and the development of experimental technique to manipulate this two- dimensional(2D) material have ignited intense interest in this system Geim ; Meyer ; Vozmediano ; Neto . One of the most attractive characters of graphene is that its low energy excitation satisfies a massless 2D Dirac equation Semenoff , and the chemical potential crosses its Dirac points(or Fermi points) in neutral graphene. These special characters lead to many unusual properties and new phenomena Gusynin ; Neto ; Novoselov2005 ; Zhang2005 , such as the anomalous integer quantum Hall effect(QHE) Novoselov2005 ; Zhang2005 . Recently, experiments have confirmed another remarkable effect that mechanical strain can induce a very strong effective pseudo-magnetic field, leading to a pseudo-QHE, which can be observed in zero magnetic field Guinea-10 ; Levy . In this paper we propose that the strain induced effective vector potential can also be explained by shift $\delta\mathbf{K}(\mathbf{x})$ of the Dirac points $\mathbf{K}(\mathbf{x})$, its effective pseudo-magnetic field is in proportion to $\nabla\times\mathbf{K}(\mathbf{x})$, only if the Dirac points $\mathbf{K}(\mathbf{x})$ can be regarded as a slowly varying function of position, and the Fermi velocity is generalized to a tensorZhu-Wang . We also prove that the Dirac points can not be arbitrarily moved, they must be confined within two triangles, each one has 1/8 the area of the Brillouin zone(BZ). Firstly, consider a tight-binding Hamiltonian describing a uniformly deformed honeycomb lattice with three different nearest-neighbor hopping energies $t_{1},t_{2},t_{3}$Hasegawa ; Montambaux08 ; Montambaux09 : $\displaystyle\hat{H}=-\sum_{<\mathbf{i}a,\mathbf{j}b>}t_{\mathbf{i}a,\mathbf{j}b}c^{{\dagger}}_{\mathbf{i}a}c_{\mathbf{j}b}+h.c.,$ (1) where $c_{\mathbf{j}b}$ ($c^{{\dagger}}_{\mathbf{i}a}$) are annihilation(creation) operators, $\mathbf{i}$($\mathbf{j}$) are position vectors of unit cells, $a$($b$) denote two inequivalent atoms in a unit cell, $t_{\mathbf{i}a,\mathbf{j}b}$ is the electronic hopping energy from the $\mathbf{j}$th unit cell $b$ atom to $\mathbf{i}$th unit cell $a$ atom. Suppose that the deformed lattice remains invariant under spatial translation, i.e., $t_{\mathbf{i}a,\mathbf{j}b}$ only depends on ${\bf i}-{\bf j}$, but the three nearest-neighbor hopping energies $t_{1,2,3}$ may be different owing to anisotropy of strains, as shown in Fig.1. Figure 1: Unit cell and hopping parameters for deformed graphene. The hopping parameters can be written as some $2\times 2$ matrixes $\mathbf{t}(\mathbf{i}-\mathbf{j})$, whose elements are defined by $[{\bf t}(\mathbf{i}-\mathbf{j})]_{a,b}\equiv t_{\mathbf{i}a,\mathbf{j}b}.$ For this nearest-neighbor tight-binding Hamiltonian, the non-vanishing hopping matrixes are $\displaystyle{\bf t}(0)=\bigg{(}\begin{array}[]{cc}0&t_{1}\\\ t_{1}&0\end{array}\bigg{)},{\bf t}({\bf a}_{1})=\bigg{(}\begin{array}[]{cc}0&t_{2}\\\ 0&0\end{array}\bigg{)},{\bf t}({\bf a}_{2})=\bigg{(}\begin{array}[]{cc}0&t_{3}\\\ 0&0\end{array}\bigg{)}$ (8) and ${\bf t}(-{\bf a}_{1})={\bf t}^{\dagger}({\bf a}_{1}),{\bf t}(-{\bf a}_{2})={\bf t}^{\dagger}({\bf a}_{2}).$ By Fourier transformation $c_{\mathbf{j},a(b)}=\frac{1}{\sqrt{N}}\sum_{\mathbf{k}}c_{\mathbf{k},a(b)}\exp(i\mathbf{k}\cdot\mathbf{j})$ with $N$ a normalization constant, the Hamiltonian (1) can be cast into the form $\displaystyle\hat{H}=-\sum_{\mathbf{k}}\big{[}c^{{\dagger}}_{\mathbf{k},a},c^{{\dagger}}_{\mathbf{k},b}\big{]}\bigg{[}\begin{array}[]{cc}h_{aa}(\mathbf{k})&h_{ab}(\mathbf{k})\\\ h_{ba}(\mathbf{k})&h_{bb}(\mathbf{k})\end{array}\bigg{]}\bigg{[}\begin{array}[]{c}c_{\mathbf{k},a}\\\ c_{\mathbf{k},b}\end{array}\bigg{]},$ (13) where $h_{aa}(\mathbf{k})=h_{bb}(\mathbf{k})=0,$ $h_{ab}(\mathbf{k})=h_{ba}^{\ast}(\mathbf{k}),$ and $\displaystyle h_{ba}(\mathbf{k})=t_{1}+t_{2}\exp(i\mathbf{k}\cdot\mathbf{a}_{1})+t_{3}\exp(i\mathbf{k}\cdot\mathbf{a}_{2}),$ (14) with $\mathbf{a}_{1},$$\mathbf{a}_{2}$ the lattice unit vectors. The energy bands obtained by diagonalizing this Hamiltonian areWallace $E_{\pm}(\mathbf{k})=\pm\|t_{1}+\tilde{t}_{2}(\mathbf{k})+\tilde{t}_{3}(\mathbf{k})\|,$ (15) where $\tilde{t}_{2}(\mathbf{k})=t_{2}e^{i\mathbf{k}\cdot\mathbf{a}_{1}},$ $\tilde{t}_{3}(\mathbf{k})=t_{3}e^{i\mathbf{k}\cdot\mathbf{a}_{2}}$, the plus sign corresponds to the upper($\pi$) and minus to the lower($\pi^{}$) band respectively. From Eq.(15) we notice that if $\mathbf{K}$ is a zero point of $h_{ba}(\mathbf{K})$, i.e., $t_{1}+\tilde{t}_{2}(\mathbf{K})+\tilde{t}_{3}(\mathbf{K})=0,$ (16) then $E_{+}(\mathbf{k})$ and $E_{-}(\mathbf{k})$ will meet at $\mathbf{K}$, i.e., $E_{+}(\mathbf{K})=E_{-}(\mathbf{K})=0$, this $\mathbf{K}$ is known as the Dirac point. The Hamiltonian (13) can be expanded up to a linear order in $\mathbf{p}=\mathbf{k}-\mathbf{K}$ in a neighborhood of point $\mathbf{K}$ $\displaystyle\bigg{[}\begin{array}[]{cc}h_{aa}(\mathbf{k})&h_{ab}(\mathbf{k})\\\ h_{ba}(\mathbf{k})&h_{bb}(\mathbf{k})\end{array}\bigg{]}\simeq\bigg{[}\begin{array}[]{cc}0&\vec{\alpha}^{}\cdot\mathbf{p}\\\ \vec{\alpha}\cdot\mathbf{p}&0\end{array}\bigg{]}=v_{\mu\nu}\sigma^{\mu}p^{\nu},$ (21) where $\mu,\nu=1,2$ denote two components of a 2D vector and a sum over the repeated indices $\mu,\nu$ is implied, $\vec{\alpha}$ is a complex vector with $\textrm{Re}(\vec{\alpha})=(v_{11},v_{12})$, $\textrm{Im}(\vec{\alpha})=(v_{21},v_{22})$, $\sigma^{1,2}$ are Pauli matrixes acting on the sublattice degree of freedom, tensor $v_{\mu\nu}$ represents the anisotropy of the dispersion near the Dirac points, it only occurs noticeable departure from $v_{F}\delta_{\mu\nu}$ in a strongly deformed grapheneZhu-Wang . However, after this modification the strain induced effective vector potential will acquire a direct physical meaning. For a graphene under nonuniform but slowly varying strain, $t_{i}(\mathbf{x})$ and hence the Dirac point $\mathbf{K}(\mathbf{x})$ as well as $v_{\mu\nu}(\mathbf{x})$ can be regarded as some smooth functions of position $\mathbf{x}$, the local linearized Hamiltonian $v_{\mu\nu}\sigma^{\mu}(k^{\nu}-K^{\nu}(\mathbf{x}))$ on the RHS of Eq.(21) can be cast into $\displaystyle v_{\mu\nu}(\mathbf{x})\sigma^{\mu}(p^{\nu}-\delta K^{\nu}(\mathbf{x})),$ (22) where $\mathbf{p}-\delta\mathbf{K}(\mathbf{x})\equiv\mathbf{k}-\mathbf{K}(\mathbf{x})$, $\delta\mathbf{K}(\mathbf{x})\equiv\mathbf{K}(\mathbf{x})-\mathbf{K}_{f}$ with $\mathbf{K}_{f}$ the corresponding Dirac point in strain-free graphene. Unlike the usual explanation of the strain induced gauge field in grapheneKane ; Neto , where the effective vector potential is an auxiliary quantity and describes the mixed effects of both anisotropy of $v_{\mu\nu}$ and the shift of Dirac point, here the vector potential only represents the relative translation of the Dirac points, $(e/c)\mathbf{A}(\mathbf{x})=\delta\mathbf{K}(\mathbf{x})$, its pseudo-magnetic field $\mathbf{B}(\mathbf{x})=(c/e)\nabla\times\mathbf{K}(\mathbf{x})$, and the physical effects are mainly determined by the pseudo-magnetic flux through a loop $(c/e)\oint_{L}\mathbf{K}(\mathbf{x})\cdot d\mathbf{x}$. In the following sections we shall discuss the properties of $\mathbf{K}(\mathbf{x})$, and illustrate how a curl field $\mathbf{K}(\mathbf{x})$ is induced by a strain. From Eq.(16) we know that the vectors representing $t_{1}$, $\tilde{t}_{2}(\mathbf{K})$, $\tilde{t}_{3}(\mathbf{K})$ in the complex plane can form a directed triangle for a Dirac point $\mathbf{K}$, as illustrated in Fig.2a. Figure 2: (color online). (a) Zero points of $h_{ba}(\mathbf{k})$ determine two directed triangles with edges $t_{1,2,3}$ in the complex plane. For a given $t_{1}$ and a fixed direction of $\tilde{t}_{2}$, the arguments of $\tilde{t}_{3}$ must satisfy conditions (29) to ensure $t_{2,3}\geq 0$. (b) Dirac points exist if $t_{i}$ satisfy inequalities (23), which describe a pyramidal domain in $(t_{1},t_{2},t_{3})$ space, if $(t_{1},t_{2},t_{3})$ goes beyond this domain, an energy gap will be opened. According to the triangle inequality, we have the following necessary and sufficient conditions for the existence of the Dirac pointsHasegawa : $t_{1}+t_{2}\geq t_{3},~{}t_{2}+t_{3}\geq t_{1},~{}t_{3}+t_{1}\geq t_{2}.$ (23) These conditions define a pyramidal domain in the $(t_{1},t_{2},t_{3})$ space, shown in Fig.2b. If $t_{1},t_{2},t_{3}$ satisfy inequalities (23), then there exists two directed triangles with the same edges $t_{1},t_{2},t_{3}$ but different possible orientations, which determine two angles $\theta_{1},\theta_{2}$ satisfying $t_{1}+t_{2}e^{i\theta_{1}}+t_{3}e^{i\theta_{2}}=0$, where $\theta_{1},\theta_{2}$ are given by the law of cosine $\displaystyle\begin{split}\theta_{\pm 1}&=\pm\big{[}\pi-\arccos\big{(}\frac{t_{1}^{2}+t_{2}^{2}-t_{3}^{2}}{2t_{1}t_{2}}\big{)}\big{]},\\\ \theta_{\pm 2}&=\pm\big{[}\arccos\big{(}\frac{t_{1}^{2}+t_{3}^{2}-t_{2}^{2}}{2t_{1}t_{3}}\big{)}-\pi\big{]}.\end{split}$ (24) Thus the Dirac points $\mathbf{K}$ can be determined by letting $\exp(i\mathbf{K}\cdot\mathbf{a}_{1})=\exp(i\theta_{1}),~{}\exp(i\mathbf{K}\cdot\mathbf{a}_{2})=\exp(i\theta_{2}),$ (25) so we have $\mathbf{K}=\frac{1}{2\pi}\big{(}\theta_{1}\mathbf{b}_{1}+\theta_{2}\mathbf{b}_{2}\big{)}+\mathbf{K}_{0},$ (26) with $\mathbf{b}_{1},\mathbf{b}_{2}$ the reciprocal lattice vectors defined by $\mathbf{a}_{i}\cdot\mathbf{b}_{j}=2\pi\delta_{ij}$, and $\mathbf{K}_{0}=n\mathbf{b}_{1}+m\mathbf{b}_{2}$ with $n,m$ are arbitrary integers. Notice that if $t_{1}+\tilde{t}_{2}+\tilde{t}_{3}=0$, then $t_{1}+\tilde{t}^{}_{2}+\tilde{t}^{}_{3}=0$, this implies that there exists two Dirac points $\mathbf{K}(\mathbf{x})$ and $-\mathbf{K}(\mathbf{x})$. However, if $(t_{1},t_{2},t_{3})$ exactly locates on the boundary surface of the pyramid, e.g., $t_{1}=t_{2}+t_{3}$, then the two triangles will mutually coincide and $\tilde{t}_{2}=\tilde{t}^{}_{2}$, $\tilde{t}_{3}=\tilde{t}^{}_{3}$(see Fig.2a), hence $\mathbf{K}(\mathbf{x})$ and $-\mathbf{K}(\mathbf{x})$ become equivalent, and $\vec{\alpha}=i(t_{2}\mathbf{a}_{1}+t_{3}\mathbf{a}_{2})$ becomes a pure imaginary vector, so the Fermi velocity in the directions perpendicular to $\vec{\alpha}$ vanishes(Fig.3b and 3d)Montambaux08 ; Montambaux09 ; Pereira . If $(t_{1},t_{2},t_{3})$ goes beyond the domain defined by Eq.(23), e.g., $t_{1}>t_{2}+t_{3}$, Eq.(16) will have no any root, an energy gap with magnitude $E_{g}=2(t_{1}-t_{2}-t_{3})$ will occur at the corresponding points $\mathbf{K}_{\pm}=\pm 1/2(\mathbf{b}_{1}+\mathbf{b}_{2})$(see Fig.3c)Zhou , and the effective Hamiltonian (22) must be further modified by adding a mass term and some second order terms. Figure 3: (color online) (a) Energy band when two Dirac points are very close, where $t_{1}=2.8,t_{2,3}=1.45,$ (b), (d) $t_{1}=2.8,t_{2,3}=1.4,$ two Dirac points are equivalent(superposed), (c) $t_{1}=2.8,t_{2,3}=1.35,$ an energy gap occurs. Another important property is the range of $\mathbf{K}(\mathbf{x})$. We shall prove that the Dirac points must be confined within some special regions of the BZ. To this end, notice that if a Dirac point $\mathbf{K}=(1/2\pi)(\theta_{1},\theta_{2})$ is given, then its associated $t_{1,2,3}$ can also be determined up to an arbitrary factor, except six special cases of $\theta_{1},\theta_{2}=0,\pm\pi$(see Fig. 2a). If $\theta_{1},\theta_{2}\neq 0,\pm\pi$. According to the law of sines we have $\displaystyle\frac{t_{2}}{t_{1}}=\frac{\sin\theta_{2}}{\sin(\theta_{1}-\theta_{2})},~{}~{}\frac{t_{3}}{t_{1}}=\frac{\sin\theta_{1}}{\sin(\theta_{2}-\theta_{1})},$ (27) or $\displaystyle(t_{1},t_{2},t_{3})\propto(\sin(\theta_{2}-\theta_{1}),-\sin\theta_{2},\sin\theta_{1}).$ (28) For the six special cases we have: if $(\theta_{1},\theta_{2})=\pm(\pi,\pi),$ $t_{1}=t_{2}+t_{3};$ if $(\theta_{1},\theta_{2})=\pm(\pi,0),$ $t_{2}=t_{1}+t_{3};$ if $(\theta_{1},\theta_{2})=\pm(0,\pi),$ $t_{3}=t_{1}+t_{2}$. From Eq.(28) we can find that the $\theta_{1},\theta_{2}$ must satisfy some constrain conditions to guarantee $t_{1,2,3}\geq 0$, as illustrated in Fig.2a. For an arbitrary $t_{2}$ and a fixed $\theta_{1}$(direction of $\tilde{t}_{2}$), $\tilde{t}_{3}$ must point in a direction between the directions of $-\tilde{t}_{2}$ and negative real axis, i.e., argument $\theta_{1},\theta_{2}$ must satisfy $\displaystyle\begin{split}\theta_{1}&+\pi<\theta_{2}<\pi,&\theta_{1}&\in(-\pi,0),\\\ -\pi&<\theta_{2}<\theta_{1}-\pi,&\theta_{1}&\in(0,\pi).\\\ \end{split}$ (29) These two inequalities respectively determine the range of $\mathbf{K}(\mathbf{x})$ and $-\mathbf{K}(\mathbf{x})$. They describe two open triangles $\bigtriangleup MM^{\prime}_{1}M^{\prime\prime}_{1}$ and $\bigtriangleup M^{\prime}M_{1}M^{\prime\prime}_{3}$ in reciprocal space, as shown in Fig.4, each one has 1/8 the area of a unit cell of the reciprocal space(the parallelogram $M^{\prime\prime}M^{\prime\prime}_{1}M^{\prime\prime}_{2}M^{\prime\prime}_{3}$), and each Dirac point is confined within a triangle, so, the Dirac points $\mathbf{K}$ and $-\mathbf{K}$ can meet(become equivalent) only at the vertexes of $\bigtriangleup MM^{\prime}_{1}M^{\prime\prime}_{1}$ and $\bigtriangleup M^{\prime}M_{1}M^{\prime\prime}_{3}$. The remaining hexagon(blue in Fig.4) is a forbidden region for the Dirac points. This confinement also limits the order of magnitude of $\nabla\times\mathbf{K}(\mathbf{x})$, i.e., the strain induced pseudo-magnetic field. Figure 4: (color inline) Rang of the Dirac points consists of six triangles in first the BZ, or $\bigtriangleup MM^{\prime}_{1}M^{\prime\prime}_{1}$ and $\bigtriangleup M^{\prime}M_{1}M^{\prime\prime}_{3}$. In order to show the underlying regularity, here we have ignored the variations of $\mathbf{b}_{1},\mathbf{b}_{2}$ with the deformation of lattice, and simply sketch all $\mathbf{K}=(k_{1},k_{2})$ in the same affine frame. After translating to the first BZ of graphene, $\bigtriangleup MM^{\prime}_{1}M^{\prime\prime}_{1}$ and $\bigtriangleup M^{\prime}M_{1}M^{\prime\prime}_{3}$ are equivalent to a ringlike region consists of six triangles $\triangle MKM^{\prime}_{1},$ $\triangle M_{1}K_{2}M^{\prime\prime}_{2},$ $\triangle M^{\prime\prime}K_{1}M^{\prime},$ etc. In order to illustrate how a non-vanishing $\nabla\times\mathbf{K}$ is induced by strain, we only need to analyze three ideal cases, in which only one $t_{i}$ is slightly changed, $t_{i}\rightarrow t_{0}+\delta t_{i}$, while the other two $t_{j,k}$ remain constant, $t_{j}=t_{k}=t_{0}$, which can also be roughly regarded as that the bond $\mathbf{c}_{i}$ is elongated(or compressed) while the other two bonds $\mathbf{c}_{j},\mathbf{c}_{k}$ and their directions remain fixed(see Fig.5b). Notice that the Dirac points only depend on the relative proportions of $t_{1},t_{2},t_{3}$, so, as an equivalent case, we can always assume that $t_{1}$ remains constant and only $t_{2}$, $t_{3}$ are variables. Moreover, in these equivalent cases the $\tilde{t}_{2}$ and $\tilde{t}_{3}$ can be determined by the end of the vector $t_{1}+\tilde{t}_{2}=-\tilde{t}_{3}$, denoted by $P$ in Fig.5a. So, we can represent the variation of the Dirac points by the shift of the point P. To this end, we have to determine the corresponding P of the three classes of characteristic points in the range of the Dirac points: (1) $K$(or $K^{\prime}$) etc.(see Fig.4), according to Eq.(27), Dirac points locate at these two points only if $t_{1}=t_{2}=t_{3}$, their corresponding P is located at $K$(or $K^{\prime}$) in Fig.5a; (2) critical points $M(M_{1})=(\pm 1/2,0)$, $M^{\prime}(M^{\prime}_{1})=(0,\pm 1/2)$, and $M^{\prime\prime}(M^{\prime\prime}_{2})=(\pm 1/2,\pm 1/2),$ in these cases there exist only one Dirac point since the points $\mathbf{K}$ and $-\mathbf{K}$ are equivalent, their corresponding $(t_{1},t_{2},t_{3})$ are located on the boundary of the pyramidal domain, while their corresponding P are located at the real axis in Fig.5a; (3) $O$, $O^{\prime}$, $O^{\prime\prime}$ etc., their corresponding P are the centers of two circles and the infinite limit points of the straight line $KK^{\prime}$ in Fig.5a, which respectively correspond to the limits of $t_{2}\rightarrow 0$, $t_{3}\rightarrow 0$ and $t_{1}\rightarrow 0$(equivalent to $t_{2}=t_{3}\rightarrow\infty$). Figure 5: (a) The trajectories of P in three ideal cases. (b) A schematic diagram of the shift of Dirac point $\delta\mathbf{K}$ and its curl, $\delta\mathbf{K}$ perpendicular to $\mathbf{c}_{i}$, if bond $\mathbf{c}_{i}$ is slightly elongated. Now we analyze the shifts of the Dirac points in the three ideal situations. (1) $t_{1},t_{2}$ remain constant and $t_{2}=t_{1}$, only $t_{3}$ is variable, the trajectory of the corresponding P is a circle with radius $t_{1}$ and centered at the point $(t_{1},0)$ in Fig.5a, so the arguments $(\theta_{1},\theta_{2})$ satisfy $\displaystyle\begin{split}\theta_{1}-2\theta_{2}-2\pi=0,&~{}~{}\theta_{1}\in(-\pi,0),\\\ \theta_{1}-2\theta_{2}+2\pi=0,&~{}~{}\theta_{1}\in(0,\pi).\end{split}$ (30) They describe line segments $M^{\prime}O_{1}$ and $M^{\prime}_{1}O$ in Fig.4; (2) $t_{3}(=t_{1})$ remain constant while $t_{2}$ is variable, the trajectory of corresponding P is another circle with radius $t_{1}$ centered at the origin, its associated $(\theta_{1},\theta_{2})$ satisfy $\displaystyle\begin{split}\theta_{2}-2\theta_{1}+2\pi=0,&~{}~{}\theta_{2}\in(-\pi,0),\\\ \theta_{2}-2\theta_{1}-2\pi=0,&~{}~{}\theta_{2}\in(0,\pi),\end{split}$ (31) which describe $MO^{\prime}$ and $M_{1}O^{\prime}_{1}$ in Fig.4; (3) $t_{1}$ remains constant while $t_{2},t_{3}$ are variable but $t_{2}=t_{3}$(or vice versa, $t_{1}$ is variable, $t_{2}(=t_{3})$ remain constant), the trajectory of P is straight line $KK^{\prime}$, $(\theta_{1},\theta_{2})$ satisfy $\theta_{1}+\theta_{2}=0,~{}~{}\frac{\pi}{2}<\|\theta_{1}\|<\pi,$ (32) which describe $M^{\prime\prime}_{1}O^{\prime\prime}$ and $M^{\prime\prime}_{3}O^{\prime\prime}_{1}$ in Fig.4. Summarizing Eqs.(30)(31)(32) and comparing with Fig.4, we observe that if a band, e.g., $\mathbf{c}_{1}$ is slightly elongated (or compressed) along its direction, $\mathbf{c}_{1}\rightarrow(1+\delta)\mathbf{c}_{1}$, while the other two bonds $\mathbf{c}_{2},\mathbf{c}_{3}$ remain fixed, then $t_{1}$ will be slightly changed while $t_{2}(=t_{3})$ remain constant, the Dirac point $\mathbf{K}$ will be slightly moved in the direction perpendicular to $\mathbf{c}_{1}$, i.e., $\delta K_{y}\neq 0$(see Fig.4, $K$ moves towards $O^{\prime\prime}$ if $t_{1}$ decreases, towards $M^{\prime\prime}_{1}$ if $t_{1}$ increases), $\mathbf{K^{\prime}}(=-\mathbf{K})$ is moved in the opposite direction. Thus, if the elongation of $\mathbf{c}_{1}$ is slowly varying in the x-direction, i.e, $\partial t_{1}/\partial x\neq 0$, then $\partial K_{y}/\partial x\neq 0$, the other two cases are similar. So, a nonuniform strain as schematically shown in Fig.5b can induce a curl field $\mathbf{K}(\mathbf{x})$, $\nabla\times\mathbf{K}\neq 0$. ###### Acknowledgements. We thank Yugui Yao, Chengshi Liu, Yichun Liu for their helpful discussions. This work was supported by the National Science Foundation of China(Grant Nos.10974027, 50725205, 50832001). ## References * (1) K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Gregorieva, and A. A. Firsov, Science 306, 666(2004); K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. M. Morozov, A. K. Geim, Proc. Natl. Acad. Sci. 102, 10451(2005). * (2) M. A. H. Vozmediano, M. P. Lopez-Sancho, and F. Guinea, Phys. Rev. Lett. 89, 166401(2002). * (3) J.C. Meyer, K. Geim, M.I. Katsnelson, K.S. Novoselov, T.J. Booth, and S. Roth, Nature 446, 60(2007). * (4) A.K. Geim, and K.S. Novoselov, Nature Materials 6, 183(2007); C.W.J. Beenakker, Rev. Mod. Phys. 80, 1337(2008). * (5) A. H. Castro Neto, F. 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B 81, 245420(2010).	arxiv-papers	2010-11-15T05:14:11	2024-09-04T02:49:14.802104	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hua-Tong Yang", "submitter": "Hua-Tong Yang", "url": "https://arxiv.org/abs/1011.3291" }
1011.3460	# Protecting Two-Qubit Quantum States by $\pi$-Phase Pulses Jia-Zhong Hu Department of Physics and the Key Laboratory of Atomic and Nanosciences, Ministry of Education, Tsinghua University, Beijing 100084, China Xiang-Bin Wang xbwang@mail.tsinghua.edu.cn Department of Physics and the Key Laboratory of Atomic and Nanosciences, Ministry of Education, Tsinghua University, Beijing 100084, China Leong Chuan Kwek kwekleongchuan@nus.edu.sg Certer for Quantum Technologies, National University of Singapore, 2 Science Driver 3, Singapore 117542 National Institute of Education and Institute of Advanced Studies, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616 ###### Abstract We study the state decay of two qubits interacted with a common harmonic oscillator reservoir. There are both decoherence error and the error caused by the amplitude change of the superradiant state. We show that frequent $\pi$-phase pulses can eliminate both typpes of errors therefore protect a two-qubit odd-parity state more effectively than the frequent measurement method. This shows that the the methods using dynamical decoupling and the quantum Zeno effects actually can give rather different results when the operation frequency is finite. ###### pacs: 03.67.Pp, 03.65.Ud ## I introduction The interaction between a quantum system and its environment inevitably leads to the decoherencePeter ; Kofman of a quantum state. Such quantum decoherence can often cause severe distortion to a quantum state rendering many quantum systems in the real world uselessEPR ; QTEL ; DiVincenzo ; DiVi ; Josephson ; Huang ; Duan ; Hu . In order to protect a quantum state, many methods against decoherence have been studied. Among the existing proposals, most of them are for single-qubit state protectionViola ; Breuer ; Vitali ; Fanchini ; Uhrig ; Yang . RecentlyMan , a scheme for the protection of quantum entanglement of two qubits at 0K temperature was proposed using quantum Zeno effect (QZE), i.e., via frequent measurement of the environment photon number for the Jaynes-Cummings (J-C) model. However, as shown below, besides the decoherence error, the amplitude of the superradiant state can also changed. It decreases with evolution. Intuitively speaking, the superradiant state changes into $\|00\rangle$ gradually in the evolution therefore the initial odd-parity state cab be severely distorted after a long time evolution. The frequent measurement method cannot eliminate such distortion efficiently because it actually removes the term $\|00\rangle$ at every step. After a long evolution time with a fixed measurement frequency, the amplitude of the superradiant state decreases a lot therefore severely distort the initial unknown state. On the other hand, given the existing technologies, it seems that the measurement of photon number of the environment of all modes remains a challenging task. It is therefore an interesting problem to study how one could protect a two- qubit state with techniques which have been demonstrated already, for example, dynamical decoupling schemeViola ; Breuer ; Vitali ; Fanchini ; Uhrig ; Yang ; West which have been demonstrated experimentally recentlyDu ; Biercuk ; Uys . It is well known that in the limit of infinitely frequent operations, QZE and dynamical decoupling are unified and can have the same resultsFacchi1 ; Facchi2 ; Busch . The two methods are not compared in the more realistic condition when the operation frequencies are finite. Here we show that the dynamical decoupling scheme achieved through a frequent application of $\pi$-phase pulses can protect a two-qubit state more effectively. The scheme not only protects the state from decoherence error, but also prevents the amplitude changing of superradiant state. This paper is arranged as following: we first review the existing results of the J-C modelScully ; Man for two qubits, in particular, the time evolution of the odd parity stateMan under zero temperature. We point out why the amplitude of superradiant state changes in a frequent measurement scheme. We then show how to protect the state by $\pi-$phase pulses and why $\pi-$phase pulses scheme can prevent the amplitude change. The consequences of the finite frequency and duration of each pulses are also presented. ## II Amplitude change of supperradiant state in J-C model Consider the following Hamiltonian for a two-qubit system and its environment as used inMan : $H=H_{s}+H_{e}+H_{i}$ (1) where $H_{s}=\omega_{0}(\sigma^{+}_{1}\sigma^{-}_{1}+\sigma^{+}_{2}\sigma^{-}_{2})$ (2) is Hamiltonian of the two-qubits (system), $H_{e}=\sum_{k}\omega_{k}b^{+}_{k}b_{k}$ (3) is the Hamiltonian of the environment, and $H_{i}=(\alpha_{1}\sigma^{+}_{1}+\alpha_{2}\sigma^{+}_{2})\sum_{k}g_{k}b_{k}+h.c.$ (4) is Hamiltonian of the interaction between the system and the environment. Notations $b_{k}$ and $b^{+}_{k}$ are the annihilation and creation operators of the environment with frequency $\omega_{k}$; $\omega_{0}$ is the atomic transition frequency between the ground state $\|0\rangle$ and the excited state $\|1\rangle$; $\sigma^{+}=\|1\rangle\langle 0\|$ and $\sigma^{-}=\|0\rangle\langle 1\|$. The solution of such a model for the case of zero environmental temperature is well known and it can be found in Ref. Scully ; Tavis . In particular, for an odd-parity two-qubit initial state, there exists a dark state $\|\mu\rangle={\alpha_{2}\over\alpha}\|1\rangle_{1}\|0\rangle_{2}-{\alpha_{1}\over\alpha}\|0\rangle_{1}\|1\rangle_{2}$ (5) and a superradiant stateYu ; Almeida ; Palma ; Zanardi . $\displaystyle\|\nu\rangle={\alpha_{1}\over\alpha}\|1\rangle_{1}\|0\rangle_{2}+{\alpha_{2}\over\alpha}\|0\rangle_{1}\|1\rangle_{2}.$ (6) The dark state $\|\mu\rangle$ does not change with time under the JC model, while the superradiant state changes with time according to $\|\nu\rangle\otimes\|0\rangle_{e}=\eta(t)\|\nu\rangle\otimes\|0\rangle_{e}+\|0\rangle_{1}\|0\rangle_{2}\otimes\sum_{k}\left(c(t)_{k}\|1_{k}\rangle_{e}\right)$ (7) and $\eta(t)=e^{-{\lambda t\over 2}}[\cosh({\Omega t\over 2})+{\lambda\over\Omega}\sinh({\Omega t\over 2})]$ (8) where $\Omega=\sqrt{\lambda^{2}-4R^{2}}$, $\alpha=\sqrt{\alpha^{2}_{1}+\alpha^{2}_{2}}$ and $R=\alpha W$. Given any odd- parity initial state $\|\psi_{0}\rangle=\beta_{1}\|\mu\rangle+\beta_{2}\|\nu\rangle$, the time evolution is $\displaystyle\beta_{1}(t)=\beta_{1}$ (9) $\displaystyle\beta_{2}(t)=\beta_{2}\eta(t)$ (10) and $\eta(t)$ is given in Eq.(8). $\|\psi(t)\rangle=(\beta_{1}\|\mu\rangle+\beta_{2}\eta(t)\|\nu\rangle)\otimes\|0\rangle_{e}+\sum_{k}\|0\rangle_{1}\|0\rangle_{2}\otimes c(t)_{k}\|1_{k}\rangle_{e}$ (11) The decoherence error come from the term $\sum_{k}\|0\rangle_{1}\|0\rangle_{2}\otimes c(t)_{k}\|1_{k}\rangle_{e}$. As shown in Ref.Man , by frequently measuring the environment, one can remove the the term $\|00\rangle$ and protect the initial state from decoherence error, as long as one does not find a photon coming from the reservoir. Intuitively speaking, such a frequent measurement works like a state filter which removes $\|00\rangle$ during the stage when its probability is small. However, even though one can always remove the term with $\|00\rangle$ successfully by measurement, one can not protect the initial state for a long time with the scheme because $\eta(t)$ decreases significantly with time. Suppose the environment is measured after every time interval $\Delta t$, and we continue to find no photon. At time $t$, the state is changed into $\|\psi(t)\rangle=(\beta_{1}\|\mu\rangle+\beta_{2}r(t)\|\nu\rangle)\otimes\|0\rangle_{e}$ (12) and $r(t)=\left[\eta(\Delta t)\right]^{t/\Delta t}.$ (13) Since each measurement removes the photon in environment, $\|\nu\rangle\otimes\|0\rangle_{e}$ restart the evolutionm from the initial state again in each time interval. To protect two-qubit state of the system more effectively, we can use the dynamical decoupling scheme through applying $\pi$ pulses frequently instead of a filtration scheme with frequent measurement. The main idea is, whenever the state decays a little bit, i.e., $\eta(t)$ decreases a little bit and the term with $\|00\rangle$ appears with a small amplitude, a $\pi$ pulse is applied and as a result, $\eta(t)$ will rise and the amplitude of term with $\|00\rangle$ decreases. That is to say, a $\pi-$pulse does not simply remove the term with $\|00\rangle$, it changes the term with $\|00\rangle$ back to the superradiant state. Therefore, it differs from the state filtration of the measurement based scheme - it is really a more effective scheme of state recovery. ## III Elimination of Decoherence with frequent $\pi$-phase pulses We show here that $\pi$-phase pulses can eliminate the decoherence on the one hand and prevent amplitude changing of the superradiant state on the other hand. A $\pi$-phase impulse take a phase-shift operation as: $\displaystyle\|0\rangle\rightarrow-\|0\rangle\;\|1\rangle\rightarrow\|1\rangle$ (14) Apply a $\pi-$pulse to each qubit, the two-qubit unitary operation is then $\displaystyle\begin{array}[]{c}\|1\rangle_{1}\|0\rangle_{2}\rightarrow\|1\rangle_{1}\|0\rangle_{2}\\\ \|0\rangle_{1}\|1\rangle_{2}\rightarrow\|0\rangle_{1}\|1\rangle_{2}\\\ \|0\rangle_{1}\|0\rangle_{2}\rightarrow-\|0\rangle_{1}\|0\rangle_{2}\end{array}$ (18) ### III.1 Some iteration formulas Our method involves applying simultaneously $\pi$-pulses to each qubit. To obtain the results of the method, we need some iteration formulas first. Suppose the interval between two consequent impulses is $\tau$ and $n=[{t\over\tau}]$, we can calculate the coefficients, $r_{1}$ and $r_{2}$, by $\begin{split}\dot{r}_{1}(t)=&-\int_{n\tau}^{t}d\kappa f(t-\kappa)[\alpha^{2}_{1}r_{1}(\kappa)+\alpha_{1}\alpha_{2}r_{2}(\kappa)]-\sum_{m=0}^{n-1}(-1)^{n-m}\int_{m\tau}^{(m+1)\tau}d\kappa f(t-\kappa)[\alpha^{2}_{1}r_{1}(\kappa)+\alpha_{1}\alpha_{2}r_{2}(\kappa)]\\\ \dot{r}_{2}(t)=&-\int_{n\tau}^{t}d\kappa f(t-\kappa)[\alpha^{2}_{2}r_{2}(\kappa)+\alpha_{1}\alpha_{2}r_{1}(\kappa)]-\sum_{m=0}^{n-1}(-1)^{n-m}\int_{m\tau}^{(m+1)\tau}d\kappa f(t-\kappa)[\alpha^{2}_{2}r_{2}(\kappa)+\alpha_{1}\alpha_{2}r_{1}(\kappa)]\end{split}$ (19) where we know $c(t)_{k}$, the coefficient of $\|1_{k}\rangle_{e}$ in Eqn.[11], changes the signal because of the two phase impulses corresponding to $t=m\tau$(m is integer.). Under the interaction with a Lorentzian spectral density, we get the similar result as the free evolution, namely that there exist one dark state $\|\mu\rangle$ and one superradiant state $\|\nu\rangle$. We next study the condition that $\beta_{1}(t)$ and $\beta_{2}(t)$ should satisfy between one time interval. When $t$ is between $m\tau$ and $(m+1)\tau$. $\displaystyle\beta_{1}(t)=\beta_{1}$ (20) $\displaystyle\ddot{\beta}_{2}(t)+\lambda\dot{\beta}_{2}(t)+R^{2}\beta_{2}(t)=0$ (21) We know that the general solution of Eqn. 21 is: $\displaystyle\beta_{2}(t)=$ $\displaystyle e^{-{\lambda(t-m\tau)\over 2}}[A_{m}\cosh({\Omega(t-m\tau)\over 2})$ (22) $\displaystyle+B_{m}\sinh({\Omega(t-m\tau)\over 2})]$ where $t\in[m\tau,(m+1)\tau]$. $A_{m}$ and $B_{m}$ are the constant coefficients of the solution $\beta_{2}(t)$ at each time interval. Our task now is to determine the relation of all $A_{m}$ and $B_{m}$. At $t=(m+1)\tau$ using the boundary condition, $\beta_{2}((m+1)\tau^{-})=\beta_{2}((m+1)\tau^{+})$ and $\dot{\beta}_{2}((m+1)\tau^{-})=-\dot{\beta}_{2}((m+1)\tau^{+})$, we can get the relation: $\begin{split}A_{m+1}=&e^{-{\lambda\tau\over 2}}[A_{m}\cosh({\Omega\tau\over 2})+B_{m}\sinh({\Omega\tau\over 2})]\\\ B_{m+1}=&-e^{-{\lambda\tau\over 2}}[A_{m}\sinh({\Omega\tau\over 2})+B_{m}\cosh({\Omega\tau\over 2})]\\\ &+{2\lambda\over\Omega}e^{-{\lambda\tau\over 2}}[A_{m}\cosh({\Omega\tau\over 2})+B_{m}\sinh({\Omega\tau\over 2})]\end{split}$ (23) Here $\beta_{2}(m\tau^{-})$ means the value of $\beta_{2}(m\tau)$ before the pulse. And $\beta_{2}(m\tau^{+})$ is the value of $\beta_{2}(m\tau)$ after the pulse. We know that if the initial state is superradiant state, $\beta_{2}(t)=1$, the initial value should be $A_{0}=1$ and $B_{0}={\lambda\over\Omega}$. By using the relationship between $A_{m}$, $B_{m}$, $A_{m+1}$ and $B_{m+1}$, we can get an analytical function $\xi(t)$ which can express the fidelity $\xi(t)=\sqrt{\langle\nu\|\rho_{\nu}(t)\|\nu\rangle}$ of superradiant state. $\displaystyle\xi(t)=$ $\displaystyle e^{-{\lambda(t-m\tau)\over 2}}[A_{m}\cosh({\Omega(t-m\tau)\over 2})$ (24) $\displaystyle+B_{m}\sinh({\Omega(t-m\tau)\over 2})]$ for $t\in[m\tau,(m+1)\tau]$. The fidelity $F(t)=\sqrt{\langle\psi\|\rho(t)\|\psi\rangle}$ is given by $\displaystyle\|\psi\rangle=(\beta_{1}\|\mu\rangle+\beta_{2}\|\nu\rangle)$ (25) $\displaystyle F(t)=\|\beta_{1}\|^{2}+\|\beta_{2}\|^{2}\xi(t)$ (26) With this iteration formula for $\xi(t)$ above, we see tah the fidelity oscillates about a horizontal line within a small range after a pulse is applied, first increasing and then descending, as shown in Fig [1]. In contrast, the fidelity based on the method of Quantum Zeno Effect descends almost monotonously with time. Figure 1: (Color online) Comparison of the fidelity evolution during $t=[0,\;1]$ of two methods for the superradiant state. We set $\tau=0.1$, $\lambda=2$ and $\Omega=1$. Line 2 is for dynamical decoupling of this work and Line 1 is for QZE of RefMan . The fidelity only oscillates in a small range around a horizontal line in the dynamical decoupling method. ### III.2 Effect of finite duration of Double-$\pi$-Phase Operation In practice, one cannot set the duration pulse time to be infinitely small. Here we consider the more realistic case that duration time of double-$\pi$-phase is finite and we study show the effect on the fidelity. We only consider the sequential pulses method and we suppose the pulse duration is $\tau\over N$ in each time interval $\tau$. We know that the decoherence coefficient of Superradiant state is Eqn. 22 when $t\in[m\tau,(m+1-{1\over N})\tau]$. So the effective Hamiltonian for the sequential double-$\pi$-phase operator is $H_{phase}={N\pi\over\tau}(\sigma^{+}_{1}\sigma^{-}_{1}+\sigma^{+}_{2}\sigma^{-}_{2})\sum_{m\geq 1}[\Theta(m\tau-{1\over N}\tau)-\Theta(m\tau)]$ and $\Theta(x)$ is the step function. Under the Hamiltonian $H_{all}=H_{s}+H_{e}+H_{i}+H_{phase}$, we can write down the state during the time of $[(m-{1\over N})\tau,m\tau]$: $\begin{split}\|\psi(t)\rangle=&e^{-i{N\pi\over\tau}(t-(m-{1\over N})\tau)}(r(t)_{1}\|1\rangle_{1}\|0\rangle_{2}\\\ &+r(t)_{2}\|0\rangle_{1}\|1\rangle_{2})\otimes\|0\rangle_{e}\\\ &+\sum_{k}\|0\rangle_{1}\|0\rangle_{2}\otimes c(t)_{k}\|1_{k}\rangle_{e}\end{split}$ (27) In the same way as shown above, we also can get a integral equation describing the coefficient $\dot{r}_{1}=-\int_{0}^{t}d\tau f^{\prime}(t-\tau)[\alpha^{2}_{1}r_{1}(\tau)+\alpha_{1}\alpha_{2}r_{2}(\tau)]$ and the new correlation function of the duration time should be $f^{\prime}(t)=W^{2}e^{(-\lambda+i{N\pi\over\tau})t}$. Using the boundary condition of $t=(m-{1\over N})\tau$ and $t=m\tau$, we eliminate the coefficient in the operator’s duration time and get coupled relationships between $A_{m}$, $B_{m}$ and $A_{m+1}$, $B_{m+1}$(Eqn.22). $\begin{split}&e^{-{\lambda\over 2}(1-{1\over N})\tau}[A_{m}cosh({\Omega\over 2}(1-{1\over N})\tau)+B_{m}sinh({\Omega\over 2}(1-{1\over N})\tau)]\\\ &=(1+\lambda{e^{\lambda\tau\over N}+1\over 4i\gamma-2\lambda})A_{m+1}-{e^{\lambda\tau\over N}+1\over 4i\gamma-2\lambda}\Omega B_{m+1}\\\ &{\Omega\over\lambda}e^{-{\lambda\over 2}(1-{1\over N})\tau}[A_{m}sinh({\Omega\over 2}(1-{1\over N})\tau)+B_{m}cosh({\Omega\over 2}(1-{1\over N})\tau)\\\ =&(1+e^{\lambda\tau\over N}+{\lambda\over 4i\gamma-2\lambda})A_{m+1}-({\Omega\over\lambda}e^{\lambda\tau\over N}+{\Omega\over 4i\gamma-2\lambda})B_{m+1}\end{split}$ (28) where $\gamma={N\pi\over 2\tau}$. The time evolution of the system is given by $\|\psi(t)\rangle=(\beta_{1}\|\mu\rangle+\beta_{2}\xi(t)\|\nu\rangle)\otimes\|0\rangle_{e}+\sum_{k}\|0\rangle_{1}\|0\rangle_{2}\otimes c(t)_{k}\|1_{k}\rangle_{e}$ and $\xi(t)$ is defined by Eqn. 22 with the new relation of $A_{m}$ and $B_{m}$ above. Also, the fidelity of the state under decoherence with the original state is $\displaystyle\|\psi\rangle=(\beta_{1}\|\mu\rangle+\beta_{2}\|\nu\rangle)$ (29) $\displaystyle F(t)=\|\beta_{1}\|^{2}+\|\beta_{2}\|^{2}\xi(t)$ (30) ### III.3 Numerical Calculation of Dynamical Decoupling and Free-evolution Figure 2: (Color online) The initial state is the superradiant state. Line 2 is the fidelity under sequential double-$\pi$-phase method (Dynamical Decoupling). Line 3 is the fidelity under QZE. Line 1 is the fidelity of free- evolution under the interaction between system and environment. From the result, we can easily find the sequential-pulse method is much better than the QZE. Figure 3: (Color online) The initial state is the superradiant state. Line 2 is the method with the instantaneous pulse. Line 4 is for $N=20$ and line 3 is for $N=10$. Line 1 is for the free-evolution. In case 1, we set $\tau=0.1$, $\lambda=2$ and $\Omega=1$ and we compare the fidelity of the free-evolution, sequential double-$\pi$-phase operator and random one. The result is shown in Fig[2]. In case 2, we set $\tau=0.2$, $\lambda=2$, $\Omega=1$, $N_{1}=10$ and $N_{2}=20$ to compare how the duration time of pulses affect the fidelity. The result is shown in Fig[3]. ## IV Concluding remark We present a strategy to protect the odd-parity states of two qubits under 0 temperature environment by frequently applying the $\pi-$pulses. Comparison between this method and the method based on frequent measurement is done, it seems that the frequent-$\pi-$pulses method is more effective in protecting the states. Our result clearly shows that the quantum Zeno effect and the dynamical decoupling can have rather different results when the operation frequency is finite, though the two methods give essentially the same results in the limit of infinite operation frequent as shown in Ref. Facchi1 ; Facchi2 ; Busch . Acknowledgments XBW is supported by the National Natural Science Foundation of China under Grant No. 60725416, the National Fundamental Research Programs of China Grant No. 2007CB807900 and 2007CB807901, and China Hi-Tech Program Grant No. 2006AA01Z420. LCK acknowledges the financial support by the National Research Foundation & Ministry of Education. ## References * (1) P. $\check{S}$telmachoni$\check{c}$ and V. Buz$\check{z}$ek, Phys. Rev. A 64, 062106 (2001) * (2) A.G. Kofman and G. Kurizki, Phys. Rev. Lett. 93, 130406 (2004) * (3) A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935) * (4) C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres and W.K. Wootters, Phys. 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Rev. 170, 379 (1968)	arxiv-papers	2010-11-15T17:22:52	2024-09-04T02:49:14.811686	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jia-Zhong Hu, Xiang-Bin Wang, and LC Kwek", "submitter": "Xiang-Bin Wang", "url": "https://arxiv.org/abs/1011.3460" }
1011.3509	# Single Crystal Study of Competing Rhombohedral and Monoclinic Order in Lead Zirconate Titanate D. Phelan1, X. Long2, Y. Xie2, Z.-G. Ye2, A. M. Glazer3, H. Yokota4, P. A. Thomas5, P. M. Gehring1 1NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA 2Department of Chemistry, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada 3Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom 4Department of Bioengineering, University of Tokyo, Tokyo 113-8656, Japan 5Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom ###### Abstract Neutron diffraction data obtained on single crystals of PbZr1-xTixO3 with x = 0.325 and x = 0.460, which lie on the pseudorhombohedral side of the morphotropic phase boundary, suggest a coexistence of rhombohedral (R3m/R3c) and monoclinic (Cm) domains and that monoclinic order is enhanced by Ti substitution. A monoclinic phase with a doubled unit cell (Cc) is ruled out as the ground state. A hallmark of highly piezoelectric perovskite materials, such as PbZr1-xTixO3 (PZT), is a morphotropic phase boundary (MPB) that segregates two compositional regimes into different ferroelectric states and pinpoints the range of substitutions for which the piezoelectric response is optimal. For PZT (and other Pb-based materials), the MPB ostensibly divides regions of Ti- poor rhombohedral (R3m) and Ti-rich tetragonal (P4mm) order. The discovery of an intermediate monoclinic phase within a narrow compositional range near the MPB [1,2] provided a mechanism whereby PZT could transform from R3m to P4mm via a common subgroup, Cm. This led to the idea that enhanced piezoelectricity at the MPB arises from the freedom of the Pb atom to move within the mirror plane of the monoclinic phase, thus allowing the electric polarization to rotate [3,4]. Alternatively, the adaptive-phases model, proposes that the lower symmetry allows for complex domain structures in which the net polarization can be made to change direction via domain populations [5]. These mechanisms remain controversial as how the phase symmetry is transformed across the phase diagram remains unclarified. The most recent measurements identify no clear phase boundary between rhombohedral and monoclinic phases [6]. Instead the anisotropic Pb displacement ellipsoid is highly flattened perpendicular to the polar displacement direction, which led to the suggestion that some kind of disorder [7,8] or local monoclinic displacements [9] are present in the rhombohedral phases. From this followed the idea that the local symmetries are monoclinic, but that the coherence lengths are limited to just a few unit cells so that diffraction methods yield only rhombohedral symmetries for Ti-poor compositions [10]. While this model is able to explain the lack of an R-M phase boundary, two different explanations were recently put forth. The first, based on refinements of synchrotron data, replaces the rhombohedral phases with a high-temperature monoclinic Cm and a low-temperature monoclinic Cc phase (the Cc phase superimposes antiferrodistortive displacements with the ferroelectric displacements of Cm, causing the unit cell to double as depicted in Ref. [11]) [11,12]. The second, based on a separate set of neutron diffraction refinements favors a coexistence of rhombohedral R3c and monoclinic Cm (but not Cc) phases at room temperature in the Ti-poor region [6]. The present results resolve this issue conclusively. A hindrance to precise symmetry determination of PZT has been the difficulty to grow single crystals. Therefore, structural studies near the MPB have been limited to powders and ceramics. A major problem with Rietveld refinement of highly pseudosymmetric polycrystalline materials is that many models refine to similar agreement factors, which makes it nearly impossible to choose between them. Very recently, single crystals were grown at Simon Fraser University using a top-seeded solution growth technique. The crystals measured here have compositions of x = 0.325 (dimensions 3.3 mm $\times$ 2.1 mm $\times$ 1.1 mm) and x = 0.460 (3.0 mm $\times$ 2.7 mm $\times$ 0.6 mm). The measurements were performed at the NIST Center for Neutron Research (NCNR) on thermal (BT-9) and cold (SPINS) triple-axis spectrometers. Various instrumental configurations were employed and are given in the figure captions. The samples were measured under vacuum in a closed-cycle helium refrigerator. Miller indices are expressed with respect to the aristotype cubic cell with a lattice parameter a $\approx$ 4.1 Å. Structural transitions in each crystal were characterized by measurements of the 200 Bragg reflection as a function of temperature (see Fig. 1). For x = 0.325, two phase transitions were identified corresponding to paraelectric- ferroelectric and ferroelectric-ferroelectric transitions. One occurs near 370 K and is marked by a change in the slope of the temperature dependence of the rocking curve width, a strong shift in the peak position of the rocking scan, and a change from positive to negative volume thermal expansion. A second near 590 K is evident from the large change in intensity resulting from a release of secondary extinction, a leveling out of the width of the rocking curve, and a return to positive volume thermal expansion. Similar measurements for x = 0.460 revealed more complicated behavior. Possible phase transitions are seen at 220, 410, 480, 540, and 650 K. Between 480 and 540 K the volume thermal expansion is $\approx$ 9.1 $\times$ 10-5 1/K, or roughly 1 order of magnitude larger than normally observed in oxides; above 540 K a steep negative volume thermal expansion ($\approx$ -4.9 $\times$ 10-5 1/K) is observed. A huge release of extinction occurs upon cooling below the cubic phase. The large number of anomalies is consistent with the fact that the crystal has a composition very close to the MPB. A similarly complex series of phase transitions has been suggested [13] based on powder neutron data [14]. One approach to identify the structural symmetry is to look for superlattice peaks that appear when the primitive unit cell is doubled. The R3c phase generates superlattice reflections that result from correlated rotations of the oxygen octahedra, but only at the $R_{2}$ points $\frac{h}{2}\frac{\overline{k}}{2}\frac{k}{2}$, where h and k are odd and h $\neq$ k (for twinned crystals, $\frac{h}{2}\frac{\overline{k}}{2}\frac{k}{2}$ and $\frac{h}{2}\frac{k}{2}\frac{k}{2}$ are not distinct). This is the only type of superlattice peak that has been observed in x-ray and neutron powder diffraction measurements [6,15,16]. Other superlattice reflections have been observed using electron diffraction methods [15,17-20]; however, these have been argued to result from surface effects or local inhomogeneities, with only $R_{2}$ attributed to the bulk [15]. On the other hand, a phase with Cc symmetry [11,12] would generate weak superlattice reflections at the $R_{1}$ points $\frac{h}{2}\frac{h}{2}\frac{h}{2}$, where h is odd. We looked for superlattice reflections in the x = 0.325 crystal in the (hk0) and (hkk) scattering planes at 35 K (see Fig. 2). No peaks were found at the $X$ points $\frac{1}{2}$00 and $\frac{3}{2}$00, or at the $M$ points $\frac{1}{2}\frac{1}{2}$0 and $\frac{3}{2}\frac{1}{2}$0, however, several $R_{2}$ peaks were observed. The order parameter measurement shown in Fig. 2(f) indicates that the $R_{2}$ peaks vanish at the low-temperature phase transition identified in Fig. 1(a) ($\approx$ 370 K). Weaker $R_{2}$ peaks were observed in the x = 0.460 crystal that vanish above the lowest transition temperature identified in Fig. 1(b) ($\approx$ 220 K). We also observed superlattice peaks at the $R_{1}$ points $\frac{1}{2}\frac{1}{2}\frac{1}{2}$ and $\frac{3}{2}\frac{3}{2}\frac{3}{2}$, which showed temperature dependences similar to those of the $R_{2}$ points. Because these peaks could be caused by double scattering as for SrTiO3 [21], we performed a standard test in which the peak intensity was measured as a function of the neutron wavelength. Figure 2(e) shows measurements performed with cold neutrons at two different wavelengths. A very strong $R_{1}$ peak was observed at $\frac{1}{2}\frac{1}{2}\frac{1}{2}$ when $\lambda$ = 4.045 Å; however, when the wavelength was increased to 5.222 Å, so that all $R_{2}$ reflections fell outside the Ewald sphere, this $R_{1}$ peak completely disappeared. Hence, we can definitively conclude that the observed $R_{1}$ peak results from double scattering. We have determined with 99% statistical confidence that the maximum ratio of the peak intensity $\frac{1}{2}\frac{1}{2}\frac{1}{2}$ compared with that of 100 is 0.002% (x = 0.325) and 0.003% (x = 0.460). These are extremely severe limits: for the refinement listed in Ref. [22], which is for a sample with x = 0.48 at 10 K, the calculated intensity ratio is 0.13% (nearly 40 times larger than our limit for x = 0.46), while refinement for a sample with x = 0:30 at 300 K [23] gives a ratio of 3.8% (more than 1000 times larger). Therefore, our superlattice survey is consistent with an R3c ground state and effectively rules out the presence of any phase, such as Cc or even triclinic symmetry, that would generate $R_{1}$ superlattice peaks for either sample. Symmetries were further studied by high resolution radial measurements of several Bragg reflections, which is possible because the crystals are multidomain. The 200 reflection can be used to distinguish between Cm and R3m/R3c phases because it splits into a doublet under Cm but remains a singlet for R3m/R3c. We achieved extraordinary wave-vector resolution for this peak by employing a perfect single crystal of Ge as analyzer and exploiting the fact that the Ge 220 reflection d spacing almost exactly matches that of the PZT 200 Bragg peak. A peculiarity of this special setup is that the radial linewidth becomes coupled to the sample mosaicity [24], particularly in the regime where the mosaic spread is less than 20’. As shown in Fig. 3(a), a narrow radial linewidth was observed at 200 in the cubic phase (640 K) for x = 0.325 that is nonetheless significantly broader than that measured on a single crystal of SrTiO3. The extra linewidth is well accounted for by the small, but non-negligible mosaic width of the x = 0.325 PZT crystal (11’) that was measured under the same conditions in a rocking scan, and which is taken into account in the resolution calculation. From Fig. 1(a) we know that the mosaic width increases as the temperature is lowered; thus the 200 radial linewidth will necessarily broaden as the temperature decreases. Indeed, the 200 radial linewidth at 450K is 12% broader than that at 640 K, but this agrees almost perfectly with the resolution calculation for the broadened mosaic width; hence the radial linewidth of 200 at 450 K is consistent with a resolution- limited peak as would be expected for R3m. However, at 40 K the peak has broadened by an additional 10% from its value at 450 K, while our resolution calculation predicts a change of only 2%. Given the larger intrinsic mosaic width of the x = 0.460 its 200 reflection was measured using very tight beam collimations and a conventional PG analyzer, for which the resolution is essentially decoupled from the mosaic width. These data, shown in Fig. 3(b), reveal a broadening of 200 as the temperature is lowered that is even larger than that for x = 0.325. The broadened 200 Bragg peaks could result from microscopic strain, finite domain sizes, or a very small Cm distortion [6]. Microscopic strain and unresolved monoclinic distortions have the same Q dependence and are impossible to distinguish in our measurements. However, the intrinsically broad 200 linewidths are consistent with recent neutron powder diffraction studies that indicate that the best refinements are obtained with a model of coexisting R3c and Cm phases [6]. It is difficult to extract quantitative information about the Cm phase because the 200 Bragg peak is only slightly broader than the instrumental resolution and can be fit in many different ways. However, we can extract important information about the R3c phase because it generates a superlattice peak that depends only on the rhombohedral structural correlations. To this end we measured radial scans through the $R_{2}$ superlattice peak for both compositions with tight beam collimations at low temperature, as shown in Fig. 4. The two crystals exhibit markedly different linewidths: the peak for x = 0.325 is narrow and close to the resolution limit whereas that for x = 0.460 is severely broadened. A superposition of two resolution-limited Gaussians cannot reproduce this broadening. Instead, the broadening implies the presence of either finite regions of correlated octahedral rotations or microstrain. Generally, there is no microstrain associated with oxygen octahedral rotations. Thus, the most likely explanation for the change in linewidth is that the coherence length of the rotations is diminished. This is an intriguing result, especially when considered in conjunction with the broadened 200 Bragg peaks, because it suggests that the structural correlations in PZT gradually evolve with Ti content from being predominantly longrange rhombohedral for compositions far from the MPB to being predominantly long-range monoclinic for compositions close to the MPB. The radial line shapes of 200, the appearance of superlattice peaks at $R_{2}$ points at low temperature, and the absence of superlattice peaks at R1 points are consistent with a purely R3m high-temperature ferroelectric phase for x = 0.325 and coexisting (R3m) and (Cm) phases for x = 0.460, and coexisting R3c and Cm ground-state ferroelectric phases for both compositions. In addition, the marked broadening of the superlattice peak at $\frac{3}{2}\frac{1}{2}\frac{1}{2}$, which is generated by R3c symmetry, suggests a scenario of competing rhombohedral and monoclinic order in highly piezoelectric compositions in which the rhombohedral correlations become increasingly short-range upon approaching the MPB. This picture lends support to the recent theoretical ideas that the tendency to form macroscopic monoclinic phases facilitates the mechanism of polarization rotation [25], achieved either by having the freedom to change the direction of Pb displacements within monoclinic unit cells or through the change in the population of twinned monoclinic components under an applied stress or electric field. The elucidation of the correct symmetries of PZT close to the MPB is critical to the theory of the origin of the high piezoelectricity in this and other materials, since it affects the discussion of the numbers and types of microdomains expected as well as the microscopic mechanisms for polarization rotation. A. M. G. and P. A. T. are grateful for funding from the Engineering and Physical Sciences Research Council (EPSRC) and from the National Science Foundation (NSF). X. L., Y. X, and Z.-G.Y. acknowledge support from the U.S. Office of Naval Research (Grant No. N00014-06-1- 0166). This work utilized facilities supported in part by the NSF under Agreement No. DMR-0944772. ## References * Noheda et al. (2000) B. Noheda, D. E. Cox, G. Shirane, R. Guo, B. Jones, and L. E. Cross, Phys. Rev. B 63, 014103 (2000). * Noheda et al. (1999) B. Noheda, D. E. Cox, G. Shirane, J. A. Gonzalo, L. E. Cross, and S. E. Park, Appl. Phys. Lett. 74, 2059 (1999). * Bellaiche et al. (2000) L. Bellaiche, A. 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Pandey, and B. J. Kennedy, Phys. Rev. B 65, 212101 (2002). * Frantii (2008) J. Frantii, J. Phys. Chem. B 112, 6521 (2008). * Frantii et al. (2003) J. Frantii, S. Eriksson, S. Hull, V. Lantto, H. Rundlof, and M. Kakihana, J. Phys. Condens. Matter 15, 6031 (2003). * Ricote et al. (1998) J. Ricote, D. L. Corker, R. W. Whatmore, S. A. Impey, A. M. Glazer, J. Dec, and K. Roleder, J. Phys. Condens. Matter 10, 1767 (1998). * Pandey and Ragini (2003) D. Pandey and Ragini, Z. Kristallogr. 218, 1 (2003). * Viehland (1995) D. Viehland, Phys. Rev. B 52, 778 (1995). * Dai et al. (1995) X. Dai, Z. Xu, and D. Viehland, J. Am. Ceram. Soc. 78, 2815 (1995). * Noheda et al. (2002) B. Noheda, L. Wu, and Y. Zhu, Phys. Rev. B 66, 060103 (2002). * Ragini et al. (2001) Ragini, S. K. Mishra, D. Pandey, H. Lemmens, and G. Van Tendeloo, Phys. Rev. B 64, 054101 (2001). * Shirane et al. (2002) G. Shirane, S. M. Shapiro, and J. M. Tranquada, _Neutron scattering with a triple-axis spectrometer_ (Cambridge University Press, Cambridge, England, 2002). * Ranjan et al. (2005) R. Ranjan, A. K. Singh, Ragini, and D. Pandey, Phys. Rev. B 71, 092101 (2005). * (23) D. Pandey, Private Communication. * Xu et al. (2004) G. Xu, P. M. Gehring, V. J. Ghosh, and G. Shirane, Acta Crystallogr. Sect. A 60, 598 (2004). * Bell (2006) A. J. Bell, J. Mater. Sci. 41, 13 (2006). Figure Captions: Fig. 1: Temperature dependence of the 200 Bragg reflection for (a) x = 0.325 and (b) x = 0.460. Shown are the integrated intensity, full-width-at-half-maximum (FWHM), and peak position ($\omega_{s}$) of Gaussian fits to rocking (transverse) scans through the Bragg peak. Also shown is the average unit cell volume (aristotype) determined from Gaussian fits to radial scans through the Bragg peak. Apparent phase transition temperatures are marked by dashed lines. Error bars correspond to uncertainties of 1$\sigma$ in the fitted parameters. Measurements were performed on BT-9 with horizontal beam collimations of 40’-47’-40’-80’ and a single PG filter. Fig. 2: (a)-(c) Radial scans through the $X$, $M_{1}$, and $M_{2}$ points reveal no superlattice reflections for x = 0.325 at 35 K. (d) Scans through one $R_{2}$ point at various temperatures for x = 0.325. (e) Scans performed using two wavelengths through an $R_{1}$ point for x = 0.325. (f) The order parameters of the superlattice phases are given by the peak intensity of the $\frac{5}{2}\frac{1}{2}\frac{1}{2}$ reflection. Error bars represent an uncertainty of 1$\sigma$ in the measured intensities. Scans in (a)-(d) and (f) were made using collimations of 40’-47’-40’-80’, 3 PG filters, and $\lambda$ = 2.359 Å on BT-9. The scans in (e) were made on SPINS with collimations of 80’-80’ and two Be filters. Fig. 3: (a) Radial scans through the 200 Bragg reflection of the x = 0.325 crystal at selected temperatures. The measurements were performed on BT-9 with $\lambda$ = 2.359 Å, collimations of 15’-47’-10’-20’, a single PG filter, and the Ge 220 analyzer. Also shown is a measurement of single crystal SrTiO3 (data from Ref. [24] represented by green triangles) obtained with collimations of 10’-40’-20’-40’. (b) Radial scans through 200 for the x = 0.460 crystal using the PG 002 analyzer, 3 PG filters, and collimations of 15’-10’-10’-10’. Fig. 4: High resolution radial scans performed through the $R_{2}$ point on SPINS using collimations of G (39’)- 40’-10’-20’, two Be filters, and with $\lambda$ = 4.045 Å.	arxiv-papers	2010-11-15T20:58:22	2024-09-04T02:49:14.820014	{ "license": "Public Domain", "authors": "D. Phelan, X. Long, Y. Xie, Z.-G. Ye, A. M. Glazer, H. Yokota, P. A.\n Thomas, P. M. Gehring", "submitter": "Daniel Phelan", "url": "https://arxiv.org/abs/1011.3509" }
1011.3534	# Fast Multiplication of Matrices with Decay Matt Challacombe and Nicolas Bock mchalla,nbock@lanl.gov Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 freeon.org ###### Abstract A fast algorithm for the approximate multiplication of matrices with decay is introduced; the Sparse Approximate Matrix Multiply (SpAMM) reduces complexity in the product space, a different approach from current methods that economize within the matrix space through truncation or rank reduction. Matrix truncation (element dropping) is compared to SpAMM for quantum chemical matrices with approximate exponential and algebraic decay. For matched errors in the electronic total energy, SpAMM is found to require fewer to far fewer floating point operations relative to dropping. The challenges and opportunities afforded by this new approach are discussed, including the potential for high performance implementations. Fast Matrix Multiplication, Electronic Structure, Matrix Function, Matrix Decay, Spectral Projection, Generalized N-Body Problem, Sparse Matrix-Matrix Multiply, Linear Scaling Quantum Chemistry ††preprint: LA-UR 10-07458 ## I Introduction For large dense linear algebra problems, the computational advantage offered by fast matrix-matrix multiplication can be substantial, even with seemingly small gains in asymptotic complexity. Relative to conventional multiplication which is $\mathcal{O}\left(n^{3}\right)$, Strassen’s algorithm achieves $\mathcal{O}\left(n^{2.8}\right)$, while the Coppersmith and Winograd method is $\mathcal{O}\left(n^{2.38}\right)$. For these dense methods, balancing the trade off between cost, complexity and error is an active area of research Demmel and Higham (1992); Demmel et al. (2007); Yuster and Zwick (2005). On the other hand, large sparse problems are typically handled with conventional sparse matrix techniques, with only small concessions between multiplication algorithms. Intermediate to these regimes, a wide class of problems exist that involve matrices with decay111A matrix $A$ is said to decay when its matrix elements decrease exponentially, as $\|a_{i,j}\|<c\lambda^{\|i-j\|}$ , or algebraically as $\|a_{i,j}\|<\frac{c}{\|i-j\|^{\lambda}+1}$ with indicial separation $\|i-j\|$. In non-synthetic cases, the separation $\|i-j\|$ typically corresponds to an underlying physical distance $\|\vec{r}_{i}-\vec{r}_{j}\|$, _e.g._ of basis functions, finite elements, _etc_. See Figure 3 as well as the excellent work by Benzi and co-authors on this topic in References Benzi and Razouk (2007); Benzi et al. (2010). where sparsity exists only asymptotically under an approximate linear algebra, historically involving matrix economization through element dropping or rank reduction. Often, problems with decay occur in the construction of matrix functions, notably the matrix inverse Benzi and Tuma (2000), the matrix exponential Iserles (1999), and in the case of electronic structure theory, the Heaviside step function Challacombe (1999, 2000); Benzi and Razouk (2007); Benzi et al. (2010). The use of an approximate matrix algebra is also an active area of interest in the solution of large eigenproblems Simoncini and Elden (2002); Simoncini and Szyld (2003, 2005); Challacombe (2010). Many approaches to a sparse approximate linear algebra exist for matrices with decay Galli (1996); Goedecker (1999); Goedecker and Scuseria (2003); Li et al. (2005), largely predicated upon the truncation of matrix elements, with the recent work of Benzi providing the most detailed analysis so far Benzi and Razouk (2007); Benzi et al. (2010). In this contribution, we develop sparse matrix multiplication as a generalized $N$-body problem Gray and Moore (2001), and introduce a fast algorithm based on hierarchical truncation in the three- dimensional space $i,\,j,\,k\in[1,n]$ of the product $C_{ij}=\sum_{k}A_{ik}B_{kj}$, where $A$ and $B$ decay exponentially or algebraically fast enough222Algebraic decay sufficient to achieve a fast $\mathcal{O}\left(n\,\lg\,n\right)$ or better complexity is an open question similar to that of conditional convergence in the shape dependent summation of dipole and quadrupole components of the Lorentz field. . Viewing the product from a length scale perspective1, if matrix elements decay as $\mathcal{O}(1/r^{\lambda})$, then the bulk of the product interactions will decay as $\mathcal{O}(1/r^{2\lambda})$. For small $\lambda$, the difference between truncation in the matrix space and the product space may be significant. ## II A Sparse Approximate Matrix Multiply The quadtree matrix representation, $A^{k}=\left(\begin{array}[]{cc}A_{11}^{k+1}&A_{12}^{k+1}\\\ A_{21}^{k+1}&A_{22}^{k+1}\end{array}\right),\>k=0,\,\ldots,\,k_{{\rm\textrm{max}}},$ (1) is the basis for recursive matrix-matrix multiplication, $C^{k}=A^{k}\cdot B^{k}.$ For conventional recursive multiplication, the operator “$\cdot$” is just the row-column product, while in fast multiplication, it represents an economized sequence of operations with reduced complexity and a more complicated error accumulation. In Reference Bini and Lotti (1980), Bini and Lotti carried out a detailed error analysis for recursive matrix multiplication schemes, and derived component-wise bounds of the form $\|\tilde{c}_{ij}-c_{ij}\|<a\,b\,\epsilon\,n\,\lg_{2}\,n$ (2) where $\tilde{c_{ij}}$ is a matrix element computed to within precision $\epsilon$, $c_{ij}$ is its exact counterpart, $a=\max_{ij}\|a_{ij}\|$ and $b=\max_{ij}\|b_{ij}\|$. While providing a sharp bound, the max norm does not immediately lend itself to the recursive separation of interaction magnitudes. Consider instead the framework provided by an arbitrary sub-multiplicative matrix norm $\\|\cdot\\|$: $\displaystyle\\|C^{k}\\|$ $\displaystyle\leq$ $\displaystyle\\|A^{k}\\|\\|B^{k}\\|$ $\displaystyle\leq$ $\displaystyle\\|A_{11}^{k+1}\\|\\|B_{11}^{k+1}\\|+\\|A_{12}^{k+1}\\|\\|B_{21}^{k+1}\\|$ $\displaystyle+$ $\displaystyle\\|A_{11}^{k+1}\\|\\|B_{12}^{k+1}\\|+\\|A_{12}^{k+1}\\|\\|A_{22}^{k+1}\\|$ $\displaystyle+$ $\displaystyle\\|A_{21}^{k+1}\\|\\|B_{11}^{k+1}\\|+\\|A_{22}^{k+1}\\|\\|B_{21}^{k+1}\\|$ $\displaystyle+$ $\displaystyle\\|A_{21}^{k+1}\\|\\|B_{12}^{k+1}\\|+\\|A_{22}^{k+1}\\|\\|A_{22}^{k+1}\\|\,.$ This structure suggests an algorithm we call the Sparse Approximate Matrix Multiply (SpAMM), which recursively tests each of the 8 contributions in Equation (II) for significance in accordance with a given numerical threshold $\tau$: ${\tt SpAMM}(A^{k},B^{k})=\begin{cases}\qquad\qquad\qquad\qquad\qquad\qquad\qquad 0&{\tt if}\quad\\|A^{k}\\|\\|B^{k}\\|<\tau\\\\[8.5359pt] \qquad\qquad\qquad\qquad\qquad\qquad\;\;A^{k}\cdot B^{k}&{\tt elseif}\quad k=k_{\textrm{max}}\\\\[8.5359pt] \left(\begin{array}[]{rr}{\tt SpAMM}(A_{11}^{k+1},B_{11}^{k+1})&{\tt SpAMM}(A_{11}^{k+1},B_{12}^{k+1})\\\ +{\tt SpAMM}(A_{12}^{k+1},B_{21}^{k+1})&+{\tt SpAMM}(A_{12}^{k+1},B_{22}^{k+1})\\\\[8.5359pt] {\tt SpAMM}(A_{21}^{k+1},B_{11}^{k+1})&{\tt SpAMM}(A_{21}^{k+1},B_{12}^{k+1})\\\ +{\tt SpAMM}(A_{22}^{k+1},B_{21}^{k+1})&+{\tt SpAMM}(A_{22}^{k+1},B_{22}^{k+1})\end{array}\right)&{\tt else}\end{cases}$ (4) The truncated product space accessed by SpAMM is shown in Figure 1 for matrices with exponential and algebraic decay. Figure 1: Hierarchical truncation of the product space ($i,\,j,\,k$) using $\tau=10^{-8}$ for synthetic matrices of dimension $n=512$, with $A_{ij}=\exp\left(-\left\|i-j\right\|\right)$ and $B_{ij}=\exp\left(-2\left\|i-j\right\|\right)$ in (A), $A_{ij}=B_{ij}=\begin{cases}1/\|i-j\|^{3}&i\neq j\\\ 0&\textrm{else}\end{cases}$ in (B). Each box above the finest ($k=k_{max}$) scale represents truncation. At each tier in SpAMM, the local truncation error is bounded by $\\|\widetilde{C}^{k}-C^{k}\\|<\tau$, with an error accumulation structurally similar to rounding in conventional recursive multiplication, except that truncation is not guaranteed to occur at each tier in the recursion, and truncation does not retain even the approximate magnitude of the avoided sub- product. To see the difference between truncation and round-off, consider the generic, norm-wise bound for recursive multiplication employed by Demmel and co-workers in Reference Demmel et al. (2007): $\displaystyle\\|\widetilde{C}-C\\|$ $\displaystyle<\mu(n)$ $\displaystyle\epsilon\,\\|A\\|\\|B\\|\,+\mathcal{O}\left(\epsilon^{2}\right),$ (5) with $\mu(n)$ a low order polynomial $\sim n^{d}$ and $d\geq 1$. Unlike rounding which is an error of commission, SpAMM creates errors of omission. If $\tau<\\|A\\|\\|B\\|$, then no work is performed and we obviously have$\\|\widetilde{C}-C\\|<\tau$, which is different than $\\|\widetilde{C}-C\\|<\epsilon\\|A\\|\\|B\\|$ corresponding to the case of comparable round-off and truncation parameters $\epsilon\sim\tau$. One could certainly bound SpAMM rigorously by decreasing $\tau$ with increasing depth, $\tau^{k+1}=\tau^{k}/8$, but that would be overly pessimistic, not taking into account signed error accumulation or the localization and attenuation of errors due to decay. SpAMM is similar to the $\mathcal{H}$-matrix algebra of Hackbusch and co- workers, where off-diagonal sub-matrices are treated as reduced rank factorizations (truncated SVD), typically structured and grouped to reflect properties of the underlying operators Grasedyck and Hackbusch (2003). For problems with rapid decay, truncated SVD may behave in a similar way to simple dropping schemes. SpAMM is different than the $\mathcal{H}$-matrix algebra as it achieves separation uniquely in the product space and does not rely on a reduced complexity representation of matrices. For very slow decay, the $\mathcal{H}$-matrix algebra may certainly offer a path for intractable problems for which SpAMM is ineffective. Figure 2: Space filling curves map atoms in Cartesian space (A) onto an index that is locality preserving (B), leading to clustering of matrix elements with respect to indicial separation (C) when interactions are short ranged. The octree generated by SpAMM in the three-dimensional product space, shown in Figure 1, is equivalent to a second space filling curve (spatial hashing) Warren and Salmon (1992, 1995); Samet (2006) that orders both matrices (C) and the product space (D), features that can be exploited to achieve domain decomposition and load balance (colors in C and D). An understanding of the error accumulation in SpAMM must account for the decay properties of $A$ and $B$, which in non-synthetic cases are intimately related to the effects of ordering and structure of the underlying physical, chemical or engineering application. Also, ordering will determine the relative efficiencies of both matrix truncation and SpAMM, particularly under blocking. The ordering used here is based on the Space Filling Curve (SFC) method described in Challacombe (2000) (Hilbert atom ordering), and shown in Figure 2, involving also a second tier of ordering in the product space. The first ordering maps atoms that are close in Cartesian-space to entries close in the index space of the matrix. The second ordering is a natural consequence of the SpAMM multiply, which recursively maps out a multi-level octree (see also Figure 1) with cuboid coordinates that are equivalent to a spatial-hash; sorting the hash produces a three-dimensional SFC in the product space Warren and Salmon (1992, 1995); Samet (2006). This two-tiered structure is novel, providing an encompassing scheme that parlays the locality of physical interactions into the data locality of matrices (element clustering), and into the irregular product space. This approach is applicable to other problems with underlying decay properties, in which the finite-elements, radial basis functions, _etc._ replace atoms, or graph theoretical methods (nested dissection, Cuthill-McKee, _etc_.) replace the first-tier SFC altogether. In either case, the correspondence between the recursive SpAMM product space and a three-dimensional SFC provides a tool to exploit both spatial and temporal locality in the distribution of work and data. ## III Results and Discussion Figure 3: Decay of normed density matrix atom-blocks $\left\\|P_{ab}\right\\|_{F}$ with Cartesian separation $\vec{\|R_{a}}-\vec{R}_{b}\|$ for the largest molecules in each sequence: a 450 atom water cluster (red), a 752 atom (4,3) nanotube (orange) and a 780 atom (3,3) nanotube (blue). In this initial contribution, we limit ourselves to exploring the numerical and computational behavior of SpAMM applied to problems in electronic structure theory, and compare its relative merits to the dropping of matrix elements. In ${\cal O}\left(n\right)$ electronic structure calculations Goedecker (1999); Goedecker and Scuseria (2003); Li et al. (2005), a primary source of error due to sparse matrix-multiplication develops from early steps in the iterative construction of the Heaviside matrix function $P=\theta[F-\mu I]$ (density matrix purification), which is a projector of the effective Hamiltonian (Fockian) $F$ Niklasson et al. (2003). Starting from the basis spanning $F$, purification drives eigenvalues to $0$s or $1$s, whereupon error accumulation due to an approximate linear algebra is quenched Niklasson et al. (2003). Under a given approximate linear algebra, the electronic energy $E_{{\rm el}}={\rm Tr}(P.F)$ is a global measure of accumulated error. In the following Section, we carry out purification on three molecular sequences of increasing size, water clusters, (4,3) nanotubes and (3,3) nanotubes. In each case, a Fockian $F$ was obtained from a fully converged Self-Consistent-Field (SCF) cycle and used as basis for the purification; our numerical experiments probe only errors within one density matrix solve (40-60 multiplies), and do not address error propagation throughout the SCF cycle. As the rate of decay slows towards SCF convergence, these calculations represent the instance of minimum decay, shown for each of the largest molecular species in Figure 3. Within each sequence, the same number of iterative steps (40-60 matrix multiplications) are taken using the TC2 purification algorithm Niklasson (2002) and either: (I) matrix element dropping and exact multiplication or (II) SpAMM. Figure 4: Average number of 4x4 matrix multiplies (MATMULs) for a sequence of (4,3) nanotubes at the RHF/STO-2G level of theory with dropping (red) and SpAMM (blue). Figure 5: Average number of 4x4 matrix multiplies (MATMULs) for a sequence of (3,3) nanotubes at the LDA/STO-2G/ level of theory with dropping (red) and SpAMM (blue). Figure 6: Average number of 4x4 matrix multiplies (MATMULs) for a sequence of water clusters at the RHF/6-31G** level of theory with dropping (red) and SpAMM (blue). In this study, we chose $k_{{\rm max}}$ to yield 4x4 blocks at the finest level of resolution, corresponding to the most aggressive use of single precision SSE on the x86 architecture. In all cases the matrix norm employed is the Frobenius norm $\left\\|\cdot\right\\|\equiv\left\\|\cdot\right\\|_{F}$. Thresholds, $\tau$, have been adjusted to roughly match relative errors in the electronic energy,$\Delta E_{\textrm{el}}=\|\tilde{E}_{{\rm el}}-E_{{\rm el}}\|/\|E_{{\rm el}}\|$ between the two schemes, (I) and (II), and the average number of 4x4 MATMULs per purification step are reported. Multiplications are only a proxy for CPU time, as neither case accounts for symbolic overheads associated with the multiply (CSR, recursive-tree _etc_.). In the case of dropping, SpAMM was also used in the matrix multiply but with zero threshold. After each multiplication in the dropping scheme, a filter was applied to the resultant, dropping blocks at the 4x4 level of resolution using the criteria $\left\\|P^{k}\right\\|_{F}<\tau$. Results for the three molecular sequences are shown in Figures 6, 4 and 5. For comparable values of $\Delta E_{\textrm{el}}$, SpAMM was found to employ from slightly fewer multiplies for the (4,3) nanotube, to dramatically less in the case of the water clusters. This result is somewhat surprising, since the spatial decay is slower in the case of the (4,3) nanotube than for the water clusters, as shown in Figure 3. While it seems reasonable to attribute this unexpected result to the effects of dimensionality, further study is required to be sure. Its also worth noting that the advantage of SpAMM relative to dropping is brought down with decreasing $\tau$; for $\tau=0$ they both revert to the same $\mathcal{O}(n^{3})$ complexity. For both systems with exponential decay, error appears tightly controlled albeit within an as yet unknown bound. Note however, that unlike matrix truncation which leads to a product error that is $\mathcal{O}\left(\tau^{2}\right)$, SpAMM leads to a truncation error that is $\mathcal{O}\left(\tau\right)$. Comparing the quasi one-dimensional metallic (3,3) nanotube with slow algebraic decay to the insulating (4,3) nanotube with exponential decay, SpAMM gains substantially over dropping in the case of slower decay. However, the ability of SpAMM to achieve a linear scaling complexity in the case of the metallic system remains in question, as in the case of the tightest threshold, the SpAMM errors do not appear to be well controlled, and the cost does not appear linear, at least not in this size regime. On the other hand, the SpAMM result does enjoy a significant reduction in cost relative to dropping, and the error increase is modest. ## IV Conclusion The Sparse Approximate Matrix Multiply (SpAMM) is a fast method for matrices with decay, which is different from element dropping or the $\mathcal{H}$-algebra in that it uniquely involves truncation of the product space rather than the matrix space. For matrices with exponential or fast algebraic decay, SpAMM can achieve stable error control comparable to element dropping, but with a greatly reduced number of floating point operations. The results presented here are preliminary, and have not yet explored the interesting problems of slow decay in the asymptotic limit, ordering, error bounds, the cost of recursion, high performance implementations or broader gauges of accuracy and efficiency, such as the use of SpAMM in the context of the Self-Consistent-Field cycle. Of particular concern is the relationship between complexity, matrix decay and error control. Based on our numerical tests, we postulate that the algorithm is at worst $\mathcal{O}(n\lg n)$ for matrices with sufficiently fast decay. In addition to similarities with the $\mathcal{H}$-algebra, SpAMM falls under the rubric of the generalized $N$-body problem Gray and Moore (2001). From this perspective, its worth noting also the connection between matrix-matrix multiplication as $N$-body problem and matrix-matrix multiplication as spatial join Amossen and Pagh (2009), as well as between $N$-body problems and data base theory in general Samet (2006). Next, we draw attention to the second tier of Space Filling Curve (SFC) shown in Figure 2, which provides a mechanism for domain decomposition and load balance that is proven for parallel irregular problems Warren and Salmon (1992, 1995); Aluru and Sevilgen (1997); Devine et al. (2005). Also, the improvement gained in Reference Buluc and Gilbert (2008) on going from a one- dimensional to a two-dimensional matrix partitioning scheme for the parallel SpMM suggests that partitioning the three-dimensional product space instead may provide an even higher degree of flexibility and granularity. The authors acknowledge support through Los Alamos LDRD award ER20110230 (computational co-design) as well as funds from the U.S. Department of Energy. Los Alamos National Laboratory is operated by the Los Alamos National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. DE-AC52-06NA25396. Special acknowledgments go to the International Ten Bar Café for tasty libations in a scientific and collegial atmosphere, and to Michele Benzi for valuable input. ## References * Demmel and Higham (1992) J. W. Demmel and N. J. Higham, ACM Trans. Math. Softw. 18, 274 (1992). * Demmel et al. (2007) J. Demmel, I. Dumitriu, O. 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Warren and J. K. Salmon, _A parallel, portable and versatile treecode_ (SIAM, Philadelphia, 1995), chap. 1. * Samet (2006) H. Samet, _Foundations of Multidimensional and Metric Data Structures_ (Morgan Kaufmann, 2006). * Niklasson et al. (2003) A. M. N. Niklasson, C. J. Tymczak, and M. Challacombe, J. Comp. Phys. 118, 8611 (2003). * Niklasson (2002) A. M. N. Niklasson, Physical Review B 66, 5 (2002). * Amossen and Pagh (2009) R. Amossen and R. Pagh, in _Proceedings of the 12th International Conference on Database Theory_ (ACM, 2009), pp. 121–126. * Aluru and Sevilgen (1997) S. Aluru and F. E. Sevilgen, in _Proceedings of the 4th IEEE Conference on High Performance Computing_ (1997), pp. 230–235. * Devine et al. (2005) K. D. Devine, E. G. Boman, R. T. Heaphy, B. A. Hendrickson, J. D. Teresco, J. Faik, J. E. Flaherty, and L. G. Gervasio, Applied Numerical Mathematics 52, 133 (2005), ADAPT ’03: Conference on Adaptive Methods for Partial Differential Equations and Large-Scale Computation. * Buluc and Gilbert (2008) A. Buluc and J. R. Gilbert, in _ICPP ’08: Proceedings of the 2008 37th International Conference on Parallel Processing_ (IEEE Computer Society, Washington, DC, USA, 2008), pp. 503–510.	arxiv-papers	2010-11-15T21:59:11	2024-09-04T02:49:14.826494	{ "license": "Public Domain", "authors": "Matt Challacombe and Nicolas Bock", "submitter": "Matt Challacombe", "url": "https://arxiv.org/abs/1011.3534" }
1011.3575	11institutetext: Purple Mountain Observatory, Chinese Academy of Sciences, 210008 Nanjing, China 11email: ywang@pmo.ac.cn 22institutetext: Max-Plank-Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany 33institutetext: Graduate University of the Chinese Academy of Sciences, 19A Yuquan Road, Shijingshan District, 100049 Beijing, China 44institutetext: Centro de Astrobiología (CSIC-INTA), Ctra. de Torrejón a Ajalvir km-4, E-28850, Torrejón de Ardoz, Madrid, Spain 55institutetext: Instituut voor Sterrenkunde, Katholieke Universiteit Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium 66institutetext: Laboratoire dAstrophysique de Marseille, UMR6110 CNRS, 38 rue F. Joliot-Curie, 13388 Marseille, France 77institutetext: National Astronomical Observatory of Japan, and GUAS, National Institutes of Natural Sciences, Japan # Different Evolutionary Stages in the Massive Star Forming Region S255 Complex Y. Wang 112233 H. Beuther 22 A. Bik 22 T. Vasyunina 22 Z. Jiang 11 E. Puga 4455 H. Linz 22 J. A. Rodón 66 Th. Henning 22 and M. Tamura 77 (Received 7 August 2010; accepted 11 November 2010) ###### Abstract Aims. Massive stars form in clusters, and they are often found in different evolutionary stages located close to each other. To understand evolutionary and environmental effects during the formation of high-mass stars, we observed three regions of massive star formation at different evolutionary stages that reside in the same natal molecular cloud. Methods. The three regions S255IR, S255N and S255S were observed at 1.3 mm with the Submillimeter Array (SMA) and follow-up short spacing information was obtained with the IRAM 30m telescope. Near infrared (NIR) $H+K$-band spectra and continuum observations were taken for S255IR with VLT-SINFONI to study the different stellar populations in this region. Results. The combination of millimeter (mm) and near infrared data allow us to characterize different stellar populations within the young forming cluster in detail. While we find multiple mm continuum sources toward all regions, their outflow, disk and chemical properties vary considerably. The most evolved source S255IR exhibits a collimated bipolar outflow visible in CO and H2 emission, the outflows from the youngest region S255S are still small and rather confined in the regions of the mm continuum peaks. Also the chemistry toward S255IR is most evolved exhibiting strong emission from complex molecules, while much fewer molecular lines are detected in S255N, and in S255S we detect only CO isotopologues and SO lines. Also, rotational structures are found toward S255N and S255IR. Furthermore, a comparison of the NIR SINFONI and mm data from S255IR clearly reveal two different (proto) stellar populations with an estimated age difference of approximately 1 Myr. Conclusions. A multi-wavelength spectroscopy and mapping study reveals different evolutionary phases of the star formation regions. We propose the triggered outside-in collapse star formation scenario for the bigger picture and the fragmentation scenario for S255IR. ###### Key Words.: stars: formation – ISM: jets and outflows – ISM: molecules – stars: early-type – Hertzsprung-Russell (HR) and C-M diagrams – stars: individual: S255IR, S255N, S255S ## 1 Introduction Massive stars ($M>8M_{\odot}$) are one of the paramount components in the evolution of the universe, yet their formation is significantly less well understood. than that of their low-mass counterparts. S255IR is a famous massive star formation complex at a distance of $1.59^{+0.07}_{-0.06}$ kpc (Rygl et al., 2010), embraced by the Sharpless regions S255 and S257 that are already evolved HII regions. The SCUBA 850 $\mu$m observation (Di Francesco et al., 2008; Klein et al., 2005) shows three main continuum sources: G192.60-MM2 (Minier et al., 2005) lies in the center region S255IR, and two additional mm continuum peaks G192.60-MM1 (Minier et al., 2005) and G192.60-MM3 (Minier et al., 2005) toward the northern region S255N and southern region S255S, respectively (Figure 1). The central region S255IR: The central IRAS source with a luminosity of $5\times 10^{4}$ L⊙ (Minier et al., 2005) harbors three UCHII regions region (Snell & Bally, 1986) that is associated with Class II CH3OH and H2O maser emission (Goddi et al., 2007), which indicate the presence of massive young stellar objects. The luminosity would be $2\times 10^{4}$ L⊙ if we apply the new distance of 1.59 kpc. The region hosts a cluster of low-mass sources that is surrounded by a shocked bubble of H2 gas (Ojha et al., 2006; Miralles et al., 1997). Tamura et al. (1991) resolved the central near infrared source into two compact sources, NIRS 1 and NIRS 3. Furthermore, Minier et al. (2007) reported HCO+ infall signatures toward this region. In the mid-infrared, Longmore et al. (2006) resolved a massive proto-binary which coincide with NIRS 1 and NIRS 3 (Figure 3). And NIRS 1 has been identified as a massive disk candidate by near-infrared polarization observations (Jiang et al., 2008, see also Simpson et al. (2009) for a different interpretation). The northern region S255N: This region with 105 L⊙ (Minier et al., 2005) also hosts a UCHII region G192.584-0.041 (Kurtz et al., 1994) as well as Class I CH3OH and H2O maser emission (Kurtz et al., 2004; Cyganowski et al., 2007), which indicate the presence of the massive young stellar objects. The luminosity would be $4\times 10^{4}$ L⊙ if we apply the new distance of 1.59 kpc. The recent Submillimeter Array observations toward that region at a spatial resolution of 3.6′′ resolved three mm sources and strong molecular line emission, and none of the sources is associated with NIR point source (Cyganowski et al., 2007). Outflow activity is evidenced by various tracers from shocked H2 emission via cm continuum and SiO emission (Miralles et al., 1997; Cyganowski et al., 2007), and global infall was reported by Minier et al. (2007). The southern region S255S: This sub-source is the least studied so far. It exhibits strong mm continuum emission (Figure 1) but no other sign of active star formation yet such as near infrared and mid infrared emissions. Therefore, it is proposed to be in a pre-stellar phase of the evolutionary sequence (Minier et al., 2007). While G192.60-MM2 in S255IR still shows signs of active star formation, it is associated with a NIR cluster and appears to be the most evolved one of the three region. G192.60-MM1 in S255N is of similar luminosity, and G192.60-MM3 is S255S is a High-Mass Starless Core candidate. Therefore, S255 complex is the ideal candidate source to simultaneously investigate several sites of massive star formation at different evolutionary stages within the same larger-scale environment. ## 2 Observations and Data reductions ### 2.1 Submillimeter Array observations The S255 complex was observed with three fields with the Submillimeter Array (SMA) on November 3rd 2008 in the compact configuration with seven antennas and on February 8th, February 17th 2009 in the extended configuration with eight antennas, and February 13th in the extended configuration with seven antennas. The phase centers of the fields, which are known as S255IR, S255N and S255S, were R.A. 06h12m54.019s Dec. $+17^{\circ}59^{\prime}23.10^{\prime\prime}$ (J2000.0), R.A. 06h12m53.669s Dec. $+18^{\circ}00^{\prime}26.90^{\prime\prime}$ (J2000.0) and R.A. 06h12m56.58s Dec. $+17^{\circ}58^{\prime}32.80^{\prime\prime}$ (J2000.0), respectively. The SMA has two spectral sidebands, both 2 GHz wide and separated by 10 GHz. The receivers were tuned to 230.538 GHz in the upper sideband ($v_{lsr}$ = 10 km s-1) with a maximum spectral resolution of 0.53 km s-1. The weather of February 8th and February 13th was mediocre with zenith opacities $\tau$(225 GHz) larger than 0.2 measured by the Caltech Submillimeter Observatory (CSO). But the weather of November 3rd 2008 and February 17th 2009 was good with zenith opacities $\tau$ (225 GHz) around 0.1. So we only used the data observed on these two days. For the compact configuration on November 3rd 2008, bandpass was derived from the quasar 3c454.3 observations. Phase and amplitude were calibrated with regularly interleaved observations of the quasar 0530+135 (11.4∘ from the source). The flux calibration was derived from Uranus observations, and the flux scale is estimated to be accurate within 20$\%$. For the extended configuration on February 17th 2009, bandpass was derived from the quasar 3c273 observations. Phase and amplitude were calibrated with regularly interleaved observations of the quasar 0530+135. Because of the lack of flux calibrator observations, the flux was estimated by the SMA calibrator database for the gain calibrator, to be accurate within 20$\%$. We merged the two configuration data set together, applied different robust parameters for the continuum and line data, and got the synthesized beam sizes between 1.4′′ $\times$1.1′′ (PA 86∘) and 1.9${}^{\prime\prime}\times 1.6^{\prime\prime}$ (PA 77∘), respectively. The 3$\sigma$ rms of 1.3 mm continuum image is $\sim$ 4.5 mJy and the 3$\sigma$ rms of the line data is 0.14 Jy/beam at 2 km s-1 spectral resolution. The flagging and calibration was done with the IDL superset MIR (Scoville et al., 1993) which was originally developed for the Owens Vally Radio Observatory and adapted for the SMA111The MIR cookbook by Chunhua Qi can be found at http://cfa-www.harvard.edu/$\sim$cqi/mircook.html.. The imaging and data analysis were conducted in MIRIAD (Sault et al., 1995). ### 2.2 Short spacing from the IRAM 30 m To complement the CO$(2-1)$ observations with the missing short spacing information and to investigate the large-scale general outflow properties to find the connection of the three regions, we observed them with the HERA array at the IRAM 30 m telescope on November 10th 2009. The 12CO$(2-1)$ line at 230.5 GHz and 13CO$(2-1)$ line at 220.4 GHz were observed in the on-the-fly mode. We mapped the whole region with a size of 5${}^{\prime}\times$ 3′ centered at R.A. 06h12m54.02s Dec. $+17^{\circ}59^{\prime}23.10^{\prime\prime}$ (J2000.0). The sampling interval was 3.5′′, a bit better than Nyquist-sampling to minimize beam-smearing effects. The region was scanned two times in the declination and right ascension direction, respectively, in order to reduce effects caused by the scanning process. The spectra were calibrated with CLASS which is part of the GILDAS software package, then the declination and right ascension scans were combined with the plait algorithm in GREG which is another component of the GILDAS package. The 12CO data has a beam size of 11′′, and the rms noise level of the corrected $T_{mb}$ is around 0.23 K at 0.6 km s-1 spectral resolution, the 13CO has the same beam size but the rms noise is 0.12 K at 0.6 km s-1 spectral resolution. After reducing the 30m data separately, single dish 12CO and 13CO data were converted to visibilities and then combined with the SMA data using MIRIAD package task UVMODEL. With different uv-range selections, the synthesized beam of the combined data varies from $2.1^{\prime\prime}\times 1.8^{\prime\prime}$ (PA $-84^{\circ}$) to $11.1^{\prime\prime}\times 10.7^{\prime\prime}$ (PA $-74^{\circ}$). ### 2.3 VLT-SINFONI integral field spectroscopy observations Near Infrared $H$\- and $K$-band observations were performed using the Integral Field instrument SINFONI (Eisenhauer et al., 2003; Bonnet et al., 2004) on UT4 (Yepun) of the VLT at Paranal, Chile. The observations of S255IR were performed in service mode on February 8th, February 12th, February 13th and March 29th 2007. The non-AO mode of SINFONI, used in combination with the setting providing the widest field of view ($8^{\prime\prime}\times 8^{\prime\prime}$) with a spatial resolution of 0.25′′ per slitlet. The typical seeing during the observations was 0.7′′ in $K$-band. The H+K grating was used, covering these bands with a resolution of R=1500 in a single exposure. S255IR was observed with a detector integration time (DIT) of 30 seconds per pointing. The observations were centered on the central near infrared source NIRS 1 at coordinates: R.A. 06h12m53.85s Dec. $+17^{\circ}59^{\prime}23.71^{\prime\prime}$ (J2000.0). We observed this area with SINFONI using a raster pattern, covering every location in the cluster at least twice, resulting in an effective integration time of 60 seconds per location in the field. The offset in the east-west direction was 4′′ (i.e. half the FOV of the detector) and the offset in the north-south direction was 6.75′′, results a total observation field of 70${}^{\prime\prime}\times 70^{\prime\prime}$. A sky frame was taken every 3 minutes using the same DIT as the science observations. The sky positions were chosen based on existing NIR imaging in order to avoid contamination. Immediately after every science observation, a telluric standard star was observed, matching as close as possible the airmass of the object. The data were reduced using the SPRED (version 1.37) software developed by the MPE SINFONI consortium (Schreiber et al., 2004; Abuter et al., 2006). The procedure described by Davies (2007) was applied to remove the OH line residuals. To calibrate the flux and remove the telluric absorption lines, the extracted spectrum of one standard star in each OB (OB1: Hip064656, OB2: Hip048128, OB3: Hip032108, OB4: Hip031899) is used. The flux calibration of the spectra uses the 2MASS (Cohen et al., 2003) magnitude of the standard stars (see Bik et al. (2010) for the details of the data reduction). ## 3 Results ### 3.1 Millimeter continuum emission We averaged the apparently line-free parts of upper and lower sideband spectra of each region (presented in Figure 2) and got the continuum images of the three regions shown in Figure 3. Assuming optical thin dust emission, we estimated the gas mass and column density of the continuum sources following the equations outlined in Hildebrand (1983) and Beuther et al. (2005a). We assumed a dust temperature of 40 K, grain size of 0.1 $\mu$m, grain mass density of 3 g cm3, gas-to-dust ratio of 100 and a grain emissivity index of 2 (corresponding to $\kappa\approx 0.3$ for comparison of Ossenkopf & Henning (1994)). The results are shown in Table 1. With the same set-up, we also calculated the gas mass and the column density of the SCUBA 850 $\mu$m (Di Francesco et al., 2008; Klein et al., 2005) continuum peaks shown in Table 2. To get the impression of how much flux we lost in the interferometer observations, we produced the SMA continuum map of the each region with the same beam size as the SCUBA 850 $\mu$m map (14${}^{\prime\prime}\times 14^{\prime\prime}$, Di Francesco et al. (2008)). We measured the continuum flux, calculated the gas mass and compare the results with the ones in Table 2; the mass we obtained from the SMA observations for S255IR, S255N and S255S is only 7.8$\%$, 8.7$\%$ and 2$\%$ of the SCUBA 850 $\mu$m measurements, respectively. Since our interferometer observations are not sensitive to spatial scales $>$ 24′′ (the shortest base line =8.5 k$\lambda$), we filter out the smoothly distributed large-scale halo and left only the compact cores. Also, the fact that we filtered out more flux for the youngest region S255S indicates that at the earliest evolutionary stages the gas is more smoothly distributed than during later stages where the collapse produces more centrally condensed structures. Nevertheless, all derived column densities are of the order or above the proposed threshold for high-mass star formation of 1 g cm-2 (Krumholz & McKee, 2008). Figure 1: SPITZER IRAC 4.5 ${\mu}$m image (Chavarría et al., 2008) overlaid with SCUBA 850 $\mu$m contours (Di Francesco et al., 2008; Klein et al., 2005). In the upper panel, the grey scale is the SPITZER IRAC 4.5 ${\mu}$m image, the black contours are the NVSS 1.4 GHz emission and the green contours are the SCUBA 850 $\mu$m continuum. In the bottom left panel, the grey scale is the SPITZER MIPS 24 ${\mu}$m image, the contours are the SCUBA 850 ${\mu}$m continuum, the three dashed circles mark the primary beam of our SMA observations. In the bottom right panel, the grey scale is the SPITZER MIPS 70 $\mu$m image, the contours and the circles are the same as the ones in the left panel. The contour levels of the NVSS start at 10$\sigma$ (1$\sigma=0.6$ mJy beam-1) with a step of 5$\sigma$. For the SCUBA 850 $\mu$m, the contour levels start at 10$\sigma$ (1$\sigma=0.1$ Jy beam-1) with a step of 10$\sigma$. The SPITZER/IRAC post-bcd data processed with pipeline version S18.7.0 and the MIPS post-bcd data processed with pipeline version S17.2.0 have been downloaded from the SPITZER archive to create these images. Figure 2: Lower and Upper sideband spectral vector-averaged over all baselines with a resolution of 2 km s-1 per channel. In the Upper sideband image, the 12CO line in S255IR and S255N panels was not fully plotted, which goes to 11.6 Jy and 5.9 Jy respectively. Table 1: Millimeter continuum peaks properties. Source \| R.A. \| Dec. \| $S_{peak}$ \| Sint. \| Mass \| N${}_{H_{2}}$ ---\|---\|---\|---\|---\|---\|--- \| (J2000.0) \| (J2000.0) \| mJy/beam \| mJy \| [$M_{\odot}$] \| cm-2 S255IR-SMA1 \| 06:12:54.01 \| +17:59:23.1 \| 95 \| 171 \| 12 \| 5.5 $\times$ 1024 S255IR-SMA2 \| 06:12:53.77 \| +17:59:26.1 \| 52 \| 150 \| 10 \| 3.0 $\times$ 1024 S255IR-SMA3 \| 06:12:53.88 \| +17:59:23.7 \| 30 \| 30 \| 2 \| 1.7 $\times$ 1024 S255N-SMA1 \| 06:12:53.71 \| +18:00:27.3 \| 91 \| 327 \| 22 \| 5.5 $\times$ 1024 S255N-SMA2 \| 06:12:52.95 \| +18:00:31.7 \| 33 \| 47 \| 3 \| 2.0 $\times$ 1024 S255N-SMA3 \| 06:12:53.65 \| +18:00:18.3 \| 18 \| 19 \| 1 \| 1.1 $\times$ 1024 S255S-SMA1 \| 06:12:56.65 \| +17:58:41.2 \| 14 \| 17 \| 1 \| 7.7 $\times$ 1023 S255S-SMA2 \| 06:12:56.85 \| +17:58:35.0 \| 12 \| 15 \| 1 \| 6.6 $\times$ 1023 Table 2: SCUBA sub-millimeter continuum data. Source \| Speak \| Sint \| Mass \| N${}_{H_{2}}$ ---\|---\|---\|---\|--- \| Jy/beam \| Jy \| [$M_{\odot}$] \| cm-2 S255IR \| 7.81 \| 31 \| 372 \| 6.5 $\times$ 1023 S255N \| 7.75 \| 30 \| 357 \| 6.4 $\times$ 1023 S255S \| 2.33 \| 9 \| 110 \| 1.9 $\times$ 1023 Figure 3: The SMA 1.3 mm continuum map of S255IR, S255N and S255S. contour levels start at 5$\sigma$ with a step of 5$\sigma$ in S255IR (1$\sigma=1.7$ mJy/beam) and S255N (1$\sigma=1.6$ mJy/beam) image and 3$\sigma$ in S255S (1$\sigma=0.9$ mJy/beam) image. The dotted contours are the negative features due to the missing flux. The synthesis beam is shown at the bottom left of each figure. In the S255IR panel, the asterisk marks the position of NIRS 3 and the right one marks NIRS 1 (Tamura et al., 1991). The inset is a zoom in the continuum image of the inner region of S255IR-SMA1 and the contour levels start at 5$\sigma$ with a step of 10$\sigma$. The triangles are the water masers (Goddi et al., 2007) and the two open diamond mark the positions of two 6.7 GHz methanol masers detected by Xu et al. (2009). In S255N panel, the crosses mark the position of the Class I 44 GHz ($7_{0}-6_{1}$) A+ methanol masers detected by Kurtz et al. (2004) and the triangle is the water maser (Cyganowski et al., 2007). The (0,0) point in each panel from left to right is R.A. 06h12m54.019s Dec. $17^{\circ}59^{\prime}23.10^{\prime\prime}$ (J2000.0), R.A. 06h12m53.669s Dec. $+18^{\circ}00^{\prime}26.90^{\prime\prime}$ (J2000.0) and R.A. 06h12m56.58s Dec. $+17^{\circ}58^{\prime}32.80^{\prime\prime}$ (J2000.0), respectively. ##### S255IR The continuum image of S255IR is presented in the left panel of Figure 3. We resolved two main continuum peaks (i.e. S255IR-SMA1 and S255IR-SMA2) and one unresolved sub peak (i.e. S255IR-SMA3) in this region. The stronger peak named as S255IR-SMA1 in the southeast coincides with the near infrared source NIRS 3 (Tamura et al., 1991). S255IR-SMA1 coincides with a UCHII region (Snell & Bally, 1986) generated by NIRS 3, and the inset in the S255IR panel in Figure 3 shows that it is also associated with Class II CH3OH and H2O maser emissions (Goddi et al., 2007; Minier et al., 2000, 2005). An unresolved peak S255IR- SMA3 that coincides with the near infrared source NIRS 1 (Tamura et al., 1991), which lies 2.4′′ west to S255IR-SMA1 and has been identified as a massive disk candidate by NIR polarization observation (Jiang et al., 2008). These two continuum sources are both respectively coincident with the mid- infrared massive proto-binary source 1 and 2 identified by Longmore et al. (2006). S255IR-SMA2, which has never been detected before, is not associated with any infrared source and likely to be in a very young source, it also shows only a few lines, which will be discussed in detail in Sec. 3.2. ##### S255N The middle panel of Figure 3 is the continuum image of S255N. The strongest peak named as S255N-SMA1 which coincides with the UCHII region G192.584-0.041 (Kurtz et al., 1994) is associate with Class I 44 GHz ($7_{0}-6_{1}$) A+ methanol masers which are detected by Kurtz et al. (2004) and water maser (Cyganowski et al., 2007). All the methanol masers which are marked with crosses in the S255N panel in Figure 3 are distributed along the direction of the outflow (Figure 10, see Sec. 3.3). S255N-SMA1 is also associated with centimeter continuum emission (Cyganowski et al., 2007) which shows an elongation aligned with the outflow (Figure 10, left panel). However, all three continuum sources in S255N have no corresponding near infrared point sources. ##### S255S The right panel of Figure 3 presents the continuum image of S255S. There are no near infrared sources or mid infrared peaks (SPITZER MIPS 24 $\mu$m and 70 $\mu$m, Figure 1) that are associated with these two sources. Meanwhile, the gas mass is only 2$\%$ of the SCUBA 850 $\mu$m measurements, which indicates that the gas in this region is at an early stage and smoothly distributed. All these features indicate that S255S is an extremely young region. ### 3.2 Spectral line emission The observed spectra of lower sideband (LSB) and upper sideband (USB) of three region are shown in Figure 2. Table 3: Observed lines in S255IR. $\nu$ \| line \| E${}_{\rm lower}/h$ ---\|---\|--- $[$GHz$]$ \| \| $[$K$]$ LSB \| \| 219.560 \| C18O(2$-$1) \| 5.3 219.734 \| HNCO(102,9$-$92,8) \| 219 219.798 \| HNCO(100,10$-$90,9) \| 48 219.949 \| SO(65$-$54) \| 24 220.038 \| HCOOH(100,10$-$90,9) \| 28.2 220.079 \| CH3OH(80,8$-$71,6)E \| 85 220.339 \| 13CO(2$-$1) \| 5.3 220.585 \| HNCO(101,9$-$91,8) \| 91 220.594 \| CH3CN(126$-$116) \| 315 220.641 \| CH3CN(125$-$115) \| 237 220.679 \| CH3CN(124$-$114) \| 173 220.709 \| CH3CN(123$-$113) \| 123 220.730 \| CH3CN(122$-$112) \| 87 220.743 \| CH3CN(121$-$111) \| 65 220.747 \| CH3CN(120$-$110) \| 58 USB \| \| 229.590 \| HCOOCH3(193,16$-$184,15)E \| 106 229.759 \| CH3OH(8-1,8$-$70,7)E \| 77 229.858 \| 34SO2(42,2$-$31,3) \| 7.7 229.864 \| CH3OH(195,15$-$204,16)A$+$ \| 568 229.939 \| CH3OH(195,14$-$204,17)A$-$ \| 568 230.027 \| CH3OH(32,2$-$41,4)E \| 28 230.538 \| 12CO(2$-$1) \| 5.5 231.061 \| OCS(19$-$18) \| 100 231.221 \| 13CS(50$-$40) \| 22 231.281 \| CH3OH(102,9$-$93,6)A$-$ \| 98.8 Table 4: Observed lines in S255N. $\nu$ \| line \| E${}_{\rm lower}/h$ ---\|---\|--- $[$GHz$]$ \| \| $[$K$]$ LSB \| \| 219.560 \| C18O(2$-$1) \| 5.3 219.949 \| SO(65$-$54) \| 24 220.079 \| CH3OH(80,8$-$71,6)E \| 85 220.339 \| 13CO(2$-$1) \| 5.3 220.743 \| CH3CN(121$-$111) \| 65 220.747 \| CH3CN(120$-$110) \| 58 USB \| \| 229.759 \| CH3OH(8-1,8$-$70,7)E \| 77 230.538 \| 12CO(2$-$1) \| 5.5 231.061 \| OCS(19$-$18) \| 100 231.221 \| 13CS(50$-$40) \| 22 Table 5: Observed lines in S255S. $\nu$ \| line \| E${}_{\rm lower}/h$ ---\|---\|--- $[$GHz$]$ \| \| $[$K$]$ LSB \| \| 219.560 \| C18O(2$-$1) \| 5.3 219.949 \| SO(65$-$54) \| 24 220.339 \| 13CO(2$-$1) \| 5.3 USB \| \| 230.538 \| 12CO(2$-$1) \| 5.5 ##### S255IR In S255IR, we detect 25 lines from 10 species. Besides three normal CO isotopologues, we also detect some sulfur-bearing species (SO, 34SO2, OCS, 13CS) and some dense gas molecules which are usually used to trace high mass hot cores, such as CH3OH, CH3CN and HCOOCH3 (Nomura & Millar, 2004; Beuther et al., 2009; Sutton et al., 1985). All the lines we detect are with lower energy levels $E_{lower}/k$ between 5.3 to 568 K (Table 3). Figures 4 and 5 present the integrated line images of all species (except 34SO2 which is blended with CH3OH and too weak for imaging, and 12CO and 13CO will be discussed in details in 3.3) with different transitions. Most lines show compact emission peaked at S255IR-SMA1, and only a few show emission at S255IR-SMA2. Several CH3OH lines (CH3OH(8${}_{0,8}-7_{1,6}$)E, CH3OH(8${}_{-1,8}-7_{0,7}$)E and CH3OH(3${}_{2,2}-4_{1,4}$)E, in Figure 4) exhibit extended emission towards S255IR-SMA2. The CH3OH(8${}_{-1,8}-7_{0,7}$)E line even extends to the northwest of S255IR-SMA1 following the direction of the blue-shifted outflow of S255IR-SMA1 (Figure 8, see Sec. 3.3). We suggest that particular the (8${}_{-1,8}-7_{0,7}$)E line is due to the shock heating excited by the outflow, which could also be mixed up with some Class I methanol maser emission (Sutton et al., 2004; Sobolev et al., 2005; Kalenskii et al., 2002; Slysh et al., 2002). SO line emission associated with S255IR-SMA1 is elongated along the outflow and could be affected by the outflow. 13CS line emission forms a shell around S255IR-SMA1 (Figure 5, similar shell-like C34S emission has also been found by Beuther et al. (2009)). Only a few lines are associated with S255IR-SMA2, SO, C18O (Figure 5) and several methanol lines (Figure 4). S255IR-SMA2 appears to be chemically younger than S2555IRmm1. We extracted C18O spectra from these two continuum peaks. The spectrum toward S255IR-SMA1 shows two peaks and can not be fitted by a single Gaussian profile, therefore, we calculated the FWZI (full width at zero intensity) of these two C18O spectra, which is 12 km s-1 and 6.8 km s-1 for S255IR-SMA1 and S255IR-SMA2, respectively. The difference of the line width also suggests that S255IR-SMA2 is at a younger evolutionary stage. Figure 4: S255IR CH3OH line integrated intensity images with the SMA 1.3 mm continuum emission in the background. Contour levels start at 3$\sigma$ with 2$\sigma/$level. For the top panels, the $\sigma$ of the contours in each panel, from left to right, is 12, 15, 11 mJy beam-1, respectively, and for the bottom panels is 12, 14, 15 mJy beam-1, respectively. The dotted contours are the negative feature. The integral velocity ranges are shown at the bottom- left of each panel in km s-1. The synthesized beams are shown at the bottom- left of each panel. The (0,0) point in each panel is R.A. 06h12m54.019s Dec. $+17^{\circ}59^{\prime}23.10^{\prime\prime}$ (J2000.0). Figure 5: S255IR molecular line integrated intensity contour with the SMA 1.3 mm continuum emission in the background. All the contour levels start at 3 $\sigma$. The contour step is 3 $\sigma$ in SO (1$\sigma=22$ mJy beam-1), HCOOH (1$\sigma=10$ mJy beam-1), HCOOCH3 (1$\sigma=12$ mJy beam-1), C18O (1$\sigma=16$ mJy beam-1), CH3CN (1$\sigma=11$ mJy beam-1) and HNCO (1$\sigma=13$ mJy beam-1) images, and 1 $\sigma$ in OCS (1$\sigma=13$ mJy beam-1)and 13CS(1$\sigma=11$ mJy beam-1) image. The dotted contours are the negative features with the same contours as the positive ones ini each panel. The integral velocity ranges are shown at the bottom-left of each panel in km s-1. The synthesized beams are shown at the bottom left of each panel. We only plotted the $k=0\&1$ line for CH3CN and $(10_{0,10}-9_{0,9})$ line for HNCO. The (0,0) point in each panel is R.A. 06h12m54.019s Dec. $+17^{\circ}59^{\prime}23.10^{\prime\prime}$ (J2000.0). ##### S255N In S255N, we detected 10 lines from 6 species (12CO, SO, CH3OH, CH3CN, OCS, 13CS) and 2 additional CO isotopologues (13CO and C18O) with lower energy levels $E_{lower}/k$ between 5.3 to 65 K (Table 4). Figure 6 shows the integrated line images of all species (except three CO isotopologues which will be discussed in Sec. 3.3 and 3.4). All the lines show a peak associated with S255N-SMA1, CH3OH and SO lines also show extended emission along the direction of the outflows (Figure 10, see Sec. 3.3). Figure 6: S255N molecular line integrated intensity contour with the SMA 1.3 mm continuum emission in the background. All the contour levels start at 3 $\sigma$. The contour step is 1$\sigma$ in CH3OH($8_{0,8}-7_{1,6}$)A- (1$\sigma=9$ mJy beam-1) and CH3CN($k=0\&1$, 1$\sigma=14$ mJy beam-1) images, 2$\sigma$ in 13CS (1$\sigma=25$ mJy beam-1) image, 3 $\sigma$ in SO (1$\sigma=13$ mJy beam-1) and CH3OH($8_{1,8}-7_{0,7}$)E images, and 0.5 $\sigma$ in OCS (1$\sigma=10$ mJy beam-1) image. The dotted contours are the negative features. The integral velocity ranges are shown at the bottom-left of each panel in km s-1. The synthesized beams are shown at the bottom left of each panel. The (0,0) point in each panel is R.A. 06h12m53.669s Dec. $+18^{\circ}00^{\prime}26.90^{\prime\prime}$ (J2000.0). ##### S255S In S255S, we detected 4 lines from two species (12CO and SO) and 2 additional CO isotopologues (13CO and C18O) with lower energy levels $E_{lower}/k$ between 5.3 to 24 K (Table 5), which again indicates that S255S is an extremely young and cold region. We measure the single dish 13CO line width toward the S255S SCUBA 850 $\mu$m continuum peak, and calculate the virial mass in this region, which is $\sim$ 80 $M_{\odot}$. The virial mass is smaller than the mass we got from the SCUBA 850 $\mu$m measurement which implies that S255S region will likely collapse. From the single dish 13CO observations, we got the vlsr of the three regions, which are shown in Table 6. However, the vlsr of the high-mass mm cores in S255IR and S255N based on the interferometer observations is different from the one we got from the single dish data. We extracted dense gas spectra, i.e. CH3OH for S255N-SMA1, CH3CN for S255IR-SMA1 and C18O for S255IR-SMA2, and then a Gaussian profile was fitted to get the vlsr of the cores. All the velocities are shown in Table 6. For S255S we can not get a proper vlsr from the SMA observations due to the missing flux problem, so only the vlsr we got from 30m observation is listed. Table 6: The vlsr measured with different telescopes. Sources \| vlsr got from 30m \| vlsr got from SMA ---\|---\|--- \| km s-1 \| km s-1 S255N-SMA1 \| 8.6 \| 10.4 S255IR-SMA1 \| 7.7 \| 5.2 S255IR-SMA2 \| 7.7 \| 9.3 S255S \| 6.6 \| … ### 3.3 The molecular outflows After combining the single dish 30m data with the interferometer data, we integrated the line-wing of 12CO and produced the outflow image of each region. Figure 7 shows the single dish only outflow map. While the northeast- southwest outflow which is associated with S255IR is clearly shown in Figure 7, the high velocity outflow is also found in S255N region, however, the direction of the outflow could not be well defined in single dish outflow map. In the youngest region S255S, the single dish outflow map does not show much emission. We combine the SMA data and 30m data together to study the outflow properties. Figure 7: The single dish outflow map of the S255 complex. The blue-shifted outflow is shown in dashed contours with a velocity-integration regime of [$-$40,0] km s-1, and the red one is shown in full contours with a velocity- integration regime of [16,56] km s-1. The contours start at 4$\sigma$ with a step of 5 $\sigma$($1\sigma$=1.2 K km s-1). The three crosses mark the position of the three SCUBA 850 $\mu$m continuum sources. ##### S255IR Figure 8 shows the combined SMA and 30m outflow images of S255IR. The velocity-integration regime of the blue-shifted part is [$-$40,0] km s-1 and [16,56] km s-1 for red-shifted part. We selected different uv-ranges resulting in different resolutions. Although at different resolution the structure of the outflow changes a lot, the northeast-southwest (NE-SW) outflow is confirmed in all images (also reported by Miralles et al. (1997)). In the bottom-left panel, the H2 jet-like sources (region (a) and (b) in Figure 19, see Sec. 3.6.1) also follow the NE-SW outflow direction. The water masers which are marked with triangles in the inset of S255IR panel in Figure 3 also follow the direction of the outflow, which confirms that S255IR-SMA1 is the driving source of this NE-SW outflow. In all panels except the bottom-right two, the red-shifted part of the outflow bends a little toward northwest, which might be a signature of the precessing jets. In the S255IR panel in Figure 3, the water masers also follow the NE-SW direction. We believe that NIRS 3 is the driving source of this outflow. As we are applying different uv- range selection, two red-shifted outflow components reveal themselves out to the north and south of the continuum sources, respectively, and these two components are both only shown in the lower velocity regime which is offset from the vlsr 10 to 20 km s-1. Finally, in the 30m-only panel and Figure 7, the red-shifted outflow component to the north of the continuum sources becomes connected with the component in S255N. Figure 9 presents the position-velocity (pv) diagram of the molecular gas along the outflow. The diagram shows that the red- and blue-shifted outflow resembles the Hubble-law with increasing velocity at longer distance from the outflow center, and the blue-shifted outflow also resembles the jet-bow-shock gas entrainment model (Arce et al., 2007). ##### S255N Figure 10 shows the combined SMA and 30m outflow images of S255N. The velocity-integration regime of the blue-shifted part is [$-$44,0] km s-1 and [16,48] km s-1 for the red-shifted part. The northeast-southwest (NE-SW) outflow is shown clearly in the SMA and SMA combined with 30m panels but is hardly seen in the 30m-only panel. In the middle panel, the crosses mark the position of the Class I 44 GHz ($7_{0}-6_{1}$) A+ methanol masers detected by Kurtz et al. (2004) and the triangle is the water maser (Cyganowski et al., 2007). In the left panel, the back ground grey scale is VLA Q-band 3.6 cm continuum (Cyganowski et al., 2007) which shows a jet-like structure. All these features follow the NE-SW direction. Based on these features we confirm the direction of the main outflow as NE-SW. The red-shifted outflow to the north of S255N-SMA1 on the line (b) top left panel in Figure 10 might be part of the outflow cavity, but we can not exclude the possibility of multiple outflows. The red-shifted gas at the bottom part of line (b) seems to be associated with S255N-SMA3 (the middle panel of Figure 10), however, another blue-shifted feature at the southern part of the map shows up as the resolution changes. These two components both can also only be detected at the relatively lower velocity regime $\sim$10 to 20 km s-1 offset from the vlsr. Figure 11 presents the position-velocity (pv) diagrams of the molecular gas along the outflows. In the top panel, the pv-plot cut follows the direction of the VLA jet-like emission (Fig. 10, left panel, line a). The diagram shows that the high-velocity gas on the blue-shifted side remains very close to the outflow center and the red-shifted part does not show the Hubble-law signature. But in the bottom panel, in which the pv-plot cut follows the direction of the two elongate red-shifted emission (Fig. 10, left panel, line b), the red-shifted northern side of the outflow resembles the Hubble-law. However, the red-shifted southern part of the outflow seems to have nothing to do with our S255N-SMA1 nor S255N-SMA3. The different blue/red pv- characteristics may be explained by the blue 12CO emission tracing predominantly the jet like component also visible in centimeter continuum emission, whereas the red 12CO emission traces mainly the cavity-like walls of the outflow or another outflow. ##### S255S Figure 12 presents the combined SMA and 30m outflow images of S255S. The velocity-integration regime of the blue-shifted part is [$-9$,1] km s-1 and [14,32] km s-1 for the red-shifted part. All the panels except the 30m-only one, show two blue-shifted components and one red-shifted component. Although the line wing emission is less pronounced than that for the other two regions, we clearly identify blue- and red-shifted 12CO emission associated with both mm peaks. The blue-shifted part in the south of the map is only detected in the 1 km s-1$\geq$v$\geq-6$ km s-1, and the red-shifted part in the north of the map is only detected in the v$\leq$16 km s-1. Since the line wing emission does not extend to very high velocities, the outflow should not be oriented directly along the line of sight. Therefore, the small spatial extent of the outflow indicates as well the youth of this region. Figure 8: S255IR 12CO$(2-1)$ outflow images observed with IRAM 30 m telescope and the SMA. The blue-shifted outflow is shown in dashed contours with a velocity-integration regime of [$-$40,0] km s-1, and the red one is shown in full contours with a velocity-integration regime of [16,56] km s-1. The top- left panel presents the SMA data only, the bottom-right presents the IRAM 30 m data only. The others are all combined SMA$+$30 m data, which are inverted with different uv-range selections (shown at the top-right of each panel in k$\lambda$) resulting in different beam sizes shown at the bottom left of each panel. All the 12CO emission contour levels start at 5 $\sigma$ with a step of 10 $\sigma$. The dotted contours are the negative features and the stars mark the position of NIRS 3. In the bottom-left panel, the back ground scale is SINFONI H2 line emission. The line in the SMA-only panel shows the pv-cut presented in Fig. 9, and the cross marks the central position of the line. In panel (0,120), the SMA 1.3 mm continuum map is over plotted (black full contours), and the contours start at 5 $\sigma$ with a step of 5 $\sigma$. In panel (0,70), the same CH3OH (8${}_{-1,8}-7_{0,7}$)E integrated intensity map in Figure 4 is over plotted in black full contours . The $\sigma$ of the outflow data is shown in Table 7. The (0,0) point in each panel is R.A. 06h12m54.019s Dec. $+17^{\circ}59^{\prime}23.10^{\prime\prime}$ (J2000.0). Figure 9: Position-velocity diagrams of S255IR for the 12CO$(2-1)$ SMA-only outflow observations with a velocity resolution of 1 km s-1. The pv-diagram is in northeast-southwest direction with a PA of 75∘ from the north (the cut is marked in Figure 8, top-left panel). The contour levels are from 10 to 90% from the peak emission (4.2 Jy beam-1) with a step of 10%. The $v_{lsr}$ at 7.7 km s-1 and the central position (marked by the cross in Fig. 8 top-left panel) is marked by vertical and horizontal line. Figure 10: S255N 12CO$(2-1)$ outflow images observed with IRAM 30 m telescope and the SMA. The blue-shifted outflow is shown in dashed contours with a velocity-integration regime of [$-$44,0] km s-1, and the red one is shown in full contours with a velocity-integration regime of [16,48] km s-1. The left panel presents the SMA data only, the right presents the IRAM 30 m data only. The middle one is combined SMA$+$30 m data, the uv-range selection is shown at the top-right of each panel in k$\lambda$. All the CO emission contour levels start at 5 $\sigma$ with a step of 6 $\sigma$. The dotted contours are the negative features and the asterisks mark the position of continuum peaks. In the middle panel, the crosses mark the position of the Class I 44 GHz ($7_{0}-6_{1}$) A+ methanol masers detected by Kurtz et al. (2004) and the triangle is the water maser (Cyganowski et al., 2007). In the top left panel, the back ground grey scale is VLA Q-band continuum (Cyganowski et al., 2007) and the line shows the pv-cut presented in Fig. 11, which follows the masers direction and crosses the continuum peak. The arrow in the 30m-only panel marks the outflow size we used to calculate the outflow physical parameters. The $\sigma$ of the outflow data is shown in Table 7. The (0,0) point in each panel is R.A. 06h12m53.669s Dec. $+18^{\circ}00^{\prime}26.90^{\prime\prime}$ (J2000.0). Figure 11: Position-velocity diagrams of S255N for the 12CO$(2-1)$ SMA-only outflow observations with a velocity resolution of 1 km s-1. The top panel is the pv-plot in northeast-southwest direction with a PA of 54∘ from the north (the cut is marked as the line a in Figure 10, left panel), and the bottom panel is the pv-plot in northwest-southeast direction with a PA of 7∘ from the north (the cut is marked as the line b in Figure 10, left panel). The contour levels are from 10 to 90% from the peak emission (4.15 Jy beam-1 in the top panel and 2.82 Jy beam-1 in the bottom panel) with a step of 10%. The $v_{lsr}$ at 8.6 km s-1 and the central position (i.e. the S255N-SMA1 position in the top panel and the cross point of line a and line b in Figure 10 top-left panel in the bottom panel) are marked by horizontal and vertical line. Figure 12: S255S 12CO$(2-1)$ outflow images observed with IRAM 30 m telescope and the SMA. The blue-shifted outflow is shown in dashed contours with a velocity-integration regime of [$-$9,1] km s-1, and the red one is shown in full contours with a velocity-integration regime of [14,32] km s-1. The left panel presents the SMA data only, the right presents the IRAM 30 m data only. The middle one is combined SMA$+$30 m data, and the uv-range selection is shown at the top-right of each panel in k$\lambda$. All the 12CO$(2-1)$ emission contour levels start at 5 $\sigma$ with a step of 4 $\sigma$, except the one in IRAM 30 m only image which starts at 5 $\sigma$ with a step of 2 $\sigma$. The dotted contours are the negative features and the stars mark the positions of continuum peak S255S-SMA1 and S255S-SMA2. The arrows in the 30m-only panel mark the outflow sizes we used to calculate the outflow physical parameters. The $\sigma$ of the outflow data is shown in Table 7. The (0,0) point is each panel corresponds to position R.A. 06h12m56.58s Dec. $+17^{\circ}58^{\prime}32.80^{\prime\prime}$ (J2000.0). Table 7: Outflow data. Region \| uv-range \| beam size \| $\sigma_{\rm red}$ \| $\sigma_{\rm blue}$ \| Data ---\|---\|---\|---\|---\|--- \| [k$\lambda$] \| [′′] \| [mJy beam-1] \| [mJy beam-1] \| S255IR \| (8, 173) \| 1.5$\times$1.5 \| 15 \| 26 \| SMA \| (0, 120) \| 2.1$\times$1.8 \| 15 \| 26 \| SMA+30m \| (0, 70) \| 3.1$\times$2.5 \| 18 \| 32 \| SMA+30m \| (0, 45) \| 4.0$\times$3.4 \| 22 \| 38 \| SMA+30m \| (0, 35) \| 6.3$\times$5.8 \| 26 \| 71 \| SMA+30m \| (0, 20) \| 9.8$\times$7.4 \| 38 \| 86 \| SMA+30m \| (0, 16) \| 11.1$\times$10.7 \| 40 \| 110 \| SMA+30m \| … \| 11.3$\times$11.3 \| 0.8 K km s-1 \| 1.1 K km s-1 \| 30m S255N \| (8, 173) \| 1.8$\times$1.4 \| 19 \| 12 \| SMA \| (0, 40) \| 4.2$\times$3.9 \| 34 \| 18 \| SMA+30m \| … \| 11.3$\times$11.3 \| 0.5 K km s-1 \| 1.1 K km s-1 \| 30m S255S \| (8, 173) \| 1.8$\times$1.4 \| 14 \| 18 \| SMA \| (0, 40) \| 4.2$\times$3.9 \| 20 \| 26 \| SMA+30m \| … \| 11.3$\times$11.3 \| 0.4 K km s-1 \| 0.5 K km s-1 \| 30m Applying the method of Cabrit & Bertout (1990, 1992) we derived properties of the outflows such as outflow mass, outflow energy and dynamical age. Since the SMA data suffer a lot from the missing flux problem, only the 30m data are used in the calculations. These calculations assume that the 13CO$(2-1)$/12CO$(2-1)$ line wing ratio is 0.1(Choi et al., 1993; Levreault, 1988; Beuther et al., 2002). The results are shown in Table 8. The dynamical age calculation depends on the tangent of the outflow’s inclination angle (with 0∘ being in the plane of the sky). Since we cannot get the inclination angle information of these sources, we assumed an inclination angle of 45∘ for all the dynamical age calculations. For S255N and S255S, the outflows are not clearly defined in the 30m-only maps (Figures 10 and 12). Therefore, we used with the SMA+30m map to determine the outflow sizes (see right panels of Figures 10 and 12). For S255S, since we just detected only one outflow lobe associated with each continuum source, we just extended the only one lobe to the opposite side of the source to get the proper size and mass of the outflow. The arrows in the 30m-only panels in Figures 10 and 12 mark the outflow size we use. Table 8: Outflow parameters. Sources \| $M_{\rm t}$ \| $p$ \| $E$ \| Size \| t \| $\dot{M}_{\rm out}$ \| $F_{\rm m}$ ---\|---\|---\|---\|---\|---\|---\|--- \| [$M_{\odot}$] \| [$M_{\odot}$ km s-1] \| [erg] \| [pc] \| [yr] \| [$M_{\odot}$/yr] \| [$M_{\odot}$/km/s/yr] S255IR-SMA1 \| 2.9 \| 137 \| 6.5$\times 10^{46}$ \| 0.7 \| 7.4$\times 10^{3}$ \| 4$\times 10^{-4}$ \| 2$\times 10^{-2}$ S255N-SMA1 \| 1.0 \| 43 \| 1.9$\times 10^{46}$ \| 0.2 \| 2.4$\times 10^{3}$ \| 4$\times 10^{-4}$ \| 2$\times 10^{-2}$ S255S-SMA1 \| 0.02 \| 0.4 \| 8$\times 10^{43}$ \| 0.1 \| 2.5$\times 10^{3}$ \| 8$\times 10^{-6}$ \| 2$\times 10^{-4}$ S255S-SMA2 \| 0.03 \| 0.8 \| 2$\times 10^{44}$ \| 0.07 \| 1.7$\times 10^{3}$ \| 2$\times 10^{-5}$ \| 5$\times 10^{-4}$ Entries include total outflow mass $M_{\rm t}$, momentum $p$, energy $E$, size, outflow dynamical age $t$, outflow rate $\dot{M}_{\rm out}$, mechanical force $F_{\rm m}$. ### 3.4 Rotational structures ##### S255IR Figure 13 shows the velocity moment maps of HCOOCH3, CH3CN ($k=2$) and C18O$(2-1)$. In the velocity maps of HCOOCH3 and CH3CN ($k=2$), we see a clear velocity gradient perpendicular to the outflow axis, which indicates the existence of a rotational structure. The C18O$(2-1)$ velocity map shows a little different picture compared to the other ones. C18O traces a more diffuse gas compared to the other two molecules. The C18O velocity map also shows a big velocity difference between S255IR-SMA1 and S255IR-SMA2. Figure 14 shows the position-velocity diagrams of the HCOOCH3 and C18O$(2-1)$ emission. The cuts, which are shown in Figure 13, go through the peak of the dust continuum and have position angles perpendicular to the direction of the outflow presented in Figure 8. The pv-diagram of HCOOCH3 shows that the rotational structure is not in Keplerian motion, hence maybe it is just a rotating and infalling core similar to the toroids described by Cesaroni et al. (2007). The approximately 2.3′′ diameter of the rotational structure corresponds to 3 700 AU at the given distance of 1.59 kpc. The pv-diagram of CH3CN ($k=2$) has a similar structure to the one of HCOOCH3, thus we do not show it. However, the picture is different for C18O, the pv-diagram shows a Keplerian- like rotation structure. The full line in the C18O panel shows a Keplerian rotation curve with a central mass of 28 $M_{\odot}$, which is the mass of the whole continuum structure in S255IR covering both mm peaks. For a more detailed discussion, see Sec. 4.3. Figure 13: HCOOCH3, CH3CN ($k=2$) and C18O$(2-1)$ velocity (1st) moment maps overlaid with the SMA 1.3 mm dust continuum of S255IR. The contours start at 5$\sigma$ and increase with a step of 10$\sigma$ in all panels ($\sigma$=1.7 mJy beam-1). The lines in each panel show the pv-diagram cut presented in Fig. 14. All moment maps were clipped at the five sigma level of the respective line channel map. The synthesized beam is shown in the lower left corner of each plot. The (0,0) point in each panel is R.A. 06h12m54.019s Dec. $+17^{\circ}59^{\prime}23.10^{\prime\prime}$ (J2000.0). Figure 14: Position- velocity diagrams derived for the cuts along the observed velocity gradient in Fig. 13. The offset refers to the distance along the cut from the dust continuum peak. The contour levels are from 3$\sigma$ with a step of 1$\sigma$ in both panels (1$\sigma$=60 mJy beam-1). The full line in the C18O panel shows a Keplerian rotation curve with a central mass of 28 $M_{\odot}$, which is the mass of the whole continuum structure in S255IR. The negative features are shown in dashed lines. ##### S255N Figure 15 shows the velocity moment maps of CH3OH(8${}_{-1,8}-7_{0,7}$), CH3OH ($8_{0,8}-7_{1,6}$) and C18O$(2-1)$. The C18O velocity map shows a complicated velocity structure, the velocity gradient is neither aligned with nor orthogonal to the outflow orientation. Because C18O is an isotopologue of 12CO, it may be influenced by infall and the outflow. Methanol is well known as a molecule that traces cores, shocks and masers in star formation regions (Jørgensen et al., 2004; Beuther et al., 2005b; Sobolev et al., 2007). It has also been reported as a low mass disk tracer (Goldsmith et al., 1999). In the velocity maps of the two methanol transitions (left and middle panel of Figure 15), we see a velocity gradient perpendicular to the outflow axis, which indicates the existence of a rotational structure. Figure 16 shows the position-velocity diagram of the (8${}_{0,8}-7_{1,6}$) methanol line emission, and the pv-diagram of the other methanol transition shows the similar velocity structure. The cuts go through the peak of the dust continuum and have position angles perpendicular to the direction of the outflow (Figure 15). The pv-diagram shows that the rotational structure is not Keplerian. The structure has a size of 4.4′′ corresponding to 7 000 AU at the given distance of 1.59 kpc. It has a narrow velocity range of 3 km s-1, and may likely be a large rotating and infalling core similar to the toroids described by Cesaroni et al. (2007). There are two low-velocity components at the northeast and southwest of the continuum peak in the methanol velocity maps which coincide with the outflow. They may be caused by the shock heating emission, and the line profile is consist with both thermal emission and maser emission (Sutton et al., 2004; Sobolev et al., 2005; Kalenskii et al., 2002; Slysh et al., 2002). Figure 15: CH3OH(8${}_{-1,8}-7_{0,7}$), CH3OH ($8_{0,8}-7_{1,6}$) and C18O$(2-1)$ velocity (1st) moment maps overlaid with the SMA 1.3 mm dust continuum of S255N. The contours start at 5$\sigma$ and increase with a step of 10$\sigma$ (1$\sigma$=1.6 mJy beam-1). All moment maps were clipped at the five sigma level of the respective line’s channel map. The lines in each panel show the pv-diagram cut presented in Fig. 14. The synthesized beam is shown in the lower left corner of each plot. The (0,0) point in each panel is R.A. 06h12m53.669s Dec. $+18^{\circ}00^{\prime}26.90^{\prime\prime}$ (J2000.0). Figure 16: Position-velocity diagrams derived for the cuts along the observed velocity gradient in Fig. 15. The offset refers to the distance along the cut from the dust continuum peak S255N-SMA1. The contour levels are all from 3$\sigma$ and with a step of 1$\sigma$ (1$\sigma$=40 mJy beam-1). ### 3.5 Temperature from CH3CN(12${}_{k}-11_{k}$) in S255IR Since we detected 7 lines of the CH3CN$(12_{k}-11_{k})$ $k$-ladder with $k$ = 0…6 in S255IR, we can utilize the varying excitation levels of the lines with lower level energies $E_{lower}/k$ between 58 to 315 K (Table 3) to estimate a temperature for the central gas core. Figure 17 shows the observed CH3CN$(12_{k}-11_{k})$ spectrum toward the continuum peak S255IR-SMA1 with only the compact configuration data. We did a simple Gaussian fitting of the spectrum and plot the level populations $N_{j,k}$ we calculated from the fitting result in Figure 18 with the assumption of optically thin emission (Zhang et al., 1998). The linear fitting result of the lower five levels is also plotted in Figure 18. It is clear that we can not fit the whole spectrum with one single temperature, which reveals a temperature gradient of the source and not optical thin emission. We modeled this spectrum in the local thermodynamic equilibrium using the XCLASS software developed by Peter Schilke (private communication). This software package uses the line catalogs from JPL and CDMS (Poynter & Pickett, 1985; Müller et al., 2001). The model spectrum with a temperature of 150 K is shown in Figure 17 in dotted line. The main difference between the model spectrum and the observed spectrum is that the model one is optically thin whereas the lower line intensity of the observed $k=3$ line indicates moderate optical depth of the CH3CN lines. Figure 17: CH3CN(12k\- 11k) spectrum toward the mm continuum peak S255IR-SMA1. The dotted line shows a model spectrum with $T_{\rm rot}$=150 K and $N_{\rm CH_{3}CN}=3.5\times 10^{14}$ cm-2. The $k=6$ line is excluded in the model because it is blurred by the HNCO(10${}_{1,9}-9_{1,8}$) line. Figure 18: From the Gaussian fitting result, we calculated the $N_{j,k}$ (Zhang et al., 1998), and plotted in the figure above. $E_{u}$ is the upper level energy of each transition. The line shows the linear fitting of the first five points. ### 3.6 SINFONI results #### 3.6.1 SINFONI line and maps The SINFONI observations are centered on the mm peak S255IR-SMA1 covering most part of the near-infrared cluster. Figure 19 shows the 3 color composite of 3 line maps (Red: Br$\gamma$ line emission, green: H2 (2.12 $\mu$m) line emission and blue: Fe II (1.64 $\mu$m) line emission). The two massive young stars (NIRS 1 and NIRS 3) are in the center of the field, surrounded with the cluster of about 120 stars down to $K=$17 mag. For all the point sources marked in Figure 24 a SINFONI $H$\- and $K$-band spectrum is available and a spectral classification is obtained of the brighter members (Sec. 3.6.2). The PDR and jet like emission is traced by the molecular H2 emission (2.12 $\mu$m, green). And to the north of the S255IR-SMA1 (the asterisk in the Figure 19), two point sources (sources #17 and #18 in Figure 24) show strong Br$\gamma$, which may indicate the existence of accreting signature (Muzerolle et al., 1998). The two strong Fe II (blue) features around S255IR-SMA1 show destructive shock (J-type, Hollenbach & McKee (1989)) emission and also follow the direction of the outflow and the H2 jets, which indicate S255IR-SMA1 is the energetic driving source of the jets. To identify the excitation mechanism of the H2 emission (the arc which is marked in Figure 19 by the red contour and the jet like emission regions (a) and (b)), we extracted the spectra of these three regions and construct the excitation diagrams for these regions. Figure 20 shows the excitation diagrams of them. In these diagrams, the measured column densities of lines are plotted against the energy of the upper level (see Martín-Hernández et al. (2008) for the detailed description of this diagram). The total column densities were calculated using the description of Zhang et al. (1998). Different symbols represent different vibrational levels (Fig. 20). The arc region (Fig. 19) has the most lines detected as it covers the largest area. For the jet-like emission region (a) and (b), the weaker lines seen in the spectrum of the arc region are not detected and 3$\sigma$ upper limits are given instead. The solid line in the diagrams is a single temperature fit to all the data points, while the dashed line in the diagram of the arc region presents a linear fit to only the $1-0$ S lines. For the arc region it is clear that the column densities are not represented by a single temperature gas, the line fluxes of the $2-1$ and $3-2$ vibrational levels are higher than expected from the $1-0$ line fluxes. A likely excitation mechanism of the H2 gas in the arc is fluorescence by non-ionizing UV photons (Davis et al., 2003). The excitation diagrams of regions (a) and (b) are well-represented by a single temperature. However the $3-2$ lines are not detected in either region, and only one $2-1$ line is detected in region (a) left most of them only upper limits. We conclude that the excitation diagrams of both regions are consistent with shock excited emission in an outflow emission. Besides, the bottom left panel of Figure 8 shows region (a) and (b) follow the direction of the outflow, also their elongated shapes that all suggest outflow origin. The linear fits of the excitation diagrams also allow us to determine the column density and temperature of the emitting gas (Table 9). Table 9: Physical properties of the H2 gas. Region \| Trot \| N(H2) ---\|---\|--- \| $[K]$ \| [cm-2] the arc \| $2416\pm 64$ \| $2.6\pm 0.2\times 10^{17}$ a \| $1655\pm 62$ \| $5.8\pm 1.7\times 10^{17}$ b \| $1091\pm 91$ \| $1.9\pm 1.2\times 10^{18}$ Figure 19: Three-color image created from SINFONI line maps. Red: Br$\gamma$ line emission, green: H2 (2.12 $\mu$m) line emission and blue: Fe II (1.64 $\mu$m) line emission. The cross marks the position of S255IR-SMA1. The contour and the dashed lines mark the regions of which the excitation diagrams are constructed to identify the nature of the emission (Fig. 20). Figure 20: The excitation diagram of the whole arc in the south-east of the Fig. 19 (the top panel) and two jet-like emissions (a) and (b). The solid line in all diagrams is a single temperature fit to all the lines detected, while for the dash-dotted line in the top panel a only the 1$-$0 S lines are included in the fit. The excitation diagram of the arc suggests that UV fluorescence is the excitation mechanism in this area. Regions (a) and (b) are most likely outflows as their excitation diagrams suggest thermal excitation. #### 3.6.2 SINFONI stellar spectral type classification Our SINFONI observations provide an $H$\- and $K$-band spectrum of each source with a spectral resolution of R=1500. SOFI $J$\- and $K$-band photometry results are taken from Bik (2004). Objects with $K<$14 have high enough S/N spectra to obtain a reliable spectral type. This reduces the sample to 39 sources, however, we can only get the spectral type of 16 sources which are listed in Table 11. For many point sources, we could not get a spectral type, some of them have very flat spectra and basically no absorption lines and also no infrared excess, they are likely to be gas clumps. Others have very red spectra, which are likely dominated by dust emission from the surrounding environment or circumstellar disks, therefore, the spectrum of the underlying objects might not be visible, such as sources #17, #18 in Table 11 and our infrared sources NIRS1 and NIRS3 which will be discussed later. The SINFONI $H+K$-band spectra of the three brightest cluster members (sources #1, #2 and #3) show Br$\gamma$ absorption in the $K$-band and Br$10-14$ absorption in the $H$-band (Figure 21). Weaker lines of He I are also visible in $H$ (1.70 $\mu$m) and K (2.058 $\mu$m). We apply the $K$-band classification scheme for B stars from Hanson et al. (1996) which links a $K$-band spectral type to an optical spectral type. For the $H$-band, we used the classification of Hanson et al. (1998) and Blum et al. (1997). Table 10 shows the measured Equivalent Widths (EW) of the relevant lines in the $H$\- and $K$-band spectra of the three B/A stars. Star #1 has a strong Br$\gamma$ absorption but does not show He I (2.11 $\mu$m), therefore is classified as B3V$-$B7V. Star #2 and #3 show much stronger Br$\gamma$ absorption and also do not have He I (2.11 $\mu$m) are therefore classified as B8V$-$A3V. The EWs of the $H$-band lines are in agreement with the $K$-band spectral type. Source #1 and source #2 are also associated with UCHII regions (Snell & Bally, 1986), which suggests that they are high to intermedia mass stars, and this consists with our spectral type classification results. The results are listed in Table 11. Table 10: Equivalent width and $K$-band spectral types of the early type stars in S255IR. Star \| Br11 (1.68 $\mu$m) \| He I (1.70 $\mu$m) \| He I (2.11$\mu$m) \| Br$\gamma$ (2.166 $\mu$m) \| $K$-Spectral type \| Optical Spectral type ---\|---\|---\|---\|---\|---\|--- 1 \| $<$0.4 Å \| 0.6$\pm 0.4$ Å \| $<$0.4 Å \| 5.9$\pm 0.4$ Å \| KB4$-$B7 \| B3V$-$B7V 2 \| 8.3$\pm 0.4$ Å \| $<$0.4 Å \| $<$0.4 Å \| 11$\pm 0.4$ Å \| KB8$-$A3 \| B8V$-$A3V 3 \| 8.9$\pm 0.4$ Å \| $<$0.4 Å \| $<$0.4 Å \| 8.9$\pm 0.4$ Å \| KB8$-$A3 \| B8V$-$A3V Besides the 3 B/A stars we found 13 stars showing absorption lines typical of later spectral type (Figure 22). The most prominent lines we used are the CO first overtone absorption bands between 2.29 and 2.45 $\mu$m, and absorption lines of Mg I and other atomic species in the $H$-band as well as Ca I and Na I in the $K$-band. To classify the late-type stars we use the reference spectra of Cushing et al. (2005) and Rayner et al. (2009). The atomic lines, such as Mg I and Na I, are used for the temperature determination, while the CO lines are used to determine the luminosity class. See Bik et al. (2010) for a detailed description. Compared with the reference spectra, the CO (2.29 $\mu$m) absorption of our SINFONI spectra is usually deeper than in dwarf reference spectra, but not as deep as in the giant spectra. In a few cases where a luminosity class IV reference spectrum was available, the spectrum provided a better match to the observed spectrum. This suggests that our late type stars are low- and intermediate PMS stars, and indeed, Luhman (1999) and Winston et al. (2009) find that PMS spectra have a surface gravity intermediate between giant and dwarf spectra. If the stars were giants, dust veiling could make the K-band CO lines weaker. However, the H-band CO and OH lines are also much weaker, while the atomic lines have the expected EWs. Therefore, this seems to be a surface gravity effect, suggesting a PMS nature of our late type stars. The ”double blind” procedure was applied during the classification of the late type stars to check the accuracy of the classification. In this procedure, Y. Wang and A. Bik did their own classification separately and independently without knowing results of each other, and their results were compared afterwards to get the differences which are the errors of the spectral type in Table 11. Our classification results showed an error of 1 to 2 subclass in spectral type. The relation from Kenyon & Hartmann (1995) was used to convert the spectral type into effective temperature. However, this relation applies only for main sequence stars, for PMS stars a different relation may hold (Hillenbrand, 1997; Winston et al., 2009). Cohen & Kuhi (1979) show that the temperature of PMS stars might be overestimated by values between 500 K (G stars) and 200 K (mid-K). We took this source of error into account when calculating the errors in the effective temperature. With the knowledge of the spectral type, we can estimate the extinction from the observed color. As the intrinsic $J-K$ color for dwarfs and giants can differ as much as 0.4 for late K stars (Koornneef, 1983), for the PMS stars, we used as the intrinsic $J-K$ color the mean of the color for dwarfs and giants. The difference between the mean value and the dwarf and giant values is used as the error in the intrinsic color. The derived extinctions are listed in Table 11. Table 11: Photometric and spectroscopic properties of the detected stars brighter than $K=14$ mag. Star \| R.A. (J2000) \| Dec. (J2000) \| $K$ \| $J-K$ \| Teff \| Sp. Type \| Lum.class \| AV ---\|---\|---\|---\|---\|---\|---\|---\|--- \| (h m s) \| (∘ ′ ′′) \| (mag) \| (mag) \| (K) \| \| \| (mag) 1 \| 06:12:54.91 \| 17:59:21.05 \| 11.26$\pm$0.03 \| 1.7$\pm$0.06 \| 15620$\pm$2850 \| KB4-B7(B3-B7) \| V \| 10.0$\pm$0.3 2 \| 06:12:55.06 \| 17:59:28.93 \| 10.60$\pm$0.02 \| 0.8$\pm$0.03 \| 9800$\pm$1590 \| KB8-A3(B8-A3) \| V \| 4.8$\pm$0.2 3 \| 06:12:54.68 \| 17:59:32.82 \| 11.68$\pm$0.04 \| 2.2$\pm$0.09 \| 9800$\pm$1590 \| KB8-A3(B8-A3) \| V \| 12.8$\pm$0.4 4 \| 06:12:54.82 \| 17:59:12.98 \| 13.39$\pm$0.08 \| 2.3$\pm$0.21 \| 3650$\pm$271 \| M1.5$\pm$1 \| V \| 8.2$\pm$1.0 5 \| 06:12:53.60 \| 17:59:28.18 \| 13.40$\pm$0.08 \| 2.9$\pm$0.26 \| 4590$\pm$460 \| K4$\pm$1 \| V \| 12.3$\pm$1.2 6 \| 06:12:55.31 \| 17:59:15.90 \| 13.17$\pm$0.07 \| 1.7$\pm$0.20 \| 4730$\pm$446 \| K3$\pm$1 \| PMS \| 5.4$\pm$1.1 7 \| 06:12:54.59 \| 17:59:17.23 \| 12.57$\pm$0.05 \| 2.2$\pm$0.19 \| 4730$\pm$446 \| K3$\pm$1 \| PMS \| 8.6$\pm$1.0 8 \| 06:12:54.13 \| 17:59:29.18 \| 13.37$\pm$0.08 \| 2.6$\pm$0.23 \| 4900$\pm$492 \| K2$\pm$1 \| V \| 11.1$\pm$1.1 9 \| 06:12:56.38 \| 17:59:32.75 \| 13.05$\pm$0.07 \| 4.0$\pm$0.39 \| 4900$\pm$492 \| K2$\pm$1 \| PMS \| 19.1$\pm$1.6 10 \| 06:12:54.55 \| 17:59:02.76 \| 12.78$\pm$0.06 \| 3.8$\pm$0.33 \| 5080$\pm$513 \| K1$\pm$1 \| PMS \| 18.2$\pm$1.4 11 \| 06:12:55.10 \| 17:59:21.34 \| 13.14$\pm$0.07 \| 2.3$\pm$0.24 \| 5080$\pm$513 \| K1$\pm$1 \| PMS \| 10.0$\pm$1.2 12 \| 06:12:53.17 \| 17:59:05.93 \| 12.50$\pm$0.05 \| 3.0$\pm$0.25 \| 5385$\pm$630 \| G8/K0$\pm$2 \| PMS \| 14.4$\pm$1.3 13 \| 06:12:55.06 \| 17:59:31.09 \| 12.24$\pm$0.05 \| 1.5$\pm$0.20 \| 5630$\pm$470 \| G7$\pm$1 \| PMS \| 6.0$\pm$1.2 14 \| 06:12:54.90 \| 17:59:40.96 \| 12.33$\pm$0.05 \| 4.7$\pm$0.42 \| 5385$\pm$630 \| G8/K0$\pm$2 \| PMS \| 24.0$\pm$1.7 15 \| 06:12:54.45 \| 17:59:37.10 \| 13.97$\pm$0.11 \| 2.4$\pm$0.29 \| 4730$\pm$446 \| K3$\pm$1 \| Giant \| 9.1$\pm$1.4 16 \| 06:12:53.29 \| 17:59:21.70 \| 13.37$\pm$0.08 \| 3.0$\pm$0.27 \| 5250$\pm$519 \| K0$\pm$1 \| S. Giant \| 13.9$\pm$1.2 17 \| 06:12:54.18 \| 17:59:47.58 \| 12.65$\pm$0.00 \| 4.9$\pm$0.00 \| … \| … \| Br$\gamma$ \| … 18 \| 06:12:54.15 \| 17:59:34.19 \| 12.29$\pm$0.00 \| 5.3$\pm$0.00 \| … \| … \| Br$\gamma$ \| … NIRS 1 \| 06:12:53.83 \| 17:59:23.71 \| 11.36$\pm$0.00 \| 6.2$\pm$0.00 \| … \| … \| … \| … NIRS 3 \| 06:12:54.01 \| 17:59:23.68 \| 12.56$\pm$0.00 \| 5.0$\pm$0.00 \| … \| … \| … \| … Figure 21: Normalized SINFONI $H$-band (left panel) and $K$-band (right panel) spectra of the three B/A stars detected in S255IR. Their spectra are characterized by absorption lines of the hydrogen Bracket series and He I (dotted vertical lines). The stars are numbered according to Table 11. Figure 22: Normalized SINFONI $H$-band (left panel) and $K$-band (right panel) spectra of the 11 objects showing late-type stellar spectral type. And most of them are PMS stars. The vertical lines show the location of some of the absorption lines used for the classification of the stars. The stars are numbered according to Table 11 #### 3.6.3 YSOs Source #17 and #18 have very red spectra and show strong Br$\gamma$ point emission, but the spectra are featureless due to the strong dust emission, which indicates that they are very young and with ongoing accretion activity (Figure 23). Source #17 also shows strong $K$-band CO line emissions (Bik & Thi, 2004), which indicates the existence of circumstellar disk. The two infrared sources NIRS3 and NIRS1 which coincide with our mm sources S255IR- SMA1 and S255IR-SMA3, respectively, also have extremely red spectra, only several H2 lines can be seen (Figure 23). Therefor we cannot get the spectral type of these sources. Figure 23: SINFONI $H$-band (left panel) and $K$-band (right panel) spectra of the 4 YSOs. The stars are numbered according to Table 11. #### 3.6.4 Cluster membership and the HR diagram For the late type stars, as discussed in Sec. 3.6.2, most of them have a surface gravity typical for PMS stars, so they are considered to be the cluster members. Source #15 and #16 show a luminosity type Giant and Super Giant, they are unlikely to be associated with this young cluster and since they show quite high extinction, they are considered to be background stars. To constrain the mass of the cluster members and the age of the whole cluster, we construct a HRD to compare the observed parameters with PMS evolutionary tracks. The derived extinctions allow us to de-redden the $K$-band magnitude, convert to absolute magnitude with the derived temperature, and plot the points in a $K$ vs. log(Teff) diagram (Fig. 25, top). We exclude the background stars #15 and #16 to enable a comparison with the isochrones. The over-plotted dashed line is the 2 Myr main sequence isochrone taken from Lejeune & Schaerer (2001). The solid lines in the left panel represent the evolutionary tracks taken from Da Rio et al. (2009) using the models of Siess et al. (2000) for 8 different masses: 5, 4, 3, 2, 1.5, 1, 0.5, 0.4 M⊙. Comparison of the location of the PMS stars with these evolutionary tracks yields an approximate mass varying from $\sim$ 5 M⊙ for the brightest stars to 0.4 M⊙. In the bottom panel of Figure 25, the over-plotted solid lines are the PMS isochrones between 0.1 and 10 Myr. The location of the stars is consistent with a range of age. Besides source #1, #14 and #4 which lie between the 0.5 and 1 Myr isochrones, most of the objects span the $1-3$ Myr isochrones, with the more massive objects closer to the 1 Myr isochrone. For the less massive objects, the spread in age is larger, most likely due to higher uncertainties in the spectral type determination. The location of the PMS stars suggests an age of $2\pm 1$ Myr. Comparison of the location of the PMS stars in the HRD with those of the Herbig AeBe stars (van den Ancker et al., 1998) shows that the late spectral type PMS stars are younger than the Herbig AeBe stars and will evolve from their present G- and K spectral type to late B, A or early F spectral type when they become main sequence stars. The three relative early spectral type sources (#1, #2 and #3) already evolved to late B early A spectral type. They are in the transit phase between our late spectral type sources and the main sequence. Figure 24: The stars listed in Table 11 marked on Br$\gamma$ line$+$continuum map overlaid with H2 line emission contours. Figure 25: $Top:$ The extinction corrected $K$ vs. log(Teff) HRD. The $K-$band magnitude has been corrected for the distance modulus. Over-plotted with a dash line is the 2 Myr main sequence isochrone from Lejeune & Schaerer (2001), and with the solid lines the pre-main-sequence evolutionary tracks from Da Rio et al. (2009). $Bottom:$ The same data but over-plotted with solid lines showing the isochrones from Da Rio et al. (2009). ## 4 Discussion ### 4.1 Different evolutionary stages and triggered star formation? The SMA and IRAM 30m data together reveal three massive star formation regions with different evolutionary stages. The SCUBA 850 $\mu$m image (Figure 1) presents three continuum peaks in the whole S255 complex region, one in each sub-region, which is S255N, S255IR, S255S, from north to south respectively. With our high resolution SMA observations, $\sim$2500 AU at the given distance of 1.59 kpc, we found that all SCUBA 850 $\mu$m sources fragment into several cores. Minier et al. (2007) suggests S255S to be at a very young stage without active star formation, however, our observations show outflows associated with the mm sources, Furthermore, the virial mass is smaller than the mass obtained from the SCUBA 850 $\mu$m measurement, which implies that the S255S region will likely collapse. Since the peak column density is also on the order of the proposed threshold for high-mass star formation of 1 g cm-2 (Krumholz & McKee, 2008), S255S is at a very early stage of ongoing star formation and may form massive stars. The single dish continuum properties of S255IR and S255N do not show much difference, however, our interferometry and NIR observations show us the different properties of S255IR and S255N. While the large scale NIR emission shows the existence of the NIR cluster in S255IR and not so many NIR point sources in S255N, which indicate S255IR is most likely in a more evolved stage compared to S255N. For the individual sub-cores, more lines are detected at S255IR-SMA1 than S255N-SMA1, which indicates a higher temperature and more evolved nature of S255IR-SMA1 compared to S255N-SMA1. Regarding the kinematic properties, S255N-SMA1 has similar outflow velocities but a much smaller size of the outflow than S255IR-SMA1, which may also suggest a younger evolutionary stage of S255N-SMA1. Among other SMA mm continuum cores, S255IR-SMA2, S255N-SMA2 and S255N-SMA3, which do not have many line emissions are considered to be at earlier evolutionary stages. Figure 1 presents the whole star formation region. The young star formation region S255 complex lies between the evolved H II regions, S255 and S257. From the morphology of the dust structure and the H II regions, it appears that the two H II regions pushed gas between them and formed the S255IR dust structure and triggered the star formation, which has also been suggested by Bieging et al. (2009); Minier et al. (2007). To inquire that, we studied the velocity map of our 30m data, however, due to the high noise level at the edge of the map, we could not find a significant velocity difference between the west and east edge of the dust structure to prove the interaction between the H II regions and the dust structure. Chavarría et al. (2008) estimated the dynamical age of the two H II regions S255 and S257, which is $\sim 1.5\times 10^{6}$ yr, similar to our NIR cluster in S255IR. So we can not prove the triggering star formation. However, we witness outflow interaction between S255IR and S255N (Sec. 3.3). Since S255N is younger compared to S255IR, S255N may be affected by S255IR. Further more, the age of the cluster around S255IR we obtained from the SINFONI data is 2$\pm$1 Myr (Sec. 3.6.4). Chavarría et al. (2008) obtained an age of the larger-scale cluster of 1 Myr which is consistent with our result. However, the dynamical age of the high-mass protostar in this region, S255IR- SMA1, is $\sim 10^{4}$ yr (Table 8). Without the knowledge of the outflow inclination angle, we could systematically underestimate the age value by a factor of 2 to 5 (Cabrit & Bertout, 1992). Therefore, S255IR-SMA1 should not be dynamically older than 105 yr and S255IR-SMA2 is even younger than S255IR- SMA1. The age difference between the NIR cluster and the massive protostellar objects indicates that the most massive sources in the cluster form last. Minier et al. (2007) suggests a collect-and-collapse and triggered star formation scenario for S255IR, which is that the B-stars (from our observations they are late B stars, i.e. star #1, #2, #3 in Figure 24) in S255IR formed through a collect-and-collpase process, and triggered the formation of NIRS 1 and NIRS 3. The NIR and mm data presented in this paper show that the cluster detected in the infrared is likely older than the mm sources detected with our SMA data. While the SMA mm sources lie in the center of the NIR cluster, the NIR cluster members are more distributed around the mm sources. Based on this information, we propose an outside-in star formation scenario, which is that the central gas filament collapses under the pressure of the two H II regions. The collapse of the clump may start outside-in under the pressure of the two H II regions, the low mass cores need lower density to form, and they formed first at the outside region of the clump and collapse to stars. And then either the low- to intermediate- mass stars may enhance the instability of the central high-mass cores and potentially trigger the high- mass star formation in S255IR, or the massive cores could build up slightly slower and start to collapse afterwards forming the most massive stars in this region. Our outflow observations show a hint that the energetic outflow from the YSOs in S255IR may again have affected the star formation in S255N, but this needs further observations to confirm. However, the different evolutionary stages of the various regions and even within each region are quite clear. This suggests a sequential star formation. ### 4.2 Multiple outflows We detected outflows in all three regions (Sec. 3.3). In the top panels of Figure 8, the outflow emission which is nearby the SMA continuum sources does not really follow the NE-SW direction. We suggest this complicated outflow environment in the nearby region of the continuum sources is due to the interaction between outflows from NIRS 1 and NIRS 3. The red-shifted lobe to the south of the continuum source (bottom panels in Figure 8) is most likely driven by NIRS 1, because the north-south bipolar reflection nebular which is associated with NIRS 1 follows this direction to the south (Jiang et al., 2008; Simpson et al., 2009), and the outflow driven by NIRS 3 should be blue- shifted at this direction. In Figure 10, the red-shifted gas at the bottom part of line (b) seems to be associated with S255N-SMA3, however, another blue-shifted feature at the southern part of the map shows up as the resolution changes. Figure 7 shows that these components are mixed together with the outflows in S255IR, which may indicate the interaction between these two regions. In the youngest region S255S, the velocity of the outflows is much smaller compared to the other two regions. Table 2 shows that the column density of this region is still relatively low, S255S is in very young evolutionary stage and likely just start collapsing. This is the likely reason we did not detect very energetic outflows like for the other two regions in this region. ### 4.3 Disk candidates in S255IR NIRS 1, which coincides with S255IR-SMA3, is reported to have a polarization disk (Jiang et al., 2008), Simpson et al. (2009) also reported a disk, however, with a slightly different interpretation. Because NIRS 1 is relatively more evolved, we detected only one unresolved mm continuum source associated with only the CO isotopologue lines (Sec. 3.2). NIRS 3, which coincides with S255IR-SMA1, is considered to be a high-mass protostar based on several signatures (e.g., UCHII region, maser emissions, hot core emissions and energetic outflows, see Sec. 1 and 3). We detected a rotational structure coinciding with this source perpendicular to the outflow (Figure 13). The HCOOCH3 position velocity diagram shows that this rotational structure is not Keplerian, so it could be a rotating toroid around NIRS 3. However, the C18O pv-diagram shows a much larger Keplerian-like velocity structure perpendicular to the outflow (Figure 14). If this source were at a much further distance or observed with worse resolution, this structure could easily be identified as a disk. However, with our resolution we resolved two mm source, and we know it is not a disk. The C18O gas size is $\sim 2\times 10^{4}$ AU and the continuum source has a size of 1.6$\times 10^{4}$ AU at the given distance of 1.59$\times 10^{3}$ pc, which is similar to the rotational structure described in Fallscheer et al. (2009). This structure is also much larger than the jeans length which is $\sim$6 000 AU for a temperature $\sim$20 K and a density $\sim 1\times 10^{6}$. Our system also exceeds the criteria to maintain a stable disk described by Kratter et al. (2010). Therefore, we suggest this structure to be a rotating toroid which fragmented into several sources, and form a multiple system. Similar structures of fragmenting pseudo-disks have also been modeled by Krumholz et al. (2009) and Peters et al. (2010). Following these, we propose a star formation scenario for this large rotational structure, which is that the rotational elongated dust structure formed first, and at the central region the massive source NIRS 1 started forming. However, this large structure is much more massive compared to NIRS 1 and is unstable, then it fragmented into two sources, S255IR-SMA1 and S255IR- SMA2. The SINFONI source #17 shows strong Br$\gamma$ and CO emission in $K$-band (Figure 23), which indicates the existence of a circumstellar disk. Source #17 might also drive the jet like emissions at the northern edge of Figure 19. It is interesting to find disk signatures toward sources with that different evolutionary stages within the same forming cluster. ### 4.4 Cores and clumps Since the single dish has a much larger beam, which can cover all the continuum sources, and 13CO may be optically thick, the observed velocity traces the outer layer of the whole clump and shows the mean line-of-sight velocity. The molecular lines we used to get the velocities for the SMA observations are all dense gas and disk tracers (Cesaroni et al., 2007), so what we obtained are the velocities of the high mass cores (Table 6). The difference between the core velocities and the clump velocities of several km s-1 (Table 6) are very different from the low-mass core cases (e.g. 0.46 km s-1 in NGC 1336 (Walsh et al., 2007), 0.17 km s-1 in Oph A (Di Francesco et al., 2004)). This difference confirms that massive star formation regions are more turbulent than low-mass ones. It further implies stronger peculiar motions of the protostars and cores within the clump/cluster gravitational potential. ## 5 Summary Combining multi-wavelength data from mm to NIR wavelength, we characterize the different (proto) stellar populations within the S255 star formation complex. S255S, S255N and S255IR show different dynamical and chemical properties, not only at mm wavelength but also at infrared wavelengths, which indicates they are in different evolutionary stages. With the SMA, IRAM 30m and VLT-SINFONI observations, we found outflows in all three regions, high velocity collimated ones in S255IR, high velocity more confined ones in S255N and lower velocity confined ones in S255S. The multiple outflows we found in S255IR and S255N suggests a potential interaction between these two regions. From the outflow maps, we estimated the dynamical age of the outflow driving sources. Although without the information of the inclination angle this dynamical age could be underestimated by a factor of 2 to 5 (Cabrit & Bertout, 1992), our mm sources should not be older than 105 yr. We detected 25 molecular lines in S255IR, including 7 lines of the CH3CN(12${}_{k}-11_{k}$) k-ladder with k = 0…6, confirming that the hot core nature of S255IR-SMA1. Only 10 molecular lines are detected in S255N, including 2 CH3CN(12${}_{k}-11_{k}$) k-ladder with k = 0, 1, which is indicative of a younger age and colder temperature. Besides the 3 CO isotopologue line emissions, only diffuse SO emission is detected in S255S. This is consistent with different chemical ages. High-density tracers like CH3CN and HCOOCH3 show rotational structures around the most prominent high-mass protostar candidates in S255IR and S255N. Furthermore in S255IR, the C18O presents an elongated rotational structure with a Keplerian-like velocity gradient perpendicular to the outflow. With a size of $\sim 2\times 10^{4}$ AU, this structure can not be a disk but may be a rotational toroid which fragments into several sources. Near infrared $H$\- and $K$-band integral field spectroscopy observations were done for S255IR. We identified the excitation mechanism of the H2 emission. We derived the spectral type of 16 stars, and 14 of them are considered to be the cluster members. With the knowledge of the spectral type and the SOFI $J$\- and $K$\- band photometry results (Bik, 2004) of the cluster members, we constructed an HR diagram to estimate the age of the cluster, which is 2$\pm$1 Myr . This age is consistent with the result Chavarría et al. (2008) obtained. The age difference between the low-mass cluster and the massive mm cores indicates different stellar populations in the cluster. This also leads to a question, do the massive stars in this cluster form last? Our data support the idea that massive stars form last. We propose the triggered outside-in collapse star formation scenario for the bigger picture and the fragmentation scenario for S255IR The whole picture of the S255 complex suggests triggered star formation, however, but we can not give hard proof at present. However, the different evolutionary stages between each region and different stellar populations in S255IR are consistent with sequential star formation. ###### Acknowledgements. The authors thank C. J. Cyganowski for providing the VLA 3.6 cm data of S255N, F. Eisenhauer for providing the data reduction software SPRED, A. Modigliani for help in the data reduction, R. Davies for providing his routines to remove the OH line residuals and N. Da Rio for providing the isochrones and evolutionary tracks, and M. Fang for the discussion. Y.W. acknowledges support by Purple Mountain Observatory, CAS and Max-Plank-Institute for Astronomy. E. 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1011.3603	# Development in the Scattering Matrix Theory: From Spin-Orbit-Coupling Affected Shot Noise to Quantum Pumping Rui Zhu**Corresponding author. Electronic address: rzhu@scut.edu.cn Department of Physics, South China University of Technology, Guangzhou 510641, People’s Republic of China ###### Abstract The review chapter starts by a pedagogical introduction to the general concept of the scattering theory: from the fundamental wave-function picture to the second-quantization language, with the aim to clear possible ambiguity in conventional textbooks. Recent progress in applying the method to current fluctuations and oscillating-parameter driven quantum pumping processes is presented with inclusion of contributions by Büttiker, Brouwer, Moskalets, Zhu, etc. In particular, the spin-orbit-coupling affected shot noise can be dealt with by taking into account the spin-dependent scattering processes. A large shot noise suppression with the Fano factor below 0.5 observed experimentally can be illustrated by effective repulsion between electrons with antiparallel spin induced by the Dresselhaus spin-orbit coupling effect. A Floquet scattering theory for quantum-mechanical pumping in mesoscopic conductors is developed by Moskalets et al., which gives a general picture of quantum pumping phenomenon, from adiabatic to non-adiabatic and from weak pumping to strong pumping. ###### pacs: 72.10.-d, 73.23.-b, 05.60.Gg, 73.50.Td, 71.70.Ej, 75.60.Ch ###### Contents 1. I Pedagogical introduction to the general concept of the scattering theory 1. I.1 Landauer-Büttiker conductance 2. I.2 Further illustration of the scattering scheme in a toy system with correlated reservoirs 3. I.3 Time-dependent scattering-matrix theory 2. II Spin-orbit coupling affected shot noise 1. II.1 Background 2. II.2 Theoretical approach 3. II.3 Numerical results in comparison with experiment 3. III Quantum pumping beyond linear response 1. III.1 Introduction to quantum pumping 2. III.2 Theoretical formulation 3. III.3 Numerical results and interpretations 4. IV Summary and future directions 5. V Acknowledgements ## I Pedagogical introduction to the general concept of the scattering theory ### I.1 Landauer-Büttiker conductance The discussion is based on the scattering approach to electrical conductance. This approach, as we will show, is conceptually simple and transparent. Nevertheless, the generality of the scattering approach and its conceptual clarity, make it the desired starting point of a discussion of noise in electrical conductors. By expanding the time-dependent scattering matrix into Fourier series, a description of the quantum pumping phenomenon can be given. We start with the wave function picture and consider an electron tunneling through a one-dimensional single barrierRef29 , which can be realized in a semiconductor heterostructure with a layer of ${\rm{Al}}_{x}{\rm{Ga}}_{1-x}{\rm{As}}$ of width $2L$ imbedded in $\rm{GaAs}$ as shown in Fig. 1. In the effective mass approximation, the electron motion in each layer of the structure is described by the stationary solution of the envelop equation in the $x$-direction. $\left[{\frac{{-\hbar^{2}}}{2}\frac{\partial}{{\partial x}}\frac{1}{{m^{}\left(x\right)}}\frac{\partial}{{\partial x}}+V_{eff}\left(x\right)}\right]\psi\left(x\right)=E\psi\left(x\right).$ (1) Here, $m^{}$ and $V_{eff}$ are the effective mass and potential in different regions with $E$ the energy of the transporting electron. The electron’s wave functions are expressible as $\Psi\left({x,t}\right)=\psi\left(x\right)e^{-iEt/\hbar}=\left\\{\begin{array}[]{l}\begin{array}[]{{20}c}{\left({Ae^{ikx}+Be^{-ikx}}\right)e^{-iEt/\hbar},}&{x\leq-L,}\\\ \end{array}\\\ \begin{array}[]{{20}c}{\left({Ce^{\kappa x}+De^{-\kappa x}}\right)e^{-iEt/\hbar},}&{-L\leq x\leq L}\\\ \end{array},\\\ \begin{array}[]{{20}c}{\left({Ee^{ikx}+Fe^{-ikx}}\right)e^{-iEt/\hbar},}&{x\geq L,}\\\ \end{array}\\\ \end{array}\right.$ (2) with $k={{\sqrt{2m^{}E}}\mathord{\left/{\vphantom{{\sqrt{2m^{}E}}\hbar}}\right.\kern-1.2pt}\hbar}$ and $\kappa={{\sqrt{2m^{}\left({V_{0}-E}\right)}}\mathord{\left/{\vphantom{{\sqrt{2m^{}\left({V_{0}-E}\right)}}\hbar}}\right.\kern-1.2pt}\hbar}$. The coefficients $A$ and $B$ are associated respectively with incoming and outgoing waves on the left side relative to the barrier. Likewise, the coefficients $E$ and $F$ are respectively outgoing and incoming waves on the right. The scattering matrix connects the incoming and outgoing fluxes as $\left[{\begin{array}[]{{20}c}B\\\ E\\\ \end{array}}\right]=\left[{\begin{array}[]{{20}c}{S_{11}}&{S_{12}}\\\ {S_{21}}&{S_{22}}\\\ \end{array}}\right]\left[{\begin{array}[]{{20}c}A\\\ F\\\ \end{array}}\right].$ (3) Ideal (i.e., without scattering) conducting leads connect the scattering region to reservoirs on the left and right characterized by quasi-Fermi energies $\mu_{1}$ and $\mu_{2}$, respectively, corresponding to the electron densities there. These reservoirs or contacts randomize the phase of the injected and absorbed electrons through inelastic processes such that there is no phase relation between particles. For such an ideal 1D system, the current injected from the left and right may be written as an integral over the fluxRef29 ; Ref33 $I_{L}=\frac{{2e}}{{2\pi}}\left[{\int_{0}^{\infty}{dkv\left(k\right)f_{1}\left(k\right)T\left(E\right)}-\int_{0}^{\infty}{dk^{\prime}v\left({k^{\prime}}\right)f_{2}\left({k^{\prime}}\right)T\left({E^{\prime}}\right)}}\right],$ (4) where $v(k)$ is the velocity, $T(E)$ is the transmission coefficient, which can be obtained from the wave function Eq. (2), and $f_{1}$ and $f_{2}$ are the reservoir distribution functions characterized by $\mu_{1}$ and $\mu_{2}$, respectively. The integrations are only over positive $k$ and $k^{\prime}$ relative to the direction of the injected charge as positive $k$ is in $+x$-direction and positive $k^{\prime}$ is in $-x$-direction. If we now assume low temperatures, electrons are injected up to an energy $\mu_{1}$ from the left lead and injected up to $\mu_{2}$ from the right one. Converting to integrals over energy, the current becomes $\begin{array}[]{l}I_{L}=\frac{{2e}}{{2\pi}}\left[{\int_{0}^{\mu_{1}}{dE\left({\frac{{dk}}{{dE}}}\right)v\left(k\right)T\left(E\right)}-\int_{0}^{\mu_{2}}{dE\left({\frac{{dk^{\prime}}}{{dE}}}\right)v\left({k^{\prime}}\right)T\left(E\right)}}\right]\\\ =\frac{{2e}}{{2\pi\hbar}}\int_{\mu_{2}}^{\mu_{1}}{dET\left(E\right)}.\\\ \end{array}$ (5) It can be seen that the first term of Eq. (5) is the flux generated by electrons injected from the left reservoir and the second term is that from the right reservoir. The integration is done in two independent ensembles. The Landauer-Büttiker conductance introduced above can be reproduced in the second-quantization language. Without loss of generality, we assume the input amplitudes from the two reservoirs in Eq. (2) to be unity. For simplicity, as a single channel is considered, the scattering matrix can be described in the relation $\hat{b}_{\alpha}=S_{\alpha\beta}\hat{a}_{\beta}$ (6) or elaborately $\left[{\begin{array}[]{{20}c}{\hat{b}_{L}}\\\ {\hat{b}_{R}}\\\ \end{array}}\right]=\left[{\begin{array}[]{{20}c}{S_{11}}&{S_{12}}\\\ {S_{21}}&{S_{22}}\\\ \end{array}}\right]\left[{\begin{array}[]{{20}c}{\hat{a}_{L}}\\\ {\hat{a}_{R}}\\\ \end{array}}\right]=\left[{\begin{array}[]{{20}c}r&{t^{\prime}}\\\ t&{r^{\prime}}\\\ \end{array}}\right]\left[{\begin{array}[]{{20}c}{\hat{a}_{L}}\\\ {\hat{a}_{R}}\\\ \end{array}}\right],$ (7) where operators $a_{L/R}$ annihilate electrons incident upon the sample from the left/right reservoir and operators $b_{L/R}$ describe electrons in the outgoing states. $t$ and $t^{\prime}$ are the transmission amplitudes of electrons incident rightward and rightward, respectively, and $r$ and $r^{\prime}$ are the corresponding reflection amplitudes, as defined conventionally. Hence, the Landauer-Büttiker formula of the current can be expressed as $I_{L}=\frac{e}{{2\pi\hbar}}\int{dE\left\langle{\left[{a_{L}^{\dagger}\left(E\right)a_{L}\left(E\right)-b_{L}^{\dagger}\left(E\right)b_{L}\left(E\right)}\right]}\right\rangle},$ (8) where $\left\langle\cdots\right\rangle$ calculates the quantum statistical average of the product of an electron creation operator and annihilation operator of a Fermi gas. As a conventional electron reservoir is considered, we have $\begin{array}[]{l}\left\langle{a_{L}^{\dagger}\left(E\right)a_{L}\left({E^{\prime}}\right)}\right\rangle=f_{L}\left(E\right)\delta\left({E-E^{\prime}}\right),\\\ \left\langle{a_{R}^{\dagger}\left(E\right)a_{R}\left({E^{\prime}}\right)}\right\rangle=f_{R}\left(E\right)\delta\left({E-E^{\prime}}\right),\\\ \left\langle{a_{L}^{\dagger}\left(E\right)a_{R}\left({E^{\prime}}\right)}\right\rangle=\left\langle{a_{R}^{\dagger}\left(E\right)a_{L}\left({E^{\prime}}\right)}\right\rangle=0.\\\ \end{array}$ (9) It should be noted from the third formula of Eq. (9) that electrons incident from the left and right reservoirs are completely incoherent and the phases of the two reservoirs are randomized and completely unrelated. Substituting Eq. (9) to Eq. (8), the formula of the current expressed in Eq. (5) can be reproduced. It is interesting to see that if we could build up a system with the conductor connected to two correlated reservoirs, in which the quantum statistical average of the cross product reads $\begin{array}[]{l}\left\langle{a_{L}^{\dagger}\left(E\right)a_{R}\left({E^{\prime}}\right)}\right\rangle=\left\langle{a_{R}^{\dagger}\left(E\right)a_{L}\left({E^{\prime}}\right)}\right\rangle=f_{3}\left(E\right)\delta\left({E-E^{\prime}}\right),\\\ f_{3}\left(E\right)=\left\\{\begin{array}[]{l}1,\begin{array}[]{{20}c}{}\hfil&{\mu<\min\left({\mu_{1},\mu_{2}}\right),}\\\ \end{array}\\\ 0,\begin{array}[]{{20}c}{}\hfil&{{\rm{others}},}\\\ \end{array}\\\ \end{array}\right.\\\ \end{array}$ (10) the system can generate a current demonstrating the interference between the electron states incident from the left reservoir and the right one. $I_{L}=-\frac{e}{{2\pi\hbar}}\int_{0}^{\mu_{1}}{dE}\left[{\int_{0}^{\mu_{1}}{\left\|r\right\|^{2}dE}+\int_{0}^{\mu_{2}}{\left\|t\right\|^{2}dE}+2\int_{0}^{\mu\left({\min}\right)}{{\mathop{\rm Re}\nolimits}\left({r^{}t^{\prime}}\right)dE}}\right].$ (11) In the next subsection, we would use a toy system based on correlated reservoirs to further illustrate the scattering scheme. Significant difference between uncorrelated and correlated reservoirs is illuminated. It is also elaborated that in scattering problems the transmission and reflection process demonstrates the quantum state spanned throughout the space. ### I.2 Further illustration of the scattering scheme in a toy system with correlated reservoirs We consider incidence only from the left in Eq. (2) and the input flux is assumed to be unity. The wave function in different scattering regions can be expressed as $\psi\left(x\right)=\left\\{\begin{array}[]{l}\begin{array}[]{{20}c}{e^{ikx}+re^{-ikx},}&{x\leq-L,}\\\ \end{array}\\\ \begin{array}[]{{20}c}{Ce^{\kappa x}+De^{-\kappa x},}&{-L\leq x\leq L}\\\ \end{array},\\\ \begin{array}[]{{20}c}{te^{ikx},}&{x\geq L.}\\\ \end{array}\\\ \end{array}\right.$ (12) An incident particle is transmitted with probability $\left\|t\right\|^{2}$ and reflected with probability $\left\|r\right\|^{2}$. It is determined by the wave function that the particle momentum has value of $\hbar k$ with probability ${{\left({1+\left\|t\right\|^{2}}\right)}\mathord{\left/{\vphantom{{\left({1+\left\|t\right\|^{2}}\right)}2}}\right.\kern-1.2pt}2}$ and $-\hbar k$ with probability ${{\left\|r\right\|^{2}}\mathord{\left/{\vphantom{{\left\|r\right\|^{2}}2}}\right.\kern-1.2pt}2}$. The mean value of the momentum $\left\|t\right\|^{2}\hbar k$ characterizes the density of the probability current $\bf{J}$, which can also be obtained from the continuity equation ${{\partial\rho}\mathord{\left/{\vphantom{{\partial\rho}{\partial{\rm{t}}}}}\right.\kern-1.2pt}{\partial{\rm{t}}}}+\nabla\cdot{\bf{J}}=0$ with $\rho$ the probability density. An interesting interference pattern can be observed if we consider the incidence from the left and the right is correlated. We consider a toy system of a one-dimensional electron gas subject to two oscillating gate volatages (see Fig. 2). The inspiration comes from the quantum pumping phenomenon. The single-particle Hamiltonian reads $H=-\frac{\hbar}{{2m^{}}}\frac{{\partial^{2}}}{{\partial x^{2}}}+U\left({x,t}\right),$ (13) with $U\left({x,t}\right)=\Theta\left({x+2L}\right)\Theta\left({-L-x}\right)U_{1}\left(t\right)+\Theta\left({x-L}\right)\Theta\left({2L-x}\right)U_{2}\left(t\right).$ The two barriers $U_{1}$ and $U_{2}$ are adiabatically modulated at the frequency $\omega$ with a phase difference $\phi$. $\begin{array}[]{l}U_{1}\left(t\right)=U_{10}+U_{1\omega}\sin\omega t,\\\ U_{2}\left(t\right)=U_{20}+U_{2\omega}\sin\left({\omega t-\phi}\right).\\\ \end{array}$ (14) We assume $\omega$ is extremely small so that a static treatment is valid. Therefore, it is tolerable to consider only the zero order of the Fourier component of the time-dependent scattering matrix without taking into account the photon-absorption/emmission processes, which is done in adiabatic quantum pumping theory. As shown in Fig. 2, the electrons are incident from the left and right reservoirs with identical amplitudes at zero bias, which is set to be unity without impairing generality. As assumed, incidence from the two correlated reservoirs characterize a coherent single-particle state. The single-particle wave function at a certain time has the following form. $\Psi\left({x,t}\right)=\left\\{\begin{array}[]{l}\begin{array}[]{{20}c}{\left[{e^{ikx}+\left({r+t^{\prime}e^{i\theta}}\right)e^{-ikx}}\right]e^{{{-iEt}\mathord{\left/{\vphantom{{-iEt}\hbar}}\right.\kern-1.2pt}\hbar}},}&{x\leq-2L,}\\\ \end{array}\\\ \begin{array}[]{{20}c}{\left[{A_{2}e^{\kappa_{2}x}+B_{2}e^{-\kappa_{2}x}}\right]e^{{{-iEt}\mathord{\left/{\vphantom{{-iEt}\hbar}}\right.\kern-1.2pt}\hbar}},}&{-2L\leq x\leq-L}\\\ \end{array},\\\ \begin{array}[]{{20}c}{\left[{A_{3}e^{ikx}+B_{3}e^{-ikx}}\right]e^{{{-iEt}\mathord{\left/{\vphantom{{-iEt}\hbar}}\right.\kern-1.2pt}\hbar}},}&{-L\leq x\leq L}\\\ \end{array},\\\ \begin{array}[]{{20}c}{\left[{A_{4}e^{\kappa_{4}x}+B_{4}e^{-\kappa_{4}x}}\right]e^{{{-iEt}\mathord{\left/{\vphantom{{-iEt}\hbar}}\right.\kern-1.2pt}\hbar}},}&{L\leq x\leq 2L}\\\ \end{array},\\\ \begin{array}[]{{20}c}{\left[{\left({t+r^{\prime}e^{i\theta}}\right)e^{ikx}+e^{-ikx}}\right]e^{{{-iEt}\mathord{\left/{\vphantom{{-iEt}\hbar}}\right.\kern-1.2pt}\hbar}},}&{x\geq 2L.}\\\ \end{array}\\\ \end{array}\right.$ (15) $k={{\sqrt{2m^{}E}}\mathord{\left/{\vphantom{{\sqrt{2m^{}E}}\hbar}}\right.\kern-1.2pt}\hbar}$ and $\kappa_{2/4}={{\sqrt{2m^{}\left({U_{1/2}-E}\right)}}\mathord{\left/{\vphantom{{\sqrt{2m^{}\left({U_{1/2}-E}\right)}}\hbar}}\right.\kern-1.2pt}\hbar}$. $t$ and $r$ quantify the transmission and reflection amplitudes of the electrons incident from the left reservoir while $t^{\prime}$ and $r^{\prime}$ quantify those incident from the right with $t^{\prime}=t$ and $r^{\prime}={{-tr^{}}\mathord{\left/{\vphantom{{-tr^{}}{t^{}}}}\right.\kern-1.2pt}{t^{}}}$ . We introduce a geometric phase $\theta$ to describe the unavoidable phase difference between the electrons injected from the two reservoirs. In the toy approach, correlated reservoirs are assumed, which justifies a particular value of $\theta$. For simplicity and without violation of the physics we take $\theta=0$. It is noted that in conventional real reservoirs mixed-ensemble integral should be applied, i.e., the probability flow of the electron incident from the left reservoir absolutely cancels out that of the incidence backward at zero bias as a result of the randomized phase distribution, which absolutely differs from our toy consideration. From the continuity equation $\frac{{\partial\rho}}{{\partial t}}+\nabla\cdot{\bf{J}}=0,$ (16) we could derive the probability current flow as functions of the transmission and reflection amplitudes as $j_{x}=-\frac{{i\hbar}}{{2m}}\left[{\psi^{\dagger}\frac{\partial}{{\partial x}}\psi-\left({\frac{\partial}{{\partial x}}\psi^{\dagger}}\right)\psi}\right]=-\frac{{4\hbar k}}{m}{\mathop{\rm Re}\nolimits}\left({r^{}t^{\prime}}\right).$ (17) We can also see from the wave function Eq. (15) that the particle momentum has value of $\hbar k$ with probability ${{\left[{1-{\mathop{\rm Re}\nolimits}\left({tr^{}}\right)}\right]}\mathord{\left/{\vphantom{{\left[{1-{\mathop{\rm Re}\nolimits}\left({tr^{}}\right)}\right]}2}}\right.\kern-1.2pt}2}$ and $-\hbar k$ with probability ${{\left[{1+{\mathop{\rm Re}\nolimits}\left({tr^{}}\right)}\right]}\mathord{\left/{\vphantom{{\left[{1+{\mathop{\rm Re}\nolimits}\left({tr^{}}\right)}\right]}2}}\right.\kern-1.2pt}2}$. The mean value of the momentum $-{\mathop{\rm Re}\nolimits}\left({tr^{}}\right)\hbar k$ characterizes the density of the probability current of Eq. (17). The net current density can be described by the period-average of the probability current density multiplied by the carrier charge and density. The accumulated contribution by electrons within the $\pm\hbar\omega$ sidebands is taken into account by an integral. Without dynamic modulation, no current occurs as the $\pm\hbar\omega$ energy channel is closed even when the probability current density is nonzero for asymmetric barrier configuration. The current density as a function of the Fermi level thus becomes $I_{L}\left(E_{F}\right)=-\frac{{4\hbar}}{{m^{}}}\int_{E_{F}-\hbar\omega}^{E_{F}+\hbar\omega}{\frac{{ek\rho_{e}N\left(E\right)f\left(E\right)}}{{\left({{{2\pi}\mathord{\left/{\vphantom{{2\pi}\omega}}\right.\kern-1.2pt}\omega}}\right)}}\int_{0}^{\frac{{2\pi}}{\omega}}{{\mathop{\rm Re}\nolimits}\left[{r^{}\left(t\right)t^{\prime}\left(t\right)}\right]dt}dE},$ (18) where the density of states of a one-dimensional electron gas is $N\left(E\right)=\frac{V}{{\pi\hbar}}\sqrt{\frac{{m_{e}}}{{2E}}}.$ (19) Here, $k$ is the wave vector of the electron. $e$ is the electron charge. $\rho_{e}$ is the carrier density of the two-dimensional electron gas in which the quantum wire is confined. $f(E)$ is the Fermi-Dirac distribution function of the leads. $V$ quantifies the volume of the one-dimensional wire and $m_{e}$ is the mass of a free electron. Here, in the toy configuration, quantum interference is remarkably demonstrated. The single-electron state Eq. (15) interferes with itself and carries the probability flow and hence the net current through oscillating cycles. We numerically calculated the current in a one-dimensional electron gas (2DEG) based on the GaAs/AlGaAs heterostructures with the average carrier density $\rho_{e}\sim 10^{11}$ cm-2 and the effective mass of the electron $m^{}\sim 0.067m_{e}$Ref73 . The width of the two gate potential barriers $L=20$ Å equally separated by a $2L=40$ Å width well. The amplitudes of the modulations $U_{1\omega}=U_{2\omega}=1$ meV. All of the above setups are not essential as we are dealing with an assumed toy structure. In Fig. 3, it is shown that the time-integrated current demonstrates a sinusoidal pattern as a function of the phase difference between two oscillating parameters, which is a result of quantum interference. The probability density flow formulated in Eq. (17) is a result of phase difference between transmission forward and backward. Quantum phase interference gives rise to nonzero probability density flow for asymmetric barrier configurations. The absolute strength of the probability flow increases as the height difference between the two potential barriers increases determined by the phase difference $\phi$. Fig. 4 presented the time variation of the net current within an oscillating cycle. Considering the time-averaged effect, the integrated asymmetry is different. When the phase difference between the two modulations approaches $\pi$, time-reversal symmetry destroys the time-integrated current to zero although the probability density flow maximizes at a certain time. i.e. the probability flow in half a pumping cycle completely offsets that of the other. Therefore a sinusoidal dependence on $\phi$ occurs in the time-integrated current. In real reservoirs, the single-particle state between reservoirs has a definite momentum direction determined by its source. When coherent reservoirs can be realized in any form, however, quantum states within the mesoscopic conductor can be expressed as Eq. (15) and double-slit interference pattern is observable in an electron device. ### I.3 Time-dependent scattering-matrix theory The scattering-matrix equation $\hat{b}_{\alpha}=S_{\alpha\beta}\hat{a}_{\beta}$ with $\alpha$ and $\beta$ indexes of lead, channel, and spin introduced in Sec. I.A. for the Landauer- Büttiker conductance characterizes the transport properties through a conductor at a certain bias. In a more general situation with dynamic processes, e.g. in quantum pumping, a time-dependent scattering matrix can be introduced as follows. $\hat{b}_{\alpha}\left(t\right)=\int_{-\infty}^{\infty}{S_{\alpha\beta}\left({t,t^{\prime}}\right)\hat{a}_{\beta}\left({t^{\prime}}\right)dt^{\prime}},\begin{array}[]{{20}c}{}\hfil&{t\geq t^{\prime},}\\\ \end{array}$ (20) with $\alpha$ and $\beta$ general indexes denoting the lead, channel, and spin. An incident state $\hat{a}_{\beta}$ at time $t^{\prime}$ is scattered into the outgoing state $\hat{b}_{\alpha}$ at time $t$ with the amplitude $S_{\alpha\beta}\left({t,t^{\prime}}\right)$. The time-dependent scattering-matrix picture described by Eq. (20) is exactly equivalent to the time-dependent Schrödinger equation with the elements of the scatting matrix amplitudes of the wave function. In usual cases, the time- dependent Schrödinger equation cannot be solved exactly, similarly to the time-dependent scattering matrix. In the static or adiabatic cases, it is advantageous to use an analog of the Wigner transform for the matrix $S_{\alpha\beta}\left({t,t^{\prime}}\right)$trs , $S_{\alpha\beta}\left({E,t}\right)=\int_{-\infty}^{\infty}{e^{iE\left({t-t^{\prime}}\right)}S_{\alpha\beta}\left({t,t^{\prime}}\right)dt^{\prime}}.$ (21) An on-time scattering process $S_{\alpha\beta}\left({t,t^{\prime}}\right)$ with $t=t^{\prime}$ is sufficient to describe the bias-driven conductance. Namely, from Eq. (21), we can use $S_{\alpha\beta}\left({E}\right)$ with $E$ labeling the energy channel to fully capture the transport physics. When the scattering time $t-t^{\prime}$ is small (i.e., the dynamic characteristic frequency is much smaller than the inverse Wigner time delay), the dynamics can be approximated into the instant-scattering picture. Physically this means that the scattering matrix changes only a little while an electron is scattered by the mesoscopic sample under dynamic modulation, in which we use the term “adiabatic”. In adiabatic dynamics, we can use low-order Fourier components of $S_{\alpha\beta}\left({E,t}\right)$ to characterize transport physics. Therefore, small $t-t^{\prime}$ is transformed into variation of the particle energy by side-band broadening around the Fermi level. In Sec. III, we would illustrate the time-dependent scattering approach in adiabatic quantum pumping beyond the linear-response approximation. ## II Spin-orbit coupling affected shot noise ### II.1 Background Current fluctuations are present in almost all kinds of conductors and have been developed into a very active and fascinating subfield of mesoscopic physics (for review see Ref. Ref20, ). At low temperatures, thermal fluctuations are extremely small, the current fluctuation properties are governed by the so-called shot noise, which is a consequence of the quantization of charge. Shot noise is useful to obtain information on a system which is not available through conductance measurements. In particular, shot noise experiments can determine the quantum correlation of electrons, the charge and statistics of the quasi-particles relevant for transport, and reveal information on the potential profile as well as internal energy scales of mesoscopic systemsRef34 ; Ref35 ; Ref36 ; Ref37 ; Ref38 ; Ref39 ; Ref40 ; Ref41 ; Ref42 ; Ref43 ; Ref44 ; Ref45 ; Ref22 ; Ref60 . Shot noise is generally more sensitive to the effects of electron-electron interactions than the average conductance. A convenient measure of shot noise is the Fano factor $F$, which is the ratio of the actual shot noise and the Poisson noise. The Poisson noise would be achieved in measurement if the transport is carried by single independent electrons. Four typical values of the Fano factor characterize the shot noise properties of different mesoscopic conductors. 1\. $F=1$ characterizes Poissonian processes. Particles are completely independent during transport corresponding to channels through which transmission is exponentially small. In diffusive transport, they are the so- called approximately-closed channels. Typical conductors featuring $F=1$ include tunneling junction, Schottky-barrier diode, and asymmetric double- barrier diode. 2\. $F=0$ characterizes ballistic transport. In ballistic transport, transmission approaches the maximum of unity. Free particles wave function extends throughout the space, i.e. particle beams exhibit full coherence. In diffusive transport, they are the so-called open channels. Typical conductors featuring $F=0$ include pure metal and free two-dimensional electron gas. 3\. $F=1/3$ characterizes diffusive transport. Open channels and closed channels are distributed randomly. As a result of ensemble average of channels, the strength shot noise falls between $F=1$ and $F=0$. Typical conductors featuring $F=1/3$ include diffusive metals, graphene, and two- dimensional electron gas modulated by magnetic barriers. 4\. $F=1/2$ characterizes ballistic transport constrained by Pauli principle. The Pauli principle forbids two electrons to be in the same channel simultaneously. As a result, the shot noise is suppressed. All kinds of conductors are subject to the Pauli principle. In the symmetric-double-barrier diode, where Pauli exclusion is the only correlation between particles, the shot noise features $F=1/2$. With increased attentionRef46 ; Ref47 ; Ref48 ; Ref49 ; Ref50 ; Ref51 to semiconductor spintronics, materials such as ${\rm{GaSb}}$, ${\rm{InAs}}$, and ${\rm{InSb}}$ with considerably strong spin-orbit coupling (SOC) constantRef51 ; Ref52 are becoming widely used in mesoscopic conductors, observationsRef34 ; Ref37 ; Ref38 beyond the formalism of earlier theory on current shot noise are reported. The scattering approach is developed to derive a general formula for the shot noise in the presence of the SOC effect and apply it to the double-barrier resonant diode (DBRD) systems. It is demonstrated that the microscopic origin of the super-suppression of the shot noise observed in experimentRef34 ; Ref37 ; Ref38 is the bunching interaction between electrons with opposite spins resulting from the Dresselhaus $k^{3}$ termsRef53 ; Ref54 in the effective Hamiltonian of the bulk semiconductor of the barriers. ### II.2 Theoretical approach In this part, the effect of the Dresselhaus SOC to the shot noise properties in the DBRD structure connected to ferromagnetic or normal metal leads is considered. The theory can be generalized to other coherent mesoscopic conductors subject to Dresselhaus and/or Rashba SOC. The Dresselhaus SOC is caused by the bulk inversion asymmetry and exists broadly in III-V compound semiconductors with zinc-blende crystal structuresRef53 . Consider the transmission of electrons with identical wave vector ${\bf{k}}=({\bf{k}}_{\parallel},k_{z})$ through a certain potential barrier grown along $z\parallel[001]$ direction. ${\bf{k}}_{\parallel}$ is the wave vector in the plane of the barrier and $k_{z}$ is the wave vector component normal to the barrier. The electron Hamiltonian of the barrier in the effective-mass approximation contains the spin-dependent $k^{3}$ Dresselhaus term $\hat{H}_{D}=\gamma(\hat{\sigma}_{x}k_{x}-\hat{\sigma}_{y}k_{y})\frac{{\partial^{2}}}{{\partial z^{2}}},$ (22) where $\gamma$ is the material constant denoting the strength of the Dresselhaus SOC, $\hat{\sigma}_{x}$ and $\hat{\sigma}_{y}$ are the Pauli matrices. The Dresselhaus term can be diagonalized by the spinors $\chi_{\pm}=\frac{1}{{\sqrt{2}}}\left({\begin{array}[]{{20}c}1\\\ {\mp e^{-i\varphi}}\\\ \end{array}}\right),$ (23) which describe the spin-up (“$+$”) and spin-down (“$-$”) electron eigenstates. Suppose the system is a layered mesoscopic conductor with its potential profile described by $V_{0}(z)$ (see Fig. 5), the electron motion in each layer of the structure is described by the Hamiltonian $\hat{H}=-\frac{{\hbar^{2}}}{{2m^{}}}\nabla^{2}+V(z)+\hat{H}_{D},$ (24) where $V(z)=V_{0}(z)-eF(z+b)\Theta(z+b)\Theta(a+c-z)$ with $F$ the magnitude of the electric field, $\Theta(z)$ the step function, and $-b$ and $a+c$ the longitudinal coordinates of surfaces in $z$ direction. Our discussion is within the framework of single electron approximation and coherent tunnelingRef55 ; Ref56 , and only zero-frequency noise at zero temperature is considered. Under the assumption that $k_{\parallel}$ is conserved during the tunneling, the wave functions for the electrons with definite longitudinal electron energy ($E_{z}$) can be obtained from Schrödinger equation based on the Hamiltonian given in Eq. (24), which can be diagonalized by spinors $\chi_{\pm}$. So, the wave functions become $\psi(\mathord{\buildrel{\lower 3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{r}})=\left\\{{\begin{array}[]{{20}c}{\exp(i{\bf{k}}_{\parallel}\cdot{\bf{\rho}})\left({\sum\limits_{j=\pm}{\sqrt{\frac{{m_{1}}}{{\hbar k_{1j}}}}\exp(ik_{1j}z)\chi_{j}}+\sum\limits_{j=\pm}{r_{j}\sqrt{\frac{{m_{1}}}{{\hbar k_{1j}}}}\exp(-ik_{1j}z)\chi_{j}}}\right),}&{z<b,}\\\ {\exp(i{\bf{k}}_{\parallel}\cdot{\bf{\rho}})\sum\limits_{j=\pm}{\Phi_{\xi j}(z)},}&{-b\leq z<a+c,}\\\ {\exp(i{\bf{k}}_{\parallel}\cdot{\bf{\rho}})\sum\limits_{j=\pm}{t_{j}\sqrt{\frac{{m_{5}}}{{\hbar k_{5j}}}}\exp(ik_{5j}z)\chi_{j}},}&{z\geq a+c,}\\\ \end{array}}\right.$ (25) where $\Phi_{\xi j}$ denotes the wave function in the conductor region, and $t_{j}$ and $r_{j}$ are the transmission and reflection amplitudes, which can be calculated using the transfer-matrix methodRef57 . The spin “$+$” and spin “$-$” components of the electron wave functions transport separately without correlation. As standard scattering method is applied, we introduce creation and annihilation operators of electrons in the scattering states. Schematics of the scattering states are shown in Fig. 6. Operators $\hat{a}_{Ln\sigma}^{\dagger}(E)$ and $\hat{a}_{Ln\sigma}(E)$ create and annihilate electrons with total energy $E$ and spin polarization $\sigma$ in the transverse channel $n$ in the left lead, which are incident upon the sample. In the same way, the creation $\hat{b}_{Ln\sigma}^{\dagger}(E)$ and annihilation $\hat{b}_{Ln\sigma}(E)$ operators describe electrons in the outgoing states. They obey anti-commutation relations. Therefore, we can write the scattering matrix of the sample as $\begin{array}[]{c}\left({\begin{array}[]{{20}c}{\hat{b}_{Ln\uparrow}}\\\ {\hat{b}_{Ln\downarrow}}\\\ {\hat{b}_{Rn\uparrow}}\\\ {\hat{b}_{Rn\downarrow}}\\\ \end{array}}\right)=\frac{1}{2}\underbrace{\left({\begin{array}[]{{20}c}1&1&0&0\\\ {e^{-i\varphi}}&{-e^{-i\varphi}}&0&0\\\ 0&0&1&1\\\ 0&0&{e^{-i\varphi}}&{-e^{-i\varphi}}\\\ \end{array}}\right)}_{M_{1}}\times\underbrace{\left({\begin{array}[]{{20}c}{r_{+}}&0&{t_{+}^{\prime}}&0\\\ 0&{r_{-}}&0&{t_{-}^{\prime}}\\\ {t_{+}}&0&{r_{+}^{\prime}}&0\\\ 0&{t_{-}}&0&{r_{-}^{\prime}}\\\ \end{array}}\right)}_{s}\\\ \times\underbrace{\left({\begin{array}[]{{20}c}1&{-e^{i\varphi}}&0&0\\\ 1&{e^{i\varphi}}&0&0\\\ 0&0&1&{-e^{i\varphi}}\\\ 0&0&1&{e^{i\varphi}}\\\ \end{array}}\right)}_{M_{2}}\times\left({\begin{array}[]{{20}c}{\hat{a}_{Ln\uparrow}}\\\ {\hat{a}_{Ln\downarrow}}\\\ {\hat{a}_{Rn\uparrow}}\\\ {\hat{a}_{Rn\downarrow}}\\\ \end{array}}\right),\\\ \end{array}$ (26) with $t^{\prime}=t,r^{\prime}=-\frac{t}{{t^{}}}r^{}$, $T_{j}(E)=\left\|{t_{j}}\right\|^{2},R_{j}(E)=\left\|{r_{j}}\right\|^{2}$, and $j=\pm$. The matrices $M_{1}$ and $M_{2}$ are unitary transformations between spin “$\uparrow\downarrow$” states and spin “$\pm$” states, $s$ is the scattering matrix connecting the incoming and outgoing spin “$\pm$” states of the $n$th channel. The current of the system can be derived as follows $\hat{I}_{L}(t)=\frac{e}{{2\pi\hbar}}\sum\limits_{\alpha\beta}{\sum\limits_{mn}{\sum\limits_{\sigma^{\prime}\sigma^{\prime\prime}}{\sum\limits_{\sigma}{\int{dE}\int{dE^{\prime}}e^{i(E-E^{\prime})t/\hbar}\hat{a}_{\alpha m\sigma^{\prime}}^{\dagger}}}}}(E)A_{\alpha,\beta,\sigma^{\prime},\sigma^{\prime\prime}}^{m,n,\sigma}(L;E,E^{\prime})\hat{a}_{\beta n\sigma^{\prime\prime}}(E^{\prime}),$ (27) where $\begin{array}[]{l}A_{\alpha,\beta,\sigma^{\prime},\sigma^{\prime\prime}}^{m,n,\sigma}(L;E,E^{\prime})=\\\ \delta_{\alpha L}\delta_{mm}\delta_{\beta L}\delta_{\sigma^{\prime}\sigma}\delta_{\sigma^{\prime\prime}\sigma}-\sum\limits_{\xi^{\prime}\xi^{\prime\prime}}{\sum\limits_{k}{M_{2\xi^{\prime\prime}\sigma^{\prime}}^{\dagger}s_{L,\alpha;m,k;\xi^{\prime\prime}}^{\dagger}}(E)M_{1\sigma\xi^{\prime\prime}}^{\dagger}M_{1\sigma\xi^{\prime}}s_{L,\beta;k,n;\xi^{\prime}}(E^{\prime})M_{2\xi^{\prime}\sigma^{\prime\prime}}}.\\\ \end{array}$ (28) For a system at thermal equilibrium, the quantum statistical average of the product of an electron creation operator and annihilation operator of a Fermi gas with spin polarization is $\begin{array}[]{{20}c}{\left\langle{a_{Ln\uparrow}^{\dagger}a_{Ln\uparrow}}\right\rangle=f_{Lp},}&{\left\langle{a_{Ln\downarrow}^{\dagger}a_{Ln\downarrow}}\right\rangle=f_{La},}&{\left\langle{a_{Ln\uparrow}^{\dagger}a_{Ln\downarrow}}\right\rangle=\left\langle{a_{Ln\downarrow}^{\dagger}a_{Ln\uparrow}}\right\rangle=\frac{1}{2}(f_{Lp}-f_{La}).}\\\ \end{array}$ (29) Without loss of generality, we set the unit vector directed along the spin orientation ${\bf{n}}_{s}=(1,0,0)$. Making use of Eqs. (24)-(26), after some algebra, we can obtain the expression for the zero-frequency noise power $\begin{array}[]{l}S_{\alpha\beta}\equiv S_{\alpha\beta}(0)=\frac{{e^{2}}}{{4\pi\hbar}}\sum\limits_{\delta\delta_{2}}{\sum\limits_{mn}{\sum\limits_{\scriptstyle\sigma^{\prime}\sigma^{\prime\prime}\hfill\atop\scriptstyle\sigma_{2}^{\prime}\sigma_{2}^{\prime\prime}\hfill}{\sum\limits_{\scriptstyle\sigma\hfill\atop\scriptstyle\sigma_{2}\hfill}{\int{dE}}}}}\\\ \times\left\\{{A_{\delta,\delta_{2},\sigma^{\prime},\sigma^{\prime\prime}}^{m,n,\sigma}(\alpha;E,E)A_{\delta_{2},\delta,\sigma_{2}^{\prime},\sigma_{2}^{\prime\prime}}^{n,m,\sigma_{2}}(\beta;E,E)+A_{\delta,\delta_{2},\sigma^{\prime},\sigma^{\prime\prime}}^{m,n,\sigma}(\beta;E,E)A_{\delta_{2},\delta,\sigma_{2}^{\prime},\sigma_{2}^{\prime\prime}}^{n,m,\sigma_{2}}(\alpha;E,E)}\right\\}\\\ \times\left\\{{\left[{\delta_{\sigma^{\prime\prime}\sigma_{2}^{\prime}}\delta_{\sigma^{\prime}\sigma_{2}^{\prime\prime}}f_{\delta}(E)+\delta_{\sigma^{\prime\prime}\sigma_{2}^{\prime}}\delta_{\bar{\sigma}^{\prime}\sigma_{2}^{\prime\prime}}\frac{1}{2}\left({f_{p\delta}(E)-f_{a\delta}(E)}\right)}\right]}\right.\\\ \left.{-\left[{\delta_{\sigma^{\prime}\sigma_{2}^{\prime\prime}}f_{\delta}(E)+\delta_{\bar{\sigma}^{\prime}\sigma_{2}^{\prime\prime}}\frac{1}{2}\left({f_{p\delta}(E)-f_{a\delta}(E)}\right)}\right]\times\left[{\delta_{\sigma^{\prime\prime}\sigma_{2}^{\prime}}f_{\delta_{2}}(E)+\delta_{\bar{\sigma}^{\prime\prime}\sigma_{2}^{\prime}}\frac{1}{2}\left({f_{p\delta_{2}}(E)-f_{a\delta_{2}}(E)}\right)}\right]}\right\\}.\\\ \end{array}$ (30) Eq. (30) gives a general formula to calculate the shot noise in mesoscopic conductors subject to the SOC effect. It can be used to predict the low- frequency noise properties of arbitrary multi-channel, multi-probe phase- coherent conductors in the presence of Dresselhaus SOC. It can be naturally extended to the system with Rashba SOC and the system with both Dresselhaus SOC and Rashba SOC. In the limit of zero SOC and when the conductor is connected to normal metal leads, there is $f_{\alpha a}(E)=f_{\alpha p}(E)$ and $T_{n+}(E)=T_{n-}(E)=T(E)$, thus Eq. (30) reconverts to the formula provided by BüttikerRef58 concerning scalar electron systems without the spin degree of freedom inducing observable difference. ### II.3 Numerical results in comparison with experiment To further demonstrate our theory, we provide numerical results based on real heterostructures in comparison with experiment. The double-barrier structure considered here is constructed of layers of ${\rm{Ga}}_{\rm{x}}{\rm{Al}}_{{\rm{1-x}}}{\rm{Sb}}$ with $x=0.15/0.3/0/0.3/0.15$ ($x=0.3$ for the barriers and $x=0$ for the well), which are known to be semiconductors with relatively strong Dresselhaus SOCRef55 ; Ref56 . Our target setup is a symmetric double-barrier structure with the thickness of the well $a=30$ ${\rm{{\AA}}}$ and the thickness of the two barriers $b=c=50$ ${\rm{{\AA}}}$. The height of the barrier $V_{b}=230$ ${\rm{meV}}$ and the depth of the well $V_{w}=200$ ${\rm{meV}}$ are given by the heterostructure propertiesRef57 (cf. Fig. 5). We assume that in the whole region the effective mass $m^{}=0.053m_{e}$Ref55 ; Ref56 . The chemical potential of the two electrodes is set to be ${\rm{12}}$ ${\rm{meV}}$. For comparison, we choose the Dresselhaus constant $\gamma=0,40,80,120$ ${\rm{eV}}$ ${\rm{{\AA}^{3}}}$. We see the current fluctuation of the system caused by Dresselhaus SOC. Fig. 7 presents results of the electric current $I$ and the shot noise $S$ versus the external bias. It is demonstrated that the peaks of the current and of the shot noise are lowered as the Dresselhaus constant $\gamma$ increases, and a concave down in the ascending side of the noise curve is obvious for non-zero $\gamma$. The concaveness gives rise to nadirs far below 0.5 in the Fano factor (see Fig. 8 (a)). To compare with experiment, $S$ vs $I$ curves normalized to unity are displayed in Fig. 8 (b). In the positive differential conductance region, the shot noise follows the value of uncorrelated electrons ($2eI$) for small tunnel currents and is significantly suppressed for larger currents and eventually increases. The suppression is above one-half for $\gamma=0$, near one-half for $\gamma=4.0\times 10^{-29}$ ${\rm{eV}}$ ${{\rm{m}}^{\rm{3}}}$, and below one-half for $\gamma$ larger than $8.0\times 10^{-29}$ ${\rm{eV}}$ ${{\rm{m}}^{\rm{3}}}$. In the negative differential conductance (NDC) region, the Coulomb interaction or charging effect in the well enhances the shot noise and overweighs the effect of SOCRef59 . Iannaccone et al. have focused on the NDC region and obtained enhanced shot noiseRef36 . The results shown in Figs. 7-8 can be understood from the following. The SOC interaction behaves like a pseudo magnetic field and induces split of different spin components of the resonant level in the barrier structureRef55 , which contribute collectively to the electric current and shot noise. Thus, the current is lowered at the peak and simultaneously lifted in both sides around the peak in the current-bias spectra. When the SOC is present, spin “$\uparrow$” electrons exclude spin “$\downarrow$” electrons as well as spin “$\uparrow$” ones. Therefore, the large noise suppressions are a consequence of the repulsion between current pulses of different spin states in addition to the consequence of the Pauli blockade and Coulomb repulsion. ## III Quantum pumping beyond linear response ### III.1 Introduction to quantum pumping Generally speaking, the transport of matter from low potential to high potential excited by absorbing energy from the environment can be described as a pump process. The driving mechanics of classic pumps is straightforward and well understoodRef61 . The concept of a quantum pump is initiated several decades agoRef62 with its mechanism involving coherent tunneling and quantum interference. Research on quantum pumping has attracted heated interest since its experimental realization in an open quantum dotRef1 ; Ref2 ; Ref3 ; Ref4 ; Ref5 ; Ref6 ; Ref7 ; Ref8 ; Ref9 ; Ref10 ; Ref11 ; Ref12 ; Ref13 ; Ref14 ; Ref15 ; Ref16 ; Ref17 ; Ref18 ; Ref19 ; Ref63 ; Ref64 ; Ref65 ; Ref66 ; Ref67 ; Ref68 ; Ref69 ; Ref70 ; Ref71 ; Ref72 . The mechanisms of an adiabatic quantum pump can be demonstrated in a mesoscopic system modulated by two oscillating barriers (see Fig. 9). To prominently picture the charge flow driven process within a cyclic period, the two potential barriers are modulated with a phase difference of $\pi/2$ in the manner of $U_{1}=U_{0}+U_{1\omega}\sin t$ and $U_{2}=U_{0}+U_{2\omega}\sin(t+\pi/2)$. Our discussion is within the framework of the single electron approximation and coherent tunneling. The Pauli principle is taken into account throughout the pumping process. The Fermi energy of the two reservoirs and the inner single-particle state energy are equalized to eliminate the external bias and secure energy-conserved tunneling [The kinetic properties (charge current, heat current, etc.) depend on the values of the scattering matrix within the energy interval of the order of ${\rm{max}}(k_{B}T,\hbar\omega)$ near the Fermi energy. In the low-frequency ($\omega\rightarrow 0$) and low-temperature ($T\rightarrow 0$) limit we assume the scattering matrix to be energy independent]. As shown in Fig. 9, the transmission strengths between one of the reservoirs and the inner single- particle state are denoted by $t_{1}$-$t_{4}$. When $t\in[0,\pi/2]$, $\sin t$ changes from 0 to 1 and $\sin(t+\pi/2)$ changes from 1 to 0. Considering the time-averaged effect, the chance of $U_{1}>U_{2}$ and $U_{1}<U_{2}$ is equal. Therefore, the probability of $t_{1}$ and $t_{3}$ balance out. The tunneling quantified by $t_{2}$ and $t_{4}$ do not occur since the inner particle state is not occupied. When $t\in[\pi/2,\pi]$, $\sin t$ changes from 1 to 0 and $\sin(t+\pi/2)$ changes from 0 to -1. $U_{1}>U_{2}$ invariably holds in this time regime. The probability of $t_{3}$ prevails and a net particle flow is driven from the right reservoir to the middle state. When $t\in[\pi,3\pi/2]$, $\sin t$ changes from 0 to -1 and $\sin(t+\pi/2)$ changes from -1 to 0. The probability of $t_{2}$ and $t_{4}$ balance out and the tunneling quantified by $t_{1}$ and $t_{3}$ are excluded from the Pauli principle. No net time- averaged tunneling occurs. When $t\in[3\pi/2,2\pi]$, $\sin t$ changes from -1 to 0 and $\sin(t+\pi/2)$ changes from 0 to 1. $U_{1}$ maintains a lower height than $U_{2}$, which drives the particle in the inner state to the left reservoir. Through one whole pumping cycle, electrons are pumped from the right reservoir to the left by absorbing energy from the two oscillating sources. The tunneling is governed by quantum coherence. In each period, the pumping process repeats and the particles are driven continuously in the same direction as time accumulates. Direction-reversed pumped current can be obtained with reversed phase difference of the two oscillating gates. The direction of the pumped current is from the phase-leading gate to the phase- lagged one without exception when we assume that higher barriers admit smaller transmission probability. It can find resemblance in its classical turnstile counterpartRef61 in which the fore-opened gate admits transmission ahead of the later-opened one driving currents in corresponding manner. The current and noise properties in various quantum pump structures and devices were investigated such as the magnetic-barrier-modulated two dimensional electron gasRef4 , mesoscopic one-dimensional wireRef6 ; Ref65 , quantum-dot structuresRef61 ; Ref5 ; Ref11 ; Ref12 ; Ref71 , mesoscopic rings with Aharonov-Casher and Aharonov-Bohm effectRef7 , magnetic tunnel junctionsRef10 , chains of tunnel-coupled metallic islandsRef68 , the nanoscale helical wireRef69 ,the Tomonaga-Luttinger liquidRef67 , and garphene-based devicesRef63 ; Ref64 . Theory also predicts that charge can be pumped by oscillating one parameter in particular quantum configurationsRef66 . A recent experimentRef70 based on two parallel quantized charge pumps offers a way forward to the potential application of quantum pumping in quantum information processing, the generation of single photons in pairs and bunches, neural networking, and the development of a quantum standard for electrical current. Correspondingly, theoretical techniques have been put forward for the treatment of the quantum pumpsRef2 ; Ref3 ; Ref18 ; Ref65 ; Ref68 ; Ref72 . One of the most prominent is the scattering approach proposed by Brouwer who presented a formula that relates the pumped current to the parametric derivatives of the scattering matrix of the system. Driven by adiabatic and weak modulation (the ac driving amplitude is small compared to the static potential), the pumped current was found to vary in a sinusoidal manner as a function of the phase difference between the two oscillating potentials. It increases linearly with the frequency in line with experimental finding. The Floquet scattering theory is developedRef72 for quantum- mechanical pumping in mesoscopic conductors. It can be used to investigate quantum pumping behavior at arbitrary pumping amplitude and frequency. As an example to demonstrate the Floquet scattering theory, we focus on the experimentally observed deviation from the weak-pumping theory with only the first-order parametric derivative of the scattering matrix considered. By expanding the scattering matrix to higher orders of the time and modulation amplitude, experimental observation can be interpreted by multi-energy- quantum-related processes. ### III.2 Theoretical formulation We use the scattering matrix approach to describe the response of a mesoscopic phase-coherent sample to two slowly oscillating (with a frequency $\omega$) external real parameters $X_{j}(t)$ (gate potential, magnetic flux, etc.), $\begin{array}[]{{20}c}{X_{j}\left(t\right)=X_{0,j}+X_{\omega,j}e^{i\left({\omega t-\varphi_{j}}\right)}+X_{\omega,j}e^{-i\left({\omega t-\varphi_{j}}\right)},}&{j=1,2.}\\\ \end{array}$ (31) $X_{0,j}$ and $X_{\omega,j}$ measure the static magnitude and ac driving amplitude of the two parameters, respectively. The phase difference between the two drivers is defined as $\phi=\varphi_{1}-\varphi_{2}$. The mesoscopic conductor is connected to two reservoirs at zero bias. The scattering matrix $\hat{s}$ being a function of parameters $X_{j}(t)$ depends on time. A time-dependent scattering matrix can be introduced as follows. $\hat{b}_{\alpha}\left(t\right)=\int_{-\infty}^{\infty}{S_{\alpha\beta}\left({t,t^{\prime}}\right)\hat{a}_{\beta}\left({t^{\prime}}\right)dt^{\prime}},\begin{array}[]{{20}c}{}\hfil&{t\geq t^{\prime}.}\\\ \end{array}$ (32) Its Wigner transform reads $S_{\alpha\beta}\left({E,t}\right)=\int_{-\infty}^{\infty}{e^{iE\left({t-t^{\prime}}\right)}S_{\alpha\beta}\left({t,t^{\prime}}\right)dt^{\prime}}.$ (33) We assume the scattering time $t-t^{\prime}$ is small. Up to corrections of order $\hbar\omega/\gamma$ ($\gamma$ measures the escape rate), the matrix $S_{\alpha\beta}\left({E,t}\right)$ is equal to the “instantaneous” scattering matrix $S_{X}(E)$, which is obtained by “freezing” all parameters $X_{j}$ to their values at time $t$. Below, we use the instant scattering matrix $\hat{s}\left(t\right)$ in place of $S_{\alpha\beta}\left({E,t}\right)$ to describe the physics for simplicity. The kinetic properties (charge current, heat current, etc.) depend on the values of the scattering matrix within the energy interval of the order of ${\rm{max}}(k_{B}T,\hbar\omega)$ near the Fermi energy. In the low-frequency ($\omega\rightarrow 0$) and low-temperature ($T\rightarrow 0$) limit we assume the scattering matrix to be energy independent. To investigate the deviation from the small amplitude $X_{\omega,j}$ limit, we expand the scattering matrix $\hat{s}(t)$ into Taylor series of $X_{j}(t)$ to second order at $X_{0,j}$ with the terms linear and quadratic of $X_{\omega,j}$ present in the expansion, $\hat{s}\left(t\right)\approx\hat{s}_{0}\left({X_{0,j}}\right)+\hat{s}_{-\omega}e^{i\omega t}+\hat{s}_{+\omega}e^{-i\omega t}+\hat{s}_{2}+\hat{s}_{-2\omega}e^{2i\omega t}+\hat{s}_{+2\omega}e^{-2i\omega t},$ (34) with $\left\\{\begin{array}[]{l}\hat{s}_{\pm\omega}=\sum\limits_{j=1,2}{X_{\omega,j}e^{\pm i\varphi_{j}}{{\partial\hat{s}}\mathord{\left/{\vphantom{{\partial\hat{s}}{\partial X_{j}}}}\right.\kern-1.2pt}{\partial X_{j}}}},\\\ \hat{s}_{2}=\sum\limits_{j=1,2}{X_{\omega,j}^{2}{{\partial^{2}\hat{s}}\mathord{\left/{\vphantom{{\partial^{2}\hat{s}}{\partial X_{j}^{2}}}}\right.\kern-1.2pt}{\partial X_{j}^{2}}}},\\\ \hat{s}_{\pm 2\omega}=\frac{1}{2}\sum\limits_{j=1,2}{X_{\omega,j}^{2}e^{\pm 2i\varphi_{j}}{{\partial^{2}\hat{s}}\mathord{\left/{\vphantom{{\partial^{2}\hat{s}}{\partial X_{j}^{2}}}}\right.\kern-1.2pt}{\partial X_{j}^{2}}}}.\\\ \end{array}\right.$ (35) It can be seen from the equations that higher orders of the Fourier spectra enter into the scattering matrix. As a result, both the nearest and next nearest sidebands are taken into account, which implies that a scattered electron can absorb or emit an energy quantum of $\hbar\omega$ or $2\hbar\omega$ before it leaves the scattering region. In principle, third or higher orders in the Taylor series can be obtained accordingly. However, the higher-order parametric derivatives of the scatter matrix diminish dramatically and approximate zero. Numerical calculation demonstrates that even in relatively large amplitude modulation, their contribution is negligible. The pumped current depends on the values of the scattering matrix within the energy interval of the order of $\max\left({k_{B}T,2\hbar\omega}\right)$ near the Fermi energy. In the low-temperature limit ($T\to 0$), an energy interval of $2\hbar\omega$ is opened during the scattering process. The mesoscopic scatterer is coupled to two reservoirs with the same temperatures $T$ and electrochemical potentials $\mu$. Electrons with the energy $E$ entering the scatterer are described by the Fermi distribution function $f_{0}(E)$, which approximates a step function at a low temperature. Due to the interaction with an oscillating scatterer, an electron can absorb or emit energy quanta that changes the distribution function. A single transverse channel in one of the leads is considered. Applying the hypothesis of an instant scattering, the scattering matrix connecting the incoming and outgoing states can be written as $\hat{b}_{\alpha}\left(t\right)=\sum\limits_{\beta}{s_{\alpha\beta}\left(t\right)\hat{a}_{\beta}\left(t\right)}.$ (36) Here $s_{\alpha\beta}$ is an element of the scattering matrix $\hat{s}$; the time-dependent operator is $\hat{a}_{\alpha}\left(t\right)=\int{dE\hat{a}_{\alpha}\left(E\right)e^{{{-iEt}\mathord{\left/{\vphantom{{-iEt}\hbar}}\right.\kern-1.2pt}\hbar}}}$, and the energy-dependent operator ${\hat{a}_{\alpha}\left(E\right)}$ annihilates particles with total energy E incident from the $\alpha$ lead into the scatter and obey the following anticommutation relations $\left[{\hat{a}_{\alpha}^{\dagger}\left(E\right),\hat{a}_{\beta}\left({E^{\prime}}\right)}\right]=\delta_{\alpha\beta}\delta\left({E-E^{\prime}}\right).$ (37) Note that above expressions correspond to single- (transverse) channel leads and spinless electrons. For the case of many-channel leads each lead index ($\alpha$, $\beta$, etc.) includes a transverse channel index and any repeating lead index implies implicitly a summation over all the transverse channels in the lead. Similarly an electron spin can be taken into account. Using Eqs. (34) and (36) and after a Fourier transformation we obtain $\begin{array}[]{l}\hat{b}_{\alpha}\left(E\right)=\sum\limits_{\beta}{\left[{\hat{s}_{0,\alpha\beta}\hat{a}_{\beta}\left(E\right)+\hat{s}_{2,\alpha\beta}\hat{a}_{\beta}\left(E\right)+\hat{s}_{-\omega,\alpha\beta}\hat{a}_{\beta}\left({E+\hbar\omega}\right)}\right.}\\\ \left.{+\hat{s}_{+\omega,\alpha\beta}\hat{a}_{\beta}\left({E-\hbar\omega}\right)+\hat{s}_{-2\omega,\alpha\beta}\hat{a}_{\beta}\left({E+2\hbar\omega}\right)+\hat{s}_{+2\omega,\alpha\beta}\hat{a}_{\beta}\left({E-2\hbar\omega}\right)}\right].\\\ \end{array}$ (38) The distribution function for electrons leaving the scatterer through the lead $\alpha$ is $f_{\alpha}^{\left({out}\right)}\left(E\right)=\left\langle{\hat{b}_{\alpha}^{\dagger}\left(E\right)\hat{b}_{\alpha}\left(E\right)}\right\rangle$, where $\left\langle\cdots\right\rangle$ means quantum-mechanical averaging. Substituting Eq. (38) we find $\begin{array}[]{l}f_{\alpha}^{\left({out}\right)}\left(E\right)=\sum\limits_{\beta}{\left[{\left\|{\hat{s}_{0,\alpha\beta}+\hat{s}_{2,\alpha\beta}}\right\|^{2}f_{0}\left(E\right)+\left\|{\hat{s}_{-\omega,\alpha\beta}}\right\|^{2}f_{0}\left({E+\hbar\omega}\right)}\right.}\\\ \left.{\left\|{\hat{s}_{+\omega,\alpha\beta}}\right\|^{2}f_{0}\left({E-\hbar\omega}\right)+\left\|{\hat{s}_{-2\omega,\alpha\beta}}\right\|^{2}f_{0}\left({E+2\hbar\omega}\right)+\left\|{\hat{s}_{+2\omega,\alpha\beta}}\right\|^{2}f_{0}\left({E-2\hbar\omega}\right)}\right].\\\ \end{array}$ (39) The distribution function for outgoing carriers is a nonequilibrium distribution function generated by the nonstationary scatterer. The Fourier amplitudes of the scattering matrix ${\left\|{\hat{s}_{-\omega,\alpha\beta}}\right\|^{2}}$ (${\left\|{\hat{s}_{+\omega,\alpha\beta}}\right\|^{2}}$) is the probability for an electron entering the scatterer through the lead $\beta$ and leaving the scatterer through the lead $\alpha$ to emit (to absorb) an energy quantum $\hbar\omega$ and ${\left\|{\hat{s}_{-2\omega,\alpha\beta}}\right\|^{2}}$ (${\left\|{\hat{s}_{+2\omega,\alpha\beta}}\right\|^{2}}$) is that of the energy quantum $2\hbar\omega$ process. ${\left\|{\hat{s}_{0,\alpha\beta}+\hat{s}_{2,\alpha\beta}}\right\|^{2}}$ is the probability for the same scattering without the change of an energy with the second-order term $\hat{s}_{2,\alpha\beta}$ much smaller than the zero-order term $\hat{s}_{0,\alpha\beta}$ in weak-modulation limit ($X_{\omega,j}\ll X_{0,j}$) and can be omitted therein. Using the distribution functions $f_{0}(E)$ for incoming electrons and $f_{\alpha}^{out}(E)$ for outgoing electrons, the pumped current measured at lead $\alpha$ reads $I_{p}=\frac{e}{{2\pi\hbar}}\int_{0}^{\infty}{\left\langle{\hat{b}_{\alpha}^{\dagger}\left(E\right)\hat{b}_{\alpha}\left(E\right)}\right\rangle-\left\langle{\hat{a}_{\alpha}^{\dagger}\left(E\right)\hat{a}_{\alpha}\left(E\right)}\right\rangle dE}.$ (40) Substituting Eqs. (34) and (30) we get $\begin{array}[]{c}I_{p}=\frac{{e\omega}}{{2\pi}}\sum\limits_{\beta,j_{1},j_{2}}{X_{\omega,j_{1}}X_{\omega,j_{2}}\frac{{\partial s_{\alpha\beta}}}{{\partial X_{j_{1}}}}\frac{{\partial s_{\alpha\beta}^{}}}{{\partial X_{j_{2}}}}2i\sin\left({\varphi_{j_{1}}-\varphi_{j_{2}}}\right)}\\\ +\frac{{e\omega}}{{2\pi}}\sum\limits_{\beta,j_{1},j_{2}}{X_{\omega,j_{1}}^{2}X_{\omega,j_{2}}^{2}\frac{{\partial^{2}s_{\alpha\beta}}}{{\partial X_{j_{1}}^{2}}}\frac{{\partial^{2}s_{\alpha\beta}^{}}}{{\partial X_{j_{2}}^{2}}}i\sin\left[{2\left({\varphi_{j_{1}}-\varphi_{j_{2}}}\right)}\right]}.\\\ \end{array}$ (41) Quantum pumping properties beyond the theory based on first-order parametric derivative of the scattering matrix are demonstrated in Eq. (41). By taking higher orders of the Fourier spectrum of the scattering matrix into consideration, double $\hbar\omega$ energy quantum (or a $2\hbar\omega$ energy quantum) emission (absorption) processes coact with single $\hbar\omega$ quantum processes. In the weak-modulation limit, the second term in the right- hand side of Eq. (41) is small, which implies that double $\hbar\omega$ quantum processes are weak and therefore not observable. As the ac driving amplitude is enlarged, this term increases markedly and contribution from double $\hbar\omega$ quantum processes takes effect. As a result, the dependence of the pumped current on the phase difference between two driving oscillations deviates from sinusoidal and changes from $\sin\phi$ to $\sin 2\phi$, which is observed in experimentRef1 . Moreover, the relation between the pumped current and the ac driving amplitude $X_{\omega,j}$ is reshaped. It is also seen that the linear dependence of the pumped current on the oscillation frequency holds for multi-quanta-related processes. ### III.3 Numerical results and interpretations Here, numerical results of the pumped current in a two-oscillating-potential- barrier modulated nanowire are presented and comparison with experiment is given. We consider a nanowire modulated by two gate potential barriers with equal width $L=20$ Å separated by a $2L=40$ Å width well (see Fig. 10). The electrochemical potential of the two reservoirs $\mu$ is set to be $60$ meV according to the resonant level within the double-barrier structure. The two oscillating parameters in Eq. (31) correspond to the two ac driven potential gates $X_{1,2}\left(t\right)\to U_{1,2}\left(t\right)$ with all the other notations correspond accordingly. We set the static magnitude of the two gate potentials $U_{0,1}=U_{0,2}=U_{0}=100$ meV and the ac driving amplitude of the modulations equal $U_{\omega,1}=U_{\omega,2}=U_{\omega}$. In Fig. 11, the dependence of the pumped current on the phase difference between the two ac oscillations is presented. In weak-modulation regime (namely $U_{\omega}$ is small), sinusoidal behavior dominates. Here, three relatively large $U_{\omega}$ is selected to reveal the deviation from the sinusoidal dependence. (The magnitude of the pumped current mounts up in power-law relation as a function of $U_{\omega}$ as shown in Fig. 12. The sinusoidal curve for small $U_{\omega}$ would be flat and invisible in the same coordinate range.) It can be seen from the figure that the $I_{p}$-$\phi$ relation varies from sinusoidal ($\sin\phi$) to double-sinusoidal ($\sin 2\phi$) as the ac oscillation amplitude is increased. The interpretation follows from Eq. (41). The single $\hbar\omega$ quantum emission (absortion) processes feature a sinusoidal behavior while the $2\hbar\omega$ quantum emission (absortion) processes feature a double-sinusoidal behavior when the Fourier index is doubled. As $U_{\omega}$ is increased, double $\hbar\omega$ quantum processes gradually parallel and outweigh the single $\hbar\omega$ quantum ones. It is also demonstrated that when the single $\hbar\omega$ quantum processes have the effect of $\sin\phi$ dependence, the double $\hbar\omega$ quantum processes induce a $-\sin 2\phi$ contribution with a sign flip, which can be understood from the sign change of the derivative of the scattering matrix. The effect of three- and higher $\hbar\omega$ quantum processes is small even for large $U_{\omega}$ comparable to $U_{0}$. The experimental observationsRef1 as a deviation from the weak-modulation limit are revealed by our theory. ExperimentRef1 also discovered that for weak pumping the dependence of the pumped current on the pumping strength obeys a power of 2 relation, as expected from the simple loop-area argumentRef2 ; for strong pumping, power of 1 and 1/2 relation is observed. We presented in Fig. 12 the numerical results based on our theory of the $I_{p}$-$U_{\omega}$ relation at a fixed $\phi$. To demonstrate its power-law dependence, natural logarithm of the variables is applied. From Eq. (41), it can be seen that for large ac driving amplitude $U_{\omega}$, contribution of double $\hbar\omega$ quantum processes (formulated in the second term on the right hand side of the equation) causes the $I_{p}$-$U_{\omega}$ relation to deviate from its weak-modulation limit, the latter of which is $I_{p}\propto U_{\omega}^{2}$. For different phase difference between the two ac drivers, the deviation is different. At $\phi=\pi$ the pumped current is invariably zero regardless of the order of approximation determined by time-reversal symmetry. At $\phi=\pi/2$, $\sin 2\phi$ is exact zero, and no difference is incurred by introducing higher order effect. If we shift the value of $\phi$ to $0.49\pi$, the abating effect of the double $\hbar\omega$ quantum processes has the order of $U_{\omega}^{4}$ with the small second-order parametric derivative of the scattering matrix smoothing that effect a bit. Consequently, a power of $2\to 1\to 1/2$ relation is obtained and visualized by the curve fit, which is analogous to experimental findings. For different values of $\phi$, sharper abating and augmental effect occurs with analogous mechanisms. It is possible that the experimentRef1 was done at the phase difference close to $\pi/2$ while trying to approach maximal pumped current in the adiabatic and weak- pumping limit. ## IV Summary and future directions The scattering matrix method is initiated by Landauer and Büttiker to investigate the conductance of multi-terminal and multi-channel mesoscopic conductors. The spin degree of freedom can be included in the formalism by enlargement of the dimension of the scattering matrix. The current-current correlation and spin-spin correlation, such as the shot noise, can be calculated from the cross products of the scattering matrix. Along this direction, higher-order correlation function can also be considered. Dynamic transport processes including the quantum pumping behavior can be dealt with by the time-dependent scattering approach. The development of the scattering theory enables its potential applications in currently open issues. 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The quantum states between two reservoirs at zero bias is indicated. Figure 2: Schematic demonstration of a quantum wire modulated by two potential barriers. The quantum state between two reservoirs at zero bias is indicated. Figure 3: Time-integrated current density as a function of the phase difference between the two modulations for different modulation frequencies. The Fermi level of the two reservoirs $E_{F}=60$ meV counting from the conduction band edge of the electron gas structure. Figure 4: Time variation of the current density within an oscillating cycle. Different phase difference between the two modulations is considered. The modulation frequency is set to be $\omega=10$ MHz and the Fermi level $E_{F}$ to be 60 meV. Figure 5: Schematics of the double-barrier resonant diode. The resonant level is sketched between the two barriers. The upper panel demonstrates the resonant level in conventional diode without the spin-orbit coupling (SOC). In the lower panel, the SOC behaves like a pseudomagnetic field and induces a split of different spin components of the resonant level in the barrier structure, which contribute collectively to the electric current and shot noise. Figure 6: Schematics of the scattering approach. Spin components of the incoming and outgoing states are indicated. Figure 7: Current $I$ (a) and shot noise $S$ (b) as functions of the applied bias $EV$ of electrons traversing a symmetric DBRD structure with different Dresselhaus constants $\gamma$. Figure 8: Fano factor as a function of the applied bias $EV$ (a) and $S/2eI_{norm}$ vs $I/I_{norm}$ (b) of electrons traversing a symmetric DBRD structure with different Dresselhaus constants $\gamma$. Two straight lines in (b) show the full shot noise value ($2eI$) and half of its value for comparison. Figure 9: The tunneling scenario of an adiabatic quantum pump. The two shadowed blocks represent the left and right electron reservoirs respectively. The two barriers oscillate adiabatically in time. The middle bar indicates the single-particle state between the two barriers. The Fermi levels of the two reservoirs are the same and are leveled to the single-particle state within the conductor. $t_{1}$-$t_{4}$ indicate the transmission amplitudes between one of the two reservoirs and the middle single-particle state. Figure 10: Schematics of the quantum pump: a nanowire modulated by two ac driven potential barriers. Figure 11: Pumped current as a function of the phase difference between the two modulations for different ac driving amplitudes. Figure 12: Pumped current as a function of the ac driving amplitude $U_{\omega}$ along with fits to $I_{p}\propto U_{\omega}^{2}$ (red solid circle) below 35 meV, $I_{p}\propto U_{\omega}$ (green upward triangle) below 41 meV, and $I_{p}\propto U_{\omega}^{1/2}$ above 41 meV (blue downward triangle). To demonstrate its power-law dependence, natural logarithm of the variables is applied. The phase difference between the two ac driver $\phi=0.49\pi$. Inset is the zoom-in of the circled region.	arxiv-papers	2010-11-16T08:02:34	2024-09-04T02:49:14.849378	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Rui Zhu", "submitter": "Rui Zhu", "url": "https://arxiv.org/abs/1011.3603" }
1011.3702	# Generalized holomorphic analytic torsion José Ignacio Burgos Gil Gerard Freixas i Montplet Răzvan Liţcanu ###### Abstract In this paper we extend the holomorphic analytic torsion classes of Bismut and Köhler to arbitrary projective morphisms between smooth algebraic complex varieties. To this end, we propose an axiomatic definition and give a classification of the theories of generalized holomorphic analytic torsion classes for projective morphisms. The extension of the holomorphic analytic torsion classes of Bismut and Köhler is obtained as the theory of generalized analytic torsion classes associated to $-R/2$, $R$ being the $R$-genus. As application of the axiomatic characterization, we give new simpler proofs of known properties of holomorpic analytic torsion classes, we give a characterization of the $R$ genus, and we construct a direct image of hermitian structures for projective morphisms. ††J. I. Burgos Gil: Instituto de Ciencias Matemáticas (CSIC-UAM-UCM-UC3), Calle Nicolás Cabrera 15 28049 Madrid, Spain; e-mail: burgos@icmat.es G. Freixas i Montplet: Institut de Mathématiques de Jussieu (IMJ), Centre National de la Recherche Scientifique (CNRS), France; e-mail: freixas@math.jussieu.fr R. Liţcanu: Faculty of Mathematics, University Al. I. Cuza Iasi, Romania; e-mail: litcanu@uaic.ro ## 1 Introduction The aim of this paper is to extend the classes of analytic torsion forms introduced by Bismut and Köhler to arbitrary projective morphisms between complex algebraic varieties. The main tool for this extension is an axiomatic characterization of all the possible theories of holomorphic analytic torsion classes. Before stating what we mean by a theory of holomorphic analytic torsion classes, we briefly recall the origin of the analytic torsion. The R-torsion is a topological invariant attached to certain euclidean flat vector bundles on a finite CW-complex. This invariant was introduced by Reidemeister and generalized by Franz in order to distinguish non-homeomorphic lens spaces that have the same homology and homotopy groups. Let $W$ be a connected CW-complex and let $K$ be an orthogonal representation of $\pi_{1}(W)$. Then $K$ defines a flat vector bundle with an euclidean inner product $E_{K}$. Assume that the chain complex of $W$ with values in $E_{K}$ is acyclic. Then the R-torsion is the determinant of this complex with respect to a preferred basis. Later, Ray and Singer introduced an analytic analogue of the R-torsion and they conjectured that, for compact riemannian manifolds, this analytic torsion agrees with the R-torsion. This conjecture was proved by Cheeger and Müller. If $W$ is a riemannian manifold and $K$ is as before, then we have the de Rham complex of $W$ with values in $E_{K}$ at our disposal. The hypothesis on $K$ implies that $(\Omega^{\ast}(W,E_{K}),\operatorname{d})$ is also acyclic. Then the analytic torsion is essentially the determinant of the de Rham complex. Here the difficulty lies in that the vector spaces $\Omega^{p}(W,E_{K})$ are infinite dimensional and therefore the “determinant” has to be defined using a zeta function regularization involving the laplacian. More details on the construction of R-torsion and analytic torsion can be found in [41]. Ray and Singer observed that, with the help of hermitian metrics, the acyclicity condition can be removed. Moreover, their definition of analytic torsion can be extended to any elliptic complex. In the paper [42], they introduced a holomorphic analogue of the analytic torsion as the determinant of the Dolbeault complex. They also studied some of its properties and computed some examples. In particular, they showed that this invariant depends on the complex structure and they gave a hint that the holomorphic analytic torsion should be interesting in number theory. This holomorphic analytic torsion and its generalizations are the main object of study of the present paper. Since this is the only kind of analytic torsion that we will consider, throughout the paper, by analytic torsion we will mean holomorphic analytic torsion. In the paper [40], Quillen, using the analytic torsion, associated to each holomorphic hermitian vector bundle on a Riemann surface a hermitian metric on the determinant of its cohomology. Furthermore, he showed that this metric varies smoothly with the holomorphic structure on the vector bundle. He also computed the curvature of the hermitian line bundle on the space of all complex structures obtained in this way. Subsequently Bismut and Freed [7], [8] generalized the construction of Quillen to families of Dirac operators on the fibers of a smooth fibration. They obtained a smooth metric and a unitary connection on the determinant bundle associated with the family of Dirac operators. Furthermore, they computed the curvature of this connection, which agrees with the degree 2 part of the differential form obtained by Bismut in his proof of the Local Index theorem [2]. Later, in a series of papers [9], [10], [11], Bismut, Gillet and Soulé considered the case of a holomorphic submersion endowed with a holomorphic hermitian vector bundle. They defined a Quillen type metric on the determinant of the cohomology of the holomorphic vector bundle. In the locally Kähler case, they showed the compatibility with the constructions of Bismut-Freed. In addition they described the variation of the Quillen metric under change of the metric on the vertical tangent bundle and on the hermitian vector bundle. The results of [9], [10], [11] represent a rigidification of [7], [8]. All in all, these works explain the relationship between analytic torsion and the Atiyah-Singer index theorem and, in the algebraic case, with Grothendieck’s relative version of the Riemann-Roch theorem. In [20], Deligne, inspired by the Arakelov formalism, gave a formula for the Quillen metric that can be seen as a very precise version of the degree one case of the Riemann-Roch theorem for families of curves. This result is in the same spirit as the arithmetic Riemann-Roch theorem of Faltings [23]. In the paper [29], Gillet and Soulé conjectured an arithmetic Riemann-Roch formula that generalizes the results of Deligne and Faltings. Besides the analytic torsion or its avatar, the Quillen metric, this Riemann-Roch formula involves a mysterious new odd additive characteristic class, the $R$-genus, that they computed with the help of Zagier. In the work [14] Bismut and Lebeau studied the behavior of the analytic torsion with respect to complex immersions. Their compatibility formula also involved the $R$-genus. Later Bost [15] and Roessler [43] explained, using geometric arguments, why the same genus appears both in the arithmetic Riemann-Roch formula and the Bismut-Lebeau compatibility formula. However these geometric arguments do not characterize the $R$-genus. Gillet and Soulé [30] proved the degree one part of the arithmetic Riemann- Roch theorem. A crucial ingredient of the proof is the compatibility formula of Bismut-Lebeau. In order to establish the arithmetic Riemann-Roch theorem in all degrees it was necessary to generalize the analytic torsion and define higher analytic torsion classes. It was clear from [30] that, once a suitable theory of higher analytic torsion classes satisfying certain properties were developed, then the arithmetic Riemann-Roch theorem would follow. A first definition of such forms was given by Gillet and Soulé in [29], but they did not prove all the necessary properties. A second equivalent definition was given in [13] by Bismut and Köhler, where some of the needed properties are proved. The compatibility of higher analytic torsion classes with complex immersions, i.e. the generalization of Bismut-Lebeau compatibility formula, was proved in [3]. As a consequence, Gillet, Soulé and Rössler [25] extended the arithmetic Riemann-Roch theorem to arbitrary degrees. In the book [24], Faltings followed a similar strategy to define direct images of hermitian vector bundles and proved an arithmetic Riemann-Roch formula up to a unique unknown odd genus. The arithmetic Riemann-Roch theorems of Gillet-Soulé and Faltings deal only with projective morphisms between arithmetic varieties such that, at the level of complex points, define a submersion. By contrast, in his thesis [50] Zha follows a completely different strategy to establish an arithmetic Riemann- Roch theorem without analytic torsion. His formula does not involve the $R$-genus. Moreover Zha’s theorem is valid for any projective morphism between arithmetic varieties. In [44], Soulé advocates for an axiomatic characterization of the analytic torsion, similar to the axiomatic characterization of Bott-Chern classes given by Bismut-Gillet-Soulé in [9]. Note that the R-torsion has also been generalized to higher degrees giving rise to different higher torsion classes. In [33], Igusa gives an axiomatic characterization of these higher torsion classes. We now explain more precisely what we mean by a theory of generalized analytic torsion classes. The central point is the relationship between analytic torsion and the Grothendieck-Riemann-Roch theorem. Let $\pi\colon X\to Y$ be a smooth projective morphism of smooth complex varieties. Let $\omega$ be a closed $(1,1)$ form on $X$ that induces a Kähler metric on the fibers of $\pi$ and, moreover, a hermitian metric on the relative tangent bundle $T_{\pi}$. We denote $\overline{T}_{\pi}$ the relative tangent bundle provided with this metric. Let $\overline{F}=(F,h^{F})$ be a hermitian vector bundle on $X$ such that for every $i\geq 0$, $R^{i}\pi_{\ast}F$ is locally free. We consider on $R^{i}\pi_{\ast}F$ the $L^{2}$ metric obtained using Hodge theory on the fibers of $\pi$ and denote the corresponding hermitian vector bundle as $\overline{R^{i}\pi_{\ast}F}$. To these data, Bismut and Köhler associate an analytic torsion differential form $\tau$ that satisfies the differential equation $\ast\partial\bar{\partial}\tau=\sum(-1)^{i}\operatorname{ch}(\overline{R^{i}\pi_{\ast}F})-\pi_{\ast}(\operatorname{ch}(\overline{F})\operatorname{Td}(\overline{T}_{\pi})),$ (1.1) where $\ast$ is a normalization factor that is irrelevant here (see 8). Moreover, if we consider the class of $\tau$ up to ${\operatorname{Im\,}}\partial+{\operatorname{Im\,}}\bar{\partial}$, then $\tau$ behaves nicely with respect to changes of metrics. The Grothendieck-Riemann-Roch theorem in de Rham cohomology says that the differential form on the right side of equation (1.1) is exact. Therefore, the existence of the higher analytic torsion classes provides us an analytic proof of this theorem. Since the Grothendieck-Riemann-Roch theorem is valid with more generality, it is natural to extend the notion of higher analytic torsion classes to non- smooth morphisms. To this end we will use the language of hermitian structures on the objects of the bounded derived category of coherent sheaves developed in [17]. In particular we will make extensive use of the category $\operatorname{\overline{\mathbf{D}}^{b}}$ introduced in _loc. cit._. Since, from now on, derived categories will be the natural framework, all functors will tacitly be assumed to be derived functors. By reasons explained in _loc. cit._ we will restrict ourselves to the algebraic category. Let $f\colon X\to Y$ be a projective morphism between smooth complex algebraic varieties. Let $\overline{F}$ be a hermitian vector bundle on $X$. Now, the relative tangent complex $T_{f}$ and the derived direct image $f_{\ast}F$ are objects of the bounded derived category of coherent sheaves on $X$ and $Y$ respectively. Since $X$ and $Y$ are smooth, using resolutions by locally free sheaves, we can choose hermitian structures on $T_{f}$ and $f_{\ast}F$. Hence we have characteristic forms $\operatorname{ch}(\overline{f_{\ast}F})$ and $\operatorname{Td}(\overline{T}_{f})$. We denote by $\overline{f}$ the morphism $f$ together with the choice of hermitian structure on $T_{f}$. Then the triple $\overline{\xi}=(\overline{f},\overline{F},\overline{f_{\ast}F})$ will be called a _relative hermitian vector bundle_. This is a particular case of the relative metrized complexes of Section 2. Then, a _generalized analytic torsion class_ for $\overline{\xi}$ is the class modulo ${\operatorname{Im\,}}\partial+{\operatorname{Im\,}}\bar{\partial}$ of a current that satisfies the differential equation $\ast\partial\bar{\partial}\tau=\operatorname{ch}(\overline{f_{\ast}F})-f_{\ast}(\operatorname{ch}(\overline{F})\operatorname{Td}(\overline{T}_{f})).$ (1.2) Note that such current $\tau$ always exists. Again, the Grothendieck-Riemann- Roch theorem in de Rham cohomology implies that the right hand side of equation (1.2) is an exact current. Thus, if $Y$ is proper, the $dd^{c}$-lemma implies the existence of such a current. When $Y$ is non-proper, a compactification argument allows us to reduce to the proper case. Of course, in each particular case, there are many choices for $\tau$. We can add to $\tau$ any closed current and obtain a new solution of equation (1.2). By a _theory of generalized analytic torsion classes_ we mean a coherent way of choosing a solution of equation (1.2) for all relative hermitian vector bundles, satisfying certain natural minimal set of properties. Each theory of generalized analytic torsion classes gives rise to a definition of direct images in arithmetic $K$-theory and therefore to an arithmetic Riemann-Roch formula. In fact, the arithmetic Riemann-Roch theorems of Gillet- Soulé and of Zha correspond to different choices of a theory of generalized analytic torsion classes. We leave for a subsequent paper the discussion of the relation with the arithmetic Riemann-Roch formula. Since each projective morphism is the composition of a closed immersion followed by the projection of a projective bundle, it is natural to study first the analytic torsion classes for closed immersions and projective bundles and then combine them in a global theory of analytic torsion classes. In [19] the authors studied the case of closed immersions (see Section 3). The generalized analytic torsion classes for closed immersions are called singular Bott-Chern classes and we will use both terms interchangeably. The definition of a _theory of singular Bott-Chern classes_ is obtained by imposing axioms analogous to those defining the classical Bott-Chern classes [26]. Namely, a theory of singular Bott-Chern classes is an assignment that, to each relative hermitian vector bundle $\overline{\xi}=(\overline{f},\overline{F},\overline{f_{\ast}F})$, with $f$ a closed immersion, assigns the class of a current $T(\overline{\xi})$ on $Y$, satisfying the following properties: 1. (i) the differential equation (1.2); 2. (ii) functoriality for morphisms that are transverse to $f$; 3. (iii) a normalization condition. A crucial observation is that, unlike the classical situation, these axioms do not uniquely characterize the singular Bott-Chern classes. Consequently there are various nonequivalent theories of singular Bott-Chern classes. They are classified by an arbitrary characteristic class of $F$ and $T_{f}$. If we further impose the condition that the theory is _transitive_ (that is, compatible with composition of closed immersions) and _compatible with the projection formula_ then the ambiguity is reduced to an arbitrary additive genus on $T_{f}$. The uniqueness can be obtained by adding to the conditions (i)–(iii) an additional homogeneity property. The theory obtained is transitive and compatible with the projection formula and agrees (up to normalization) with the theory introduced in [12]. Similarly, one can define a theory of analytic torsion classes for projective spaces (Section 5). This is an assignment that, to each relative hermitian vector bundle $\overline{\xi}=(\overline{f},\overline{F},\overline{f_{\ast}F})$, where $f\colon{\mathbb{P}}^{n}_{Y}\to Y$ is the projection of a trivial projective bundle, assigns the class of a current $T(\overline{\xi})$ satisfying the properties analogous to (i)-(iii) below, plus the additivity and the compatibility with the projection formula. The theories of analytic torsion classes for projective spaces are classified by their values in the cases $Y=\operatorname{Spec}{\mathbb{C}}$, $n\geq 0$, $F=\mathcal{O}(k)$, $0\leq k\leq n$ for one particular choice of metrics (see Theorem 5.9). We say that a theory of analytic torsion classes for closed immersions and one for projective spaces are compatible if they satisfy a compatibility equation similar to Bismut-Lebeau compatibility formula for the diagonal immersion $\Delta\colon{\mathbb{P}}^{n}_{{\mathbb{C}}}\to{\mathbb{P}}^{n}_{{\mathbb{C}}}\times{\mathbb{P}}^{n}_{{\mathbb{C}}}$, $n\geq 0$. Given a theory of singular Bott-Chern classes that is transitive and compatible with the projection formula, there exists a unique theory of analytic torsion classes for projective spaces that is compatible with it (Theorem 6.4). The central result of this paper (Theorem 7.7) is that, given a theory of singular Bott-Chern classes and a compatible theory of analytic torsion classes for projective spaces, they can be combined to produce a unique theory of generalized analytic torsion classes (Definition 7.1). Moreover, every theory of analytic torsion classes arises in this way. Thus we have a complete classification of the theories of generalized analytic torsion classes by additive genera. Once we have proved the classification theorem, we derive several applications. The first consequence of Theorem 7.7 is that the classes of the analytic torsion forms of Bismut-Köhler arise as the restriction to Kähler fibrations of the theory of generalized analytic torsion classes associated to minus one half of the $R$-genus (Theorem 8.8). In particular, we have succeeded to extend Bismut-Köhler analytic torsion classes to arbitrary projective morphisms in the algebraic category. Moreover, we reprove and generalize the theorems of Berthomieu-Bismut [1] and Ma [35], [36] on the compatibility of analytic torsion with the composition of submersions (Corollary 8.11). The second application of the classification theorem is a characterization of the $R$-genus. From the axiomatic point of view, the role played by the $R$-genus is mysterious. It would seem more natural to consider the generalized analytic torsion classes associated to the trivial genus $0$. This is the choice made implicitly by Zha in his thesis [50]. In fact, with our point of view, one of the main results of Zha’s thesis is the existence of a theory of analytic torsion classes associated to the trivial genus. This theory leads to an arithmetic Riemann-Roch formula identical to the classical one without any correction term. Thus, one is tempted to consider the $R$-genus as an artifact of the analytic definition of the analytic torsion. Nevertheless, by the work of several authors, the $R$-genus seems to have a deeper meaning. A paradigmatic example is the computation by Bost and Kühn [34] of the arithmetic self-intersection of the line bundle of modular forms on a modular curve, provided with the Petersson metric. This formula gives an arithmetic meaning to the first term of the $R$-genus. Thus it is important to characterize the $R$-genus from an axiomatic point of view and to understand its role in the above computations. From a theorem of Bismut [5] we know that the Bismut-Köhler analytic torsion classes of the relative de Rham complex of a Kähler fibration (with the appropriate hermitian structures) vanish. This result is important because one of the main difficulties to apply the arithmetic Riemann-Roch theorem is precisely the estimation of the analytic torsion. Moreover, this result explains why the terms of the $R$-genus appear in different arithmetic computations. For instance, the equivariant version of this result (due to Maillot and Roessler in degree 0 and to Bismut in general) allows Maillot and Roessler [37] to prove some cases of a conjecture of Gross-Deligne. The above vanishing property characterizes the analytic torsion classes of Bismut and Köhler. In order to show this, we first construct the dual theory $T^{\vee}$ to a given theory $T$ of generalized analytic torsion classes (Theorem Definition 9.10). A theory is self-dual ($T=T^{\vee}$) if and only if the even coefficients of the associated genus vanish (Corollary 9.14). In particular, Bismut-Köhler’s theory is self-dual. Self-duality can also be characterized in terms of the de Rham complex of smooth morphisms (Theorem 9.18). A theory $T$ is self-dual if its components of bidegree $(2p-1,p)$, $p$ odd, in the Deligne complex, vanish on the relative de Rham complexes of Kähler fibrations. Finally, in Theorem 9.24 we show that, if it exists, a theory of analytic torsion classes that vanishes, on all degrees, on the relative de Rham complexes of Kähler fibrations is unique, hence it agrees with Bismut-Köhler’s one. In fact, to characterize this theory, it is enough to assume the vanishing of the analytic torsion classes for the relative de Rham complexes of Kähler fibrations of relative dimension one. To establish this characterization we appeal to the non-vanishing of the tautological class $\kappa_{g-2}$ on the moduli stack $\mathcal{M}_{g}$ of smooth curves of genus $g\geq 2$. The third application of generalized analytic torsion classes is the construction of direct images of hermitian structures. We consider the category $\operatorname{\overline{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$ introduced in [17]. The objects of this category are smooth complex varieties, and the morphisms are projective morphisms equipped with a hermitian structure on the relative tangent complex. Assume that we have chosen a theory of generalized analytic torsion classes. Let $\overline{f}\colon X\to Y$ be a morphism in $\operatorname{\overline{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. One would like to define a direct image functor $f_{\ast}\colon\operatorname{\overline{\mathbf{D}}^{b}}(X)\to\operatorname{\overline{\mathbf{D}}^{b}}(Y)$. It turns out that, using analytic torsion, we can not define the direct image functor on the category $\operatorname{\overline{\mathbf{D}}^{b}}$ and we have to introduce a new category $\operatorname{\mathbf{\widehat{D}}^{b}}$. Roughly speaking, the relation between $\operatorname{\mathbf{\widehat{D}}^{b}}$ and $\operatorname{\mathbf{D}^{b}}$, is the same as the relation between the arithmetic $K$-groups and the usual $K$-groups ([28]). Then we are able to define a direct image functor $f_{\ast}\colon\operatorname{\mathbf{\widehat{D}}^{b}}(X)\to\operatorname{\mathbf{\widehat{D}}^{b}}(Y)$, that satisfies the composition rule, projection formula and base change. Moreover, if the theory of generalized analytic torsion is self-dual (as the Bismut-Köhler theory) this functor satisfies a Grothendieck duality theorem. In a forthcoming paper, the direct image functor will be the base of an arithmetic Grothendieck-Riemann-Roch theorem for projective morphisms. The last application that we discuss is a new proof of a theorem of Bismut- Bost on the singularity of the Quillen metric for degenerating families of curves, whose singular fibers have at most ordinary double points [6]. In contrast with _loc. cit._ , where the spectral definition of the Ray-Singer analytic torsion is required, our arguments rely on the existence of a generalized theory for arbitrary projective morphisms and some elementary computations of Bott-Chern classes. This theorem has already been generalized by Bismut [4] and Yoshikawa [49] to families of varieties of arbitrary dimension. In fact, our approach is very similar to the one in [4] and [49]. One of the main ingredients of their proof is Bismut-Lebeau immersion formula, while our approach uses implicitly Bismut’s generalization of the immersion formula in the comparison between Bismut-Köhler analytic torsion and a theory of generalized analytic torsion classes. But what we want to emphasize is that, once we have identified Bismut-Köhler as (part of) a theory of generalized analytic torsion classes, many arguments can be simplified considerably because the theory has been extended to non-smooth projective morphism. For simplicity, we treat only the case of families of curves and the Quillen metric, but the methods can be applied to higher dimensional families and analytic torsion forms of higher degree. A few words about notations. The normalizations of characteristic classes and Bott-Chern classes in this paper differ from the ones used by Bismut, Gillet- Soulé and other authors. The first difference is that they work with real valued characteristic classes, while we use characteristic classes in Deligne cohomology, that naturally include the algebro-geometric twist. The second difference is a factor $1/2$ in Bott-Chern classes, that explains the factor $1/2$ that appears in the characteristic class associated to the torsion classes of Bismut-Köhler. This change of normalization appears already in [16] and its objective is to avoid the factor $1/2$ that appears in the definition of arithmetic degree in [27, §3.4.3] and the factor $2$ that appears in [27, Theorem 3.5.4] when relating Green currents with Beilinson regulator. The origin of this factor is that the natural second order differential equation that appears when defining Deligne-Beilinson cohomology is $\operatorname{d}_{\mathcal{D}}=-2\partial\bar{\partial}$, while the operator used when dealing with real valued forms is $\operatorname{d}\operatorname{d}^{c}=\frac{1}{4\pi i}\operatorname{d}_{\mathcal{D}}.$ Thus the characteristic classes that appear in the present article only agree with the ones in the papers of Bismut, Gillet and Soulé after renormalization. With respect to the work of these authors we have also changed the sign of the differential equation that characterizes singular Bott-Chern classes. In this way, the same differential equation appears when considering both, singular Bott-Chern classes and analytic torsion classes. This change is necessary to combine them. We point out that our construction of generalized analytic torsion classes is influenced by the thesis of Zha [50], where the author uses implicitly a theory of analytic torsion classes different from that of Bismut-Köhler. In the unpublished e-print [46], L. Weng gives another axiomatic approach to analytic torsion classes, only for smooth morphisms between Kähler fibrations. This forces him to include a continuity condition with respect to the deformation to the normal cone as one of the axioms. The remaining axioms he uses are: the differential equation, functoriality with respect to cartesian squares, compatibility with respect to the projection formula and two anomaly formulas. A collection of differential forms satisfying these axioms are called relative Bott-Chern secondary characteristic classes. These characteristic classes are not unique. The main result of Weng’s paper is that any two such theories are related by an additive genus. Moreover he is able to obtain a weak form of the existence theorem for relative Bott-Chern secondary characteristic classes. Further applications of the generalized analytic torsion classes are left for future work. We plan to prove generalizations of the arithmetic Grothendieck- Riemann-Roch theorem of Gillet-Soulé [30] and Gillet-Rössler-Soulé [25] to arbitrary projective morhisms, along the lines of [19]. It is possible to compute explicitly the characteristic numbers of the unique theory of analytic torsion classes for projective spaces compatible with the homogeneous theory for closed immersions. This computation makes the characterization of generalized analytic torsion classes more precise. Nevertheless, since this computation is much more transparent when written in terms of properties of arithmetic Chow groups and the Riemann-Roch theorem, we leave it to the paper devoted to the arithmetic Riemann-Roch theorem. We also plan to study the possible axiomatic characterization of equivariant analytic torsion classes. Note that the characterization of equivariant singular Bott-Chern forms has already been obtained by Tang in [45]. ## 2 Deligne complexes, transverse morphisms and relative metrized complexes In this section we fix the notations and conventions used through the article, we also recall the definition of transverse morphisms and we review some basic properties. Finally we introduce the notion of relative metrized complex, and explain some basic constructions. The natural context where one can define the Bott-Chern classes and the analytic torsion classes is that of Deligne complexes. For the convenience of the reader we will summarize in this section the basic facts about the Deligne complexes we will use in the sequel. For more details the reader is referred to [16] and [18]. ###### Definition 2.1. A _Dolbeault complex_ $A=(A^{\ast}_{\mathbb{R}},\operatorname{d}_{A})$ is a bounded below graded complex of real vector spaces equipped with a compatible bigrading on $A_{\mathbb{C}}=A_{\mathbb{R}}\otimes_{\mathbb{R}}{\mathbb{C}}$, i.e., $A^{n}_{\mathbb{C}}=\bigoplus_{p+q=n}A^{p,q},$ satisfying the following properties: 1. (i) The differential $\operatorname{d}_{A}$ can be decomposed as the sum $\operatorname{d}_{A}=\partial+\bar{\partial}$ of operators $\partial$ of type $(1,0)$, respectively $\bar{\partial}$ of type $(0,1)$. 2. (ii) The symmetry property $\overline{A^{p,q}}=A^{q,p}$ holds, where $\overline{\phantom{M}}$ denotes complex conjugation. The basic example of Dolbeault complex is the complex of differential forms on a smooth variety $X$ over ${\mathbb{C}}$, denoted $E^{\ast}(X)_{{\mathbb{R}}}$. Following [18, §5.2], to a Dolbeault complex one assigns a Deligne complex denoted $\mathcal{D}^{\ast}(A,\ast)$. In this paper we will only use the following pieces of this complex: $\displaystyle\mathcal{D}^{2p+1}(A,p)$ $\displaystyle=(A^{p,p+1}\oplus A^{p+1,p})\cap(2\pi i)^{p}A^{2p+1}_{{\mathbb{R}}},$ $\displaystyle\mathcal{D}^{2p}(A,p)$ $\displaystyle=A^{p,p}\cap(2\pi i)^{p}A^{2p}_{{\mathbb{R}}},$ $\displaystyle\mathcal{D}^{2p-1}(A,p)$ $\displaystyle=A^{p-1,p-1}\cap(2\pi i)^{p-1}A^{2p-2}_{{\mathbb{R}}},$ $\displaystyle\mathcal{D}^{2p-2}(A,p)$ $\displaystyle=(A^{p-2,p-1}\oplus A^{p-1,p-2})\cap(2\pi i)^{p-1}A^{2p-3}_{{\mathbb{R}}}.$ The differential of the Deligne complex, denoted by $\operatorname{d}_{\mathcal{D}}\colon\mathcal{D}^{n}(A,p)\to\mathcal{D}^{n+1}(A,p)$ is given, in the above degrees by $\displaystyle\text{if }\eta$ $\displaystyle\in\mathcal{D}^{2p}(A,p),$ $\displaystyle\quad\operatorname{d}_{\mathcal{D}}\eta$ $\displaystyle=\operatorname{d}\eta,$ $\displaystyle\text{if }\eta$ $\displaystyle\in\mathcal{D}^{2p-1}(A,p),$ $\displaystyle\quad\operatorname{d}_{\mathcal{D}}\eta$ $\displaystyle=-2\partial\bar{\partial}\eta,$ $\displaystyle\text{if }\eta=(u,v)$ $\displaystyle\in\mathcal{D}^{2p-2}(A,p),$ $\displaystyle\quad\operatorname{d}_{\mathcal{D}}\eta$ $\displaystyle=-\partial u-\bar{\partial}v.$ When $A$ is a Dolbeault algebra, that is, $A$ is a graded commutative real differential algebra and the product is compatible with the bigrading, then $\mathcal{D}^{\ast}(A,\ast)$ has a product $\bullet\colon\mathcal{D}^{n}(A,p)\otimes\mathcal{D}^{m}(A,q)\longrightarrow\mathcal{D}^{n+m}(A,p+q)$ that is graded commutative with respect to the first degree, it is associative up to homotopy and satisfies the Leibnitz rule. The only case where we will need the explicit formula for the product is for $\omega\in\mathcal{D}^{2p}(A,p)$ and $\eta\in\mathcal{D}^{m}(A,q)$: $\omega\bullet\eta=\omega\land\eta.$ The _Deligne algebra of differential forms_ on $X$ is defined to be $\mathcal{D}^{\ast}(X,\ast):=\mathcal{D}^{\ast}(E^{\ast}(X)_{{\mathbb{R}}},\ast).$ If $X$ is equi-dimensional of dimension $d$, there is a natural trace map given by $\int\colon H^{2d}_{c}(X,{\mathbb{R}}(d))\to{\mathbb{R}},\quad\omega\longmapsto\frac{1}{(2\pi i)^{d}}\int_{X}\omega.$ To take this trace map into account the Dolbeault complex of currents is constructed as follows. Denote by $E^{\ast}_{c}(X)_{{\mathbb{R}}}$ the space of differential forms with compact support. Then $D_{p,q}(X)$ is the topological dual of $E^{p,q}_{c}(X)$ and $D_{n}(X)_{{\mathbb{R}}}$ is the topological dual of $E^{n}_{c}(X)_{{\mathbb{R}}}$. In this complex the differential is given by $\operatorname{d}T(\eta)=(-1)^{n}T(\operatorname{d}\eta)$ for $T\in D_{n}(X)_{{\mathbb{R}}}$. For $X$ equi-dimensional of dimension $d$ we write $D^{p,q}(X)=D_{d-p,d-q}(X),\qquad D^{n}(X)_{{\mathbb{R}}}=(2\pi i)^{-d}D_{2d-n}(X).$ With these definitions, $D^{\ast}(X)_{{\mathbb{R}}}$ is a Dolbeault complex and it is a Dolbeault module over $E^{\ast}(X)_{{\mathbb{R}}}$. We will denote $\mathcal{D}_{D}^{\ast}(X,\ast):=\mathcal{D}^{\ast}(D^{\ast}(X)_{{\mathbb{R}}},\ast).$ for the Deligne complex of currents on $X$. The trace map above defines an element $\delta_{X}\in\mathcal{D}_{D}^{0}(X,0).$ More generally, if $Y\subset X$ is a subvariety of pure codimension $p$, then the current integration along $Y$, denoted $\delta_{Y}\in\mathcal{D}_{D}^{2p}(X,p)$ is given by $\delta_{Y}(\omega)=\frac{1}{(2\pi i)^{d-p}}\int_{Y}\omega.$ Moreover, if $S\subset T^{\ast}X_{0}$ is a closed conical subset of the cotangent bundle of $X$ with the zero section removed, we will denote by $(\mathcal{D}_{D}^{\ast}(X,S,\ast),\operatorname{d}_{\mathcal{D}})$ the Deligne complex of currents on $X$ whose wave front set is contained in $S$. For instance, if $N^{\ast}_{Y}$ is the conormal bundle to $Y$, then $\delta_{Y}\in\mathcal{D}^{2p}_{D}(X,N^{\ast}_{Y},p).$ If $\omega$ is a locally integrable differential form, we associate to it a current $[\omega](\eta)=\frac{1}{(2\pi i)^{\dim X}}\int_{X}\eta\land\omega.$ This map induces an isomorphism $\mathcal{D}^{\ast}(X,\ast)\to\mathcal{D}_{D}^{\ast}(X,\emptyset,\ast)$ that we use to identify them. For instance, when in a formula sums of currents and differential forms appear, we will tacitly assume that the differential forms are converted into currents by this map. Note also that, if $f\colon X\to Y$ is a proper morphism of smooth complex varieties of relative dimension $e$, then there are direct image morphisms $f_{\ast}\colon\mathcal{D}^{n}_{D}(X,p)\longrightarrow\mathcal{D}^{n-2e}_{D}(X,p-e).$ If $f$ is smooth, the direct image of differential forms is defined by, first converting them into currents and then applying the above direct image of currents. If $f$ is a smooth morphism of relative dimension $e$ we can convert them back into differential forms. This procedure gives us $1/(2\pi i)^{e}$ times the usual integration along the fiber. We shall use the notations and definitions of [19]. In particular, we write $\displaystyle\widetilde{\mathcal{D}}^{n}(X,p)$ $\displaystyle=\left.\mathcal{D}^{n}(X,p)\right/\operatorname{d}_{\mathcal{D}}\mathcal{D}^{n-1}(X,p),$ $\displaystyle\widetilde{\mathcal{D}}^{n}_{D}(X,p)$ $\displaystyle=\left.\mathcal{D}^{n}_{D}(X,p)\right/\operatorname{d}_{\mathcal{D}}\mathcal{D}^{n-1}_{D}(X,p).$ We now recall the definition of the set of normal direction of a map and the definition of transverse morphisms. ###### Definition 2.2. Let $f\colon X\to Y$ be a morphism of smooth complex varieties. Let $T^{\ast}Y_{0}$ be the cotangent bundle to $Y$ with the zero section removed. The _set of normal directions of_ $f$ is the conic subset of $T^{\ast}Y_{0}$ given by $N_{f}=\\{(y,v)\in T^{\ast}Y_{0}\|\operatorname{d}f^{t}v=0\\}.$ ###### Definition 2.3. Let $f\colon X\to Y$ and $g\colon Z\to Y$ be morphisms of smooth complex varieties. We say that $f$ and $g$ are _transverse_ if $N_{f}\cap N_{g}=\emptyset.$ It is easily seen that, if $f$ is a closed immersion, this definition of transverse morphisms agrees with that given in [31, IV-17.13]. If $f$ and $g$ are transverse, then the cartesian product $X\underset{Y}{\times}Z$ is smooth. For lack of a good reference we prove the following result. ###### Proposition 2.4. Let $f\colon X\to Y$ and $g\colon Z\to Y$ be transverse morphisms of smooth complex varieties. Then they are tor-independent. ###### Proof. Since the conditions of being transverse and being tor-independent are both local on $Y$, $X$ and $Z$ we may assume that the map $f$ factorizes as $X\overset{i}{\to}Y\times\mathbb{A}^{n}\overset{p}{\to}Y$, where $i$ is a closed immersion and $p$ is the projection. Let $g^{\prime}\colon Z\times\mathbb{A}^{n}\to Y\times\mathbb{A}^{n}$ be the morphism $g\times\operatorname{id}$. If $f$ and $g$ are transverse then $i$ and $g^{\prime}$ are transverse. While, if $i$ and $g^{\prime}$ are tor- independent then $f$ and $g$ are tor-independent. Hence we may suppose that $f$ is a closed immersion. Since every closed immersion between smooth schemes is regular, we may assume that $Y=\operatorname{Spec}A$, $X=\operatorname{Spec}A/I$, where $I$ is an ideal generated by a regular sequence $(s_{1},\dots,s_{k})$ and $Z=\operatorname{Spec}B$. The transversality condition implies that $(s_{1},\dots,s_{k})$ is a regular sequence generating $IB$. Let $K$ be the Koszul resolution of $A/I$ attached to the above sequence. Then $K\otimes_{A}B$ is the Koszul resolution of $B/IB$, hence exact. Therefore, $\operatorname{Tor}_{A}^{i}(A/I,B)=0$ for all $i\geq 1$. Thus $f$ and $g$ are tor-independent. ∎ Let now $Y^{\prime\prime}\overset{h}{\to}Y^{\prime}\overset{g}{\to}Y$ be morphisms of smooth complex varieties such that $g$ and $g\circ h$ are smooth. We form the cartesian diagram $\textstyle{X^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime\prime}}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Y.}$ The smoothness of $g$ implies that $N_{f^{\prime}}=g^{\ast}N_{f}$. Then the smoothness of $g\circ h$ implies that $h$ and $f^{\prime}$ are transverse. Therefore, any current $\eta\in\mathcal{D}_{D}^{\ast}(Y^{\prime},N_{f^{\prime}},\ast)$ can be pulled back to a current $h^{\ast}\eta\in\mathcal{D}_{D}^{\ast}(Y^{\prime\prime},N_{f^{\prime\prime}},\ast)$. The following result will be used to characterize several Bott-Chern classes and analytic torsion classes. ###### Lemma 2.5. Let $f\colon X\to Y$ be a morphism of smooth complex varieties. Let $\widetilde{\varphi}$ be an assignment that, to each smooth morphism of complex varieties $g\colon Y^{\prime}\to Y$ and each acyclic complex $\overline{A}$ of hermitian vector bundles on $X^{\prime}:=X\underset{Y}{\times}Y^{\prime}$ assigns a class $\widetilde{\varphi}(\overline{A})\in\bigoplus_{n,p}\widetilde{\mathcal{D}}_{D}^{n}(Y^{\prime},g^{\ast}N_{f},p)$ fulfilling the following properties: 1. (i) (Differential equation) the equality $\operatorname{d}_{\mathcal{D}}\widetilde{\varphi}(\overline{A})=0$ holds; 2. (ii) (Functoriality) for each morphism $h\colon Y^{\prime\prime}\to Y^{\prime}$ of smooth complex varieties with $g\circ h$ smooth, the relation $h^{\ast}\widetilde{\varphi}(\overline{A})=\widetilde{\varphi}(h^{\ast}\overline{A})$ holds; 3. (iii) (Normalization) if $\overline{A}$ is orthogonally split, then $\widetilde{\varphi}(\overline{A})=0$. Then $\widetilde{\varphi}=0$. ###### Proof. The argument of the proof of [19, Thm. 2.3] applies _mutatis mutandis_ to the present situation. One only needs to observe that all the operations with differential forms of that argument can be extended to the currents that appear in the present situation thanks to the hypothesis about their wave front sets. ∎ In the paper [17] we defined and studied hermitian structures on objects of the bounded derived category of coherent sheaves on a smooth complex variety. The language and the results of _loc. cit._ will be used extensively in this paper. We just mention here that a hermitian metric on an object $\mathcal{F}$ of $\operatorname{\mathbf{D}^{b}}(X)$ is an isomorphism $E\dashrightarrow\mathcal{F}$ in $\operatorname{\mathbf{D}^{b}}(X)$, with $E$ a bounded complex of vector bundles, together with a choice of a hermitian metric on each constituent vector bundle of $E$. Such an isomorphism always exists due to the fact that we work in the algebraic category. A hermitian structure is an equivalence class of hermitian metrics. To each smooth complex variety $X$, we associated the category $\operatorname{\overline{\mathbf{D}}^{b}}(X)$ ([17, § 3]) whose objects are objects of $\operatorname{\mathbf{D}^{b}}(X)$ provided with a hermitian structure. We introduced the hermitian cone ([17, Def. 3.14]), denoted $\operatorname{\overline{cone}}$, of a morphism in $\operatorname{\overline{\mathbf{D}}^{b}}(X)$. We also defined Bott-Chern classes for isomorphisms ([17, Thm. 4.11]) and distinguished triangles ([17, Thm. 4.18]) in $\operatorname{\overline{\mathbf{D}}^{b}}(X)$. We introduced a universal abelian group for additive Bott-Chern classes. Namely, the set of hermitian structures on a zero object of $\operatorname{\mathbf{D}^{b}}(X)$ is an abelian group that we denote $\operatorname{\mathbf{KA}}(X)$ ([17, Def. 2.31]). Finally, we defined the category $\operatorname{\overline{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$ ([17, § 5]) whose objects are smooth complex varieties and whose morphisms are projective morphisms together with a hermitian structure on the relative tangent complex. We introduce now one of the central objects of the paper. ###### Definition 2.6. Let $f\colon X\to Y$ be a projective morphism of smooth complex varieties and $\overline{f}\in\operatorname{Hom}_{\operatorname{\overline{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}}(X,Y)$ a morphism over $f$. Let $\overline{\mathcal{F}}\in\operatorname{Ob}\operatorname{\overline{\mathbf{D}}^{b}}(X)$ and let $\overline{f_{\ast}\mathcal{F}}\in\operatorname{Ob}\operatorname{\overline{\mathbf{D}}^{b}}(Y)$ be an object over $f_{\ast}\mathcal{F}$. The triple $\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$ will be called _a relative metrized complex_. When $f$ is a closed immersion we will also call it an _embedded metrized complex_. When $\overline{\mathcal{F}}$ and $\overline{f_{\ast}\mathcal{F}}$ are clear from the context we will denote the relative metrized complex $\overline{\xi}$ by the morphism $\overline{f}$. Let $\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$ be a relative metrized complex and let $g\colon Y^{\prime}\to Y$ be a morphism of smooth complex varieties that is transverse to $f$. Consider the cartesian diagram $\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f^{\prime}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Y.}$ (2.7) Then $f^{\prime}$ is still projective. Moreover, the transversality condition implies that the canonical morphism ${g^{\prime}}^{\ast}T_{\overline{f}}\dashrightarrow T_{f^{\prime}}$ is a hermitian structure on $T_{f^{\prime}}$. We define $g^{\ast}\overline{f}=(f^{\prime},{g^{\prime}}^{\ast}T_{\overline{f}})\in\operatorname{Hom}_{\operatorname{\overline{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}}(X^{\prime},Y^{\prime}).$ (2.8) By tor-independence, there is a canonical isomorphism $g^{\ast}f_{\ast}\mathcal{F}\dashrightarrow f^{\prime}_{\ast}{g^{\prime}}^{\ast}\mathcal{F}.$ Therefore $g^{\ast}\overline{f_{\ast}\mathcal{F}}$ induces a hermitian structure on $f^{\prime}_{\ast}{g^{\prime}}^{\ast}\mathcal{F}$. ###### Definition 2.9. The _pull-back_ of $\overline{\xi}$ by $g$ is the relative metrized complex $g^{\ast}\overline{\xi}=(g^{\ast}\overline{f},{g^{\prime}}^{\ast}\overline{\mathcal{F}},g^{\ast}\overline{f_{\ast}\mathcal{F}}).$ ###### Definition 2.10. Let $\overline{\xi}=(\overline{f}\colon X\to Y,\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$ be a relative metrized complex. Let $\overline{\mathcal{G}}$ be an object of $\operatorname{\overline{\mathbf{D}}^{b}}(Y)$. The hermitian structures on $\overline{f_{\ast}\mathcal{F}}$ and $\overline{\mathcal{G}}$ induce a natural hermitian structure on $f_{\ast}(\mathcal{F}\otimes f^{\ast}\mathcal{G})$ that we denote $\overline{f_{\ast}\mathcal{F}}\otimes\overline{\mathcal{G}}$. The _tensor product of $\overline{\xi}$ by $\overline{\mathcal{G}}$_ is then defined to be the relative metrized complex $\overline{\xi}\otimes\overline{\mathcal{G}}=(\overline{f},\overline{\mathcal{F}}\otimes f^{\ast}\overline{\mathcal{G}},\overline{f_{\ast}\mathcal{F}}\otimes\overline{\mathcal{G}}).$ ###### Definition 2.11. Let $\overline{\xi}_{i}=(\overline{f},\overline{\mathcal{F}_{i}},\overline{f_{\ast}\mathcal{F}_{i}})$, $i=1,2$ be relative metrized coherent complexes on $X$. Then _the direct sum relative metrized complex_ is $\overline{\xi}_{1}\oplus\overline{\xi}_{2}:=(\overline{f},\overline{\mathcal{F}_{1}}\oplus\overline{\mathcal{F}_{2}},\overline{f_{\ast}\mathcal{F}_{1}}\oplus\overline{f_{\ast}\mathcal{F}_{2}}).$ We now introduce a notation for Todd-twisted direct images of currents and differential forms, that will simplify many formulas involving the Todd genus. Let $\overline{f}=(f,\overline{T}_{f})$ be a morphism in $\operatorname{\overline{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. To $\overline{f}$ we associate a Todd differential form $\operatorname{Td}(\overline{f}):=\operatorname{Td}(\overline{T}_{f})\in\bigoplus_{p}\mathcal{D}^{2p}(X,p)$ [17, (5.15)]. Let $S$ be a closed conic subset of $T^{\ast}X_{0}$. Then we denote $f_{\ast}(S)=\\{(f(x),\eta)\in T^{\ast}Y_{0}\mid(x,(\operatorname{d}f)^{t}\eta)\in S\\}\cup N_{f}.$ (2.12) If $g\colon Y\to Z$ is another morphism of smooth complex varieties, it is easy to see that we have $(g\circ f)_{\ast}(S)\subseteq g_{\ast}f_{\ast}(S)$. ###### Definition 2.13. Let $\overline{f}\colon X\to Y$ be a morphism in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$ of relative dimension $e$. For each closed conical subset $S\subset T^{\ast}X_{0}$ and each pair of integers $n$, $p$, we define the map $\overline{f}_{\flat}\colon\mathcal{D}^{n}_{D}(X,S,p)\to\mathcal{D}^{n-2e}_{D}(Y,f_{\ast}S,p-e),\quad\overline{f}_{\flat}(\omega)=f_{\ast}(\omega\bullet\operatorname{Td}(\overline{f})).$ ###### Proposition 2.14. Let $\overline{f}\colon X\to Y$ and $\overline{g}\colon Y\to Z$ be morphisms in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$ of relative dimensions $e_{1}$ and $e_{2}$ respectively. Let $S\subset T^{\ast}X_{0}$, $T\subset T^{\ast}Y_{0}$ be closed conical subsets and let $\overline{h}=\overline{f}\circ\overline{g}$, of relative dimension $e=e_{1}+e_{2}$. 1. (i) The following diagram is commutative $\textstyle{\mathcal{D}_{D}^{n}(X,S,p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{f}_{\flat}}$$\scriptstyle{\overline{h}_{\flat}}$$\textstyle{\mathcal{D}_{D}^{n-2e_{1}}(Y,f_{\ast}S,p-e_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{g}_{\flat}}$$\textstyle{\mathcal{D}_{D}^{n-2e}(Z,h_{\ast}S,p-e)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{D}_{D}^{n-2e}(Z,g_{\ast}f_{\ast}S,p-e).}$ 2. (ii) Let $\theta\in\mathcal{D}_{D}^{m}(X,S,q)$ and $\omega\in\mathcal{D}_{D}^{n}(Y,T,p)$. Assume $T\cap N_{f}=\emptyset$ and that $T+f_{\ast}S$ is disjoint with the zero section in $T^{\ast}Y_{0}$. Then $f^{\ast}T+S$ is disjoint with the zero section and there is an equality of currents $\overline{f}_{\flat}(f^{\ast}(\omega)\bullet\theta)=\omega\bullet\overline{f}_{\flat}(\theta)$ in $\mathcal{D}_{D}^{n+m}(Y,W,p+q)$, with $W=f_{\ast}(S+f^{\ast}T)\cup f_{\ast}S\cup f_{\ast}f^{\ast}T.$ ###### Proof. For the first assertion, it is enough to notice the equality of currents $\overline{g}_{\flat}(\overline{f}_{\flat}(\omega))=(g\circ f)_{\ast}(\omega\bullet f^{\ast}\operatorname{Td}(\overline{g})\bullet\operatorname{Td}(\overline{f}))).$ For the second, it is easy to see that $f^{\ast}T+S$ does not cross the zero section, and hence both sides of the equality are defined. It then suffices to establish the equality of currents $f_{\ast}(f^{\ast}\omega\bullet\theta)=f_{\ast}(\omega)\bullet\theta.$ If $\theta$ and $\omega$ are smooth, then the equality follows from the definitions. The general case follows by approximation of $\theta$ and $\omega$ by smooth currents and the continuity of the operators $f^{\ast}$ and $f_{\ast}$. ∎ ###### Proposition 2.15. Let $\overline{f}$ be a morphism in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$ of relative dimension $e$ and $S$ a closed conical subset of $T^{\ast}X_{0}$. Let $g\colon Y^{\prime}\to Y$ be a morphism of smooth complex varieties transverse to $f$. Consider the cartesian diagram (2.7) and let $\overline{f}^{\prime}=g^{\ast}\overline{f}$. Suppose that $N_{g^{\prime}}$ is disjoint with $S$. Then: 1. (i) $N_{g}$ and $f_{\ast}S$ are disjoint and $g^{\ast}f_{\ast}S\subset f^{\prime}_{\ast}{g^{\prime}}^{\ast}S$; 2. (ii) the following diagram commutes: $\textstyle{\mathcal{D}_{D}^{n}(X,S,p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{f}_{\flat}}$$\scriptstyle{{g^{\prime}}^{\ast}}$$\textstyle{\mathcal{D}_{D}^{n-2e}(Y,f_{\ast}S,p-e)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\ast}}$$\textstyle{\mathcal{D}_{D}^{n}(X^{\prime},{g^{\prime}}^{\ast}S,p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{f}^{\prime}_{\flat}}$$\textstyle{\mathcal{D}_{D}^{n-2e}(Y^{\prime},f^{\prime}_{\ast}{g^{\prime}}^{\ast}S,p-e)}$ ###### Proof. The first claim follows from the definitions. In particular the diagram makes sense. For the commutativity of the diagram, we observe that, since $g^{\prime}{}^{\ast}\operatorname{Td}(\overline{f})=\operatorname{Td}(\overline{f}^{\prime}),$ it suffices to check the equality of currents $g^{\ast}f_{\ast}(\theta)={f^{\prime}}_{\ast}{g^{\prime}}^{\ast}(\theta)$ for $\theta\in\mathcal{D}_{D}^{n}(X,S,p)$. By the continuity of the operators $g^{\ast}$, ${g^{\prime}}^{\ast}$, $f_{\ast}$ and ${f^{\prime}}_{\ast}$, it is enough to prove the relation whenever $\theta$ is smooth. Moreover, using a partition of unity argument we are reduced to the following local analytic statement. ###### Lemma 2.16. Let $f\colon X\to Y$ and $g\colon Y^{\prime}\to Y$ be transverse morphisms of complex manifolds. Let $\theta$ be a smooth differential form on $X$ with compact support. Consider the diagram (2.7). Then $g^{\ast}f_{\ast}(\theta)={f^{\prime}}_{\ast}{g^{\prime}}^{\ast}(\theta).$ (2.17) _Proof._ The map $f$ can be factored as $X\overset{\varphi}{\longrightarrow}X\times Y\overset{p_{2}}{\longrightarrow}Y$, where $\varphi(x)=(x,f(x))$ is a closed immersion and $p_{2}$, the second projection, is smooth. Using again the continuity of the operators $g^{\ast}$ (respectively ${g^{\prime}}^{\ast}$) and $f_{\ast}$ (respectively ${f^{\prime}}^{\ast}$), we are reduced to prove the equation (2.17) in the case when $f$ is smooth and in the case when $f$ is a closed immersion. The case when $f$ is smooth is clear. Assume now that $f$ is a closed immersion. By transversality, $f^{\prime}$ is also a closed immersion of complex manifolds. We may assume that $\theta=f^{\ast}\widetilde{\theta}$ for some smooth form $\widetilde{\theta}$ on $Y$. Then equation (2.17) follows from the chain of equalities $g^{\ast}f_{\ast}\theta=g^{\ast}f_{\ast}f^{\ast}\widetilde{\theta}=g^{\ast}(\widetilde{\theta}\land\delta_{X})=g^{\ast}(\widetilde{\theta})\land\delta_{X^{\prime}}=f^{\prime}_{\ast}f^{\prime}{}^{\ast}g^{\ast}\widetilde{\theta}=f^{\prime}_{\ast}g^{\prime}{}^{\ast}f^{\ast}\widetilde{\theta}=f^{\prime}_{\ast}g^{\prime}{}^{\ast}\theta.$ This concludes the proof of the lemma and the proposition. $\square$ ∎ ## 3 Analytic torsion for closed immersions In the paper [19] the authors study the singular Bott-Chern classes associated to closed immersions of smooth complex varieties. The singular Bott-Chern classes are the analogue, for closed immersions, of the analytic torsion for smooth morphisms. For this reason, we will call them also analytic torsion classes. The aim of this section is to recall the main results of [19] and to translate them into the language of derived categories. ###### Definition 3.1. A _theory of analytic torsion classes for closed immersions_ is a map that, to each embedded metrized complex $\overline{\xi}=(\overline{f}\colon X\to Y,\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$ assigns a class $T(\overline{\xi})\in\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(Y,N_{f},p)$ satisfying the following conditions. 1. (i) (Differential equation) The equality $\operatorname{d}_{\mathcal{D}}T(\overline{\xi})=\operatorname{ch}(\overline{f_{\ast}\mathcal{F}})-\overline{f}_{\flat}[\operatorname{ch}(\overline{\mathcal{F}})]$ holds. 2. (ii) (Functoriality) For every morphism $h\colon Y^{\prime}\to Y$ of smooth complex varieties that is transverse to $f$ we have the equality $h^{\ast}T(\overline{\xi})=T(h^{\ast}\overline{\xi}).$ 3. (iii) (Normalization) If $X=\emptyset$ (hence $\overline{\mathcal{F}}=\overline{0}$), $Y=\operatorname{Spec}{\mathbb{C}}$, and $\overline{f_{\ast}\mathcal{F}}=\overline{0}$, then $T(\overline{f},\overline{0},\overline{0})=0.$ ###### Definition 3.2. Let $T$ be a theory of analytic torsion classes for closed immersions. 1. (i) We say that $T$ is _compatible with the projection formula_ if, for every embedded metrized complex $\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$, and every object $\overline{\mathcal{G}}\in\operatorname{\overline{\mathbf{D}}^{b}}(Y)$, we have $T(\overline{\xi}\otimes\overline{\mathcal{G}})=T(\overline{\xi})\bullet\operatorname{ch}(\overline{\mathcal{G}}).$ (3.3) 2. (ii) We say that $T$ is _additive_ if, given $\overline{\xi}_{i}=(\overline{f},\overline{\mathcal{F}_{i}},\overline{f_{\ast}\mathcal{F}_{i}})$, $i=1,2$, two embedded metrized complexes, we have $T(\overline{\xi}_{1}\oplus\overline{\xi}_{2})=T(\overline{\xi}_{1})+T(\overline{\xi}_{2}).$ (3.4) 3. (iii) We say that $T$ is _transitive_ if, given a embedded metrized complex $\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$, a closed immersion of smooth complex varieties $g\colon Y\to Z$, a morphism $\overline{g}$ over $g$, and an object $\overline{(g\circ f)_{\ast}\mathcal{F}}\in\operatorname{Ob}\operatorname{\overline{\mathbf{D}}^{b}}(Z)$ over $(g\circ f)_{\ast}\mathcal{F}$, we have $T(\overline{g}\circ\overline{f})=T(\overline{g})+\overline{g}_{\flat}(T(\overline{f})).$ (3.5) ###### Remark 3.6. 1. (i) If $T$ is well defined for objects of $\operatorname{\overline{\mathbf{D}}^{b}}$, then the normalization condition in Definition 3.1 and the normalization condition in [19, Def. 6.9] are equivalent. The compatibility with the projection formula implies the normalization condition and the additivity (see [19, Prop. 10.9]) 2. (ii) To check that a theory is compatible with the projection formula, it is enough to consider complexes consisting of a single hermitian vector bundle in degree 0. Let $X$ be a smooth complex variety and let $\overline{N}$ be a hermitian vector bundle of rank $r$. We denote by $P={\mathbb{P}}(N\oplus\mathbf{1})$ the projective bundle obtained by completing $N$. Let $\pi_{P}\colon P\to X$ be the projection and let $s\colon X\to P$ be the zero section. Since $N$ can be identified with the normal bundle to $X$ on $P$, the hermitian metric of $\overline{N}$ induces a hermitian structure on $s$. We will denote it by $\overline{s}$. On $P$ we have a tautological quotient vector bundle with an induced metric $\overline{Q}$. For each hermitian vector bundle $\overline{F}$ on $X$ we have the Koszul resolution $K(F,N)$ of $s_{\ast}F$. We denote by $K(\overline{F},\overline{N})$ the Koszul resolution with the induced metrics. See [19, Def. 5.3] for details. ###### Definition 3.7. Let $T$ be a theory of analytic torsion classes for closed immersions. We say that $T$ is _homogeneous_ if, for every pair of hermitian vector bundles $\overline{N}$ and $\overline{F}$ with $\operatorname{rk}N=r$, there exists a homogeneous class of bidegree $(2r-1,r)$ in the Deligne complex $\widetilde{e}(\overline{F},\overline{N})\in\widetilde{\mathcal{D}}^{2r-1}_{D}(P,N_{s},r)$ such that $T(\overline{s},\overline{F},K(\overline{F},\overline{N}))\bullet\operatorname{Td}(\overline{Q})=\widetilde{e}(\overline{F},\overline{N})\bullet\operatorname{ch}(\pi_{P}^{\ast}\overline{F}).$ (3.8) ###### Remark 3.9. Observe that Definition 3.7 is equivalent to [19, Def. 9.2]. The advantage of the definition in this paper is that it treats on equal footing the case when $\operatorname{rk}F=0$. Let ${\mathbb{D}}$ denote the base ring for Deligne cohomology (see [19] before Definition 1.5). A consequence of [19, Thm. 1.8] is that there is a bijection between the set of additive genus in Deligne cohomology and the set of power series in one variable ${\mathbb{D}}[[x]]$. To each power series $\varphi\in{\mathbb{D}}[[x]]$ it corresponds the unique additive genus such that $\varphi(L)=\varphi(c_{1}(L))$ for every line bundle $L$. ###### Definition 3.10. A _real additive genus_ is an additive genus such that the corresponding power series belongs to ${\mathbb{R}}[[x]]$. Let $\mathbf{1}_{1}\in{\mathbb{D}}$ be the element represented by the constant function 1 of $\mathcal{D}^{1}(\operatorname{Spec}\mathbb{C},1)={\mathbb{R}}$. The main result of [19] can be written as follows. ###### Theorem 3.11. 1. (i) There is a unique homogeneous theory of analytic torsion classes for closed immersions, that we denote $T^{h}$. This theory is compatible with the projection formula, additive and transitive. 2. (ii) Let $T$ be any transitive theory of analytic torsion classes for closed immersions, that is compatible with the projection formula. Then there is a unique real additive genus $S_{T}$ such that, for any embedded metrized complex $\overline{\xi}:=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$, we have $T(\overline{\xi})-T^{h}(\overline{\xi})=-f_{\ast}[\operatorname{ch}(\mathcal{F})\bullet\operatorname{Td}(T_{f})\bullet S_{T}(T_{f})\bullet\mathbf{1}_{1}].$ (3.12) 3. (iii) Conversely, any real additive genus $S$ defines, by means of equation (3.12), a unique transitive theory of analytic torsion classes $T_{S}$ for closed immersions, that is compatible with the projection formula and additive. ###### Proof. Existence and uniqueness for both $T^{h}$ and $T_{S}$ is the content of [19] when restricting to triples $\overline{\xi}$ with $T_{\overline{f}}=\overline{N}_{X/Y}[-1]$, $\overline{\mathcal{F}}$ a hermitian vector bundle placed in degree 0 and $\overline{f_{\ast}\mathcal{F}}$ given by a finite locally free resolution. For the general case, we thus need to prove that the theories of analytic torsion classes for closed immersions in the sense of _loc. cit._ uniquely extend to arbitrary $\overline{\xi}$, fulfilling the desired properties. Assume given a theory $T$ in the sense of [19], compatible with the projection formula and transitive. We will call $T$ the initial theory. We consider a triple $\overline{\xi}$ with $T_{\overline{f}}=\overline{N}_{X/Y}[-1]$ and $\overline{\mathcal{F}}\in\operatorname{Ob}\operatorname{\overline{\mathbf{D}}^{b}}(X)$. Choose a representative $\overline{F}\dashrightarrow\mathcal{F}$ of the hermitian structure on $\overline{\mathcal{F}}$. We then define $T(\overline{\xi})$ by induction on the length of the complex $F$. First suppose that $F=F^{d}[-d]$ consists of a single vector bundle placed in degree $d$. Choose a finite locally free resolution $\dots\to E^{1}\to E^{0}\to f_{\ast}F^{d}\to 0.$ Endow the vector bundles $E^{i}$ with smooth hermitian metrics. Observe that there is an induced isomorphism $\overline{E}[-d]\overset{\sim}{\dashrightarrow}\overline{f_{\ast}\mathcal{F}},$ in $\operatorname{\overline{\mathbf{D}}^{b}}(Y)$, whose Bott-Chern classes $\operatorname{\widetilde{ch}}$ are defined in [17, § 4]. We then put $T(\overline{\xi})=(-1)^{d}T(\overline{N}_{X/Y},\overline{F}^{d},\overline{E})+\widetilde{\operatorname{ch}}(\overline{E}[-d]\overset{\sim}{\dashrightarrow}\overline{f_{\ast}\mathcal{F}}).$ (3.13) This definition does not depend on the choice of representative of the hermitian structure on $\overline{\mathcal{F}}$, nor on the choice of $\overline{E}$, due to [17, Thm. 4.11, Prop. 4.13], and [19, Cor. 6.14]. The differential equation is satisfied as a consequence of the differential equations for $T(\overline{N}_{X/Y},\overline{F}^{d},\overline{E})$ and $\widetilde{\operatorname{ch}}(\overline{E}[-d]\overset{\sim}{\dashrightarrow}\overline{f_{\ast}\mathcal{F}})$. The compatibility with pull-back by morphisms $h\colon Y^{\prime}\to Y$ transverse to $f$ is immediate as well. Finally, notice that by construction, if $\overline{\xi}^{\prime}=(\overline{N}_{X/Y},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}}^{\prime})$, then $T(\overline{\xi^{\prime}})=T(\overline{\xi})+\widetilde{\operatorname{ch}}(\overline{f_{\ast}\mathcal{F}}^{\prime},\overline{f_{\ast}\mathcal{F}}).$ (3.14) Now suppose that $T(\overline{\xi})$ has been defined for $F$ of length $l$, satisfying in addition (3.14). If $F$ has length $l+1$, let $F^{d}$ be the first non-zero vector bundle of $F$. Consider the exact sequence of complexes $(\overline{\varepsilon})\qquad 0\to\sigma^{>d}\overline{F}\to\overline{F}\to\overline{F}^{d}[-d]\to 0,$ where $\sigma^{>d}$ is the bête filtration. Observe that as a distinguished triangle, $(\overline{\varepsilon})$ is tightly distinguished, hence $\widetilde{\operatorname{ch}}(\overline{\varepsilon})=0$. Choose hermitian metrics on $f_{\ast}\sigma^{>d}F$ and $f_{\ast}F^{d}[-d]$. We thus have a distinguished triangle in $\operatorname{\overline{\mathbf{D}}^{b}}(Y)$ $(\overline{\tau})\qquad\overline{f_{\ast}\sigma^{>d}F}\to\overline{f_{\ast}F}\to\overline{f_{\ast}F^{d}}[-d]\to\overline{f_{\ast}\sigma^{>d}F}[1]\to\dots.$ We define $T(\overline{\xi})=T(\overline{N}_{X/Y},\sigma^{>d}\overline{F},\overline{f_{\ast}\sigma^{>d}F})+(-1)^{d}T(\overline{N}_{X/Y},\overline{F}^{d},\overline{f_{\ast}F^{d}})-\widetilde{\operatorname{ch}}(\overline{\tau}).$ (3.15) This does not depend on the choice of hermitian structures on $f_{\ast}\sigma^{>d}F$ and $f_{\ast}F^{d}$, by the analogue to [17, Thm. 3.33 (vii)] for $\widetilde{\operatorname{ch}}$ and because (3.14) holds by assumption for $T(\overline{N}_{X/Y},\sigma^{>d}\overline{F},\overline{f_{\ast}\sigma^{>d}F})$ and $T(\overline{N}_{X/Y},\overline{F}^{d},\overline{f_{\ast}F^{d}})$. Similarly, (3.14) holds for $T(\overline{\xi})$ defined in (3.15). The differential equation and compatibility with pull-back are proven as in the first case. This concludes the proof of the existence in case that $T_{\overline{f}}=\overline{N}_{X/Y}[-1]$. To conclude with the existence, we may now consider a general $\overline{\xi}$. Choose a hermitian metric on the normal bundle $N_{X/Y}$. Put $\overline{\xi}^{\prime}=(\overline{N}_{X/Y}[-1],\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$. We define $T(\overline{\xi})=T(\overline{\xi}^{\prime})+\overline{f}_{\flat}[\operatorname{ch}(\overline{\mathcal{F}})\bullet\widetilde{\operatorname{Td}}_{m}(T_{\overline{f}}\dashrightarrow\overline{N}_{X/Y}[-1])],$ (3.16) where $\widetilde{\operatorname{Td}}_{m}$ is the multiplicative Todd secondary class defined in [17, § 5]. It is straightforward from the definition that $T(\overline{\xi})$ satisfies the differential equation and is compatible with pull-back by morphisms transverse to $f$. We call $T$ the extended theory. We now proceed to prove that the extended theory $T$ is transitive and compatible with the projection formula. For the projection formula, it suffices by Remark 3.6 (ii) to prove $T(\overline{\xi}\otimes\overline{G})=T(\overline{\xi})\bullet\operatorname{ch}(\overline{G})$ for $\overline{G}$ a hermitian vector bundle placed in degree 0. This readily follows from the inductive construction of the extended theory $T$ and the assumptions on the initial theory $T$. One similarly establishes the transitivity and the additivity We conclude by observing that, since Lemma 2.5 implies that the equations (3.13), (3.14), (3.15) and (3.16) hold, the theory $T(\overline{\xi})$ thus constructed for arbitrary $\overline{\xi}$ is completely determined by the values $T(\overline{\xi}^{\prime})$, with $\overline{\xi}^{\prime}$ of the form $(\overline{N}_{X/Y},\overline{F},\overline{E})$ where $\overline{F}$ is a hermitian vector bundle and $E\to f_{\ast}F$ is a finite locally free resolution. Once we have seen that any theory of singular Bott-Chern classes as in [19] can be uniquely extended, then statements (ii) and (iii) follow combining equation (7.3) and Corollary 9.43 in [19]. Note that the minus sign in equation (3.12) comes from the fact that $S(T_{f})=-S(N_{X/Y})$. ∎ In [19, §6] several anomaly formulas are proved. We now indicate the translation of these formulas to the current setting. Recall that we are using the notation of [17] with respect to secondary characteristic classes. ###### Proposition 3.17. Let $T$ be a theory of analytic torsion classes for closed immersions. Let $\overline{\xi}=(\overline{f}\colon X\to Y,\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$ be an embedded metrized complex. 1. (i) If $\overline{\mathcal{F}}^{\prime}$ is another choice of metric on $\mathcal{F}$ and $\overline{\xi}_{1}=(\overline{f}\colon X\to Y,\overline{\mathcal{F}}^{\prime},\overline{f_{\ast}\mathcal{F}})$, then $T(\overline{\xi}_{1})=T(\overline{\xi})+\overline{f}_{\flat}[\operatorname{\widetilde{ch}}(\overline{\mathcal{F}}^{\prime},\overline{\mathcal{F}})].$ 2. (ii) If $\overline{f}^{\prime}$ is another hermitian structure on $f$ and $\overline{\xi}_{2}=(\overline{f}^{\prime}\colon X\to Y,\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$, then $T(\overline{\xi}_{2})=T(\overline{\xi})+\overline{f}^{\prime}_{\flat}[\operatorname{ch}(\overline{\mathcal{F}})\bullet\widetilde{\operatorname{Td}}_{m}(\overline{f}^{\prime},\overline{f})].$ (3.18) 3. (iii) If $\overline{f_{\ast}\mathcal{F}}^{\prime}$ is another choice of metric on $f_{\ast}\mathcal{F}$, and $\overline{\xi}_{3}=(\overline{f}\colon X\to Y,\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}}^{\prime})$, then $T(\overline{\xi}_{3})=T(\overline{\xi})-\operatorname{\widetilde{ch}}(\overline{f_{\ast}\mathcal{F}}^{\prime},\overline{f_{\ast}\mathcal{F}}).$ ###### Proof. We first prove the second assertion. Let $\overline{E}\dashrightarrow T_{f}$ be a representative of the hermitian structure on $T_{\overline{f}}$. By [17, Thm. 3.13 (ii)], we may assume the hermitian structure on $T_{\overline{f}^{\prime}}$ is represented by the composition $\overline{E}\oplus\overline{A}\overset{\operatorname{pr}_{1}}{\longrightarrow}\overline{E}\dashrightarrow T_{f}$ for some bounded acyclic complex $\overline{A}$ of hermitian vector bundles on $X$. For every smooth morphism $g\colon Y^{\prime}\to Y$ of complex varieties, consider the cartesian diagram (2.7). We introduce the assignment that, to every such $g$ and each bounded acyclic complex of hermitian vector bundles $\overline{A}$ on $X^{\prime}$, assigns the class $\begin{split}\widetilde{\varphi}(\overline{A})=&T({g^{\prime}}^{\ast}\overline{\xi})-T\left((f^{\prime},{g^{\prime}}^{\ast}T_{\overline{f}}+[\overline{A}]),{g^{\prime}}^{\ast}\overline{\mathcal{F}},g^{\ast}\overline{f_{\ast}\mathcal{F}}\right)\\\ &+{f^{\prime}}_{\ast}\left[\operatorname{ch}({g^{\prime}}^{\ast}\overline{\mathcal{F}})\widetilde{\operatorname{Td}}_{m}\left(({g^{\prime}}^{\ast}T_{\overline{f}}+[\overline{A}]),{g^{\prime}}^{\ast}T_{\overline{f}}\right)\operatorname{Td}({g^{\prime}}^{\ast}T_{\overline{f}}+[\overline{A}])\right].\end{split}$ Here $[\overline{A}]$ stands for the class of $\overline{A}$ in $\operatorname{\mathbf{KA}}(X^{\prime})$ ([17, Def. 2.31]) and $+$ denotes the action of $\operatorname{\mathbf{KA}}(X^{\prime})$ on $\operatorname{\overline{\mathbf{D}}^{b}}(X^{\prime})$ ([17, Thm. 3.13]). Since $\widetilde{\varphi}$ satisfies the hypothesis of Lemma 2.5, we have $\widetilde{\varphi}=0$. This concludes the proof of (ii). To prove (i), we observe that $\overline{\mathcal{F}}^{\prime}=\overline{\mathcal{F}}+[\overline{A}]$ for some bounded acyclic complex $\overline{A}$ of hermitian vector bundles on $X$. For each cartesian diagram as (2.7), we set $\overline{f}^{\prime}=g^{\ast}\overline{f}$. Let $\widetilde{\varphi}_{1}$ be the assignment that, to each such diagram and each bounded acyclic complex of hermitian vector bundles $\overline{A}$ on $X^{\prime}$, assigns the class $\widetilde{\varphi}_{1}(\overline{A})=T({g^{\prime}}^{\ast}\overline{\xi})-T\left(\overline{f}^{\prime},{g^{\prime}}^{\ast}\overline{\mathcal{F}}+[A],g^{\ast}\overline{f_{\ast}\mathcal{F}}\right)-{\overline{f}^{\prime}}_{\flat}[\operatorname{\widetilde{ch}}(A)].$ The hypothesis of Lemma 2.5 are satisfied, hence $\widetilde{\varphi}_{1}=0$. This concludes the proof of (i). Finally, to prove (iii), to each morphism $g\colon Y^{\prime}\to Y$, transverse to $f$, we associate the cartesian diagram (2.7) and we consider the assignment $\widetilde{\varphi}_{2}$ that, to each bounded acyclic complex of hermitian vector bundles $\overline{B}$ on $Y^{\prime}$, assigns the class $\widetilde{\varphi}_{2}(\overline{B})=T({g^{\prime}}^{\ast}\overline{\xi})-T\left(\overline{f}^{\prime},{g^{\prime}}^{\ast}\overline{\mathcal{F}},g^{\ast}\overline{f_{\ast}\mathcal{F}}+[B]\right)+\operatorname{\widetilde{ch}}(B).$ By Lemma 2.5 applied to $\operatorname{id}_{Y}$, we have $\widetilde{\varphi}_{2}=0$. This concludes the proof of (iii). ∎ The following result provides a compatibility relation for analytic torsion classes for closed immersions with respect to distinguished triangles. The statement is valid for additive theories, in particular the ones we are concerned with. ###### Proposition 3.19. Let $T$ be an additive theory of analytic torsion classes for closed immersions. Let $f\colon X\to Y$ be a closed immersion of smooth complex varieties. Consider distinguished triangles in $\operatorname{\overline{\mathbf{D}}^{b}}(X)$ and $\operatorname{\overline{\mathbf{D}}^{b}}(Y)$ respectively, $(\overline{\tau}):\ \overline{\mathcal{F}}_{2}\to\overline{\mathcal{F}}_{1}\to\overline{\mathcal{F}}_{0}\to\overline{\mathcal{F}}_{2}[1],\quad(\overline{f_{\ast}\tau}):\ \overline{f_{\ast}\mathcal{F}}_{2}\to\overline{f_{\ast}\mathcal{F}}_{1}\to\overline{f_{\ast}\mathcal{F}}_{0}\to\overline{f_{\ast}\mathcal{F}}_{2}[1],$ and the relative hermitian complexes $\overline{\xi}_{i}=(\overline{f},\overline{\mathcal{F}}_{i},\overline{f_{\ast}\mathcal{F}}_{i}),$ $i=0,1,2$. Then we have: $\sum_{j}(-1)^{j}T(\overline{\xi}_{j})=\widetilde{\operatorname{ch}}(\overline{f_{\ast}\tau})-\overline{f}_{\flat}(\widetilde{\operatorname{ch}}(\overline{\tau})).$ ###### Proof. We can assume that the distinguished triangles $\overline{\tau}$ and $\overline{f_{\ast}\tau}$ can be represented by short exact sequences of complexes of hermitian vector bundles $\displaystyle\overline{\varepsilon}:\quad$ $\displaystyle 0\longrightarrow\overline{E}_{2}\longrightarrow\overline{E}_{1}\longrightarrow\overline{E}_{0}\longrightarrow 0,$ $\displaystyle\overline{\nu}:\quad$ $\displaystyle 0\longrightarrow\overline{V}_{2}\longrightarrow\overline{V}_{1}\longrightarrow\overline{V}_{0}\longrightarrow 0.$ Applying the explicit construction at the beginning of the proof of [19, Theorem 2.3] to each row of the above exact sequences, we obtain exact sequences $\displaystyle\widetilde{\varepsilon}^{i}:\quad$ $\displaystyle 0\longrightarrow\widetilde{E}_{2}^{i}\longrightarrow\widetilde{E}_{1}^{i}\longrightarrow\widetilde{E}_{0}^{i}\longrightarrow 0,$ $\displaystyle\widetilde{\nu}^{i}:\quad$ $\displaystyle 0\longrightarrow\widetilde{V}_{2}^{i}\longrightarrow\widetilde{V}_{1}^{i}\longrightarrow\widetilde{V}_{0}^{i}\longrightarrow 0$ over $X\times{\mathbb{P}}^{1}$ and $Y\times{\mathbb{P}}^{1}$ respectively. The restriction of $\widetilde{\varepsilon}^{i}$ (respectively $\widetilde{\nu}^{i}$) to $X\times\\{0\\}$ (respectively $Y\times\\{0\\}$) agrees with $\overline{\varepsilon}$ (respectively $\overline{\nu}$). Whereas the restriction to $X\times\\{\infty\\}$ (respectively $Y\times\\{\infty\\}$) is orthogonally split. The sequences $\widetilde{\varepsilon}^{i}$ and $\widetilde{\nu}^{i}$ form exact sequences of complexes that we denote $\widetilde{\varepsilon}$ and $\widetilde{\nu}$. It is easy to verify that the restriction to $X\times\\{\infty\\}$ (respectively $Y\times\\{\infty\\}$) are orthogonally split as sequences of complexes. Moreover, there are isomorphisms $\widetilde{V}_{j}\dashrightarrow f_{\ast}\widetilde{E}_{j}$, $j=0,1,2$. We denote $\widetilde{\xi}_{j}=(\overline{f}\times\operatorname{id}_{{\mathbb{P}}^{1}},\widetilde{E}_{j},\widetilde{V}_{j}).$ Then, in the group $\bigoplus_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(Y,N_{f},p)$, we have $\displaystyle 0$ $\displaystyle=\operatorname{d}_{\mathcal{D}}\frac{1}{2\pi i}\int_{{\mathbb{P}}^{1}}\frac{-1}{2}\log t\bar{t}\bullet\sum_{j}(-1)^{j}T(\widetilde{\xi}_{j})$ $\displaystyle=T(\overline{\xi}_{1})-T(\overline{\xi}_{0}\oplus\overline{\xi}_{2})-\frac{1}{2\pi i}\int_{{\mathbb{P}}^{1}}\frac{-1}{2}\log t\bar{t}\bullet\sum_{j}(-1)^{j}\operatorname{ch}(\widetilde{V}_{j})$ $\displaystyle+\frac{1}{2\pi i}\int_{{\mathbb{P}}^{1}}\frac{-1}{2}\log t\bar{t}\bullet\sum_{j}(-1)^{j}(f\times\operatorname{id}_{{\mathbb{P}}^{1}})_{\ast}(\operatorname{ch}(\widetilde{E}_{j})\bullet\operatorname{Td}(\overline{f}\times\operatorname{id}_{{\mathbb{P}}^{1}}))$ $\displaystyle=T(\overline{\xi}_{1})-T(\overline{\xi}_{0}\oplus\overline{\xi}_{2})+\widetilde{\operatorname{ch}}(\overline{f_{\ast}\tau})-f_{\ast}(\widetilde{\operatorname{ch}}(\overline{\tau})\operatorname{Td}(\overline{f})).$ Thus the result follows from the additivity. ∎ We end this chapter with the relation between the singular Bott-Chern classes of Bismut-Gillet-Soulé [12] and the theory of homogeneous analytic torsion classes. We draw attention to the difference of normalizations. Let us momentarily denote by $\tau$ the theory of singular Bott-Chern classes of Bismut-Gillet-Soulé. By the anomaly formulas, it may be extended to arbitrary embedded metrized complexes. Let $\overline{\xi}=(\overline{f}:X\to Y,\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$ be a relative metrized complex, with $Y$ of dimension $d$. If $\tau^{(p-1,p-1)}$ denotes the component of degree $(p-1,p-1)$ of the current $\tau$, we define $T^{BGS}(\overline{\xi})^{(2p-1,p)}=-\frac{1}{2(2\pi i)^{d-(p-1)}}\tau^{(p-1,p-1)}\in\widetilde{\mathcal{D}}_{D}^{2p-1}(Y,N_{f},p).$ (3.20) In the above equation, the factor $(2\pi i)^{(p-1)}$ comes from the difference in the normalization of characteristic classes. In [12] the authors use real valued classes while we use twisted coefficients. The factor $(2\pi i)^{d}$ comes from our convention about the Deligne complex of currents. The factor $2$ comes from the fact that the second order differential operator that appears in the Deligne complex is $-2\partial\bar{\partial}=2(2\pi i)dd^{c}$, while the second order differential operator that appears in the differential equation considered by Bismut, Gillet and Soulé is $dd^{c}$. The main reason behind this change is that we want the Bott-Chern classes to be related to Beilinson’s regulator and not to twice Beilinson’s regulator (see [27] Theorem 3.5.4). Finally, the minus sign comes from the discrepancy of the differential equations of the singular Bott-Chern forms of Bismut-Gillet-Soulé and the analytic torsion forms of Bismut-Köhler. Note that we are forced to change this sign because we want to merge singular Bott-Chern forms and analytic torsion forms on a single theory. We put $T^{BGS}(\overline{\xi})=\sum_{p\geq 1}T^{BGS}(\overline{\xi})^{(2p-1,p)}\in\bigoplus_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(Y,N_{f},p).$ We have the following comparison theorem [19, Thm. 9.25]. ###### Theorem 3.21. For every embedded metrized complex $\xi$ we have $T^{BGS}(\overline{\xi})=T^{h}(\overline{\xi}).$ ## 4 Regular coherent sheaves In this section we recall some properties of regular sheaves. Let $X$ be a scheme and let ${\mathbb{P}}^{n}_{X}={\mathbb{P}}_{X}(V)$ be the projective space of lines of the trivial bundle $V$ of rank $n+1$ on $X$. Let $\pi\colon{\mathbb{P}}^{n}_{X}\rightarrow X$ be the natural projection. By abuse of notation, if ${\mathcal{G}}$ is a sheaf on $X$, we will denote also by ${\mathcal{G}}$ the inverse image $\pi^{\ast}{\mathcal{G}}$. ###### Definition 4.1 ([38], Lecture 14). A quasi-coherent sheaf ${\mathcal{F}}$ on ${\mathbb{P}}^{n}_{X}$ is called regular if $R^{q}\pi_{\ast}{\mathcal{F}}(-q)=0$ for all $q>0$. Recall the following properties of regular sheaves (see [39]). 1. (i) If ${\mathcal{G}}$ is a quasi-coherent sheaf on $X$, then ${\mathcal{G}}\otimes_{X}\mathcal{O}_{{\mathbb{P}}^{n}_{X}}(k)$ is regular for $k\geq 0$. 2. (ii) If the scheme $X$ is noetherian and ${\mathcal{F}}$ is a coherent sheaf on ${\mathbb{P}}^{n}_{X}$, then Serre’s vanishing theorem implies that for $d$ large enough ${\mathcal{F}}(d)$ is regular. 3. (iii) Let $0\rightarrow{\mathcal{F}}_{2}\rightarrow{\mathcal{F}}_{1}\rightarrow{\mathcal{F}}_{0}\rightarrow 0$ be an exact sequence of quasi-coherent sheaves on ${\mathbb{P}}^{n}_{X}$ and $d$ be an integer. Then 1. (a) if ${\mathcal{F}}_{2}(d)$ and ${\mathcal{F}}_{0}(d)$ are regular, then ${\mathcal{F}}_{1}(d)$ is regular; 2. (b) if ${\mathcal{F}}_{2}(d+1)$ and ${\mathcal{F}}_{1}(d)$ are regular, then ${\mathcal{F}}_{0}(d)$ is regular; 3. (c) if ${\mathcal{F}}_{0}(d)$ and ${\mathcal{F}}_{1}(d+1)$ are regular and the map $R^{0}\pi_{\ast}({\mathcal{F}}_{1}(d))\rightarrow R^{0}\pi_{\ast}({\mathcal{F}}_{0}(d))$ is surjective, then ${\mathcal{F}}_{2}(d+1)$ is regular. 4. (iv) If ${\mathcal{F}}$ is regular, then ${\mathcal{F}}(k)$ is regular for $k>0$. 5. (v) If ${\mathcal{F}}$ is regular, then the canonical map $R^{0}\pi_{\ast}{\mathcal{F}}\otimes_{X}\mathcal{O}_{{\mathbb{P}}^{n}_{X}}\rightarrow{\mathcal{F}}$ is surjective. ###### Theorem 4.2 ([39, §8.1]). Let ${\mathcal{F}}$ be a regular quasi-coherent sheaf on ${\mathbb{P}}^{n}_{X}$. Then there exists a canonical resolution $\gamma_{\text{{\rm can}}}({\mathcal{F}})\ :\ 0\rightarrow{\mathcal{G}}_{n}(-n)\rightarrow{\mathcal{G}}_{n-1}(-n+1)\rightarrow\dots\rightarrow{\mathcal{G}}_{0}\rightarrow{\mathcal{F}}\rightarrow 0$ where ${\mathcal{G}}_{i}$ ($i=0,\dots,n$) are quasi-coherent sheaves on $X$. Moreover, for every $k\geq 0$, the sequence $0\rightarrow{\mathcal{G}}_{k}\rightarrow{\mathcal{G}}_{k-1}\otimes\operatorname{Sym}^{1}V^{\vee}\rightarrow\dots\rightarrow{\mathcal{G}}_{0}\otimes\operatorname{Sym}^{k}V^{\vee}\rightarrow R^{0}\pi_{\ast}({\mathcal{F}}(k))\rightarrow 0$ is exact. Hence the sheaves ${\mathcal{G}}_{k}$ are determined by ${\mathcal{F}}$ up to unique isomorphism. ###### Corollary 4.3. Let $X$ be a noetherian scheme and ${\mathcal{F}}$ a coherent sheaf on ${\mathbb{P}}^{n}_{X}$. Then, for $d$ large enough, we have a resolution $\gamma_{d}({\mathcal{F}})\ :\ 0\rightarrow{\mathcal{G}}_{n}(-n-d)\rightarrow{\mathcal{G}}_{n-1}(-n-d+1)\rightarrow\dots\rightarrow{\mathcal{G}}_{0}(-d)\rightarrow{\mathcal{F}}\rightarrow 0$ where ${\mathcal{G}}_{i}$, $i=0,\dots,n$ are coherent sheaves on $X$. ###### Example 4.4. The sheaf $\mathcal{O}(1)$ is regular. Its canonical resolution is $0\to\Lambda^{n+1}V^{\vee}(-n)\to\Lambda^{n}V^{\vee}(-n+1)\to\dots\to\Lambda^{2}V^{\vee}(-1)\to V^{\vee}\to\mathcal{O}(1)\to 0.$ Twisting this exact sequence by $\mathcal{O}(-1)$ we obtain the Koszul exact sequence $0\to\Lambda^{n+1}V^{\vee}(-n-1)\to\Lambda^{n}V^{\vee}(-n)\to\dots\to\Lambda^{2}V^{\vee}(-2)\to V^{\vee}(-1)\to\mathcal{O}\to 0,$ that we denote $K$. We will denote by $K(k)$ its twist by $\mathcal{O}(k)$. ###### Theorem 4.5 ([50]). 1. (i) Let ${\mathcal{F}}$ be a regular coherent sheaf on ${\mathbb{P}}^{n}_{X}$, and let $\gamma_{\text{{\rm can}}}({\mathcal{F}})$ be the canonical resolution of ${\mathcal{F}}$ as in Theorem 4.2. Let $\varepsilon_{1}\ :\ 0\rightarrow{\mathcal{F}}_{n+k}(-n-k)\rightarrow\dots\rightarrow{\mathcal{F}}_{1}(-1)\rightarrow{\mathcal{F}}_{0}\rightarrow{\mathcal{F}}\rightarrow 0$ be an exact sequence of coherent sheaves, where the ${\mathcal{F}}_{i}$ are sheaves on $X$. Then there exist natural surjective morphisms of sheaves ${\mathcal{F}}_{i}\rightarrow{\mathcal{G}}_{i}$ on $X$, $0\leq i\leq n$ making commutative the diagram $\textstyle{{\mathcal{F}}_{n+1}(-n-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}_{n}(-n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{G}}_{n}(-n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{G}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$ 2. (ii) Let ${\mathcal{F}}$ be a regular coherent sheaf on $X$, and $\gamma_{\text{{\rm can}}}({\mathcal{F}})$ the canonical resolution. There exists a resolution of ${\mathcal{F}}(1)$ of the form $\varepsilon_{2}\ :\ 0\rightarrow{\mathcal{S}}_{n+k}(-n-k)\rightarrow\dots\rightarrow{\mathcal{S}}_{1}(-1)\rightarrow{\mathcal{S}}_{0}\rightarrow{\mathcal{F}}(1)\rightarrow 0$ such that ${\mathcal{S}}_{0}\dots,{\mathcal{S}}_{n+k}$ are coherent sheaves on $X$ and the following diagram of exact sequences with surjective vertical arrows is commutative: $\textstyle{{\mathcal{S}}_{n+1}(-n-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{S}}_{n}(-n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{S}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{G}}_{n}(-n+1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{G}}_{0}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$ ###### Proof. We introduce the sheaves ${\mathcal{N}}_{j}$ and ${\mathcal{K}}_{j}$ defined as the kernels at each term of the sequences $\gamma_{\text{{\rm can}}}$ and $\varepsilon_{1}$, respectively. Hence, there are exact sequences $\displaystyle 0\to{\mathcal{N}}_{j+1}(j+1)\to{\mathcal{G}}_{j+1}\to{\mathcal{N}}_{j}(j+1)\to 0,$ $\displaystyle 0\to{\mathcal{K}}_{j+1}(j+1)\to{\mathcal{F}}_{j+1}\to{\mathcal{K}}_{j}(j+1)\to 0.$ With these notations, observe that ${\mathcal{N}}_{-1}={\mathcal{K}}_{-1}={\mathcal{F}}$. By induction, starting from the left hand side of the long exact sequences, it is easily checked that ${\mathcal{N}}_{j}(j+1)$ and ${\mathcal{K}}_{j}(j+1)$ are regular sheaves, for $j\geq-1$. Also, by Theorem 4.2, we find that ${\mathcal{G}}_{j+1}=\pi_{\ast}({\mathcal{N}}_{j}(j+1))$ for $j\geq-1$. We next prove by induction that, for each $k\geq-1$, there is a commutative diagram of exact sequences $\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{P}}_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{H}}_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{P}}_{k}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{K}}_{k+1}(k+1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{K}}_{k}(k+1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{N}}_{k+1}(k+1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{G}}_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{N}}_{k}(k+1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0,}$ (4.6) where ${\mathcal{H}}_{k+1}$, ${\mathcal{P}}_{k}$ and ${\mathcal{P}}_{k+1}$ are defined as the kernels of the corresponding morphisms, and that ${\mathcal{P}}_{k}(1)$ is regular. Assume that this is true for a fixed $k\geq 1$. In order to proceed with the induction, we need to prove: (a) the map ${\mathcal{K}}_{k+1}(k+2)\to{\mathcal{N}}_{k+1}(k+2)$ is surjective, (b) the sheaf ${\mathcal{P}}_{k+1}(1)$ is regular, and (c) there is an induced surjective map ${\mathcal{F}}_{k+2}\to{\mathcal{G}}_{k+2}$. We first prove (a). If we apply $\pi_{\ast}$ to the last two columns of diagram (4.6). Observing that ${\mathcal{F}}_{k+1}$, ${\mathcal{G}}_{k+1}$ and ${\mathcal{H}}_{j+1}$ are actually sheaves on $X$ and recalling that ${\mathcal{K}}_{k+1}(k+2)$ is regular (so that $R^{1}\pi_{\ast}{\mathcal{K}}_{k+1}(k+1)=0$), we find a commutative diagram of exact sequences $\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{H}}_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{G}}_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{\ast}({\mathcal{P}}_{k}(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{\ast}({\mathcal{K}}_{k}(k+1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{\ast}({\mathcal{N}}_{k}(k+1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$ It follows that the map ${\mathcal{H}}_{k+1}\twoheadrightarrow\pi_{\ast}({\mathcal{P}}_{k}(1))$ is a surjection. Since ${\mathcal{P}}_{k}(1)$ is regular, we have that $\pi_{\ast}({\mathcal{P}}_{k}(1))\otimes{\mathcal{O}}_{{\mathbb{P}}^{n}_{X}}\twoheadrightarrow{\mathcal{P}}_{k}(1)$ is also a surjection. Thus the map $\mathcal{H}_{k+1}\to\mathcal{P}_{k}(1)$ is surjective. The diagram (4.6) implies that the map ${\mathcal{K}}_{k+1}(k+1)\to{\mathcal{N}}_{k+1}(k+1)$ is also surjective. Twisting by ${\mathcal{O}}(1)$, we obtain (a). Now the regularity of ${\mathcal{H}}_{k+1}$ and ${\mathcal{P}}_{k}(1)$, and the surjectivity of ${\mathcal{H}}_{k+1}\twoheadrightarrow\pi_{\ast}({\mathcal{P}}_{k}(1))$ ensure the regularity of ${\mathcal{P}}_{k+1}(1)$. In its turn, this shows that the sequence $0\to\pi_{\ast}({\mathcal{P}}_{k+1}(1))\to\pi_{\ast}({\mathcal{K}}_{k+1}(k+2))\to\pi_{\ast}({\mathcal{N}}_{k+1}(k+2))\to 0$ (4.7) is exact. Finally, we observe that there is a surjective map $\textstyle{{\mathcal{F}}_{k+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{\ast}({\mathcal{K}}_{k+1}(k+2)),}$ (4.8) by the regularity of ${\mathcal{K}}_{k+2}(k+3)$. From (4.7) and (4.8), we obtain a surjection $\textstyle{{\mathcal{F}}_{k+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{\ast}({\mathcal{N}}_{k+1}(k+2))={\mathcal{G}}_{k+2}.}$ This completes the proof of the inductive step. Note that the first step of the induction ($k=-1$) is part of the data. From the existence of the diagrams we deduce (i). To prove the second item we construct the resolution $\mathcal{S}_{\ast}$ inductively. We will denote by ${\mathcal{K}}_{k}$ the kernel of any map ${\mathcal{S}}_{k}(-k)\to{\mathcal{S}}_{k-1}(-k+1)$ already defined and by ${\mathcal{N}}_{k}$ the successive kernels of the canonical resolution of ${\mathcal{F}}$ as in the proof of the first statement. Assume that we have constructed the sequence $\varepsilon_{2}$ up to ${\mathcal{S}}_{k}(-k)$ with the further conditions that ${\mathcal{K}}_{k}(k+1)$ is regular and that there is an exact sequence $0\to{\mathcal{P}}_{k}(1)\to{\mathcal{K}}_{k}(k+1)\to{\mathcal{N}}_{k}(k+2)\to 0$ with ${\mathcal{P}}_{k}(1)$ regular. We have to show that we can extend the resolution one step satisfying the same conditions. Recall that we already know that ${\mathcal{N}}_{k}(k+1)$ is regular. We consider as well the surjection $\textstyle{{\mathcal{G}}_{k+1}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{N}}_{k}(k+2).}$ We form the fiber product ${\mathcal{T}}_{k+1}:=\operatorname{Ker}({\mathcal{K}}_{k}(k+1)\oplus{\mathcal{G}}_{k+1}(1)\to{\mathcal{N}}_{k}(k+2)).$ Observe that ${\mathcal{T}}_{k+1}$ is regular, because both ${\mathcal{N}}_{k}(k+1)$, ${\mathcal{K}}_{k}(k+1)\oplus{\mathcal{G}}_{k+1}(1)$ are regular and the morphism $\textstyle{\pi_{\ast}({\mathcal{K}}_{k}(k)\oplus{\mathcal{G}}_{k+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{G}}_{k+1}=\pi_{\ast}({\mathcal{N}}_{k}(k+1))}$ is surjective. So are the arrows ${\mathcal{T}}_{k+1}\to{\mathcal{G}}_{k+1}(1)$ and ${\mathcal{T}}_{k+1}\to{\mathcal{K}}_{k}(k+1)$. Therefore, if we define ${\mathcal{S}}_{k+1}=\pi_{\ast}({\mathcal{T}}_{k+1})$, we have a commutative diagram of exact sequences $\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{P}}_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{H}}_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{P}}_{k}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{K}}_{k+1}(k+1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{S}}_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{K}}_{k}(k+1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{N}}_{k+1}(k+2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{G}}_{k+1}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{N}}_{k}(k+2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0,}$ where ${\mathcal{H}}_{k+1}$ and ${\mathcal{P}}_{k+1}$ are defined as the kernels of the corresponding morphisms. Thus we have been able to extend the resolution one step further. We still need to show that this extension satisfies the extra properties. We observe that, by the definition of ${\mathcal{S}}_{k+1}$ and the left exactness of direct images, the map $\pi_{\ast}({\mathcal{S}}_{k+1})\to\pi_{\ast}({\mathcal{G}}_{k+1}(1))$ is surjective. Therefore ${\mathcal{H}}_{k+1}(1)$ is regular. Moreover, one can check that ${\mathcal{S}}_{k+1}$ is the fiber product ${\mathcal{S}}_{k+1}=\operatorname{Ker}\left(\pi_{\ast}({\mathcal{G}}_{k+1}(1))\oplus\pi_{\ast}({\mathcal{K}}_{k}(k+1))\to\pi_{\ast}({\mathcal{N}}_{k}(k+2))\right).$ This implies easily that $\pi_{\ast}({\mathcal{H}}_{k+1})=\pi_{\ast}({\mathcal{P}}_{k}(1))$. We also observe that, by definition of fiber product, ${\mathcal{P}}_{k}(1)=\operatorname{Ker}({\mathcal{T}}_{k+1}\to{\mathcal{G}}_{k+1}(1))$. Since ${\mathcal{S}}_{k+1}$ surjects onto ${\mathcal{T}}_{k+1}$, we deduce that the morphism ${\mathcal{H}}_{k+1}\to{\mathcal{P}}_{k}(1)$ is surjective. From this we conclude that the morphism ${\mathcal{K}}_{k+1}(k+2)\to{\mathcal{N}}_{k+1}(k+3)$ is surjective and that the sheaf ${\mathcal{P}}_{k+1}(1)$ is regular. Since ${\mathcal{N}}_{k+1}(k+3)$ is regular, we deduce that ${\mathcal{K}}_{k+1}(k+2)$ is regular. Therefore $\mathcal{S_{k+1}}$ satisfies all the required properties, concluding the proof of (ii). ∎ We end this section recalling the notion of generating class of a triangulated category. ###### Definition 4.9. Let $\mathbf{D}$ be a triangulated category. A _generating class_ is a subclass $\mathbf{C}$ of $\mathbf{D}$ such that the smallest triangulated subcategory of $\mathbf{D}$ that contains $\mathbf{C}$ is equivalent to $\mathbf{D}$ via the inclusion. A well-known direct consequence of Theorem 4.2 is the following result. ###### Corollary 4.10. The class of objects of the form $\mathcal{G}(k)$, with $\mathcal{G}$ a coherent sheaf in $X$ and $-n\leq k\leq 0$, is a generating class of $\operatorname{\mathbf{D}^{b}}({\mathbb{P}}^{n}_{X})$. ## 5 Analytic torsion for projective spaces Let $n$ be a non-negative integer, $V$ the $n+1$ dimensional vector space ${\mathbb{C}}^{n+1}$ and ${\mathbb{P}}^{n}:={\mathbb{P}}^{n}(V)$ the projective space of lines in $V$. We write $\overline{V}$ for the vector space $V$ together with the trivial metric. We will denote by $V$ the trivial vector bundle of fiber $V$ over any base. We may construct natural relative hermitian complexes that arise by considering the sheaves $\mathcal{O}(k)$, their cohomology and the Fubini- Study metric. If we endow the trivial sheaf with the trivial metric and $\mathcal{O}(1)$ with the Fubini-Study metric, then the tangent bundle $T_{\pi}$ carries a quotient hermitian structure via the short exact sequence $0\to\mathcal{O}_{{\mathbb{P}}^{n}_{{\mathbb{C}}}}\to\mathcal{O}(1)^{n+1}\to T_{\pi}\to 0.$ (5.1) We will denote the resulting hermitian vector bundle by $\overline{T}_{\pi}^{\text{\rm FS}}$ and call it the Fubini-Study metric of $T_{\pi}$. The arrow $(\pi,\overline{T}_{\pi}^{\text{\rm FS}})$ in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$ will be written $\overline{\pi}^{\text{\rm FS}}$. We endow the invertible sheaves ${\mathcal{O}}(k)$ with the $k$-th tensor power of the Fubini-Study metric on ${\mathcal{O}}(1)$. We refer to them by $\overline{{\mathcal{O}}(k)}$. We now describe natural hermitian structures on the complexes $\pi_{\ast}\mathcal{O}(k)$. First assume $k\geq 0$. The sheaf ${\mathcal{O}}(k)$ is $\pi$-acyclic, hence $\pi_{\ast}\mathcal{O}(k)=\text{H}^{0}({\mathbb{P}}^{n}_{{\mathbb{C}}},\mathcal{O}(k))$ as a complex concentrated in degree 0. This space is naturally equipped with the $L^{2}$ metric with respect to the Fubini-Study metric on ${\mathcal{O}}(k)$ and the volume form $\mu=c_{1}(\overline{{\mathcal{O}}(1)})^{\land n}/n!$ on ${\mathbb{P}}^{n}_{{\mathbb{C}}}$. Namely, given global sections $s,t$ of ${\mathcal{O}}(k)$, $\langle s,t\rangle_{L^{2}}=\int_{{\mathbb{P}}^{n}_{{\mathbb{C}}}}\langle s(x),t(x)\rangle_{x}\mu(x).$ If $-n\leq k<0$, then $\pi_{\ast}\mathcal{O}(k)=0$ and we put the trivial metric on it. Finally, let $k\leq-n-1$. Then the cohomology of $\pi_{\ast}\mathcal{O}(k)$ is concentrated in degree $n$ and there is an isomorphism, $\pi_{\ast}\mathcal{O}(k)\cong\text{H}^{0}({\mathbb{P}}^{n}_{{\mathbb{C}}},{\mathcal{O}}(-k-n-1))^{\vee}[-n].$ Notice that this isomorphism is canonical due to Grothendieck duality and to the natural identification $\omega_{{\mathbb{P}}^{n}_{{\mathbb{C}}}}=\mathcal{O}(-n-1)$. Hence we may endow $\pi_{\ast}\mathcal{O}(k)$ with the dual of the $L^{2}$ metric on $\text{H}^{0}({\mathbb{P}}^{n}_{{\mathbb{C}}},{\mathcal{O}}(-k-n-1))$. ###### Notation 5.2. For every integer $k$, we introduce the relative metrized complex $\overline{\xi_{n}}(k)=(\overline{\pi}^{\text{\rm FS}},\overline{\mathcal{O}(k)},\overline{\pi_{\ast}\mathcal{O}(k)}).$ (5.3) If $X$ is a smooth complex variety, we will also denote by $\overline{\xi}_{n}(k)$ its pull-back to ${\mathbb{P}}^{n}_{X}$. Let $\overline{\mathcal{F}}$ be a metrized coherent sheaf on $X$. Then we define $\overline{\mathcal{F}}(k)$ and $\overline{\pi_{\ast}{\mathcal{F}}(k)}$ by the equality $\overline{\xi}_{n}(k)\otimes\overline{\mathcal{F}}=(\overline{\pi}^{\text{\rm FS}},\overline{\mathcal{F}}(k),\overline{\pi_{\ast}{\mathcal{F}}(k)}).$ ###### Definition 5.4. Let $X$ be a complex smooth variety and $\pi\colon{\mathbb{P}}^{n}_{X}\to X$ the projection. An _analytic torsion class_ for the relative hermitian complex $\overline{\xi}=(\overline{\pi},\overline{\mathcal{F}},\overline{\pi_{\ast}\mathcal{F}})$ is a class $\widetilde{\eta}\in\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}(X,p)$ such that $\operatorname{d}_{\mathcal{D}}\widetilde{\eta}=\operatorname{ch}(\overline{\pi_{\ast}\mathcal{F}})-\overline{\pi}_{\flat}[\operatorname{ch}(\overline{\mathcal{F}})].$ (5.5) The existence of this class is guaranteed by the Grothendieck-Riemann-Roch theorem, which implies that the two currents at the right hand side of equation (5.5) are cohomologous. Since the map $\pi$ is smooth, the analytic torsion class is the class of a smooth form. ###### Definition 5.6. Let $n$ be a non-negative integer. A _theory of analytic torsion classes for projective spaces of dimension $n$_ is an assignment that, to each relative metrized complex $\overline{\xi}=(\overline{\pi}\colon{\mathbb{P}}^{n}_{X}\rightarrow X,\overline{\mathcal{F}},\overline{\pi_{\ast}\mathcal{F}})$ of relative dimension $n$, assigns a class of differential forms $T(\overline{\xi})\in\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}(X,p),$ satisfying the following properties. 1. (i) (Differential equation) $\operatorname{d}_{\mathcal{D}}T(\overline{\xi})=\operatorname{ch}(\overline{\pi_{\ast}\mathcal{F}})-\overline{\pi}_{\flat}[\operatorname{ch}(\overline{\mathcal{F}})].$ 2. (ii) (Functoriality) Given a morphism $f\colon Y\longrightarrow X$, we have $T(f^{\ast}\overline{\xi})=f^{\ast}T(\overline{\xi}).$ 3. (iii) (Additivity and normalization) If $\overline{\xi}_{1}$ and $\overline{\xi}_{2}$ are relative metrized complexes on $X$, then $T(\overline{\xi}_{1}\oplus\overline{\xi}_{2})=T(\overline{\xi}_{1})+T(\overline{\xi}_{2}).$ 4. (iv) (Projection formula) For any hermitian vector bundle $\overline{G}$ on $X$, and any integer $k\in[-n,0]$, we have $T(\overline{\xi}_{n}(k)\otimes\overline{G})=T(\overline{\xi}_{n}(k))\bullet\operatorname{ch}(\overline{G}).$ A _theory of analytic torsion classes for projective spaces_ is an assignment as before, for all non-negative integers $n$. ###### Definition 5.7. Let $T$ be a theory of analytic torsion classes for projective spaces of dimension $n$. Fix as base space the point $\operatorname{Spec}{\mathbb{C}}$. The _characteristic numbers_ of $T$ are $t_{n,k}(T):=T(\overline{\xi}_{n}(k))\in\widetilde{\mathcal{D}}^{1}(\operatorname{Spec}{\mathbb{C}},1)={\mathbb{R}},\ k\in\mathbb{Z}.$ (5.8) The numbers $t_{n,k}(T)$, $-n\leq k\leq 0$ will be called the _main characteristic numbers_ of $T$. The central result of this section is the following classification theorem. ###### Theorem 5.9. Let $n$ be a non-negative integer and let $\mathfrak{t}=(t_{n,k})_{k=-n,\dots,0}$ be a family of arbitrary real numbers. Then there exists a unique theory $T_{\mathfrak{t}}$ of analytic torsion classes for projective spaces of dimension $n$, such that $t_{n,k}(T_{\mathfrak{t}})=t_{n,k}$. Before proving Theorem 5.9, we show some consequences of the definition of the analytic torsion classes. First we state some anomaly formulas that determine the dependence of the analytic torsion classes with respect to different choices of metrics. ###### Proposition 5.10. Let $T$ be a theory of analytic torsion classes for projective spaces of dimension $n$. Let $\overline{\xi}=(\overline{\pi}\colon{\mathbb{P}}^{n}_{X}\rightarrow X,\overline{\mathcal{F}},\overline{\pi_{\ast}\mathcal{F}})$ be a relative metrized complex. 1. (i) If $\overline{\mathcal{F}}^{\prime}$ is another choice of metric on $\mathcal{F}$ and $\overline{\xi}_{1}=(\overline{\pi}\colon{\mathbb{P}}^{n}_{X}\rightarrow X,\overline{\mathcal{F}}^{\prime},\overline{\pi_{\ast}\mathcal{F}})$, then $T(\overline{\xi}_{1})=T(\overline{\xi})+\overline{\pi}_{\flat}[\operatorname{\widetilde{ch}}(\overline{\mathcal{F}}^{\prime},\overline{\mathcal{F}})].$ 2. (ii) If $\overline{\pi}^{\prime}$ is another hermitian structure on $\pi$ and $\overline{\xi}_{2}=(\overline{\pi}^{\prime}\colon{\mathbb{P}}^{n}_{X}\rightarrow X,\overline{\mathcal{F}},\overline{\pi_{\ast}\mathcal{F}})$, then $T(\overline{\xi}_{2})=T(\overline{\xi})+\overline{\pi}^{\prime}_{\flat}[\operatorname{ch}(\overline{\mathcal{F}})\bullet\widetilde{\operatorname{Td}}_{m}(\overline{\pi}^{\prime},\overline{\pi})].$ (5.11) 3. (iii) If $\overline{\pi_{\ast}\mathcal{F}}^{\prime}$ is another choice of metric on $\pi_{\ast}\mathcal{F}$, and $\overline{\xi}_{3}=(\overline{\pi}\colon{\mathbb{P}}^{n}_{X}\rightarrow X,\overline{\mathcal{F}},\overline{\pi_{\ast}\mathcal{F}}^{\prime})$, then $T(\overline{\xi}_{3})=T(\overline{\xi})-\operatorname{\widetilde{ch}}(\overline{\pi_{\ast}\mathcal{F}}^{\prime},\overline{\pi_{\ast}\mathcal{F}}).$ ###### Proof. The proof is the same as the proof of Proposition 3.17. ∎ Next we state the behavior of analytic torsion classes for projective spaces with respect to distinguished triangles. ###### Proposition 5.12. Let $T$ be a theory of analytic torsion classes for projective spaces of dimension $n$. Let $X$ be a smooth complex variety and $\pi\colon{\mathbb{P}}^{n}_{X}\to X$ the projection. Consider distinguished triangles in $\operatorname{\overline{\mathbf{D}}^{b}}({\mathbb{P}}^{n}_{X})$ and $\operatorname{\overline{\mathbf{D}}^{b}}(X)$ respectively: $(\overline{\tau}):\ \overline{\mathcal{F}}_{2}\to\overline{\mathcal{F}}_{1}\to\overline{\mathcal{F}}_{0}\to\overline{\mathcal{F}}_{2}[1]\ \text{ and }\ (\overline{\pi_{\ast}\tau}):\ \overline{\pi_{\ast}\mathcal{F}}_{2}\to\overline{\pi_{\ast}\mathcal{F}}_{1}\to\overline{\pi_{\ast}\mathcal{F}}_{0}\to\overline{\pi_{\ast}\mathcal{F}}_{2}[1],$ and define relative metrized complexes $\overline{\xi}_{i}=(\overline{\pi},\overline{\mathcal{F}}_{i},\overline{\pi_{\ast}\mathcal{F}}_{i}),$ $i=0,1,2$. Then $\sum_{j}(-1)^{j}T(\overline{\xi}_{j})=\widetilde{\operatorname{ch}}(\overline{\pi_{\ast}\tau})-\overline{\pi}_{\flat}(\widetilde{\operatorname{ch}}(\overline{\tau})).$ ###### Proof. The proof is similar to that of 3.19. ∎ In view of this proposition, we see that the additivity axiom is equivalent to the apparently stronger statement of the next corollary. ###### Corollary 5.13. With the assumptions of Proposition 5.12, if $\overline{\tau}$ and $\overline{\pi_{\ast}\tau}$ are tightly distinguished, then $T(\overline{\xi}_{1})=T(\overline{\xi}_{0})+T(\overline{\xi}_{2}).$ ###### Corollary 5.14. Let $\overline{\xi}=(\overline{\pi},\overline{\mathcal{F}},\overline{\pi_{\ast}\mathcal{F}})$ be a relative metrized complex and let $\overline{\xi}[i]=(\overline{\pi},\overline{\mathcal{F}}[i],\overline{\pi_{\ast}\mathcal{F}}[i])$ be the shifted relative metrized complex. Then $T(\overline{\xi})=(-1)^{i}T(\overline{\xi}[i]).$ ###### Proof. It is enough to treat the case $i=1$. We consider the tightly distinguished triangle $\overline{\mathcal{F}}\dashrightarrow\operatorname{\overline{cone}}(\operatorname{id}_{\overline{\mathcal{F}}})\dashrightarrow\overline{\mathcal{F}}[1]\dashrightarrow$ and the analogous triangle for direct images. Since $\operatorname{\overline{cone}}(\operatorname{id}_{\overline{\mathcal{F}}})$ and $\operatorname{\overline{cone}}(\operatorname{id}_{\overline{\pi_{\ast}\mathcal{F}}})$ are meager, we have, by the anomaly formulas and the additivity axiom, $T(\overline{\pi},\operatorname{\overline{cone}}(\operatorname{id}_{\overline{\mathcal{F}}}),\operatorname{\overline{cone}}(\operatorname{id}_{\overline{\pi_{\ast}\mathcal{F}}}))=T(\overline{\pi},\overline{0},\overline{0})=0.$ Hence, the result follows from Corollary 5.13. ∎ Next we rewrite Proposition 5.12 in the language of complexes of metrized coherent sheaves. Let $\overline{\varepsilon}:\quad 0\to\overline{\mathcal{F}}_{m}\to\dots\to\overline{\mathcal{F}}_{l}\to 0$ be a bounded complex of metrized coherent sheaves on ${\mathbb{P}}^{n}_{X}$ and assume that hermitian structures on the complexes $\pi_{\ast}\mathcal{F}_{j}$, $j=l,\dots,m$ are chosen. Let $[\overline{\varepsilon}],[\overline{\pi_{\ast}\varepsilon}]\in\operatorname{Ob}\operatorname{\overline{\mathbf{D}}^{b}}({\mathbb{P}}^{n}_{X})$ be the associated objects as in [17, Def. 3.37, Def. 3.39]. ###### Corollary 5.15. With the above hypothesis, $T(\overline{\pi},[\overline{\varepsilon}],[\overline{\pi_{\ast}\varepsilon}])=\sum_{j=l}^{m}(-1)^{j}T(\overline{\pi},\overline{\mathcal{F}}_{j},\overline{\pi_{\ast}\mathcal{F}_{j}}).$ Moreover, if $\varepsilon$ is acyclic, then $T(\overline{\pi},[\overline{\varepsilon}],[\overline{\pi_{\ast}\varepsilon}])=\widetilde{\operatorname{ch}}(\overline{\pi_{\ast}\varepsilon})-\overline{\pi}_{\flat}[\widetilde{\operatorname{ch}}(\overline{\varepsilon})].$ Finally, we show that the projection formula holds in greater generality: ###### Proposition 5.16. Let $T$ be a theory of analytic torsion classes for projective spaces of dimension $n$. Let $X$ be a smooth complex variety, let $\overline{\xi}=(\overline{\pi},\overline{\mathcal{F}},\overline{\pi_{\ast}\mathcal{F}})$ be a relative metrized complex and let $\overline{\mathcal{G}}$ be an object in $\operatorname{\overline{\mathbf{D}}^{b}}(X)$. Then $T(\overline{\xi}\otimes\overline{\mathcal{G}})=T(\overline{\xi})\bullet\operatorname{ch}(\overline{\mathcal{G}}).$ (5.17) ###### Proof. By the anomaly formulas, if equation (5.17) holds for a particular choice of hermitian structures on $\pi$, $\mathcal{F}$ and $\pi_{\ast}\mathcal{F}$ then it holds for any other choice. Moreover, if we are in the situation of Proposition 5.12 and equation (5.17) holds for two of $\overline{\xi}_{0}$, $\overline{\xi}_{1}$, $\overline{\xi}_{2}$, then it holds for the third. Using that the objects of the form $\mathcal{H}(k)$, where $\mathcal{H}$ is a coherent sheaf on $X$ and $k=-n,\dots,0$, constitute a generating class of $\operatorname{\mathbf{D}^{b}}({\mathbb{P}}^{n}_{X})$, we are reduced to prove that $T(\overline{\xi}_{n}(k)\otimes\overline{\mathcal{G}})=T(\overline{\xi}_{n}(k))\bullet\operatorname{ch}(\overline{\mathcal{G}}).$ for $k=-n,\dots,0$. Now, if $\overline{\mathcal{G}}_{2}\dashrightarrow\overline{\mathcal{G}}_{1}\dashrightarrow\overline{\mathcal{G}}_{0}\dashrightarrow$ is a distinguished triangle in $\operatorname{\overline{\mathbf{D}}^{b}}(X)$ and equation (5.17) is satisfied for two of $\overline{\mathcal{G}}_{2}$, $\overline{\mathcal{G}}_{1}$, $\overline{\mathcal{G}}_{0}$, then it is satisfied also by the third. Therefore, since the complexes of vector bundles concentrated in a single degree constitute a generating class of $\operatorname{\mathbf{D}^{b}}(X)$, the projection formula axiom implies the proposition. ∎ ###### Proof of Theorem 5.9 . To begin with, we prove the uniqueness assertion. Assume a theory of analytic torsion classes $T$, with main characteristic numbers $t_{n,k}$, $-n\leq k\leq 0$, exists. Then, the anomaly formulas (Proposition 5.10) imply that, if $T(\overline{\pi},\overline{\mathcal{F}},\overline{\pi_{\ast}\mathcal{F}})$ is known for a particular choice of hermitian structures on $\pi$, $\mathcal{F}$ and $\pi_{\ast}\mathcal{F}$ then the value of $T(\overline{\pi}^{\prime},\overline{\mathcal{F}}^{\prime},\overline{\pi_{\ast}\mathcal{F}}^{\prime})$ for any other choice of hermitian structures is fixed. By Proposition 5.12, if we know the value of $T(\overline{\pi},\overline{\mathcal{F}},\overline{\pi_{\ast}\mathcal{F}})$, for $\mathcal{F}$ in a generating class, then $T$ is determined. By the projection formula (Proposition 5.16), the characteristic numbers determine the values of $T(\overline{\xi}(k)\otimes\mathcal{G})$, $k=-n,\dots,0$. Finally, since by Corollary 4.10, the objects of the form $\mathcal{G}(k)$, $k=-n,\dots,0$ form a generating class, we deduce that the characteristic numbers determine the theory $T$. Thus, if it exists, the theory $T_{\mathfrak{t}}$ is unique. In particular, from the above discussion we see that the main characteristic numbers determine all the characteristic numbers. We now derive an explicit inductive formula for them. Consider the metrized Koszul resolution $\overline{K}:0\to\Lambda^{n+1}\overline{V}^{\vee}(-n-1)\to\dots\to\Lambda^{1}\overline{V}^{\vee}(-1)\to\overline{\mathcal{O}}_{{\mathbb{P}}^{n}_{{\mathbb{C}}}}\to 0,$ (5.18) where $\mathcal{O}(k)$, for $k\not=0$, has the Fubini-Study metric and $\overline{\mathcal{O}}_{{\mathbb{P}}^{n}_{{\mathbb{C}}}}$ has the trivial metric. We will denote by $\overline{K}(k)$ the above exact sequence twisted by $\overline{\mathcal{O}(k)}$, $k\in\mathbb{Z}$, again with the Fubini-Study metric. Recall the definition of the relative metrized complexes $\overline{\xi}_{n}(k)$ (5.3). In particular, for every $k$, we have fixed natural hermitian structures on the objects $\pi_{\ast}{\mathcal{O}}(k-j)$. According to [17, Def. 3.37, Def. 3.39], we may consider the classes $[\overline{K}(k)]$ and $[\overline{\pi_{\ast}K(k)}]$ in $\operatorname{\overline{\mathbf{D}}^{b}}({\mathbb{P}}^{n}_{{\mathbb{C}}})$ and $\operatorname{\overline{\mathbf{D}}^{b}}(\operatorname{Spec}{\mathbb{C}})$, respectively. By Corollary 5.15, for each $k\in\mathbb{Z}$ we find $\sum_{j=0}^{n+1}(-1)^{j}T(\overline{\xi}_{n}(k-j)\otimes\Lambda^{j}\overline{V}^{\vee})=\widetilde{\operatorname{ch}}(\overline{\pi_{\ast}K(k)})-\overline{\pi}_{\flat}^{\text{\rm FS}}[\widetilde{\operatorname{ch}}(\overline{K}(k))].$ Because $\Lambda^{j}\overline{V}^{\vee}$ is isometric to ${\mathbb{C}}^{\binom{n+1}{j}}$ with the trivial metric, the additivity axiom for the theory $T$ and the definition of the characteristic numbers $t_{n,k-j}$ provide $T(\overline{\xi}_{n}(k-j)\otimes\Lambda^{j}\overline{V}^{\vee})=t_{n,k-j}\binom{n+1}{j}.$ Therefore we derive $\sum_{j=0}^{n+1}(-1)^{j}\binom{n+1}{j}t_{n,k-j}=\widetilde{\operatorname{ch}}(\overline{\pi_{\ast}K(k)})-\overline{\pi}^{\text{\rm FS}}_{\flat}[\widetilde{\operatorname{ch}}(\overline{K}(k))].$ (5.19) This equation gives us an inductive formula for all the characteristic numbers $t_{n,k}$ once we have fixed $n+1$ consecutive characteristic numbers and, in particular, once we have fixed the main characteristic numbers. To prove the existence, we follow the proof of the uniqueness to obtain a formula for $T(\overline{\xi})$. We start with the main characteristic numbers $\mathfrak{t}=(t_{n,k})_{-n\leq k\leq 0}$. We define the characteristic numbers $t_{n,k}$ for $k\in\mathbb{Z}$ inductively using equation (5.19). We will need the following results. ###### Lemma 5.20. Let $\overline{\eta}:0\to\overline{\mathcal{F}}_{2}\to\overline{\mathcal{F}}_{1}\to\overline{\mathcal{F}}_{0}\to 0$ be a short exact sequence of metrized coherent sheaves on $X$. Let $k$ be an integer, and $\overline{\mathcal{F}}(k)$ and $\overline{\pi_{\ast}{\mathcal{F}}(k)}$ be as in Notation 5.2. Thus we have an exact sequence $\overline{\eta}(k)$ of metrized coherent sheaves on ${\mathbb{P}}^{n}_{X}$ and a distinguished triangle $\overline{\pi_{\ast}\eta(k)}$. Then $\operatorname{\widetilde{ch}}(\overline{\pi_{\ast}\eta(k)})=\overline{\pi}^{\text{\rm FS}}_{\flat}(\operatorname{\widetilde{ch}}(\overline{\eta}(k))).$ (5.21) ###### Proof. By the Riemann-Roch theorem for the map ${\mathbb{P}}^{n}_{\mathbb{C}}\to\operatorname{Spec}\mathbb{C}$ we have $\operatorname{ch}(\overline{\pi_{\ast}\mathcal{O}(k)})=\pi_{\ast}(\operatorname{ch}(\overline{\mathcal{O}(k)})\operatorname{Td}(\overline{\pi}^{\text{\rm FS}})).$ (5.22) Hence, by the properties of Bott-Chern classes and the choice of metrics $\displaystyle\operatorname{\widetilde{ch}}(\overline{\pi_{\ast}\eta(k)})$ $\displaystyle=\operatorname{\widetilde{ch}}(\overline{\eta})\bullet\operatorname{ch}(\overline{\pi_{\ast}\mathcal{O}(k)})$ $\displaystyle=\operatorname{\widetilde{ch}}(\overline{\eta})\bullet\pi_{\ast}(\operatorname{ch}(\overline{\mathcal{O}(k)})\operatorname{Td}(\overline{\pi}^{\text{\rm FS}}))$ $\displaystyle=\pi_{\ast}\left(\operatorname{\widetilde{ch}}(\overline{\eta}(k))\bullet\operatorname{Td}(\overline{\pi}^{\text{\rm FS}})\right)$ $\displaystyle=\overline{\pi}^{\text{\rm FS}}_{\flat}(\operatorname{\widetilde{ch}}(\overline{\eta}(k))).\qed$ ###### Lemma 5.23. Let $\overline{\mu}:0\to\overline{\mathcal{M}}_{m}(-m-d)\to\dots\to\overline{\mathcal{M}}_{l}(-l-d)\to 0$ (5.24) be an exact sequence of metrized coherent sheaves on ${\mathbb{P}}^{n}_{X}$, where, for each $i=l,\dots,m$, $\overline{\mathcal{M}}_{i}$ is a metrized coherent sheaf on $X$, and $\overline{\mathcal{M}}_{i}(k)$ is as in Notation 5.2. On $\pi_{\ast}{\mathcal{M}}_{i}(k)$ we consider the hermitian structures given also by Notation 5.2. Then $\sum_{i=l}^{m}(-1)^{i}t_{n,-d-i}\operatorname{ch}(\overline{\mathcal{M}}_{i})=\operatorname{\widetilde{ch}}(\overline{\pi_{\ast}\mu})-\overline{\pi}^{\text{\rm FS}}_{\flat}(\operatorname{\widetilde{ch}}(\overline{\mu})).$ (5.25) ###### Proof. We consider a commutative diagram of exact sequences $\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mu}^{\prime}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathcal{M}}_{m}^{\prime}(-m-d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathcal{M}}_{l}^{\prime}(-l-d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\overline{\mu}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathcal{M}}_{m}(-m-d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathcal{M}}_{l}(-l-d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\overline{\mu}^{\prime\prime}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathcal{M}}_{m}^{\prime\prime}(-m-d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathcal{M}}_{l}^{\prime\prime}(-l-d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{\overline{\xi}_{m}}$$\textstyle{\dots}$$\textstyle{\overline{\xi}_{l}.}$ _Claim._ If equation (5.25) holds for two of $\overline{\mu}$, $\overline{\mu}^{\prime}$ and $\overline{\mu}^{\prime\prime}$, then it holds for the third. _Proof of the claim._ On the one hand we have $\sum_{i=l}^{m}(-1)^{i}t_{n,-d-i}\left(\operatorname{ch}(\overline{\mathcal{M}}^{\prime}_{i})-\operatorname{ch}(\overline{\mathcal{M}}_{i})+\operatorname{ch}(\overline{\mathcal{M}}^{\prime\prime}_{i})\right)=\sum_{i=l}^{m}(-1)^{i}t_{n,-d-i}\operatorname{d}_{\mathcal{D}}\operatorname{\widetilde{ch}}(\overline{\xi}_{i}).$ But, if $t\in\mathcal{D}^{1}(\operatorname{Spec}\mathbb{C},1)=\mathbb{R}$ is a real number, in the group $\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}(X,p)$, we have $t\operatorname{d}_{\mathcal{D}}\operatorname{\widetilde{ch}}(\overline{\xi}_{i})=-\operatorname{d}_{\mathcal{D}}(t\bullet\operatorname{\widetilde{ch}}(\overline{\xi}_{i}))=0.$ On the other hand, by Lemma 5.20 $\operatorname{\widetilde{ch}}(\overline{\pi_{\ast}\mu^{\prime}})-\operatorname{\widetilde{ch}}(\overline{\pi_{\ast}\mu})+\operatorname{\widetilde{ch}}(\overline{\pi_{\ast}\mu^{\prime\prime}})=\pi_{\flat}^{\text{\rm FS}}(\operatorname{\widetilde{ch}}(\overline{\mu}^{\prime}))-\pi_{\flat}^{\text{\rm FS}}(\operatorname{\widetilde{ch}}(\overline{\mu}))+\pi_{\flat}^{\text{\rm FS}}(\operatorname{\widetilde{ch}}(\overline{\mu}^{\prime\prime})).$ The proof of the lemma is done by induction on the length $r=m-l$ of the complex. If $r\leq n$ then $\mu(d+l)$ has the same shape as the canonical resolution of the zero sheaf. By the uniqueness of the canonical resolution, we have ${\mathcal{M}}_{i}=0$, for $i=l,\dots,m$. Using the above claim when ${\mathcal{M}}_{i}=0$ has a non-trivial metric, we obtain the lemma for $r\leq n$. Assume now that $r>n$. Let $K$ be the Koszul exact sequence (5.18). Then $K(1)\otimes{\mathcal{M}}_{l}$ is the canonical resolution of the regular coherent sheaf ${\mathcal{M}}_{l}(1)$. By Theorem 4.5 (i) there is a surjection of exact sequences $\mu\to K(-l-d)\otimes{\mathcal{M}}_{l}$ whose kernel is an exact sequence $\mu^{\prime}:0\to{\mathcal{M}}^{\prime}_{m}(-m-d)\to\dots\to{\mathcal{M}}^{\prime}_{l+1}(-d-l-1)\to 0.$ We consider on $K$ the metrics of (5.18), for $i=l+1,\dots,m$, we choose arbitrary metrics on ${\mathcal{M}}^{\prime}_{i}$ and denote by $\overline{\mu}^{\prime}$ the corresponding exact sequence of metrized coherent sheaves. By induction hypothesis, $\overline{\mu}^{\prime}$ satisfies equation (5.25). Moreover, since the characteristic numbers $t_{n,k}$ for $k\not\in[0,n]$ are defined using equation (5.19), the exact sequence $\overline{K}(-l-d)\otimes\overline{\mathcal{M}}_{l}$ also satisfies equation (5.25). Hence the lemma follows from the previous claim. ∎ We now treat the case of complexes concentrated in a single degree. Let $\overline{\mathcal{F}}$ be a coherent sheaf on ${\mathbb{P}}^{n}_{X}$ with a hermitian structure and let $\overline{\pi_{\ast}\mathcal{F}}$ be a choice of a hermitian structure on the direct image complex. Write $\overline{\xi}=(\overline{\pi}^{\text{\rm FS}},\overline{\mathcal{F}},\overline{\pi_{\ast}\mathcal{F}})$ for the corresponding relative metrized complex. Choose an integer $d$ such that $\mathcal{F}(d)$ is regular. Then we have the resolution $\gamma_{d}(\mathcal{F})$ of Corollary 4.3. More generally, let $\mu$ be an exact sequence of the form $0\to\mathcal{S}_{m}(-d-m)\to\dots\to\mathcal{S}_{1}(-d-1)\to\mathcal{S}_{0}(-d)\to\mathcal{F}\to 0,$ where the $\mathcal{S}_{i}$, $i=0,\dots,m$ are coherent sheaves on $X$. Assume that we have chosen hermitian structures on the sheaves $\mathcal{S}_{i}$. Using Notation 5.2 and [17, Def. 3.37, Def. 3.39] we have objects $[\overline{\mu}]$ in $\operatorname{\mathbf{KA}}({\mathbb{P}}^{n}_{X})$ and $[\overline{\pi_{\ast}\mu}]$ in $\operatorname{\mathbf{KA}}(X)$. Then we write $T_{\mathfrak{t},\overline{\mu}}(\overline{\xi})=\sum_{j=0}^{m}(-1)^{j}t_{n,j-d}\operatorname{ch}(\overline{\mathcal{S}}_{j})-\operatorname{\widetilde{ch}}(\overline{\pi_{\ast}\mu})+\overline{\pi}^{\text{\rm FS}}_{\flat}(\operatorname{\widetilde{ch}}{(\overline{\mu})})$ (5.26) ###### Lemma 5.27. Given any choice of metrics on the sheaves $\mathcal{G}_{i}$, (respectively ${\mathcal{G}}_{i}^{\prime}$) $i=0,\dots,n$, that appear in the resolution $\gamma_{d}(\mathcal{F})$ (respectively $\gamma_{d+1}(\mathcal{F})$), denote by $\overline{\gamma}_{d}$ and $\overline{\gamma}_{d+1}$ the corresponding exact sequences of metrized coherent sheaves. Then $T_{\mathfrak{t},\overline{\gamma}_{d+1}}(\overline{\xi})=T_{\mathfrak{t},\overline{\gamma}_{d}}(\overline{\xi}).$ In particular, $T_{\mathfrak{t},\overline{\gamma}_{d}}(\overline{\xi})$ does not depend on the choice of metrics on the sheaves $\mathcal{G}_{i}$. ###### Proof. By Theorem 4.5 (ii), there is an exact sequence $\overline{\mu}:0\to\overline{\mathcal{S}}_{n+k}(-n-k-d-1)\to\dots\to\overline{\mathcal{S}}_{0}(-d-1)\to\overline{\mathcal{F}}\to 0,$ (5.28) and a surjection of exact sequences $f\colon\overline{\mu}\to\overline{\gamma}_{d}$ extending the identity on $\overline{\mathcal{F}}$. Here $\overline{\mathcal{S}}_{i}$, $i=0,\dots,n+k$ are coherent sheaves on $X$ with hermitian structures. By Theorem 4.5 (i) there is a surjection of exact sequences $\overline{\mu}\longrightarrow\overline{\gamma}_{d+1}$ extending the identity on $\overline{\mathcal{F}}$, whose kernel is an exact sequence $\overline{\varepsilon}:0\to\overline{\mathcal{M}}_{n+k}(-n-k-d-1)\to\dots\to\overline{\mathcal{M}}_{0}(-d-1)\to 0,$ (5.29) where $\overline{\mathcal{M}}_{i}$, $i=0,\dots,n+k$ are coherent sheaves on $X$, and we have chosen arbitrarily an hermitian structure on them. Denote by $\overline{\eta}_{i}$ the rows of the exact sequence $0\to\overline{\varepsilon}\to\overline{\mu}\to\overline{\gamma}_{d+1}\to 0.$ Observe that $\overline{\eta}_{i}=\overline{\eta}^{\prime}_{i}(-i-d-1)$ for some short exact sequence $\overline{\eta}_{i}^{\prime}$ on $X$. When $j\geq n$ we denote $\overline{\mathcal{G}}^{\prime}_{j}=\overline{0}$. Then, we have $\sum_{j=0}^{n+k}(-1)^{j}t_{n,j-d-1}\left(\operatorname{ch}(\overline{\mathcal{G}}^{\prime}_{j})-\operatorname{ch}(\overline{\mathcal{S}}_{j})+\operatorname{ch}(\overline{\mathcal{M}}_{j})\right)=\\\ \sum_{j=0}^{n+k}(-1)^{j}t_{n,j-d-1}\operatorname{d}_{\mathcal{D}}\operatorname{\widetilde{ch}}(\overline{\eta}_{i}^{\prime})=0.$ (5.30) By [17, Prop. 3.41], we have $\displaystyle\operatorname{\widetilde{ch}}(\overline{\pi_{\ast}\gamma_{d+1}})-\operatorname{\widetilde{ch}}(\overline{\pi_{\ast}\mu})+\operatorname{\widetilde{ch}}(\overline{\pi_{\ast}\varepsilon})$ $\displaystyle=\sum_{j=0}^{n+k}(-1)^{j}\operatorname{\widetilde{ch}}(\overline{\pi_{\ast}\eta_{j}}),$ (5.31) $\displaystyle\operatorname{\widetilde{ch}}(\overline{\gamma}_{d+1})-\operatorname{\widetilde{ch}}(\overline{\mu})+\operatorname{\widetilde{ch}}(\overline{\varepsilon})$ $\displaystyle=\sum_{j=0}^{n+k}(-1)^{j}\operatorname{\widetilde{ch}}(\overline{\eta}_{j}).$ (5.32) Combining equations (5.30), (5.31) and (5.32) and lemmas 5.20 and 5.23 we obtain $T_{\mathfrak{t},\overline{\mu}}(\overline{\xi})=T_{\mathfrak{t},\overline{\gamma}_{d+1}}(\overline{\xi}).$ (5.33) We consider now $\operatorname{cone}(\mu,\gamma_{d})$. On it we put the obvious hermitian structure induced by $\overline{\mu}$ and $\overline{\gamma_{d}}$, $\overline{\operatorname{cone}(\mu,\gamma_{d})}$. On $\pi_{\ast}\operatorname{cone}(\mu,\gamma_{d})$, we put the obvious family of hermitian metrics induced by $\overline{\pi_{\ast}\mu}$ and $\overline{\pi_{\ast}\gamma_{d}}$, and denote it as $\overline{\pi_{\ast}\operatorname{cone}(\mu,\gamma_{d})}$. By [17, Cor. 3.42] we have $\displaystyle\operatorname{\widetilde{ch}}(\overline{\operatorname{cone}(\mu,\gamma_{d})})$ $\displaystyle=\operatorname{\widetilde{ch}}(\overline{\gamma}_{d})-\operatorname{\widetilde{ch}}(\overline{\mu}),$ (5.34) $\displaystyle\operatorname{\widetilde{ch}}(\overline{\pi_{\ast}\operatorname{cone}(\mu,\gamma_{d})})$ $\displaystyle=\operatorname{\widetilde{ch}}(\overline{\pi_{\ast}\gamma_{d}})-\operatorname{\widetilde{ch}}(\overline{\pi_{\ast}\mu}).$ (5.35) Observe that $\overline{\operatorname{cone}(\mu,\gamma_{d})}^{i}=\overline{\mathcal{S}}_{-i-1}(i-d)\oplus\overline{\mathcal{G}}_{-i}(i-d)$. Combining Lemma 5.23 for $\overline{\operatorname{cone}(\mu,\gamma_{d})}$ with equations (5.34) and (5.35), we obtain $T_{\mathfrak{t},\overline{\mu}}(\overline{\xi})=T_{\mathfrak{t},\overline{\gamma}_{d}}(\overline{\xi}),$ (5.36) Together with equation (5.33) this proves the lemma. ∎ Now we are in position to prove the existence of $T_{\mathfrak{t}}$. Let $n$ and $\mathfrak{t}$ be as in Theorem 5.9. We define the numbers $t_{n,k}$, for $k<0$ and $k>n$ by equation (5.19). Let $\overline{\xi}=(\overline{\pi}^{\text{\rm FS}},\overline{\mathcal{F}},\overline{\pi_{\ast}\mathcal{F}})$ be a relative metrized complex. We construct $T_{\mathfrak{t}}(\overline{\xi})$ by induction on the length of the cohomology of $\mathcal{F}$. If it has at most a single non zero coherent sheaf $\mathcal{H}$ sitting at degree $j$, then $\overline{\mathcal{F}}$ and $\overline{\pi_{\ast}\mathcal{F}}$ determine hermitian structures on $\mathcal{H}[-j]$ and $\pi_{\ast}\mathcal{H}[-j]$ respectively. We choose an integer $d$ such that $\mathcal{H}(d)$ is regular and we write $T_{\mathfrak{t}}(\overline{\xi})=(-1)^{j}T_{\mathfrak{t},\overline{\gamma}_{d}(\mathcal{H})}(\overline{\pi}^{\text{\rm FS}},\overline{\mathcal{H}},\overline{\pi_{\ast}\mathcal{H}}).$ (5.37) By Lemma 5.27, this does not depend on the choice of $d$ nor on the choice of metrics on $\overline{\gamma}_{d}(\mathcal{H})$. Assume that we have defined the analytic torsion classes for all complexes whose cohomology has length less than $l$ and that the cohomology of $\mathcal{F}$ has length $l$. Let $\mathcal{H}$ be the highest cohomology sheaf of $\mathcal{F}$, say of degree $j$. Choose auxiliary hermitian structures on $\mathcal{H}[-j]$ and $\pi_{\ast}\mathcal{H}[-j]$. There is a unique natural map $\mathcal{H}[-j]\dashrightarrow\mathcal{F}$. Then we define $T_{\mathfrak{t}}(\overline{\xi})=T_{\mathfrak{t}}(\overline{\pi}^{\text{\rm FS}},\overline{\mathcal{H}[-j]},\overline{\pi_{\ast}\mathcal{H}[-j]})\\\ +T_{\mathfrak{t}}(\overline{\pi}^{\text{\rm FS}},\operatorname{\overline{cone}}(\overline{\mathcal{H}[-j]},\overline{\mathcal{F}}),\operatorname{\overline{cone}}(\overline{\pi_{\ast}\mathcal{H}[-j]},\overline{\pi_{\ast}\mathcal{F}})).$ (5.38) It follows from [17, Thm. 2.27 (iv)] that the right hand side of this equality does not depend on the choice of the auxiliary hermitian structures. Finally, we consider the case when $\overline{\pi}$ has a metric different from the Fubini-Study metric. Thus, let $\overline{\xi}=(\overline{\pi},\overline{\mathcal{F}},\overline{\pi_{\ast}\mathcal{F}})$ and write $\overline{\xi}^{\prime}=(\overline{\pi}^{\text{\rm FS}},\overline{\mathcal{F}},\overline{\pi_{\ast}\mathcal{F}})$. Then we put $T_{\mathfrak{t}}(\overline{\xi})=T_{\mathfrak{t}}(\overline{\xi}^{\prime})+\overline{\pi}_{\flat}[\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}}_{m}(\overline{\pi},\overline{\pi}^{\text{\rm FS}})].$ (5.39) ###### Definition 5.40. Let $n$ and $\mathfrak{t}$ be as in Theorem 5.9. Then $T_{\mathfrak{t}}$ is the assignment that to each relative metrized complex $\overline{\xi}$ associates $T_{\mathfrak{t}}(\overline{\xi})$ given by equations (5.37), (5.38) and (5.39). It remains to prove that $T_{\mathfrak{t}}$ satisfies axioms (i) to (iv). Axiom (i) follows from the differential equations satisfied by the Bott-Chern classes. Axiom (ii) follows from the functoriality of the canonical resolution, the Chern forms and the Bott-Chern classes. Axiom (iii) follows from the additivity of the canonical resolution and of the Chern character. Finally Axiom (iv) follows from the multiplicativity of the Chern character. This concludes the proof of Theorem 5.9. ∎ We finish this section showing the compatibility of analytic torsion classes with the composition of projective bundles. Let $X$ be a smooth complex variety. Consider the commutative diagram with cartesian square $\textstyle{{\mathbb{P}}^{n_{1}}_{X}\underset{X}{\times}{\mathbb{P}}^{n_{2}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\scriptstyle{p}$$\textstyle{{\mathbb{P}}^{n_{1}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\textstyle{{\mathbb{P}}^{n_{2}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2}}$$\textstyle{X}$ On $\pi_{1}$ and $\pi_{2}$ we introduce arbitrary hermitian structures and on $p_{1}$ and $p_{2}$ the hermitian structures induced by the cartesian diagram. ###### Proposition 5.41. Let $\overline{\mathcal{F}}$ be an object of $\operatorname{\overline{\mathbf{D}}^{b}}({\mathbb{P}}^{n_{1}}_{X}\underset{X}{\times}{\mathbb{P}}^{n_{2}}_{X})$. Put arbitrary hermitian structures on $(p_{1})_{\ast}\mathcal{F}$, $(p_{2})_{\ast}\mathcal{F}$, and $p_{\ast}\mathcal{F}$. Then $T(\overline{\pi}_{1})+(\overline{\pi}_{1})_{\flat}(T(\overline{p}_{1}))=T(\overline{\pi}_{2})+(\overline{\pi}_{2})_{\flat}(T(\overline{p}_{2})),$ (5.42) where we are using the convention at the end of Definition 2.6. ###### Proof. By the anomaly formulas (Proposition 5.10), if equation (5.42) holds for a particular choice of hermitian structures on $\mathcal{F}$, $(p_{1})_{\ast}\mathcal{F}$, $(p_{2})_{\ast}\mathcal{F}$, and $p_{\ast}\mathcal{F}$, then it holds for any other choice. Let $\overline{\mathcal{F}}_{2}\dashrightarrow\overline{\mathcal{F}}_{1}\dashrightarrow\overline{\mathcal{F}}_{0}\dashrightarrow$ be a distinguished triangle and put hermitian structures on the direct images as before. Then Proposition 5.12 implies that, if equation (5.42) holds for two of them, then it also holds for the third. Since the objects of the form $\mathcal{G}(k,l):=p^{\ast}\mathcal{G}\otimes p_{1}^{\ast}\mathcal{O}(k)\otimes p_{2}^{\ast}\mathcal{O}(l)$ are a generating class of $\operatorname{\mathbf{D}^{b}}({\mathbb{P}}^{n_{1}}_{X}\underset{X}{\times}{\mathbb{P}}^{n_{2}}_{X})$, the previous discussion shows that it is enough to prove the case $\mathcal{F}=\mathcal{G}(k,l)$, with the hermitian structure of $\mathcal{F}$ induced by a hermitian structure of $\mathcal{G}$ and the Fubini-Study metric on $\mathcal{O}(k)$ and $\mathcal{O}(l)$, and the hermitian structures on the direct images defined as in (5.3). In this case the result follows easily from the functoriality and the projection formula. ∎ ## 6 Compatible analytic torsion classes In this section we study the compatibility between analytic torsion classes for closed immersions and analytic torsion classes for projective spaces. It turns out that, once the compatibility between the diagonal embedding of ${\mathbb{P}}^{n}$ into ${\mathbb{P}}^{n}\times{\mathbb{P}}^{n}$ and the second projection of ${\mathbb{P}}^{n}\times{\mathbb{P}}^{n}$ onto ${\mathbb{P}}^{n}$ is established, then all the other possible compatibilities follow. Essentially this observation can be traced back to [15]. Let $n$, $V$, $\overline{V}$ and ${\mathbb{P}}^{n}(V)$ be as in the previous section. We consider the diagram $\textstyle{{\mathbb{P}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\Delta\ \ \ }$$\textstyle{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\textstyle{{\mathbb{P}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{{\mathbb{P}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\textstyle{\operatorname{Spec}{\mathbb{C}}\ .}$ On ${\mathbb{P}}^{n}$ we have the tautological short exact sequence $0\rightarrow\mathcal{O}(-1)\rightarrow V\rightarrow Q\rightarrow 0\ .$ This induces on ${\mathbb{P}}^{n}\times{\mathbb{P}}^{n}$ the exact sequence $0\rightarrow p_{2}^{\ast}\mathcal{O}(-1)\rightarrow V\rightarrow p_{2}^{\ast}Q\rightarrow 0\ .$ By composition with the injection $p_{1}^{\ast}\mathcal{O}(-1)\hookrightarrow V$, we obtain a morphism $p_{1}^{\ast}\mathcal{O}(-1)\to p_{2}^{\ast}Q,$ hence a section of $p_{2}^{\ast}Q\otimes p_{1}^{\ast}\mathcal{O}(1)$. The zero locus of this section is the image of the diagonal. Moreover, the associated Koszul complex is quasi-isomorphic to $\Delta_{\ast}\mathcal{O}_{{\mathbb{P}}^{n}}$. That is, the sequence $0\rightarrow\Lambda^{n}(p_{2}^{\ast}Q^{\vee})\otimes p_{1}^{\ast}\mathcal{O}_{{\mathbb{P}}^{n}}(-n)\rightarrow\dots{\\\ }\dots\rightarrow\Lambda^{1}(p_{2}^{\ast}Q^{\vee})\otimes p_{1}^{\ast}\mathcal{O}_{{\mathbb{P}}^{n}}(-1)\rightarrow\mathcal{O}_{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}}\rightarrow\Delta_{\ast}\mathcal{O}_{{\mathbb{P}}^{n}}\rightarrow 0$ (6.1) is exact. On $T_{{\mathbb{P}}^{n}}$ and $T_{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}}$ we consider the Fubini-Study metrics. We denote by $\overline{\Delta}$ and $\overline{p}_{2}$ the morphisms of $\operatorname{\overline{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$ determined by these metrics. As in [17, Ex. 5.7], we have that $\overline{p_{2}}\circ\overline{\Delta}=\overline{\operatorname{id}}_{{\mathbb{P}}^{n}}$, where $T_{\overline{\operatorname{id}_{{\mathbb{P}}^{n}}}}=\overline{0}$. The Fubini-Study metric on $\mathcal{O}(-1)$ and the metric induced by the tautological exact sequence on $Q$ induce a metric $\overline{K}(\Delta)$ on the Koszul complex. This is a hermitian structure on $\Delta_{\ast}\mathcal{O}_{{\mathbb{P}}^{n}}$. Finally on $\mathcal{O}_{{\mathbb{P}}^{n}}$ we consider the trivial metric. This is a hermitian structure on $(p_{2})_{\ast}K(\Delta)$. Fix a real additive genus $S$ and denote by $T_{S}$ the theory of analytic torsion classes for closed immersions that is compatible with the projection formula and transitive, associated to $S$ (Theorem 3.11). Moreover, fix a family of real numbers $\mathfrak{t}=\\{t_{nk}\ \|\ n\geq 0,\ -n\leq k\leq 0\\}$ and denote $T_{\mathfrak{t}}$ the theory of generalized analytic torsion classes for projective spaces associated to this family. Compatible analytic torsion classes for closed immersions and for projective spaces should combine to provide analytic torsion classes for arbitrary projective morphisms, and these classes should be transitive. The transitivity condition for the composition $\operatorname{id}_{{\mathbb{P}}^{n}}=p_{2}\circ\Delta$ should give us $0=T(\overline{\operatorname{id}}_{{\mathbb{P}}^{n}},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})=T_{\mathfrak{t}}(\overline{p_{2}},\overline{K}(\Delta),\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})+(\overline{p}_{2})_{\flat}(T_{S}(\overline{\Delta},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}},\overline{K}(\Delta))).$ In general we define ###### Definition 6.2. The theories of analytic torsion classes $T_{S}$ and $T_{\mathfrak{t}}$ are called _compatible_ if $T_{\mathfrak{t}}(\overline{p_{2}},\overline{K}(\Delta),\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})+(\overline{p}_{2})_{\flat}(T_{S}(\overline{\Delta},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}},\overline{K}(\Delta)))=0.$ (6.3) ###### Theorem 6.4. Let $S$ be a real additive genus. Then there exists a unique family of real numbers $\mathfrak{t}=\\{t_{n,k}\ \|\ n\geq 0,\ -n\leq k\leq 0\\}$ such that the theories of analytic torsion classes $T_{S}$ and $T_{\mathfrak{t}}$ are compatible. The theory $T_{\mathfrak{t}}$ will also be denoted $T_{S}$. ###### Proof. The first step is to make explicit equation (6.3) in terms of the main characteristic numbers $\mathfrak{t}$. To this end, first observe that, since the exact sequence $0\to T_{p_{2}}\to T_{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}}\to p_{2}^{\ast}T_{{\mathbb{P}}^{n}}\to 0$ (6.5) is split and the hermitian metric on $T_{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}}$ is the orthogonal direct sum metric, $\overline{p_{2}}=\pi^{\ast}_{1}(\overline{\pi}^{\text{\rm FS}})$. Next, we denote by $\overline{K}(\Delta)_{i}$ the component of degree $i$ of the Koszul complex, and we define $\overline{(p_{2})_{\ast}K(\Delta)_{i}}=\begin{cases}\overline{\mathcal{O}}_{{\mathbb{P}}^{n}},&\text{for }i=0,\\\ \overline{0},&\text{for }i>0.\end{cases}$ Finally using Corollary 5.15, functoriality and the compatibility with the projection formula, we derive $\displaystyle T_{\mathfrak{t}}(\overline{p_{2}},\overline{K}(\Delta),\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})$ $\displaystyle=\sum_{i=0}^{n}(-1)^{i}T_{\mathfrak{t}}(\overline{p_{2}},\overline{K}(\Delta)_{i},\overline{(p_{2})_{\ast}K(\Delta)_{i}})$ $\displaystyle=\sum_{i=0}^{n}(-1)^{i}T_{\mathfrak{t}}(\pi_{1}^{\ast}\overline{\xi}_{n}(-i)\otimes\Lambda^{i}\overline{Q}^{\vee})$ $\displaystyle=\sum_{i=0}^{n}(-1)^{i}t_{n,-i}\operatorname{ch}(\Lambda^{i}\overline{Q}^{\vee}).$ Thus, the second and last step is to solve the equation $\sum_{i=0}^{n}(-1)^{i}t_{n,-i}\operatorname{ch}(\Lambda^{i}\overline{Q}^{\vee})=-(p_{2})_{\ast}(T_{S}(\overline{\Delta},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}},\overline{K}(\Delta))\bullet\operatorname{Td}(\overline{p_{2}})).$ (6.6) Since the left hand side of equation (6.6) is closed, in order to be able to solve this equation we have to show that the right hand side is also closed. We compute $\displaystyle\operatorname{d}_{\mathcal{D}}(p_{2})_{\ast}$ $\displaystyle(T_{S}(\overline{\Delta},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}},\overline{K}(\Delta))\bullet\operatorname{Td}(\overline{p_{2}}))$ $\displaystyle=(p_{2})_{\ast}\left(\sum_{i=0}^{n}(-1)^{i}\operatorname{ch}(\overline{K}(\Delta)_{i})\operatorname{Td}(\overline{p}_{2})-\Delta_{\ast}(\operatorname{ch}(\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})\operatorname{Td}(\overline{\Delta}))\operatorname{Td}(\overline{p}_{2})\right)$ $\displaystyle=(p_{2})_{\ast}\left(\sum_{i=0}^{n}(-1)^{i}p_{2}^{\ast}(\operatorname{ch}(\Lambda^{i}\overline{Q}^{\vee}))p_{1}^{\ast}(\operatorname{ch}(\overline{\mathcal{O}}(-i)))\operatorname{Td}(\overline{p}_{2})\right)-1$ $\displaystyle=\sum_{i=0}^{n}(-1)^{i}\operatorname{ch}(\Lambda^{i}\overline{Q}^{\vee})(p_{2})_{\ast}\left(p_{1}^{\ast}(\operatorname{ch}(\overline{\mathcal{O}}(-i)))\operatorname{Td}(\overline{p}_{2})\right)-1$ $\displaystyle=\sum_{i=0}^{n}(-1)^{i}\operatorname{ch}(\Lambda^{i}\overline{Q}^{\vee})\pi_{1}^{\ast}\pi_{\ast}\left(\operatorname{ch}(\overline{\mathcal{O}}(-i))\operatorname{Td}(\overline{\pi})\right)-1$ $\displaystyle=1-1=0.$ In the first equality we have used the differential equation of $T_{S}$. In the second one we have used the definition of the Koszul complex, the equation $\operatorname{ch}(\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})=1$ and the fact that, by the choice of hermitian structures on $T_{\overline{\Delta}}$ and $T_{\overline{p}_{2}}$ we have $\operatorname{Td}(\overline{\Delta})\bullet\Delta^{\ast}(\operatorname{Td}(\overline{p}_{2}))=1$. The third equality is the projection formula and the fourth is base change for cohomology. For the last equality we have used equation (5.22). Both sides of equation (6.6) are closed and defined up to boundaries, hence this is an equation in cohomology classes. The tautological exact sequence induces exact sequences $0\to\Lambda^{k}Q^{\vee}\to\Lambda^{k}V^{\vee}\to\Lambda^{k-1}Q^{\vee}\otimes\mathcal{O}(1)\to 0,$ that give us equations $\operatorname{ch}(\Lambda^{k}Q^{\vee})=\binom{n+1}{k}-\operatorname{ch}(\Lambda^{k-1}Q^{\vee})\operatorname{ch}(\mathcal{O}(1)).$ Hence $\operatorname{ch}(\Lambda^{k}Q^{\vee})=\sum^{k}_{i=0}(-1)^{i}\binom{n+1}{k-i}\operatorname{ch}(\mathcal{O}(i)).$ Since the classes $\operatorname{ch}(\mathcal{O}(i))$, $i=0,\dots,n$, form a basis of $\bigoplus_{p}H^{2p}_{\mathcal{D}}({\mathbb{P}}^{n},{\mathbb{R}}(p))$, the same is true for the classes $\operatorname{ch}(\Lambda^{i}Q^{\vee})$, $i=0,\dots,n$. Therefore, if $\mathbf{1}_{1}\in H^{1}_{\mathcal{D}}({\mathbb{P}}^{n},{\mathbb{R}}(1))$ is the class represented by the constant function $1$, the classes $\mathbf{1}_{1}\bullet\operatorname{ch}(\Lambda^{i}Q^{\vee})$, $i=0,\dots,n$ form a basis of $\bigoplus_{p=1}^{n+1}H^{2p-1}_{\mathcal{D}}({\mathbb{P}}^{n},{\mathbb{R}}(p))$. This implies that equation (6.6) has a unique solution. ∎ ###### Remark 6.7. Given a theory $T$ of analytic torsion classes for projective spaces, obtained from an arbitrary choice of characteristic numbers, in general, it does not exist an additive genus such that the associated theory of singular Bott-Chern classes is compatible with $T$. It would be interesting to characterize the collections of characteristic numbers that arise from Theorem 6.4. By definition, compatible analytic torsion classes for closed immersions and projective spaces satisfy a compatibility condition for the trivial vector bundle and the diagonal embedding. When adding the functoriality and the projection formula, we obtain compatibility relations for arbitrary sections of the trivial projective bundle and arbitrary objects. Let $X$ be a smooth complex variety, let $\pi\colon{\mathbb{P}}_{X}^{n}\to X$ be the projective space over $X$ and let $s\colon X\to{\mathbb{P}}_{X}^{n}$ be a section. Choose any hermitian structure on $T_{\pi}$. Since we have an isomorphism $T_{s}\dashrightarrow s^{\ast}T_{\pi}[-1]$, this hermitian structure induces a hermitian structure on $s$. Denote by $\overline{\pi}$ and $\overline{s}$ the corresponding morphisms in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. With this choice of hermitian structures, we have $\overline{\pi}\circ\overline{s}=(\pi\circ s,\operatorname{\overline{cone}}(s^{\ast}T_{\overline{\pi}}[-1],s^{\ast}T_{\overline{\pi}}[-1]))=(\operatorname{id}_{X},\overline{0}),$ because the cone of the identity is meager. ###### Proposition 6.8. Let $S$ be a real additive genus. Let $T_{S}$ denote both, the theory of analytic torsion classes for closed immersions determined by $S$, and the theory of analytic torsion classes for projective spaces compatible with it. Let $\overline{\mathcal{F}}$ be an object of $\operatorname{\overline{\mathbf{D}}^{b}}(X)$. Put a hermitian structure on $s_{\ast}\mathcal{F}$. Then $T_{S}(\overline{\pi},\overline{s_{\ast}\mathcal{F}},\overline{\mathcal{F}})+\overline{\pi}_{\flat}(T_{S}(\overline{s},\overline{\mathcal{F}},\overline{s_{\ast}\mathcal{F}}))=0.$ (6.9) ###### Proof. By the anomaly formulas Proposition 3.17 and Proposition 5.10, if equation (6.9) holds for a particular choice of hermitian structure on $s_{\ast}\mathcal{F}$ then it holds for any other choice. Therefore we can assume that the hermitian structure on $s_{\ast}\mathcal{F}$ is given by $\overline{K}(s)\otimes\pi^{\ast}\overline{\mathcal{F}}$, where $\overline{K}(s)$ is the Koszul complex associated to the section $s$. By the projection formulas, if (6.9) holds for the trivial bundle $\mathcal{O}_{X}$ then it holds for arbitrary objects of $\operatorname{\overline{\mathbf{D}}^{b}}(X)$. We now prove that, if equation (6.9) holds for a particular choice of hermitian structure $\overline{\pi}$, then it holds for any other choice. Thus, assume that equation (6.9) is satisfied for $\overline{\pi}$ and $\overline{s}$. Let $\overline{\pi}^{\prime}$ be another choice of hermitian structure on $\pi$ and $\overline{s}^{\prime}$ be the hermitian structure induced on $s$. On one hand, we have $T_{S}(\overline{\pi}^{\prime},\overline{K}(s),\overline{\mathcal{O}}_{X})=T_{S}(\overline{\pi},\overline{K}(s),\overline{\mathcal{O}}_{X})+\pi_{\ast}\left(\operatorname{ch}(\overline{K}(s)\bullet\widetilde{\operatorname{Td}}_{m}(\overline{\pi}^{\prime},\overline{\pi})\bullet\operatorname{Td}(\overline{\pi}^{\prime}))\right).$ (6.10) On the other hand, we have $\displaystyle T_{S}(\overline{s}^{\prime},\overline{\mathcal{F}},\overline{s_{\ast}\mathcal{F}})\bullet\operatorname{Td}(\overline{\pi}^{\prime})$ (6.11) $\displaystyle\phantom{A}=\left(T_{S}(\overline{s},\overline{\mathcal{F}},\overline{s_{\ast}\mathcal{F}})+s_{\ast}(\widetilde{\operatorname{Td}}_{m}(\overline{s}^{\prime},\overline{s})\operatorname{Td}(\overline{s}^{\prime}))\right)\bullet\left(\operatorname{Td}(\overline{\pi})-\operatorname{d}_{\mathcal{D}}(\widetilde{\operatorname{Td}}_{m}(\overline{\pi}^{\prime},\overline{\pi})\bullet\operatorname{Td}(\overline{\pi}^{\prime}))\right)$ $\displaystyle\phantom{A}=T_{S}(\overline{s},\overline{\mathcal{F}},\overline{s_{\ast}\mathcal{F}})\bullet\operatorname{Td}(\overline{\pi})+s_{\ast}(\widetilde{\operatorname{Td}}_{m}(\overline{s}^{\prime},\overline{s})\operatorname{Td}(\overline{s}^{\prime}))\bullet\operatorname{Td}(\overline{\pi}^{\prime})$ $\displaystyle\phantom{AAAAAAAAAAAAAAA}-T_{S}(\overline{s},\overline{\mathcal{F}},\overline{s_{\ast}\mathcal{F}})\bullet\operatorname{d}_{\mathcal{D}}(\widetilde{\operatorname{Td}}_{m}(\overline{\pi}^{\prime},\overline{\pi})\bullet\operatorname{Td}(\overline{\pi}^{\prime}))$ In the group $\bigoplus_{p}\widetilde{D}_{D}^{2p-1}({\mathbb{P}}^{n}_{X},N_{s},p)$ we have $T_{S}(\overline{s},\overline{\mathcal{F}},\overline{s_{\ast}\mathcal{F}})\bullet\operatorname{d}_{\mathcal{D}}(\widetilde{\operatorname{Td}}_{m}(\overline{\pi}^{\prime},\overline{\pi})\bullet\operatorname{Td}(\overline{\pi}^{\prime}))\\\ =\left(\operatorname{ch}(\overline{K}(s))-s_{\ast}(\operatorname{Td}(\overline{s}))\right)\bullet\left(\widetilde{\operatorname{Td}}_{m}(\overline{\pi}^{\prime},\overline{\pi})\bullet\operatorname{Td}(\overline{\pi}^{\prime})\right).$ (6.12) Observe that, by the definition of the hermitian structure of $\overline{s}$ and $\overline{s}^{\prime}$ we have $\operatorname{Td}(\overline{s})\bullet s^{\ast}\widetilde{\operatorname{Td}}_{m}(\overline{\pi}^{\prime},\overline{\pi})=-\widetilde{\operatorname{Td}}_{m}(\overline{s}^{\prime},\overline{s})\bullet\operatorname{Td}(\overline{s}^{\prime}).$ (6.13) By combining equations (6.9), (6.10), (6.12) and (6.13) we obtain $T_{S}(\overline{\pi}^{\prime},\overline{s_{\ast}\mathcal{F}},\overline{\mathcal{F}})=-\pi_{\ast}\left(T_{S}(\overline{s}^{\prime},\overline{\mathcal{F}},\overline{s_{\ast}\mathcal{F}})\bullet\operatorname{Td}(\overline{\pi}^{\prime})\right).$ (6.14) We now prove (6.9) for a particular choice of hermitian structures. Let $f\colon X\to{\mathbb{P}}^{n}$ denote the composition of $s$ with the projection ${\mathbb{P}}^{n}_{X}\to{\mathbb{P}}^{n}$. Then we have a commutative diagram with cartesian squares $\textstyle{{\mathbb{P}}^{n}\times X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\times f}$$\scriptstyle{\pi}$$\textstyle{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{f}$$\textstyle{{\mathbb{P}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\scriptstyle{\operatorname{id}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{{\mathbb{P}}^{n}}$ Let $\overline{\Delta}$ and $\overline{p}_{2}$ be as in Definition 6.2. On $\overline{\pi}$ and $\overline{s}$ we put the hermitian structures induced by $\overline{\Delta}$. Since the Koszul complex $\overline{K}(s)=(\operatorname{id}_{{\mathbb{P}}^{n}}\times f)^{\ast}\overline{K}(\Delta)$, by Proposition 2.15 and functoriality, equation (6.9) in this case follows from equation (6.3). ∎ We now study another compatibility between analytic torsion classes for closed immersions and projective spaces. Let $\iota\colon X\to Y$ be a closed immersion of smooth complex varieties. Consider the cartesian square $\textstyle{{\mathbb{P}}^{n}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\iota_{1}}$$\textstyle{{\mathbb{P}}^{n}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{Y}$ Choose hermitian structures on $\pi$ and $\iota$ and put on $\pi_{1}$ and $\iota_{i}$ the induced ones. ###### Proposition 6.15. Let $S$ be a real additive genus. Let $T_{S}$ denote both, the theory of analytic torsion classes for closed immersions determined by $S$, and the theory of analytic torsion classes for projective spaces compatible with it. Let $\overline{\mathcal{F}}$ be and object of $\operatorname{\overline{\mathbf{D}}^{b}}({\mathbb{P}}^{n}_{X})$. Put hermitian structures on $(\pi_{1})_{\ast}\mathcal{F}$, $(\iota_{1})_{\ast}\mathcal{F}$ and $(\pi\circ\iota_{1})_{\ast}\mathcal{F}$. Then $T_{S}(\overline{\pi})+\overline{\pi}_{\flat}(T_{S}(\overline{\iota_{1}}))=T_{S}(\overline{\iota})+\overline{\iota}_{\flat}(T_{S}(\overline{\pi_{1}})).$ (6.16) ###### Proof. By the anomaly formulas, if equation (6.16) holds for a particular choice of metrics on $(\pi_{1})_{\ast}\mathcal{F}$, $(\iota_{1})_{\ast}\mathcal{F}$ and $(\pi\circ\iota_{1})_{\ast}\mathcal{F}$, then it holds for any choice. Because the sheaves $\mathcal{G}(k)$, with $\mathcal{G}$ a coherent sheaf on $X$, constitute a generating class of $\operatorname{\mathbf{D}^{b}}({\mathbb{P}}^{n}_{X})$ and by propositions 3.19 and 5.12, we reduce to the case when $\mathcal{F}$ is of the form $\mathcal{G}(k)$. We choose arbitrary hermitian structures on $\mathcal{G}$ and $\iota_{\ast}\mathcal{G}$. Furthermore, we assume ${\mathcal{O}}(k)$, $(\pi_{1})_{\ast}{\mathcal{O}}(k)$ and $\pi_{\ast}{\mathcal{O}}(k)$ endowed with the hermitian structures of Notation 5.2. From these choices and the projection formula, the objects $(\pi_{1})_{\ast}\mathcal{F}$, $(\iota_{1})_{\ast}\mathcal{F}$ and $(\pi\circ\iota_{1})_{\ast}\mathcal{F}$ automatically inherit hermitian structures. Indeed, it is enough to observe the natural isomorphisms $\displaystyle(\pi_{1})_{\ast}\mathcal{F}\cong\mathcal{G}\otimes(\pi_{1})_{\ast}{\mathcal{O}}(k)$ (6.17) $\displaystyle(\iota_{1})_{\ast}(\pi_{1}^{\ast}\mathcal{G}\otimes\iota_{1}^{\ast}{\mathcal{O}}(k))\cong\pi^{\ast}(\iota_{\ast}\mathcal{G})\otimes{\mathcal{O}}(k)$ (6.18) $\displaystyle(\pi\circ\iota_{1})_{\ast}\mathcal{F}\cong\pi_{\ast}(\pi^{\ast}\iota_{\ast}\mathcal{G}\otimes{\mathcal{O}}(k))\cong\iota_{\ast}\mathcal{G}\otimes\pi_{\ast}{\mathcal{O}}(k).$ (6.19) We now work out the left hand side of equation (6.16). Using the projection formula for the theory $T_{S}$ for projective spaces, and equations (6.17)–(6.19), we find $T_{S}(\overline{\pi})=t_{n,k}\bullet\operatorname{ch}(\overline{\iota_{\ast}\mathcal{G}}).$ (6.20) Using the functoriality of $T_{S}$ for closed immersions and the projection formula we have $\displaystyle T_{S}(\overline{\iota}_{1})=$ $\displaystyle\pi^{\ast}T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota_{\ast}\mathcal{G}})\bullet\operatorname{ch}(\overline{{\mathcal{O}}(k)})$ $\displaystyle\overline{\pi}_{\flat}(T_{S}(\overline{\iota}_{1}))=$ $\displaystyle T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota_{\ast}\mathcal{G}})\bullet\pi_{\ast}(\operatorname{ch}(\overline{{\mathcal{O}}(k)})\bullet\operatorname{Td}(\overline{\pi})).$ (6.21) Now for the right hand side of (6.16). The projection formula for $T_{S}$ for closed immersions implies $T_{S}(\overline{\iota})=T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota\mathcal{G}})\bullet\operatorname{ch}(\overline{\pi_{\ast}{\mathcal{O}}(k)}).$ (6.22) Similarly, we obtain $T_{S}(\overline{\pi}_{1})=t_{n,k}\bullet\operatorname{ch}(\overline{\mathcal{G}}),$ and hence $\overline{\iota}_{\flat}(T_{S}(\overline{\pi}_{1}))=t_{n,k}\bullet\iota_{\ast}(\operatorname{ch}(\overline{\mathcal{G}})\bullet\operatorname{Td}(\overline{\iota})).$ (6.23) Using (6.20)–(6.23), the difference of the two sides of (6.16) becomes $t_{n,k}\bullet\operatorname{d}_{\mathcal{D}}T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota_{\ast}\mathcal{G}})-T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota_{\ast}\mathcal{G}})\bullet\operatorname{d}_{\mathcal{D}}t_{n,k}=-\operatorname{d}_{\mathcal{D}}(t_{n,k}\bullet T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota_{\ast}\mathcal{G}}))=0$ in the group $\oplus_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(Y,N_{\iota},p)$. ∎ ## 7 Generalized analytic torsion classes In this section we will extend the definition of analytic torsion classes to arbitrary morphisms of smooth complex varieties. Our construction is based on the construction of analytic torsion classes by Zha in [50]. ###### Definition 7.1. A theory of generalized analytic torsion classes is an assignment that, to each morphism $\overline{f}\colon X\to Y$ in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$ and relative metrized complex $\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}}),$ assigns a class of currents $T(\overline{\xi})\in\bigoplus_{p=1}^{n+1}\widetilde{\mathcal{D}}^{2p-1}_{D}(Y,N_{f},p)$ satisfying the following properties: 1. (i) (Differential equation) For any current $\eta\in T(\overline{\xi})$, we have $\operatorname{d}_{\mathcal{D}}\eta=\operatorname{ch}(\overline{f_{\ast}\mathcal{F}})-\overline{f}_{\flat}[\operatorname{ch}(\overline{\mathcal{F}})].$ (7.2) 2. (ii) (Functoriality) If $g\colon Y^{\prime}\to Y$ is a morphism transverse to $f$, then $g^{\ast}T(\overline{\xi})=T(g^{\ast}\overline{\xi}).$ 3. (iii) (Additivity and normalization) If $\overline{\xi}_{1}$, $\overline{\xi}_{2}$ are relative metrized complexes on $X$, then $T(\overline{\xi}_{1}\oplus\overline{\xi}_{2})=T(\overline{\xi}_{1})+T(\overline{\xi}_{2}).$ 4. (iv) (Projection formula) If $\overline{\xi}$ is a relative metrized complex, and $\overline{\mathcal{G}}\in\operatorname{Ob}\operatorname{\overline{\mathbf{D}}^{b}}(Y)$, then $T(\overline{\xi}\otimes\overline{\mathcal{G}})=T(\overline{\xi})\bullet\operatorname{ch}(\overline{\mathcal{G}}).$ 5. (v) (Transitivity) If $\overline{f}\colon X\to Y$, $\overline{g}\colon Y\to Z$ are morphisms in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$, and $(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$ and $(\overline{g},\overline{f_{\ast}\mathcal{F}},\overline{(g\circ f)_{\ast}\mathcal{F}})$ are relative metrized complexes, then $T(\overline{g}\circ\overline{f})=T(\overline{g})+\overline{g}_{\flat}(T(\overline{f})).$ (7.3) Propositions 7.4 and 7.6 below contain several anomaly and compatibility formulas satisfied by an arbitrary theory of generalized analytic torsion classes. They follow from properties (i)–(iii) and are analogous to those in propositions 3.17 and 5.10, 3.19 and 5.12 respectively. The proofs are omitted, as they are similar to those of the analogous statements referred to hereinbefore. ###### Proposition 7.4. Let $T$ be a theory of generalized analytic torsion classes. Let $\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$ be a relative metrized complex. 1. (i) If $\overline{\mathcal{F}}^{\prime}$ is another choice of metric on $\mathcal{F}$ and $\overline{\xi}_{1}=(\overline{f},\overline{\mathcal{F}}^{\prime},\overline{f_{\ast}\mathcal{F}})$, then $T(\overline{\xi}_{1})=T(\overline{\xi})+\overline{f}_{\flat}[\operatorname{\widetilde{ch}}(\overline{\mathcal{F}}^{\prime},\overline{\mathcal{F}})].$ 2. (ii) If $\overline{f}^{\prime}$ is another choice of hermitian structure on $f$ and $\overline{\xi}_{2}=(\overline{f}^{\prime},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$, then $T(\overline{\xi}_{2})=T(\overline{\xi})+\overline{f}^{\prime}_{\flat}[\operatorname{ch}(\overline{\mathcal{F}})\bullet\widetilde{\operatorname{Td}}_{m}(\overline{f}^{\prime},\overline{f})].$ (7.5) 3. (iii) If $\overline{f_{\ast}\mathcal{F}}^{\prime}$ is a different choice of metric on $f_{\ast}\mathcal{F}$, and $\overline{\xi}_{3}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}}^{\prime})$, then $T(\overline{\xi}_{3})=T(\overline{\xi})-\operatorname{\widetilde{ch}}(\overline{f_{\ast}\mathcal{F}}^{\prime},\overline{f_{\ast}\mathcal{F}}).$ ###### Proposition 7.6. Let $T$ be a theory of generalized analytic torsion classes. Let $\overline{f}\colon X\to Y$ be a morphism in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. Consider the distinguished triangles in $\operatorname{\overline{\mathbf{D}}^{b}}(X)$ and $\operatorname{\overline{\mathbf{D}}^{b}}(Y)$ respectively: $(\overline{\tau}):\ \overline{\mathcal{F}}_{2}\to\overline{\mathcal{F}}_{1}\to\overline{\mathcal{F}}_{0}\to\overline{\mathcal{F}}_{2}[1],\ \text{ and }\ (\overline{f_{\ast}\tau}):\ \overline{f_{\ast}\mathcal{F}}_{2}\to\overline{f_{\ast}\mathcal{F}}_{1}\to\overline{f_{\ast}\mathcal{F}}_{0}\to\overline{f_{\ast}\mathcal{F}}_{2}[1],$ and the relative metrized complexes $\overline{\xi}_{i}=(\overline{f},\overline{\mathcal{F}}_{i},\overline{f_{\ast}\mathcal{F}}_{i})$, $i=0,1,2$. Then we have: $\sum_{j=0,1,2}(-1)^{j}T(\overline{\xi}_{j})=\widetilde{\operatorname{ch}}(\overline{\pi_{\ast}\tau})-\overline{f}_{\flat}(\widetilde{\operatorname{ch}}(\overline{\tau})).$ The main result of this section is the following classification theorem. ###### Theorem 7.7. Let $S$ be a real additive genus. Then there exists a unique theory of generalized analytic torsion classes that agrees with $T_{S}$ when restricted to the class of closed immersions. Moreover, if $T$ is a theory of generalized analytic torsion classes, then there exists a real additive genus $S$ such that $T=T_{S}$. We will denote the theory associated to the additive genus $S$, whose existence is guaranteed by the preceding theorem, by $T_{S}$. In particular, there is a unique theory of generalized analytic torsion classes that agrees with $T^{h}$ when restricted to the class of closed immersions. This theory will be called homogeneous. ###### Proof. We first prove the uniqueness. Let $T$ be a theory of analytic torsion classes that agrees with $T_{S}$ for the class of closed immersions. Since the restriction of $T$ to projective spaces, by the transitivity axiom, is compatible with $T_{S}$, by Theorem 6.4, it also agrees with $T_{S}$. Finally, the transitivity axiom implies that $T$ is determined by its values for closed immersions and projective spaces. We now prove the existence. For the moment, let $T_{S}$ be the theory of analytic torsion classes for closed immersions and projective spaces determined by $S$. Let $\overline{f}\colon X\to Y$ be a morphism in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$, and let $\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$ be a relative metrized complex. Since $f$ is assumed to be projective, there is a factorization $f=\pi\circ\iota$, where $\iota\colon X\to{\mathbb{P}}^{n}_{Y}$ is a closed immersion and $\pi\colon{\mathbb{P}}_{Y}^{n}\to Y$ is the projection. Choose auxiliary hermitian structures on $\iota$, $\pi$ and $\iota_{\ast}\mathcal{F}$. Then we define $T_{S}(\overline{\xi})=T_{S}(\overline{\pi})+\overline{\pi}_{\flat}(T_{S}(\overline{\iota}))+\overline{f}_{\flat}\left[\operatorname{ch}(\overline{\mathcal{F}})\bullet\widetilde{\operatorname{Td}}_{m}(\overline{f},\overline{\pi}\circ\overline{\iota})\right]$ (7.8) To simplify the notations, in the sequel we will also refer to it simply by $T(\overline{\xi})$. The anomaly formulas easily imply that this definition does not depend on the choice of hermitian structures on $\iota$, $\pi$ and $\iota_{\ast}\mathcal{F}$. We next show that this definition is independent of the factorization of $f$. Let $f=\pi_{1}\circ\iota_{1}=\pi_{2}\circ\iota_{2}$ be two different factorizations, being ${\mathbb{P}}^{n_{i}}$, the target of $\iota_{i}$, $i=1,2$. Since equation (7.8) is independent of the choice of auxiliary hermitian structures, by [17, Lem. 5.12], we may assume that $\overline{f}=\overline{\pi}_{1}\circ\overline{\iota}_{1}=\overline{\pi}_{2}\circ\overline{\iota}_{2}$. We consider the commutative diagram with cartesian square $\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{1}}$$\scriptstyle{\operatorname{id}_{X}}$$\textstyle{X\underset{Y}{\times}{\mathbb{P}}_{Y}^{n_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k_{1}}$$\scriptstyle{q_{1}}$$\textstyle{{\mathbb{P}}_{Y}^{n_{1}}\underset{Y}{\times}{\mathbb{P}}_{Y}^{n_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{1}}$$\scriptstyle{f}$$\textstyle{{\mathbb{P}}_{Y}^{n_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\textstyle{Y}$ where $j_{1}(x)=(x,\iota_{2}(x))$, $p_{1}$ is the first projection and $q_{1}$ and $k_{1}$ are defined by the cartesian square. The hermitian structure of $\overline{\pi}_{2}$ induces a hermitian structure on $p_{1}$ that, in turn, induces a hermitian structure on $q_{1}$. The hermitian structure of $\iota_{1}$ induces a hermitian structure on $k_{1}$ and the hermitian structure of $\iota_{2}$ induces one on $j_{1}$. We will denote the corresponding morphisms of $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$ by $\overline{p}_{1}$, $\overline{q}_{1}$, $\overline{k}_{1}$ and $\overline{j}_{1}$. We consider also the analogous diagram obtained swapping 1 and 2. Finally, we write $\overline{p}=\overline{\pi}_{1}\circ\overline{p}_{1}=\overline{\pi}_{2}\circ\overline{p}_{2}$ and $\overline{j}=\overline{k}_{1}\circ\overline{j}_{1}=\overline{k}_{2}\circ\overline{j}_{2}$. Then we have $\displaystyle T(\overline{\pi}_{1})+(\overline{\pi}_{1})_{\flat}(T(\overline{\iota}_{1}))$ $\displaystyle=T(\overline{\pi}_{1})+(\overline{\pi}_{1})_{\flat}(T(\overline{\iota}_{1}))+\overline{f}_{\flat}\left(T(\overline{q}_{1})+(\overline{q}_{1})_{\flat}(T(\overline{j}_{1}))\right)$ $\displaystyle=T(\overline{\pi}_{1})+(\overline{\pi}_{1})_{\flat}\big{(}T(\overline{\iota}_{1})+(\overline{\iota}_{1})_{\flat}(T(\overline{q}_{1}))\big{)}+\overline{p}_{\flat}(\overline{k}_{1})_{\flat}(T(\overline{j}_{1}))$ $\displaystyle=T(\overline{\pi}_{1})+(\overline{\pi}_{1})_{\flat}\big{(}T(\overline{p}_{1})+(\overline{p}_{1})_{\flat}(T(\overline{k}_{1}))\big{)}+\overline{p}_{\flat}(\overline{k}_{1})_{\flat}(T(\overline{j}_{1}))$ $\displaystyle=T(\overline{p})+\overline{p}_{\flat}(T(\overline{j})).$ Analogously, we obtain $T(\overline{\pi}_{2})+(\overline{\pi}_{2})_{\flat}(T(\overline{\iota}_{2}))=T(\overline{p})+\overline{p}_{\flat}(T(\overline{j})).$ Hence $T_{S}$ is well defined for all relative metrized complexes. It remains to prove that it satisfies the properties of a theory of analytic torsion classes. The properties (i) to (iv) are clear. We thus focus on property (v). Let $\overline{f}\colon X\to Y$ and $\overline{g}\colon Y\to Z$ be morphisms in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. We choose factorizations of $\overline{g}\circ\overline{f}$ and $\overline{g}$: where the hermitian structures on $\overline{p}$ and $\overline{r}$ come from fixed hermitan structures on the tangent bundles $T_{{\mathbb{P}}^{m}_{{\mathbb{C}}}}$ and $T_{{\mathbb{P}}^{n}_{{\mathbb{C}}}}$, and the hermitian structures $\overline{i}$ and $\overline{\ell}$ are obtained by using [17, Lem. 5.12]. We define $\varphi:X\to{\mathbb{P}}^{m}_{{\mathbb{C}}}$ to be the morphism obtained from $i$ by composing with the projection to ${\mathbb{P}}^{m}_{{\mathbb{C}}}$. Then we see that the morphism $j:=(\varphi,f)\colon X\to{\mathbb{P}}^{m}_{Y}$ is a closed immersion. Indeed, it is enough to realize that the composition $\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\varphi,f)}$$\textstyle{{\mathbb{P}}^{m}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\operatorname{id},g)}$$\textstyle{{\mathbb{P}}^{m}_{Z}}$ agrees with the closed immersion $i$ and that $G:=(\operatorname{id},g)$ is separated (since proper). We can thus decompose $\overline{f}$ as $\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{j}}$$\scriptstyle{\overline{f}}$$\textstyle{{\mathbb{P}}^{m}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{q}}$$\textstyle{Y.}$ Again, in this factorization the hermitian structure $\overline{q}$ comes from the previously fixed hermitian structure on $T_{{\mathbb{P}}^{m}_{{\mathbb{C}}}}$ and the hermitian structure $\overline{j}$ is obtained by using [17, Lem. 5.12]. Because $\overline{g}\circ\overline{f}=\overline{p}\circ\overline{i}$ and by the very construction of $T$ for arbitrary projective morphisms (7.8), we have $T(\overline{g}\circ\overline{f})=T(\overline{p})+\overline{p}_{\flat}(T(\overline{i})).$ (7.9) We proceed to work on $T(\overline{i})$. For this we write the commutative diagram $\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\scriptstyle{i}$$\textstyle{{\mathbb{P}}^{m}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\scriptstyle{G}$$\textstyle{{\mathbb{P}}^{m}_{{\mathbb{P}}^{n}_{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{P}}^{n}_{{\mathbb{P}}^{m}_{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{{\mathbb{P}}^{m}_{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$$\textstyle{{\mathbb{P}}^{m}_{Z}.}$ We recall that $G=(\operatorname{id},g)$ and $k=(\operatorname{id},\ell)$. Below, $G$, $k$ and $\pi$ will be endowed with the obvious hermitian structures. With these choices, we observe that $\overline{i}=\overline{G}\circ\overline{j}$ and $\overline{G}=\overline{\pi}\circ\overline{k}$. Taking also into account the construction of $T$ and the fact that $T=T_{S}$ is transitive for compositions of closed immersions, we find $T(\overline{i})=T(\overline{\pi}\circ\overline{k}\circ\overline{j})=T(\overline{\pi})+\overline{\pi}_{\flat}(T(\overline{k}))+\overline{G}_{\flat}(T(\overline{j}))=T(\overline{G})+\overline{G}_{\flat}(T(\overline{j})).$ (7.10) Therefore, from equations (7.9), (7.10) and the identity $\overline{p}_{\flat}\overline{G}_{\flat}=\overline{g}_{\flat}\overline{q}_{\flat}$ we derive $T(\overline{g}\circ\overline{f})=T(\overline{p})+\overline{p}_{\flat}(T(\overline{G}))+\overline{g}_{\flat}\overline{q}_{\flat}(T(\overline{j})).$ (7.11) We claim that $T(\overline{p})+\overline{p}_{\flat}(T(\overline{G}))=T(\overline{g})+\overline{g}_{\flat}(T(\overline{q})).$ (7.12) Assuming this for a while, we combine (7.11) and (7.12) into $T(\overline{g}\circ\overline{f})=T(\overline{g})+\overline{g}_{\flat}(T(\overline{q})+\overline{q}_{\flat}(T(\overline{j})))\\\ =T(\overline{g})+\overline{g}_{\flat}(T(\overline{f})).$ (7.13) Hence we are lead to prove (7.12). For this we construct the commutative diagram with cartesian squares $\textstyle{{\mathbb{P}}^{m}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\ell}}$$\scriptstyle{\overline{q}}$$\textstyle{{\mathbb{P}}^{m}_{Z}\times_{Z}{\mathbb{P}}^{n}_{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip 14.22636pt\tilde{r}}$$\scriptstyle{\tilde{p}}$$\textstyle{{\mathbb{P}}^{m}_{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{p}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\ell}}$$\textstyle{{\mathbb{P}}^{n}_{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{r}}$$\textstyle{Z.}$ Observe that $\overline{G}=\tilde{r}\circ\tilde{\ell}$. Recall now Proposition 5.41 and Proposition 6.15. We then have the chain of equalities $T(\overline{p})+\overline{p}_{\flat}(T(\overline{G}))=T(\overline{p})+\overline{p}_{\flat}(T(\tilde{r})+\tilde{r}_{\flat}(T(\tilde{\ell})))=T(\overline{r})+\overline{r}_{\flat}(T(\tilde{p})+\tilde{p}_{\flat}(T(\tilde{\ell}))\\\ =T(\overline{r})+\overline{r}_{\flat}(T(\overline{\ell})+\overline{\ell}_{\flat}(T(\overline{q})))=T(\overline{g})+\overline{g}_{\flat}(T(\overline{q})).$ This proves the claim. The last assertion of the statement of the theorem follows from the uniqueness. ∎ ###### Theorem 7.14. 1. (i) Let $T$ be a theory of generalized analytic torsion classes. Then there is a unique real additive genus $S$ such that, for any relative metrized complex $\overline{\xi}:=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$, we have $T(\overline{\xi})-T^{h}(\overline{\xi})=-f_{\ast}[\operatorname{ch}(\mathcal{F})\bullet\operatorname{Td}(T_{f})\bullet S(T_{f})\bullet\mathbf{1}_{1}].$ (7.15) 2. (ii) Conversely, any real additive genus $S$ defines, by means of equation (7.15), a unique theory of generalized analytic torsion classes $T_{S}$. ###### Proof. We prove the first item, the second being immediate. Let $S$ be the real additive genus corresponding to $T$, provided by Theorem 7.7. Then (7.15) holds for embedded metrized complexes. Because $T$ and $T^{h}$ are both transitive, it suffices to show that (7.15) holds whenever $f\colon{\mathbb{P}}^{n}_{X}\to X$ is a trivial projective bundle. Observe $T$ and $T^{h}$ satisfy the same anomaly formulas. Then, since the sheaves $\mathcal{G}(k)$, $k=-n,\dots,0$ form a generating class for $\operatorname{\mathbf{D}^{b}}({\mathbb{P}}^{n}_{X})$, and by the projection formula for $T$ and $T^{h}$, we easily reduce to the case $\overline{\xi}=\overline{\xi}(k)$. Let $t_{n,k}$, $t_{n,k}^{h}$ be the characteristic numbers of $T$, $T^{h}$ respectively. We have to establish the equality $t_{n,-i}-t^{h}_{n,-i}=-\pi_{\ast}(\operatorname{ch}(\overline{{\mathcal{O}}}(-i))\operatorname{Td}(\overline{\pi})S(T_{\overline{\pi}})),\quad i=-n,\dots,0.$ (7.16) This is an equation of real numbers. By functoriality, this equation is equivalent to the analogous equation in $\oplus_{p}H^{2p-1}_{\mathcal{D}}({\mathbb{P}}^{n}_{{\mathbb{C}}},{\mathbb{R}}(p))$, for the second projection $p_{2}:{\mathbb{P}}^{n}_{{\mathbb{C}}}\times{\mathbb{P}}^{n}_{{\mathbb{C}}}\to{\mathbb{P}}^{n}_{{\mathbb{C}}}$ instead of $\pi$. Because the classes $\operatorname{ch}(\Lambda^{i}Q^{\vee})$ constitute a basis for $\oplus_{p}H^{2p-1}_{\mathcal{D}}({\mathbb{P}}^{n}_{{\mathbb{C}}},{\mathbb{R}}(p))$, (7.16) is equivalent to the equation in cohomology $\sum_{i}(-1)^{i}(t_{n,-i}-t^{h}_{n,-i})\operatorname{ch}(\Lambda^{i}\overline{Q}^{\vee})=\\\ -p_{2\ast}(\sum_{i}(-1)^{i}\operatorname{ch}(p_{1}^{\ast}\overline{{\mathcal{O}}}(-i)\otimes\Lambda^{i}p_{2}^{\ast}\overline{Q}^{\vee})\operatorname{Td}(\overline{p}_{2})S(T_{\overline{p}_{2}})\bullet\mathbf{1}_{1}).$ (7.17) Recalling the exact sequence (6.1), minus the right hand side of (7.17) becomes $p_{2\ast}(\operatorname{ch}(\overline{\Delta_{\ast}{\mathcal{O}}_{{\mathbb{P}}^{n}}})\operatorname{Td}(\overline{p}_{2})S(T_{\overline{p}_{2}})\bullet\mathbf{1}_{1})=\\\ p_{2\ast}(\Delta_{\ast}(\operatorname{ch}(\overline{{\mathcal{O}}}_{{\mathbb{P}}^{n}})\operatorname{Td}(\overline{\Delta}))\operatorname{Td}(\overline{p}_{2})S(T_{\overline{p}_{2}})\bullet\mathbf{1}_{1})=S(T_{{\mathbb{P}}^{n}})\bullet\mathbf{1}_{1}.$ On the other hand, using the compatibility condition (Definition 6.2), the left hand side of (7.17) can be equivalently written as $T(\overline{p}_{2},\overline{\Delta_{\ast}{\mathcal{O}}_{{\mathbb{P}}^{n}}},\overline{{\mathcal{O}}_{{\mathbb{P}}^{n}}})-T^{h}(\overline{p}_{2},\overline{\Delta_{\ast}{\mathcal{O}}_{{\mathbb{P}}^{n}}},\overline{{\mathcal{O}}_{{\mathbb{P}}^{n}}})=\\\ -p_{2\flat}(T(\Delta,\overline{{\mathcal{O}}}_{{\mathbb{P}}^{n}},\overline{\Delta_{\ast}{\mathcal{O}}_{{\mathbb{P}}^{n}}})-T^{h}(\Delta,\overline{{\mathcal{O}}}_{{\mathbb{P}}^{n}},\overline{\Delta_{\ast}{\mathcal{O}}_{{\mathbb{P}}^{n}}})).$ (7.18) The genus $S$ is additive, so in Deligne cohomology we have the relation $S(T_{\overline{\Delta}})=S(T_{{\mathbb{P}}^{n}})-\Delta^{\ast}S(T_{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}})=S(T_{{\mathbb{P}}^{n}})-\Delta^{\ast}p_{1}^{\ast}S(T_{{\mathbb{P}}^{n}})-\Delta^{\ast}p_{2}^{\ast}S(T_{{\mathbb{P}}^{n}})=-S(T_{{\mathbb{P}}^{n}}).$ Hence, since the statement is known for closed immersions, the right hand side of (7.18) becomes $p_{2\ast}(\Delta_{\ast}(\operatorname{ch}(\overline{{\mathcal{O}}_{{\mathbb{P}}^{n}}})\operatorname{Td}(T_{\overline{\Delta}})S(T_{\overline{\Delta}})\bullet\mathbf{1}_{1})\operatorname{Td}(\overline{p}_{2}))=-S(T_{{\mathbb{P}}^{n}})\bullet\mathbf{1}_{1}.$ This concludes the proof. ∎ ## 8 Higher analytic torsion forms of Bismut and Köhler We now explain the relationship between the theory of analytic torsion forms of Bismut-Köhler [13] and the theory of generalized analytic torsion classes developed so far. Let $\pi\colon X\to Y$ be a smooth projective morphism (a projective submersion) of smooth complex varieties. Let $\omega$ be a closed $(1,1)$-form on $X$ that induces a Kähler metric on the fibers of $\pi$. Then $(\pi,\omega)$ is called a Kähler fibration. The form $\omega$ defines a hermitian structure on $T_{\pi}$, and we will abusively write $\overline{\pi}=(\pi,\omega)$ for the corresponding morphism in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. Let $\overline{F}$ be a hermitian vector bundle on $X$ such that for every $i\geq 0$, $R^{i}\pi_{\ast}F$ is locally free. We consider on $R^{i}\pi_{\ast}F$ the $L^{2}$ metric obtained using Hodge theory on the fibers of $\pi$. Using [17, Def. 3.47] we obtain a hermitian structure on $\pi_{\ast}F$, denoted by $\overline{\pi_{\ast}F}_{L^{2}}$. Then $\overline{\xi}=(\overline{\pi},\overline{F},\overline{\pi_{\ast}F}_{L^{2}})$ is a relative metrized complex. The relative metrized complexes that arise in this way will be said to be _Kähler_. In the paper [13], Bismut and Köhler associate to every Kähler relative metrized complex $\overline{\xi}$ a differential form, that we temporarily denote by $\tau(\overline{\xi})$. Since in [13] the authors use real valued characteristic classes, while we use characteristic classes in the Deligne complex, we have to change the normalization of this form. To this end, if $\tau(\overline{\xi})^{(p-1,p-1)}$ is the component of degree $(p-1,p-1)$ of $\tau(\overline{\xi})$, then we put $T^{BK}(\overline{\xi})^{(2p-1,p)}=\frac{1}{2}(2\pi i)^{p-1}[\tau(\overline{\xi})^{(p-1,p-1)}]\in\widetilde{\mathcal{D}}_{D}^{2p-1}(Y,\emptyset,p).$ We recall that $[\cdot]$ converts differential forms into currents according with the conventions in [19, §1] (compare with equation (3.20)). We define $T^{BK}(\overline{\xi})=\sum_{p\geq 1}T^{BK}(\overline{\xi})^{(2p-1,p)}.$ The first main result of [13] is that this class satisfies the differential equation $d_{\mathcal{D}}T^{BK}(\overline{\xi})=\operatorname{ch}(\overline{\pi_{\ast}F}_{L^{2}})-\overline{\pi}_{\flat}[\operatorname{ch}(\overline{F})].$ Thus, $T^{BK}(\overline{\xi})$ is an example of analytic torsion class. Let now $\omega^{\prime}$ be another closed $(1,1)$-form on $X$ that induces a Kähler metric on the fibers of $\pi$. We denote $\overline{\pi}^{\prime}=(\pi,\omega^{\prime})$. Let $\overline{F}^{\prime}$ be the vector bundle $F$ with another choice of metric and define $\overline{\pi_{\ast}F}^{\prime}_{L^{2}}$ to be the object $\pi_{\ast}F$ with the $L^{2}$ metric induced by $\omega^{\prime}$ and $\overline{F}^{\prime}$. We write $\overline{\xi}^{\prime}$ for the Kähler relative metrized complex $(\overline{\pi}^{\prime},\overline{F}^{\prime},\overline{\pi_{\ast}F}^{\prime}_{L^{2}})$. The second main result of [13] is the following anomaly formula. ###### Theorem 8.1 ([13] Theorem 3.10). The following formula holds: $T^{BK}(\overline{\xi}^{\prime})-T^{BK}(\overline{\xi})=\operatorname{\widetilde{ch}}(\overline{\pi_{\ast}F}_{L^{2}},\overline{\pi_{\ast}F}^{\prime}_{L^{2}})+\overline{\pi}^{\prime}_{\flat}\left[\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}}_{m}(\overline{\pi}^{\prime},\overline{\pi})-\operatorname{\widetilde{ch}}(\overline{F},\overline{F}^{\prime})\right].$ In the book [3], Bismut studies the compatibility of higher analytic torsion forms with complex immersions. Before stating his result we have to recall the definition of the $R$-genus of Gillet and Soulé [29]. It is the additive genus attached to the power series $R(x)=\sum_{\begin{subarray}{c}m\text{ odd}\\\ m\geq 1\end{subarray}}\left(2\zeta^{\prime}(-m)+\left(1+\frac{1}{2}+\dots+\frac{1}{m}\right)\zeta(-m)\right)\frac{x^{m}}{m!}.$ (8.2) Let $T_{-R/2}$ be the theory of analytic torsion classes for closed immersions associated to $\frac{-1}{2}R$. ###### Remark 8.3. The fact that we obtain the additive genus $-R/2$ instead of $R$ is due to two facts. The signs comes from the minus sign in equation (3.12), while the factor $1/2$ comes from the difference of the normalization of Green forms used in this paper and the one used in [27]. Note however that the arithmetic intersection numbers computed using both normalizations agree, because the definition of arithmetic degree in [27, §3.4.3] has a factor $1/2$ while the definition of arithmetic degree in [18, (6.24)] does not. Consider a commutative diagram of smooth complex varieties $\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\iota}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Z}$ where $f$ and $g$ are projective submersions and $\iota$ is a closed immersion. Let $\overline{F}$ be a hermitian vector bundle on $X$ such that the sheaves $R^{i}f_{\ast}F$ are locally free and let $0\to\overline{E}_{n}\to\dots\to\overline{E}_{0}\to\iota_{\ast}F\to 0$ be a resolution of $\iota_{\ast}F$ by hermitian vector bundles. We assume that for all $i,j$, $R^{i}g_{\ast}E_{j}$ is locally free. We will denote by $E$ the complex $E_{n}\to\dots\to E_{0}$. Let $\omega^{X}$ and $\omega^{Y}$ be closed $(1,1)$ forms that define a structure of Kähler fibration on $f$ and $g$ respectively. As before we write $\overline{f}=(f,\omega^{X})$ and $\overline{g}=(g,\omega^{Y})$. The exact sequence $0\longrightarrow T_{f}\longrightarrow f^{\ast}T_{g}\longrightarrow N_{X/Y}\longrightarrow 0$ induces a hermitian structure on $N_{X/Y}$. We will denote $\overline{\iota}$ the inclusion $\iota$ with this hermitian structure. Finally we denote by $\overline{f_{\ast}F}_{E}$ the hermitian structure on $f_{\ast}F$ induced by the hermitian structures $\overline{g_{\ast}E_{j}}_{L^{2}}$, $j=0,\dots,n$. Then, adapted to our language, the main result of [3] can be stated as follows. ###### Theorem 8.4 ([3] Theorems 0.1 and 0.2). The following equation holds in the group $\bigoplus_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(Z,\emptyset,p)$: $T^{BK}(\overline{f},\overline{F},\overline{f_{\ast}F}_{L_{2}})=\sum_{j=0}^{n}(-1)^{j}T^{BK}(\overline{g},\overline{E}_{j},\overline{f_{\ast}E_{j}}_{L_{2}})\\\ +\overline{g}_{\flat}(T_{-R/2}(\overline{\iota},\overline{F},\overline{E}))+\operatorname{\widetilde{ch}}(\overline{f_{\ast}F}_{E},\overline{f_{\ast}F}_{L^{2}}).$ We can particularize the previous result to the case when $F=0$. Then $E$ and $g_{\ast}E$ are acyclic objects. The hermitian structures of $\overline{E}_{j}$ and $\overline{g_{\ast}E_{j}}_{L^{2}}$ induce hermitian structures on them. We denote these hermitian structures as $\overline{E}$ and $\overline{g_{\ast}E}_{L^{2}}$. ###### Corollary 8.5. Let $\overline{E}$ be a bounded acyclic complex of hermitian vector bundles on $Y$ such that the direct images $R^{i}g_{\ast}E_{j}$ are locally free on $Z$. Then $\sum_{j=0}^{n}(-1)^{j}T^{BK}(\overline{g},\overline{E_{j}},\overline{g_{\ast}E_{j}}_{L^{2}})=\operatorname{\widetilde{ch}}(\overline{g_{\ast}E}_{L^{2}})-\overline{g}_{\flat}(\operatorname{\widetilde{ch}}(\overline{E}))$ in $\bigoplus_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(Z,\emptyset,p)$. We will also need a particular case of functoriality and projection formula for the higher analytic torsion forms of Bismut-Köhler proved by Rössler [43]. The relative metrized complexes $\overline{\xi}_{n}(k)$ of Notation 5.2 are Kähler. Therefore we can apply the construction of Bismut-Köhler to them. We denote $t^{BK}_{n,k}=T^{BK}(\overline{\xi}_{n}(k)).$ (8.6) By Corollary 8.5, the numbers $t^{BK}_{n,k}$ satisfy the relation (5.19). Hence they are determined by the main characteristic numbers $t^{BK}_{n,k}$ for $-n\leq k\leq 0$. ###### Theorem 8.7 ([43] Lemma 7.15). Let $\pi\colon{\mathbb{P}}^{n}_{X}\to X$ be a trivial projective bundle. Let $\overline{G}$ be a hermitian vector bundle on $X$. Then $T^{BK}(\overline{\xi}_{n}(k)\otimes\overline{G})=t^{BK}_{n,k}\bullet\operatorname{ch}(\overline{G}).$ ###### Proof. In [43] this result is proved for $k\gg 0$. Using Corollary 8.5 and the Koszul resolution (5.18) one can extend the result to all $k\in{\mathbb{Z}}$. ∎ We have all the ingredients we need to prove the main result of this section. ###### Theorem 8.8. Let $T_{-R/2}$ be the theory of generalized analytic torsion classes associated to the additive genus $\frac{-1}{2}R$. Then, for every Kähler relative metrized complex $\overline{\xi}$, we have $T^{BK}(\overline{\xi})=T_{-R/2}(\overline{\xi}).$ In particular $T_{-R/2}$ extends the construction of Bismut-Köhler to arbitrary projective morphisms of smooth complex varieties and arbitrary smooth metrics. ###### Proof. Let $\mathfrak{t}^{BK}=\\{t^{BK}_{n,k}\mid n\geq 0,-n\leq k\leq 0\\}$ and let $T_{\mathfrak{t}^{BK}}$ be the theory of analytic torsion classes for projective spaces associated to it. Let $\pi\colon{\mathbb{P}}^{n}_{X}\to X$ be a relative projective space and let $\overline{\xi}=(\overline{\pi},\overline{E},\overline{\pi_{\ast}E}_{L^{2}})$ be a Kähler relative metrized complex. By choosing $d\gg 0$ we may assume that all the coherent sheaves of the resolution $\gamma_{d}(F)$ of Corollary 4.3 are locally free. Using theorems 8.7 and 8.1, Proposition 5.10 and corollaries 5.15 and 8.5 we obtain that $T^{BK}(\overline{\xi})=T_{\mathfrak{t}^{BK}}(\overline{\xi}).$ By Theorem 8.4, the theories $T_{\mathfrak{t}^{BK}}$ and $T_{-R/2}$ are compatible in the sense of Definition 6.2. Therefore, $T^{BK}=T_{-R/2}$ when restricted to projective spaces. Finally, by factoring a smooth projective morphism as a closed immersion followed by the projection of a relative projective space, Theorem 8.4 implies that $T^{BK}=T_{-R/2}$ for all smooth projective morphisms. ∎ ###### Remark 8.9. 1. (i) The construction of Bismut-Köhler applies to a wider class of varieties and morphisms: complex analytic manifolds and proper Kähler submersions. However for the comparison we have to restrict to smooth algebraic varieties and smooth projective morphisms. 2. (ii) The results of Bismut and his coworkers are more precise. Here the class $T^{BK}(\overline{\xi})$ is well defined up to the image of $\operatorname{d}_{\mathcal{D}}$. In contrast, the higher analytic torsion form of Bismut and Köhler is a well defined differential form, local on the base and whose class modulo $\operatorname{d}_{\mathcal{D}}$ agrees with $T^{BK}(\overline{\xi})$. As a consequence of Theorem 8.8, we obtain the following results that, although they should follow from the definition of higher analytic torsion classes, we have not been able to find them explicitly in the literature. ###### Corollary 8.10. Let $f\colon X\to Y$ be a smooth projective morphism of smooth complex varieties, and let $\overline{\xi}=(\overline{f},\overline{E},\overline{f_{\ast}E}_{L^{2}})$ be a Kähler relative metrized complex. 1. (i) Let $g\colon Y^{\prime}\to Y$ be a morphism of smooth complex varieties. Then $T^{BK}(g^{\ast}\overline{\xi})=g^{\ast}T^{BK}(\overline{\xi}).$ 2. (ii) Let $\overline{G}$ be a hermitian vector bundle on $Y$. Then $T^{BK}(\overline{\xi}\otimes\overline{G})=T^{BK}(\overline{\xi})\bullet\operatorname{ch}(\overline{G}).$ The last consequence we want to discuss generalizes results already proved by Berthomieu-Bismut [1, Thm 3.1] and Ma [35, Thm. 0.1], [36, Thm. 0.1]. However we note that while we stay within the algebraic category and work with projective morphisms, these authors deal with proper Kähler holomorphic submersions of complex manifolds. Let $\overline{g}\colon X\to Y$ and $\overline{h}\colon Y\to Z$ be morphisms in the category $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$, such that the composition $\overline{f}=\overline{h}\circ\overline{g}$ is a smooth morphism. We choose a structure of Kähler fibration on $f$, that we denote $\overline{f}^{\prime}$. Let $\overline{E}$ be a hermitian vector bundle on $X$ and assume that the higher direct images $R^{i}f_{\ast}E$ are locally free. Then we may consider the analytic torsion $T^{BK}(\overline{f}^{\prime})$ attached to the Kähler relative metrized complex $(\overline{f}^{\prime},\overline{E},\overline{f_{\ast}E}_{L^{2}})$. Also, we choose an auxiliary hermitian structure on $g_{\ast}E$. We can consider the torsion classes $T_{R/2}(\overline{g})$ and $T_{R/2}(\overline{h})$ of the relative metrized complexes $(\overline{g},\overline{E},\overline{g_{\ast}E})$ and $(\overline{h},\overline{g_{\ast}E},\overline{f_{\ast}E}_{L^{2}})$. We make the following additional assumption in some particular situations: () The morphisms $g$ and $h$ are Kähler fibrations, the higher direct images $R^{i}g_{\ast}E$ and $R^{j}h_{\ast}R^{i}g_{\ast}E$ are locally free and the auxiliary hermitian structure on $g_{\ast}E$ is the $L^{2}$ hermitian structure. When the hypothesis () is satisfied we denote by $\overline{h_{\ast}g_{\ast}E}_{L^{2}}$ the $L^{2}$ hermitian structure attached to the Kähler structure on $\overline{h}$ and the $L^{2}$ metric on $\overline{g_{\ast}E}_{L^{2}}$. Observe that this last structure may differ from the $L^{2}$ structure on $\overline{f_{\ast}E}_{L^{2}}$. In this situation we can consider the torsion classes $T^{BK}(\overline{g})$ and $T^{BK}(\overline{h}^{\prime})$ attached to $(\overline{g},\overline{E},\overline{g_{\ast}E}_{L^{2}})$ and $(\overline{h},\overline{g_{\ast}E}_{L^{2}},\overline{h_{\ast}g_{\ast}E}_{L^{2}})$. By Proposition 5.10, we have the relation $T^{BK}(\overline{h}^{\prime})=T_{R/2}(\overline{h})-\widetilde{\operatorname{ch}}(\overline{h_{\ast}g_{\ast}E}_{L^{2}},\overline{f_{\ast}E}_{L^{2}}).$ The properties of the generalized analytic torsion classes imply immediately: ###### Corollary 8.11. Under these assumptions, we have the equality $T^{BK}(\overline{f}^{\prime})=T_{-R/2}(\overline{h})+\overline{h}_{\flat}(T_{-R/2}(\overline{g}))+\overline{f}^{\prime}_{\flat}(\operatorname{ch}(\overline{E})\bullet\widetilde{\operatorname{Td}}_{m}(\overline{f}^{\prime},\overline{f})).$ If in addition the hypothesis () is satisfied, then we have $T^{BK}(\overline{f}^{\prime})=T^{BK}(\overline{h}^{\prime})+\overline{h}_{\flat}(T^{BK}(\overline{g}))+\overline{f}^{\prime}_{\flat}(\operatorname{ch}(\overline{E})\bullet\widetilde{\operatorname{Td}}_{m}(\overline{f}^{\prime},\overline{f}))+\widetilde{\operatorname{ch}}(\overline{h_{\ast}g_{\ast}E}_{L^{2}},\overline{f_{\ast}E}_{L^{2}}).$ Since $T_{-R/2}$ extends the theory of analytic torsion classes $T^{BK}$, we will denote $T_{-R/2}$ by $T^{BK}$ for arbitrary relative metrized complexes. ## 9 Grothendieck duality and analytic torsion We will study now the compatibility of the analytic torsion with Grothendieck duality. ###### Definition 9.1. Let $\overline{\mathcal{F}}=(\mathcal{F},\overline{E}\dashrightarrow\mathcal{F})$ be an object of $\operatorname{\overline{\mathbf{D}}^{b}}(X)$. Then the _rank_ of $\overline{\mathcal{F}}$ is $\operatorname{rk}(\overline{\mathcal{F}})=\sum_{i}(-1)^{i}\dim(E_{i}).$ This is just the Euler characteristic of the complex. _The determinant_ of $\overline{\mathcal{F}}$ is the complex $\det(\overline{\mathcal{F}})=\bigotimes_{i}\left(\Lambda^{\dim E^{i}}\overline{E}^{i}\right)^{(-1)^{i}}[-\operatorname{rk}(\overline{\mathcal{F}})].$ It consists of a single line bundle concentrated in degree $\operatorname{rk}(\overline{\mathcal{F}})$. ###### Definition 9.2. Let $\overline{f}\colon X\to Y$ be a morphism in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$ of relative dimension $e$. The _metrized dualizing complex_ , is the complex given by $\boldsymbol{\omega}_{\overline{f}}=(\det T_{\overline{f}})^{\vee}.$ This complex is concentrated in degree $-e$. The underlying object of $\operatorname{\mathbf{D}^{b}}(X)$ will be denoted by $\boldsymbol{\omega}_{f}$. If we are interested in the dualizing sheaf as a sheaf and not as an element of $\operatorname{\mathbf{D}^{b}}(X)$ we will denote it by $\omega_{f}$ or $\omega_{X/Y}$. Finally, if $Y=\operatorname{Spec}{\mathbb{C}}$, we will denote $\boldsymbol{\omega}_{f}$ (respectively $\omega_{f}$) by $\boldsymbol{\omega}_{X}$ (respectively $\omega_{X}$). ###### Definition 9.3. Let $\mathcal{D}^{\ast}(\ast)$ be the Deligne complex associated to a Dolbeault complex. The _sign operator_ is $\sigma\colon\mathcal{D}^{\ast}(\ast)\longrightarrow\mathcal{D}^{\ast}(\ast),\quad\omega\in\mathcal{D}^{n}(p)\mapsto\sigma(\omega)=(-1)^{p}\omega.$ The sign operator satisfies the following compatibilities. ###### Proposition 9.4. 1. (i) Let $(\mathcal{D}^{\ast}(\ast),\operatorname{d}_{\mathcal{D}})$ be a Deligne algebra. Then the sign operator is a morphism of differential algebras. That is $\operatorname{d}_{\mathcal{D}}\circ\,\sigma=\sigma\circ\operatorname{d}_{\mathcal{D}},\quad\sigma(\omega\bullet\eta)=\sigma(\omega)\bullet\sigma(\eta).$ 2. (ii) Let $\overline{\mathcal{F}}$ be an object of $\operatorname{\overline{\mathbf{D}}^{b}}(X)$. Then the following equalities are satisfied $\displaystyle\sigma\operatorname{ch}(\overline{\mathcal{F}})$ $\displaystyle=\operatorname{ch}(\overline{\mathcal{F}}^{\vee}),$ (9.5) $\displaystyle\sigma\operatorname{ch}(\det(\overline{\mathcal{F}}))$ $\displaystyle=\operatorname{ch}(\det(\overline{\mathcal{F}})^{\vee})=\operatorname{ch}(\det(\overline{\mathcal{F}}))^{-1},$ (9.6) $\displaystyle\sigma\operatorname{Td}(\overline{\mathcal{F}})$ $\displaystyle=(-1)^{\operatorname{rk}(\overline{\mathcal{F}})}\operatorname{Td}(\overline{\mathcal{F}})\bullet\operatorname{ch}(\det(\overline{\mathcal{F}})^{\vee}.$ (9.7) ###### Proof. The first statement is clear because if $\omega\in\mathcal{D}^{n}(p)$ and $\eta\in\mathcal{D}^{m}(q)$ then $\operatorname{d}_{\mathcal{D}}\omega\in\mathcal{D}^{n+1}(p)$ and $\omega\bullet\eta\in\mathcal{D}^{n+m}(p+q)$. For the second statement, let $\overline{E}\dashrightarrow\mathcal{F}$ be the hermitian structure of $\mathcal{F}$. Write $\overline{E}^{+}=\bigoplus_{i\text{ even}}\overline{E}^{i},\qquad\overline{E}^{-}=\bigoplus_{i\text{ odd}}\overline{E}^{i}.$ Since this statement is local on $X$, we can chose trivializations of $\overline{E}^{+}$ and $\overline{E}^{-}$ over an open subset $U$. Let $H^{+}$ and $H^{-}$ be the matrices of the hermitian metrics on $\overline{E}^{+}$ and $\overline{E}^{-}$. The curvature matrices of $\overline{E}^{+}$ and $\overline{E}^{-}$, whose entries are elements of $\mathcal{D}^{2}(U,1)$, are $K^{\pm}=K^{\pm}(\overline{\mathcal{F}})=-\overline{\partial}(H^{\pm})^{-1}\partial H^{\pm}.$ The characteristic forms can be computed from the curvature matrix: $\displaystyle\operatorname{ch}(\overline{\mathcal{F}})$ $\displaystyle=\operatorname{tr}(\exp(K^{+}))-\operatorname{tr}(\exp(K^{-})),$ $\displaystyle\operatorname{ch}(\det(\overline{\mathcal{F}}))$ $\displaystyle=(-1)^{\operatorname{rk}(\overline{\mathcal{F}})}\det(\exp(K^{+}))\bullet\det(\exp(K^{-}))^{-1},$ $\displaystyle\operatorname{Td}(\overline{\mathcal{F}})$ $\displaystyle=\det\left(\frac{K^{+}}{1-\exp(-K^{+})}\right)\bullet\det\left(\frac{K^{-}}{1-\exp(-K^{-})}\right)^{-1}.$ The sign in the second equation comes from the fact that $\det(\overline{\mathcal{F}})$ is concentrated in degree $\operatorname{rk}(\overline{\mathcal{F}})$. Therefore, since $\sigma(K^{\pm})=-K^{\pm}=K^{\pm}(\overline{\mathcal{F}}^{\vee})$, we have $\displaystyle\sigma\operatorname{ch}(\overline{\mathcal{F}})$ $\displaystyle=\sigma\operatorname{tr}(\exp(K^{+}))-\sigma\operatorname{tr}(\exp(K^{-}))$ $\displaystyle=\operatorname{tr}(\exp(K^{+}(\overline{\mathcal{F}}^{\vee})))-\operatorname{tr}(\exp(K^{-}(\overline{\mathcal{F}}^{\vee})))=\operatorname{ch}(\overline{\mathcal{F}}^{\vee}),$ $\displaystyle\sigma\operatorname{ch}(\det(\overline{\mathcal{F}}))$ $\displaystyle=\det(\exp(-K^{+}))\bullet\det(\exp(-K^{-}))^{-1}=\operatorname{ch}(\det(\overline{\mathcal{F}}))^{-1},$ $\displaystyle\sigma\operatorname{Td}(\overline{\mathcal{F}})$ $\displaystyle=\det\left(\frac{-K^{+}}{1-\exp(K^{+})}\right)\bullet\det\left(\frac{-K^{-}}{1-\exp(K^{-})}\right)^{-1}$ $\displaystyle=\det\left(\frac{K^{+}}{1-\exp(-K^{+})}\right)\bullet\det(\exp(-K^{+}))$ $\displaystyle\phantom{AAA}\bullet\det\left(\frac{K^{-}}{1-\exp(-K^{-})}\right)^{-1}\bullet\det(\exp(-K^{-}))^{-1}$ $\displaystyle=\operatorname{Td}(\overline{\mathcal{F}})\bullet\operatorname{ch}(\det(\overline{\mathcal{F}}))^{-1}.$ ∎ ###### Corollary 9.8. Let $[\overline{E}]\in\operatorname{\mathbf{KA}}(X)$. Then $\widetilde{\operatorname{ch}}(\overline{E}^{\vee})=\sigma\widetilde{\operatorname{ch}}(\overline{E}).$ ###### Proof. Due to Proposition 9.4, the assignment sending $[\overline{E}]$ to $\sigma\widetilde{\operatorname{ch}}(\overline{E})$ satisfies the characterizing properties of $\widetilde{\operatorname{ch}}$. ∎ In the particular case of a projective morphism between smooth complex varieties or, more generally, smooth varieties over a field, Grothendieck duality takes a very simple form (see for instance [32, §3.4] and the references therein). If $\mathcal{F}$ is an object of $\operatorname{\mathbf{D}^{b}}(X)$ and $f\colon X\to Y$ is a projective morphism of smooth complex varieties, then there is a natural functorial isomorphism $f_{\ast}(\mathcal{F}^{\vee}\otimes\boldsymbol{\omega}_{f})\cong(f_{\ast}\mathcal{F})^{\vee}.$ (9.9) The compatibility between analytic torsion and Grothendieck duality is given by the following result. ###### Theorem Definition 9.10. Let $T$ be a theory of generalized analytic torsion classes. Then the assignment that, to a relative metrized complex $\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$, associates the class $T^{\vee}(\overline{\xi})=\sigma T(\overline{f},\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}},\overline{f_{\ast}\mathcal{F}}^{\vee})$ is a theory of generalized analytic torsion classes that we call the _theory dual to_ $T$. ###### Proof. We have to show that, if $T$ satisfies the conditions of Definition 7.1, then the same is true for $T^{\vee}$. We start with the differential equation. Let $e$ be the relative dimension of $f$. $\displaystyle\operatorname{d}_{\mathcal{D}}T^{\vee}(\overline{\xi})$ $\displaystyle=\operatorname{d}_{\mathcal{D}}\sigma T(\overline{f},\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}},\overline{f_{\ast}\mathcal{F}}^{\vee})$ $\displaystyle=\sigma\operatorname{d}_{\mathcal{D}}T(\overline{f},\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}},\overline{f_{\ast}\mathcal{F}}^{\vee})$ $\displaystyle=\sigma\operatorname{ch}(\overline{f_{\ast}\mathcal{F}}^{\vee})-\sigma f_{\ast}\left[\operatorname{ch}(\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}})\bullet\operatorname{Td}(\overline{f})\right]$ $\displaystyle=\operatorname{ch}(\overline{f_{\ast}\mathcal{F}})-(-1)^{e}f_{\ast}\left[\sigma\operatorname{ch}(\overline{\mathcal{F}}^{\vee})\bullet\sigma(\operatorname{ch}(\det(T_{\overline{f}})^{\vee})\bullet\operatorname{Td}(T_{\overline{f}}))\right]$ $\displaystyle=\operatorname{ch}(\overline{f_{\ast}\mathcal{F}})-f_{\ast}\left[\operatorname{ch}(\overline{\mathcal{F}})\bullet\operatorname{Td}(\overline{f})\right]$ The functoriality and the additivity are clear. We next check the projection formula. Let $\overline{\mathcal{G}}$ be an object of $\operatorname{\overline{\mathbf{D}}^{b}}(Y)$. Then $T^{\vee}(\overline{\xi}\otimes\overline{\mathcal{G}})=\sigma T(\overline{f},\overline{\mathcal{F}}^{\vee}\otimes f^{\ast}\overline{\mathcal{G}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}},\overline{f_{\ast}\mathcal{F}}^{\vee}\otimes\overline{\mathcal{G}}^{\vee})\\\ =\sigma\left(T(\overline{f},\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}},\overline{f_{\ast}\mathcal{F}}^{\vee})\bullet\operatorname{ch}(\overline{\mathcal{G}}^{\vee})\right)=T^{\vee}(\overline{\xi})\bullet\operatorname{ch}(\overline{\mathcal{G}}).$ Finally we check the transitivity. Let $\overline{g}\colon Y\to Z$ be another morphism in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. By the definition of $\overline{g}\circ\overline{f}$ we have $\boldsymbol{\omega}_{\overline{g}\circ\overline{f}}=f^{\ast}\boldsymbol{\omega}_{\overline{g}}\otimes\boldsymbol{\omega}_{\overline{f}}.$ Therefore, $f_{\ast}\left(\mathcal{F}^{\vee}\otimes\boldsymbol{\omega}_{g\circ f}\right)=f_{\ast}\left(\mathcal{F}^{\vee}\otimes f^{\ast}\boldsymbol{\omega}_{g}\otimes\boldsymbol{\omega}_{f})\right)\\\ =f_{\ast}\left(\mathcal{F}^{\vee}\otimes\boldsymbol{\omega}_{f})\right)\otimes\boldsymbol{\omega}_{g}=(f_{\ast}\mathcal{F})^{\vee}\otimes\boldsymbol{\omega}_{g}.$ On $f_{\ast}\left(\mathcal{F}^{\vee}\otimes\boldsymbol{\omega}_{g\circ f}\right)$ we put the hermitian structure of $\overline{f_{\ast}\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{g}}$. Then we have $\displaystyle T^{\vee}(\overline{g}\circ\overline{f})$ $\displaystyle=\sigma T(\overline{g}\circ\overline{f},\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{g}\circ\overline{f}},\overline{(g\circ f)_{\ast}\mathcal{F}}^{\vee})$ $\displaystyle=\sigma T(\overline{g},\overline{f_{\ast}\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{g}},\overline{(g\circ f)_{\ast}\mathcal{F}}^{\vee})$ $\displaystyle\phantom{AAA}+\sigma\overline{g}_{\flat}T(\overline{f},\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}}\otimes f^{\ast}\boldsymbol{\omega}_{\overline{g}},\overline{f_{\ast}\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{g}})$ $\displaystyle=T^{\vee}(\overline{g},\overline{f_{\ast}\mathcal{F}},\overline{(g\circ f)_{\ast}\mathcal{F}})$ $\displaystyle\phantom{AAA}+\sigma g_{\ast}(T(\overline{f},\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}},\overline{f_{\ast}\mathcal{F}}^{\vee})\bullet\operatorname{ch}(\boldsymbol{\omega}_{\overline{g}})\bullet\operatorname{Td}(\overline{g}))$ $\displaystyle=T^{\vee}(\overline{g})+\overline{g}_{\flat}T^{\vee}(\overline{f}).$ Therefore, $T^{\vee}$ satisfies also the transitivity property. Hence it is a generalized theory of analytic torsion classes. ∎ ###### Definition 9.11. A theory of generalized analytic torsion classes $T$ is called _self-dual_ when $T^{\vee}=T$. We want to characterize the self-dual theories of generalized analytic torsion classes. ###### Theorem 9.12. The homogeneous theory of generalized analytic torsion classes is self-dual. ###### Proof. By the uniqueness of the homogeneous theory, it is enough to prove that, if $T$ is homogeneous then $T^{\vee}$ is homogeneous. Let $X$ be a smooth complex variety and let $\overline{N}$ be a hermitian vector bundle of rank $r$ on $X$. Put $P=\mathbb{P}(N\oplus 1)$ and let $s\colon X\to P$ be the zero section and $\pi\colon P\to X$ the projection. Let $\overline{Q}$ be the tautological quotient bundle with the induced metric and $\overline{K}(s)$ the Koszul resolution associated to the section $s$. Since the normal bundle $N_{X/P}$ can be identified with $N$, on the map $s$ we can consider the hermitian structure given by the hermitian metric on $N$. Then $\det\overline{Q}$ is a complex concentrated in degree $r$. Moreover $s^{\ast}\det\overline{Q}=\det\overline{N}=\boldsymbol{\omega}_{\overline{s}}.$ The Koszul resolution satisfies the duality property $\overline{K}(s)^{\vee}=\overline{K}(s)\otimes\det\overline{Q}.$ The theory $T$ is homogeneous if and only if the class $T(\overline{s},\overline{\mathcal{O}}_{X},\overline{K}(s))\bullet\operatorname{Td}(\overline{Q})$ is homogeneous of bidegree $(2r-1,r)$ in the Deligne complex. Then $\displaystyle T^{\vee}(\overline{s},\overline{\mathcal{O}}_{X},\overline{K}(s))\bullet\operatorname{Td}(\overline{Q})$ $\displaystyle=\sigma T(\overline{s},\boldsymbol{\omega}_{\overline{s}},\overline{K}(s)^{\vee})\bullet\operatorname{Td}(\overline{Q})$ $\displaystyle=\sigma T(\overline{s},s^{\ast}\det\overline{Q},\overline{K}(s)\otimes\det\overline{Q})\bullet\operatorname{Td}(\overline{Q})$ $\displaystyle=\sigma(T(\overline{s},\overline{\mathcal{O}}_{X},\overline{K}(s))\bullet\operatorname{ch}(\det\overline{Q}))\bullet\operatorname{Td}(\overline{Q})$ $\displaystyle=\sigma T(\overline{s},\overline{\mathcal{O}}_{X},\overline{K}(s))\bullet\operatorname{ch}(\det\overline{Q}^{\vee})\bullet\operatorname{Td}(\overline{Q})$ $\displaystyle=(-1)^{r}\sigma(T(\overline{s},\overline{\mathcal{O}}_{X},\overline{K}(s))\bullet\operatorname{Td}(\overline{Q}))$ is homogeneous of bidegree $(2r-1,r)$ in the Deligne complex. ∎ ###### Proposition 9.13. Let $S(x)=\sum_{n=0}^{\infty}a_{n}x^{n}\in{\mathbb{R}}[[x]]$ be a power series in one variable with real coefficients. Denote by $S$ the corresponding real additive genus and by $T_{S}$ the associated theory of analytic torsion classes. Then the dual theory $T^{\vee}_{S}$ has corresponding real additive genus $S^{\sigma}(x):=-S(-x)$. ###### Proof. Let $\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$ be a relative metrized complex. If $e$ is the relative dimension of $f$, then we have $\sigma f_{\ast}=(-1)^{e}f_{\ast}\sigma$. Then the proposition readily follows from the definition of $T^{\vee}_{S}$, the self-duality of $T^{h}$ and Proposition 9.4. ∎ We can now characterize the self-dual theories of analytic torsion classes. ###### Corollary 9.14. The theory of analytic torsion classes $T_{S}$ attached to the real additive genus $S(x)=\sum_{n\geq 0}a_{n}x^{n}$ is self-dual if and only if $a_{n}=0$ for $n$ even. ###### Proof. By the proposition, $T_{S}^{\vee}=T_{S^{\sigma}}$, hence $T$ is self-dual if, and only if, $S^{\sigma}=S$. The corollary follows. ∎ In particular we recover the following fact, which is well known if we restrict to Kähler relative metrized complexes. ###### Corollary 9.15. The theory of analytic torsion classes of Bismut-Köhler $T^{BK}$ is self-dual. ###### Proof. We just remark that the even coefficients of the $R$-genus vanish (8.2). ∎ We now elaborate on an intimate relation between self-duality phenomena and the analytic torsion of de Rham complexes. Let $f\colon X\to Y$ be a smooth projective morphism of smooth algebraic varieties, of relative dimension $e$. Let $\overline{T}_{X/Y}$ denote the vertical tangent bundle, endowed with a hermitian metric. Write $\overline{f}$ for the corresponding morphism in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. On the locally free sheaves $\Omega_{X/Y}^{p}=\Lambda^{p}\Omega_{X/Y}$ we put the induced hermitian structures. The metrized de Rham complex is $0\to\overline{{\mathcal{O}}}_{X}\overset{0}{\to}\overline{\Omega}_{X/Y}\overset{0}{\to}\overline{\Omega}_{X/Y}^{2}\overset{0}{\to}\dots\overset{0}{\to}\overline{\Omega}_{X/Y}^{e}\to 0$ with 0 differentials. In fact, we are really considering the de Rham graded sheaf and converting it into a complex in a trivial way. We refer to the corresponding object of $\operatorname{\overline{\mathbf{D}}^{b}}(X)$ by $\overline{\Omega}_{X/Y}^{\bullet}$ ([17, Def. 3.37]). The individual terms $\overline{\Omega}_{X/Y}^{p}$ will be considered as complexes concentrated in degree $p$. We then obviously have: ###### Lemma 9.16. The objects $(\overline{\Omega}_{X/Y}^{\bullet})^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}}$ and $\overline{\Omega}_{X/Y}^{\bullet}[2e]$ are tightly isomorphic. For every $p$, $q$, the cohomology sheaf $R^{q}f_{\ast}\Omega_{X/Y}^{p}$ is locally free, because the Hodge numbers $h^{p,q}$ of the fibers of $f$ (which are projective, hence Kähler) are known to be locally constant. Every stalk of this sheaf is endowed with the usual $L^{2}$ metric of Hodge theory. This family of $L^{2}$ metrics on $R^{q}f_{\ast}\Omega_{X/Y}^{p}$ glue into a smooth metric. Because the Hodge star operators $\ast$ act by isometries, it is easily shown that Serre duality becomes an isometry for the $L^{2}$ structures: the isomorphism $(R^{q}f_{\ast}\Omega_{X/Y}^{p})^{\vee}\overset{\sim}{\longrightarrow}R^{e-q}f_{\ast}((\Omega_{X/Y}^{p})^{\vee}\otimes\boldsymbol{\omega}_{f})=R^{e-q}f_{\ast}\Omega_{X/Y}^{e-p}$ preserves the $L^{2}$ hermitian structures. For every $p$, let $\overline{f_{\ast}\Omega_{X/Y}^{p}}$ denote the object of $\operatorname{\overline{\mathbf{D}}^{b}}(Y)$ with the metric induced by the $L^{2}$ metrics on its cohomology pieces ([17, Def. 3.47]). Here $f_{\ast}$ stands for the derived direct image. By [17, Prop. 3.48], Grothendieck duality $(f_{\ast}\Omega_{X/Y}^{p})^{\vee}\overset{\sim}{\longrightarrow}f_{\ast}\Omega_{X/Y}^{e-p}[2e]$ is a tight isomorphism. Finally, let $[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}]$ be the object of $\operatorname{\overline{\mathbf{D}}^{b}}(Y)$ provided by [17, Def. 3.39]. The next lemma follows easily from the construction of [17, Def. 3.39]. ###### Lemma 9.17. Grothendieck duality defines a tight isomorphism in $\operatorname{\overline{\mathbf{D}}^{b}}(Y)$ $[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}]^{\vee}\cong[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}][2e].$ ###### Theorem 9.18. Let $T$ be a theory of analytic torsion classes. The following assertions are equivalent: 1. (i) the theory $T$ is self-dual; 2. (ii) for every $f$, $\overline{T}_{f}$, $\overline{\Omega}_{X/Y}^{\bullet}$ and $[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}]$ as above and for every odd integer $p\geq 1$, the part of bidegree $(2p-1,p)$ (in the Deligne complex) of $T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}])$ vanishes. ###### Proof. Assume first of all that $T$ is self-dual. We apply the definition of $T^{\vee}$, the self-duality assumption and lemmas 9.16 and 9.17. We find the equality $\begin{split}T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}])&=\sigma T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet}[2e],[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}][2e])\\\ &=(-1)^{2e}\sigma T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}])\\\ &=\sigma T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}]).\end{split}$ The sign operator $\sigma$ changes the sign of the components of bidegree $(2p-1,p)$ for odd $p$. Hence $T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}])^{(2p-1,p)}$ vanishes for $p\geq 1$ odd. For the converse implication, let $S(x)=\sum_{n\geq 0}a_{n}x^{n}$ be the real additive genus attached to $T$ via Theorem 7.14. By Corollary 9.14, we have to show that the coefficients $a_{n}$ with $n$ even vanish. Let us look at a smooth morphism $f\colon X\to Y$ of relative dimension $1$, with an arbitrary metric on $T_{f}$. Then, developing the power series of $\operatorname{ch}$ and $\operatorname{Td}$ and taking into account that $\Omega^{1}_{X/Y}=T_{f}^{\vee}=\omega_{X/Y}$, we compute $f_{\ast}[\operatorname{ch}(\Omega_{X/Y}^{\bullet})\operatorname{Td}(T_{f})S(T_{f})\bullet\mathbf{1}_{1}]=\sum_{n\geq 0}(-1)^{n+1}a_{n}f_{\ast}[c_{1}(\omega_{X/Y})^{n+1}\bullet\mathbf{1}_{1}].$ Therefore, for $p\geq 1$ odd, we have $(-1)^{p}a_{p-1}f_{\ast}[c_{1}(\omega_{X/Y})^{p}\bullet\mathbf{1}_{1}]=\\\ (T(\overline{f},\overline{\Omega_{X/Y}^{\bullet}},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}])-T^{h}(\overline{f},\overline{\Omega_{X/Y}^{\bullet}},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}]))^{(2p-1,p,p)}=0.$ (9.19) Hence it is enough that for every odd integer $p\geq 1$, we find a relative curve $f\colon X\to Y$ such that $f_{\ast}(c_{1}(\omega_{X/Y})^{p})\neq 0$ in the cohomology group $H^{2p}(Y,{\mathbb{C}})$. Let $d=p-1$ and choose $Y$ to be a smooth projective variety of dimension $d$. Let $L$ be an ample line bundle on $Y$ and take $X={\mathbb{P}}(L\oplus{\mathcal{O}}_{Y})$. Consider the tautological exact sequence $0\longrightarrow{\mathcal{O}}(-1)\longrightarrow f^{\ast}(L\oplus{\mathcal{O}}_{Y})\longrightarrow Q\longrightarrow 0.$ We easily derive the relations $\displaystyle\pi^{\ast}c_{1}(L)=c_{1}(Q)-c_{1}({\mathcal{O}}(1))$ (9.20) $\displaystyle c_{1}({\mathcal{O}}(-1))c_{1}(Q)=0.$ (9.21) Moreover we have $c_{1}(\omega_{X/Y})=-c_{1}(Q)-c_{1}({\mathcal{O}}(1)).$ (9.22) From (9.20)–(9.22) and because $d=p-1$ is even, we compute $c_{1}(\omega_{X/Y})^{d}=c_{1}(Q)^{d}+c_{1}({\mathcal{O}}(1))^{d}=\pi^{\ast}c_{1}(L)^{d}.$ Therefore we find $c_{1}(\omega_{X/Y})^{p}=\pi^{\ast}c_{1}(L)^{d}c_{1}(\omega_{X/Y}).$ (9.23) Finally, $f$ is a fibration in curves of genus $0$, hence $f_{\ast}(c_{1}(\omega_{X/Y}))=-2$. We infer that (9.23) leads to $f_{\ast}(c_{1}(\omega_{X/Y})^{p})=-2c_{1}(L)^{d}.$ This class does not vanish, since $Y$ is projective of dimension $d$ and $L$ is ample. ∎ We end with a characterization of the theory of analytic torsion classes of Bismut-Köhler. ###### Theorem 9.24. The theory of analytic torsion classes of Bismut-Köhler $T^{BK}$ is the unique theory of generalized analytic torsion classes such that, for every $\overline{f}\colon X\to Y$, a Kähler fibration in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$, we have the vanishing $T^{BK}(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}])=0.$ ###### Proof. That the theory $T^{BK}$ vanishes for de Rham complexes of Kähler fibrations is a theorem of Bismut [5]. For the uniqueness, let $T$ be a theory of generalized analytic torsion classes vanishing on de Rham complexes of Kähler fibrations. Denote by $S(x)=\sum_{k\geq 0}a_{k}x^{k}$ its corresponding genus. If $\overline{f}$ is a relative curve with a structure of Kähler fibration, then by Theorem 7.14 $T^{h}(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}])=\sum_{k\geq 0}(-1)^{k}a_{k}f_{\ast}[c_{1}(\omega_{X/Y})^{k+1}\bullet\mathbf{1}_{1}].$ (9.25) It is enough to find, for every $k\geq 0$, a relative curve $f$ such that $f_{\ast}(c_{1}(\omega_{X/Y})^{k+1})$ does not vanish. The elementary construction in the proof of Theorem 9.18 works whenever $k$ is even, but one easily sees it fails for $k$ odd. Fortunately, there is an alternative argument. Let $g\geq 2$ and $n\geq 3$ be integers. Consider the fine moduli scheme of smooth curves of genus $g$ with a Jacobi structure of level $n$ [21, Def. 5.4], to be denoted $\mathcal{M}_{g}^{n}$. Let $\pi\colon\mathcal{C}_{g}^{n}\to\mathcal{M}_{g}^{n}$ be the universal curve. An example of Kähler fibration structure on $\pi$ is provided by Teichmüller theory (see for instance [48, Sec. 5]). By [22, Thm. 1], the tautological class $\kappa_{g-2}:=\pi_{\ast}(c_{1}(\omega_{\pi})^{g-1})\in H^{2(g-2)}(\mathcal{M}_{g}^{n},{\mathbb{C}})$ does not vanish. Taking $g=k+2$ and $f=\pi$, we conclude the proof of the theorem. ∎ We note that in the previous theorem, the existence is provided by Bismut’s theorem. It would be interesting to have a proof of the existence of a theory satisfying the condition of Theorem 9.24 from the axiomatic point of view. ## 10 Direct images of hermitian structures As another application of a theory of generalized analytic torsion classes, we construct direct images of metrized complexes. It turns out that the natural place to define direct images is not the category $\operatorname{\overline{\mathbf{D}}^{b}}(\cdot)$ but a new category $\operatorname{\mathbf{\widehat{D}}^{b}}(\cdot)$ that is the analogue to the arithmetic $K$-theory of Gillet and Soulé [28]. Let $X$ be a smooth complex variety. The fibers of the forgetful functor $\operatorname{\overline{\mathbf{D}}^{b}}(X)\to\operatorname{\mathbf{D}^{b}}(X)$ have a structure of $\operatorname{\mathbf{KA}}(X)$-torsor, for the action of $\operatorname{\mathbf{KA}}(X)$ by translation of the hermitian structures ([17, Thm. 3.13]). At the same time, the group $\operatorname{\mathbf{KA}}(X)$ acts on the group $\oplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,p)$ by translation, via the Bott-Chern character $\widetilde{\operatorname{ch}}$ ([17, Prop. 4.6]). Observe that all Bott-Chern classes live in these groups, as for the analytic torsion classes. It is thus natural to build a product category over $\operatorname{\mathbf{KA}}(X)$. ###### Definition 10.1. Let $S\subset T^{\ast}X_{0}$ be a closed conical subset. We define $\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)=\operatorname{\overline{\mathbf{D}}^{b}}(X)\times_{\operatorname{\mathbf{KA}}(X)}\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,S,p)$ to be the category whose objects are equivalence classes $[\overline{\mathcal{F}},\eta]$ of pairs $(\overline{\mathcal{F}},\eta)$ belonging to $\operatorname{Ob}\operatorname{\overline{\mathbf{D}}^{b}}(X)\times\oplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,S,p)$, under the equivalence relation $(\overline{\mathcal{F}},\eta)\sim(\overline{\mathcal{F}}+[\overline{E}],\eta-\widetilde{\operatorname{ch}}(\overline{E}))$ for $[\overline{E}]\in\operatorname{\mathbf{KA}}(X)$, and with morphisms $\operatorname{Hom}_{\operatorname{\mathbf{\widehat{D}}^{b}}(X)}([\overline{\mathcal{F}},\eta],[\overline{\mathcal{G}},\nu])=\operatorname{Hom}_{\operatorname{\mathbf{D}^{b}}(X)}(\mathcal{F},\mathcal{G}).$ If $S\subset T$ are closed conical subsets of $T^{\ast}X_{0}$, then $\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$ is naturally a full subcategory of $\operatorname{\mathbf{\widehat{D}}^{b}}(X,T)$. In the sequel, we extend the main operations in $\operatorname{\mathbf{D}^{b}}(X)$ to the categories $\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$. In particular, we use the theory of generalized analytic torsion classes to construct push-forward morphisms attached to morphisms in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. The category $\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$ has a natural additive structure. More generally, if $S,T$ are closed conical subsets of $T^{\ast}X_{0}$, then there is an obvious addition functor $\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)\times\operatorname{\mathbf{\widehat{D}}^{b}}(X,T)\overset{\oplus}{\longrightarrow}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S\cup T).$ The object $[\overline{0},0]$ is a neutral element for this operation. If $S+T$ is disjoint with the zero section in $T^{\ast}X$, then there is a product defined by the functor $\begin{split}\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)\times\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,T)&\overset{\otimes}{\longrightarrow}\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,(S+T)\cup S\cup T)\\\ ([\overline{\mathcal{F}},\eta],[\overline{\mathcal{G}},\nu])&\longmapsto[\overline{\mathcal{F}}\otimes\overline{\mathcal{G}},\operatorname{ch}(\overline{\mathcal{F}})\bullet\nu+\eta\bullet\operatorname{ch}(\overline{\mathcal{G}})+\operatorname{d}_{D}\eta\bullet\nu]\end{split}$ (10.2) and the obvious assignment for morphisms. This product is commutative up to natural isomorphism. It induces on $\operatorname{\mathbf{\widehat{D}}^{b}}(X,\emptyset)$ a structure of commutative and associative ring category. Also, $[\overline{\mathcal{O}}_{X},0]$ is a unity object for the product structure. More generally, the category $\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$ becomes a left and right $\operatorname{\mathbf{\widehat{D}}^{b}}(X,\emptyset)$ module. Under the same assumptions on $S$, $T$ we may define an internal $\operatorname{Hom}$. For this, let $[\overline{\mathcal{F}},\eta]\in\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$ and $[\overline{\mathcal{G}},\nu]\in\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,T)$. Then we put $\underline{\operatorname{Hom}}([\overline{\mathcal{F}},\eta],[\overline{\mathcal{G}},\nu])=[\underline{\operatorname{Hom}}(\overline{\mathcal{F}},\overline{\mathcal{G}}),(\sigma\operatorname{ch}(\overline{\mathcal{F}}))\bullet\nu+(\sigma\eta)\bullet\operatorname{ch}(\overline{\mathcal{G}})+(\operatorname{d}_{D}\sigma\eta)\bullet\nu],$ where we recall that $\sigma$ is the sign operator (Definition 9.3). Using Corollary 9.8, it is easily seen this is well defined. In particular, we put $[\overline{\mathcal{F}},\eta]^{\vee}:=\underline{\operatorname{Hom}}([\overline{\mathcal{F}},\eta],[\overline{{\mathcal{O}}}_{X},0])=[\overline{\mathcal{F}}^{\vee},\sigma\eta].$ The shift $[1]$ on $\operatorname{\overline{\mathbf{D}}^{b}}(X)$ induces a well defined shift functor on $\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$, whose action on objects is $[\overline{\mathcal{F}},\eta][1]=[\overline{\mathcal{F}}[1],-\eta].$ There is a Chern character $\operatorname{ch}:\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)\longrightarrow\bigoplus_{p}\widetilde{\mathcal{D}}_{D}^{2p}(X,S,p),\quad[\overline{\mathcal{F}},\eta]\longmapsto\operatorname{ch}(\overline{\mathcal{F}})+\operatorname{d}_{D}\eta,$ which is well defined because $\operatorname{d}_{D}\widetilde{\operatorname{ch}}(\overline{E})=\operatorname{ch}(\overline{E})$ for $[\overline{E}]\in\operatorname{\mathbf{KA}}(X)$. The Chern character is additive and compatible with the product structure: $\operatorname{ch}([\overline{\mathcal{F}},\eta]\otimes[\overline{\mathcal{G}},\nu])=\operatorname{ch}([\overline{\mathcal{F}},\eta])\bullet\operatorname{ch}([\overline{\mathcal{G}},\nu]).$ Notice the relations $\displaystyle\operatorname{ch}([\overline{\mathcal{F}},\eta]^{\vee})=\sigma\operatorname{ch}([\overline{\mathcal{F}},\eta]),$ $\displaystyle\operatorname{ch}([\overline{\mathcal{F}},\eta][1])=-\operatorname{ch}([\overline{\mathcal{F}},\eta]).$ We may also define Bott-Chern classes for isomorphisms and distinguished triangles. Let $\widehat{\varphi}\colon[\overline{\mathcal{F}},\eta]\dashrightarrow[\overline{\mathcal{G}},\nu]$ be an isomorphism in $\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$, whose underlying morphism in $\operatorname{\mathbf{D}^{b}}(X)$ we denote $\varphi$. While the class $\widetilde{\operatorname{ch}}(\varphi\colon\overline{\mathcal{F}}\dashrightarrow\overline{\mathcal{G}})$ depends on the representatives $(\overline{\mathcal{F}},\eta)$, $(\overline{\mathcal{G}},\nu)$, the class $\widetilde{\operatorname{ch}}(\widehat{\varphi}):=\widetilde{\operatorname{ch}}(\varphi\colon\overline{\mathcal{F}}\dashrightarrow\overline{\mathcal{G}})+\nu-\eta$ is well defined. ###### Lemma 10.3. Let $\widehat{\varphi}:[\overline{\mathcal{F}},\eta]\dashrightarrow[\overline{\mathcal{G}},\nu]$ be an isomorphism in $\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$, with underlying morphism $\varphi$ in $\operatorname{\mathbf{D}^{b}}(X)$. Then, the following conditions are equivalent: 1. (i) there exists $[\overline{E}]\in\operatorname{\mathbf{KA}}(X)$ such that $\varphi$ induces a tight isomorphism between $\overline{\mathcal{F}}+[\overline{E}]$ and $\overline{\mathcal{G}}$, and $\nu=\eta-\widetilde{\operatorname{ch}}(\overline{E})$; 2. (ii) $\widetilde{\operatorname{ch}}(\widehat{\varphi})=0$. ###### Proof. This is actually a tautology. Because $\operatorname{\mathbf{KA}}(X)$ acts freely and transitively on the possible hermitian structures on $\mathcal{F}$, there exists a unique $[\overline{E}]\in\operatorname{\mathbf{KA}}(X)$ such that $\overline{\mathcal{F}}+[\overline{E}]$ is tightly isomorphic to $\overline{\mathcal{G}}$ via the morphism $\varphi$. Then we have $\widetilde{\operatorname{ch}}(\widehat{\varphi})=\widetilde{\operatorname{ch}}(\overline{E})+\nu-\eta.$ The lemma follows. ∎ ###### Definition 10.4. Let $\widehat{\varphi}$ be an isomorphism in $\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$. We say that $\widehat{\varphi}$ is tight if the equivalent conditions of Lemma 10.3 are satisfied. In particular, if $\varphi\colon\overline{\mathcal{F}}\dashrightarrow\overline{\mathcal{G}}$ is a tight isomorphism in $\operatorname{\overline{\mathbf{D}}^{b}}(X)$, then $\varphi$ induces a tight isomorphism $[\overline{\mathcal{F}},\eta]\dashrightarrow[\overline{\mathcal{G}},\nu]$ if and only if $\eta=\nu$. The following lemma provides an example involving the notion of tight isomorphism. ###### Lemma 10.5. Let $[\overline{\mathcal{F}},\eta]\in\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$ and $[\overline{\mathcal{G}},\nu]\in\operatorname{\mathbf{\widehat{D}}^{b}}(X,T)$. Assume that $S+T$ does not cross the zero section. Then there is a functorial tight isomorphism $[\overline{\mathcal{F}},\eta]^{\vee}\otimes[\overline{\mathcal{G}},\nu]\cong\underline{\operatorname{Hom}}([\overline{\mathcal{F}},\eta],[\overline{\mathcal{G}},\nu]).$ Assume now given a distinguished triangle $\widehat{\tau}\colon\quad[\overline{\mathcal{F}},\eta]\dashrightarrow[\overline{\mathcal{G}},\nu]\dashrightarrow[\overline{\mathcal{H}},\mu]\dashrightarrow[\overline{\mathcal{F}},\eta][1].$ Let $\overline{\tau}$ denote the distinguished triangle $\overline{\mathcal{F}}\dashrightarrow\overline{\mathcal{G}}\dashrightarrow\overline{\mathcal{H}}\dashrightarrow$ in $\operatorname{\overline{\mathbf{D}}^{b}}(X)$. Then we put $\widetilde{\operatorname{ch}}(\widehat{\tau})=\widetilde{\operatorname{ch}}(\overline{\mathcal{\tau}})+\eta-\nu+\mu.$ By [17, Thm. 3.33 (vii)], this class does not depend on the representatives and is thus well defined. We study now the functoriality of $\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$ with respect to inverse and direct images. Let $f\colon X\to Y$ be a morphism of smooth complex varieties. Let $T\subset T^{\ast}Y_{0}$ be a closed conical subset disjoint with $N_{f}$. The action of the left inverse image functor on objects is $f^{\ast}\colon\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(Y,T)\longrightarrow\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,f^{\ast}T),\qquad[\overline{\mathcal{F}},\eta]\longmapsto[f^{\ast}\overline{\mathcal{F}},f^{\ast}\eta].$ That this assignment is well defined amounts to the functoriality of $\widetilde{\operatorname{ch}}$. Let $\overline{f}$ be a morphism in the category $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. The definition of a direct image functor attached to $\overline{f}$ depends upon the choice of a theory of generalized analytic torsion classes. Let $T$ be such a theory. Then we define a functor $\overline{f}_{\ast}$ whose action on objects is $\begin{split}\overline{f}_{\ast}\colon\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)&\longrightarrow\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(Y,f_{\ast}S)\\\ [\overline{\mathcal{F}},\eta]&\longmapsto[\overline{f_{\ast}\mathcal{F}},\overline{f}_{\flat}(\eta)-T(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})],\end{split}$ (10.6) where $\overline{f_{\ast}\mathcal{F}}$ is an arbitrary choice of hermitian structure on $f_{\ast}\mathcal{F}$. By the anomaly formulas, this definition does not depend on the representative $(\overline{\mathcal{F}},\eta)$ nor on the choice of hermitian structure on $\overline{f_{\ast}\mathcal{F}}$. ###### Theorem 10.7. Let $\overline{f}:X\to Y$ and $\overline{g}:Y\to Z$ be morphisms in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. Let $S\subset T^{\ast}X_{0}$ and $T\subset T^{\ast}Y_{0}$ be closed conical subsets. 1. (i) Let $[\overline{\mathcal{F}},\eta]\in\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$. Then there is a functorial tight isomorphism $(\overline{g}\circ\overline{f})_{\ast}([\overline{\mathcal{F}},\eta])\cong\overline{g}_{\ast}\overline{f}_{\ast}([\overline{\mathcal{F}},\eta]).$ 2. (ii) (Projection formula) Assume that $T\cap N_{f}=\emptyset$ and that $T+f_{\ast}S$ does not cross the zero section of $T^{\ast}Y$. Let $[\overline{\mathcal{F}},\eta]\in\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$ and $[\overline{\mathcal{G}},\nu]\in\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(Y,T)$. Then there is a functorial tight isomorphism $\overline{f}_{\ast}([\overline{\mathcal{F}},\eta]\otimes f^{\ast}[\overline{\mathcal{G}},\nu])\cong\overline{f}_{\ast}[\overline{\mathcal{F}},\eta]\otimes[\overline{\mathcal{G}},\nu]$ in $\operatorname{\mathbf{\widehat{D}}^{b}}(Y,W)$, where $W=f_{\ast}(S+f^{\ast}T)\cup f_{\ast}S\cup f_{\ast}f^{\ast}T.$ 3. (iii) (Base change) Consider a cartesian diagram $\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\prime}}$$\scriptstyle{f^{\prime}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{Y.}$ Suppose that $f$ and $h$ are transverse and that $N_{h^{\prime}}$ is disjoint with $S$. Equip $f^{\prime}$ with the hermitian structure induced by the natural isomorphism $h^{\ast}T_{f}\dashrightarrow T_{f^{\prime}}$. Let $[\overline{\mathcal{F}},\eta]\in\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$. Then there is a functorial tight isomorphism $h^{\ast}\overline{f}_{\ast}[\overline{\mathcal{F}},\eta]\cong\overline{f}^{\prime}_{\ast}{h^{\prime}}^{\ast}[\overline{\mathcal{F}},\eta]$ in $\operatorname{\mathbf{\widehat{D}}^{b}}(Y^{\prime},f^{\prime}_{\ast}{h^{\prime}}^{\ast}S)$. ###### Proof. The first and the second assertions follow from Proposition 2.14, the transitivity and the projection formula for $T$. For the third item, one uses the functoriality of the analytic torsion classes and Proposition 2.15. ∎ We close this section with an extension of Grothendieck duality to $\operatorname{\mathbf{\widehat{D}}^{b}}$. Let $\overline{f}:X\to Y$ be a morphism is $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. To enlighten notations, we denote by $\boldsymbol{\omega}_{\overline{f}}$ the object $[\boldsymbol{\omega}_{\overline{f}},0]$ in $\operatorname{\mathbf{\widehat{D}}^{b}}(X,\emptyset)$ (Definition 9.2). Suppose given a closed conical subset $T\subset T^{\ast}Y_{0}$ such that $T\cap N_{f}=\emptyset$. Then we define the functor $\overline{f}^{!}$ whose action on objects is $\overline{f}^{!}:\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(Y,T)\longrightarrow\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,f^{\ast}T),\qquad[\overline{\mathcal{F}},\eta]\longmapsto f^{\ast}[\overline{\mathcal{F}},\eta]\otimes\boldsymbol{\omega}_{\overline{f}}.$ Observe the equality $[\overline{\mathcal{G}},\nu]\otimes\boldsymbol{\omega}_{\overline{f}}=[\overline{\mathcal{G}}\otimes\boldsymbol{\omega}_{\overline{f}},\nu\bullet\operatorname{ch}(\boldsymbol{\omega}_{\overline{f}})].$ (10.8) Now fix a theory of generalized analytic torsion classes. To the morphism $\overline{f}$ we have attached the direct image functor $\overline{f}_{\ast}$. We denote by $\overline{f}^{\vee}_{\ast}$ the direct image functor associated to $\overline{f}$ and the dual theory (Theorem Definition 9.10). ###### Theorem 10.9 (Grothendieck duality for $\operatorname{\mathbf{\widehat{D}}^{b}}$). Let $\overline{f}:X\to Y$ be a morphism in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. Let $S\subset T^{\ast}X_{0}$ and $T\subset T^{\ast}Y_{0}$ be closed conical subsets such that $T\cap N_{f}=\emptyset$ and $T+f_{\ast}S$ is disjoint with the zero section. Let $[\overline{\mathcal{F}},\eta]\in\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$ and $[\overline{\mathcal{G}},\sigma]\in\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(Y,T)$. Then there is a functorial tight isomorphism $\underline{\operatorname{Hom}}(\overline{f}_{\ast}[\overline{\mathcal{F}},\eta],[\overline{\mathcal{G}},\nu])\cong\overline{f}^{\vee}_{\ast}\underline{\operatorname{Hom}}([\overline{\mathcal{F}},\eta],\overline{f}^{!}[\overline{\mathcal{G}},\nu])$ in $\operatorname{\mathbf{\widehat{D}}^{b}}(Y,W)$, where $W=f_{\ast}(S+f^{\ast}T)\cup f_{\ast}S\cup f_{\ast}f^{\ast}T.$ In particular, we have $(\overline{f}_{\ast}[\overline{\mathcal{F}},\eta])^{\vee}\cong\overline{f}^{\vee}_{\ast}([\overline{\mathcal{F}},\eta]^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}}).$ (10.10) ###### Proof. By Lemma 10.5 and Proposition 10.7, we are reduced to establish the functorial tight isomorphism (10.10). The proof follows readily from the definitions, Grothendieck duality and the following two observations. First of all, if $T$ is the theory of analytic torsion classes, then by the very definition of $T^{\vee}$ we find $\sigma T(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})=T^{\vee}(\overline{f},\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}},\overline{f_{\ast}(\mathcal{F}^{\vee}\otimes\boldsymbol{\omega}_{f})}),$ where the metric on $\overline{f_{\ast}(\mathcal{F}^{\vee}\otimes\boldsymbol{\omega}_{f})}$ is chosen so that Grothendieck duality provides a tight isomorphism $\overline{f_{\ast}\mathcal{F}}^{\vee}\cong\overline{f_{\ast}(\mathcal{F}^{\vee}\otimes\boldsymbol{\omega}_{f})}.$ Secondly, for direct images of currents, we compute $\sigma\overline{f}_{\flat}(\eta)=\sigma f_{\ast}(\eta\bullet\operatorname{Td}(T_{\overline{f}}))=(-1)^{e}f_{\ast}(\sigma\eta\bullet\sigma\operatorname{Td}(T_{\overline{f}}))=f_{\ast}(\sigma\eta\bullet\operatorname{ch}(\boldsymbol{\omega}_{\overline{f}})\bullet\operatorname{Td}(T_{\overline{f}})).$ Here $e$ is the relative dimension of $f$, and to derive the last equality we appeal to Proposition 9.4. To conclude, we recall equation (10.8). ∎ ###### Corollary 10.11. Let $T$ be a self-dual theory of generalized analytic torsion classes. 1. (i) Then there is a functorial isomorphism $(\overline{f}_{\ast}[\overline{\mathcal{F}},\eta])^{\vee}\cong\overline{f}_{\ast}([\overline{\mathcal{F}},\eta]^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}}).$ 2. (ii) If the hermitian structure of $\overline{f}$ comes from chosen metrics on $T_{X}$, $T_{Y}$ and $\boldsymbol{\omega}_{X}$, $\boldsymbol{\omega}_{Y}$ are equipped with the induced metrics, then we have a commutative diagram $\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{f}_{\ast}}$$\scriptstyle{(\cdot)^{\vee}\otimes\overline{\boldsymbol{\omega}}_{X}}$$\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{f}_{\ast}}$$\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(Y,f_{\ast}S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\cdot)^{\vee}\otimes\overline{\boldsymbol{\omega}}_{Y}}$$\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(Y,f_{\ast}S).}$ ###### Proof. The first claim is immediate from Theorem 10.9. The second item follows from the first one and the projection formula (Proposition 10.7). ∎ ## 11 Analytic torsion for degenerating families of curves As a second example of application of the theory developed in this article, we describe the singularities of the analytic torsion for degenerating families of curves. The results we prove are particular instances of those obtained by Bismut-Bost [6], Bismut [4] and Yoshikawa [49]. Although the methods of this section can be extended to recover the results of Yoshikawa in [49], for simplicity, we will restrict ourselves to fibrations in curves over a curve. In fact, our proof is not very different from the one in [4] and [49]. For instance, one of the main ingredients of the proof of the results in [4] and [49] is the Bismut-Lebeau immersion formula. Our approach implicitly uses Bismut’s generalization of the immersion formula, encoded in the existence of analytic torsion theories for arbitrary projective morphisms. We expect that the techniques of this section can be used to generalize the above results to situations more general than the ones considered by Yoshikawa. Let $S$ be a smooth complex curve and $f\colon X\to S$ a projective morphism of smooth complex varieties, whose fibers are reduced curves with at most ordinary double singular points. We assume that $f$ is generically smooth. Following Bismut-Bost [6, Sec. 2(b)], we call such a family an f.s.o. (_famille à singularités ordinaires_). The singular locus of $f$, to be denoted $\Sigma$, is a zero dimensional reduced closed subset of $X$. Its direct image $\Delta=f_{\ast}(\Sigma)$ is the Weil divisor $\Delta=\sum_{p\in S}n_{p}p,$ where $n_{p}$ is the number of singular points of the fiber $f^{-1}(p)$. We will abusively identify $\Delta$ with its support. With these notations, we put $V=S\setminus\Delta$. Locally for the analytic topology, the morphism $f$ can be written in complex coordinates either as $f(z_{0},z_{1})=z_{0}$ or $f(z_{0},z_{1})=z_{0}z_{1}$ [6, Sec. 3(a)]. In the second case, the point of coordinates $(z_{0},z_{1})=(0,0)$ belongs to the singular locus $\Sigma$. For a vector bundle $F$ over $X$, let ${\mathbb{P}}(F)$ be the projective space of lines in $F$. The differential $df\colon T_{X}\to f^{\ast}T_{S}$ induces a section ${\mathcal{O}}_{X}\to\Omega_{X}\otimes f^{\ast}T_{S}$. Because $f$ is smooth over $X\setminus\Sigma$, this section does not vanish on $X\setminus\Sigma$. Therefore there is an induced map $\mu\colon X\setminus\Sigma\longrightarrow{\mathbb{P}}(\Omega_{X}\otimes f^{\ast}T_{S})\cong{\mathbb{P}}(\Omega_{X}),$ called the _Gauss map_. Notice that this map was already used in [4] and [49]. We next study the blow-up $\widetilde{X}=\text{Bl}_{\Sigma}(X)$ of $X$ at $\Sigma$ and relate it to the Gauss map. Let $\pi\colon\widetilde{X}\to X$ be the natural projection. Let $E$ be the exceptional divisor of $\pi$, $E=\bigsqcup_{p\in\Sigma}E_{p},\quad E_{p}\cong{\mathbb{P}}(T_{p}X),$ with the reduced scheme structure. For every $p\in\Sigma$, there is an identification $T_{p}X\cong\Omega_{X,p}$ provided by the hessian of $f$, which is a non-degenerate bilinear form on $T_{p}X$. The local description of the blow-up at a point implies: ###### Lemma 11.1. There is a commutative diagram $\textstyle{E_{p}={\mathbb{P}}(T_{p}X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip 8.5359pt\sim}$$\textstyle{{\mathbb{P}}(\Omega_{X,p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\mu}}$$\scriptstyle{\pi}$$\textstyle{{\mathbb{P}}(\Omega_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{X}$$\textstyle{X\setminus\Sigma.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$ Denote by ${\mathcal{O}}(-1)$ the tautological divisor either on ${\mathbb{P}}(\Omega_{X})$ or on $E_{p}$. Then there is a natural isomorphism $\widetilde{\mu}^{\ast}{\mathcal{O}}(-1)\mid_{E_{p}}\cong{\mathcal{O}}(-1).$ Consider now the short exact sequence of vector bundles on ${\mathbb{P}}(\Omega_{X})$ $0\to{\mathcal{O}}(-1)\to p^{\ast}\Omega_{X}\to Q\to 0,$ where $Q$ is the universal quotient bundle. Observe that $Q$ is of rank 1. The dual exact sequence is $0\to U\to p^{\ast}T_{X}\to{\mathcal{O}}(1)\to 0,$ $U$ being the universal vector subsheaf. We denote by $\eta$ the induced exact sequence on $\widetilde{X}$ $\eta\colon 0\to\widetilde{\mu}^{\ast}U\to\pi^{\ast}T_{X}\to\widetilde{\mu}^{\ast}{\mathcal{O}}(1)\to 0,$ (11.2) From (11.2) and the definition $\omega_{X/S}=\omega_{X}\otimes f^{\ast}T_{S}$, we derive a natural isomorphism $\widetilde{\mu}^{\ast}U\otimes\pi^{\ast}\omega_{X/S}\cong\widetilde{\mu}^{\ast}{\mathcal{O}}(-1)\otimes\widetilde{f}^{\ast}T_{S}.$ (11.3) ###### Lemma 11.4. We have $\widetilde{\mu}^{\ast}{\mathcal{O}}(-1)\otimes\widetilde{f}^{\ast}T_{S}={\mathcal{O}}(E).$ (11.5) ###### Proof. First of all we observe that $\widetilde{\mu}^{\ast}U\otimes\pi^{\ast}\omega_{X/S}$ is trivial on the open $W=\widetilde{X}\setminus E$. Indeed, by construction of the Gauss map we have $\widetilde{\mu}^{\ast}U\mid_{W}=\ker(df\colon T_{X}\to f^{\ast}T_{S})\mid_{W}=\omega_{X/S}^{\vee}\mid W.$ Hence by equation (11.3) we can write $\widetilde{\mu}^{\ast}{\mathcal{O}}(-1)\otimes\widetilde{f}^{\ast}T_{S}={\mathcal{O}}(\sum_{p\in\Sigma}m_{p}E_{p}).$ To compute the multiplicities $m_{p}$ we use that $\widetilde{\mu}^{\ast}{\mathcal{O}}(-1)\mid_{E_{p}}={\mathcal{O}}(-1)$, $(E_{p}\cdot\widetilde{f}^{\ast}T_{S})=0$ and $(E_{p}\cdot E_{p})=-1$: $-m_{p}=\deg(\widetilde{\mu}^{\ast}{\mathcal{O}}(-1)\otimes\widetilde{f}^{\ast}T_{S})\mid_{E_{p}}=-1+0=-1.$ The lemma follows. ∎ Later we will need the commutative diagram of exact sequences $\textstyle{\eta\mid_{W}\colon}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mu}^{\ast}U\mid_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{T_{X}\mid_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{\widetilde{\mu}^{\ast}{\mathcal{O}}(1)\mid_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{0}$$\textstyle{\varepsilon\colon}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\omega_{X/S}^{\vee}\mid_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{X}\mid_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f^{\ast}T_{S}\mid_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$ (11.6) After the identification $\widetilde{\mu}^{\ast}{\mathcal{O}}(-1)\otimes\widetilde{f}^{\ast}T_{S}={\mathcal{O}}(E)$ provided by the lemma, the morphism $\gamma$ is the restriction to $W$ of the natural inclusion $\widetilde{\mu}^{\ast}{\mathcal{O}}(1)\to\widetilde{\mu}^{\ast}{\mathcal{O}}(1)\otimes{\mathcal{O}}(E)$. This fact will be used below. We now proceed to introduce the hermitian vector bundles and the analytic torsion classes we aim to study. We fix a theory of generalized analytic torsion classes $T$. Let $f\colon X\to S$, $\widetilde{f}\colon\widetilde{X}\to S$ be f.s.o. as above. Recall that we write $W=X\setminus\Sigma=\widetilde{X}\setminus E$ and $V=S\setminus\Delta$, so that $f^{-1}(V)\subset W$. We endow the tangent spaces $T_{X}$ and $T_{S}$ with smooth hermitian metrics. We will denote by $\overline{f}$ the corresponding morphism in the category $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. On the open subset $W$, there is a quasi-isomorphism $\omega_{X/S}^{\vee}\mid_{W}=\boldsymbol{\omega}_{X/S}^{\vee}[1]\mid_{W}\to T_{f}$ induced by the identification $\omega_{X/S}^{\vee}\mid_{W}=\ker(T_{X}\mid_{W}\to f^{\ast}T_{S})$. On $\omega_{X/S}^{\vee}\mid_{W}$, and in particular on $\omega_{f^{-1}(V)/V}^{\vee}$, we will put the metric induced by $\overline{T_{X}}\mid_{W}$. We will write $\overline{f}^{\prime}\colon f^{-1}(V)\to V$ for the corresponding morphism in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. Observe that the restriction of $f$ to $W$, and hence to $f^{-1}(V)$, may be identified with the restriction of $\widetilde{f}$. Let $\overline{\mathcal{F}}$ be an object in $\operatorname{\overline{\mathbf{D}}^{b}}(X)$ and fix a hermitian structure on $f_{\ast}\mathcal{F}$. Then we consider the relative metrized complexes $\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}}),\qquad\overline{\xi}^{\prime}=(\overline{f}^{\prime},\overline{\mathcal{F}}\mid_{f^{-1}(V)},\overline{f_{\ast}\mathcal{F}}\mid_{V}),$ and the corresponding analytic torsion classes $T(\overline{\xi})\in\bigoplus_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(S,N_{f},p),\qquad T(\overline{\xi}^{\prime})\in\bigoplus_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(V,\emptyset,p).$ By the functoriality of analytic torsion classes and the anomaly formulas, we have $T(\overline{\xi}^{\prime})=T(\overline{\xi})\mid_{V}-\overline{f}_{\flat}[\operatorname{ch}(\overline{\mathcal{F}}\mid_{f^{-1}(V)})\widetilde{\operatorname{Td}}_{m}(\overline{\varepsilon}\mid_{f^{-1}(V)})].$ (11.7) Here $\overline{\varepsilon}$ is the exact sequence in (11.6), with the hermitian metrics we have just defined. From now on we will omit the reference to $f^{-1}(V)$ and $V$ in the formulas. We consider the hermitian structures on the sheaves $U$ and ${\mathcal{O}}(1)$ on ${\mathbb{P}}(\Omega_{X})$ induced by $p^{\ast}\overline{T_{X}}$. We will write $\overline{\eta}$ for the exact sequence in (11.2) and $\overline{\alpha}$, $\overline{\beta}$ and $\overline{\gamma}$ for the vertical isomorphisms in diagram (11.6), all provided with the corresponding metrics. Notice that $\overline{\alpha}$ and $\overline{\beta}$ are isometries. By the properties of the Bott-Chern class $\widetilde{\operatorname{Td}}_{m}$, we have $\widetilde{\operatorname{Td}}_{m}(\overline{\varepsilon})=\widetilde{\operatorname{Td}}_{m}(\overline{\eta})+\operatorname{Td}(\overline{\eta})\widetilde{\operatorname{Td}}_{m}(\overline{\gamma}).$ (11.8) Hence, from (11.7)–(11.8) and identifying $f$ with $\widetilde{f}$ over $V$, we have $T(\overline{\xi}^{\prime})=T(\overline{\xi})-\widetilde{f}_{\ast}[\pi^{\ast}\operatorname{ch}(\overline{\mathcal{F}})\pi^{\ast}\operatorname{Td}(\overline{f})\widetilde{\operatorname{Td}}_{m}(\overline{\eta})]\\\ -\widetilde{f}_{\ast}[\pi^{\ast}\operatorname{ch}(\overline{\mathcal{F}})\pi^{\ast}\operatorname{Td}(\overline{f})\operatorname{Td}(\overline{\eta})\widetilde{\operatorname{Td}}_{m}(\overline{\gamma})].$ (11.9) It will be convenient to have a precise description of $\widetilde{\operatorname{Td}}_{m}(\overline{\gamma})$ at our disposal. For shorthand, we write $L:=\widetilde{\mu}^{\ast}{\mathcal{O}}(1)$ and $\\|\cdot\\|_{0}$ for its hermitian structure constructed before. We denote by $\\|\cdot\\|_{1}$ the metric on ${\mathcal{O}}(E)$ such that the isomorphism $\overline{{\mathcal{O}}(E)}_{1}=\overline{L}_{0}^{-1}\otimes\widetilde{f}^{\ast}\overline{T_{S}}$ (Lemma 11.4) is an isometry. Recall that $\gamma$ gets identified with the restriction to $W$ of the natural inclusion $L\to L\otimes{\mathcal{O}}(E)$. We let $\\|\cdot\\|_{\infty}$ be the hermitian metric on $L\mid_{W}$ such that $\gamma$ is an isometry. Hence, if $\mathbf{1}$ denotes the canonical section of ${\mathcal{O}}(E)$ and $\ell$ is any section of $L\mid_{W}$, then we have $\\|\ell\\|_{\infty}=\\|\ell\\|_{0}\\|\mathbf{1}\\|_{1}.$ To simplify the notations, we will skip the reference to $W$. We then have on $W$ $\widetilde{\operatorname{Td}}_{m}(\overline{\gamma})=\widetilde{\operatorname{Td}}_{m}(\overline{L}_{0}\overset{\operatorname{id}}{\to}\overline{L}_{\infty}).$ To compute a representative of this class, we fix a smooth function $h\colon{\mathbb{P}}^{1}_{{\mathbb{C}}}\to{\mathbb{R}}$ such that $h(0)=0$ and $h(\infty)=1$. Then we proceed by a deformation argument. Let $q\colon W\times{\mathbb{P}}^{1}_{{\mathbb{C}}}\to W$ be the projection to the first factor. On the line bundle $q^{\ast}L$ we put the metric that, on the fiber at the point $(w,t)\in W\times{\mathbb{P}}^{1}_{{\mathbb{C}}}$, is determined by the formula $\\|\ell\\|_{(w,t)}=\\|\ell\\|_{0,w}\\|\mathbf{1}\\|_{1,w}^{h(t)}.$ We will write $\\|\cdot\\|_{t}$ for this family of metrics parametrized by ${\mathbb{P}}^{1}_{{\mathbb{C}}}$. Define $\overline{\operatorname{Td}}(\overline{L}_{0}\to\overline{L}_{\infty})=\frac{1}{2\pi i}\int_{{\mathbb{P}}^{1}_{{\mathbb{C}}}}\frac{-1}{2}\log(t\overline{t})(\operatorname{Td}(\overline{q^{\ast}L}_{t})-\operatorname{Td}(\overline{q^{\ast}L}_{0})).$ Then $\overline{\operatorname{Td}}_{m}(\overline{\gamma})=\operatorname{Td}^{-1}(\overline{L}_{0})\overline{\operatorname{Td}}(\overline{L}_{0}\to\overline{L}_{\infty})$ (11.10) represents the class $\widetilde{\operatorname{Td}}_{m}(\gamma)$. Let us develop $\overline{\operatorname{Td}}_{m}(\overline{\gamma})$. If $\overline{{\mathcal{O}}}_{t}$ denotes the trivial line bundle on $W\times{\mathbb{P}}^{1}_{{\mathbb{C}}}$ with the norm $\\|\mathbf{1}\\|_{t}=\\|\mathbf{1}\\|_{1}^{h(t)}$, then we compute $\operatorname{Td}(\overline{q^{\ast}L}_{t})-\operatorname{Td}(\overline{q^{\ast}L}_{0})=\frac{1}{2}c_{1}(\overline{{\mathcal{O}}}_{t})+\frac{1}{6}c_{1}(\overline{{\mathcal{O}}}_{t})q^{\ast}c_{1}(\overline{L}_{0})+\frac{1}{12}c_{1}(\overline{{\mathcal{O}}}_{t})^{2}.$ By the very definition of $c_{1}$, we find $c_{1}(\overline{{\mathcal{O}}}_{t})=\operatorname{\partial}\operatorname{\overline{\partial}}\log\\|\mathbf{1}\\|_{t}^{2}=\operatorname{\partial}\operatorname{\overline{\partial}}(h(t)\log\\|\mathbf{1}\\|_{1}^{2})\\\ =h(t)c_{1}(\overline{{\mathcal{O}}(E)}_{1})+\log\\|\mathbf{1}\\|_{1}^{2}\operatorname{\partial}\operatorname{\overline{\partial}}h(t)+\operatorname{\partial}h(t)\wedge\operatorname{\overline{\partial}}\log\\|\mathbf{1}\\|_{1}^{2}+\operatorname{\partial}\log\\|\mathbf{1}\\|_{1}^{2}\wedge\operatorname{\overline{\partial}}h(t).$ We easily obtain $\displaystyle\frac{1}{2\pi i}\int_{{\mathbb{P}}^{1}_{{\mathbb{C}}}}\frac{-1}{2}\log(t\overline{t})\frac{1}{2}c_{1}(\overline{{\mathcal{O}}}_{t})=-\frac{1}{2}\log\\|\mathbf{1}\\|_{1},$ (11.11) $\displaystyle\frac{1}{2\pi i}\int_{{\mathbb{P}}^{1}_{{\mathbb{C}}}}\frac{-1}{2}\log(t\overline{t})\frac{1}{6}q^{\ast}c_{1}(\overline{L}_{0})c_{1}(\overline{{\mathcal{O}}}_{t})=-\frac{1}{6}\log\\|\mathbf{1}\\|_{1}c_{1}(\overline{L}_{0}).$ (11.12) With some more work, we have $\begin{split}\frac{1}{2\pi i}\int_{{\mathbb{P}}^{1}_{{\mathbb{C}}}}\frac{-1}{2}\log(t\overline{t})\frac{1}{12}c_{1}(\overline{{\mathcal{O}}}_{t})^{2}=&-\frac{a}{6}\log\\|\mathbf{1}\\|_{1}c_{1}(\overline{{\mathcal{O}}(E)}_{1})\\\ &+\frac{b}{3}\operatorname{\partial}(\log\\|\mathbf{1}\\|_{1}\ \operatorname{\overline{\partial}}\log\\|\mathbf{1}\\|_{1}),\end{split}$ (11.13) where $a=\frac{1}{2\pi i}\int_{{\mathbb{P}}^{1}_{{\mathbb{C}}}}\log(t\overline{t})\frac{1}{2}\operatorname{\partial}\operatorname{\overline{\partial}}(h(t)^{2}),\qquad b=\frac{1}{2\pi i}\int_{{\mathbb{P}}^{1}_{{\mathbb{C}}}}\log(t\overline{t})\operatorname{\partial}h(t)\wedge\operatorname{\overline{\partial}}h(t).$ (11.14) We observe that $a=\frac{1}{2\pi i}\int_{{\mathbb{P}}^{1}_{{\mathbb{C}}}}\log(t\overline{t})\frac{1}{2}\operatorname{\partial}\operatorname{\overline{\partial}}(h(t)^{2})=\frac{1}{2},$ which is independent of $h$. All in all, equations (11.10)–(11.14) provide the following expression for the representative $\overline{\operatorname{Td}}_{m}(\overline{\gamma})$ of $\widetilde{\operatorname{Td}}_{m}(\overline{\gamma})$: $\begin{split}\overline{\operatorname{Td}}_{m}(\overline{\gamma})=\operatorname{Td}^{-1}(\overline{L}_{0})\Big{(}&-\frac{1}{2}\log\\|\mathbf{1}\\|_{1}-\frac{1}{6}\log\\|\mathbf{1}\\|_{1}c_{1}(\overline{L}_{0})\\\ &-\frac{1}{12}\log\\|\mathbf{1}\\|_{1}c_{1}(\overline{{\mathcal{O}}(E)}_{1})+\frac{b}{3}\operatorname{\partial}(\log\\|\mathbf{1}\\|_{1}\operatorname{\overline{\partial}}\log\\|\mathbf{1}\\|_{1})\Big{)}.\end{split}$ (11.15) Given a current $\eta\in\mathcal{D}_{D}^{n}(X,p)$, we will call $(n,p)$ its _Deligne bidegree_ , while we will call the _Dolbeault bidegree_ to the bidegree in the Dolbeault complex. When it is clear from the context to which bidegree we are referring, we call it bidegree. We now study the singularities of the component of Deligne bidegree $(1,1)$ of $T(\overline{\xi}^{\prime})$ near the divisor $\Delta$. For this we first recall the decomposition of equation (11.9). Observe that $\widetilde{\mathcal{D}}_{D}^{1}(V,\emptyset,1)$ gets identified with the space of smooth real functions on $V$. In the sequel, for an element $\vartheta\in\oplus_{p}\widetilde{D}_{D}^{2p-1}(\ast,p)$, we write $\vartheta^{(2r-1,r)}$ to refer to its component of bidegree $(2r-1,r)$. By construction of the Deligne complex, an element of Deligne bidegree $(2r-1,r)$ is just a current of Dolbeault bidegree $(r-1,r-1)$. The following assertion is well-known. See for instance [47, Lemma 2.1, Cor. 2.2]. ###### Lemma 11.16. Let $\Omega\subset{\mathbb{C}}$ be an open subset and $\vartheta$ a current of Dolbeault bidegree $(0,0)$ on $\Omega$. Let $\Delta$ be the standard laplacian. If the current $\Delta\vartheta$ is represented by a locally bounded measurable function, then $\vartheta$ is represented by a continuous function. ###### Proposition 11.17. The current $T(\overline{\xi})^{(1,1)}\in\widetilde{\mathcal{D}}_{D}^{1}(S,N_{f},1)$ is represented by a continuous function on $S$. ###### Proof. The differential equation satisfied by $T(\overline{\xi})^{(1,1)}$ is $\operatorname{d}_{\mathcal{D}}T(\overline{\xi})^{(1,1)}=\operatorname{ch}(\overline{f_{\ast}\mathcal{F}})^{(2,1)}-f_{\ast}[\operatorname{ch}(\overline{\mathcal{F}})\operatorname{Td}(\overline{f})]^{(2,1)}.$ (11.18) In local coordinates, the operator $\operatorname{d}_{\mathcal{D}}=-2\operatorname{\partial}\operatorname{\overline{\partial}}$ is a rescaling of the laplacian $\Delta$. By the lemma, it is enough we prove that the current at the right hand side of (11.18) is represented by a locally bounded measurable differential form. Because $\operatorname{ch}(\overline{f_{\ast}\mathcal{F}})^{(2,1)}$ and $\operatorname{ch}(\overline{\mathcal{F}})\operatorname{Td}(\overline{f})$ are smooth differential forms, we are reduced to study currents of the form $f_{\ast}[\theta]^{(2,1)}$, where $\theta$ is a smooth differential form. By a partition of unity argument, we reduce to the case where $f\colon{\mathbb{C}}^{2}\to{\mathbb{C}}$ is the morphism $f(z_{0},z_{1})=z_{0}z_{1}$ and $\theta$ is a differential form of Dolbeault bidegree (2,2) with compact support. Then we need to prove that the fiber integral $G(w)=\int_{z_{0}z_{1}=w}\theta$ is a bounded form in a neighborhood of $w=0$. Write $\theta=h(z_{0},z_{1})dz_{0}\wedge d\overline{z}_{0}\wedge dz_{1}\wedge d\overline{z}_{1}.$ We reduce to study integrals of the form $G(w)=\left(\int_{\|w\|<\|z_{0}\|<1}h(z_{0},z_{0}/w)\frac{\|w\|^{2}}{\|z_{0}\|^{4}}dz_{0}\wedge\overline{z}_{0}\right)dw\wedge d\overline{w}.$ The property follows from an easy computation in polar coordinates. ∎ ###### Proposition 11.19. Let $\theta$ be a differential form of Dolbeault bidegree (1,1) on $\widetilde{X}$. Then the current $\widetilde{f}_{\ast}[\theta]$ is represented by a bounded function on $S$. ###### Proof. The proof is the same as in [6, Prop. 5.2]. One only has to show that the argument in _loc. cit._ carries over to the case of the non-reduced fibres that have appeared when blowing up the nodes. ∎ ###### Corollary 11.20. The current $\widetilde{f}_{\ast}[\pi^{\ast}\operatorname{ch}(\overline{\mathcal{F}})\pi^{\ast}\operatorname{Td}(\overline{f})\widetilde{\operatorname{Td}}_{m}(\overline{\eta})]$ is represented by a bounded function on $S$. ###### Proof. It suffices to observe that the differential form $\pi^{\ast}\operatorname{ch}(\overline{\mathcal{F}})\pi^{\ast}\operatorname{Td}(\overline{f})\widetilde{\operatorname{Td}}_{m}(\overline{\eta})$ is actually smooth on the whole $\widetilde{X}$. ∎ According to (11.9), it remains to study the current $\widetilde{f}_{\ast}[\pi^{\ast}\operatorname{ch}(\overline{\mathcal{F}})\pi^{\ast}\operatorname{Td}(\overline{f})\operatorname{Td}(\overline{\eta})\widetilde{\operatorname{Td}}_{m}(\overline{\gamma})]\mid_{V}.$ The main difference with the situation in Corollary 11.20 is that the class $\widetilde{\operatorname{Td}}_{m}(\overline{\gamma})$ is not defined on the whole $\widetilde{X}$, but only on $W=\widetilde{X}\setminus E$. In the following discussion we will use the representative $\overline{\operatorname{Td}}_{m}(\overline{\gamma})$ defined in (11.10) at the place of $\widetilde{\operatorname{Td}}_{m}(\overline{\gamma})$. In view of equations (11.11)–(11.13), the first result we need is the following statement. ###### Proposition 11.21. Let $\theta$ be a smooth and $\operatorname{\partial},\operatorname{\overline{\partial}}$ closed differential form on $\widetilde{X}$, of Dolbeault bidegree $(1,1)$. Let $w$ be an analytic coordinate in a neighborhood of $p\in\Delta$ with $w(p)=0$. Write $D_{p}=E\cap\widetilde{f}^{-1}(p)$. Then, the current $\widetilde{f}_{\ast}[\log\\|\mathbf{1}\\|_{1}\theta]-\left(\frac{1}{2\pi i}\int_{D_{p}}\theta\right)[\log\|w\|]$ is represented by a continuous function in a neighborhood of $p$. In particular, if $\theta$ is cohomologous to a form $\pi^{\ast}\vartheta$, where $\vartheta$ is a smooth and $\operatorname{\partial},\operatorname{\overline{\partial}}$ closed differential form on $X$, then $\widetilde{f}_{\ast}[\log\\|\mathbf{1}\\|_{1}\theta]$ is represented by a continuous function on $S$. ###### Proof. Recall that the Poincaré-Lelong formula provides the equality of currents $\operatorname{d}_{\mathcal{D}}[\log\\|\mathbf{1}\\|_{1}^{-1}]=[c_{1}(\overline{{\mathcal{O}}(E)}_{1})]-\delta_{E}.$ Moreover, the operator $\operatorname{d}_{\mathcal{D}}$ commutes with proper push-forward. Therefore, taking into account that $\theta$ is $\operatorname{\partial}$ and $\operatorname{\overline{\partial}}$ closed, the equation $\operatorname{d}_{\mathcal{D}}\widetilde{f}_{\ast}[\log\\|\mathbf{1}\\|_{1}\theta]=\left(\frac{1}{2\pi i}\int_{D_{p}}\theta\right)\delta_{p}-\widetilde{f}_{\ast}[c_{1}(\overline{{\mathcal{O}}(E)}_{1})\theta].$ (11.22) holds in a neighborhood of $p$. On the other hand, the Poincaré-Lelong equation also gives $\operatorname{d}_{\mathcal{D}}[\log\|w\|]=\delta_{p}$. Using (11.22), we see that $\operatorname{d}_{\mathcal{D}}\left(\widetilde{f}_{\ast}[\log\\|\mathbf{1}\\|_{1}\theta]-\left(\frac{1}{2\pi i}\int_{D_{p}}\theta\right)[\log\|w\|]\right)=-\frac{1}{2}\widetilde{f}_{\ast}[c_{1}(\overline{{\mathcal{O}}(E)}_{1})\theta].$ Finally, by Proposition 11.19, the current $\widetilde{f}_{\ast}[c_{1}(\overline{{\mathcal{O}}(E)}_{1})\theta]$ is represented by a continuous function on $S$. Hence the first assertion follows from Lemma 11.16. For the second assertion, we just observe that, in this case, $\int_{D_{p}}\theta=\int_{D_{p}}\pi^{\ast}\vartheta=0.$ The proof is complete. ∎ ###### Corollary 11.23. Let $n_{p}$ be the multiplicity of $\Delta$ at $p$ and $O(1)$ the current represented by a locally bounded function. The following estimates hold in a neighborhood of $p$ $\displaystyle\widetilde{f}_{\ast}[\log\\|\mathbf{1}\\|_{1}c_{1}(\pi^{\ast}\overline{T_{X}})]=O(1),$ $\displaystyle\widetilde{f}_{\ast}[\log\\|\mathbf{1}\\|_{1}c_{1}(\overline{{\mathcal{O}}(E)}_{1})]=-n_{p}[\log\|w\|]+O(1),$ $\displaystyle\widetilde{f}_{\ast}[\log\\|\mathbf{1}\\|_{1}c_{1}(\overline{L}_{0})]=n_{p}[\log\|w\|]+O(1),$ $\displaystyle\widetilde{f}_{\ast}[\log\\|\mathbf{1}\\|_{1}c_{1}(\widetilde{\mu}^{\ast}\overline{U})]=-n_{p}[\log\|w\|]+O(1).$ ###### Proof. We use (11.3)–(11.5) and the intersection numbers $(D_{p}\cdot D_{p})=(D_{p}\cdot E)=-n_{p}$. ∎ ###### Corollary 11.24. With the notations above, the development $\widetilde{f}_{\ast}[\pi^{\ast}\operatorname{ch}(\overline{\mathcal{F}})\pi^{\ast}\operatorname{Td}(\overline{f})\operatorname{Td}(\overline{\eta})\widetilde{\operatorname{Td}}_{m}(\overline{\gamma})]^{(3,2)}=\operatorname{rk}(\overline{\mathcal{F}})\frac{n_{p}}{6}[\log\|w\|]+O(1)$ holds in a neighborhood of $p$. ###### Proof. We take into account the expression (11.15) for the representative $\overline{\operatorname{Td}}_{m}(\overline{\gamma})$, the developments of the smooth differential forms $\operatorname{ch}(\overline{\mathcal{F}})$, $\operatorname{Td}(\overline{f})$, $\operatorname{Td}(\overline{\eta})$ and $\operatorname{Td}^{-1}(\overline{L}_{0})$, and then apply Corollary 11.23. We find $\widetilde{f}_{\ast}[\pi^{\ast}\operatorname{ch}(\overline{\mathcal{F}})\pi^{\ast}\operatorname{Td}(\overline{f})\operatorname{Td}(\overline{\eta})\widetilde{\operatorname{Td}}_{m}(\overline{\gamma})]^{(3,2)}=\\\ \operatorname{rk}(\overline{\mathcal{F}})\frac{n_{p}}{6}[\log\|w\|]+\operatorname{rk}(\overline{\mathcal{F}})\frac{b}{3}\widetilde{f}_{\ast}[\operatorname{\partial}(\log\\|\mathbf{1}\\|_{1}\operatorname{\overline{\partial}}\log\\|\mathbf{1}\\|_{1})]+O(1).$ To conclude we observe that, on $V$, the term $\widetilde{f}_{\ast}[\operatorname{\partial}(\log\\|\mathbf{1}\\|_{1}\operatorname{\overline{\partial}}\log\\|\mathbf{1}\\|_{1})]$ vanishes. Indeed, the morphism $\widetilde{f}_{\ast}$ is smooth on $V$ with one dimensional fibers. Hence this current is represented by the function $V\ni s\mapsto\frac{1}{2\pi i}\int_{\widetilde{f}^{-1}(s)}\operatorname{\partial}(\log\\|\mathbf{1}\\|_{1}\operatorname{\overline{\partial}}\log\\|\mathbf{1}\\|_{1})=\frac{1}{2\pi i}\int_{\widetilde{f}^{-1}(s)}d(\log\\|\mathbf{1}\\|_{1}\operatorname{\overline{\partial}}\log\\|\mathbf{1}\\|_{1})=0.$ This ends the proof. ∎ The results of this section are summarized in the following statement. ###### Theorem 11.25. Let $p\in\Delta$ and let $n_{p}$ be the number of singular points of $f\colon X\to S$ lying above $p$. Let $w$ be a local coordinate on $S$, centered at $p$. Then, in a neighborhood of $p$, we have the estimate $T(\overline{\xi}^{\prime})^{(1,1)}=-\frac{\operatorname{rk}{\overline{\mathcal{F}}}}{6}n_{p}[\log\|w\|]+O(1).$ ###### Proof. It is enough to join (11.9), Proposition 11.17, Corollary 11.20 and 11.24. ∎ ###### Corollary 11.26. Assume that $\overline{\mathcal{F}}=\overline{E}$ is a vector bundle placed in degree $0$, and that $R^{1}f_{\ast}E=0$ on $S$. Endow $f_{\ast}E$ with the $L^{2}$ metric on $V$ depending on $\overline{E}$ and the metric on $\overline{\omega_{f^{-1}(V)/V}}$. Write $\xi^{\prime\prime}=(\overline{f}^{\prime},\overline{E},\overline{f_{\ast}E}_{L^{2}})$ for the corresponding relative metrized complex on $V$. Let $p$ and $w$ be as in the theorem. Then we have $T(\overline{\xi}^{\prime\prime})^{(1,1)}=-\frac{\operatorname{rk}(\overline{\mathcal{F}})}{6}n_{p}[\log\|w\|]+O(\log\log\|w\|^{-1})$ as $w\to 0$. ###### Proof. Introduce an auxiliary smooth hermitian metric on the vector bundle $f_{\ast}E$ on $S$, and let $\overline{\xi}^{\prime}=(\overline{f}^{\prime},\overline{E},\overline{f_{\ast}E})$ be the corresponding relative metrized complex. Then the theorem applies to $\overline{\xi}^{\prime}$. By the anomaly formulas, on $V$ we have $T(\overline{\xi}^{\prime\prime})^{(1,1)}=T(\overline{\xi}^{\prime})^{(1,1)}+\widetilde{\operatorname{ch}}(\overline{f_{\ast}E},\overline{f_{\ast}E}_{L^{2}})^{(1,1)}.$ By [6, Prop. 7.1], the $L^{2}$ metric has logarithmic singularities near $w=0$ and $\widetilde{\operatorname{ch}}(\overline{f_{\ast}E},\overline{f_{\ast}E}_{L^{2}})=O(\log\log\|w\|^{-1})$ as $w\to 0$. This proves the corollary. ∎ ###### Remark 11.27. The corollary is to be compared with [6, Thm. 9.3]. The difference of sign is due to the fact that Bismut and Bost work with the inverse of the usual determinant line bundle. The approach of _loc. cit._ is more analytic and requires the spectral description of the Ray-Singer analytic torsion. Acknowledgments. During the elaboration of this paper we have benefited from conversations with many colleagues, that helped us to understand some points, to clarify others or to find the relevant bibliography. Our thanks to J.-M. Bismut, J.-B. Bost, D. Burghelea, D. Eriksson, J. Kramer, U. Kühn, X. Ma, V. Maillot, D. Rössler, C. Soulé. We would like to thank the following institutions where part of the research conducting to this paper was done: the CRM in Bellaterra (Spain), the CIRM in Luminy (France), the Morningside Institute of Beijing (China), the University of Barcelona and the IMUB, the Alexandru Ioan Cuza University of Iasi, the Institut de Mathématiques de Jussieu and the ICMAT (Madrid). Burgos and Freixas were partially supported by grant MTM2009-14163- C02-01, Burgos was partially supported by CSIC research project 2009501001, Liţcanu was partially supported by CNCSIS -UEFISCSU, project number PNII - IDEI 2228/2008 . ## References [1] A. Berthomieu and J.-M. Bismut, _Quillen metrics and higher analytic torsion forms_ , J. Reine Angew. Math. 457 (1994), 85–184. * [2] J.-M. Bismut, _The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs_ , Invent. 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Wolpert, _On obtaining a positive line bundle from the Weil-Petersson class_ , Am. J. Math. 107 (1985), 1485–1507. * [48] , _Chern forms and the riemann tensor for the moduli space of curves_ , Invent. math. 85 (1986), 119–145. * [49] K.-I. Yoshikawa, _On the singularity of Quillen metrics_ , Math. Ann. 337 (2007), 61–89. * [50] Y. Zha, _A general arithmetic Riemann-Roch theorem_ , Ph.D. thesis, University of Chicago, 1998.	arxiv-papers	2010-11-16T14:16:08	2024-09-04T02:49:14.866948	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "J. I. Burgos Gil, G. Freixas i Montplet and R. Litcanu", "submitter": "Jos\\'e Ignacio Burgos Gil", "url": "https://arxiv.org/abs/1011.3702" }
1011.3714	# Semipurity of tempered Deligne cohomology José Ignacio Burgos Gil Facultad de Matemáticas Universidad de Barcelona Gran Vía C. C. 585 Barcelona, Spain ###### Abstract. In this paper we define the formal and tempered Deligne cohomology groups, that are obtained by applying the Deligne complex functor to the complexes of formal differential forms and tempered currents respectively. We then prove the existence of a duality between them, a vanishing theorem for the former and a semipurity property for the latter. The motivation of these results comes from the study of covariant arithmetic Chow groups. The semi-purity property of tempered Deligne cohomology implies, in particular, that several definitions of covariant arithmetic Chow groups agree for projective arithmetic varieties. Partially supported by Grants BFM2003-02914 and MTM2006-14234-C02-01 ###### Contents 1. 1 Introduction 2. 2 Complexes of forms and currents 1. 2.1 Flat forms and Whitney forms 2. 2.2 Currents with support in a subvariety 3. 2.3 Formal and tempered Deligne cohomology 4. 2.4 Semi-purity of tempered Deligne cohomology 3. 3 Arithmetic Intersection Theory 1. 3.1 Definition of Covariant arithmetic Chow groups 2. 3.2 Properties of Covariant arithmetic Chow groups ## 1\. Introduction The aim of this note is to study some properties of formal and tempered Deligne cohomology (with real coefficients). These cohomology groups are defined by applying the Deligne complex functor to the complexes of formal differential forms and tempered currents respectively. Let $X$ be a complex projective manifold and let $W$ be a Zariski locally closed subset of $X$. Let $i:W\longrightarrow X$ denote the inclusion and let $i^{},i^{!},i_{\ast},i_{!}$ be the induced functors in the derived category of abelian sheaves. Then the complex of formal differential forms of $W$ computes the cohomology of $W$ with compact supports. That is, it computes the groups $H^{\ast}(X,i_{!}i^{\ast}\underline{\mathbb{R}})$. The complex of tempered currents on $W$ compute the cohomology of $X$ with supports on $W$, that is, it computes the groups $H^{\ast}(X,i_{}i^{!}\underline{\mathbb{R}})$. Following Deligne, the previous groups have a mixed Hodge structure, hence a Hodge filtration that we will call the Deligne-Hodge filtration. The complexes of formal differential forms and tempered currents are examples of Dolbeault complexes (see [6]). Therefore they have a Hodge filtration obtained from the bigrading of differential forms. In general, this Hodge filtration does not induce the Deligne-Hodge filtration in cohomology. Moreover, the spectral sequence associated to this Hodge filtration does not degenerate at the $E^{1}$-term. This implies that formal and tempered Deligne cohomology groups with real coefficients will not have, in general, the same properties as Deligne- Beilinson cohomology. For instance they do not need to be finite dimensional. They have a structure of topological vector spaces, but they may be non- separated. Note however that, in the particular case when $W=X$, the formal and tempered Deligne cohomology groups with real coefficients, agree with the usual real Deligne cohomology groups. In this note we will construct a (Poincaré like) duality between formal Deligne cohomology and tempered Deligne cohomology, that induce a perfect pairing between the corresponding separated vector spaces. In particular, applying this duality to the case $W=X$ we obtain an exceptional duality for real Deligne Beilinson cohomology (corollary 2.28) of smooth projective varieties that, to my knowledge, is new. The shape of this exceptional duality reminds very much the functional equation of $L$-functions. It would be interesting to know whether this duality has any arithmetic meaning. The second result is a vanishing result for formal Deligne cohomology. Thanks to the previous duality, the vanishing result of formal Deligne cohomology implies a semipurity property of tempered Deligne cohomology (corollary 2.34). The motivation for these results comes from the study of covariant arithmetic Chow groups introduced in [3] and [6]. The covariant arithmetic Chow groups are a variant of the arithmetic Chow groups defined by Gillet and Soulé, that are covariant for arbitrary proper morphism. By contrast, the groups defined by Gillet and Soulé are only covariant for proper morphisms between arithmetic varieties that induce smooth maps between the corresponding complex varieties. The covariant arithmetic Chow groups do not have a product structure, but they are a module over the contravariant arithmetic Chow groups (see [6] for more details). Similar definitions of covariant Chow groups have been given by Kawaguchi and Moriwaki [13] and by Zha [16]. These two definitions are equivalent except for the fact that Zha neglects the structure of real manifold induced on the complex manifold associated to an arithmetic variety. Although not explicitly stated, in the paper [6], the covariant arithmetic Chow groups are defined by means of tempered Deligne cohomology. The semi- purity property of tempered Deligne cohomology was announced and used in [6]. Hence this paper can be seen as a complement of [6]. A new consequence of the semipurity property is that, for an arithmetic variety that is generically projective, the covariant Chow groups introduced in [3] and [6] are isomorphic to the covariant Chow groups introduced by Kawaguchi and Moriwaki. Acknowledgments. In the course of preparing this manuscript, I had many stimulating discussions with many colleagues. We would like to thank them all. In particular, I would like to express my gratitude to J.-B. Bost, U. Kühn, J. Kramer, K. Kühnemann, V. Maillot, D. Roessler, C. Schapira, J Wildeshaus. Furthermore, I would like to thank the CRM (Bellaterra, Barcelona), for partial support of this work. ## 2\. Complexes of forms and currents By a complex algebraic manifold we will mean the analytic manifold associated to a smooth scheme over $\mathbb{C}.$ Let $X$ be a projective complex algebraic manifold. We will consider the following situation: let $Z\subset Y$ be closed subvarieties of $X$, let $U$ and $V$ be the open subsets $U=X\setminus Y$, $V=X\setminus Z$ and let $W$ be the locally closed subset $W=Y\setminus Z$. ### 2.1. Flat forms and Whitney forms The complex of Whitney forms. Let $\mathscr{E}^{\ast}_{X}$ denote the sheaf of smooth differential forms on $X$. We will denote by $E^{\ast}(U)$ the complex of global differential forms over $U$ and by $E^{\ast}_{c}(U)$ the complex of differential forms with compact support. Let $\mathscr{E}^{\ast}_{X}(\operatorname{flat}Y)$ denote the ideal sheaf of differential forms that are flat along $Y$. Recall that a differential form on $X$ is called flat along $Y$ if its Taylor expansion vanishes at all points of $Y$. We write $\mathscr{E}^{\ast}_{Y^{\infty}}=\mathscr{E}^{\ast}_{X}/\mathscr{E}^{\ast}_{X}(\operatorname{flat}Y).$ The sections of this complex of sheaves are called Whitney forms on $Y$. Whitney’s extension theorem ([15] IV theorem 3.1), gives us a precise description of the space of Whitney forms in terms of jets over $Y$. For instance, if $Y$ is the smooth subvariety of $\mathbb{C}^{n}$ defined by the equations $z_{1}=\dots=z_{k}=0$, then the germ of the sheaf of Whitney functions on $Y$ at the point $x=(0,\dots,0)$ is $\mathscr{E}^{0}_{Y^{\infty},x}=\mathscr{E}^{0}_{Y,x}[[z_{k+1},\dots,z_{n},\bar{z}_{k+1},\dots,\bar{z}_{n}]].$ We will write $\mathscr{E}^{\ast}_{Y^{\infty}}(\operatorname{flat}Z)=\mathscr{E}^{\ast}_{X}(\operatorname{flat}Z)/\mathscr{E}^{\ast}_{X}(\operatorname{flat}Y).$ Observe that $\mathscr{E}^{\ast}_{Y^{\infty}}(\operatorname{flat}Z)$ can also be defined as the kernel of the morphism $\mathscr{E}^{\ast}_{Y^{\infty}}\longrightarrow\mathscr{E}^{\ast}_{Z^{\infty}}.$ The sheaf $\mathscr{E}^{\ast}_{Y^{\infty}}(\operatorname{flat}Z)$ agrees with the sheaf denoted $\mathbb{C}_{W}\overset{W}{\otimes}\mathcal{C}_{X}^{\infty}$ in [12]. The complex $\mathscr{E}^{\ast}_{Y^{\infty}}(\operatorname{flat}Z)$ is a complex of fine sheaves. We will denote the corresponding complex of global sections by $E^{\ast}_{X^{\mathcal{W}}}(W):=\Gamma(X,\mathscr{E}^{\ast}_{Y^{\infty}}(\operatorname{flat}Z))$. Note that the complex $E^{\ast}_{X^{\mathcal{W}}}(W)$ depends only on the locally closed subspace $W\subset X$ and not on a particular choice of closed subsets $Y$ and $Z$. Observe also that $E^{\ast}_{X^{\mathcal{W}}}(X)=E^{\ast}(X)$ is the usual complex of smooth differential forms on $X$. We will denote by $E^{\ast}_{X^{\mathcal{W}},\mathbb{R}}(W)$ the real subcomplex underlying $E^{\ast}_{X^{\mathcal{W}}}(W)$. By the acyclicity of fine sheaves, there is a diagram of short exact sequences (2.1) $\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{X^{\mathcal{W}}}^{\ast}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{X^{\mathcal{W}}}^{\ast}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\ast}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{X^{\mathcal{W}}}^{\ast}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{X^{\mathcal{W}}}^{\ast}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\ast}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\ast}_{X^{\mathcal{W}}}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{E_{X^{\mathcal{W}}}^{\ast}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$ The complex $E^{\ast}(X)$ is a topological vector space with the $C^{\infty}$ topology. With this topology $E^{\ast}(X)$ is a Fréchet topological vector space ([1] III p. 9). Moreover $E^{\ast}_{X^{\mathcal{W}}}(U)$ is a closed subspace. In fact, by [15] V corollaire 1.6, it is the closure of the complex of differential forms that have compact support contained in $U$, that we denote $E^{\ast}_{c}(U)$. More generally, all the monomorphisms in diagram (2.1) are closed immersions. The following result states that, being $U$ an algebraic open subset of $X$, the complex $E^{\ast}_{X^{\mathcal{W}}}(U)$ does not depend on $X$ but only on $U$. ###### Proposition 2.2. Let $\pi:\widetilde{X}\longrightarrow X$ be a proper birational morphism with $D=\pi^{-1}(Y)$, that induces an isomorphism between $\widetilde{X}\setminus D$ and $U$, then the natural map $\pi^{\ast}:E^{\ast}(X)\longrightarrow E^{\ast}(\widetilde{X})$ induces an isomorphism $\pi^{\ast}:\Gamma(X,\mathscr{E}^{\ast}_{X}(\operatorname{flat}Y))\longrightarrow\Gamma(\widetilde{X},\mathscr{E}^{\ast}_{\widetilde{X}}(\operatorname{flat}D))$. ###### Proof. By [14] the morphism $\pi^{\ast}:E^{\ast}(X)\longrightarrow E^{\ast}(\widetilde{X})$ is a closed immersion. Since $\Gamma(X,\mathscr{E}^{\ast}_{X}(\operatorname{flat}Y))$ and $\Gamma(\widetilde{X},\mathscr{E}^{\ast}_{\widetilde{X}}(\operatorname{flat}D))$ are the closure of $E^{\ast}_{c}(U)$ in $E^{\ast}(X)$ and $E^{\ast}(\widetilde{X})$ respectively, then they are identified by $\pi^{\ast}$. ∎ The cohomology of the complex of Whitney forms. By [14] (see also [2] for a more general statement) we have ###### Proposition 2.3. The complex $\mathscr{E}^{\ast}_{Y^{\infty}}$ is a resolution of the constant sheaf $\underline{\mathbb{C}}$ on $Y$ by fine sheaves. Therefore $H^{\ast}(E_{X^{\mathcal{W}}}^{\ast}(W))=H^{\ast}_{c}(W,\mathbb{C}),$ where $H^{\ast}_{c}$ denotes cohomology with compact supports. $\square$ ### 2.2. Currents with support in a subvariety The complex of currents. We first recall the definition of the complex of currents and we fix the sign convention and some normalizations. We will follow the conventions of [6] §5.4 but with the homological grading. Let $\mathscr{D}_{n}^{X}$ be the sheaf of degree $n$ currents on $X$. That is, for any open subset $V$ of $X$, the group $\mathscr{D}_{n}^{X}(V)$ is the topological dual of the group of sections with compact support $E^{n}_{c}(V)$. The differential $\operatorname{d}:\mathscr{D}_{n}^{X}\longrightarrow\mathscr{D}_{n-1}^{X}$ is defined by $\operatorname{d}T(\varphi)=(-1)^{n}T(\operatorname{d}\varphi);$ here $T$ is a current and $\varphi$ a test form. Note that we are using the sign convention of, for instance [11], instead of the sign convention of [9]. The bigrading $\mathscr{E}^{n}_{X}=\bigoplus_{p+q=n}\mathscr{E}^{p,q}_{X}$ induces a bigrading $\mathscr{D}_{n}^{X}=\bigoplus_{p+q=n}\mathscr{D}_{p,q}^{X},$ with $\mathscr{D}_{p,q}^{X}(V)$ the topological dual of $\Gamma_{c}(V,\mathscr{E}^{p,q}_{X})$. The real structure of $\mathscr{E}^{n}_{X}$ induces a real structure $\mathscr{D}_{n}^{X,\mathbb{R}}\subset\mathscr{D}_{n}^{X}.$ We will denote $\mathscr{D}_{n}^{X,\mathbb{R}}(p)=\frac{1}{(2\pi i)^{p}}\mathscr{D}_{n}^{X,\mathbb{R}}\subset\mathscr{D}_{n}^{X}.$ If $X$ is equidimensional of dimension $d$ we will write (2.4) $\mathscr{D}^{n}_{X}=\mathscr{D}_{2d-n}^{X},\quad\mathscr{D}^{p,q}_{X}=\mathscr{D}_{d-p,d-q}^{X},\quad\text{and}\quad\mathscr{D}^{n}_{X,\mathbb{R}}(p)=\mathscr{D}_{2d-n}^{X,\mathbb{R}}(d-p).$ We will use all the conventions of [6] §5.4. In particular, if $y$ is an algebraic cycle of $X$ of dimension $e$, we will write $\delta_{y}\in\mathscr{D}_{e,e}^{X}\cap\mathscr{D}_{2e}^{X,\mathbb{R}}(e)$ for the current $\delta_{y}(\eta)=\frac{1}{(2\pi i)^{e}}\int_{y}\eta.$ Furthermore, there is an action $\begin{matrix}\mathscr{E}^{n}_{X}\otimes\mathscr{D}_{m}^{X}&\longrightarrow&\mathscr{D}_{m-n}^{X},\\\ \omega\otimes T&\longmapsto&\omega\land T\end{matrix}$ where the current $\omega\land T$ is defined by $(\omega\land T)(\eta)=T(\eta\land\omega).$ This action induces actions $\mathscr{E}^{p,q}_{X}\otimes\mathscr{D}_{r,s}^{X}\longrightarrow\mathscr{D}_{r-p,s-q}^{X},\quad\text{and}\quad\mathscr{E}^{n}_{X,\mathbb{R}}(p)\otimes\mathscr{D}_{m}^{X,\mathbb{R}}(q)\longrightarrow\mathscr{D}_{m-n}^{X,\mathbb{R}}(q-p).$ Finally, if $X$ is equidimensional of dimension $d$, there is a fundamental current $\delta_{X}\in\mathscr{D}_{d,d}^{X}\cap\mathscr{D}_{2d}^{X,\mathbb{R}}(d)$, and a morphism (2.5) $\mathscr{E}^{\ast}_{X}\longrightarrow\mathscr{D}_{2d-\ast}^{X}=\mathscr{D}^{\ast}_{X},\quad\omega\longmapsto[\omega]=\omega\land\delta_{X}.$ This morphism sends $\mathscr{E}^{n}_{X\mathbb{R}}(p)$ to $\mathscr{D}_{2d-n}^{X,\mathbb{R}}(d-p)=\mathscr{D}^{n}_{X,\mathbb{R}}(p)$. Currents with support on a subvariety and tempered currents. As in the previous section let $Z\subset Y$ denote two closed subvarieties of $X$ and put $U=X\setminus Y$, $V=X\setminus Z$ and $W=Y\setminus Z$. We denote by $\mathscr{D}_{\ast}^{Y^{\infty}}$ the subcomplex of $\mathscr{D}_{\ast}^{X}$ formed by currents with support on $Y$. In other words, for any open subset $U^{\prime}$ of $X$ we have $\mathscr{D}_{n}^{Y^{\infty}}(U^{\prime})=\\{T\in\mathscr{D}_{n}^{X}(U^{\prime})\mid T(\eta)=0,\ \forall\eta\in\Gamma_{c}(U^{\prime}\cap U,\mathscr{E}^{n}_{X})\\}.$ Observe that, by continuity, the sections of $\mathscr{D}_{n}^{Y^{\infty}}(U^{\prime})$ vanish on the subgroup $\Gamma_{c}(U^{\prime},\mathscr{E}^{\ast}_{X}(\operatorname{flat}Y))$. We write $\mathscr{D}_{n}^{X/Y^{\infty}}=\mathscr{D}_{n}^{X}\left/\mathscr{D}_{n}^{Y^{\infty}}\right.$ and $\mathscr{D}_{n}^{Y^{\infty}/Z^{\infty}}=\mathscr{D}_{n}^{Y^{\infty}}/\mathscr{D}_{n}^{Z^{\infty}}.$ As in the case of differential forms, the complex $\mathscr{D}_{n}^{Y^{\infty}/Z^{\infty}}$ can also be defined as the kernel of the morphism $\mathscr{D}_{n}^{X/Z^{\infty}}\longrightarrow\mathscr{D}_{n}^{X/Y^{\infty}}.$ All the above sheaves inherit a bigrading and a real structure. Observe that, except for the fact that we are using here the homological grading, the complex of sheaves $\mathscr{D}_{n}^{X/Y^{\infty}}$ agrees with the complex denoted by $\mathcal{TH}om(\mathbb{C}_{W},\mathcal{D}b_{X})$ in [12]. The complex $\mathscr{D}_{n}^{Y^{\infty}/Z^{\infty}}$ is a complex of fine sheaves. We will denote the complex of global sections by $D_{\ast}^{X^{\mathcal{T}}}(W^{\infty})=\Gamma(X,\mathscr{D}_{\ast}^{Y^{\infty}/Z^{\infty}})$. Thus the complex $D_{\ast}^{X^{\mathcal{T}}}(W^{\infty})$ is defined for any Zariski locally closed subset $W\subset X$. The corresponding real complex will be denoted by $D_{\ast}^{X^{\mathcal{T}},\mathbb{R}}(W^{\infty})$. By [14], the complex $D_{\ast}^{X^{\mathcal{T}}}(U)$ can be identified with the image of the morphism $D^{\ast}(X)\longrightarrow D^{\ast}(U).$ That is, it is the complex of currents on $U$ that can be extended to a current on the whole $X$. The elements of $D_{\ast}^{X^{\mathcal{T}}}(U)$ will be called tempered currents. In the literature they are called also moderate, temperate or extendable currents. Moreover, as was the case with the complex $E^{\ast}_{X^{\mathcal{W}}}(U)$, being $U$ a Zariski open subset, the complex $D_{\ast}^{X^{\mathcal{T}}}(U)$ only depends on $U$ and not on $X$. The pairing between forms and currents. We have already introduced an action (2.6) $E^{n}(X)\otimes D_{m}(X)\longrightarrow D_{m-n}(X),\quad\omega\otimes T\longmapsto\omega\land T,$ where the current $\omega\land T$ is defined by $(\omega\land T)(\eta)=T(\eta\land\omega).$ The subspace $D_{\ast}^{X^{\mathcal{T}}}(Y)$ is invariant under this action and annihilates the subspace $E_{X^{\mathcal{W}}}^{\ast}(U)$. Therefore we obtain induced actions (2.7) $E^{n}_{X^{\mathcal{W}}}(Y)\otimes D^{X^{\mathcal{T}}}_{m}(Y)\longrightarrow D^{X^{\mathcal{T}}}_{m-n}(Y),\qquad E^{n}_{X^{\mathcal{W}}}(U)\otimes D_{m}^{X^{\mathcal{T}}}(U)\longrightarrow D_{m-n}^{X^{\mathcal{T}}}(U)$ and, more generally, an action (2.8) $E^{n}_{X^{\mathcal{W}}}(W)\otimes D_{m}^{X^{\mathcal{T}}}(W)\longrightarrow D_{m-n}^{X^{\mathcal{T}}}(W).$ Since $X$ is proper, there is a canonical morphism $\deg:D_{0}(X)\longrightarrow\mathbb{C}$ given by $\deg(T)=T(1)$. Observe that $\deg(D_{0}^{\mathbb{R}}(X))\subset\mathbb{R}$. Combining the degree and the above actions, we recover the pairing $E^{n}(X)\otimes D_{n}(X)\longrightarrow\mathbb{C},$ that identifies $D_{n}(X)$ with the topological dual of $E^{n}(X)$. Under this identification, the subspace $E^{n}_{X^{\mathcal{W}}}(U)$ is the orthogonal to the subspace $D_{n}^{X^{\mathcal{T}}}(Y)$. Therefore $D_{n}^{X^{\mathcal{T}}}(U)$ is the topological dual of $E^{n}_{X^{\mathcal{W}}}(U)$ and $D_{n}^{X^{\mathcal{T}}}(Y)$ is the topological dual of $E^{n}_{X^{\mathcal{W}}}(Y)$. More generally $D_{n}^{X^{\mathcal{T}}}(W)$ is the topological dual of $E^{n}_{X^{\mathcal{W}}}(W)$. Note that here, the key point is the fact that $E^{n}_{X^{\mathcal{W}}}(U)$ is the closure of $\Gamma_{c}(U,\mathscr{E}^{n}_{X})$ and hence a closed subspace. The above pairings induce a pairing $E^{n}_{\mathbb{R}}(X)(p)\otimes D_{n}^{\mathbb{R}}(X)(p)\longrightarrow\mathbb{R},$ and similar pairings for the other complexes of forms and currents. Finally, observe that there is a commutative diagram with exact rows and columns (2.9) $\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D^{X^{\mathcal{T}}}_{\ast}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{\ast}^{X^{\mathcal{T}}}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{\ast}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{\ast}^{X^{\mathcal{T}}}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{\ast}^{X^{\mathcal{T}}}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{\ast}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{\ast}^{X^{\mathcal{T}}}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{D^{X^{\mathcal{T}}}_{\ast}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$ that is the topological dual of the diagram (2.1). The homology of the complexes of currents. By [14] we have ###### Proposition 2.10. The homology of the complexes $D_{\ast}^{X^{\mathcal{T}}}(W)$ is given by $H_{\ast}(D^{X^{\mathcal{T}}}_{\ast}(W))=H_{\ast}^{BM}(W,\mathbb{C}),$ where $H_{\ast}^{BM}$ denote Borel-Moore homology. In particular, since we are assuming $Y$ proper, $H_{\ast}(D_{\ast}^{X^{\mathcal{T}}}(Y))=H_{\ast}(Y,\mathbb{C}).$ $\square$ ### 2.3. Formal and tempered Deligne cohomology Formal Deligne cohomology. The complex $E^{\ast}_{X^{\mathcal{W}},\mathbb{R}}(W)$ is an example of a Dolbeault algebra (see [6]). Recall that, following Deligne, the cohomology of any complex variety has a mixed Hodge structure. We will call the Hodge filtration of this mixed Hodge structure the Deligne-Hodge filtration. From the structure of Dolbeault algebra of $E^{\ast}_{X^{\mathcal{W}}}(W)$ we can define a Hodge filtration. It is the filtration associated to the bigrading. In general, this Hodge filtration does not induce the Deligne-Hodge filtration in cohomology. Moreover, the spectral sequence associated to this Hodge filtration does not need to degenerate at the $E_{1}$ term. Therefore, the Dolbeault cohomology groups $H^{p,q}_{\overline{\partial}}(E^{\ast}(Y^{\infty}))$ are not, in general, direct summands of $H^{p+q}(Y,\mathbb{C})$. In fact, they can be infinite dimensional as can be seen in the easiest example: Put $X=\mathbb{P}^{1}_{\mathbb{C}}$. Let $t$ be the absolute coordinate and let $Y$ be the point $t=0$. Then $H^{0,0}_{\bar{\partial}}(E^{\ast}(Y^{\infty}))=\mathbb{C}[[t]]$, the ring of formal power series in one variable. Following [4] and [6], to every Dolbeault algebra we can associate a Deligne algebra. We refer the reader to [4] and [6] §5 for the definition and properties of Dolbeault algebras, Dolbeault complexes and the associated Deligne complexes. We will use freely the notation therein. In particular the Deligne algebra associated to the above Dolbeault algebra will be denoted $\mathcal{D}^{\ast}(E^{\ast}_{X^{\mathcal{W}}}(W),\ast)$. ###### Definition 2.11. The real formal Deligne cohomology of $W$ (with compact supports) is defined by $\displaystyle H_{\mathcal{D}^{f},c}^{\ast}(W^{\infty},\mathbb{R}(p))$ $\displaystyle=H^{\ast}(\mathcal{D}^{\ast}(E_{X^{\mathcal{W}}}(W),p)).$ When $W$ is proper we will just write $H_{\mathcal{D}^{f}}^{\ast}(W^{\infty},\mathbb{R}(p))$. The notation $W^{\infty}$ is a reminder that this cohomology depends, not only on $W$ but on an infinitesimal neighborhood of infinite order of $W$ in $X$. ###### Remark 2.12. Since we are assuming that $X$ is smooth and proper, the formal Deligne cohomology of $X$, $H_{\mathcal{D}^{f}}^{\ast}(X^{\infty},\mathbb{R}(p))$, given in the previous definition, agrees with the usual Deligne cohomology of $X$. Nevertheless, by the discussion before the definition, the formal Deligne cohomology with compact supports of $U$ or the formal Deligne cohomology of $Y$, do not agree, in general, with the usual Deligne-Beilinson cohomology. For instance the groups $H_{\mathcal{D}^{f}}^{\ast}(U,\mathbb{R}(p))$ can be infinite dimensional. Homological Dolbeault complexes and homological Deligne complexes. In order to define formal Deligne homology we first translate the notions of [6] §5.2 to the homological grading. ###### Definition 2.13. A _homological Dolbeault complex_ $A=(A_{\ast}^{\mathbb{R}},\operatorname{d}_{A})$ is a graded complex of real vector spaces, which is bounded from above and equipped with a bigrading on $A^{\mathbb{C}}=A^{\mathbb{R}}\otimes_{\mathbb{R}}{\mathbb{C}}$, i.e., $A_{n}^{\mathbb{C}}=\bigoplus_{p+q=n}A_{p,q},$ satisfying the following properties: 1. (i) The differential $\operatorname{d}_{A}$ can be decomposed as the sum $\operatorname{d}_{A}=\partial+\bar{\partial}$ of operators $\partial$ of type $(-1,0)$, resp. $\bar{\partial}$ of type $(0,-1)$. 2. (ii) It satisfies the symmetry property $\overline{A_{p,q}}=A_{q,p}$, where $\overline{\phantom{M}}$ denotes complex conjugation. ###### Notation 2.14. Given a homological Dolbeault complex $A=(A_{\ast}^{\mathbb{R}},\operatorname{d}_{A})$, we will use the following notations. The Hodge filtration $F$ of $A$ is the increasing filtration of $A^{\mathbb{C}}_{\ast}$ given by $F_{p}A_{n}=F_{p}A_{n}^{\mathbb{C}}=\bigoplus_{p^{\prime}\leq p}A_{p^{\prime},n-p^{\prime}}.$ The filtration $\overline{F}$ of $A$ is the complex conjugate of $F$, i.e., $\overline{F}_{p}A_{n}=\overline{F}_{p}A_{n}^{\mathbb{C}}=\overline{F_{p}A_{n}^{\mathbb{C}}}.$ For an element $x\in A^{\mathbb{C}}$, we write $x_{i,j}$ for its component in $A_{i,j}$. For $k,k^{\prime}\in\mathbb{Z}$, we define an operator $F_{k,k^{\prime}}:A^{\mathbb{C}}\longrightarrow A^{\mathbb{C}}$ by the rule $F_{k,k^{\prime}}(x):=\sum_{l\leq k,l^{\prime}\leq k^{\prime}}x_{l,l^{\prime}}.$ We note that the operator $F_{k,k^{\prime}}$ is the projection of $A^{\ast}_{\mathbb{C}}$ onto the subspace $F_{k}A_{\ast}\cap\overline{F}_{k^{\prime}}A_{\ast}$. This subspace will be denoted $F_{k,k^{\prime}}A_{\ast}$. We will also denote by $F_{k}$ the operator $F_{k,\infty}$. We denote by $A_{n}^{\mathbb{R}}(p)$ the subgroup $(2\pi i)^{-p}\cdot A_{n}^{\mathbb{R}}\subseteq A_{n}^{\mathbb{C}}$, and we define the operator $\pi_{p}:A^{\mathbb{C}}\longrightarrow A^{\mathbb{R}}(p)$ by setting $\pi_{p}(x):=\frac{1}{2}(x+(-1)^{p}\bar{x})$. To any homological Dolbeault complex we can associate a homological Deligne complex. ###### Definition 2.15. Let $A$ be a homological Dolbeault complex. We denote by $A_{\ast}(p)^{\mathcal{D}}$ the complex $s(A^{\mathbb{R}}(p)\oplus F_{p}A\overset{u}{\longrightarrow}A^{\mathbb{C}})$, where $u(a,f)=-a+f$ and $s(\ )$ denotes the simple complex of a morphism of complexes. ###### Definition 2.16. Let $A$ be a homological Dolbeault complex. Then, the _(homological) Deligne complex $(\mathcal{D}^{\ast}(A,\ast),\operatorname{d}_{\mathcal{D}})$ associated to $A$_ is the graded complex given by $\displaystyle\mathcal{D}_{n}(A,p)=\begin{cases}A^{\mathbb{R}}_{n+1}(p+1)\cap F_{n-p,n-p}A_{n+1}^{\mathbb{C}},&\qquad\text{if}\quad n\geq 2e+1,\\\ A^{\mathbb{R}}_{n}(p)\cap F_{p,p}A_{n}^{\mathbb{C}},&\qquad\text{if}\quad n\leq 2p,\end{cases}$ with differential given, for $x\in\mathcal{D}_{n}(A,p)$, by $\displaystyle\operatorname{d}_{\mathcal{D}}x=\begin{cases}-F_{n-p+1,n-p+1}\operatorname{d}_{A}x,&\qquad\text{if}\quad n>2p+1,\\\ -2\partial\bar{\partial}x,&\qquad\text{if}\quad n=2p+1,\\\ \operatorname{d}_{A}x,&\qquad\text{if}\quad n\leq 2p.\end{cases}$ For instance, let $A$ be a Dolbeault complex satisfying $A_{p,q}=0$ for $p<0$, $q<0$, $p>n$, or $q>n$. Then, for $p\geq n$, the complex $\mathcal{D}(A,p)$ agrees with the real complex $A_{\ast}^{\mathbb{R}}(p)$. For $0\leq p<n$, we have represented $\mathcal{D}(A,p)$ in figure 1, where the upper right square is shifted by one; this means in particular that $A_{n,n}$ sits in degree $2n-1$ and $A_{p+1,p+1}$ sits in degree $2p+1$. For $p<0$ the complex $\mathcal{D}(A,p)$ agrees with the real complex $A_{\ast}^{\mathbb{R}}(p+1)[1]$. $\textstyle{\makebox{$\begin{pmatrix}A_{p+1,n}&\leftarrow&\cdots&\leftarrow&A_{n,n}\\\ \downarrow&&&&\downarrow\\\ \vdots&&&&\vdots\\\ \downarrow&&&&\downarrow\\\ A_{p+1,p+1}&\leftarrow&\cdots&\leftarrow&A_{n,p+1}\end{pmatrix}$}_{\makebox[0.0pt]{$\mathbb{R}$}}\makebox[0.0pt][l]{$(p+1)$}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-2\partial\overline{\partial}}$$\textstyle{\makebox{$\begin{pmatrix}A_{0,p}&\leftarrow&\cdots&\leftarrow&A_{p,p}\\\ \downarrow&&&&\downarrow\\\ \vdots&&&&\vdots\\\ \downarrow&&&&\downarrow\\\ A_{0,0}&\leftarrow&\cdots&\leftarrow&A_{p,0}\end{pmatrix}$}_{\makebox[0.0pt]{$\mathbb{R}$}}\makebox[0.0pt][l]{$(p)$}}$ Figure 1. $\mathcal{D}(A,p)$ ###### Remark 2.17. It is clear from the definition that, for all $p\in\mathbb{Z}$, the functor $\mathcal{D}(\cdot,p)$ is exact. The main property of the Deligne complex is expressed by the following proposition; for a proof in the cohomological case see [4]. ###### Proposition 2.18. The complexes $A_{\ast}(p)^{\mathcal{D}}$ and $\mathcal{D}_{\ast}(A,p)$ are homotopically equivalent. The homotopy equivalences $\psi:A_{n}(p)^{\mathcal{D}}\longrightarrow\mathcal{D}_{n}(A,p)$, and $\varphi:\mathcal{D}_{n}(A,p)\longrightarrow A_{n}(p)^{\mathcal{D}}$ are given by $\psi(a,f,\omega)=\begin{cases}\pi(\omega),\qquad&\text{if }n\geq 2p+1,\\\ F_{p,p}a+2\pi_{p}(\partial\omega_{p+1,n-p-1}),\quad&\text{if }n\leq 2p,\end{cases}$ where $\pi(\omega)=\pi_{p+1}(F_{n-p,n-p}\omega)$, i.e., $\pi$ is the projection of $A_{\mathbb{C}}$ over the cokernel of $u$, and $\varphi(x)=\begin{cases}(\partial x_{p+1,n-p}-\bar{\partial}x_{n-p,p+1},2\partial x_{p+1,n-p},x),\quad&\text{if }n\geq 2p+1,\\\ (x,x,0),&\text{if }n\leq 2p.\end{cases}$ Moreover, $\psi\circ\varphi=\operatorname{id}$, and $\varphi\circ\psi-\operatorname{id}=\operatorname{d}h+h\operatorname{d}$, where $h:A_{n}(p)^{\mathcal{D}}\longrightarrow A_{n+1}(p)^{\mathcal{D}}$ is given by $h(a,f,\omega)=\begin{cases}(\pi_{p}(\overline{F}_{p}\omega+\overline{F}_{n-p}\omega),-2F_{p}(\pi_{p+1}\omega),0),\quad&\text{if }n\geq 2p+1,\\\ (2\pi_{p}(\overline{F}_{n-p}\omega),-F_{p,p}\omega-2F_{n-p}(\pi_{p+1}\omega),0),\quad&\text{if }n\leq 2p.\end{cases}$ $\square$ Tempered Deligne homology. Applying the above discussion to the complex of currents $D_{\ast}^{X^{\mathcal{T}},\mathbb{R}}(W)$ we define the homological Deligne complex $\mathcal{D}_{\ast}(D_{\ast}^{X^{\mathcal{T}}}(W),\ast).$ ###### Definition 2.19. _The tempered Deligne (Borel-Moore) homology of $W$_ is defined by $H^{\mathcal{D}^{\mathcal{T}}}_{\ast}(W^{\infty},\mathbb{R}(p))=H_{\ast}(\mathcal{D}_{\ast}(D^{X^{\mathcal{T}}}_{\ast}(W),p)).$ ###### Remark 2.20. 1. (i) Again, since $X$ is smooth and proper, the tempered Deligne homology of $X$ agrees with the Deligne homology of $X$. In particular, the group $H^{\mathcal{D}}_{n}(X,\mathbb{R}(p))$ agrees with the group denoted ${}^{\prime}H^{-n}_{\mathcal{D}}(X,\mathbb{R}(-p))$ in [11]. But, since the Hodge filtration of the complex of currents with support on $Y$ does not induce the Deligne-Hodge filtration in the homology of $Y$, the tempered Deligne homology does not agree in general with Deligne-Beilinson homology. 2. (ii) As in the case of formal cohomology, the notation $H^{\mathcal{D}^{\mathcal{T}}}_{\ast}(W^{\infty},\mathbb{R}(p))$ reminds us that these groups do not depend only on $W$ but on an infinitesimal neighborhood of $W$ of infinite order. Equidimensional manifolds. If $X$ is equidimensional of dimension $d$ the morphism (2.5) induces morphisms (2.21) $\mathcal{D}^{n}(E^{\ast}(X),p)\longrightarrow\mathcal{D}_{2d-n}(D_{\ast}(X),d-p),\quad p\in\mathbb{Z},$ that, in turn, induce the Poincaré duality isomorphisms (2.22) $H^{n}_{\mathcal{D}}(X,\mathbb{R}(p))\longrightarrow H_{2d-n}^{\mathcal{D}}(X,\mathbb{R}(d-p)),\quad n,p\in\mathbb{Z}.$ By analogy, we can define tempered Deligne cohomology groups as follows $\displaystyle H_{\mathcal{D}^{\mathcal{T}}}^{n}(U,\mathbb{R}(p))$ $\displaystyle=H^{\mathcal{D}^{\mathcal{T}}}_{2d-n}(U,\mathbb{R}(d-p)),$ $\displaystyle H_{\mathcal{D}^{\mathcal{T}},W}^{n}(V,\mathbb{R}(p))$ $\displaystyle=H^{\mathcal{D}^{\mathcal{T}}}_{2d-n}(W^{\infty},\mathbb{R}(d-p)).$ In general, if $X$ is a disjoint union of equidimensional algebraic manifolds, then we define the tempered Deligne cohomology of $X$ as the direct sum of the tempered Deligne cohomology of its components. The module structure of tempered Deligne homology. The notion of Dolbeault module over a Dolbeault algebra introduced in [6] can be easily modified to define homological Dolbeault modules over a Dolbeault algebra. The actions (2.6), (2.7) and (2.8) provide the basic examples. Modifying the construction of [6] 5.17 and 5.18 we obtain ###### Proposition 2.23. There is a pseudo-associative action $\mathcal{D}^{n}(E_{X^{\mathcal{W}}}(W),p)\otimes\mathcal{D}_{m}(D_{\ast}^{X^{\mathcal{T}}}(W),q)\longrightarrow\mathcal{D}_{m-n}(D_{\ast}^{X^{\mathcal{T}}}(W),q-p)$ that induces an associative action $H_{\mathcal{D}^{f},c}^{n}(W^{\infty},\mathbb{R}(p))\otimes H^{\mathcal{D}^{\mathcal{T}}}_{m}(W^{\infty},\mathbb{R}(q))\longrightarrow H^{\mathcal{D}^{\mathcal{T}}}_{m-n}(W^{\infty},\mathbb{R}(q-p)).$ $\square$ The exceptional duality. In general, Poincaré duality for Deligne cohomology is not given by a bilinear pairing, but by the isomorphism (2.22) between Deligne cohomology and Deligne homology (see for instance [11]). Nevertheless, in the case of real Deligne cohomology, there is an exceptional duality that comes from the symmetry of the Deligne complex associated with a Dolbeault complex. This duality can be generalized to a pairing between formal Deligne cohomology and tempered Deligne homology. ###### Proposition 2.24. For every pair of integers $n,p$, there is a pairing $\mathcal{D}^{n}(E_{X^{\mathcal{W}}}(W),p)\otimes\mathcal{D}_{n-1}(D^{X^{\mathcal{T}}}(W),p-1)\longrightarrow\mathbb{R}$ given by $\omega\otimes T\longmapsto T(\omega)$. This pairing identifies $\mathcal{D}_{n-1}(D^{X^{\mathcal{T}}}(W),p-1)$ with the topological dual of $\mathcal{D}^{n}(E_{X_{\mathcal{W}}}(W),p)$. Moreover, it is compatible, up to the sign, with the differential in the Deligne complex: $T(\operatorname{d}_{\mathcal{D}}\omega)=\begin{cases}(-1)^{n+1}(\operatorname{d}_{\mathcal{D}}T)(\omega),&\text{ if }n\leq 2p-1,\\\ (-1)^{n}(\operatorname{d}_{\mathcal{D}}T)(\omega),&\text{ if }n\geq 2p.\\\ \end{cases}$ It is also compatible, up to the sign, with the action of $\mathcal{D}^{\ast}(E_{X^{\mathcal{W}}}(W),\ast)$. That is, if the forms $\omega\in\mathcal{D}^{n}(E_{X^{\mathcal{W}}}(W^{\infty}),p)$ and $\eta\in\mathcal{D}^{l}(E_{X^{\mathcal{W}}}(W),r)$, and the current $T\in\mathcal{D}_{m}(D^{X^{\mathcal{T}}}(W),q)$, with $n-m+l=1$ and $p-q+r=1$ then $(\omega\bullet T)(\eta)=\begin{cases}(-1)^{n}T(\eta\bullet\omega),&\text{ if }m>2q,\ l\geq 2r,\\\ T(\eta\bullet\omega),&\text{ if }m\leq 2q,\ l<2r,\\\ (-1)^{m-1}T(\eta\bullet\omega),&\text{ if }m>2q,\ l<2r,\\\ (-1)^{l}T(\eta\bullet\omega),&\text{ if }m\leq 2q,\ l\geq 2r.\\\ \end{cases}$ ###### Proof. Assume that $n<2p$. Put $q=p-1$ and $m=n-1$. Then $\displaystyle\mathcal{D}^{n}(E_{X^{\mathcal{W}}}(W),p)$ $\displaystyle\phantom{A}=E^{n-1}_{X^{\mathcal{W}},\mathbb{R}}(W)(p-1)\left/(F^{p}E^{n-1}_{X^{\mathcal{W}}}(W)+\bar{F}^{p}E^{n-1}_{X^{\mathcal{W}}}(W))\cap E^{n-1}_{X^{\mathcal{W}}}(W)_{\mathbb{R}}(p-1)\right.$ $\displaystyle\phantom{A}=E^{n-1}_{X^{\mathcal{W}},\mathbb{R}}(W)(p-1)\cap\bar{F}^{n-p}E^{n-1}_{X^{\mathcal{W}}}(W))\cap F^{n-p}E^{n-1}_{X^{\mathcal{W}}}(W),$ $\displaystyle\mathcal{D}_{m}(D^{X^{\mathcal{T}}}(W),q)$ $\displaystyle\phantom{A}=D_{m}^{X^{\mathcal{T}},\mathbb{R}}(W^{\infty})(q)\cap F_{q}D_{m}^{X^{\mathcal{T}}}(W)\cap\bar{F}_{q}D_{m}^{X^{\mathcal{T}}}(W))$ $\displaystyle\phantom{A}=D_{n-1}^{X^{\mathcal{T}},\mathbb{R}}(W)(p-1)\cap F_{p-1}D_{n-1}^{X^{\mathcal{T}}}(W)\cap\bar{F}_{p-1}D_{n-1}^{X^{\mathcal{T}}}(W)).$ Therefore, the first statement follows from the duality between $E_{X^{\mathcal{W}}}(W)$ and $D^{X^{\mathcal{T}}}(W)$ and the fact that, under this duality, $D_{n-1}^{X^{\mathcal{T}},\mathbb{R}}(W)(p-1)$ is identified with the dual of $E^{n-1}_{X^{\mathcal{W}},\mathbb{R}}(W)(p-1)$ and $F_{p-1}D_{n-1}^{X^{\mathcal{T}}}(W)$ is identified with the dual of $\bar{F}^{n-p}E^{n-1}_{X^{\mathcal{T}}}(W)$. The compatibility with the differential is a straightforward computation using the formulas for the differential given in [4] theorem 2.6. For instance, if $\omega\in\mathcal{D}^{n}(E_{X^{\mathcal{W}}}(W),p)$, with $n<2p-1$ and $T\in\mathcal{D}_{m}(D^{X^{\mathcal{T}}}(W),q)$, with $m=n$ and $q=p-1$, then we have $\displaystyle(\operatorname{d}_{\mathcal{D}}T)(\omega)$ $\displaystyle=(\operatorname{d}T)(\omega)$ $\displaystyle=(-1)^{n}T(\operatorname{d}\omega)$ $\displaystyle=(-1)^{n}T(F^{n-p+1,n-p+1}\operatorname{d}\omega)$ $\displaystyle=(-1)^{n}T(-\operatorname{d}_{\mathcal{D}}\omega).$ In the third equality we have used that $T\in F_{q}\cap\bar{F}_{q}=F_{p-1,p-1}$, which implies that, for any form $\eta$, we have $T(\eta)=T(F^{n-p+1,n-p+1}\eta)$. The other cases are analogous. Similarly, the compatibility with the product follows from [4] theorem 2.6. For instance, let $\omega\in\mathcal{D}^{n}(E_{X^{\mathcal{W}}}(W),p)$, $T\in\mathcal{D}_{m}(D^{X^{\mathcal{T}}}(W),q)$ and $\eta\in\mathcal{D}^{l}(E_{X^{\mathcal{W}}}(W),r)$, with $n-m+l=1$ and $p-q+r=1$. Assume that $n<2p$, $m>2q$, $l\geq 2r$, then $(\omega\bullet T)(\eta)=((-1)^{n}r_{p}(\omega)\land T+\omega\land r_{q}(T))(\eta),$ where $r_{p}(\omega)=2\pi_{p}(F^{p}\operatorname{d}\omega)$ and $r_{q}(T)=2\pi_{q}(F_{q}\operatorname{d}T)$. But $(-1)^{n}r_{p}(\omega)\land T(\eta)=(-1)^{n}T(\eta\land r_{p}(\omega)),$ and $\displaystyle(\omega\land r_{q}(T))(\eta)$ $\displaystyle=r_{q}(T)(\eta\land\omega)$ $\displaystyle=2\pi_{q}F_{q}(\operatorname{d}T)(\eta\land\omega)$ $\displaystyle=2F_{q}(\operatorname{d}T)(\eta\land\omega)$ $\displaystyle=2\partial T_{q+1,m-q}(\eta\land\omega)$ $\displaystyle=T\left(2(-1)^{m-1}\partial(\eta\land\omega)^{q,m-q}\right)$ $\displaystyle=T\left(2(-1)^{n+l}\partial(\eta\land\omega)^{p+r-1,n+l-p-r}\right).$ On the other hand $T(\eta\bullet\omega)=T\left(\eta\land r_{p}(\omega)+(-1)^{l}2\partial(\omega\land\eta)^{p+r-1,n+l-p-r}\right).$ The other cases are analogous. ∎ Duality. We summarize in the next proposition the basic properties of formal Deligne cohomology and tempered Deligne homology that follow from the previous discussions. ###### Proposition 2.25. For every pair of integers $n$ and $p$, by applying the exact functors $\mathcal{D}^{\ast}(\underline{\phantom{A}},p)$ and $\mathcal{D}_{\ast}(\underline{\phantom{A}},p-1)$ to the diagrams (2.1) and (2.9) respectively, we obtain the corresponding diagrams of Deligne complexes that are the topological dual of each other. In particular we obtain long exact sequences (2.26) $H^{n}_{\mathcal{D}^{f},c}(W^{\infty},\mathbb{R}(p))\rightarrow H^{n}_{\mathcal{D}^{f}}(Y^{\infty},\mathbb{R}(p))\rightarrow H^{n}_{\mathcal{D}^{f}}(Z^{\infty},\mathbb{R}(p))\rightarrow\\\ H^{n+1}_{\mathcal{D}^{f},c}(W^{\infty},\mathbb{R}(p))\rightarrow$ and (2.27) $\leftarrow H_{n-1}^{\mathcal{D}^{\mathcal{T}}}(W^{\infty},\mathbb{R}(p-1))\leftarrow H_{n-1}^{\mathcal{D}^{\mathcal{T}}}(Y^{\infty},\mathbb{R}(p-1))\leftarrow\\\ H_{n-1}^{\mathcal{D}^{\mathcal{T}}}(Z^{\infty},\mathbb{R}(p-1))\leftarrow H_{n}^{\mathcal{D}^{\mathcal{T}}}(W^{\infty},\mathbb{R}(p-1))$ and pairings $\displaystyle H^{n}_{\mathcal{D}^{f}}(Y^{\infty},\mathbb{R}(p))\otimes H_{n-1}^{\mathcal{D}^{\mathcal{T}}}(Y^{\infty},\mathbb{R}(p-1))$ $\displaystyle\longrightarrow\mathbb{R},$ $\displaystyle H^{n}_{\mathcal{D}^{f},c}(W^{\infty},\mathbb{R}(p))\otimes H_{n-1}^{\mathcal{D}^{\mathcal{T}}}(W^{\infty},\mathbb{R}(p-1))$ $\displaystyle\longrightarrow\mathbb{R},$ $\displaystyle H^{n}_{\mathcal{D}^{f}}(Z^{\infty},\mathbb{R}(p))\otimes H_{n-1}^{\mathcal{D}^{\mathcal{T}}}(Z^{\infty},p-1)$ $\displaystyle\longrightarrow\mathbb{R}.$ that are compatible with the above sequences. Moreover, the topologies of the space of differential forms and of the space of currents induce structures of topological vector spaces on the real formal Deligne cohomology groups and the tempered Deligne homology groups. The above pairings induce a perfect pairing of the corresponding separated vector spaces. ###### Proof. This is a direct consequence of the exactness of the functors $\mathcal{D}^{\ast}(\underline{\phantom{A}},p)$ and $\mathcal{D}_{\ast}(\underline{\phantom{A}},p-1)$ and proposition 2.24. ∎ The image of $\operatorname{d}_{\mathcal{D}}$ in the complex $\mathcal{D}^{\ast}(E_{\flat}(U),p)$ does not need to be closed. Therefore the pairing between formal cohomology and tempered homology do not need to be perfect. Only the induced pairing in the corresponding separated vector spaces is perfect. Nevertheless, in the case of a proper algebraic complex manifold $X$, by Hodge theory, we obtain a perfect pairing between Deligne-Beilinson cohomology and homology. ###### Corollary 2.28 (Exceptional duality for Deligne cohomology). Let $X$ be a proper complex algebraic manifold, equidimensional of dimension $d$. Then there is a perfect duality $H_{\mathcal{D}}^{n}(X,\mathbb{R}(p))\otimes H_{\mathcal{D}}^{2d-n+1}(X,\mathbb{R}(d-p+1))\longrightarrow\mathbb{R}$ which is compatible, up to a sign, with the product in Deligne cohomology. ###### Proof. By Poincaré duality in Deligne cohomology (cf. [11] 1.5) there is a natural isomorphism $H_{\mathcal{D}}^{2d-n+1}(X,\mathbb{R}(d-p+1))\cong H^{\mathcal{D}}_{n-1}(X,\mathbb{R}(p-1)).$ By Hodge theory we know that $H_{\mathcal{D}}^{n}(X,\mathbb{R}(p))=\begin{cases}H^{n-1}(X,\mathbb{R}(p-1))\cap\overline{F}^{n-p}\cap F^{n-p},&\text{ if }n<2p,\\\ H^{n}(X,\mathbb{R}(p))\cap\overline{F}^{p}\cap F^{p},&\text{ if }n\geq 2p.\end{cases}$ Moreover, the pairing is given, up to a sign, by the wedge product of differential forms followed by the integral along $X$. Therefore, by Serre’s duality, the pairing of proposition 2.25 is perfect. ∎ ### 2.4. Semi-purity of tempered Deligne cohomology Vanishing theorems. The aim of this section is to prove the following result ###### Theorem 2.29. (Semi-purity of tempered Deligne homology) Let $X$ be a projective complex algebraic manifold, $W$ a locally closed subvariety, of dimension at most $p$. Then $H_{n}^{\mathcal{D}^{\mathcal{T}}}(W^{\infty},\mathbb{R}(e))=0,\text{ for all }n>\max(e+p,2p-1).$ ###### Proof. We will prove the result by ascending induction over $p$. The result is trivially true for $p<0$. Then, by the exact sequence (2.27) and induction, one is reduced to the case $W$ closed. We will deduce the theorem by duality from the following proposition ###### Proposition 2.30. Let $Y$ be a closed subvariety of a projective complex algebraic manifold. Let $p$ be the dimension of $Y$. Then $H^{n+1}_{\mathcal{D}^{f}}(Y^{\infty},\mathbb{R}(e+1))=0,\text{ for all }n>max(e+p,2p-1)$ ###### Proof. Let $\mathscr{I}_{Y}$ be the ideal of holomorphic functions on $X$ vanishing at $Y$. We denote $\Omega^{q}_{Y^{\infty}}=\lim_{\begin{subarray}{c}\longleftarrow\\\ k\end{subarray}}\Omega^{q}_{X}\left/\mathscr{I}_{Y}^{k}\Omega^{q}_{X}.\right.$ By [12] theorem 5.12 we have ###### Lemma 2.31. The complex of sheaves $\mathscr{E}^{q,\ast}_{Y,\mathbb{R}}$ is a fine resolution of $\Omega^{q}_{Y^{\infty}}$. Since, by [14], the sheaf $\mathscr{E}^{\ast}_{Y^{\infty},\mathbb{R}}$ is an acyclic resolution of the constant sheaf $\underline{\mathbb{R}}_{Y}$, from lemma 2.31 and the techniques of [4], we deduce that $H^{\ast}_{\mathcal{D}^{f}}(Y^{\infty},\mathbb{R}(e+1))$ is isomorphic to the hypercohomology of the complex of sheaves (2.32) $\underline{\mathbb{R}}_{\mathcal{D}^{f},Y^{\infty}}(e):=\underline{\mathbb{R}}_{Y}(f)\longrightarrow\Omega^{0}_{Y^{\infty}}\longrightarrow\dots\longrightarrow\Omega^{e}_{Y^{\infty}}.$ ###### Lemma 2.33. If $n>p$ then $H^{n}(Y,\Omega_{Y^{\infty}}^{q})=0$. ###### Proof. By [10] proposition I.6.1 $H^{n}(Y,\Omega_{Y^{\infty}}^{q})=H^{n}(Y^{\text{{\rm alg}}},\hat{\Omega}_{Y}^{q}),$ where $Y^{\text{{\rm alg}}}$ is the corresponding algebraic variety and $\hat{\Omega}_{Y}^{q}$ is the completion of the sheaf of algebraic differentials. But now $Y^{\text{{\rm alg}}}$ is a noetherian topological space of dimension $p$, hence the lemma. ∎ Using lemma 2.33 we obtain that the $E_{1}^{s,t}$ term of the spectral sequence of the hypercohomology of the complex (2.32) can be non zero only for $s=0$, $0\leq t\leq 2p$ and $1\leq s\leq e+1$, $0\leq t\leq p$, which implies proposition 2.30. ∎ We finish now the proof of the theorem. By proposition 2.30, for every $n>\max(p+e,2p-1)$, the morphism $\operatorname{d}_{\mathcal{D}}^{n}:\mathcal{D}^{n}(E_{X^{\mathcal{W}}}(Y),e+1)\longrightarrow\mathcal{D}^{n+1}(E_{X^{\mathcal{W}}}(Y),e+1)$ satisfies $\operatorname{Im}(\operatorname{d}_{\mathcal{D}}^{n})=\operatorname{Ker}(\operatorname{d}_{\mathcal{D}}^{n+1})$, hence the image of $\operatorname{d}_{\mathcal{D}}^{n}$ is a closed subspace. Therefore, by [1] IV.2 theorem 1, we have that the dual morphism $\operatorname{d}_{\mathcal{D}}:\mathcal{D}_{n}(D^{X^{\mathcal{T}}}(Y),e)\longrightarrow\mathcal{D}_{n-1}(D^{X^{\mathcal{T}}}(Y),e)$ has closed image. This implies that, for $n\geq\max(p+e,2p-1)$, the vector space $H_{n}^{\mathcal{D}^{\mathcal{T}}}(Y^{\infty},\mathbb{R}(e))$ is separated. Therefore, by proposition 2.25, for $n>\max(p+e,2p-1)$ the pairing $H^{n+1}_{\mathcal{D}^{f}}(Y^{\infty},\mathbb{R}(e+1))\otimes H_{n}^{\mathcal{D}^{\mathcal{T}}}(Y^{\infty},\mathbb{R}(e))\longrightarrow\mathbb{R}$ is perfect. Hence by proposition 2.30 we obtain the theorem. ∎ semi-purity of tempered Deligne cohomology. The semi-purity theorem can be stated in terms of tempered Deligne cohomology as follows. ###### Corollary 2.34. Let $X$ be a complex quasi-projective manifold and $Y$ a closed subvariety of codimension at least $p$. Then $H^{n}_{\mathcal{D}^{\mathcal{T}},Y}(X,\mathbb{R}(e))=0,\text{ for all }n<\min(e+p,2p+1),$ In particular $H^{n}_{\mathcal{D}^{\mathcal{T}},Y}(X,\mathbb{R}(p))=0,\text{ for all }n<2p.$ This is the weak purity property used in [6] 6.4. ## 3\. Arithmetic Intersection Theory ### 3.1. Definition of Covariant arithmetic Chow groups In [3], the author introduced a variant of the arithmetic Chow groups that are covariant with respect to arbitrary proper morphisms. In the paper [6] these groups are further studied as an example of cohomological arithmetic Chow groups. These groups are denoted by $\operatorname{\widehat{CH}}^{\ast}(X,\mathcal{D}_{\text{{\rm cur}}})$. The semi-purity property (corollary 2.34) was announced in [6] and has consequences in the behavior of the covariant arithmetic Chow groups. On the other hand, Kawaguchi and Moriwaki [13] have given another definition of covariant arithmetic Chow groups called $D$-arithmetic Chow groups. A consequence of Corollary 2.34 is that, when $X$ is equidimensional and generically projective, both definitions of covariant arithmetic Chow groups agree. We note that Zha [16] has also introduced a notion of covariant arithmetic Chow groups that only differs from the definition of [13] on the fact that he neglects the anti-linear involution $F^{\infty}$. In this section we will summarize the properties of the covariant arithmetic Chow groups. We will follow the notations and terminology of [6], but we will use the grading by dimension that is more natural when dealing with covariant Chow groups. Arithmetic rings and arithmetic varieties. Let $A$ be an arithmetic ring (see [7]) with fraction field $F$. In particular $A$ is provided with a non empty set of complex embeddings $\Sigma$ and a conjugate linear involution $F_{\infty}$ of $\mathbb{C}^{\Sigma}$ that commutes with the diagonal embedding of $A$ in $\mathbb{C}^{\Sigma}$. Since we will be working with dimension of cycles, following [8] we will further impose that $A$ is equicodimensional and Jacobson. Let $S=\operatorname{Spec}A$ and let $e=\dim S$. An arithmetic variety $X$ is a flat quasi-projective scheme over $A$, that has smooth generic fiber $X_{F}$. To every arithmetic variety $X$ we can associate a complex algebraic manifold $X_{\Sigma}$ and a real algebraic manifold $X_{\mathbb{R}}=(X_{\Sigma},F_{\infty})$. The arithmetic complex of tempered Deligne homology. To every pair of integers $n,p$, and every open Zariski subset $U$ of $X_{\mathbb{R}}$ we assign the group $\mathcal{D}_{n}^{\text{{\rm cur}},X}(U,p)=\mathcal{D}_{n}\left(D_{\ast}^{X_{\Sigma}^{\mathcal{T}}}(U),p\right)^{\sigma},$ where $\sigma$ is the involution that acts as complex conjugation on the space and on the currents. That is, if $T\in D_{n}(X_{\mathbb{C}})$ then $\sigma(T)=\overline{(F_{\infty})_{\ast}T}$. And $(\phantom{A})^{\sigma}$ denote the elements that are fixed by $\sigma$. Then $\mathcal{D}_{n}^{\text{{\rm cur}},X}(\underline{\phantom{A}},p)$ is a totally acyclic sheaf (in the sense of [6]) for the real scheme underlying $X_{\mathbb{R}}$. When $X$ is fixed, $\mathcal{D}_{\ast}^{\text{{\rm cur}},X}$ will be denoted by $\mathcal{D}_{\ast}^{\text{{\rm cur}}}$. If $U$ is a Zariski open subset of $X_{\mathbb{R}}$ and $Y=X\setminus U_{\mathbb{R}}$ we write (3.1) $\displaystyle H^{\mathcal{D}^{\mathcal{T}}}_{\ast}(U,\mathbb{R}(p))$ $\displaystyle=H_{\ast}(\mathcal{D}^{\text{{\rm cur}}}(U,p)),$ (3.2) $\displaystyle H^{\mathcal{D}^{\mathcal{T}},Y}_{\ast}(X_{\mathbb{R}},\mathbb{R}(p))$ $\displaystyle=H_{\ast}(s(\mathcal{D}^{\text{{\rm cur}}}(U,p),\mathcal{D}^{\text{{\rm cur}}}(X_{\mathbb{R}},p))),$ (3.3) $\displaystyle\widetilde{\mathcal{D}}_{2p-1}^{\text{{\rm cur}}}(X_{\mathbb{R}},p)$ $\displaystyle=\mathcal{D}_{2p-1}^{\text{{\rm cur}}}(X_{\mathbb{R}},p)\left/\operatorname{Im}\operatorname{d}_{\mathcal{D}},\right.$ (3.4) $\displaystyle{\rm Z}\mathcal{D}_{2p}^{\text{{\rm cur}}}(X_{\mathbb{R}},p)$ $\displaystyle=\operatorname{Ker}(\operatorname{d}_{\mathcal{D}}:\mathcal{D}_{2p}^{\text{{\rm cur}}}(X_{\mathbb{R}},p)\longrightarrow\mathcal{D}_{2p+1}^{\text{{\rm cur}}}(X_{\mathbb{R}},p)).$ Let $\mathcal{Z}_{p}=\mathcal{Z}_{p}(X_{\mathbb{R}})$ be the set of dimension $p$ Zariski closed subsets of $X_{\mathbb{R}}$ ordered by inclusion. Then we will write $\displaystyle\mathcal{D}_{\ast}^{\text{{\rm cur}}}(X_{\mathbb{R}}\setminus\mathcal{Z}_{p},p)$ $\displaystyle=\lim_{\begin{subarray}{c}\longrightarrow\\\ Y\in\mathcal{Z}^{p}\end{subarray}}\mathcal{D}_{\ast}^{\text{{\rm cur}}}(X_{\mathbb{R}}\setminus Y,p),$ $\displaystyle\widetilde{\mathcal{D}}_{\ast}^{\text{{\rm cur}}}(X_{\mathbb{R}}\setminus\mathcal{Z}_{p},p)$ $\displaystyle=\mathcal{D}_{\ast}^{\text{{\rm cur}}}(X_{\mathbb{R}}\setminus\mathcal{Z}_{p},p)\left/\operatorname{Im}\operatorname{d}_{\mathcal{D}}\right.,$ $\displaystyle H^{\mathcal{D}^{\mathcal{T}},\mathcal{Z}_{p}}_{\ast}(X_{\mathbb{R}},\mathbb{R}(p))$ $\displaystyle=H_{\ast}(s(\mathcal{D}^{\text{{\rm cur}}}(X_{\mathbb{R}}\setminus\mathcal{Z}_{p},p),\mathcal{D}^{\text{{\rm cur}}}(X_{\mathbb{R}},p))).$ Green objects. We recall the definition of Green object for a cycle given in [6] but adapted to the grading by dimension. Let $y$ be a dimension $p$ algebraic cycle of $X_{\mathbb{R}}$. Let $Y$ be the support of $y$. The class of $y$ in $H^{\mathcal{D}^{\mathcal{T}},Y}_{2p}(X_{\mathbb{R}},\mathbb{R}(p))$, denoted $\operatorname{cl}(y)$, is represented by the pair $(\delta_{y},0)\in s(\mathcal{D}^{\text{{\rm cur}}}(X_{\mathbb{R}},p),\mathcal{D}^{\text{{\rm cur}}}(U_{\mathbb{R}},p))$. We denote also by $\operatorname{cl}(y)$ the image of this class in $H^{\mathcal{D}^{\mathcal{T}},\mathcal{Z}_{p}}_{2p}(X_{\mathbb{R}},\mathbb{R}(p))$. In this setting, the truncated homology classes can be written as $\widehat{H}^{\mathcal{D}^{\mathcal{T}},\mathcal{Z}_{p}}_{\ast}(X_{\mathbb{R}},\mathbb{R}(p))=\\\ \\{(\omega_{y},\widetilde{g}_{y})\in{\rm Z}\mathcal{D}_{2p}^{\text{{\rm cur}}}(X,p)\oplus\widetilde{\mathcal{D}}_{2p-1}^{\text{{\rm cur}}}(X_{\mathbb{R}}\setminus\mathcal{Z}_{p},p)\mid\operatorname{d}_{\mathcal{D}}\widetilde{g}_{y}=\omega_{y}\\}.$ There is an obvious class map $\operatorname{cl}:\widehat{H}^{\mathcal{D}^{\mathcal{T}},\mathcal{Z}_{p}}_{\ast}(X_{\mathbb{R}},\mathbb{R}(p))\longrightarrow H^{\mathcal{D}^{\mathcal{T}},\mathcal{Z}_{p}}_{\ast}(X_{\mathbb{R}},\mathbb{R}(p)).$ Then a Green object for $y$ is an element $\mathfrak{g}_{y}=(\omega_{y},\widetilde{g}_{y})\in\widehat{H}^{\mathcal{D}^{\mathcal{T}},\mathcal{Z}_{p}}_{2p}(X_{\mathbb{R}},\mathbb{R}(p))$ such that $\operatorname{cl}(\mathfrak{g}_{y})=\operatorname{cl}(y)$. The following result follows directly from the definition ###### Lemma 3.5. An element $\mathfrak{g}_{y}=(\omega_{y},\widetilde{g}_{y})\in\widehat{H}^{\mathcal{D}^{\mathcal{T}},\mathcal{Z}_{p}}_{2p}(X_{\mathbb{R}},\mathbb{R}(p))$ is a Green object for $y$ if and only if there exists a current $\widetilde{\gamma}\in\widetilde{\mathcal{D}}_{2p-1}^{\text{{\rm cur}}}(X_{\mathbb{R}},p)$ such that $\displaystyle\widetilde{g}_{y}$ $\displaystyle=\widetilde{\gamma}\|_{X\setminus\mathcal{Z}_{p}}$ $\displaystyle\operatorname{d}_{\mathcal{D}}\widetilde{\gamma}+\delta_{y}$ $\displaystyle=\omega_{y}.$ Arithmetic Chow groups. Every dimension $p$ algebraic cycle $y$ on $X$ defines a dimension $(p-e)$ algebraic cycle $y_{\mathbb{R}}$ on $X_{\mathbb{R}}$, where $e$ is the dimension of the base scheme $S$. ###### Definition 3.6. The group of arithmetic cycles of dimension $p$ is defined as $\operatorname{\widehat{Z}}_{p}(X,\mathcal{D}^{\text{{\rm cur}}})=\\{(y,\mathfrak{g}_{y})\in{\rm Z}_{p}(X)\oplus\widehat{H}^{\mathcal{D}^{\mathcal{T}},\mathcal{Z}_{p-e}}_{2p-2e}(X_{\mathbb{R}},\mathbb{R}(p-e))\mid\operatorname{cl}(y_{\mathbb{R}})=\operatorname{cl}(\mathfrak{g}_{y})\\}.$ Let $W$ be a dimension $p+1$ irreducible subvariety of $X$ and $f\in K(W)^{\ast}$ be a rational function. Let $\widetilde{W}_{\mathbb{R}}$ be a resolution of singularities of $W_{\mathbb{R}}$ and let $\iota:\widetilde{W}_{\mathbb{R}}\longrightarrow X_{\mathbb{R}}$ be the induced map. Then we write $\operatorname{\widehat{div}}f=(\operatorname{div}f,(0,\iota_{\ast}(-\frac{1}{2}\log f\bar{f})).$ The group of cycles rationally equivalent to zero is the subgroup $\operatorname{\widehat{Rat}}_{p}(X,\mathcal{D}^{\text{{\rm cur}}})\subset\operatorname{\widehat{Z}}_{p}(X,\mathcal{D}^{\text{{\rm cur}}})$ generated by the elements of the form $\operatorname{\widehat{div}}f$. The _homological arithmetic Chow groups_ of $X$ are defined as $\operatorname{\widehat{CH}}_{p}(X,\mathcal{D}^{\text{{\rm cur}}})=\operatorname{\widehat{Z}}_{p}(X,\mathcal{D}^{\text{{\rm cur}}})\left/\operatorname{\widehat{Rat}}_{p}(X,\mathcal{D}^{\text{{\rm cur}}})\right.$ There are well-defined maps $\displaystyle\zeta$ $\displaystyle:\operatorname{\widehat{CH}}_{p}(X,\mathcal{D}^{\text{{\rm cur}}})\longrightarrow\operatorname{CH}_{p}(X),$ $\displaystyle\quad\zeta[y,\mathfrak{g}_{y}]=[y],$ $\displaystyle\rho$ $\displaystyle:\operatorname{CH}_{p,p+1}(X)\longrightarrow H_{2p-2e+1}^{\mathcal{D}^{\mathcal{T}}}(X,p-e)\subseteq\widetilde{\mathcal{D}}_{2p+1}^{\text{{\rm cur}}}(X,p),$ $\displaystyle\quad\rho[f]=\operatorname{cl}(f),$ $\displaystyle\operatorname{a}$ $\displaystyle:\widetilde{\mathcal{D}}_{2p-2e+1}(X,p-e)\longrightarrow\operatorname{\widehat{CH}}_{p}(X,\mathcal{D}^{\text{{\rm cur}}}),$ $\displaystyle\quad\operatorname{a}(\widetilde{a})=[0,\operatorname{a}(\widetilde{a})],$ $\displaystyle\omega$ $\displaystyle:\operatorname{\widehat{CH}}_{p}(X,\mathcal{D}^{\text{{\rm cur}}})\longrightarrow{\rm Z}\mathcal{D}_{2p-2e}^{\text{{\rm cur}}}(X,p-e),$ $\displaystyle\quad\omega[y,\mathfrak{g}_{y}]=\omega(\mathfrak{g}_{y}),$ $\displaystyle h$ $\displaystyle:{\rm Z}\mathcal{D}_{2p}^{\text{{\rm cur}}}(X,p)\longrightarrow H_{2p}^{\mathcal{D}^{\mathcal{T}}}(X,p),$ $\displaystyle\quad h(\alpha)=[\alpha].$ ### 3.2. Properties of Covariant arithmetic Chow groups Basic properties. Recall that in [6], there are defined contravariant arithmetic Chow groups denoted by $\operatorname{\widehat{CH}}^{\ast}(X,\mathcal{D}_{\log})$. The following result follows from the theory developed [6] and corollary 2.34 (semi-purity property). ###### Theorem 3.7. With the above notations, we have the following statements: 1. (i) There are exact sequences $\operatorname{CH}_{p,p+1}(X)\overset{\rho}{\longrightarrow}\widetilde{\mathcal{D}}_{2p-2e+1}^{\text{{\rm cur}}}(X,p-e)\overset{\operatorname{a}}{\longrightarrow}\operatorname{\widehat{CH}}_{p}(X,\mathcal{D}^{\text{{\rm cur}}})\overset{\zeta}{\longrightarrow}\operatorname{CH}_{p}(X)\longrightarrow 0.$ $\displaystyle\operatorname{CH}_{p,p+1}(X)\overset{\rho}{\longrightarrow}H_{2p-2e+1}^{\mathcal{D}^{\mathcal{T}}}(X_{\mathbb{R}},\mathbb{R}(p-e))\overset{\operatorname{a}}{\longrightarrow}\operatorname{\widehat{CH}}_{p}(X,\mathcal{D}^{\text{{\rm cur}}})\overset{(\zeta,-\omega)}{\longrightarrow}$ $\displaystyle\phantom{CH_{p,p+1}}\operatorname{CH}_{p}(X)\oplus{\rm Z}\mathcal{D}_{2p-2e}^{\text{{\rm cur}}}(X,p-e)\overset{\operatorname{cl}+h}{\longrightarrow}H_{2p-2e}^{\mathcal{D}^{f}}(X_{\mathbb{R}},\mathbb{R}(p-e))\longrightarrow 0.$ In particular, if $X_{F}$ is projective, then there is an exact sequence $\displaystyle\operatorname{CH}_{p,p+1}(X)\overset{\rho}{\longrightarrow}H_{2p-2e+1}^{\mathcal{D}}(X_{\mathbb{R}},\mathbb{R}(p-e))\overset{\operatorname{a}}{\longrightarrow}\operatorname{\widehat{CH}}_{p}(X,\mathcal{D}^{\text{{\rm cur}}})\overset{(\zeta,-\omega)}{\longrightarrow}$ $\displaystyle\phantom{CH_{p,p+1}}\operatorname{CH}_{p}(X)\oplus{\rm Z}\mathcal{D}_{2p-2e}^{\text{{\rm cur}}}(X,p-e)\overset{\operatorname{cl}+h}{\longrightarrow}H_{2p-2e}^{\mathcal{D}}(X_{\mathbb{R}},\mathbb{R}(p-e))\longrightarrow 0.$ 2. (ii) For any regular arithmetic variety $X$ over $A$ there are defined contravariant arithmetic Chow groups $\operatorname{\widehat{CH}}^{p}(X,\mathcal{D}_{\log})$. Furthermore, if $X$ is equidimensional of dimension $d$, then there is a morphism of arithmetic Chow groups $\operatorname{\widehat{CH}}^{p}(X,\mathcal{D}_{\log})\longrightarrow\operatorname{\widehat{CH}}_{d-p}(X,\mathcal{D}^{\text{{\rm cur}}}).$ When $X_{F}$ is projective this morphism is a monomorphism. Moreover, if $X_{F}$ has dimension zero, this morphism is an isomorphism. 3. (iii) For any proper morphism $f:X\longrightarrow Y$ of arithmetic varieties over $A$, there is a morphism of covariant arithmetic Chow groups $f_{\ast}:\operatorname{\widehat{CH}}_{p}(X,\mathcal{D}_{\text{{\rm cur}}})\longrightarrow\operatorname{\widehat{CH}}_{p}(Y,\mathcal{D}_{\text{{\rm cur}}}).$ If $g:Y\longrightarrow Z$ is another such morphism, the equality $(g\circ f)_{\ast}=g_{\ast}\circ f_{\ast}$ holds. Moreover, if $X$ and $Y$ are regular and $f_{F}:X_{F}\longrightarrow Y_{F}$ is a smooth proper morphism of projective varieties, then $f_{\ast}$ is compatible with the direct image of contravariant arithmetic Chow groups. 4. (iv) If $f:X\longrightarrow Y$ is a flat morphism, equidimensional of relative dimension $d$, and such that $f_{F}$ is smooth, then there is a pull-back map $f^{\ast}:\operatorname{\widehat{CH}}_{p}(Y,\mathcal{D}^{\text{{\rm cur}}})\longrightarrow\operatorname{\widehat{CH}}_{p+d}(X,\mathcal{D}^{\text{{\rm cur}}}).$ If $X$ and $Y$ are regular and equidimensional, this map is equivalent with the pullback map defined in the contravariant Chow groups. 5. (v) Let $f:X\longrightarrow Y$ be a flat map between arithmetic varieties, which is smooth over $F$ and let $g:P\longrightarrow Y$ be a proper map. Let $Z$ be the fiber product of $X$ and $P$ over $Y$, with $p:Z\longrightarrow P$ and $q:Z\longrightarrow X$ the two projections. Thus $p$ is flat and smooth over $F$ and $q$ is proper. Then for any $x\in\operatorname{\widehat{CH}}_{\ast}(P,\mathcal{D}^{\text{{\rm cur}}})$, it holds $q_{\ast}p^{\ast}(x)=f^{\ast}g_{\ast}(x)\in\operatorname{\widehat{CH}}_{\ast}(X,\mathcal{D}^{\text{{\rm cur}}}).$ ###### Proof. Part (i) follows from the standard exact sequences of [6] Theorem 4.13 adapted to the grading by dimension and corollary 2.34. For (ii) we first note that, if $M$ is an equidimensional complex algebraic manifold, $D\subset X$ is a normal crossing divisor, $\omega$ is a differential form with logarithmic singularities along $D$ and $\eta$ is a form that is flat along $D$, then $\eta\wedge\omega$ is flat along $D$. In particular, if $M$ is proper and $U=M\setminus D$, then the associated current $[\omega]$ belongs to $D^{\text{{\rm extd}}}_{\ast}(U)$. Therefore, if $y$ is a codimension $p$ cycle on $X$ then, by the assumptions on $X$ and on the arithmetic ring, $y$ is a dimension $d-p$ algebraic algebraic cycle. Moreover, if $(\omega_{y},\widetilde{g}_{y})$ is a Green form for $y$ (i.e. a $\mathcal{D}_{\log}$-Green object for $y$) then, by lemma 3.5 and [6] Proposition 6.5 we have that $([\omega_{y}],[\widetilde{g}_{y}])$ is a $\mathcal{D}^{\text{{\rm cur}}}$-Green object for $y$. Thus we have a well defined map $\operatorname{\widehat{Z}}^{p}(X,\mathcal{D}_{\log})\longrightarrow\operatorname{\widehat{Z}}_{d-p}(X,\mathcal{D}^{\text{{\rm cur}}}).$ By definition this map is compatible with rational equivalence, hence we obtain a map at the level of Chow groups. To prove (iii) we first observe that, if $Z\subset X_{\Sigma}$ is a closed subset, then $f_{\ast}D_{\ast}^{X_{\Sigma}^{\mathcal{T}}}(Z)\subset D^{X_{\Sigma}^{\mathcal{T}}}(f(Z))$. Therefore, the push-forward of currents define a covariant $f$-morphism $f_{\\#}:f_{\ast}\mathcal{D}_{\ast}^{\text{{\rm cur}},X}\longrightarrow\mathcal{D}^{\text{{\rm cur}},Y}_{\ast}.$ Here we are using the terminology of [6] 3.67 but adapted to the grading by dimension. Therefore applying [6] §4.5 we obtain the push-forward map for covariant arithmetic Chow groups. More concretely this map is defined as $f_{\ast}(y,(\omega_{y},\widetilde{g}_{y}))=(f_{\ast}y,(f_{\ast}\omega_{y},(f_{\ast}g_{y})\widetilde{\phantom{A}})).$ It is straightforward to check that it is compatible with the direct image of $\mathcal{D}_{\log}$-arithmetic Chow groups when $Y$ is projective and $f_{F}$ smooth. We now prove (iv). Since $f_{F}$ is smooth, for any Zariski closed subset $Z\subset Y_{\mathbb{R}}$ equidimensional of dimension $p$, there is a well defined morphism $f^{\ast}D_{n}(Y_{\Sigma})\longrightarrow D_{n+2d}(X_{\Sigma})$ that sends $D_{n}^{Y_{\Sigma}^{\mathcal{T}}}(Z)$ to $D^{X_{\Sigma}^{\mathcal{T}}}_{n+2d}(f^{-1}(Z))$. Therefore we obtain well defined morphisms $\begin{matrix}f^{\\#}:\mathcal{D}^{\text{{\rm cur}}}_{n}(Y_{\mathbb{R}},p)&\longrightarrow&\mathcal{D}^{\text{{\rm cur}}}_{n+2d}(X_{\mathbb{R}},p+d),\\\ f^{\\#}:\mathcal{D}^{\text{{\rm cur}}}_{n}(Y_{\mathbb{R}}\setminus Z,p)&\longrightarrow&\mathcal{D}^{\text{{\rm cur}}}_{n+2d}(X_{\mathbb{R}}\setminus f^{-1}Z,p+d),\end{matrix}$ that send $T$ to $f^{\ast}T/(2\pi i)^{d}$. Then the proof of (iv) is straightforward using the theory of [6] 4.4 adapted to the grading by dimension. (v) Follows as [8] Lemma 11. ∎ Multiplicative properties. In the next result we state the multiplicative properties between covariant and contravariant Chow groups. The proofs are simple modification of [8] Theorem 3. First, for a form $\eta\in\widetilde{\mathcal{D}}^{2p-1}_{\log}(X_{\mathbb{R}},p)$ and an element $x\in\operatorname{\widehat{CH}}_{q}(X,\mathcal{D}^{\text{{\rm cur}}})$ we define $\eta\cap x=\operatorname{a}(\eta\bullet\omega(x))=\operatorname{a}(\eta\wedge\omega(x)).$ ###### Theorem 3.8. Given a map $f:X\longrightarrow Y$ of arithmetic varieties, with $Y$ regular, there is a cap product $\begin{matrix}\operatorname{\widehat{CH}}^{p}(Y,\mathcal{D}_{\log})\otimes\operatorname{\widehat{CH}}_{q}(X,\mathcal{D}^{\text{{\rm cur}}})&\longrightarrow&\operatorname{\widehat{CH}}_{q-p}(X,\mathcal{D}^{\text{{\rm cur}}})_{\mathbb{Q}}\\\ y\otimes x&\longmapsto&y._{f}x\end{matrix}$ which is also denoted $y\cap X$ if $X=Y$. This product satisfies the following properties 1. (i) $\omega(y._{f}x)=f^{\ast}\omega(y)\land\omega(x)$, and, for any $\eta\in\widetilde{\mathcal{D}}_{\log}^{2p-1}(Y_{\mathbb{R}},p)$, it holds $\operatorname{a}(\eta)._{f}x=\operatorname{a}(f^{\ast}(\eta))\cap x$. 2. (ii) $\operatorname{\widehat{CH}}_{\ast}(X,\mathcal{D}^{\text{{\rm cur}}})_{\mathbb{Q}}$ is a graded $\operatorname{\widehat{CH}}^{\ast}(Y,\mathcal{D}_{\log})$-module. 3. (iii) If $g:Y\longrightarrow Y^{\prime}$ is a map of arithmetic varieties with $Y^{\prime}$ also regular, $y^{\prime}\in\operatorname{\widehat{CH}}^{p}(Y^{\prime},\mathcal{D}_{\log})$ and $x\in\operatorname{\widehat{CH}}_{q}(X,\mathcal{D}^{\text{{\rm cur}}})$, then $y^{\prime}._{gf}x=g^{\ast}(y^{\prime})._{f}x$. 4. (iv) If $h:X^{\prime}\longrightarrow X$ is a projective morphism, $x^{\prime}\in\operatorname{\widehat{CH}}_{q}(X^{\prime},\mathcal{D}^{\text{{\rm cur}}})$ and $y\in\operatorname{\widehat{CH}}^{p}(Y,\mathcal{D}_{\log})$, then $y._{f}(h_{\ast}(x^{\prime}))=h_{\ast}(y._{fh}x^{\prime})$. 5. (v) If $h:X^{\prime}\longrightarrow X$ is flat and smooth over $F$, $x\in\operatorname{\widehat{CH}}_{q}(X,\mathcal{D}^{\text{{\rm cur}}})$, $y\in\operatorname{\widehat{CH}}^{p}(Y,\mathcal{D}_{\log})$, then $h^{\ast}(y._{f}x)=y._{f}(h^{\ast}(x))$. 6. (vi) Let $f:X\longrightarrow Y$ be a flat map between arithmetic varieties, with $Y$ regular and projective, and let $g:P\longrightarrow Y$ be a proper smooth map of arithmetic varieties of relative dimension $d$. Let $Z$ be the fiber product of $X$ and $P$ over $Y$, with $p:Z\longrightarrow P$ and $q:Z\longrightarrow X$ the two projections. Then, for all $x\in\operatorname{\widehat{CH}}_{p}(X,\mathcal{D}^{\text{{\rm cur}}})$ and $\gamma\in\operatorname{\widehat{CH}}^{q}(P,\mathcal{D}_{\log})$, it holds the equality $q_{\ast}(\gamma._{p}q^{\ast}(x))=g_{\ast}\gamma._{f}\alpha.$ ###### Proof. To define $y._{f}x$ we follow closely [8]. We may assume that $Y$ is equidimensional, that $x=(V,\mathfrak{g}_{V})$ with $V$ a prime algebraic cycle and $y=(W,\mathfrak{g}_{W})$ with each component of $W$ meeting $V$ properly on the generic fiber $X_{F}$. As in [8] we can define a cycle $[V]._{f}[W]\in CH_{q-p}(V\cap f^{-1}(\|W\|))_{\mathbb{Q}}$ that gives us a well defined cycle $([V]._{f}[W])_{F}\in{\rm Z}_{q-p}(X_{F})$. Our task now is to construct the Green object for this cycle. Let $\mathfrak{g}_{W}=(\omega_{W},\widetilde{g}_{W})$ and $\mathfrak{g}_{V}=(\omega_{V},\widetilde{g}_{V})$. We write $U_{V}=X_{\mathbb{R}\setminus\|V\|}$, $U_{W}=X_{\mathbb{R}\setminus f^{-1}\|W\|}$ and $r=q-p$. We now define, in analogy with [6] theorem 3.37, $\displaystyle\mathfrak{g}_{W}$ $\displaystyle\ast_{f}\mathfrak{g}_{V}=f^{\ast}\mathfrak{g}_{W}\ast\mathfrak{g}_{V}$ $\displaystyle=\left(f^{\ast}(\omega_{W})\bullet\omega_{V},((f^{\ast}(g_{W})\bullet\omega_{V},f^{\ast}(\omega_{W})\bullet g_{V}),-f^{\ast}(g_{W})\bullet g_{V})^{\widetilde{\phantom{=}}}\right)$ $\displaystyle=(f^{\ast}(\omega_{W})\wedge\omega_{V},((f^{\ast}(g_{W})\wedge\omega_{V},f^{\ast}(\omega_{W})\wedge g_{V}),$ $\displaystyle\ \partial f^{\ast}(g_{W})\land g_{V}-\bar{\partial}f^{\ast}(g_{W})\land g_{V}-f^{\ast}(g_{W})\land\partial g_{V}+f^{\ast}(g_{W})\land\bar{\partial}g_{V})^{\widetilde{\phantom{=}}}$ $\displaystyle\in\widehat{H}_{2e}(\mathcal{D}^{\text{{\rm cur}}}_{\ast}(X_{\mathbb{R}},e),s(\mathcal{D}^{\text{{\rm cur}}}_{\ast}(U_{W},e)\oplus\mathcal{D}^{\text{{\rm cur}}}_{\ast}(U_{V},e)\rightarrow\mathcal{D}^{\text{{\rm cur}}}_{\ast}(U_{W}\cap U_{V},e)))$ $\displaystyle\cong\widehat{H}_{2e}(\mathcal{D}^{\text{{\rm cur}}}_{\ast}(X_{\mathbb{R}},e),\mathcal{D}^{\text{{\rm cur}}}_{\ast}(U_{W}\cup U_{V},e)).$ Now the proof follows as in [8] Theorem 3 and Lemma 12. ∎ ###### Remark 3.9. 1. (i) The main difference between the arithmetic Chow groups introduced here and the arithmetic Chow groups used in [8] is that, if $x\in\operatorname{\widehat{CH}}_{\ast}(X,\mathcal{D}^{\text{{\rm cur}}})$ then $\omega(x)$ is an arbitrary current instead of a smooth differential form. This allows us to define direct images for arbitrary proper morphisms. But the price we have to pay is that there ire defined inverse images only for morphisms that are smooth over $F$. 2. (ii) The fact that the compatibility of direct images for the covariant Chow groups and direct images for the contravariant Chow groups in theorem 3.7 is stated only for varieties that are generically projective, is due to the fact that the latter is only defined when the base is proper. There are two ways to overcome this difficulty. One is to allow arbitrary singularities at infinity in the spirit of [5] 3.5, but then, one will have to allow also arbitrary singularities at infinity for currents. This means that we will have to consider currents that are tempered in some components of the boundary but are not tempered in the other. The second option would be to use a different notion of logarithmic singularities that has better properties with respect to direct images. Relationship with other arithmetic Chow groups. Let us assume now that $X_{F}$ is projective and let $\operatorname{\widehat{CH}}^{\ast}(X)$ denote the arithmetic Chow groups introduced in [7] and $\operatorname{\widehat{CH}}_{\ast}(X)$ denote the arithmetic Chow groups introduced in [8]. In [6] it is shown that there is an isomorphism $\psi:\operatorname{\widehat{CH}}^{\ast}(X,\mathcal{D}_{\log})\longrightarrow\operatorname{\widehat{CH}}^{\ast}(X),$ that is compatible with products, inverse images with respect to arbitrary morphisms and direct images with respect to proper morphism that are smooth over $F$. We shall state the analogous result for covariant arithmetic Chow groups. ###### Proposition 3.10. Let $X$ be an arithmetic variety with $X_{F}$ projective. Then there is a short exact sequence $0\longrightarrow\operatorname{\widehat{CH}}_{\ast}(X)\overset{\phi}{\longrightarrow}\operatorname{\widehat{CH}}_{\ast}(X,\mathcal{D}^{\text{{\rm cur}}})\\\ \longrightarrow\bigoplus_{p}{\rm Z}\mathcal{D}_{2p}^{\text{{\rm cur}}}(X_{\mathbb{R}},p)\left/{\rm Z}\mathcal{D}_{2p}^{\text{{\rm smooth}}}(X_{\mathbb{R}},p)\longrightarrow 0\right.,$ where $\mathcal{D}_{2p}^{\text{{\rm smooth}}}(X_{\mathbb{R}},p)$ denotes the subspace of currents that can be represented by smooth differential forms. Moreover $\phi$ satisfies the following properties 1. (i) If $f:X\longrightarrow Y$ is a proper morphism of arithmetic varieties that is smooth over $F$ and with $Y_{F}$ projective, then $f_{\ast}\circ\phi=\phi\circ f_{\ast}$. 2. (ii) If $f:X\longrightarrow Y$ is a flat morphism of arithmetic varieties that is smooth over $F$, with $X_{F}$ and $Y_{F}$ projective, then $f^{\ast}\circ\phi=\phi\circ f^{\ast}$. 3. (iii) If $f:X\longrightarrow Y$ is a morphism of arithmetic varieties, with $X_{F}$ and $Y_{F}$ projective and $Y$ regular then, for $y\in\operatorname{\widehat{CH}}^{p}(Y,\mathcal{D}_{\log})$ and $x\in\operatorname{\widehat{CH}}_{q}(Y)$, it holds the equality $y._{f}\phi(x)=\psi(y)._{f}x.$ ###### Proof. Let $y$ be a dimension $p$ algebraic cycle of $X$ and let $g_{y}$ be a Green current for $y$ in the sense of [8]. Recall that the normalization used here for the current $\delta_{y}$ differs with the normalization used in [8] by a factor $\frac{1}{(2\pi i)^{p}}$. Then, by 3.5, the pair $\left(\frac{1}{2(2\pi i)^{p+1}}g_{y}\|_{X_{\mathbb{R}}\setminus\mathcal{Z}_{p}},\frac{1}{2(2\pi i)^{p+1}}(-2\partial\bar{\partial})g_{y}+\delta_{y}\right)$ is a $\mathcal{D}^{\text{{\rm cur}}}$-Green object for $y$. Therefore we obtain a well defined morphism $\operatorname{\widehat{Z}}_{p}(X)\longrightarrow\operatorname{\widehat{Z}}_{p}(X,\mathcal{D}^{\text{{\rm cur}}})$. It is straightforward to check that this map preserves rational equivalence, the exactness of the above exact sequence and properties (i), (ii) and (iii). ∎ ###### Corollary 3.11. With the hypothesis of the proposition, every element $x\in\operatorname{\widehat{CH}}_{p}(X,\mathcal{D}^{\text{{\rm cur}}})$ can be represented as $x=\phi(x_{1})+\operatorname{a}(\eta)$ where $x_{1}\in\operatorname{\widehat{CH}}_{p}(X)$ and $\eta\in\widetilde{\mathcal{D}}_{2p+1}^{\text{{\rm cur}}}(X_{\mathbb{R}},p)$. Moreover, if $x=\phi(x_{1})+\operatorname{a}(\eta)=\phi(x^{\prime}_{1})+\operatorname{a}(\eta^{\prime})$ are two such representations, then $\eta-\eta^{\prime}\in\widetilde{\mathcal{D}}_{2p+1}^{\text{{\rm smooth}}}(X_{\mathbb{R}},p).$ ###### Proof. This follows from the previous proposition and the fact that the map $\operatorname{d}_{\mathcal{D}}:\widetilde{\mathcal{D}}_{2p+1}^{\text{{\rm cur}}}(X_{\mathbb{R}},p)\longrightarrow{\rm Z}\mathcal{D}_{2p}^{\text{{\rm cur}}}(X_{\mathbb{R}},p)\left/{\rm Z}\mathcal{D}_{2p}^{\text{{\rm smooth}}}(X_{\mathbb{R}},p)\right.$ is surjective due to the projectivity of $X$. The last statement follows from [7] Theorem 1.2.2. ∎ The following result follows now easily from the previous corollary. ###### Corollary 3.12. Assume furthermore that $X$ is equidimensional of dimension $d$ and let $\operatorname{\widehat{CH}}^{\ast}_{D}(X)$ denote the $D$-arithmetic Chow groups introduced in [13]. Then there is a natural isomorphism $\bigoplus_{p}\operatorname{\widehat{CH}}^{p}_{D}(X)\longrightarrow\bigoplus_{p}\operatorname{\widehat{CH}}_{d-p}(X,\mathcal{D}^{\text{{\rm cur}}}).$ Moreover this isomorphism is compatible with push-forwards and the structure of module over the contravariant arithmetic Chow groups. $\square$ ## References * [1] N. Bourbaki, _Topological vector spaces, chapters 1-5_ , Springer Verlag, 1987\. * [2] J.P. Brasselet and M. Pflaum, _On the homology of algebras of Whitney functions on subanalytic sets_ , Institut de Mathématiques de Luminy, Prétirage no 2002-02, March 2002. * [3] J.I. Burgos Gil, _Arithmetic Chow rings_ , Ph.D. thesis, University of Barcelona, 1994. * [4] by same author, _Arithmetic Chow rings and Deligne-Beilinson cohomology_ , J. Alg. Geom. 6 (1997), 335–377. * [5] J.I. Burgos Gil, J. Kramer, and U. Kühn, _Arithmetic characteristic classes of automorphic vector bundles_ , Documenta Math. 10 (2005), 619–716. * [6] by same author, _Cohomological arithmetic Chow rings_ , J. Inst. Math. Jussieu 6 (2007), no. 1, 1–172. MR MR2285241 * [7] H. Gillet and C. Soulé, _Arithmetic intersection theory_ , Publ. Math. IHES 72 (1990), 94–174. * [8] by same author, _An arithmetic Riemann-Roch theorem_ , Invent. Math. 110 (1992), 473–543. * [9] P. Griffiths and J. Harris, _Principles of algebraic geometry_ , John Wiley & Sons, Inc., 1994. * [10] R. Hartshorne, _On the de Rham cohomology of algebraic varieties_ , Publ. math. IHES 45 (1975), 5–99. * [11] U. Jannsen, _Deligne homology, Hodge-D-conjecture and motives_ , Beilinson’s Conjectures on Special Values of $L$-Functions (M. Rapoport, N. Schappacher, and P. Schneider, eds.), Perspectives in Math., vol. 4, Academic Press, 1988, pp. 305–372. * [12] Masaki Kashiwara and Pierre Schapira, _Moderate and formal cohomology associated with constructible sheaves_ , Mém. Soc. Math. France (N.S.) (1996), no. 64, iv+76. MR MR1421293 (97m:32052) * [13] S. Kawaguchi and A. Moriwaki, _Inequalities for semistable families of arithmetic varieties_ , Preprint alg-geom 9710007, 1998. * [14] J.P. Poly, _Sur l’homologie des courants à support dans un ensemble semi- analytique_ , Mém. Soc. Math. France 38 (1974), 35–43. * [15] J.C. Tougeron, _Idéaux de fonctions différentiables_ , Springer-Verlag, 1972\. * [16] Yuhan Zha, _A general arithmetic Riemann-Roch theorem_ , Ph.D. thesis, University of Chicago, 1998.	arxiv-papers	2010-11-16T14:51:34	2024-09-04T02:49:14.895595	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "J. I. Burgos Gil", "submitter": "Jos\\'e Ignacio Burgos Gil", "url": "https://arxiv.org/abs/1011.3714" }
1011.3862	We propose a strategy to demonstrate the transition from the quantum Zeno effect (QZE) to the anti-Zeno effect (AZE) using a superconducting qubit coupled to a transmission line cavity, by varying the central frequency of the cavity mode. Our results are obtained without the rotating wave approximation (RWA), and the initial state (a dressed state) is easy to prepare. Moreover, we find that in the presence of both qubit’s intrinsic bath and the cavity bath, the emergence of the QZE and the AZE behaviors relies not only on the match between the qubit energy level spacing and the central frequency of the cavity mode, but also on the coupling strength between the qubit and the cavity mode. Keywords: Quantum Zeno effect, anti-Zeno effect, qubit, cavity # The transition from quantum Zeno to anti-Zeno effects for a qubit in a cavity by varying the cavity frequency Xiufeng Cao111Email: xfcao@xmu.edu.cn Fax: 080-0592-2189426 Department of Physics and Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen, 361005,China Advanced Science Institute, RIKEN, Wako-shi 351-0198, Japan Qing Ai Advanced Science Institute, RIKEN, Wako-shi 351-0198, Japan Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China C. P. Sun Advanced Science Institute, RIKEN, Wako-shi 351-0198, Japan Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China Franco Nori Advanced Science Institute, RIKEN, Wako-shi 351-0198, Japan Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA ###### pacs: 03.65.Xp, 42.50.Ct, 03.65.Yz ## I Introduction The QZE predicts that the decay rate of a system can be slowed down by measuring it frequently enough pra-41-2295 ; pra-42-5720 ; pra-44-1466 ; pascazio . However some systems are predicted to have an enhancement of the decay due to the frequent measurements, namely the AZE or inverse Zeno effect nature-405-546 ; p ; q . The QZE and AZE have been observed in an unstable system prl-87-040402 . Recently, the QZE-AZE crossover in quantum Brownian motion model was investigated prl-97-130402 , where a system of damped harmonic oscillator interacts with a bosonic reservoir in thermal equilibrium. It was found prl-97-130402 that controlling the system-environment coupling by an artificially-controllable engineered environment (e.g., nature-407-57 ; nature-403-269 ) would allow one to monitor the transition from the QZE to the AZE dynamics. The QZE and AZE of a nanomechanical resonator measured by a quantum point contact detector (non-equilibrium fermionic reservoir) also was studied prb-81-115307 . Therefore, modulating the system and reservoir parameters can induce the QZE-AZE crossover. In cavity QED, the coupling between the qubit and the cavity, in which the electromagnetic field modes are concentrated around the cavity resonant frequency, depends on the cavity frequency. For an excited qubit located in a cavity, the cavity mode is the dominant one available for the qubit to emit photons. If the qubit energy level spacing is resonant with the cavity mode, the rate of decay into the particular cavity mode is enhanced. Otherwise, it is inhibited. Therefore, one may manipulate the qubit decay rate by varying the central frequency of the cavity mode in or off resonance with the qubit level energy spacing. The variation of the qubit decay pra-81-062131 ; pra-54-R3750 in the cavity is an increasingly important topic for experimental and theoretical studies prl-85-2272 ; prl-101-180402 ; pra-81-062131 . In this paper, we propose to modulate the qubit’s decay rate in cavity QED by the QZE, which means invoke the frequent measurements in the qubit and achieve the transition between QZE and AZE. We study a model of a qubit in a cavity, and investigate the occurrence of either the QZE or AZE by varying the cavity central frequency. We insert frequent projection measurements in the qubit decay process and find that the normalized decay rate depends on whether the central frequency of the cavity mode is in resonance with the qubit energy level spacing or not. In the resonant case, the normalized decay rate is lower than 1, so the QZE of the qubit occurs. However, when the cavity mode is detuned from the energy level spacing of the qubit, the normalized decay rate is larger than 1 and the qubit exhibits AZE. The variation from the QZE to the AZE, by varying the central frequency of the cavity mode, should help distinguishing these two kinds of effects. Moreover, we consider the case when both the qubit’s intrinsic bath and the cavity bath are simultaneously present. And find the dependence of the behaviors (the QZE and the AZE) on the coupling strength of the qubit-cavity and the cavity central frequency. The QZE-AZE crossover may be achieved in a superconducting qubit coupled to a transmission line cavity jqyou ; prl-103-147003 ; pra-81-042304 ; pra-69-062320 . This is because there are two physical mechanisms to tune the resonant frequency of the transmissionline resonator. One method is to change the boundary condition of the electromagnetic field in the transmission line prb-74-224506 ; apl-92-203905 ; apl-93-042510 , as shown in the Fig. 1(a). Another method is to construct a transmission line resonator by using a series of magnetic-flux biased SQUIDs, as shown in the Fig. 1(b). Because the effective inductor of a magnetic-flux-biased SQUID can be tuned by changing the applied magnetic flux nature-4-929 ; apl-91-083509 , the inductance per unit length of the SQUID array is controllable. Figure 1: (Color online) (a) Superconducting circuit model of a frequency- tunable transmission line resonator, which is archived by changing the boundary condition, coupled with a qubit. (b) Superconducting circuit model (1) of the effective tunable inductors, which are consisted of a series array of SQUIDs (2). ## II Hamiltonian of a qubit in a cavity beyond the rotating wave approximation Including the qubit dissipation environment, the Hamiltonian of a qubit in a lossy cavity can be written as $\displaystyle H$ $\displaystyle=$ $\displaystyle\frac{1}{2}\Delta\sigma_{z}+\sum_{k}\omega_{k,1}b_{k}^{\dagger}b_{k}+\sum_{k}f_{k}(b_{k}^{\dagger}+b_{k})\sigma_{x}$ (1) $\displaystyle+\sum_{k}\omega_{k,2}a_{k}^{\dagger}a_{k}+\sum_{k}g_{k}(a_{k}^{\dagger}+a_{k})\sigma_{x}.$ The Pauli operators, $\sigma_{z}$ and $\sigma_{x},$ describe the qubit level energy spacing and tunneling. The operators $b_{k}$ and $b_{k}^{\dagger}$ are the annihilation and creation operators characterizing the qubit’s intrinsic bath with frequencies $\omega_{k,1}$. The lossy cavity is modeled as a collection of harmonic oscillators with frequencies $\omega_{k,2},$ with the creation operators $a_{k}^{\dagger}$ and the annihilation operators $a_{k}$. Figure 2(a) schematically shows the model considered here. Figure 2: (Color online) (a) Sketch of a qubit with the spontaneous dissipation rate $\gamma$ coupled to a cavity with the loss rate $\kappa$ via a coupling strength $g.$ (b) and (c) schematically show the bath density spectrum of the qubit environment: (b) the Ohmic qubit’s intrinsic bath (green dashed) and the Lorentzian cavity bath (red solid), (c) the low-frequency qubit’s intrinsic bath (green dashed) and the Lorentzian cavity bath (red solid). Notice that no RWA is invoked in the Hamiltonian $H$ and thus it can not be diagonalized exactly. Now let us solve the Schrödinger equation of the Hamiltonian (1). We take the anti-rotating terms into account, which guarantees that our discussions extend to the off-resonant regime and also the case when there is a strong qubit- cavity interaction. Due to the anti-rotating terms, we apply a unitary transformation to the Hamiltonian $H$, $H^{\prime}=\exp(S)H\exp\left(-S\right),$ (2) with $S=\sum_{k}\left[\frac{f_{k}}{\omega_{k,1}}\xi_{k,1}(b_{k}^{\dagger}-b_{k})+\frac{g_{k}}{\omega_{k,2}}\xi_{k,2}(a_{k}^{\dagger}-a_{k})\right]\sigma_{x}.$ (3) Here, the $k$-dependent variables $\xi_{k,1}=\omega_{k,1}/(\omega_{k,1}+\eta_{1}\;\Delta),$ (4) and $\xi_{k,2}=\omega_{k,2}/(\omega_{k,2}+\eta_{2}\;\Delta),$ (5) are introduced in the transformation. The transformed Hamiltonian $H^{\prime}$ can be written as $\displaystyle H^{\prime}$ $\displaystyle\approx$ $\displaystyle\frac{1}{2}\eta\;\Delta\sigma_{z}+\sum_{k}\omega_{k,1}b_{k}^{\dagger}b_{k}+\sum_{k}\omega_{k,2}a_{k}^{\dagger}a_{k}$ (6) $\displaystyle+\sum_{k}V_{k,1}(b_{k}^{\dagger}\sigma_{-}+b_{k}\sigma_{+})$ $\displaystyle+\sum_{k}V_{k,2}(a_{k}^{\dagger}\sigma_{-}+a_{k}\sigma_{+}),$ with $\sigma_{\pm}=\left(\sigma_{x}\pm i\sigma_{y}\right)/2$ and $\eta=\eta_{1}\>\eta_{2}.$ (7) Then, the qubit energy-level-spacing $\Delta$ is renormalized to $\eta\;\Delta$ because of its coupling to the qubit’s intrinsic bath and the cavity bath. These factors $\eta_{1}$ and $\eta_{2},$ are respectively denoted by $\displaystyle\eta_{1}$ $\displaystyle=$ $\displaystyle\exp\left(-\sum_{k}2f_{k}^{2}\xi_{k,1}^{2}{}/\omega_{k,1}^{2}\right),$ (8) $\displaystyle\eta_{2}$ $\displaystyle=$ $\displaystyle\exp\left(-\sum_{k}2g_{k}^{2}\xi_{k,2}^{2}{}/\omega_{k,2}^{2}\right).$ (9) The coupling constants $f_{k}$ and $g_{k},$ of the qubit-environment interaction are also renormalized. The renormalized factors are respectively denoted by $\displaystyle V_{k,1}$ $\displaystyle=$ $\displaystyle 2\eta_{1}\;\Delta\;f_{k}/\left(\omega_{k,1}+\eta_{1}\;\Delta\right),$ (10) $\displaystyle V_{k,2}$ $\displaystyle=$ $\displaystyle 2\eta_{2}\;\Delta\;g_{k}/\left(\omega_{k,2}+\eta_{2}\;\Delta\right),$ (11) owing to the anti-rotating coupling terms. In Eq. (6), we drop the higher- order terms, which include the induced effect of the two baths by the coupling to the same qubit $\mathcal{O}$$\left(f_{k}\cdot g_{k}\right)$, whose contributions to the physical quantities are of the order $\mathcal{O}$$\left(g_{k}^{4}\right)$ $\left[\text{or }\mathcal{O}\left(f_{k}^{4}\right),\text{ or }\mathcal{O}\left(f_{k}^{2}g_{k}^{2}\right)\right]$ and higher. ## III Equation of motion of a qubit in a cavity beyond the rotating wave approximation Below, we will solve the equation of motion of the wave function, beyond the RWA, in the transformed Hamiltonian $H^{\prime}$ in Eq. (6). Since the total excitation number operator $N=\sum_{k}\left(a_{k}^{\dagger}a_{k}+b_{k}^{\dagger}b_{k}\right)+\left(1+\sigma_{z}\right)/2,$ (12) of the dissipative qubit-cavity system is a conserved observable, i.e., $\left[N,H^{\prime}\right]=0,$ it is reasonable to restrict our discussion to the single-particle excitation subspace. A general state in this subspace can be written as $\left\|\Phi(t)\right\rangle=\chi(t)\left\|\uparrow\right\rangle\left\|\\{0_{k}\;0_{k}\\}\right\rangle+\sum_{k,i}\beta_{k,i}(t)\left\|\downarrow\right\rangle\left\|\\{0_{k,\overline{i}}\;1_{k,i}\\}\right\rangle,$ (13) where $\left\|\uparrow\right\rangle$ and $\left\|\downarrow\right\rangle$ are the eigenstates of $\sigma_{z}$ ($\sigma_{z}\left\|\uparrow\right\rangle=\left\|\uparrow\right\rangle$ and $\sigma_{z}\left\|\downarrow\right\rangle=-\left\|\downarrow\right\rangle$), the state $\left\|\\{0_{k,\overline{i}}\;1_{k,i}\\}\right\rangle$ ($i$ can be $1,2$) means that either the cavity bath or the qubit’s intrinsic bath has one quantum excitation. Substituting $\left\|\Phi(t)\right\rangle$ into the Schrödinger equation, we have $i\frac{d\chi(t)}{dt}=\frac{\eta\;\Delta}{2}\chi(t)+\sum_{k,i}V_{k,i}\;\beta_{k,i}(t),$ (14) $i\frac{d\beta_{k,i}(t)}{dt}=\left(\omega_{k,i}-\frac{\eta\;\Delta}{2}\right)\beta_{k,i}(t)+\sum_{k,i}V_{k,i}\;\chi(t).$ (15) Applying the transformation $\displaystyle\chi(t)\\!$ $\displaystyle\\!=\\!$ $\displaystyle\\!\widetilde{\chi}(t)\exp\left(-i\frac{\eta\;\Delta}{2}t\right),$ (16) $\displaystyle\beta_{k,i}(t)\\!$ $\displaystyle\\!=\\!$ $\displaystyle\\!\widetilde{\beta}_{k,i}(t)\exp\left[-i\left(\omega_{k,i}-\frac{\eta\;\Delta}{2}\right)t\right],$ (17) Eqs. (14) and (15) is simplified as $\frac{d\widetilde{\chi}(t)}{dt}\\!=\\!-i\sum_{k,i}V_{k,i}\;\widetilde{\beta}_{k,i}(t)\exp\left[-i(\omega_{k,i}-\eta\;\Delta)t\right],$ (18) $\frac{d\widetilde{\beta}_{k,i}(t)}{dt}=-iV_{k,i}\;\widetilde{\chi}(t)\exp\left[i(\omega_{k,i}-\eta\;\Delta)t\right].$ (19) Integrating Eq. (19) and substituting it into Eq. (18), we obtain $\frac{d\widetilde{\chi}(t)}{dt}=-\int\limits_{0}^{t}\sum_{k,i}V_{k,i}^{2}\exp\left[-i(\omega_{k,i}-\eta\;\Delta)(t-t^{\prime})\right]\;\widetilde{\chi}(t^{\prime})\;dt^{\prime}.$ (20) This integro-differential equation can be solved exactly by a Laplace transformation, $\overline{\widetilde{\chi}(p)}=\frac{\widetilde{\chi}(0)}{p+\sum_{k,i}V_{k,i}^{2}/\left[p-i(\eta\;\Delta-\omega_{k,i})\right]},$ (21) with $\overline{\widetilde{\chi}(p)}=\int\widetilde{\chi}(t)\exp(-pt)dt.$ (22) Inversing of the Laplace transformation, we obtain the amplitude in the excited-state $\widetilde{\chi}(t)=\frac{1}{2\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty}\frac{\widetilde{\chi}(0)\exp(pt)}{p+\sum_{k,i}V_{k,i}^{2}/\left[p-i(\eta\;\Delta-\omega_{k,i})\right]}dp$ (23) Then replace $p$ to $i\omega+0^{+},$ $\widetilde{\chi}(t)=\frac{1}{2\pi i}\int_{-\infty}^{\infty}\frac{\widetilde{\chi}(0)\exp(i\omega t)}{\omega-\sum_{k,i}V_{k,i}^{2}/\left[(\omega+\eta\;\Delta)-\omega_{k,i}-i0^{+}\right]}d\omega$ (24) Denote $R(\omega)$ and $\Gamma(\omega)$ as the real and imaginary parts of the summation term $\sum_{k,i}V_{k,i}^{2}/(\omega-\omega_{k,i}-i0^{+}),$ then $\displaystyle R(\omega)$ $\displaystyle=$ $\displaystyle\wp\sum_{k,i}V_{k,i}^{2}/(\omega-\omega_{k,i}-i0^{+})$ (25) $\displaystyle\Gamma(\omega)$ $\displaystyle=$ $\displaystyle\pi\sum_{k,i}V_{k,i}^{2}\delta(\omega-\omega_{k,i}-i0^{+})$ (26) where $\wp$ is the Cauchy principal value. Applying the pole approximation, $\widetilde{\chi}(t)=\widetilde{\chi}(0)\sum_{j}\exp(i\omega_{j}t)Q_{j}(\omega_{j})$ (27) where $\omega_{j}$ corresponds to the singularity of the quantity $\overline{\widetilde{\chi}(p)}$ and $Q_{j}(\omega_{j})$ is the normalized factor. Before doing further calculations, let us now focus on the initial state of the system $\widetilde{\chi}(0)$, since different initial states may result in distinct predictions about the QZE and the AZE PRL-101-200404 ; pra-ai . Indeed, these two effects can strongly depend on the initial conditions. Through the unitary transformation in Eq. (2), the Hamiltonian (1), which contains the anti-rotating terms, is reduced to $H^{\prime}$ in Eq. (6), which has the similar form of the Hamiltonian under the RWA, with the parameters renormalized. Under energy conservation, the ground state of $H^{\prime}$ is $\left\|g^{\prime}\right\rangle=\left\|\downarrow\right\rangle\left\|\\{0_{k}\;0_{k}\\}\right\rangle$ and the corresponding ground-state energy is $-\eta\;\Delta/2$. Therefore, through inversing the unitary transformation, we obtain the ground state of the original Hamiltonian $H$ as $\left\|g\right\rangle=\exp[-S]\left\|\downarrow\right\rangle\left\|\\{0_{k}\;0_{k}\\}\right\rangle,$ which is a dressed state of the qubit and its environment due to the anti- rotating terms prl-103-147003 ; pra-80-053810 . In this paper, we choose the excited state $\exp[-S]\left\|\uparrow\right\rangle\left\|\\{0_{k}\;0_{k}\\}\right\rangle$ as the initial state, which can be achieved by acting the operator $\sigma_{x}$ on the ground state, $\left\|\psi(0)\right\rangle=\sigma_{x}\left\|g\right\rangle=\exp[-S]\left\|\uparrow\right\rangle\left\|\\{0_{k}\;0_{k}\\}\right\rangle.$ (28) Thus, the initial state after the transformation is $\left\|\psi^{\prime}(0)\right\rangle=\left\|\uparrow\right\rangle\left\|\\{0_{k}\;0_{k}\\}\right\rangle$, correspondingly the excited-state probability amplitude $\chi(0)=1.$ To obtain the final result, we need the knowledge of the interacting spectra of the qubit’s intrinsic bath and also the cavity bath. From the quasi-mode approach, the qubit-cavity coupling density spectrum is a Lorentzian density spectrum pra-71-032302 ; a $J_{\mathrm{cav}}(\omega)=\sum g_{k,2}^{2}\delta(\omega-\omega_{k})=\frac{g^{2}\;\lambda}{\pi[(\omega-\omega_{\mathrm{cav}})^{2}+\lambda^{2}]},$ (29) where $g$ is the coupling constant between the cavity and the qubit, $\omega_{\mathrm{cav}}$ the central frequency of the cavity mode, and $\lambda$ is the frequency width of the cavity bath density spectrum and is related to the cavity bath correlation time. The physical quantity $\omega_{\mathrm{cav}}/\lambda$ denotes the quality factor $Q$ of the cavity. Experiments in some superconducting qubits indicate that the noise chiefly comes from the low-frequency region. The density spectrum of the low-frequency bath can be approximately written as $J_{\mathrm{qb}}^{\mathrm{low}}(\omega)=\sum_{k}g_{k,2}^{2}\delta(\omega-\omega_{k,1})=\frac{2\;\alpha_{\mathrm{low}}\;\omega}{\left(\omega/\Delta\right)^{2}+\left(\omega_{\mathrm{low}}/\Delta\right)^{2}}.$ (30) where $\omega_{\mathrm{low}}$ is an energy lower than the qubit energy spacing $\Delta$, and $\alpha_{\mathrm{low}}$ a dimensionless coupling strength between the qubit and the intrinsic bath. In semiconductor quantum dot qubits, the qubit spontaneous dissipation bath, mainly the phonon bath, is usually described by an Ohmic density spectrum. Thus, the density spectrum $J_{\mathrm{qb}}(\omega)$ of the Ohmic bath with Drude cutoff can be given as $J_{\mathrm{qb}}^{\mathrm{Ohm}}(\omega)=\sum_{k}g_{k,1}^{2}\delta(\omega-\omega_{k,1})=\frac{2\;\alpha_{\mathrm{Ohm}}\;\omega}{1+\left(\omega/\omega_{\mathrm{Ohm}}\right)^{2}},$ (31) where $\omega_{\mathrm{Ohm}}$ is the high-frequency cutoff, which is typically assumed to be larger than the qubit energy level spacing, and $\alpha_{\mathrm{Ohm}}$ is the dimensionless coupling strength. So we consider three kinds of interacting density spectra: Lorentzian cavity bath, low-frequency qubit’s intrinsic bath and Ohmic qubit’s intrinsic bath, and present a sketch of the density spectra of the qubit environment in Fig. 2(b, c): showing the same cavity bath and different qubit’s intrinsic baths (a low-frequency bath in (b) and an Ohmic bath in (c)). Figure 3: (Color online) Time dependence of the probability for the qubit at its excited state. In the resonant case, the parameters are $\omega_{\mathrm{cav}}=\Delta=100~{}g$ and $\tau=0.1~{}g^{-1}$. In the detuning case, the cavity mode frequency is varied to $\omega_{\mathrm{cav}}=80~{}g$. Note that the successive measurements slow down the decay rate of excited state in the resonant case, which is the QZE. While in the detuning case, the measurements speed up the qubit decay rate, which is the AZE. Before illustrate our results, let us recall the standard master equation of a qubit coupled to a single-mode cavity under the RWA and Markov approximation pra-40-5516 $\displaystyle\dot{\rho}$ $\displaystyle=$ $\displaystyle-i\left[H_{\mathrm{RWA}},\rho\right]+\gamma\left(2\sigma_{-}\rho\sigma_{+}-\sigma_{+}\sigma_{-}\rho-\rho\sigma_{+}\sigma_{-}\right)$ (32) $\displaystyle+\kappa\left(2a\rho a^{\dagger}-a^{\dagger}a\rho-\rho a^{\dagger}a\right),$ where $H_{\mathrm{RWA}}=g(\sigma_{-}a^{\dagger}+\sigma_{+}a),$ $g$ is the qubit-cavity coupling strength, $a^{\dagger}$ and $a$ are the creation and annihilation operators for the single-mode cavity. The two parameters $\kappa$ and $\gamma$ correspond to the decay rates induced by the two baths: the qubit’s intrinsic bath and the cavity bath, respectively. Then the survival probability of the qubit in the excited state is approximately pra-40-5516 $P_{\mathrm{e}}(t)=\left\|\chi(t)\right\|^{2}\;=\cos\left(gt\right)\exp\left[-\left(\kappa+\gamma\right)t/2\right],$ (33) where the subscript “$\mathrm{e}$” refers to the initial and final excited states. The exponential factor $\left(\kappa+\gamma\right)/2$ can be considered as an effective decay rate. In the RWA case, the qubit energy is splitting to $\Delta\pm g/2.$ While in our results beyond the RWA, the qubit energy splitting depends on the qubit environment. Assume $\lambda=0.1~{}g,$ $\omega_{\mathrm{cav}}=100~{}g.$ If the qubit in the low-frequency bath with $\omega_{\mathrm{low}}=10~{}g$ and $\alpha_{\mathrm{low}}=10^{-4}$, the qubit energy is splitting to $\Delta-0.4786~{}g$ and $\Delta+0.5011~{}g.$ While, if the qubit in the Ohmic bath with $\omega_{\mathrm{Ohm}}=10^{3}~{}g$ and $\alpha_{\mathrm{Ohm}}=10^{-4},$ the qubit energy level is splitting to $\Delta-0.5018~{}g$ and $\Delta+0.4782~{}g.$ Figure 3 shows the probability $P_{\mathrm{e}}(t)$ for the qubit to be in the excited state in the region $0<t<\pi$. When the qubit and the cavity mode is resonant, the qubit decay with the measurements, whose interval between successive measurements is $\tau=0.1~{}g^{-1},$ is slowed down compared to the case without measurement (the interval $\tau$ extends to infinite), which means QZE. While tune the cavity mode to $\omega_{\mathrm{cav}}=80~{}g$ and fix the energy level spacing of the qubit $\Delta=100~{}g$, the decay with the measurements ($\tau=0.1~{}g^{-1}$) is speeded up contrast to the case without the measurements, which means AZE. ## IV The effective decay rate of a qubit in a cavity with successive measurements In the following, we will solve the Eq. (20) iteratively and obtain the effective decay rate with successive measurement pra-54-R3750 ; A10 . When the interval between measurements is sufficiently short, the evolution of the qubit after measurements can be approximately expressed by an exponential form. So the discussion in nature-405-546 can be extended to damped oscillations. Namely, if the exponential factor is larger or smaller than the effective decay rate $\left(\kappa+\gamma\right)/2,$ then the measurements reduce or enhance the decay rate. After the first iteration, Eq. (20) is solved as $\widetilde{\chi}(t)\simeq 1-\int\limits_{0}^{t}(t-t^{{}^{\prime}})\sum_{k,i}V_{k,i}^{2}\exp[-i(\omega_{k,i}-\eta\;\Delta)t^{{}^{\prime}}]\;dt^{{}^{\prime}}.$ (34) For a small $t$, we can approximately write $\widetilde{\chi}(t)$ in an exponential form: $\displaystyle\widetilde{\chi}(t)$ $\displaystyle=$ $\displaystyle\exp\left[-\int\limits_{0}^{t}(t-t^{{}^{\prime}})\sum_{k,i}V_{k,i}^{2}\exp\left[-i(\omega_{k,i}-\eta\;\Delta)t^{{}^{\prime}}\right]dt^{{}^{\prime}}\right]$ (35) $\displaystyle=$ $\displaystyle\exp\left\\{-t\left[-\frac{1}{t}\sum_{k,i}V_{k,i}^{2}\frac{\exp\left[-i\left(\omega_{k,i}-\eta\;\Delta\right)t\right]-1+i\left(\omega_{k,i}-\eta\;\Delta\right)t}{\left(\omega_{k,i}-\eta\;\Delta\right)^{2}}\right]\right\\}$ $\displaystyle=$ $\displaystyle\exp\left\\{-t\left[\sum_{k,i}V_{k,i}^{2}\left(\frac{2\sin\left(\frac{\omega_{k,i}-\eta\;\Delta}{2}t\right)^{2}}{t\left(\omega_{k,i}-\eta\;\Delta\right)^{2}}-i\frac{\left(\omega_{k,i}-\eta\;\Delta\right)t-\sin\left[\left(\omega_{k,i}-\eta\;\Delta\right)t\right]}{t\left(\omega_{k,i}-\eta\;\Delta\right)^{2}}\right)\right]\right\\}.$ Note that only when $\tau\ll g^{-1},$ the qubit evolution can be approximately described as an exponential decay pra-40-5516 ; pra-54-R3750 ; pra-81-062131 , which has been reflected in Fig. 2. Assume now that the instantaneously-ideal projection measurement is performed periodically, separated by time intervals $\tau$. For a single measurement, the probability amplitude of the qubit maintaining in the initial state is $\widetilde{\chi}(t=\tau).$ After a sufficiently large number of measurements, the survival probability of the initial state becomes $P_{\mathrm{e}}(t=n\tau)\;=\;\left\|\widetilde{\chi}(t=n\tau)\right\|^{2}\;=\;\exp[-\gamma(\tau)t].$ (36) And the exponential decay constant $\gamma(\tau)$ is obtained $\displaystyle\gamma(\tau)$ $\displaystyle=$ $\displaystyle 2\pi\int_{0}^{\infty}d\omega\sum_{k,i}V_{k,i}^{2}\;\frac{2\sin^{2}(\frac{\eta\;\Delta-\omega}{2}\tau)}{\pi(\eta\;\Delta-\omega)^{2}\tau}$ (37) $\displaystyle=$ $\displaystyle 2\pi\int_{0}^{\infty}d\omega J(\omega)f(\omega)F(\omega-\eta\;\Delta,\tau),$ where $f(\omega)=\left(1-\frac{\omega-\eta\;\Delta}{\omega+\eta\;\Delta}\right)^{2},$ (38) $\displaystyle J(\omega)$ $\displaystyle=$ $\displaystyle\sum_{k}\left[f_{k}^{2}\delta(\omega-\omega_{k,1})+g_{k}^{2}\delta(\omega-\omega_{k,2})\right]$ (39) $\displaystyle=$ $\displaystyle J_{\mathrm{cav}}(\omega)+J_{\mathrm{qb}}(\omega),$ (40) and $F(\omega-\eta\;\Delta,\tau)=\frac{2\sin^{2}\left[\left(\eta\;\Delta-\omega\right)\tau/2\right]}{\pi(\eta\;\Delta-\omega)^{2}\tau}.$ (41) In Eq. (40), $J(\omega)$ is the entire interacting density spectrum with $J_{\mathrm{cav}}(\omega)$ from the cavity bath and $J_{\mathrm{qb}}(\omega)$ the qubit’s intrinsic bath. The function $F(\omega-\eta\;\Delta,\tau)$ comes from the projection measurements and can be called a modulating function of the measurements. The decay rate $\gamma(\tau)$, in Eq. (37), depends on the renormalization factor $\eta$ and $f(\omega)$ in Eq. (38), which are mainly from the anti- rotating terms. If we use the RWA, $\eta=1$ and $f(\omega)=1,$ which is consistent with the case of weak interaction. Therefore, our results can apply to not only the weak coupling case, but also to the case of strong coupling between the qubit and the environment. Furthermore, since the function $F(\omega-\Delta,\tau)$ becomes $\delta(\omega-\eta\;\Delta)$ in the long-time limit, we obtain the effective decay rate under the Weisskopf-Wigner approximation $\gamma_{0}=\gamma(\tau\rightarrow\infty)=2\pi J(\eta\;\Delta).$ (42) The normalized decay rate, which characterizes the QZE and the AZE, is determined by $\frac{\gamma(\tau)}{\gamma_{0}}=\frac{\int_{0}^{\infty}d\omega J(\omega)f(\omega)F(\omega-\eta\;\Delta,\tau)}{J(\eta\;\Delta)}.$ (43) For a finite time $\tau,$ and when $\gamma(\tau)/\gamma_{0}<1$ holds, we have the QZE, i.e., measurements hinder the decay. However, when $\gamma(\tau)/\gamma_{0}>1,$ this implies the AZE, i.e., measurements enhance the decay. To see the contribution of each bath to the decay rate, Eq. (43) can be reexpressed as $\displaystyle\frac{\gamma(\tau)}{\gamma_{0}}$ (44) $\displaystyle=$ $\displaystyle\frac{J_{\mathrm{cav}}(\eta\;\Delta)}{J(\eta\;\Delta)}\frac{\int_{0}^{\infty}d\omega J_{\mathrm{cav}}(\omega)f(\omega)F(\omega-\eta\;\Delta,\tau)}{J_{\mathrm{cav}}(\eta\;\Delta)}$ $\displaystyle+\frac{J_{\mathrm{qu}}(\eta\;\Delta)}{J(\eta\;\Delta)}\frac{\int_{0}^{\infty}d\omega J_{\mathrm{qu}}(\omega)f(\omega)F(\omega-\eta\;\Delta,\tau)}{J_{\mathrm{qu}}(\eta\;\Delta)}.$ From this Eq. (44), we see that the normalized decay rate due to the two baths is combined by the normalized decay rate from each bath by the weights $J_{\mathrm{qb}}(\eta\;\Delta)/J(\eta\;\Delta)$ and $J_{\mathrm{cav}}(\eta\;\Delta)/J(\eta\;\Delta),$ respectively. ## V Results and Discussion In this section, we will show the normalized decay rate of the qubit-cavity system in three cases: $\left(i\right)$ only the cavity bath, $\left(ii\right)$ both the cavity bath and the low-frequency qubit spontaneous dissipation bath coexist, as well as both the cavity bath and the Ohmic qubit’s intrinsic bath coexistence. According to the experiment nature-431-162 , we consider the qubit weakly coupled to the qubit intrinsic bath with coupling constants $\alpha_{\mathrm{Ohm}}=10^{-4}$ and $\alpha_{\mathrm{low}}=10^{-4}.$ The quality factor $Q$ of the cavity is assumed in the range of $2\times 10^{2}\sim 10^{4}$. ### V.1 Only cavity bath Let us first consider the case of a qubit only in a cavity bath. For example, when the qubit-cavity interaction $g\gg\alpha_{\mathrm{low}}\Delta,$ or $g\gg\alpha_{\mathrm{Ohm}}\Delta$, which has been realized in a superconducting qubit coupled to a transmission line cavity nature-458-178 ; add8 . For such strong coupling between the qubit and cavity, the normalized decay rate mainly depends on the cavity bath. Then in this case, the decay rate can be approximately written as $\gamma(\tau)=2\pi\int_{0}^{\infty}d\omega\;J_{\mathrm{cav}}(\omega)\;f(\omega)\;F(\omega-\eta\;\Delta,\tau).$ (45) From the normalized decay rate in Eq. (43), we see that the qubit-cavity coupling strength $g$ is in both, the numerator and denominator, so it cancels out. Therefore, the normalized decay rate $\gamma(\tau)$ is independent of the qubit-cavity coupling strength $g.$ However we still note that only in the case when $\tau\ll g^{-1},$ the qubit evolution can be approximately described by an exponential decay. This means that if there is a strong qubit-cavity coupling $g=0.1\Delta,$ the measurement interval becomes $\tau\ll g^{-1}\sim 10\Delta^{-1}.$ When the qubit-cavity is not so strong, $g=10^{-2}\Delta,$ the measurement interval could be $\tau\ll g^{-1}\sim 10^{2}\Delta^{-1}.$ Figure 4 displays the normalized decay rate as a function of the measurement interval $\tau$ and the cavity central frequency $\omega_{\mathrm{cav}}.$ Figures 4(a) and (b) correspond to two quality factors of the cavity: $Q=10^{4}$ and $Q=2\times 10^{3}$, respectively. We can see that in the limit when $\tau\rightarrow 0,$ only the QZE occurs. For a finite interval, the normalized decay rate of the qubit exhibits a transition from the QZE to the AZE, by modulating the central frequency of the cavity mode $\omega_{\mathrm{cav}}$ in and off resonance with the qubit energy level spacing $\Delta$. The variation should be useful to distinguish the QZE and the AZE. Let us now estimate the condition for the transition between the QZE and the AZE. From Fig. 4(a), the crossover from QZE to AZE, by varying the cavity frequency, appears only for the measurement interval $\tau>0.6\Delta^{-1}$. Using the condition $\tau\ll g^{-1}$, we obtain the qubit-cavity coupling strength $g\ll 1.7\Delta.$ Similarly, for the cavity quality factor $Q=2\times 10^{3},$ we obtain the qubit-cavity coupling strength $g\ll 0.38\Delta.$ In Fig. 5, we plot the normalized decay rate with the cavity frequency in resonance with the qubit, $\omega_{\mathrm{cav}}=\Delta$, versus the time interval $\tau$ between successive measurements, and the cavity spectral width $\lambda$. It is obvious that only the QZE exists in the resonant case. The normalized decay rate $\gamma(\tau)/\gamma_{0}$ becomes smaller as the cavity spectral width $\lambda$ decreases. This indicates that the transition from the QZE to the AZE becomes sharper as the cavity spectral width $\lambda$ reduces. Figure 4: (Color online) Contour plots of the normalized decay rate $\gamma(\tau)/\gamma_{0}$ of the qubit only in the cavity bath, versus the time interval $\tau$ between successive measurements, and the central frequency $\omega_{\mathrm{cav}}$ of the cavity mode. (a) The width of the cavity frequency is $\lambda=10^{-4}\Delta$, and accordingly the cavity quality factor $Q=10^{4}.$ (b) The width of the cavity frequency $\lambda=5\times 10^{-3}\Delta$, corresponding to the cavity quality factor $Q=2\times 10^{3}$. The region $1\leq\gamma(\tau)/\gamma_{0}\leq 1.05$ is shown as light magenta. The QZE region corresponds to $\gamma(\tau)/\gamma_{0}<1$. The AZE region covers the rest, when $\gamma(\tau)/\gamma_{0}>1$. Evidently, a transition from the QZE to the AZE is observed by varying the central frequency of the cavity mode at finite $\tau$ ($\tau>0.6\Delta^{-1}$ when $Q=10^{4}$, and $\tau>2.6\Delta^{-1}$ when $Q=2\times 10^{3}$). Figure 5: (Color online) Contour plots of the normalized effective decay rate $\gamma(\tau)/\gamma_{0}$ of the qubit in the resonant case $\Delta=\omega_{\mathrm{cav}}$. To better understand the transition from the QZE to the AZE, we discuss the results in two regimes of near-resonance (including on-resonance) and off- resonance (the central frequency of the cavity mode higher and lower than the qubit spacing $\Delta$): 1\. In the case of on-resonance $\Delta=\omega_{\mathrm{cav}},$ and near- resonance $\left\|\Delta-\omega_{\mathrm{cav}}\right\|<\lambda$, without measurements, the effective decay rate of the qubit is given by $J(\Delta).$ Moreover, the qubit is resonant with the cavity mode, $\Delta=\omega_{\mathrm{cav}}$. Note that $\omega_{\mathrm{cav}}$ is the peak of the density spectrum of the cavity-bath, where the probability of energy transfer from the qubit to the cavity bath is maximum. In this case, the qubit strongly decays in its evolution. Every measurement to project the qubit on the initial state protects the qubit from decay, i.e., protects the qubit from exchanging energy with the cavity. From Eq. (41), the modulating function $F(\omega-\eta\Delta,\tau)$ of the measurements is a periodically oscillating function versus energy $\omega$ for a fixed time interval $\tau$. Moreover, its integral over all energies is $1$. Thus we consider each oscillator peak as a decay channel induced by measurements. Without measurements, $F(\omega-\eta\;\Delta,\tau)$ becomes $\delta(\omega-\eta\;\Delta,\tau)$. Only one channel $\omega=\eta\;\Delta$ exists. With measurements, more channels will appear, but the probability of qubit-energy-decay via every channel decreased to less than 1. Among these channels, the largest one is still $\omega=\eta\;\Delta$, which is less than the non-measurement one. Therefore, the superposition of the density spectrum function $J_{\mathrm{cav}}(\omega)$ of the cavity-bath and the modulating function $F(\omega-\eta\;\Delta,\tau)$ of the measurements reduces the effective decay rate and protects the qubit energy from leaking to the cavity-bath when the qubit is resonantly coupled to the cavity. 2\. For the case of off-resonance $\left\|\Delta-\omega_{\mathrm{cav}}\right\|>\lambda,$ especially for the large- detuning limit $\left\|\Delta-\omega_{\mathrm{cav}}\right\|\gg\lambda,$ the effective interaction between the qubit and the cavity becomes very weak. For example, the ratio of the effective decay rate in $\omega_{\mathrm{cav}}=0.98\Delta$ (or $\omega_{\mathrm{cav}}=1.02\Delta$) to $\omega_{\mathrm{cav}}=\Delta$ is about $2\times 10^{-5}.$ In most quantum optics papers, large-detuning means that the qubit is free from decay. Thus, the probability of the qubit maintaining its initial state is close to 1. After introducing frequent measurements, the qubit suffers from AZE, i.e., measurements enhance the decay, which is opposite to the on-resonant case. The reason for this also comes from the modulating function of the measurements $F(\omega-\eta\;\Delta,\tau),$ a periodic oscillation function of the energy. As long as one of the oscillation peaks of $F(\omega-\eta\;\Delta,\tau)$ is located in the effective region of the density spectrum $J(\omega)$, especially the half-width of the maximum, the product of these two functions will lead to an enhancement of the effective decay rate. In other words, the periodic oscillations of $F(\omega-\eta\;\Delta,\tau)$ connect the qubit energy with the density spectrum of the cavity bath and open the decay channels of the qubit energy to the cavity bath. From this point of view, the measurements act as a new decay element, besides the cavity and the qubit intrinsic bath. The AZE becomes more obvious as the detuning increases. In the above discussion, we have investigated the qubit decay dynamics subject to measurements mainly induced by the cavity bath. Also, we have studied the qubit decay dynamics subject to measurements due to either the low-frequency qubit spontaneous dissipation bath or the Ohmic qubit intrinsic bath in Ref. [arkiv, ]. In the next two subsections, we will show the normalized effective decay rate of the qubit in the presence of both the cavity bath and the qubit’s intrinsic bath. ### V.2 Coexistence of the cavity bath and the low-frequency qubit intrinsic bath Figure 6: (Color online) Contour plots of the normalized effective decay rate $\gamma(\tau)/\gamma_{0}$ in the presence of both the cavity bath and the low- frequency qubit’s intrinsic bath. The interaction strength $\alpha_{\mathrm{low}}=10^{-4}$, between the qubit and qubit’s intrinsic bath, and the qubit-cavity coupling $g=10^{-2}\Delta$. (a) Results for the cavity quality factor $Q=10^{4}.$ (b) Results for the cavity quality factor $Q=2\times 10^{3}.$ The region $1\leq\gamma(\tau)/\gamma_{0}\leq 1.05$ is shown as light magenta. The QZE region corresponds to $\gamma(\tau)/\gamma_{0}<1$. The AZE region covers the rest, when $\gamma(\tau)/\gamma_{0}>1$. Evidently, a transition from the QZE to the AZE is observed by varying the central frequency $\omega_{\mathrm{cav}}$ of the cavity mode at finite $\tau$. Figure 7: (Color online) The qubit-cavity coupling is $g=10^{-3}\Delta$. The other caption is the same as Fig. 6. In Figs. 6 and 7, we plot the normalized effective decay rate, when the cavity bath and the low-frequency qubit spontaneous dissipation bath coexist, versus the time interval $\tau$ between measurements in the regime of strong ($g=10^{-2}\Delta$) and weak ($g=10^{-3}\Delta$) cavity-qubit coupling with the cavity central frequency around the qubit energy-level-spacing $\Delta.$ Figures 6(a) and (b) correspond to the cavity quality factor $Q=10^{4}$ and $Q=2\times 10^{3}$, respectively. From Fig. 6, we see that for a strong qubit- cavity coupling, by modulating the cavity central frequency from in-resonance to off-resonance with the qubit energy-level-spacing, the normalized effective decay rate grows and becomes larger than $1,$ which clearly displays the transition from the QZE to the AZE. Comparing Figs. 6(a) and (b), as the width $\lambda$ of the cavity frequency decreases (or the quality factor $Q=\omega_{\mathrm{cav}}/\lambda$ increases), the region of the cavity frequency for the QZE becomes narrower. For example, Table I presents the normalized effective decay rate $\gamma(\tau)/\gamma_{0}$ for two quality factors $Q$ and different central frequencies $\omega_{\mathrm{cav}}$ of the cavity, when $\tau=5\;\Delta^{-1}$. Table 1: The normalized effective decay rate $\gamma(\tau)/\gamma_{0}$ of the qubit for two quality factors $Q$ when $\tau=5\;\Delta^{-1}$, in the presence of both the cavity bath and the low-frequency qubit’s intrinsic bath. Cavity quality factor \| Central frequency of the cavity ---\|--- $0.98\Delta$ \| $\;0.99\Delta$ \| $0.999\Delta$ \| $\Delta$ \| $1.001\Delta$ \| $1.01\Delta$ \| $1.02\Delta$ $Q=10^{4}$ \| 1.994 \| 1.784 \| 0.122 \| 0.001 \| 0.122 \| 1.777 \| 1.979 $Q=2\times 10^{3}$ \| 1.727 \| 1.149 \| 0.032 \| 0.006 \| 0.032 \| 1.145 \| 1.714 For weak qubit-cavity coupling in Fig. 7, only in the short-time regime (about $2\Delta^{-1}<\tau<6\Delta^{-1}$), the normalized effective decay rate of the qubit shows obviously the transition from the QZE to the AZE. When the measurement interval $\tau$ increases to $\tau>10\Delta^{-1}$, although the transition still exists, $\gamma(\tau)/\gamma_{0}$ for the AZE is slightly larger than 1, which is mainly in the region between $1.0$ and $1.1.$ In Figs. 6 and 7, there appear distinct oscillations in the qubit’s QZE-AZE transition processes. From the results of Ref. [arkiv, ], we have known that the AZE always occurs for a qubit in the low-frequency bath. While, for a qubit in the cavity bath, the transition from QZE to AZE takes place by varying the cavity frequency. These oscillations in Figs. 6-7 come from the different impacts of the cavity-bath and the low-frequency bath on the qubit’s measurement dynamics. ### V.3 Coexistence of the cavity bath and the Ohmic qubit spontaneous dissipation bath Figure 8: (Color online) Contour plots of the normalized effective decay rate $\gamma(\tau)/\gamma_{0}$ in the presence of both: the cavity bath and the Ohmic qubit’s intrinsic bath. The interaction strength $\alpha_{\mathrm{Ohm}}=10^{-4}$, between the qubit and the qubit’s intrinsic bath. Also the qubit-cavity coupling $g=10^{-2}\Delta$. (a) The cavity quality factor of the cavity $Q=10^{4}.$ (b) The cavity quality factor $Q=2\times 10^{3}.$ The region $1\leq\gamma(\tau)/\gamma_{0}\leq 1.05$ is shown as light magenta. The QZE region corresponds to $\gamma(\tau)/\gamma_{0}<1$. The AZE region is the rest, when $\gamma(\tau)/\gamma_{0}>1$. Evidently, a transition from the QZE to the AZE is observed by varying the central frequency $\omega_{\mathrm{cav}}$ of the cavity mode at finite $\tau$. Figure 9: (Color online) The qubit-cavity coupling $g=10^{-3}\Delta$. The other caption is the same as Fig. 8 In Figs. 8 and 9, we show the normalized effective decay rate $\gamma(\tau)/\gamma_{0}$ in the presence of both the Ohmic intrinsic bath and cavity bath. Comparing Figs. 8 with 6, we find that for the strong qubit- cavity coupling $g=10^{-2}\Delta,$ the time interval $\tau$ for the QZE increases in the short-time region. In the long-time region, the features of Figs. 6 and 8 are almost identical. From Fig. 9 we can see that in the short- time region, ($0<\tau<30\Delta^{-1})$, only the QZE exists, regardless of the central frequency of cavity. For $\tau>30\Delta^{-1},$ the normalized effective decay rate $\gamma/\gamma_{0}$ for the AZE is in the small region of $1.0\sim 1.02$, which is not conducive to observe the transition from the QZE to the AZE. ## VI Summary We investigated the QZE and AZE of a qubit in a cavity when both the cavity bath and the qubit’s intrinsic bath (either low-frequency or Ohmic bath) are simultaneously present. We find that in the case of strong qubit-cavity coupling, modulating the cavity central frequency from on-resonance ($\omega_{\mathrm{cav}}=\Delta)$ to off-resonance ($\omega_{\mathrm{cav}}$ larger or smaller than $\Delta)$ with the qubit energy-level-spacing, the transition from the QZE to the AZE occurs. Thus, our results provide a proposal to observe the QZE and the AZE in the qubit-cavity system. Acknowledgements We thank A. G. Kofman for comments on the manuscript. FN acknowledges partial support from DARPA, AFOSR, the National Security Agency (NSA), Laboratory for Physical Sciences (LPS), Army Research Office (USARO), National Science Foundation (NSF) under Grant No. 0726909, JSPS-RFBR under Contract No. 09-02-92114, MEXT Kakenhi on Quantum Cybernetics, and FIRST (Funding Program for Innovative R&D on S&T). X.-F. Cao acknowledges support from the National Natural Science Foundation of China under Grant No. 10904126 and Fujian Province Natural Science Foundation under Grant No. 2009J05014. ——————– ## References * (1) W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland , Phys. Rev. A 41, 2295 (1990). * (2) A. Peres and A. Ron, Phys. Rev. A 42, 5720 (1990). * (3) E. Block and P. R. Berman, Phys. Rev. A 44, 1466 (1991). * (4) M. Namiki, S. Pascazio and H. Nakazato, Decoherence and Quantum measurements (World Scientific, Singapore, 1997). * (5) A. G. Kofman and G. Kurizki, Nature 405, 546 (2000). * (6) P. Facchi, H. Nakazato, and S. 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1011.3923	# Interaction anisotropy and random impurities effects on the critical behaviour of ferromagnets H Chamati 1 and S Romano 2 1 Institute of Solid State Physics, Bulgarian Academy of Sciences, 72 Tzarigradsko Chaussée, 1784 Sofia, Bulgaria chamati@issp.bas.bg 2 Dipartimento di Fisica “A. Volta“, Università di Pavia, Via A. Bassi 6, I-27100 Pavia, Italy Silvano.Romano@pv.infn.it ###### Abstract The theory of phase transitions is based on the consideration of "idealized" models, such as the Ising model: a system of magnetic moments living on a cubic lattice and having only two accessible states. For simplicity the interaction is supposed to be restricted to nearest–neighbour sites only. For these models, statistical physics gives a detailed description of the behaviour of various thermodynamic quantities in the vicinity of the transition temperature. These findings are confirmed by the most precise experiments. On the other hand, there exist other cases, where one must account for additional features, such as anisotropy, defects, dilution or any effect that may affect the nature and/or the range of the interaction. These features may have impact on the order of the phase transition in the ideal model or smear it out. Here we address two classes of models where the nature of the transition is altered by the presence of anisotropy or dilution. To appear in Journal of Physics: Conference Series ## 1 Introduction Materials in Nature can be found in qualitatively different phases having distinct properties. The change from one phase to another is the consequence of a variation of an intensive thermodynamic quantity, e.g., the temperature $T$, the pressure $P$, external electric or magnetic fields $E$ or $H$. Phase transitions are accompanied by abrupt changes in a number of macroscopic thermodynamic quantities. Some familiar examples of phase transitions include the gas–liquid transition (condensation), the liquid–solid transition (freezing), the normal–superconducting transition in conductors, the paramagnet–ferromagnet transition in magnetic materials, and the superfluid transition in liquid helium. Further examples are transitions involving amorphous or glassy structures, spin glasses, liquid crystals, charge–density waves, and spin–density waves. In many cases, the two phases above and below the transition point, say $T_{c}$ for a temperature driven phase transition, may be discerned from each other in terms of some ordering that occurs in the phase below $T_{c}$. For example in the liquid–solid transition the molecules of the liquid get “ordered” in space when they form the solid phase. In a paramagnet, the magnetic moments of each atom can point in any random direction (in the absence of an external magnetic field), but in the ferromagnetic phase the moments are lined up in a particular direction of ordering. Thus in the high temperature phase (above $T_{c}$), the degree of ordering is smaller than in the low temperature phase (below $T_{c}$). To quantify the amount of ordering in a system one uses the so called order parameter, which is usually a vanishing quantity in the high temperature (disordered) phase. The phase diagram shows regions within which homogeneous equilibrium states exist as a function of temperature and other thermodynamic variables like $P$, $E$, $H$. For some physical systems the chemical potential $\mu$ or composition variables are also involved. The different regions of the phase diagram are delimited by phase boundaries that mark conditions under which multiple phases can coexist at equilibrium. Phase transitions take place along phase boundaries marked by lines of equilibrium. The theoretical framework that aims at describing phase transitions and related phenomena as a result from cooperative effects over macroscopic scales is a part of the realm of equilibrium statistical physics [1, 2, 3]. Statistical physics is based on probabilistic models of the interactions of microscopic entities forming large assemblages (macroscopic bodies). The probability a macroscopic system of volume $V$, having $N$ particles, in a state $\mathfrak{S}$ with energy $E$ at a temperature $T$, is given by $\mathscr{P}(\mathfrak{S})=\frac{e^{-E(\mathfrak{S})/(k_{B}T)}}{\mathscr{Z}(T)},$ (1) where $k_{B}$ is the Boltzmann constant, and $\mathscr{Z}(T)$ is the partition function $\mathscr{Z}(T)=\sum_{\mathfrak{S}}e^{-E(\mathfrak{S})/(k_{B}T)},$ (2) relating microscopic degrees of freedom to macroscopic thermodynamic quantities. The function $\mathscr{Z}(T)$ depends upon any parameters that might affect the value of $E(\mathfrak{S})$. The expectation value of any statistical operator $\mathfrak{O}$ is defined via $\langle\mathfrak{O}(\mathfrak{S})\rangle=\sum_{\mathfrak{S}}\mathfrak{O}(\mathfrak{S})\mathscr{P}(\mathfrak{S}).$ (3) For example the internal energy $U$ is obtained by averaging $E(\mathfrak{S})$ over all the accessible states of the system. This is given by $U=\langle E(\mathfrak{S})\rangle.$ A fundamental thermodynamic quantity related to the partition function is the free energy $f(T)=-k_{B}T\ln\mathscr{Z}(T),$ (4) which contains all the information on the thermodynamics of the considered system. According to the Ehrenfest classification scheme there are different kinds of phase transitions depending on the nature of the singularities of the thermodynamic quantities at the transition. Such transitions are categorized as first–, second–, or higher order transitions if the lowest derivative of the free energy that exhibit nonanalytic behaviour with a finite jump is the first, second or higher one. Within this classification the Berezinskiǐ–Kosterlitz–Thouless (BKT) [4, 5] may be considered as being of infinite order. To describe universal features of phase transitions one uses relatively “simple” microscopic models such as the magnetic model involving interaction between magnetic degrees of freedom. In a classical model for a magnet, the spins (magnetic moments) may be represented by $n$–component unit vectors $\bm{s}_{i}$ located at sites $i$, with coordinates $\bm{x}_{i}$, belonging to a finite subset of a generic $d$–dimensional lattice $\Lambda_{d}$, i.e. $\mathfrak{L}\subset\Lambda_{d}$, with $\|\mathfrak{L}\|=N$. We are considering here saturated lattice models, where each lattice site hosts one spin. The simplest and probably most extensively studied cases considered in the literature assume a hypercubic lattice $\Lambda_{d}=\mathbb{Z}^{d}$ and isotropically interacting spins, thus the (nearest–neighbour) Hamiltonian $\mathscr{H}_{\mathfrak{L}}=-\frac{J}{2}\sum_{\langle i,j\rangle}\bm{s}_{i}\cdot\bm{s}_{j}-\bm{H}\sum_{i}\bm{s}_{i},$ (5) where the coupling $J$ is restricted to nearest–neighbouring sites $i$ and $j$, with each distinct pair being counted once. $\bm{H}$ stands for an uniform magnetic field. In the absence of the magnetic field, i.e. $\bm{H}=\bm{0}$, a ferromagnetic coupling, $J>0$, favours a parallel orientation of the spins in the ground state, whereas thermal fluctuations tend to create an orientational disorder. On the other hand an interaction, $J<0$, would be the precursor of an antiferromagnetic order at low temperatures. Actually, in the specific case considered here, i.e. nearest–neighbour coupling and bipartite lattice, and with $\bm{H}=\bm{0}$, the sign of the coupling constant is immaterial, i.e. models defined by $+J$ and $-J$ yield the same partition function, whereas the two correlation functions are connected by suitable sign factors. For $n=1,2,3$, the model corresponds to the Ising, planar rotator (PR) and Heisenberg (He), respectively. The Ising model is known to have a $Z_{2}$ discrete symmetry, while systems with $n\geq 2$, like PR and He, are said to possess a continuous $O(n)$ symmetry. It has become customary to refer to the number of component $n$ of the order parameter as symmetry index. The interaction with an external field breaks this symmetry and establishes a preferred direction for spin alignment. By reducing the external field to zero in the thermodynamic limit, the system may exhibit spontaneous magnetization pointing in the initial direction of the field. It has been shown [6] that the limit $n\to\infty$ leads to the spherical model [7] obtained by requiring the spins to be continuous variables subject to a global relaxed constraint ($\sum_{i=1}^{N}\bm{s}_{i}^{2}=N$) rather than forcing them to take unit lengths i.e. $\|\bm{s}_{i}\|^{2}=1$. The formal limit $n\to 0$ is relevant to the study of self–avoiding walk problem, which can be applied to polymers. By now a number of rigorous results, assuming translational invariance, have been worked out, entailing existence or absence of a phase transition in the thermodynamic limit, depending on lattice dimensionality $d$ and number of spins components $n$ [8, 9]. Model (5) with different values of $n$ and $d$ is extensively studied in the literature via different methods and its behaviour as a function of the temperature is very well known. For a review with a rich list of references see [10]. At $H=0$, it exhibits a second order phase transition,111Here and below the temperature will be measured in units of $J/k_{B}$. for any $d>1$ and with discrete spin variables, i.e. $n=1$. Such a transition is characterized by a significant growth of the nearest–neighbour correlations for orientational fluctuations, and also the onset for long–range orientational correlations. On the other hand, according to the Mermin–Wegner theorem [11] in $O(n)$ symmetric models ($n\geq 2$) there can be no spontaneous symmetry breaking at finite temperatures for $d\leq 2$ meaning that the system remains orientationally disordered at any finite temperature. For $d>2$ and $n\geq 2$ a second order phase transition takes place in the system. In the two–dimensional case $d=2$ and $n=2$ the system exhibits a BKT transition from a high temperature disordered phase to a low temperature phase with slow decay of the correlation function and an infinite magnetic susceptibility [12]. In the vicinity of a continuous phase transition, such as second order or BKT transition, there is only one dominating length scale related to the growth of fluctuations: The correlation length. Because of the diverging nature of the correlation length as the critical point is approached the microscopic details of the system becomes irrelevant. Thus the description of the singular behaviour of many thermodynamic observables requires a small number of universal variables: critical exponents, amplitudes and functions. This allows the arrangement of a great variety of different microscopic systems in universality classes of equivalent critical behaviour. A universality class depends upon the number of components of the order parameter and the dimensionality of the system. For more details the reader is invited to consult references [13, 14]. The model (5) can describe the properties of a wide variety of physical situations in the vicinity of their transition point, however, it may happen that the very experimental situation at hand requests the interaction model to be complicated (made physically richer and theoretically more challenging) in various ways. On the one hand, there exist different possible lattice types $\Lambda_{d}$, in addition to $\mathbb{Z}^{d}$. On the other hand, as for the orientational dependence, more elaborate potential models involve (in some combination or other) isotropic or anisotropic linear couplings between spin components, sometimes higher powers of scalar products among the interacting spins, multipolar (usually dipolar) interactions, Dzyaloshinski–Moriya terms, single–site anisotropy fields. Notice also that more distant neighbours are sometimes involved, or even, in principle, all neighbours may be coupled by long–range interactions. In some specific favourable cases one has even been able to match the model and its potential parameters to a specific experimental system [15]. The ferromagnetic ordering transition observed experimentally in the absence of an external field is more frequently second order, but first–order transitions are also known. They might, for example, result from doping by nonmagnetic impurities, anisotropy of interactions in spin space, or coupling to the lattice [16]. Other interaction models for different physical systems can be found in [17]. This review is devoted to the description and analysis of effects related to the nature of the interaction. This aims at gaining insights in the thermodynamics of some specific systems. The models considered here are some of the most popular models in the theory of phase transitions. They involve different kind of interactions that hopefully might be adapted to the characteristics of a given material. They describe ferromagnetism with anisotropic coupling, systems with random dilution and may to some extent be used to investigate fluids. Special attention is paid to the construction of the phase diagrams of these models that are determined from the investigation of different thermodynamic quantities such as: the free energy, the susceptibility, specific heat etc. The review is organised as follows: In Section 2 we discuss the transitional behaviour of generalised XY models introduced in reference [18]. These are generalization of the XY model with a nontrivial coupling along the $z$ components of the spins. We construct the phase diagrams of the models in two and three dimensions and determine the effect of the nontrivial coupling on the nature and location of transition temperature. In Section 3 we review the effect of dilution of ferromagnets by the introduction of random impurities and present the phase diagram of the diluted Heisenberg in three dimensions and the diluted plane rotator in two dimensions. We conclude with Section 4, where we discuss the results and comment on other models sharing similar transitional behaviour. ## 2 Generalised XY models Let us first consider interactions being anisotropic in spin space with nonvanishing and equal ferromagnetic couplings involving $m<n$ components of the partner spins only, while still keeping the interaction restricted to nearest neighbours. In this case the model reads $\mathscr{H}_{\mathrm{anis.}}=-\frac{J}{2}\sum_{\langle i,j\rangle}\sum_{\alpha,\beta\leq m}\delta_{\alpha,\beta}s_{i}^{\alpha}s_{j}^{\beta}.$ (6) Among these models we may mention the continuous Ising ($m=1$ and $n$ arbitrary) and the various versions of XY models ($m=2$ and $n$ arbitrary), where the interaction is restricted to only one component and two components of the magnetic moments, respectively In general the transitional behaviour of such models is analogous to that of their $O(m)$ isotropic counterparts: On the one hand, the universal critical behaviour of these models in the vicinity of their corresponding transition temperature is equivalent to their isotropic $O(m)$ counterparts i.e. the models share the same universal features (critical exponents and amplitudes). on the other hand, the transition temperature is a typical non–universal quantity, and is recognizably affected by the anisotropy. A more general class of anisotropic spin models may be constructed by introducing some kind of extreme anisotropy coupling only a part of the spin components in some nontrivial manner in addition to the anisotropic interaction in the spin space. A model that will be discussed in this study is the so called “generalised” XY model introduced in reference [18]. This model involves $3-$component unit vectors, and is defined by $\mathscr{H}_{XY}^{p}=-\frac{J}{2}\sum_{\langle i,j\rangle}(\sin\theta_{i}\sin\theta_{j})^{p}\cos(\phi_{i}-\phi_{j}),$ (7) where $p\in\mathbb{N}$ is a parameter controlling the strength of anisotropy along the $z$–spin direction, and the spins are expressed in terms of the usual spherical coordinates $\theta_{i}$ and $\phi_{i}$ i.e. $\bm{s}_{i}=(\sin\theta_{i}\cos\phi_{i},\sin\theta_{i}\sin\phi_{i},\cos\theta_{i})$. Notice that by setting $p=1$ ($p=0$) we recover the familiar XY model (planar rotator). In this case of the planar rotator model, the $\theta$–dependence only survives in the free–spin measure. The Hamiltonian (7) can also be written in terms of spin components, in the more complicated form $\mathscr{H}_{XY}^{p}=-\frac{J}{2}\sum_{\langle i,j\rangle}\left[1-(s_{i}^{z})^{2}-(s_{j}^{z})^{2}+(s_{i}^{z}s_{j}^{z})^{2}\right]^{(p-1)/2}\left(s_{i}^{x}s_{j}^{x}+s_{i}^{y}s_{j}^{y}\right).$ (8) As for the role of $p$ in Eq. (7), notice that it could be taken to be a real positive number, say ranging between $0$ and $1$ (and hence continuously interpolating between planar rotator and XY models). On the other hand, larger values of $p$ reinforce the out–of–plane fluctuations. This makes it possible to widely vary the anchoring of spins with respect to the horizontal plane which might have direct experimental relevance. As for the present model, this change of anchoring is ultimately reflected by the significant changes in transition behaviour. The transitional behaviour of the model in two and three dimensions has been investigated by different approaches. First, it was proven rigorously [18], on the basis of the known behaviour of PR, that when $d=2$ and for all values of $p$, the named potential models produce orientational disorder at all finite temperatures, and support a BKT–like transition. On the other hand, when $d=3$, these models support ordering transitions taking place at finite temperatures. In both cases, the transition temperatures are bounded from above by the corresponding values for the PR counterpart. It was later proven [19], again rigorously, that the transition turns first–order for sufficiently large $p$. Notice that the threshold values had to be estimated by other means. ### 2.1 Two dimensions Using Monte Carlo simulations, the thermodynamics of model (7) has been investigated in details in reference [20] for $d=2$ and values of $p$ ranging from 2 to 5. Analysis of Monte Carlo data, showed that the model produces a BKT–(like) transition, possibly changing to a first–order transition for larger $p$, due to the large number of vortices and strong out–of–plane fluctuations. It has been found that the transition temperature is indeed decreasing with increasing $p$. To gain insight into the transitional behaviour of the model for large values of $p$ we performed further Monte Carlo simulations at $p=6$. Our analysis shows that the transition is most likely to have a weak first order nature. Unfortunately more simulations are required and different approaches are needed to be more conclusive. Figure 1: ($p,T$) phase diagram of the two–dimensional generalised XY model. Full circles indicate a Berezinskiǐ–Kosterlitz–Thouless like phase transition, while the square is believed to corresponds to a weak first order phase transition. In figure 1 we show the phase diagram of the two dimensional generalised XY model in the $(p,T)$ plane. It is seen that the model exhibits a phase transition from a paramagnetic phase to a BKT–like phase as the temperature decreases. The transition temperature is evaluated using Monte Carlo simulations according to reference [20]. The value of the transition temperature corresponding to $p=1$ is taken from reference [21]. ### 2.2 Three dimensions For $d=3$, different analytical approximations, such as a Mean Field (MF) approach, as well as its Two–Site Cluster (TSC) refinement have been used in reference [18] to estimate transition temperatures for $p=2,3,4$, and then for higher values of $p$ in subsequent papers (see below). Notice also that $(\sin\theta)^{p}$, and hence the absolute value of the interaction potential, decreases with increasing $p$, and this aspect is reflected by the $p$–dependence of the estimated transition temperature. Transition temperatures have been estimated in reference [22] by self–consistent harmonic approximation, both for $d=2$ and $d=3$, and it was found that the transition temperature is decreasing against $p$. A study of the model in its continuum limit, carried out in reference [22] also showed that out–of–plane fluctuations, and consequently the magnon density, decrease with increasing $p$. We have also addressed the three dimensional generalised XY models for various values of $p$, by means of Monte Carlo (MC) simulation. We have investigated the transitional behaviour of various thermodynamic functions, such as the susceptibility and the specific heat, and made comparisons with MF and TSC predictions. MF yielded a tricritical behaviour with tricritical points having real (non–integer) values of the parameter $p$. As for simulation results, transitional behaviour characetristic of the XY model was found for $p=2,3,4,8$, the case $p=12$ suggested tricritical behaviour, whereas evidence of first–order transitions was obtained for $p=16,20$. In Figure 2 we present the $(p,T)$ phase diagram, where we can read off the behaviour of the transition temperature as a function of $p$ for $1\leq p\leq 20$. MF and TSC data can be found in references [18, 23, 24, 19], while Monte carlo data are taken from references [25, 23, 24]. Notice that, for $p\leq 4$, TSC gives better estimates of $T_{c}$ than MF, then the two roles are exchanged for $p=8,12$, and finally the three methods give very similar answers when $p\geq 16$, i.e. where the transition has a pronounced first–order character. Figure 2: Phase diagram of the three dimensional generalised XY model. The solid and dashed lines stand for transition temperatures estimated via mean–field and two–site–cluster treatments, respectively. Full circles and squares indicate second and first order phase transitions temperatures evaluated via Monte Carlo simulations, respectively. Full diamonds shows the locations of tricritical points. ## 3 Annealed dilution in ferromagnets Another class of systems that has been extensively studied in the literature spans those with random impurities (disorder). For a review on diluted magnetism see reference [26]. The interest in these systems stems from the fact that in nature no system is really pure. Indeed, the presence of cavities, grain boundaries, lattice defects, chemical impurities (i.e. of other chemical individuals) or some other kind of disorder may affect the properties of the pure system, and result in different effects. To consider a simple but rather important example, in a system of magnetically coupled spins, some lattice sites may be occupied by nonmagnetic constituents (site–dilution). Both in experimental and theoretical terms, one can distinguish between annealed and quenched disorder. In quenched systems, impurities are held frozen in randomly distributed fixed positions, without the possibility of overcoming potential barriers for diffusing into the host material: in this case the relaxation time is very long and thermal equilibrium between impurities and the constituents of the host is never reached. In annealed materials, impurities are allowed to diffuse randomly and to reach thermal equilibrium with the other constituents of the host material. One can also think of a two–component solution being very dilute with respect to one of the components. When the system is in the liquid phase, molecules of the two types can exchange their positions, and diffuse throughout the sample. Let then the system crystallize, at sufficiently low temperature. In the resulting solid, particles of the minority component are fixed in certain lattice sites only. The simplest extension of equation (5) taking these situations into account reads $\mathscr{H}_{\mathrm{dis.}}=-\frac{J}{2}\sum_{\langle i,j\rangle}\nu_{i}\nu_{j}\ \bm{s}_{i}\cdot\bm{s}_{j}.$ (9) where the occupation numbers $\nu_{i}$ equal “zero” for a site $i$ hosting a nonmagnetic impurity and “one” for a magnetic particle. The density of magnetic particles in the system is defined via $\rho=\frac{1}{N}\sum_{i}^{N}\langle\nu_{i}\rangle.$ Within this notation the pure system, i.e. without impurities, corresponds to $\rho=1$. In the annealed case, Hamiltonian (9) can be interpreted as describing a two–component system consisting of interconverting “real” ($\nu_{i}=1$) and “ghost”, “virtual” or ideal–gas particles ($\nu_{i}=0$). Both kinds of particles have the same kinetic energy, and the total number of particles equals the number of available lattice sites. In this case one works in the Grand–Canonical Ensemble where the probability (1) for a configuration, now involving the occupation numbers $\\{\nu_{i}\\}$, as well as the spins $\\{\bm{s}_{i}\\}$, is defined by $\mathscr{P}\propto\exp\left[-\beta\left(\mathscr{H}_{\mathrm{dis.}}-\mu\sum_{i}\nu_{i}\right)\right],$ (10) where $\mu$ denotes the excess chemical potential of “real” particles over “ideal” ones. Of course, the interaction may be anisotropic in spin space, as in the cases outlined above, or involve more distant neighbours. The system remains translationally invariant on average. This model bears some similarity with the Blume–Emery–Griffiths model [27] for 3He impurities in superfluid 4He. More precisely, notice also that, starting from an assigned model, its lattice gas extensions can be written in general as $\mathscr{H}_{LG}=-\frac{J}{2}\sum_{\langle i,j\rangle}\nu_{i}\nu_{j}\ \bm{s}_{i}\cdot\bm{s}_{j}+\frac{\lambda}{2}\sum_{\langle i,j\rangle}\nu_{i}\nu_{j}\ ,$ (11) where the purely positional $\lambda$ term only becomes immaterial in the pure limit $\mu\rightarrow+\infty$. In equation (9) we have chosen the simplest case $\lambda=0$. Lattice gas models can be used to model adsorption, and, in general, the variable occupation numbers produce some fluidity of the system. One can also set the coupling $J$ term to zero, so that the resulting model becomes isomorph with an Ising model in external field. In general, the interplay between $J$ and $\lambda$ can produce a richer phase diagram: for example, when $\lambda>0$ and $\mu$ is sufficiently large, the ground state may exhibit checkerboard positional order but no orientational one. It is very well known that a small amount of annealed disorder does not affect the way the singularities take place in pure systems i.e. the phase transition remains a second order one. The $\rho$–dependent critical temperature is shifted towards $T_{c}(\rho)<T_{c}(1)$. If the chemical potential $\mu$ is held fixed, the properties of the phase transition are the same as those known for the pure system, corresponding to $\mu\to\infty$ and $\rho=1$. If, however, the concentration $\rho$ is kept fixed during the transition, care must taken in characterising the transition. For details see reference [28]. A significant amount of impurities may alter the order of the phase transition or make it disappear. Investigations of the XY model, as a protype for He3 – He4 mixtures, in three dimensions, via high temperature series expansion of the partition function, show that by increasing the density of randomness the transition temperature decreases. At a certain value of $\rho$ the transition changes its nature and turns into a first order one [29]. The second order phase transition line ends at a tricritical point, which marks the begining of a line of fisrt–oder phase transition temperature that keeps decreasing as the concentration of impurities increases. In the three–dimensional case, the topology of the phase diagram of model (9) had been investigated by MF and TSC approximations for the Ising [30], as well as PR cases [31] in the presence of a magnetic field, and for He at zero magnetic field [32]. These investigations were later extended [33] to the extremely anisotropic (Ising–like) two–dimensional model, and in the absence of a magnetic field, as well. The studied models were found to exhibit a tricritical behaviour i.e. the ordering transition turned out to be of first order for $\mu$ below an appropriate threshold, and of second order above it. When the transition is of first order, the orientationally ordered phase is also denser than the disordered one. To check the predictions of the molecular–field–like treatments used to construct the phase diagrams, extensive Monte Carlo simulations has been performed [31, 32, 33] for particular values of the chemical potential. A number of thermodynamic and structural properties had been investigated. It had been found that there is a second order ferromagnetic phase transition manifested by a significant growth of magnetic and density fluctuations. The transition temperatures were found to be smaller than those of the corresponding values for the pure systems and the critical behaviour of the investigated models to be consistent with that of their pure counterparts. Furthermore it had been found that MF yields a qualitatively correct picture, and the quantitative agreement with simulation could be improved by TSC, which has the advantage of predicting two–site correlations. In general we found that simulations results are consistent with the molecular–field like treatments. Figure 3: Phase diagrams of the three–dimensional Heisenberg model in the chemical potential – temperature and density – temperature planes. Solid and dashed lines correspond to mean–field and two–site–cluster results, respectively. Circles and squares stand, respectively, for estimates of the second and first order transition temperatures using Monte Carlo simulations. Diamonds mark the locations of tricritical points. The dotted line marks the transition temperature of the model in the absence of impurities. In figure 3 we report the phase diagrams in the chemical potential – temperature and density – temperature planes. We show the behaviour of the transition temperature $T_{c}$ versus chemical potential $\mu$. The plot also shows a fast, approximately linear, increase of $T_{c}$ with $\mu$ up to $\mu\approx 0$, and a slower one above this value. Moreover, MF and TSC results essentially coincide up to $\mu\approx 0$. In the phase diagram in the ($\rho,T$) plane the existence of a first–order transition is reflected by a biphasic region. We found that the system exhibits a first–order phase transition from a dense BKT phase to a paramagnetic one. In the temperature–density phase diagram, both phases are expected to coexist over some range of densities and temperatures. Two–dimensional annealed lattice models were investigated [34, 35] as well, and the obtained results for $\mu=0$ or a moderately negative $\mu$ were found to support a BKT phase transition with a transition temperature lower than that of the pure parent due to the presence of impurities. For negative and sufficiently large in magnitude $\mu$ we found evidence of a first order phase transition in agreement with renormalization group treatments [36, 37]. Figure 4: Phase diagrams of the two–dimensional planar rotator. Circles and squares stand, respectively, for estimates of the second and first order transition temperatures using Monte Carlo simulations. Diamonds mark the locations of tricritical points. The dotted line marks the transition temperature of the model in the absence of impurities. In figure 4 we show the phase diagram in the $(\rho,T)$ plane. The existence of a first–order transition is reflected here by a biphasic (binodal) region. The phase diagram obtained here is similar to the one resulting from the study of the diluted planar rotator model [31]. So far, we have discussed isotropic models. As we have mentioned, in addition to dilution one can consider anisotropic interactions as well. In reference [33] we have investigated two–dimensional continuous Ising spin models with two and tree component spins. The phase diagrams of these models were found to be topologically similar to those found for the Heisenberg model shown in figure 3. The main differences come from the locations of the phase boundaries. ## 4 Conclusion The Statistical Mechanics of lattice systems is essentially based on the study of a number of relatively “simple” models which are gradually made more complicated. We have shown here some examples, involving generalized XY models and annealed magnetic systems, where rigorous results have been produced to prove existence and type of transition, supplemented by a variety of techniques (MF or TSC approximation and Monte Carlo simulation) for elucidating the resulting physical behaviour and estimating numerical values. We obtained the phase diagram in two and three dimensions of a generalized version of the lattice XY model, where the out–of–plane fluctuations of the spins are controlled by a parameter $p$. Our investigation shows that the nature of the transition is highly affected by the strength of the out–of–plane fluctuations. For $p\geq 12$ at three dimensions and $p\gtrsim 6$ at two dimensions, it is first order, whereas for small values its transitional behaviour coincides with that of the original XY model. The transition temperature is found to decrease as $p$ increases at any dimension. We constructed the phase diagrams of the annealed lattice plane rotator in two dimensions and the Heisenberg model in three dimensions. It is found that the transition changes from a second order at three dimensions and Berezinskiǐ–Kosterlitz–Thouless at two dimensions into a first order transition as the concentration of impurities is increased. In turn the transition temperature is found to decrease with increasing impurity density. This work was supported by the Ministry of Education and Science of Bulgaria under Grant No $\Phi$–1517. ## References * [1] Yeomans J M 1992 Statistical Mechanics of Phase Transitions (New York: Oxford University Press) * [2] Pathria R K 1996 Statistical mechanics 2nd ed (Oxford, England: Butterworth–Heinemmann) * [3] Mazenko G F 2003 Fluctuations, Order and Defects (Hoboken: Wiley) * [4] Berezinskiǐ V L 1971 Sov. Phys. JETP 32 493–500 [Zh. Eksp. Teor. Fiz. 59, 907–920] * [5] Kosterlitz J M and Thouless D J 1973 J. Phys. C: Solid State Phys. 6 1181–1203 * [6] Stanley H E 1968 Phys. Rev. 176 718–722 * [7] Berlin T H and Kac M 1952 Phys. Rev. 86 821–835 * [8] Georgii H O 1988 Gibbs Measures and Phase Transitions (de Gruyter Studies in Mathematics vol 9) (Berlin: Walter de Gruyter) * [9] Sinai Y G 1982 Theory of Phase Transitions: Rigorous Results (Oxford: Pergamon) * [10] Pelissetto A and Vicari E 2002 Phys. Rep. 368 549–727 * [11] Mermin N D and Wagner H 1966 Phys. Rev. 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B 19 2488–2503	arxiv-papers	2010-11-17T10:09:36	2024-09-04T02:49:14.920189	{ "license": "Public Domain", "authors": "H Chamati and S Romano", "submitter": "Hassan Chamati", "url": "https://arxiv.org/abs/1011.3923" }
1011.3947	# Covariant Transform Vladimir V. Kisil School of Mathematics, University of Leeds, Leeds LS2 9JT, UK (On leave from the Odessa University) kisilv@maths.leeds.ac.uk ###### Abstract. The paper develops theory of covariant transform, which is inspired by the wavelet construction. It was observed that many interesting types of _wavelets_ (or _coherent states_) arise from group representations which _are not_ square integrable or vacuum vectors which _are not_ admissible. Covariant transform extends an applicability of the popular wavelets construction to classic examples like the Hardy space $H_{2}{}$, Banach spaces, covariant functional calculus and many others. Keywords: Wavelets, coherent states, group representations, Hardy space, Littlewood–Paley operator, functional calculus, Berezin calculus, Radon transform, Möbius map, maximal function, affine group, special linear group, numerical range, characteristic function, functional model. _Dedicated to the memory of Cora Sadosky_ ###### Contents 1. 1 Covariant Transform 2. 2 Examples of Covariant Transform 3. 3 Induced Covariant Transform 4. 4 Inverse Covariant Transform A general group-theoretical construction [Perelomov86, FeichGroech89a, Kisil98a, AliAntGaz00, Fuhr05a, ChristensenOlafsson09a, KlaSkag85] of _wavelets_ (or _coherent states_) starts from an irreducible square integrable representation—in the proper sense or modulo a subgroup. Then a mother wavelet is chosen to be admissible. This leads to a wavelet transform which is an isometry to $L_{2}{}$ space with respect to the Haar measure on the group or (quasi)invariant measure on a homogeneous space. The importance of the above situation shall not be diminished, however an exclusive restriction to such a setup is not necessary, in fact. Here is a classical example from complex analysis: the Hardy space $H_{2}{}(\mathbb{T}{})$ on the unit circle and Bergman spaces $B_{2}^{n}{}(\mathbb{D}{})$ in the unit disk produce wavelets associated with representations $\rho_{1}$ and $\rho_{n}$ of the group $SL_{2}{}(\mathbb{R}{})$ respectively [Kisil97c]. While representations $\rho_{n}$ are from square integrable discrete series, the mock discrete series representation $\rho_{1}$ is not square integrable [Lang85]§ VI.5 [MTaylor86]§ 8.4. However it would be natural to treat the Hardy space in the same framework as Bergman ones. Some more examples will be presented below. ## 1\. Covariant Transform To make a sharp but still natural generalisation of wavelets we give the following definition. ###### Definition 1. [Kisil09d] Let ${\rho}$ be a representation of a group $G$ in a space $V$ and $F$ be an operator from $V$ to a space $U$. We define a _covariant transform_ $\mathcal{W}$ from $V$ to the space $L{}(G,U)$ of $U$-valued functions on $G$ by the formula: (1) $\mathcal{W}:v\mapsto\hat{v}(g)=F({\rho}(g^{-1})v),\qquad v\in V,\ g\in G.$ Operator $F$ will be called _fiducial operator_ in this context. We borrow the name for operator $F$ from fiducial vectors of Klauder and Skagerstam [KlaSkag85]. ###### Remark 2. We do not require that fiducial operator $F$ shall be linear. Sometimes the homogeneity, i.e. $F(tv)=tF(v)$ for $t>0$, alone can be already sufficient, see Example 12. ###### Remark 3. Usefulness of the covariant transform is in the reverse proportion to the dimensionality of the space $U$. The covariant transform encodes properties of $v$ in a function $\mathcal{W}v$ on $G$. For a low dimensional $U$ this function can be ultimately investigated by means of harmonic analysis. Thus $\dim U=1$ (scalar-valued functions) is the ideal case, however, it is unattainable sometimes, see Example 9 below. We may have to use a higher dimensions of $U$ if the given group $G$ is not rich enough. As we will see below covariant transform is a close relative of wavelet transform. The name is chosen due to the following common property of both transformations. ###### Theorem 4. The covariant transform (1) intertwines ${\rho}$ and the left regular representation $\Lambda$ on $L{}(G,U)$: $\mathcal{W}{\rho}(g)=\Lambda(g)\mathcal{W}.$ Here $\Lambda$ is defined as usual by: (2) $\Lambda(g):f(h)\mapsto f(g^{-1}h).$ ###### Proof. We have a calculation similar to wavelet transform [Kisil98a]Prop. 2.6. Take $u={\rho}(g)v$ and calculate its covariant transform: $\displaystyle{}[\mathcal{W}({\rho}(g)v)](h)$ $\displaystyle=$ $\displaystyle[\mathcal{W}({\rho}(g)v)](h)=F({\rho}(h^{-1}){\rho}(g)v)$ $\displaystyle=$ $\displaystyle F({\rho}((g^{-1}h)^{-1})v)$ $\displaystyle=$ $\displaystyle[\mathcal{W}v](g^{-1}h)$ $\displaystyle=$ $\displaystyle\Lambda(g)[\mathcal{W}v](h).$ ∎ The next result follows immediately: ###### Corollary 5. The image space $\mathcal{W}(V)$ is invariant under the left shifts on $G$. ## 2\. Examples of Covariant Transform We start from the classical example of the group-theoretical wavelet transform: ###### Example 6. Let $V$ be a Hilbert space with an inner product $\left\langle\cdot,\cdot\right\rangle$ and ${\rho}$ be a unitary representation of a group $G$ in the space $V$. Let $F:V\rightarrow\mathbb{C}{}$ be a functional $v\mapsto\left\langle v,v_{0}\right\rangle$ defined by a vector $v_{0}\in V$. The vector $v_{0}$ is oftenly called the _mother wavelet_ in areas related to signal processing or the _vacuum state_ in quantum framework. Then the transformation (1) is the well-known expression for a _wavelet transform_ [AliAntGaz00, (7.48)] (or _representation coefficients_): (3) $\mathcal{W}:v\mapsto\hat{v}(g)=\left\langle{\rho}(g^{-1})v,v_{0}\right\rangle=\left\langle v,{\rho}(g)v_{0}\right\rangle,\qquad v\in V,\ g\in G.$ The family of vectors $v_{g}={\rho}(g)v_{0}$ is called _wavelets_ or _coherent states_. In this case we obtain scalar valued functions on $G$, thus the fundamental rôle of this example is explained in Rem. 3. This scheme is typically carried out for a square integrable representation ${\rho}$ and $v_{0}$ being an admissible vector [Perelomov86, FeichGroech89a, AliAntGaz00, Fuhr05a, ChristensenOlafsson09a]. In this case the wavelet (covariant) transform is a map into the square integrable functions [DufloMoore] with respect to the left Haar measure. The map becomes an isometry if $v_{0}$ is properly scaled. However square integrable representations and admissible vectors does not cover all interesting cases. ###### Example 7. Let $G=\mathrm{Aff}$ be the “$ax+b$” (or _affine_) group [AliAntGaz00, § 8.2]: the set of points $(a,b)$, $a\in\mathbb{R}_{+}{}$, $b\in\mathbb{R}{}$ in the upper half-plane with the group law: (4) $(a,b)(a^{\prime},b^{\prime})=(aa^{\prime},ab^{\prime}+b)$ and left invariant measure $a^{-2}\,da\,db$. Its isometric representation on $V=L_{p}{}(\mathbb{R}{})$ is given by the formula: (5) $[{\rho_{p}}(g)\,f](x)=a^{\frac{1}{p}}f\left(ax+b\right),\qquad\text{where }g^{-1}=(a,b).$ We consider the operators $F_{\pm}:L_{2}{}(\mathbb{R}{})\rightarrow\mathbb{C}{}$ defined by: (6) $F_{\pm}(f)=\frac{1}{2\pi i}\int_{\mathbb{R}{}}\frac{f(t)\,dt}{x\mp\mathrm{i}}.$ Then the covariant transform (1) is the Cauchy integral from $L_{p}{}(\mathbb{R}{})$ to the space of functions $\hat{f}(a,b)$ such that $a^{-\frac{1}{p}}\hat{f}(a,b)$ is in the Hardy space in the upper/lower half- plane $H_{p}{}(\mathbb{R}^{2}_{\pm}{})$. Although the representation (5) is square integrable for $p=2$, the function $\frac{1}{x\pm\mathrm{i}}$ used in (6) is not an admissible vacuum vector. Thus the complex analysis become decoupled from the traditional wavelet theory. As a result the application of wavelet theory shall relay on an extraneous mother wavelets [Hutnik09a]. Many important objects in complex analysis are generated by inadmissible mother wavelets like (6). For example, if $F:L_{2}{}(\mathbb{R}{})\rightarrow\mathbb{C}{}$ is defined by $F:f\mapsto F_{+}f+F_{-}f$ then the covariant transform (1) reduces to the _Poisson integral_. If $F:L_{2}{}(\mathbb{R}{})\rightarrow\mathbb{C}^{2}{}$ is defined by $F:f\mapsto(F_{+}f,F_{-}f)$ then the covariant transform (1) represents a function $f$ on the real line as a jump: (7) $f(z)=f_{+}(z)-f_{-}(z),\qquad f_{\pm}(z)\in H_{p}{}(\mathbb{R}^{2}_{\pm}{})$ between functions analytic in the upper and the lower half-planes. This makes a decomposition of $L_{2}{}(\mathbb{R}{})$ into irreducible components of the representation (5). Another interesting but non-admissible vector is the Gaussian $e^{-x^{2}}$. ###### Example 8. For the group $G=SL_{2}{}(\mathbb{R}{})$ [Lang85] let us consider the unitary representation ${\rho}$ on the space of square integrable function $L_{2}{}(\mathbb{R}^{2}_{+}{})$ on the upper half-plane through the Möbius transformations: (8) ${\rho}(g):f(z)\mapsto\frac{1}{(cz+d)^{2}}\,f\left(\frac{az+b}{cz+d}\right),\qquad g^{-1}=\ \begin{pmatrix}a&b\\\ c&d\end{pmatrix}.$ This is a representation from the discrete series and $L_{2}{}(\mathbb{D}{})$ and irreducible invariant subspaces are parametrised by integers. Let $F_{k}$ be the functional $L_{2}{}(\mathbb{R}^{2}_{+}{})\rightarrow\mathbb{C}{}$ of pairing with the lowest/highest $k$-weight vector in the corresponding irreducible component $B_{k}{}(\mathbb{R}^{2}_{\pm}{})$, $k\geq 2$ of the discrete series [Lang85, Ch. VI]. Then we can build an operator $F$ from various $F_{k}$ similarly to the previous Example. In particular, the jump representation (7) on the real line generalises to the representation of a square integrable function $f$ on the upper half-plane as a sum $f(z)=\sum_{k}a_{k}f_{k}(z),\qquad f_{k}\in B_{n}{}(\mathbb{R}^{2}_{\pm}{})$ for prescribed coefficients $a_{k}$ and analytic functions $f_{k}$ in question from different irreducible subspaces. Covariant transform is also meaningful for principal and complementary series of representations of the group $SL_{2}{}(\mathbb{R}{})$, which are not square integrable [Kisil97c]. ###### Example 9. A straightforward generalisation of Ex. 6 is obtained if $V$ is a Banach space and $F:V\rightarrow\mathbb{C}{}$ is an element of $V^{}$. Then the covariant transform coincides with the construction of wavelets in Banach spaces [Kisil98a]. ###### Example 10. The next stage of generalisation is achieved if $V$ is a Banach space and $F:V\rightarrow\mathbb{C}^{n}{}$ is a linear operator. Then the corresponding covariant transform is a map $\mathcal{W}:V\rightarrow L{}(G,\mathbb{C}^{n}{})$. This is closely related to M.G. Krein’s works on _directing functionals_ [Krein48a], see also _multiresolution wavelet analysis_ [BratJorg97a], Clifford-valued Bargmann spaces [CnopsKisil97a] and [AliAntGaz00, Thm. 7.3.1]. ###### Example 11. Let $F$ be a projector $L_{p}{}(\mathbb{R}{})\rightarrow L_{p}{}(\mathbb{R}{})$ defined by the relation $(Ff)\hat{\ }(\lambda)=\chi(\lambda)\hat{f}(\lambda)$, where the hat denotes the Fourier transform and $\chi(\lambda)$ is the characteristic function of the set $[-2,-1]\cup[1,2]$. Then the covariant transform $L_{p}{}(\mathbb{R}{})\rightarrow C{}(\mathrm{Aff},L_{p}{}(\mathbb{R}{}))$ generated by the representation (5) of the affine group from $F$ contains all information provided by the _Littlewood–Paley operator_ [Grafakos08]§ 5.1.1. ###### Example 12. A step in a different direction is a consideration of non-linear operators. Take again the “$ax+b$” group and its representation (5). We define $F$ to be a homogeneous but non-linear functional $V\rightarrow\mathbb{R}_{+}{}$: $F(f)=\frac{1}{2}\int\limits_{-1}^{1}\left\|f(x)\right\|\,dx.$ The covariant transform (1) becomes: (9) $[\mathcal{W}_{p}f](a,b)=F({\rho_{p}}(a,b)f)=\frac{1}{2}\int\limits_{-1}^{1}\left\|a^{\frac{1}{p}}f\left(ax+b\right)\right\|\,dx=a^{\frac{1}{p}}\frac{1}{2a}\int\limits^{b+a}_{b-a}\left\|f\left(x\right)\right\|\,dx.$ Obviously $M_{f}(b)=\max_{a}[\mathcal{W}_{\infty}f](a,b)$ coincides with the Hardy _maximal function_ , which contains important information on the original function $f$. From the Cor. 5 we deduce that the operator $M:f\mapsto M_{f}$ intertwines ${\rho_{p}}$ with itself ${\rho_{p}}M=M{\rho_{p}}$. Of course, the full covariant transform (9) is even more detailed than $M$. For example, $\left\\|f\right\\|=\max_{b}[\mathcal{W}_{\infty}f](\frac{1}{2},b)$ is the shift invariant norm [Johansson08a]. ###### Example 13. Let $V=L_{c}{}(\mathbb{R}^{2}{})$ be the space of compactly supported bounded functions on the plane. We take $F$ be the linear operator $V\rightarrow\mathbb{C}{}$ of integration over the real line: $F:f(x,y)\mapsto F(f)=\int_{\mathbb{R}{}}f(x,0)\,dx.$ Let $G$ be the group of Euclidean motions of the plane represented by ${\rho}$ on $V$ by a change of variables. Then the wavelet transform $F({\rho}(g)f)$ is the _Radon transform_. ###### Example 14. Let a representation ${\rho}$ of a group $G$ act on a space $X$. Then there is an associated representation ${\rho_{B}}$ of $G$ on a space $V=B{}(X,Y)$ of linear operators $X\rightarrow Y$ defined by the identity: (10) $({\rho_{B}}(g)A)x=A({\rho}(g^{-1})x),\qquad x\in X,\ g\in G,\ A\in B{}(X,Y).$ Following the Remark 3 we take $F$ to be a functional $V\rightarrow\mathbb{C}{}$, for example $F$ can be defined from a pair $x\in X$, $l\in Y^{}$ by the expression $F:A\mapsto\left\langle Ax,l\right\rangle$. Then the covariant transform is: $\mathcal{W}:A\mapsto\hat{A}(g)=F({\rho_{B}}(g)A).$ This is an example of _covariant calculus_ [Kisil98a, Kisil02a]. ###### Example 15. A modification of the previous construction is obtained if we have two groups $G_{1}$ and $G_{2}$ represented by ${\rho_{1}}$ and ${\rho_{2}}$ on $X$ and $Y^{}$ respectively. Then we have a covariant transform $B{}(X,Y)\rightarrow L{}(G_{1}\times G_{2},\mathbb{C}{})$ defined by the formula: $\mathcal{W}:A\mapsto\hat{A}(g_{1},g_{2})=\left\langle A{\rho_{1}}(g_{1})x,{\rho_{2}}(g_{2})l\right\rangle.$ This generalises _Berezin functional calculi_ [Kisil98a]. ###### Example 16. Let us restrict the previous example to the case when $X=Y$ is a Hilbert space, ${\rho_{1}}{}={\rho_{2}}{}={\rho}$ and $x=l$ with $\left\\|x\right\\|=1$. Than the range of the covariant transform: $\mathcal{W}:A\mapsto\hat{A}(g)=\left\langle A{\rho}(g)x,{\rho}(g)x\right\rangle$ is a subset of the _numerical range_ of the operator $A$. As a function on a group $\hat{A}(g)$ provides a better description of $A$ than the set of its values—numerical range. ###### Example 17. The group $SU(1,1)\simeq SL_{2}{}(\mathbb{R}{})$ consists of $2\times 2$ matrices of the form $\begin{pmatrix}\alpha&\beta\\\ \bar{\beta}&\bar{\alpha}\end{pmatrix}$ with the unit determinant [Lang85, § IX.1]. Let $T$ be an operator with the spectral radius less than $1$. Then the associated Möbius transformation (11) $g:T\mapsto g\cdot T=\frac{\alpha T+\beta I}{\bar{\beta}T+\bar{\alpha}I},\qquad\text{where}\quad g=\begin{pmatrix}\alpha&\beta\\\ \bar{\beta}&\bar{\alpha}\end{pmatrix}\in SL_{2}{}(\mathbb{R}{}),\ $ produces a well-defined operator with the spectral radius less than $1$ as well. Thus we have a representation of $SU(1,1)$. A choice of an operator $F$ will define the corresponding covariant transform. In this way we obtain generalisations of _Riesz–Dunford functional calculus_ [Kisil02a]. ###### Example 18. Consider again the action (11) of the Moebius transformations on operators from the previous Example. Let us introduce the defect operators $D_{T}=(I-T^{}T)^{1/2}$ and $D_{T^{}}=(I-TT^{})^{1/2}$. For the case $F=D_{T^{}}$ the covariant transform is, cf. [NagyFoias70]§ VI.1, (1.2): ${}[\mathcal{W}T](g)=F(g\cdot T)=-e^{\mathrm{i}\phi}\,\Theta_{T}(z)\,D_{T},\qquad\text{for }g=\begin{pmatrix}e^{\mathrm{i}\phi/2}&0\\\ 0&e^{-\mathrm{i}\phi/2}\end{pmatrix}\begin{pmatrix}1&-z\\\ -\bar{z}&1\end{pmatrix},$ where the _characteristic function_ $\Theta_{T}(z)$ [NagyFoias70]§ VI.1, (1.1) is: $\Theta_{T}(z)=-T+D_{T^{}}\,(I-zT^{})^{-1}\,z\,D_{T}.$ Thus we approached the _functional model_ of operators from the covariant transform. In accordance with Remark 3 the model is most fruitful for the case of operator $F=D_{T^{*}}$ being one-dimensional. ## 3\. Induced Covariant Transform The choice of fiducial operator $F$ can significantly influence the behaviour of the covariant transform. Let $G$ be a group and ${H}$ be its closed subgroup with the corresponding homogeneous space $X=G/{H}$. Let ${\rho}$ be a representation of $G$ by operators on a space $V$, we denote by ${\rho_{H}}$ the restriction of ${\rho}$ to the subgroup $H$. ###### Definition 19. Let $\chi$ be a representation of the subgroup ${H}$ in a space $U$ and $F:V\rightarrow U$ be an intertwining operator between $\chi$ and the representation ${\rho_{H}}$: $F({\rho}(h)v)=F(v)\chi(h),\qquad\text{ for all }h\in{H},\ v\in V.$ Then the covariant transform (1) generated by $F$ is called the _induced covariant transform_. The following is the main motivating example. ###### Example 20. Consider the traditional wavelet transform as outlined in Ex. 6. Chose a vacuum vector $v_{0}$ to be a joint eigenvector for all operators ${\rho}(h)$, $h\in H$, that is ${\rho}(h)v_{0}=\chi(h)v_{0}$, where $\chi(h)$ is a complex number depending of $h$. Then $\chi$ is obviously a character of $H$. The image of wavelet transform (3) with such a mother wavelet will have a property: $\hat{v}(gh)=\left\langle v,{\rho}(gh)v_{0}\right\rangle=\left\langle v,{\rho}(g)\chi(h)v_{0}\right\rangle=\chi(h)\hat{v}(g).$ Thus the wavelet transform is uniquely defined by cosets on the homogeneous space $G/H$. In this case we can speak about the _reduced wavelet transform_ [Kisil97a]. A representation ${\rho_{0}}$ is _square integrable_ $\mod H$ if the induced wavelet transform $[\mathcal{W}f_{0}](w)$ of the vacuum vector $f_{0}(x)$ is square integrable on $X$. The image of induced covariant transform have the similar property: (12) $\hat{v}(gh)=F({\rho}((gh)^{-1})v)=F({\rho}(h^{-1}){\rho}(g^{-1})v)=F({\rho}(g^{-1})v)\chi{}{}(h^{-1}).$ Thus it is enough to know the value of the covariant transform only at a single element in every coset $G/H$ in order to reconstruct it for the entire group $G$ by the representation $\chi$. Since coherent states (wavelets) are now parametrised by points homogeneous space $G/H$ they are referred sometimes as coherent states which are not connected to a group [Klauder96a], however this is true only in a very narrow sense as explained above. ###### Example 21. To make it more specific we can consider the representation of $SL_{2}{}(\mathbb{R}{})$ defined on $L_{2}{}(\mathbb{R}{})$ by the formula, cf. (8): ${\rho}(g):f(z)\mapsto\frac{1}{(cx+d)}\,f\left(\frac{ax+b}{cx+d}\right),\qquad g^{-1}=\ \begin{pmatrix}a&b\\\ c&d\end{pmatrix}.$ Let $K\subset SL_{2}{}(\mathbb{R}{})$ be the compact subgroup of matrices $h_{t}=\begin{pmatrix}\cos t&\sin t\\\ -\sin t&\cos t\end{pmatrix}$. Then for the fiducial operator $F_{\pm}$ (6) we have $F_{\pm}\circ{\rho}(h_{t})=e^{\mp\mathrm{i}t}F_{\pm}$. Thus we can consider the covariant transform only for points in $SL_{2}{}(\mathbb{R}{})/K$, however this set can be naturally identified with the $ax+b$ group. Thus we do not obtain any advantage of extending the group in Ex. 7 from $ax+b$ to $SL_{2}{}(\mathbb{R}{})$ if we will be still using the fiducial operator $F_{\pm}$ (6). Functions on the group $G$, which have the property $\hat{v}(gh)=\hat{v}(g)\chi(h)$ (12), provide a space for the representation of $G$ induced by the representation $\chi$ of the subgroup $H$. This explains the choice of the name for induced covariant transform. ###### Remark 22. Induced covariant transform uses the fiducial operator $F$ which passes through the action of the subgroup ${H}$. This reduces information which we obtained from this transform in some cases. There is also a simple connection between a covariant transform and right shifts: ###### Proposition 23. Let $G$ be a Lie group and ${\rho}$ be a representation of $G$ in a space $V$. Let $[\mathcal{W}f](g)=F({\rho}(g^{-1})f)$ be a covariant transform defined by the fiducial operator $F:V\rightarrow U$. Then the right shift $[\mathcal{W}f](gg^{\prime})$ by $g^{\prime}$ is the covariant transform $[\mathcal{W^{\prime}}f](g)=F^{\prime}({\rho}(g^{-1})f)]$ defined by the fiducial operator $F^{\prime}=F\circ{\rho}(g^{-1})$. In other words the covariant transform intertwines right shifts with the associated action ${\rho_{B}}$ (10) on fiducial operators. Although the above result is obvious, its infinitesimal version has interesting consequences. ###### Corollary 24. Let $G$ be a Lie group with a Lie algebra $\mathfrak{g}$ and ${\rho}$ be a smooth representation of $G$. We denote by $d{\rho_{B}}$ the derived representation of the associated representation ${\rho_{B}}$ (10) on fiducial operators. Let a fiducial operator $F$ be a null-solution, i.e. $AF=0$, for the operator $A=\sum_{J}a_{j}d{\rho^{X_{j}}_{B}}$, where $X_{j}\in\mathfrak{g}$ and $a_{j}$ are constants. Then the wavelet transform $[\mathcal{W}f](g)=F({\rho}(g^{-1})f)$ for any $f$ satisfies: $DF(g)=0,\qquad\text{where}\quad D=\sum_{j}\bar{a}_{j}\mathfrak{L}^{X_{j}}.$ Here $\mathfrak{L}^{X_{j}}$ are the left invariant fields (Lie derivatives) on $G$ corresponding to $X_{j}$. ###### Example 25. Consider the representation ${\rho}$ (5) of the $ax+b$ group with the $p=1$. Let $A$ and $N$ be the basis of the corresponding Lie algebra generating one- parameter subgroups $(e^{t},0)$ and $(0,t)$. Then the derived representations are: $[d{\rho^{A}}f](x)=f(x)+xf^{\prime}(x),\qquad[d{\rho^{N}}f](x)=f^{\prime}(x).$ The corresponding left invariant vector fields on $ax+b$ group are: $\mathfrak{L}^{A}=a\partial_{a},\qquad\mathfrak{L}^{N}=a\partial_{b}.$ The mother wavelet $\frac{1}{x+\mathrm{i}}$ is a null solution of the operator $d{\rho^{A}}+\mathrm{i}d{\rho^{N}}=I+(x+\mathrm{i})\frac{d}{dx}$. Therefore the covariant transform with the fiducial operator $F_{+}$ (6) will consist with the null solutions to the operator $\mathfrak{L}^{A}-\mathrm{i}\mathfrak{L}^{N}=-\mathrm{i}a(\partial_{b}+\mathrm{i}\partial_{a})$, that is in the essence the Cauchy-Riemann operator in the upper half-plane. There is a statement which extends the previous Corollary from differential operators to integro-differential ones. We will formulate it for the wavelets setting. ###### Corollary 26. Let $G$ be a Lie group with a Lie algebra $\mathfrak{g}$ and ${\rho}$ be a unitary representation of $G$, which can be extended to a vector space $V$ of functions or distributions on $G$. Let a mother wavelet $w\in V^{\prime}$ satisfy the equation $\int_{G}a(g)\,{\rho}(g)w\,dg=0,$ for a fixed distribution $a(g)\in V$. Then any wavelet transform $F(g)=\mathcal{W}f(g)=\left\langle f,{\rho}(g)w_{0}\right\rangle$ obeys the condition: $DF=0,\qquad\text{where}\quad D=\int_{G}\bar{a}(g)\,R(g)\,dg,$ with $R$ being the right regular representation of $G$. Clearly the Corollary 24 is a particular case of Corollary 26. ## 4\. Inverse Covariant Transform An object invariant under the left action $\Lambda$ (2) is called _left invariant_. For example, let $L$ and $L^{\prime}$ be two left invariant spaces of functions on $G$. We say that a pairing $\left\langle\cdot,\cdot\right\rangle:L\times L^{\prime}\rightarrow\mathbb{C}{}$ is _left invariant_ if (13) $\left\langle\Lambda(g)f,\Lambda(g)f^{\prime}\right\rangle=\left\langle f,f^{\prime}\right\rangle,\quad\textrm{ for all }\quad f\in L,\ f^{\prime}\in L^{\prime}.$ ###### Remark 27. 1. (1) We do not require the pairing to be linear in general. 2. (2) If the pairing is invariant on space $L\times L^{\prime}$ it is not necessarily invariant (or even defined) on the whole $C{}(G)\times C{}(G)$. 3. (3) In a more general setting we shall study an invariant pairing on a homogeneous spaces instead of the group. However due to length constraints we cannot consider it here beyond the Example 30. 4. (4) An invariant pairing on $G$ can be obtained from an invariant functional $l$ by the formula $\left\langle f_{1},f_{2}\right\rangle=l(f_{1}\bar{f}_{2})$. For a representation ${\rho}$ of $G$ in $V$ and $v_{0}\in V$ we fix a function $w(g)={\rho}(g)v_{0}$. We assume that the pairing can be extended in its second component to this $V$-valued functions, say, in the weak sense. ###### Definition 28. Let $\left\langle\cdot,\cdot\right\rangle$ be a left invariant pairing on $L\times L^{\prime}$ as above, let ${\rho}$ be a representation of $G$ in a space $V$, we define the function $w(g)={\rho}(g)v_{0}$ for $v_{0}\in V$. The _inverse covariant transform_ $\mathcal{M}$ is a map $L\rightarrow V$ defined by the pairing: (14) $\mathcal{M}:f\mapsto\left\langle f,w\right\rangle,\qquad\text{ where }f\in L.$ ###### Example 29. Let $G$ be a group with a unitary square integrable representation $\rho$. An invariant pairing of two square integrable functions is obviously done by the integration over the Haar measure: $\left\langle f_{1},f_{2}\right\rangle=\int_{G}f_{1}(g)\bar{f}_{2}(g)\,dg.$ For an admissible vector $v_{0}$ [DufloMoore], [AliAntGaz00, Chap. 8] the inverse covariant transform is known in this setup as a _reconstruction formula_. ###### Example 30. Let $\rho$ be a square integrable representation of $G$ modulo a subgroup $H\subset G$ and let $X=G/H$ be the corresponding homogeneous space with a quasi-invariant measure $dx$. Then integration over $dx$ with an appropriate weight produces an invariant pairing. The inverse covariant transform is a more general version [AliAntGaz00, (7.52)] of the _reconstruction formula_ mentioned in the previous example. Let $\rho$ be not a square integrable representation (even modulo a subgroup) or let $v_{0}$ be inadmissible vector of a square integrable representation $\rho$. An invariant pairing in this case is not associated with an integration over any non singular invariant measure on $G$. In this case we have a _Hardy pairing_. The following example explains the name. ###### Example 31. Let $G$ be the “$ax+b$” group and its representation ${\rho}$ (5) from Ex. 7. An invariant pairing on $G$, which is not generated by the Haar measure $a^{-2}da\,db$, is: (15) $\left\langle f_{1},f_{2}\right\rangle=\lim_{a\rightarrow 0}\int\limits_{-\infty}^{\infty}f_{1}(a,b)\,\bar{f}_{2}(a,b)\,db.$ For this pairing we can consider functions $\frac{1}{2\pi i(x+i)}$ or $e^{-x^{2}}$, which are not admissible vectors in the sense of square integrable representations. Then the inverse covariant transform provides an _integral resolutions_ of the identity. Similar pairings can be defined for other semi-direct products of two groups. We can also extend a Hardy pairing to a group, which has a subgroup with such a pairing. ###### Example 32. Let $G$ be the group $SL_{2}{}(\mathbb{R}{})$ from the Ex. 8. Then the “$ax+b$” group is a subgroup of $SL_{2}{}(\mathbb{R}{})$, moreover we can parametrise $SL_{2}{}(\mathbb{R}{})$ by triples $(a,b,\theta)$, $\theta\in(-\pi,\pi]$ with the respective Haar measure [Lang85, III.1(3)]. Then the Hardy pairing (16) $\left\langle f_{1},f_{2}\right\rangle=\lim_{a\rightarrow 0}\int\limits_{-\infty}^{\infty}f_{1}(a,b,\theta)\,\bar{f}_{2}(a,b,\theta)\,db\,d\theta.$ is invariant on $SL_{2}{}(\mathbb{R}{})$ as well. The corresponding inverse covariant transform provides even a finer resolution of the identity which is invariant under conformal mappings of the Lobachevsky half-plane. A further study of covariant transform shall be continued elsewhere. Acknowledgement. Author is grateful to the anonymous referee for many helpful suggestions. ## References	arxiv-papers	2010-11-17T11:31:27	2024-09-04T02:49:14.928795	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Vladimir V. Kisil", "submitter": "Vladimir V Kisil", "url": "https://arxiv.org/abs/1011.3947" }
1011.4080	Spatial Guilds in the Serengeti Food Web Revealed by a Bayesian Group Model Edward B. Baskerville1,∗, Andy P. Dobson2, Trevor Bedford1,3, Stefano Allesina4, Mercedes Pascual1,3 1 Department of Ecology and Evolutionary Biology, University of Michigan, Ann Arbor, Michigan, United States of America 2 Department of Ecology and Evolutionary Biology, Princeton University, Princeton, New Jersey, United States of America 3 Howard Hughes Medical Institute, University of Michigan, Ann Arbor, Michigan, United States of America 4 Department of Ecology and Evolution, Computation Institute, The University of Chicago, Chicago, Illinois, United States of America $\ast$ E-mail: ebaskerv@umich.edu ## Abstract Food webs, networks of feeding relationships among organisms, provide fundamental insights into mechanisms that determine ecosystem stability and persistence. Despite long-standing interest in the compartmental structure of food webs, past network analyses of food webs have been constrained by a standard definition of compartments, or modules, that requires many links within compartments and few links between them. Empirical analyses have been further limited by low-resolution data for primary producers. In this paper, we present a Bayesian computational method for identifying group structure in food webs using a flexible definition of a group that can describe both functional roles and standard compartments. The Serengeti ecosystem provides an opportunity to examine structure in a newly compiled food web that includes species-level resolution among plants, allowing us to address whether groups in the food web correspond to tightly-connected compartments or functional groups, and whether network structure reflects spatial or trophic organization, or a combination of the two. We have compiled the major mammalian and plant components of the Serengeti food web from published literature, and we infer its group structure using our method. We find that network structure corresponds to spatially distinct plant groups coupled at higher trophic levels by groups of herbivores, which are in turn coupled by carnivore groups. Thus the group structure of the Serengeti web represents a mixture of trophic guild structure and spatial patterns, in contrast to the standard compartments typically identified in ecological networks. From data consisting only of nodes and links, the group structure that emerges supports recent ideas on spatial coupling and energy channels in ecosystems that have been proposed as important for persistence. Our Bayesian approach provides a powerful, flexible framework for the study of network structure; we believe it will prove instrumental in a variety of biological contexts. ## Introduction Food webs, networks of feeding relationships in ecosystems, connect the biotic interactions among organisms with energy flows, thus linking together population dynamics, ecosystem function, and network topology. Ecologists have been using this powerful conceptual tool for more than a century [1, 2, 3]. One element of structure of particular relevance to large food webs is the subdivision of species into compartments or groups, a feature that has been proposed to contribute to food web stability by constraining the propagation of disturbances through a network [4]. Although a large literature has considered the presence and dynamic significance of compartments in food webs, on the whole, evidence for the importance of compartments has been inconclusive [5, 6, 7, 8]. In this literature, compartments are alternately referred to as modules, clusters, or “communities” [9], and are defined by high link density within groups and low link density between them. Recent work with a probabilistic model considers a more flexible notion of groups, allowing link density to be high or low within any group or between any pair of groups [8]. Groups can thus represent compartments in the previous sense, but can also represent trophic guilds or roles [10, 11], sets of species that feed on, and are fed on, by similar sets of species. By fitting models of this type to data, the dominant topological pattern in the network can be found, which may include (spatial) compartments, or trophic guilds, or some combination of the two. The initial application of this model to empirical food webs from different ecosystems has revealed a predominance of trophic guilds rather than compartments [8]. Two major challenges limit the application of this model in resolving the group structure of large food webs and interpreting its biological basis. First, most food webs have poor resolution of primary producers; plants in terrestrial systems and phytoplankton in aquatic ones are typically represented by a few nodes that are highly aggregated taxonomically. This is unfortunate, because their position at the base of a food web means that primary producers are essential for understanding how the web is spatially organized, and how this spatial organization percolates up through the trophic relationships of other species. Fully resolved primary producers are also essential in examining how the structure of other trophic levels cuts across their spatial distribution, and, in so doing, couples different habitats. Second, some technical problems have hindered the use of probabilistic models in analyzing group structure. Early food web models served as null models for food web structure and were tested by generating model webs and comparing summary statistics against data from real webs [12, 13]. More recently, a more rigorous approach for measuring the goodness of fit of a model has been provided by maximum likelihood and model selection [14, 8]; two problems still remain within this framework. One is technical: standard model-selection criteria are not applicable to “discrete parameters” such as group membership. The second problem is more fundamental: there are many almost equally good arrangements, and it is desirable to extract information not just from a single best arrangement, but also from the rest of the ensemble. The Bayesian approach is gaining popularity in ecological modeling due to the philosophical and conceptual appeal of explicitly considering uncertainty in parameter estimation as well as its methodological flexibility [15]. This approach is especially well-suited for handling uncertainty in complex food web models, and allows us to overcome the limitations of the previous implementation of the group model. In network inference, there are only a few examples of complete Bayesian models [16, 17] and a few examples of MCMC for maximum-likelihood inference [18, 19], but Bayesian inference in phylogenetics has been long established [20, 21], and provides a clear methodological analogue. In this paper, we address the group structure of a newly assembled food web for the mammals and plants of the Serengeti grassland ecosystem of Tanzania; we use a new and novel approach to the identification of groups based on Bayesian inference. We specifically ask whether the structure that emerges reflects the underlying spatial dimension, as delineated by the different plant communities that characterize different sub-habitats within the ecosystem, or whether it is determined by trophic dimensions in the form of species guilds that share functional roles. The Serengeti food web is emerging as the most highly resolved terrestrial web to date [22]. The Serengeti has been studied as an integrated ecosystem for almost five decades [23, 24, 25], and because of widespread popular familiarity with the consumer-resource dynamics of lions, hyenas, wildebeest, zebra and grasses, it provides a strong intuitive test for probabilistic food web models. Most importantly, at the primary producer level, the Serengeti food web includes a number of distinct grass and woodland plant communities on different soils and across a rainfall gradient [26]. The sequential and well- documented changes in the underlying plant diversity allows us to examine the extent to which grassland communities define network topology at higher trophic levels. The high resolution of the species that comprise the mammal and plant communities of the ecosystem allow us to address the role of space in the group structure of the food web. From a complex network of only nodes and links that represent species and their interactions, the groups that emerge from an otherwise blind classification of species make remarkable biological sense. We find that the group structure identified in the Serengeti food web represents a mixture of trophic guild structure and spatial patterns, an arrangement that differs significantly from the compartmental structure typically identified in ecological networks. ## Results/Discussion ### Bayesian Inference and Model Selection for Food Webs #### Probabilistic Models for Food Webs In this paper, we use probabilistic modeling as a tool for formalizing hypotheses about food web structure. We treat a food web, an observed network of who eats whom in an ecosystem, as data. We start with the basic question: assuming a probabilistic model of food web structure, what is the probability of observing this particular real-world food web? This probability is referred to as the likelihood of observing the data, given model parameters. In a maximum-likelihood framework, the mechanical part of the inference process is to find the set of model parameters that makes the likelihood as great as possible, with the interpretation that this represents the best point estimate of the underlying process. We begin with the group model of Allesina and Pascual [8], which was originally treated in a maximum-likelihood framework. Conceptually, this model encodes the simple hypothesis that species can be divided into groups, and species in the same group have statistically similar behavior: they tend to consume species in certain groups and tend to be consumed by species in certain groups. Specifically, the probability that a species belonging to group $i$ is eaten by a species belonging to group $j$ is given by $p_{ij}$, and conversely, the probability of a link being absent is $(1-p_{ij})$. If there are $K$ groups, then a matrix $\mathbf{P}$ of $K^{2}$ link probabilities is required to completely describe the relationships among all groups. The likelihood for the whole network is the product over all pairs of species of the probability of a link being present (if present) or absent (if absent). In the statistical literature, this model structure is known as a stochastic block model [27]. The assignment of species to groups is also an unobserved parameter in this model, which adds a layer of difficulty to parameter estimation. For example, in a network of $100$ species, there are approximately $5\times 10^{115}$ different ways to partition the network into groups (see Methods). That is, if you had a computer that could process $10^{80}$ partitions (as many partitions as there are atoms in the universe) every femtosecond ($10^{-15}$ s), it would take $1.5\times 10^{13}$ years to process them all. (By comparison, the universe is only $1.4\times 10^{10}$ years old.) The group model allows for a more flexible definition of groups than standard approaches to network clustering, which find groups that have large numbers of internal connections and relatively few connections between groups [9]. Because each $p_{ij}$ parameter may take any value between $0$ and $1$, good model fits may result from other relationships, such as high link density between groups and low link density within groups, and may accommodate different relationships in different parts of the network. In general, the best-fitting partitions will try to maximize or minimize the number of links within specific groups and between specific pairs of groups. #### Bayesian Inference and Priors for the Group Model In a Bayesian framework, rules of probability are taken to govern both the data and model parameters. Rather than finding the set of parameter values that maximize the likelihood, the goal becomes to estimate a probability distribution over parameters based on observed data. In this way, we can directly quantify the uncertainty in our parameters in terms of probabilities. This permits questions such as: what is the probability that a parameter lies in a particular range? The name “Bayesian” comes from Bayes’ rule, which tells us how to use conditional probability statements to infer a posterior distribution, in this case, the probability distribution over parameter values conditional on having observed the data, $\mathrm{Pr}(\theta\|D)$. If we are dealing entirely with discrete probability distributions, Bayes’ rule takes its most intuitive form: $\displaystyle\mathrm{Pr}(\theta\|D)$ $\displaystyle=\frac{\mathrm{Pr}(\theta)\mathrm{Pr}(D\|\theta)}{\mathrm{Pr}(D)}.$ (1) The numerator of the right-hand-side is the probability of producing the data from the given parameters: the prior probability of those parameters, $\mathrm{Pr}(\theta)$, times the probability of producing the data given those parameters, $\mathrm{Pr}(D\|\theta)$, the likelihood. The denominator is the marginal probability of observing the data unconditional on the particular parameter values at play, which is simply the sum of the probabilities of all the different ways of producing the data using all possible parameter values, $\mathrm{Pr}(D)=\sum_{\theta}\mathrm{Pr}(\theta)\mathrm{Pr}(D\|\theta)$. In other words, in order to calculate the posterior probability of parameters $\theta$, we add up all the different ways of producing the data weighted by their probability, and then calculate what fraction of that probability came from parameters $\theta$. From here, we will write these quantities in more general notation, suitable for a mix of discrete and continuous probability distributions: $\displaystyle f(\theta\|D)$ $\displaystyle=\frac{f(\theta)f(D\|\theta)}{\int_{\theta}f(\theta)f(D\|\theta)\,d\theta}$ (2) where the integral sign represents a multiple integral over discrete and continuous parameters. In the Bayesian framework, the model includes not only the formulation of the likelihood but also a prior distribution over parameters. With the group model, this means defining a prior distribution over both link probabilities and arrangements into groups (“partitions”). In general, priors may incorporate informed knowledge about the system, but in this case we simply use them to encode different variants of the same basic model. We use two distributions for partitions and two distributions for link probabilities, which are combined to form four different model variants. The two alternative distributions for elements $p_{ij}$ of the link probability matrix $\mathbf{P}$ are (1) a uniform distribution between 0 and 1, and (2) a beta distribution with shape parameters $\alpha$ and $\beta$, which are in turn governed by exponential distributions with mean $1$. With $\alpha$ and $\beta$ fixed at their means, alternative (2) reduces to a uniform distribution; at other values, the distribution may take a uniform, convex, concave, or skewed shape. Alternative (2) is thus structured hierarchically, with exponential hyperpriors for $\alpha$ and $\beta$ governing the beta prior for elements of $\mathbf{P}$. For partitions, we consider (1) a uniform distribution and (2) a distribution generated by the Dirichlet process, sometimes referred to as the “Chinese restaurant process” [28]. Alternative (2) is controlled by an aggregation parameter $\chi$ that is in turn drawn from an exponential distribution with mean $1$. The uniform distribution assigns equal prior probability to each possible partition, irrespective of the number of groups. Because there are far more ways to partition the network at an intermediate, but relatively high, number of groups, the uniform prior implicitly biases the model toward that number. For example, in the Serengeti food web, there are $170$ nodes, yielding an a priori expectation of $43.8$ groups. In contrast, the hierarchically structured Dirichlet process prior provides flexibility via the aggregation parameter $\chi$. When $\chi$ is large, partitions tend to have many small groups; when $\chi$ is small, partitions tend to have fewer groups, with a skewed group size distribution. We also consider two simple models without groups as null comparisons: (1) a directed random graph model (i.e., one group) with a uniform prior on a single link probability parameter $p$, and (2) a fully parameterized model, with each species in its own group (and a $170\times 170$ link probability parameter matrix $\mathbf{P}$, also with a uniform link probability prior. For a fuller discussion of models and prior distributions, in particular the properties of the distribution generated by the Dirichlet process, see Methods. #### Bayesian Model Selection via Marginal Likelihood The Bayesian framework provides a natural way to make probabilistic inferences based on a particular model. However, we also want to be able to choose between different models by quantifying their relative goodness of fit. One approach to Bayesian model selection can be framed directly in terms of Bayes’ rule, mirroring the process for estimating the posterior distribution over parameters for a single model. Consider two models, $M_{1}$ and $M_{2}$, to which we assign prior weight $\mathrm{Pr}(M_{1})$ and $\mathrm{Pr}(M_{2})$. After the data has been observed, we can calculate the posterior probability of the models using Bayes’ rule: $\displaystyle\mathrm{Pr}(M_{1}\|D)$ $\displaystyle=\frac{\mathrm{Pr}(M_{1})\mathrm{Pr}(D\|M_{1})}{\mathrm{Pr}(D)},$ (3) $\displaystyle\mathrm{Pr}(M_{2}\|D)$ $\displaystyle=\frac{\mathrm{Pr}(M_{2})\mathrm{Pr}(D\|M_{2})}{\mathrm{Pr}(D)},$ (4) where the denominator is equal to the probability of observing the data unconditional of the particular model at play, $\mathrm{Pr}(D)=\mathrm{Pr}(M_{1})P(D\|M_{1})+\mathrm{Pr}(M_{2})P(D\|M_{2})$. The probabilities $\mathrm{Pr}(D\|M_{1})=\int_{\theta_{1}}f(\theta_{1})f(D\|\theta_{1})\,d\theta_{1}$ and $\mathrm{Pr}(D\|M_{2})=\int_{\theta_{2}}f(\theta_{2})f(D\|\theta_{2})\,d\theta_{2}$ are the marginal likelihoods of the two models, corresponding to the denominator in Equation 2. If we give the two models equal prior weight, then the relative posterior weight of the two models is simply given by the marginal likelihoods. This reasoning extends naturally to any number of models. The ratio of the marginal likelihoods is often called the Bayes factor [29, 30, 31], and is equal to the posterior odds ratio of the two models, assuming equal prior weight: $\displaystyle B_{12}$ $\displaystyle=\frac{\mathrm{Pr}(D\|M_{1})}{\mathrm{Pr}(D\|M_{2})}$ (5) The Bayes factor provides a convenient way to compare models: if $B_{12}=10$, then we consider support for model $M_{1}$ to be ten times stronger than model $M_{2}$. In AIC-based model selection, the Bayes factor is analogous to a ratio of Akaike weights [32]. #### Consensus Partitions The output of an MCMC simulation includes a long sequence of network partitions representing draws from the posterior distribution over partitions. As these partitions are potentially all distinct from each other, but represent similar tendencies of species to be grouped together, it is useful to try to summarize the information contained in all the samples in a more compact form. One approach is to construct a pairwise group-membership matrix for species in the food web, with entries equal to the posterior probability that two species are in the same group. A visual representation of this matrix can illuminate the group structure, and a consensus partition can then be constructed from this matrix using a simple clustering algorithm. (For more details, see Methods.) ### Bayesian analysis of the Serengeti food web #### The Serengeti Data Set We compiled the food web from published accounts of feeding links in the literature [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 26, 43, 44, 45]. The compiled Serengeti food web (Tables S1 and S2 and Figure 1) consists of $L=667$ feeding links among $S=170$ species ($136$ plants, $25$ herbivores, and $9$ carnivores). $550$ of the links are herbivorous, and $117$ are predatory. The fraction of all possible links or connectance ($C=L/S^{2}$), ignoring all biological constraints, is equal to $0.0231$. #### Performance of Model Variants We find unequivocal support for the use of group-based models in describing the Serengeti food web. Models with group structure have vastly greater marginal likelihoods than simple null models that ignore group structure (Table Tables). Furthermore, the use of both the Dirichlet process prior for network partitions and the beta prior for link probability parameters vastly improved the fit of the basic model. The best model variant as measured by marginal likelihood included both the beta prior on link probabilities and the Dirichlet process prior on partitions. The next best variant included a uniform partition prior and beta link probability prior, followed by the variant with Dirichlet process prior and uniform link probability prior. The strongest variant surpassed its closest competitor by $106$ log-orders in likelihood and surpassed the model with two uniform priors by $470$ log-orders in likelihood, providing unequivocal support for including both flexible priors in the model specification. Accordingly, in the remaining analysis we consider only the best model variant. #### Identification of Model Parameters The posterior mean number of groups $K$ is $14.9$ (95% credible interval $12,18$), and the mean value of the Dirichlet process parameter $\chi$ is $3.2$ ($1.6,5.3$) (Figure 2). The prior expectation of $\chi$ was 1.0 and the prior expectation of $K$ was $5.4$. The finding of posterior values substantially greater than prior values strongly supports the presence of detailed group structure in the Serengeti food web. Mean values for beta distribution parameters are $\alpha=0.046$ ($0.029,0.068$) and $\beta=0.89$ ($0.50,1.43$) (Figure S2). The corresponding beta prior has support concentrated near 0, since most species do not feed on most other species (Figure S3). #### Groups Identified in the Serengeti Food Web The structure of the Serengeti food web is best represented by groups containing trophically similar species, subdivided by specialization on different components of prey species, with plant species corresponding to spatially distinct habitats. The best consensus partition, with $16$ groups, is shown in Table 2. There are 6 groups of plant species (groups 11–16), 7 groups of herbivore species (groups 4–10) and 3 groups of carnivores species (groups 1–3). On average, plant groups contain more species than herbivore and carnivore groups ($22.7$, $3.6$ and $3.0$, respectively). As evident in the pairwise group-membership matrix (Figure 3), the carnivore and herbivore groups are well-defined, including several individual species or pairs of species with distinct diets. Plant groups demonstrate mild overlap, indicating a partially hierarchical relationship between smaller groups and larger groups. Figures 4, 6, and 5 show three alternate views of the food web, organized by the $16$-group consensus partition. Groups $11$–$13$, $14$–$15$, and $16$ consist of plants located in grassland, woodland, and kopje habitat, respectively. Group $11$ includes short grasses that are grazed by a wide variety of species, distinguished from group $12$, which includes short grasses fed on only by the large grazers ($4$). The 1-species groups (2, golden jackal Canis aureus; 6, African buffalo Syncerus caffer; 8, agama lizard Agama planiceps; 10, elephant Loxodenta africana; and 13, Microchloa kunthii) represent individual species with distinct diets or consumers. Plant groups are coupled by groups of herbivores, which are in turn coupled by groups of carnivores. Large migratory grazers ($4$, wildebeest, zebra, and gazelles) feed primarily on plant groups consisting of grasses ($11$ and $12$), and are the exclusive consumers of a group of grasses adapted to seasonal rainfall on sandy or volcanic soils ($12$) that dominate the short- grass plains in the southern part of the ecosystem. Herbivores feeding in the longer grasslands and woodlands and in riparian habitats (group $5$) couple the first grass group ($11$) with woodland plants (group $14$), which are also consumed to a lesser extent by the large grazers. The hyraxes (group $7$) and group $9$ (giraffe, olive baboon, and dik-dik) couple kopje habitat (group $16$) with both woodland and grassland plants. At the highest trophic level, the large carnivores ($1$) integrate across all the herbivore groups; smaller carnivores ($2$, $3$) show more specialized diets, reflecting the more distinct habitats in which they are usually found. ### Discussion #### Spatial Guilds in the Serengeti Food Web In order to analyze the group structure of the Serengeti food web, we used a flexible Bayesian model of network structure that includes no biological information aside from a set of nodes representing species and links representing their interactions. The groups that emerge from an otherwise blind classification of species make remarkable biological sense. Species are divided into trophic guilds that reveal a clear relationship between the spatial organization of plant, herbivore, and carnivore groups and the structure of the network. At the coarsest scale, the groups in the Serengeti food web correspond to carnivores, herbivores, and plants. The further subdivisions that emerge within carnivores, herbivores, and plants reveal a spatial dimension to feeding structure that is only evident because of high species resolution at the plant level. Ultimately, the group structure we have derived mirrors the flow of energy up the food web from different spatial locations, with herbivores integrating spatially separated groups of plants, and carnivores integrating spatially widespread herbivores. Although the addition of birds, reptiles, invertebrates, and pathogens will certainly add a significant number of new groups, we do not expect them to significantly modify the derived structure for the mammal and plant community. Nor will they modify the larger tendency for groups to be assembled in ways that reflect the underlying spatial and trophic structure of the species in the web. Recently, interesting theoretical and empirical work has highlighted the relationship between observed patterns of food-web structure and energy flow that seemingly mirrors the trophic guild structure in the Serengeti. Rooney and colleagues [46] give evidence that real ecosystems may be dominated by nested sets of fast and slow “energy channels,” each of which represents a food chain of trophic guilds. They suggest that this pattern may have a strong stabilizing effect, based on theoretical work by McCann on spatially coupled food webs [47]. The group structure for the Serengeti web that emerges from our analysis supports a pattern of spatial coupling at multiple trophic levels: the grasslands have very high turnover rates compared to those of the kopjes and woodlands. This suggests a similar pattern of fast and slow energy channels to those described by Rooney and colleagues, with fast energy flow up through the highly seasonal but very productive grasses of the short-grass plains. These are almost completely consumed by wildebeest and zebra during their peak calving season, which are then in turn consumed by large predators (lions and hyenas). In contrast, the resident herbivore species living on kopjes or in the woodlands reproduce at slower rates and are consumed less frequently by large carnivores, except during the time when they are unable to feed on migratory wildebeest and zebra. The group model and the inference approach presented here allow the examination of the dynamical consequences of this type of structure to be fully rooted in an empirical pattern, complementing more theoretical considerations of its central importance for preserving biodiversity [47]. These patterns emerge directly from the topology of the food web without being explicitly labeled as different habitats upfront as was done in previous empirical work [46], showing that topological analysis can reveal structures that may be very significant for food-web dynamics. They are subtly different, however, from the proposed pure fast and slow chains, in that they incorporate the migration of the keystone species in the ecosystem, so the fastest energy chain is seasonally ephemeral and may only operate for three to four months in any year. We suspect that even within the sub-habitats of kopjes and woodlands there are similarly nested faster and slower chains that involve species for which we are still collating data (e.g., birds, small mammals, and insects). #### Bayesian Analysis of Food-Web Structure In this paper, we used a probabilistic model to analyze the structure of a single food web, an approach we have seen in only one other study based on a probabilistic version of the niche model [19]. This approach has proved fruitful in Bayesian phylogenetics, where the combinatorial challenges are similar. Moreover, we view the group model as only a starting point for richer modeling efforts to help identify relevant processes that influence the structure of ecological communities. In fact, the Bayesian approach described here provides a powerful general framework for encoding hypotheses about the structure of food webs and comparing models against each other, and we see it as a natural next step in the current trend of representing food-web models in a common way. Simple abstract models such as the niche model and the group model used here act as proxies for the high-dimensional trait space that determines feeding relationships in an ecosystem. The identification of actual traits that correspond to groups (or niche dimensions) is another valuable direction, so far followed primarily by finding correlations between compartments/groups [48] or niche values [13] and traits such as body size or phylogenetic relatedness. Another approach is to directly incorporate these traits into the probabilistic models, either as model predictors or as informed priors. Both kinds of analyses are valuable, but the second kind becomes more approachable in a general Bayesian modeling framework. The use of flexible hierarchical priors for model parameters is one straightforward innovation possible in the Bayesian framework. The number of groups identified by the model increases dramatically with the use of a flexible beta prior distribution for link probability parameters. In that model variant, we effectively introduce two degrees of freedom to the model (the beta distribution parameters) but dramatically reduce the effective degrees of freedom of the link probability parameters. Note that we properly penalize parameters by using the marginal likelihood for model selection, so that the model selection rep- resents a balance between goodness of fit and model complexity. Moreover, this structure makes intuitive sense: since most link probability parameters are simply zero, they should not be penalized. An alternate approach is to remove and add parameters to the model, but this hierarchical technique is much easier to implement in practice. Advanced Markov-chain Monte Carlo methods make it possible to accurately estimate marginal likelihoods for probabilistic network models. Unlike information criteria such as AIC, BIC, or DIC, an accurate estimate of the marginal likelihood provides a direct measurement of goodness of fit that takes into account the degrees of freedom in a model without making any asymptotic assumptions about parameter distributions [49], and can handle discrete parameters such as partitioning into groups that are not properly handled by AIC and BIC. The Bayesian approach also serves as a means to avoid fundamental issues inherent in network models with a large parameter space. In a recent study, Good and colleagues [50] examined the properties of module-finding in networks using the pervasive modularity-maximization approach [51], finding that even in relatively small networks a large number of good solutions exist. A maximization algorithm is thus guaranteed to find a single local maximum of many—possibly even the best one, but certainly not one that captures the full range of good solutions. This problem arises whether the quantity to be maximized is a heuristic such as modularity or a likelihood value. The group model and other parameter-rich models presumably suffer from similar degeneracy problems. In the present case, we find that every partition sampled from the posterior distribution for the best-fitting group model variant is unique. Although MCMC sampling cannot reproduce the full posterior distribution, it is an important step in the right direction. Philosophical arguments aside, one of the main reasons for maximizing likelihood or modularity is simply that a single solution is far more tractable than a distribution. The consensus partitioning heuristic used here is one attempt at recovering a simpler object of study (see Methods); more sophisticated approaches will be welcomed. The group model, based on the simple notion that groups of species may have similar feeding relationships to other groups, reveals that trophic guilds are the topologically dominant type of group in the Serengeti food web. The model also reveals an interesting relationship between spatial structure and network structure that corroborates recent ideas on spatial coupling in food webs. A theoretical study with a dynamical model suggests that this type of structure may contribute to ‘stability’ in the sense of the persistence of species [47]. We are now in a position to examine different aspects of stability, including robustness to secondary extinctions, based on structures directly inferred from empirical networks. Although the Bayesian modeling approach is not new to network analysis in general [16, 17], it remains relatively rare. The Bayesian group model, and, more importantly, the general framework for modeling and model selection, naturally extend to other kinds of biological networks, such as metabolic and regulatory networks [52] and networks describing other ecological interactions such as pollination [53]. We advocate this framework as a way to build stronger ties between hypothesis formulation, model building, and data analysis. ## Methods ### Group Model We use as a starting point the group model of Allesina and Pascual [8], in which a network of $N$ nodes is partitioned into $K$ groups. The groups to which a potential prey and to which a potential predator belong completely determine the probability that a feeding relationship exists between them. The assignment of species to groups is given by the vector $\mathbf{G}=(g_{1},\dots,g_{n})$, with $g_{i}\in\\{1,\dots,K\\}$. We refer to this assignment as a set ‘partition,’ in keeping with standard mathematical terminology. The probability that a species assigned to group $i$ is consumed by a species assigned to group $j$ is equal to $p_{ij}$. This gives a matrix $\mathbf{P}$ of $K^{2}$ probabilities, containing the probabilities of observing directed links between members of each pair of groups, and within members of each group. If we take $\mathbf{A}$ to be the directed adjacency matrix of a network, with entries $a_{ij}$ equal to 1 if a link exists from node $i$ to node $j$, 0 otherwise, then the probability of the network being generated by partition $\mathbf{G}$ and link probabilities $\mathbf{P}$ is given by $\displaystyle f(\mathbf{A}\|\mathbf{G},\mathbf{P})$ $\displaystyle=\prod_{i=1}^{K}\prod_{j=1}^{K}p_{ij}^{Y_{ij}}(1-p{ij})^{Z_{ij}}\,,$ (6) where $Y_{ij}$ and $Z_{ij}$ are the number of 1-entries and 0-entries in the submatrix of $\mathbf{A}$ containing entries from rows $r$ satisfying $g_{r}=i$ and columns $c$ satisfying $g_{c}=j$. In the simplest case, all nodes are assigned to the same group, and the likelihood simplifies to $\displaystyle f(\mathbf{A}\|p)$ $\displaystyle=p^{Y}(1-p)^{Z}$ (7) where $Y$ and $Z$ are the total number of 1-entries and 0-entries in $\mathbf{A}$. In order to use the group model for Bayesian inference, we want to infer the posterior distribution over partitions and parameters, $\displaystyle f(\mathbf{G},\mathbf{P}\|\mathbf{A})\propto f(\mathbf{G},\mathbf{P})f(\mathbf{A}\|\mathbf{G},\mathbf{P}).$ (8) This requires specifying a prior distribution over partitions $\mathbf{G}$ and link probabilities $\mathbf{P}$. We consider two priors over $\mathbf{G}$ and two priors over $\mathbf{P}$. ### Model Priors #### Priors for Partitions The simplest prior over partitions assigns equal probability to each possible assignment of nodes into groups. For a network of $N$ nodes, the number of possible partitions is given by the $N$th Bell number, $\displaystyle\mathcal{B}(N)$ $\displaystyle=\sum_{K=1}^{N}\mathcal{S}_{2}(N,K),$ (9) where $\mathcal{S}_{2}(N,K)$ is the Stirling number of the second kind, the number of ways to partition $N$ objects into exactly $K$ groups, $\displaystyle\mathcal{S}_{2}(N,K)$ $\displaystyle=\frac{1}{K!}\sum_{j=0}^{K}(-1)^{K-j}{\binom{K}{j}}j^{N}.$ (10) Therefore, the prior probability of a particular partition is uniform across all possible partitions $\displaystyle f(\mathbf{G})$ $\displaystyle=\frac{1}{\mathcal{B}(N)},$ (11) and the prior probability of having exactly $K$ groups is $\displaystyle f(K)$ $\displaystyle=\frac{\mathcal{S}_{2}(N,K)}{\mathcal{B}(N)}.$ (12) For partitions, the choice of a uniform prior, although simple, includes hidden assumptions. In particular, there are far more possible partitions for an intermediate number of groups than a small or large number, so the prior will implicitly bias results toward that number. For example, with 100 nodes, the distribution is peaked at $K=28$ (Figure S1). An alternate prior for partitioning objects into groups comes from the Dirichlet process, also known as the “Chinese restaurant process,” which is becoming a standard Bayesian prior for related problems [28, 54, 55]. Consider a restaurant with an infinitely large number of infinitely large tables, all initially empty. The first patron sits alone at the first table, and subsequent patrons may either sit at an occupied table or a new table. They choose occupied tables with weight equal to the number of current occupants, or a new table with weight equal to an aggregation parameter $\chi$. For example, the second patron will sit at the same table as the first patron with probability $1/(1+\chi)$. In fact, because the process is exchangeable, the probability of any pair of patrons sitting at the table is also $1/(1+\chi)$. If $\chi$ is small, there will tend to be a small number of occupied tables and a skewed distribution of table sizes; if $\chi$ is large, there will be a larger number of tables occupied by few patrons. Interpreting tables of patrons as groups of nodes, under the Dirichlet process the prior probability of a particular partition $\mathbf{G}$ is $\displaystyle f(\mathbf{G}\|\chi)$ $\displaystyle=\chi^{K}\frac{\prod_{j=1}^{K}(\eta_{j}-1)!}{\prod_{i=1}^{N}(\chi+i-1)},$ (13) where $N$ is the number of nodes in the network, $K$ is the number of groups in the partition, and $\eta_{j}$ is the number of nodes in group $j$. The prior probability of $K$ groups is $\displaystyle f(K\|\chi)$ $\displaystyle=\frac{\|\mathcal{S}_{1}(N,K)\|\chi^{K}}{\prod_{i=1}^{N}(\chi+i-1)},$ (14) where $\mathcal{S}_{1}(N,K)$ is a Stirling number of the first kind, equal to the coefficients on $x_{K}$ in the expansion $x(x-1)(x-2)\ldots(x-K+1)$. Rather than choosing a fixed value of $\chi$ for the prior, we give $\chi$ an exponential hyperprior distribution with mean $1$: $\displaystyle f(\chi)$ $\displaystyle=e^{-\chi}\qquad\chi\geq 0.$ (15) #### Priors for Link Probabilities Similarly, the elements of link probability matrix $\mathbf{P}$ may be given a simple uniform prior over $[0,1]$: $\displaystyle f(p_{ij})$ $\displaystyle=1\qquad 0\leq p_{ij}\leq 1.$ (16) As there may be some regularity in the values of the link probabilities, we also tried a beta prior: $\displaystyle f(p_{ij}\|\alpha,\beta)$ $\displaystyle=\frac{1}{B(\alpha,\beta)}p_{ij}^{\alpha-1}(1-p_{ij})^{\beta-1},$ (17) where $B(\alpha,\beta)$ is the beta function, $\displaystyle B(\alpha,\beta)$ $\displaystyle=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}\,dt.$ (18) The parameters $\alpha$ and $\beta$ control the shape of the distribution, which may be convex, concave, or skewed toward $0$ or $1$. When $\alpha=\beta=1$, the beta prior becomes a uniform distribution. We use $\alpha$ and $\beta$ exponential hyperpriors with mean $1$: $\displaystyle f(\alpha)$ $\displaystyle=e^{-\alpha}\qquad\alpha\geq 0,$ (19) $\displaystyle f(\beta)$ $\displaystyle=e^{-\beta}\qquad\beta\geq 0.$ (20) ### Markov-chain Monte Carlo Sampling For networks of any appreciable size, the number of possible partitions is far too large to enumerate, so we must use a Markov-chain Monte Carlo (MCMC) technique to sample from the posterior distribution. Here we describe the sampling procedure and the details of the Metropolis-Hastings proposal distribution used. #### Sampling Procedure We employ the standard Metropolis-Hastings algorithm to sample from the posterior distribution over partitions and model hyperparameters [56, 57]. The general idea of an MCMC method is to set up a sequence of dependent samples $\theta_{1},\theta_{2},\ldots$ that is guaranteed to converge to a target distribution, in this case the posterior distribution of our model. Starting from the current sample, a change is proposed, drawn from a proposal distribution over possible changes, $q(\theta\rightarrow\theta^{})$. This sample is either rejected, in which case the current sample is repeated, or the proposed sample is accepted as the new sample. The Metropolis-Hastings acceptance probability, $\displaystyle r(\theta\rightarrow\theta^{})$ $\displaystyle=\min\left\\{1,\frac{f(\theta^{})}{f(\theta)}\frac{q(\theta^{}\rightarrow\theta)}{q(\theta\rightarrow\theta^{})}\right\\},$ guarantees that the sequence of samples will converge to the posterior distribution, $f(\theta\|D)\propto f(\theta)f(D\|\theta)$, the prior times the likelihood. For the group model, the samples $\theta$ consist of hyperparameters for the model variant—the Dirichlet process prior parameter $\chi$ and the beta prior parameters $\alpha$ and $\beta$—as well as the group count $K$ and assignment vector $\mathbf{G}$. The link probabilities $\mathbf{P}$ governing links between groups are not included, because the likelihood function would not be compatible between partitions with different values of $K$. One possible solution to this problem would be to include $\mathbf{P}$ in the sampling procedure, restrict $K$ to a particular number for a particular run, and then appropriately weight runs with different values of $K$. Another approach is reversible-jump MCMC [58], which appropriately handles a mapping between two different parameter spaces as part of the Metropolis-Hastings proposal ratio. (We tried a reversible-jump scheme, but chains tended to get stuck at local maxima.) Instead of trying to sample values of $\mathbf{P}$, we use the marginal likelihood of a partition given model hyperparameters—that is, the posterior distribution, conditional on values of $\alpha$, $\beta$, and $\mathbf{G}$, integrated over all possible values of $\mathbf{P}$—directly in the Metropolis-Hastings procedure. This is possible because the marginal likelihood of a single partition can be calculated analytically. For a beta prior over link probabilities, the likelihood of $\mathbf{G}$, $\alpha$ and $\beta$ marginalized over all possible values of $\mathbf{P}$ is $\displaystyle f(\mathbf{A}\|\mathbf{G},\alpha,\beta)$ $\displaystyle=\int_{\mathbf{P}}f(\mathbf{P\|\alpha,\beta})f(\mathbf{A}\|\mathbf{G},\mathbf{P})\,d\mathbf{P}$ (21) $\displaystyle=\prod_{i=1}^{K}\prod_{j=1}^{K}\int_{0}^{1}\frac{1}{B(\alpha,\beta)}p_{ij}^{\alpha-1}(1-p_{ij})^{\beta-1}p_{ij}^{Y_{ij}}(1-p_{ij})^{Z_{ij}}\,dp_{ij}$ (22) $\displaystyle=\prod_{i=1}^{K}\prod_{j=1}^{K}\int_{0}^{1}\frac{1}{B(\alpha,\beta)}p_{ij}^{Y_{ij}+\alpha-1}(1-p_{ij})^{Z_{ij}+\beta-1}\,dp_{ij}$ (23) $\displaystyle=\prod_{i=1}^{K}\prod_{j=1}^{K}\frac{B(Y_{ij}+\alpha,Z_{ij}+\beta)}{B(\alpha,\beta)}.$ (24) Similarly, for a uniform prior over link probabilities, the marginal likelihood of a particular partition is simply $\displaystyle f(\mathbf{A}\|\mathbf{G})$ $\displaystyle=\prod_{i=1}^{K}\prod_{j=1}^{K}B(Y_{ij}+1,Z_{ij}+1).$ (25) #### Proposal Distribution For the case of uniform priors on partitions and link probabilities, the proposal distribution only allows changes to the partition. With the Dirichlet process prior on partitions and the beta prior on link probabilities, hyperparameters $\chi$, $\alpha$, and $\beta$ can also be changed. The proposal distribution is described as follows: 1. 1. Each hyperparameter $h$ ($\alpha$, $\beta$, and $\chi$) is chosen for update with probability $p_{h}$, where $p_{h}$ are tuned to improve convergence. A proposed new value $h^{\prime}$ is drawn from a uniform distribution between $\max(0,h-r_{h})$ and $h+r_{h}$, where $r_{h}$ is a proposal radius manually tuned to improve convergence. (A scale-free proposal could easily be used instead, and may require less tuning.) 2. 2. With probability $\left(1-\sum_{h}p_{h}\right)$, a group-change move is proposed: 1. (a) A node $i$ is chosen uniformly at random as the species to be moved. 2. (b) Another node $j\neq i$ is chosen uniformly at random. 3. (c) If $i$ and $j$ are in different groups, node $i$ is moved into the group of node $j$. If $i$ and $j$ are in the same group, node $i$ is moved into a new group. ### Metropolis-coupled MCMC Although the Metropolis-Hastings algorithm is guaranteed to converge to the target distribution at some point, local maxima in the likelihood surface can cause a chain to become stuck for long periods of time. One approach to avoiding this problem, known as “Metropolis coupling,” involves running multiple chains in parallel. One chain, the “cold chain,” explores the target distribution, while the other chains, “hot chains,” explore low-likelihood configurations more freely. Periodically, swaps are proposed between chains, allowing good configurations discovered on hot chains to propagate toward the cold chain. Rather than exploring the target distribution $f(\theta\|D)\propto f(\theta)f(D\|\theta)$, heated chains explore $\displaystyle f_{\tau}(\theta\|D)\propto f(\theta)\left[f(D\|\theta)\right]^{\tau}\qquad\tau\in[0,1],$ (26) where $\tau$ is a heating parameter. We use linearly spaced values of $\tau$, with the hottest chain exploring the prior ($\tau=0$) and the coldest chain exploring the posterior ($\tau=1$). Swap moves are standard Metropolis-Hastings proposals, but rather than considering a change to a single chain, they consider a change to the joint distribution of two chains. The acceptance probability is thus the ratio of the joint distribution after and before the move: $\displaystyle r\left((\theta_{i},\theta_{j})\rightarrow(\theta_{j},\theta_{i})\right)$ $\displaystyle=\frac{f(\theta_{j})\left[f(D\|\theta_{j})\right]^{\tau_{i}}f(\theta_{i})\left[f(D\|\theta_{i})\right]^{\tau_{j}}}{f(\theta_{i})\left[f(D\|\theta_{i})\right]^{\tau_{i}}f(\theta_{j})\left[f(D\|\theta_{j})\right]^{\tau_{j}}}$ (27) $\displaystyle=\left[\frac{f(D\|\theta_{i})}{f(D\|\theta_{j})}\right]^{\tau_{j}-\tau_{i}},$ (28) where $\theta_{i},\theta_{j}$ are the configurations that begin in chains $i$ and $j$, and $\tau_{i},\tau_{j}$ are the heat parameters of the two chains. The use of multiple heated chains has the side effect of drastically improving estimates of marginal likelihoods for model selection, as described in the next section. ### Model Selection via Marginal Likelihood The marginal likelihood of a model is the likelihood averaged over the prior distribution. That is, it is the likelihood one would expect by randomly sampling parameters from the prior distribution: $\displaystyle f(D\|M)$ $\displaystyle=\int_{\theta}f(\theta)f(D\|\theta)\,d\theta\,.$ (29) This value serves as a useful measure of model fit because it directly incorporates the dependence of the likelihood on uncertainty in parameter values, implicitly penalizing extra degrees of freedom [49]. If an additional parameter improves the maximum likelihood but decreases the average likelihood, the model suffers from overfitting relative to the simpler model. #### Marginal Likelihood for the Group Model The marginal likelihood for the group model involves integrating over all hyperparameters, partitions, and link probabilities. For the model with uniform distributions over partitions and link probabilities, the marginal likelihood is $\displaystyle f(\mathbf{A}\|M_{u,u})$ $\displaystyle=\sum_{\mathbf{G}}f(\mathbf{G})f(\mathbf{A}\|\mathbf{G})$ (30) $\displaystyle=\sum_{\mathbf{G}}\frac{1}{\mathcal{B}(N)}\left[\prod_{i=1}^{K}\prod_{j=1}^{K}B(Y_{ij}+1,Z_{ij}+1)\right].$ (31) With a Dirichlet process prior over partitions and a uniform distribution over link probabilities, the marginal likelihood is similarly $\displaystyle f(\mathbf{A}\|M_{d,u})$ $\displaystyle=\int_{0}^{\infty}f(\chi)\sum_{\mathbf{G}}f(\mathbf{G}\|\chi)f(\mathbf{A}\|\mathbf{G})\,d\chi.$ (32) Using a uniform prior over partitions and a beta prior over link probabilities yields $\displaystyle f(\mathbf{A}\|M_{u,b})$ $\displaystyle=\sum_{\mathbf{G}}f(\mathbf{G})\int_{0}^{\infty}f(\alpha)\int_{0}^{\infty}f(\beta)f(\mathbf{A}\|\mathbf{G},\alpha,\beta)\,d\beta\,d\alpha.$ (33) Combining both gives $\displaystyle f(\mathbf{A}\|M_{d,b})$ $\displaystyle=\int_{0}^{\infty}f(\chi)\sum_{\mathbf{G}}f(\mathbf{G}\|\chi)\int_{0}^{\infty}f(\alpha)\int_{0}^{\infty}f(\beta)f(\mathbf{A}\|\mathbf{G},\alpha,\beta)\,d\beta\,d\alpha\,d\chi.$ (34) #### Thermodynamic Integration for Marginal Likelihood Estimation As enumeration across all possible partitions is impossible for networks of any significant size, we would like to use MCMC to estimate the marginal likelihood for the sake of comparison among different models. Marginal likelihood estimates derived from a single chain, such as the harmonic mean estimator of Raftery [31], converge very slowly, because MCMC fails to sample sufficiently from low-likelihood areas. However, it is possible to use the information gathered about low-likelihood areas in heated chains using a technique called thermodynamic integration [59, 60], or path sampling [61]. Assuming a continuum of heated chains, the thermodynamic estimator for the log-marginal likelihood is $\displaystyle\log\hat{\mathcal{L}}(M)$ $\displaystyle=\int_{0}^{1}\frac{1}{m}\sum_{i=1}^{m}\pi(\theta_{i,\tau})\log\mathcal{L}(\theta_{i,\tau})\,d\tau$ (35) where $m$ is the number of samples in the MCMC output, and $\theta_{i,\tau}$ is a single sample from the output in a chain with heat parameter $\tau$ [60]. With a finite number of chains, we estimate this integral using cubic spline interpolation as implemented in the splinefun function in the R software package [62]. ### Consensus Partitions The full output of an MCMC chain from the group model includes an extremely large number of different partitions, and, for the sake of interpretation, it is desirable to seek a consensus partition that does a reasonable job of summarizing the distribution. We use a simple, computationally inexpensive method to accomplish this task: in short, clustering the nodes in the network based on a pairwise group-membership matrix. The group-membership matrix $\mathbf{M}$ is the posterior probability that two nodes are in the same group and 0 otherwise, that is, $\displaystyle\mathbf{M}$ $\displaystyle=\sum_{\mathbf{G}}P(\mathbf{G}\|\mathbf{A})\mathbf{M}_{\mathbf{G}}\,,$ (36) where an entry $\mathbf{M}_{\mathbf{G}}$ is 1 if nodes $i$ and $j$ are in the same group, that is, $\displaystyle\mathbf{M}_{\mathbf{G},ij}$ $\displaystyle=\delta_{\mathbf{G}_{i},\mathbf{G}_{j}}\,,$ (37) where $\delta$ is the Kronecker delta and $\mathbf{G}$ is the assignment vector for the partition. This matrix is estimated from MCMC output as the fraction of MCMC samples in which the corresponding species are in the same group: $\displaystyle\mathbf{\hat{M}}$ $\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\mathbf{M}_{\mathbf{G}_{i}}\,.$ (38) A consensus partition is formed by applying a hierarchical clustering algorithm to the group-membership matrix estimate $\mathbf{\hat{M}}$, and then cutting the dendrogram at some number of groups $K$, forming a consensus partition with assignment vector $\mathbf{G}_{K}$ and group-membership matrix $\mathbf{M}_{K}$. The goodness of fit of a consensus partition is simply measured as the correlation between $\mathbf{\hat{M}}$ and $\mathbf{M}_{K})$. The best consensus partition is thus identified using the value of $K$ that gives the highest correlation. We use the average-linkage clustering algorithm [63] as implemented by the hclust function in the R software package [62], treating $\mathbf{1}-\mathbf{\hat{M}}$ as distance matrix. We find that the average- linkage algorithm produces higher correlations than the other algorithms implemented as well as ideal $K$ close to the mean $K$ in the MCMC output. Furthermore, we find that consensus partitions produce higher correlations with the $\mathbf{\hat{M}}$ than any individual partition in the MCMC output. ## Acknowledgments We acknowledge the support of NSF grant EF-0827493 (Program on Theory in Biology) to S.A. and M.P., and of the DOE Computational Science Graduate Fellowship (grant DE-FG02-97ER25308) to E.B. T.B. was supported by the Howard Hughes Medical Institute, and M.P. is a Howard Hughes Medical Institute Investigator. A.D. acknowledges the McDonnell Foundation for financial support, the Frankfurt Zoological Society for logistical support in the Serengeti for his work on food webs, and the members of Serengeti Biocomplexity Project for many interesting discussions about the Serengeti food web. 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Journal of Statistical Software 21: 1–16. ## Figures Figure 1: The Serengeti food web. The network is shown using a spring-layout algorithm without clustering. Plant, herbivore, and carnivore nodes are green, orange, and red, respectively. Figure 2: Posterior distributions and prior expectations of aggregation parameter $\chi$ and group count $K$. Figure 3: Pairwise group membership matrix. Species are identically ordered top to bottom and left to right according to the $16$-group consensus partition as listed in Table 2. Hue indicates group identity; color saturation indicates the fraction of partitions in which species occupy the same group. Note that this image conveys information about group membership, not network connectivity. Figure 4: Adjacency matrix ordered by groups. Species are identically ordered top to bottom and left to right according to the $16$-group consensus partition as listed in Table 2. Black matrix entries indicate that the species in the column feeds on the species in the row. Columns that would indicate prey of plant groups are omitted. Note that in a modular network according to the standard definition, links would be concentrated on the diagonal of the adjacency matrix, since they occur within groups. By contrast, here links are concentrated in off-diagonal blocks. Figure 5: Network layout of groups. The network is shown organized and colored by group according to the $16$-group consensus partition listed in Table 2. Figure 6: Network layout of aggregated groups. Nodes in the network are aggregated and colored by group according to the $16$-group consensus partition listed in Table 2, and arranged vertically by trophic level. Line thickness indicates the link density between groups. Node area is proportional to the number of species in a group. ## Supporting Information Figure S1: Implicit prior on number of groups with uniform partition prior. The prior distribution on number of groups $K$ is shown for a uniform partition prior for a network with 100 nodes. The mode of the distribution is at $K=28$. Figure S2: Posterior distributions of link density parameters $\alpha$ and $\beta$. Color brightness indicates posterior density, estimated using the ks multivariate kernel density estimation package for R [64]. Contours indicate cumulative density. The $\alpha$ parameter is significantly lower than 1, indicating departure from a uniform distribution. Figure S3: Distribution of link probability parameters. The prior distribution for link probability parameters, integrated over the priors for beta distribution parameters $\alpha$ and $\beta$, is indicated with a dotted line. The heat map shows beta distributions corresponding to the posterior distribution for $\alpha$ and $\beta$, with lightness indicating the posterior density of the parameter values. (See supporting file tab_S1_species_list.csv for data.) Table S1: Species in the Serengeti food web. (See supporting file tab_S2_edge_list.csv for data.) Table S2: Feeding links in the Serengeti food web. (See supporting file tab_S3_consensus_partition.csv for data.) Table S3: $\mathbf{16}$-group consensus partition. (See supporting file tab_S4_link_density.csv for data.) Table S4: Link densities between groups in the $\mathbf{16}$-group consensus partition. ## Tables Table 1: Marginal likelihood estimates for model variants calculated via thermodynamic integration. Partition prior \| Link prior \| Log marginal likelihood estimate \| 95% bootstrap confidence int. ---\|---\|---\|--- Uniform \| Uniform \| $-1826.87$ \| $(-1826.94,-1826.82)$ Uniform \| Beta \| $-1463.22$ \| $(-1463.39,-1463.03)$ Dirichlet process \| Uniform \| $-1547.41$ \| $(-1547.45,-1547.37)$ Dirichlet process \| Beta \| $-1356.96$ \| $(-1357.02,-1356.90)$ One group \| Uniform \| $-2870.58$ \| (exact) $170$ groups \| Uniform \| $-20031.95$ \| (exact) $170$ groups \| Beta \| $-2870.58$ \| Table 2: Groups identified in the Serengeti food web using a $\mathbf{16}$-group consensus partition. Group 1 \| Crocuta crocuta, Lycaon pictus, Panthera leo, Panthera pardus, Acinonyx jubatus ---\|--- Group 2 \| Canis aureus Group 3 \| Canis mesomelas, Leptailurus serval, Caracal caracal Group 4 \| Connochaetus taurinus, Gazella granti, Gazella thomsoni, Equus burchelli, Alcelaphus buselaphus, Aepyceros melampus, Damaliscus korrigum Group 5 \| Kobus ellipsiprymnus, Phacochaerus aethiopicus, Tragelaphus scriptus, Ourebia ourebi, Redunca redunca, Pedetes capensis, Taurotragus oryx, Rhabdomys pumilio, Hippopotamus amphibus, Cercopithecus aethiops Group 6 \| Syncerus caffer Group 7 \| Heterohyrax brucei, Procavia capensis Group 8 \| Agama planiceps Group 9 \| Papio anubis, Giraffa camelopardalis, Madoqua kirkii Group 10 \| Loxodenta africana Group 11 \| Panicum coloratum, Sporobolus pyramidalis, Hyparrhenia filipendula, Harpachne schimperi, Digitaria macroblephara, Eragrostis tenuifolia, Grewia bicolor, Aristida adoensis, Brachiaria semiundulata, Pennisetum mezianum, Bothriochloa insculpta, Panicum maximum, Sida spp., Eustachys paspaloides, Croton macrostachyus, Solanum incanum, Indigofera hochstetteri, Hibiscus spp., Heteropogon contortus, Cynodon dactylon, Themeda triandra, Balanites aegytiaca Group 12 \| Digitaria scalarum/abysinnica, Dinebra retroflexa, Ischaemum afrum, Eragostris cilianensis, Hyparrhenia rufa, Sporobolus fimbriatus, Sporobolus spicatus Group 13 \| Microchloa kunthii Group 14 \| Echinochloa haploclada, Digitaria milanjiana, Panicum deustum, Digitaria ternata, Andropogon schirensis, Cymbopogon excavatus, Setaria sphacelata, Typha capensis, Setaria pallidifusca, Phragmites mauritianus, Eragrostis exasperata, Andropogon greenwayi, Lonchocarpus eriocalyx, Sporobolus centrifugus, Hyparrhenia dissoluta, Chloris roxburghiana, Aristidia hordacea, Chloris pycnothrix, Panicum repens, Aristidia kenyensis, Combretum molle, Acacia xanthophloea, Disperma kilimandscharica, Vossia cuspida, Odyssea jaegeri, Sporobolus ioclados, Euphorbia candelabrum, Sorghum versicolor, Kigelia africana, Olea spp., Sporobolus festivus, Acacia pallens, Crotalaria spinosa, Digitaria diagonalis, Boscia augustifolia, Acacia robusta, Acacia seyal/hockii, Chloris gayana, Pennisetum stramineum, Commiphora africana/trothae Group 15 \| Acacia senegal, Acacia tortilis Group 16 \| Grewia fallax, Cissus quadrangularis, Cissus rotundifolia, Commelina africana, Allophylus rubifolus, Sensevieria ehrenbergiana, Pavetta assimilis, Phyllanthus sepialis, Acalypha fructicosa, Maerua triphllya, Ficus glumosa, Croton dichogamus, Sclerocarya birrea, Capparis tomentosa, Ximenia caffra, Cordia ovalis, Grewia trichocarpa, Abutilon angulatum, Pappaea capensis, Commiphora schimperi, Albuca spp., Ficus ingens, Hoslundia opposita, Ocinum suave, Cenchrus ciliaris, Solanum dennekense, Aloe macrosiphon, Indigofera basiflora, Ipomoea obscura, Albizia harveyi, Ficus thinningii, Emilia coccinea, Cyphostemma nierensis, Spirocarpa spp., Sensevieria suffruticosa, Pupalia lappacea, Aloe secundiflora, Turreae fischeri, Pavonia patens, Jasminum fluminense, Acacia clavigera, Cassia didymobotrya, Kedrotis foetidissima, Hypoestes forskalii, Zisiphus mucronata, Commiphora merkeri, Blepharis acanthoides, Iboza sp., Rhoicissus revoilii, Kalanchoe sp., Solanum nigrum, Achyranthes aspera, Digitaria velutina, Tricholaena eichingeri, Lippia ukambensis, Heliotropium steudneri, Kyllinga nervosa, Sporobolus stapfianus, Cyperus kilimandscharica, Pellaea calomelanos, Sporobolus pellucidus, Eragrostis aspera, Eriochloa nubica, Diheteropogon amplectus	arxiv-papers	2010-11-17T21:23:29	2024-09-04T02:49:14.941622	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Edward B. Baskerville, Andy P. Dobson, Trevor Bedford, Stefano\n Allesina, Mercedes Pascual", "submitter": "Edward Baskerville", "url": "https://arxiv.org/abs/1011.4080" }
1011.4141	# On Artinianness of formal local cohomology, colocalization and coassociated primes Majid Eghbali Martin-Luther-Universität Halle-Wittenberg, Institut für Informatik, D – 06099 Halle (Saale), Germany.-And- Islamic Azad University Branch Sofian, Sofian, Iran m.eghbali@yahoo.com ###### Abstract. This paper at first concerns some criteria on Artinianness and vanishing of formal local cohomology modules. Then we consider the cosupport and the set of coassociated primes of these modules more precisely. ###### Key words and phrases: formal local cohomology, local cohomology, Artinian modules, cosupport, coassociated primes ###### 2000 Mathematics Subject Classification: 13D45, 13C14. ## 1\. Introduction Throughout, $\mathfrak{a}$ is an ideal of a commutative Noetherian ring $R$ and $M$ an $R$-module. Let $V(\mathfrak{a})$ be the set of prime ideals in $R$ containing $\mathfrak{a}$. For an integer $i$, let $H^{i}_{\mathfrak{a}}(M)\ $ denote the $i$-th local cohomology module of $M$. We have the isomorphism of $H^{i}_{\mathfrak{a}}(M)\ $ to ${\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{i}_{R}(R/\mathfrak{a}^{n},M)$ for every $i\in\mathbb{Z}$, see [2] for more details. Consider the family of local cohomology modules $\\{H^{i}_{\mathfrak{m}}(M/{\mathfrak{a}}^{n}M)\\}_{n\in\mathbb{N}}\ $. For every $n$ there is a natural homomorphism $H^{i}_{\mathfrak{m}}(M/{\mathfrak{a}}^{n+1}M)\rightarrow H^{i}_{\mathfrak{m}}(M/{\mathfrak{a}}^{n}M)$ such that the family forms a projective system. The projective limit $\mathfrak{F}^{i}_{\mathfrak{a}}(M):={\operatornamewithlimits{\varprojlim}}_{n}H^{i}_{\mathfrak{m}}(M/\mathfrak{a}^{n}M)$ is called the $i$-th formal local cohomology of $M$ with respect to $\mathfrak{a}$. Formal local cohomology modules were used by Peskine and Szpiro in [12] when $R$ is a regular ring in order to solve a conjecture of Hartshorne in prime characteristic. It is noteworthy to mention that if $U=\operatorname{Spec}(R)\setminus\\{\mathfrak{m}\\}$ and $(\widehat{U},\mathcal{O}_{\widehat{u}})$ denote the formal completion of $U$ along $V(\mathfrak{a})\setminus\\{\mathfrak{m}\\}$ and also $\widehat{\mathcal{F}}$ denotes the $\mathcal{O}_{\widehat{u}}$-sheaf associated to ${\operatornamewithlimits{\varprojlim}}_{n}M/\mathfrak{a}^{n}M$, they have described the formal cohomology modules $H^{i}(\widehat{U},\mathcal{O}_{\widehat{u}})$ via the isomorphisms $H^{i}(\widehat{U},\mathcal{O}_{\widehat{u}})\cong\mathfrak{F}^{i}_{\mathfrak{a}}(M)$, $i\geq 1$. See also [11, proposition (2.2)] when $R$ is a Gorenstein ring. Let $\textbf{x}=\\{x_{1},...,x_{r}\\}$ denote a system of elements such that $\mathfrak{m}=\operatorname{Rad}(\textbf{x})$. In [15], Schenzel has studied formal local cohomology module via following isomorphism $\begin{array}[]{ll}\ {\operatornamewithlimits{\varprojlim}}_{n}H^{i}_{\mathfrak{m}}(M/\mathfrak{a}^{n}M)\cong H^{i}({\operatornamewithlimits{\varprojlim}}_{n}(\check{C_{\textbf{x}}}\otimes M/\mathfrak{a}^{n}M))\end{array}$ where $\ \check{C_{\textbf{x}}}\ $ denotes the $\check{C}$ech complex of $R$ with respect to x . When the local ring $(R,\mathfrak{m})$ is a quotient of a local Gorenstein ring $(S,\mathfrak{n})$, we have $\begin{array}[]{ll}\ \mathfrak{F}^{i}_{\mathfrak{a}}(M)\cong\operatorname{Hom}_{R}(H^{\dim S-i}_{\mathfrak{a}^{{}^{\prime}}}(M,S),E),\ i\in\mathbb{Z}\ \ \ \ \ \ \ \ \ (1.1)\end{array}$ where $E$ denotes the injective hull of $R/\mathfrak{m}$ and $\mathfrak{a}^{{}^{\prime}}$ is the preimage of $\mathfrak{a}$ in $S$ (cf. [15, Remark 3.6]). Important problems concerning local cohomology modules are vanishing, finiteness and Artinianness results (see, e.g., [6]). In Section 2 we examine the vanishing and Artinianness of formal local cohomology modules. In the next theorem we give some criteria for vanishing and Artinianness of formal local cohomology modules: ###### Theorem 1.1. Let $(R,\mathfrak{m})$ be a local ring and $M$ be a finitely generated $R$-module. For given integers $i$ and $t>0$, the following statements are equivalent: 1. (1) $\operatorname{Supp}_{\widehat{R}}(\mathfrak{F}^{i}_{\mathfrak{a}}(M))\subseteq V(\mathfrak{m}\widehat{R})$ for all $i<t$; 2. (2) $\mathfrak{F}^{i}_{\mathfrak{a}}(M)$ is Artinian for all $i<t$; 3. (3) $\operatorname{Supp}_{\widehat{R}}(\mathfrak{F}^{i}_{\mathfrak{a}}(M))\subseteq V(\mathfrak{a}\widehat{R})$ for all $i<t$; 4. (4) $\mathfrak{a}\subseteq\operatorname{Rad}(\operatorname{Ann}_{R}(\mathfrak{F}^{i}_{\mathfrak{a}}(M)))$ for all $i<t$; Suppose that $t\leq 0ptM$, then the above conditions are equivalent to 5. (5) $\mathfrak{F}^{i}_{\mathfrak{a}}(M)=0$ for all $i<t$; where $\widehat{R}$ denotes the $\mathfrak{m}$-adic completion of $R$. It should be noted that it has been shown independently in [8] that statements (2) and (4) are equivalent. Note that as we see in Theorem 1.1, we have the equivalence between $\operatorname{Supp}_{\widehat{R}}(\mathfrak{F}^{i}_{\mathfrak{a}}(M))\subseteq V(\mathfrak{a}\widehat{R})$ for all $i<t$ and $\mathfrak{a}\subseteq\operatorname{Rad}(\operatorname{Ann}_{R}(\mathfrak{F}^{i}_{\mathfrak{a}}(M)))$ for all $i<t$, which is not true in general for an arbitrary module. In Section 3, we study the cosupport of formal local cohomology via Richardson’s definition of colocalization (cf. Definition 3.1). We show that when $(R,\mathfrak{m})$ is a local ring, $M$ is a finite $R$-module and $\mathfrak{F}^{i}_{\mathfrak{a}}(M)$ is Artinian $(i\in\mathbb{Z})$, then $\operatorname{CoSupp}(\mathfrak{F}^{i}_{\mathfrak{a}}(M))\subseteq V(\mathfrak{a})$ (cf. 3.5). As a further result we reduce to the case $M=R$ when considering the cosupport of top formal local cohomology modules which is the analogue for formal local cohomology of the result due to Huneke-Katz- Marley in [7, Proposition 2.1]: ###### Theorem 1.2. Let $(R,\mathfrak{m})$ be a local ring. Let $M$ be a finitely generated $R$-module. Then 1. (1) $\operatorname{CoSupp}(\mathfrak{F}^{c}_{\mathfrak{a}}(M))=\operatorname{CoSupp}(\mathfrak{F}^{c}_{\mathfrak{a}}(R/J))$, 2. (2) $\operatorname{Supp}(\mathfrak{F}^{c}_{\mathfrak{a}}(M))=\operatorname{Supp}(\mathfrak{F}^{c}_{\mathfrak{a}}(R/J))$, where $J$ is $\operatorname{Ann}_{R}M$ and $c:=\dim R/\mathfrak{a}$. For a representable module $M$, $\operatorname{CoSupp}M=V(\operatorname{Ann}M)$ (cf. [13, Theorem 2.7]). It motivates us to see when the cosupport of formal local cohomology module is a closed subset of $\operatorname{Spec}R$ in Zariski topology. For this reason in Section 4 we study the set of coassociated primes of formal local cohomology more precisely. In this direction when $(R,\mathfrak{m})$ is a local ring and $M$ is an $R$-module, the set of minimal primes in $\operatorname{CoSupp}(M)$ is finite if and only if $\operatorname{CoSupp}(M)$ is a closed subset of $\operatorname{Spec}R$ (lemma 4.2). Hence, it is enough to ask when the $\operatorname{Coass}M$ is finite. We give affirmative answers to this question in some cases, see Proposition 4.4 and Theorem 1.4 below. It is noteworthy that for a finitely generated module $M$ over a local ring $(R,\mathfrak{m})$, $\operatorname{Coass}_{\widehat{R}}\mathfrak{F}^{0}_{\mathfrak{a}}(M)$ is finite since $\mathfrak{F}^{0}_{\mathfrak{a}}(M)$ is a finite $\widehat{R}$-module (cf. [15, Lemma 4.1]) and $\operatorname{Coass}\mathfrak{F}^{\dim}_{\mathfrak{a}}(M)$ is finite as $\mathfrak{F}^{\dim M}_{\mathfrak{a}}(M)$ is an Artinian $R$-module (cf. [1, Lemma 2.2] or Proposition 2.1). As final results in Section 4, we give the following results for top formal local cohomology modules: ###### Theorem 1.3. Let $(R,\mathfrak{m})$ be a local ring of dimension $d>1$. Let $\mathfrak{F}^{d}_{\mathfrak{a}}(R)=0$. Then: 1. (1) If $\mathfrak{p}\in\operatorname{Coass}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)$, then it implies that $\dim(R/(\mathfrak{a},\mathfrak{p}))=d-1$. 2. (2) $\operatorname{Assh}(R)\cap\operatorname{Coass}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)\subseteq\\{\mathfrak{p}\in\operatorname{Spec}R:\dim R/\mathfrak{p}=d,\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})\neq\mathfrak{m}\\}$. 3. (3) If $\operatorname{Coass}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)\subseteq\operatorname{Assh}(R)$, then $\\{\mathfrak{p}\in\operatorname{Spec}R:\dim(R/(\mathfrak{a},\mathfrak{p}))=d-1\\}\subseteq\operatorname{Coass}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)$. Next result shows that for a one dimensional ideal $\mathfrak{a}$ of a complete local ring $R$ of dimension $d$, $\operatorname{Cosupp}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)$ is closed. ###### Theorem 1.4. Let $(R,\mathfrak{m})$ be a local complete ring of dimension $d$. Let $\mathfrak{a}$ be an ideal of dimension one. Then $\begin{array}[]{ll}\ \mathfrak{F}^{d-1}_{\mathfrak{a}}(R)=0,\ when\ \ d>2,\end{array}$ in particular $\operatorname{Coass}_{R}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)=\emptyset$. $\begin{array}[]{ll}\ \operatorname{Coass}_{R}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)\subseteq\\{\mathfrak{m}\\},\ when\ \ d=1,\end{array}$ and in the case $d=2$ we have $\begin{array}[]{ll}\ \operatorname{Coass}_{R}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)=\bigcup^{r}_{i=1}\operatorname{Coass}_{R}(\widehat{R_{\mathfrak{p}_{i}}})=\\{\mathfrak{p}_{1},...,\mathfrak{p}_{r}\\}\cup(\bigcup^{s}_{j=1}\\{\mathfrak{q}_{j}:R_{\mathfrak{p}_{i}}/\mathfrak{q}_{j}R_{\mathfrak{p}_{i}}\text{ is not complete }\\}),\end{array}$ where $\mathfrak{p}_{1},...\mathfrak{p}_{r}$ are minimal prime ideals of $\mathfrak{a}$ and $\mathfrak{q}_{1},...\mathfrak{q}_{s}$ are minimal prime ideals of $R$ with $\mathfrak{q}_{j}\subseteq\mathfrak{p}_{i}$ for $i\in\\{1,...,r\\}$. In particular $\operatorname{Cosupp}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)$ is closed for all $d>0$. We denote by $\max(R)$ the set of maximal ideals of $R$. ## 2\. On Artinianness of $\mathfrak{F}^{i}_{\mathfrak{a}}(M)$ Important problems concerning local cohomology modules are vanishing, finiteness and Artinianness results. In the present section we study the vanishing and Artinianness conditions of formal local cohomology modules as our main result. Not so much is known about the mentioned properties. In [1] Asgharzadeh and Divani-Aazar have investigated some properties of formal local cohomology modules. For instance they showed that $\mathfrak{F}^{d}_{\mathfrak{a}}(M)$ is Artinian for $d:=\dim M$. Here we give an alternative proof of it with more information on the attached primes of $\mathfrak{F}^{d}_{\mathfrak{a}}(M)$: ###### Proposition 2.1. Let $\mathfrak{a}$ be an ideal of a local ring $(R,\mathfrak{m})$ and $M$ a finitely generated $R$-module of dimension $d$. Then $\mathfrak{F}^{d}_{\mathfrak{a}}(M)$ is Artinian. Furthermore $\begin{array}[]{ll}\operatorname{Att}_{R}(\mathfrak{F}^{d}_{\mathfrak{a}}(M))=\\{\mathfrak{p}\in\operatorname{Ass}M:\dim R/\mathfrak{p}=d\\}\cap V(\mathfrak{a}).\end{array}$ Proof. By Independence Theorem we may assume that $\operatorname{Ann}M=0$ and so $d=\dim R$. As $H^{d}_{\mathfrak{m}}(M/\mathfrak{a}^{n}M)$ is right exact ($n\in\mathbb{N}$), we have $\begin{array}[]{ll}H^{d}_{\mathfrak{m}}(M/\mathfrak{a}^{n}M)&\cong H^{d}_{\mathfrak{m}}(R)\otimes_{R}M/\mathfrak{a}^{n}M\\\ &\cong H^{d}_{\mathfrak{m}}(M)\otimes_{R}R/\mathfrak{a}^{n}\\\ &\cong H^{d}_{\mathfrak{m}}(M)/\mathfrak{a}^{n}H^{d}_{\mathfrak{m}}(M).\end{array}$ Since $H^{d}_{\mathfrak{m}}(M)$ is an Artinian module so there exists an integer $n_{0}$ such that for all integer $t\geq n_{0}$ we have $\mathfrak{a}^{t}H^{d}_{\mathfrak{m}}(M)=\mathfrak{a}^{n_{0}}H^{d}_{\mathfrak{m}}(M)$. Then one can see that $\begin{array}[]{ll}\mathfrak{F}^{d}_{\mathfrak{a}}(M)\cong H^{d}_{\mathfrak{m}}(M)/\mathfrak{a}^{n_{0}}H^{d}_{\mathfrak{m}}(M),\end{array}$ which is an Artinian module. By virtue of above equations and [2, Theorem 7.3.2], the second claim is clear. $\ \ \ \ \ \ \ \Box$ ###### Lemma 2.2. Let $(R,\mathfrak{m})$ be a complete local ring and $M$ a finitely generated $R$-module. Then $\operatorname{Supp}(\mathfrak{F}^{0}_{\mathfrak{a}}(M))=\bigcup_{\mathfrak{p}\in\operatorname{Ass}_{R}\mathfrak{F}^{0}_{\mathfrak{a}}(M)}V(\mathfrak{p})$. Moreover $\operatorname{Supp}(\mathfrak{F}^{0}_{\mathfrak{a}}(M))\cap V(\mathfrak{a})\subseteq V(\mathfrak{m})$ . Proof. To prove the claim, it is enough to consider that $\operatorname{Ass}_{R}\mathfrak{F}^{0}_{\mathfrak{a}}(M)=\\{\mathfrak{p}\in\operatorname{Ass}_{R}M:\dim R/(\mathfrak{a}+\mathfrak{p})=0\\}$ (cf. [15, lemma 4.1]) . $\ \ \ \ \ \ \ \Box$ Using Lemma 2.2 we are now able to prove Theorem 1.1: Proof of Theorem 1.1. $(1)\Rightarrow(3)$ and $(2)\Rightarrow(1)$ are obvious. $(3)\Rightarrow(2):$ By passing to the completion, we may assume that $R$ is complete (cf. [15, Proposition 3.3]). We argue by induction on $t$. When $\ t=1$, there is nothing to prove, since Lemma 2.2 and the assumptions imply that $\begin{array}[]{ll}\operatorname{Supp}(\mathfrak{F}^{0}_{\mathfrak{a}}(M))=\operatorname{Supp}(\mathfrak{F}^{0}_{\mathfrak{a}}(M))\cap V(\mathfrak{a})\subseteq V(\mathfrak{m}).\end{array}$ Hence $\ \mathfrak{F}^{0}_{\mathfrak{a}}(M)\ $ is Artinian. To this end note that $\mathfrak{F}^{0}_{\mathfrak{a}}(M)$ is a finitely generated submodule of $M$. So suppose that $t>1$ and the result has been proved for smaller values of $t$. Put $\overline{M}=M/H^{0}_{\mathfrak{a}}(M)$. From the short exact sequences $\begin{array}[]{ll}0\longrightarrow H^{0}_{\mathfrak{a}}(M)\longrightarrow M\longrightarrow\overline{M}\longrightarrow 0\end{array}$ and by [15, Proposition 3.11]), we get the following long exact sequence $\begin{array}[]{ll}...\longrightarrow\mathfrak{F}^{i}_{\mathfrak{a}}(H^{0}_{\mathfrak{a}}(M))\longrightarrow\mathfrak{F}^{i}_{\mathfrak{a}}(M)\longrightarrow\mathfrak{F}^{i}_{\mathfrak{a}}(\overline{M})\longrightarrow\mathfrak{F}^{i+1}_{\mathfrak{a}}(H^{0}_{\mathfrak{a}}(M))\longrightarrow....\end{array}$ As $\mathfrak{F}^{j}_{\mathfrak{a}}(H^{0}_{\mathfrak{a}}(M))=H^{j}_{\mathfrak{m}}(H^{0}_{\mathfrak{a}}(M))$ is an Artinian $R$-module for every $j\in\mathbb{Z}$ ([2, Theorem 7.1.3]) then, one can see that $\ \operatorname{Supp}(\mathfrak{F}^{i}_{\mathfrak{a}}(\overline{M}))\subseteq V(\mathfrak{a})$ for all $i<t$. Hence, it is enough to show that $\mathfrak{F}^{i}_{\mathfrak{a}}(\overline{M})$ is Artinian, so we may assume that $H^{0}_{\mathfrak{a}}(M)=0$. Thus, there exists an $M$-regular element $x$ in ${\mathfrak{a}}$ such that from the short exact sequence $\begin{array}[]{ll}0\longrightarrow M\stackrel{{\scriptstyle x}}{{\rightarrow}}M\longrightarrow M/xM=\widetilde{M}\longrightarrow 0\end{array}$ we deduce the next long exact sequence $\begin{array}[]{ll}...\longrightarrow\mathfrak{F}^{i}_{\mathfrak{a}}(M)\stackrel{{\scriptstyle x}}{{\rightarrow}}\mathfrak{F}^{i}_{\mathfrak{a}}(M)\longrightarrow\mathfrak{F}^{i}_{\mathfrak{a}}(\widetilde{M})\longrightarrow\mathfrak{F}^{i+1}_{\mathfrak{a}}(M)\longrightarrow....\ \ \ \ \ \ \ \ \ \ (\ast)\end{array}$ Since $\ \operatorname{Supp}(\mathfrak{F}^{i}_{\mathfrak{a}}(M))\subseteq V(\mathfrak{a})\ $ for all $i<t$, it follows from the above long exact sequence that $\ \operatorname{Supp}(\mathfrak{F}^{i}_{\mathfrak{a}}(\widetilde{M}))\subseteq V(\mathfrak{a})\ $ for all $\ i<{t-1}\ $. Hence, by induction hypothesis we have $\ \mathfrak{F}^{i}_{\mathfrak{a}}(\widetilde{M})\ $ is Artinian for all $\ i<{t-1}$. Therefore in the view of $(\ast)$, $\ (0:_{\mathfrak{F}^{i}_{\mathfrak{a}}(M)}x)\ $ is Artinian for all $\ i<t$ . On the other hand since $\ \operatorname{Supp}(\mathfrak{F}^{i}_{\mathfrak{a}}(M))\subseteq V(\mathfrak{a})\ $ for all $i<t\ $, one can see that $\begin{array}[]{ll}\ \mathfrak{F}^{i}_{\mathfrak{a}}(M)=\bigcup(0:_{\mathfrak{F}^{i}_{\mathfrak{a}}(M)}{\mathfrak{a}^{\alpha}})\subseteq\bigcup(0:_{\mathfrak{F}^{i}_{\mathfrak{a}}(M)}{x^{\alpha}})\subseteq\mathfrak{F}^{i}_{\mathfrak{a}}(M)\end{array}$ so $\ \mathfrak{F}^{i}_{\mathfrak{a}}(M)=\bigcup(0:_{\mathfrak{F}^{i}_{\mathfrak{a}}(M)}{x^{\alpha}})\ $. Therefore by [9, Theorem 1.3], $\ \mathfrak{F}^{i}_{\mathfrak{a}}(M)\ $ will be Artinian for all $i<t$. $(2)\Rightarrow(4):$ Since $\mathfrak{F}^{i}_{\mathfrak{a}}(M)$ is $\mathfrak{a}$-adically complete for every $i\in\mathbb{Z}\ $(cf. [15, Theorem 3.9] or [5, Remark 3.1] ), we get $\bigcap_{n}\mathfrak{a}^{n}\mathfrak{F}^{i}_{\mathfrak{a}}(M)=0$. Moreover for all $i<t$, $\ \mathfrak{F}^{i}_{\mathfrak{a}}(M)$ is Artinian. Hence, there is an integer $u$ such that $\mathfrak{a}^{u}\mathfrak{F}^{i}_{\mathfrak{a}}(M)=0$. $(4)\Rightarrow(3)$ is obvious. $(1)\Rightarrow(5):$ By passing to the completion we may assume that $R$ is complete. We use induction on $t$. Let $t=1$. As $\operatorname{Supp}(\mathfrak{F}^{0}_{\mathfrak{a}}(M))\subseteq V(\mathfrak{m})$ so $\mathfrak{F}^{0}_{\mathfrak{a}}(M)$ must be zero. Otherwise since $\begin{array}[]{ll}\ \emptyset\neq\operatorname{Ass}(\mathfrak{F}^{0}_{\mathfrak{a}}(M))\subseteq\operatorname{Supp}(\mathfrak{F}^{0}_{\mathfrak{a}}(M))\subseteq V({\mathfrak{m}})\end{array}$ then, $\begin{array}[]{ll}\ \mathfrak{m}\in\operatorname{Ass}(\mathfrak{F}^{0}_{\mathfrak{a}}(M))=\\{\mathfrak{p}\in\operatorname{Ass}M;\dim(R/\mathfrak{a}+\mathfrak{p})=0\\},\end{array}$ this is contradiction to $0ptM>0$. Now suppose that $0ptM\geq t>1$ and that the result has been proved for smaller values of $t$. By this inductive assumption, $\mathfrak{F}^{i}_{\mathfrak{a}}(M)=0$ for $i=0,1,...,t-2$ and it only remains for us to prove that $\mathfrak{F}^{t-1}_{\mathfrak{a}}(M)=0$. Since $0ptM>1$ then, there exists $x\in{\mathfrak{m}}\ $ that is an $M$-regular element. Consider the short exact sequence $\begin{array}[]{ll}\ 0\rightarrow M\stackrel{{\scriptstyle x^{l}}}{{\rightarrow}}M\rightarrow M/x^{l}M=\bar{M}\rightarrow 0\end{array}$ for every $l$. Thus, we have the following long exact sequence $\begin{array}[]{ll}\ ...\rightarrow\mathfrak{F}^{i-1}_{\mathfrak{a}}(\bar{M})\rightarrow\mathfrak{F}^{i}_{\mathfrak{a}}(M)\stackrel{{\scriptstyle x^{l}}}{{\rightarrow}}\mathfrak{F}^{i}_{\mathfrak{a}}(M)\rightarrow\mathfrak{F}^{i}_{\mathfrak{a}}(\bar{M})\rightarrow...\end{array}$ for every $l$. As $\ 0pt\bar{M}=0ptM-1>0\ $ and for all $i<t-1$, $\ \operatorname{Supp}(\mathfrak{F}^{i}_{\mathfrak{a}}(\bar{M}))\subseteq V(\mathfrak{m})$ then, by inductive hypothesis $\ \mathfrak{F}^{i}_{\mathfrak{a}}(\bar{M})=0\ $ for all $\ i<t-1$ . Thus, for every $l$, $\ (0:_{\mathfrak{F}^{t-1}_{\mathfrak{a}}(M)}x^{l})$ is a homomorphic image of $\mathfrak{F}^{t-2}_{\mathfrak{a}}(\bar{M})$. Hence, $\ (0:_{\mathfrak{F}^{t-1}_{\mathfrak{a}}(M)}x^{l})=0\ $ for every $l$. Take into account that by assumption $\operatorname{Supp}(\mathfrak{F}^{i}_{\mathfrak{a}}(M))\subseteq V(\mathfrak{m})$ for every $i<t$. Then, $\ {\mathfrak{F}^{t-1}_{\mathfrak{a}}(M)}=\cup(0:_{\mathfrak{F}^{t-1}_{\mathfrak{a}}(M)}x^{l})=0\ $ . This completes the proof. $\ \ \ \ \ \ \ \Box$ ## 3\. Cosupport In this section we examine the cosupport of formal local cohomology. The notion of cosupport was introduced by S. Yassemi in [17]. He defined the $\operatorname{CoSupp}_{R}M$ as the set of prime ideals $\mathfrak{q}$ such that there exists a cocyclic homomorphic image $L$ of $M$ with $\mathfrak{p}\supseteq\operatorname{Ann}(L)$. His definition is equivalent to Melkersson-Schenzel’s definition for Artinian $R$-modules. Melkersson- Schenzel’s definition of colocalization does not map Artinian $R$-module to Artinian $S^{-1}R$-module through colocalization at a multiplicative closed subset of $R$ (cf. [10]). In this note we use the concept of cosupport has been introduced by A. Richardson [13]. It maps Artinian $R$-modules to Artinian $S^{-1}R$-modules (when $R$ is complete). Also it is suitable to investigate formal local cohomology modules. ###### Definition 3.1. (cf. [13]) Let $R$ be a ring and $M$ an $R$-module. 1. (1) Let $S$ be a multiplicative closed subset of $R$ and $D_{R}(-):=\operatorname{Hom}_{R}(-,E_{R})$, where $E_{R}$ is the injective hull of $\oplus R/\mathfrak{m}$, the sum running over all maximal ideals $\mathfrak{m}$ of $R$. The colocalization of $M$ relative to $S$ is the $S^{-1}R$-module $S_{-1}M=D_{S^{-1}R}(S^{-1}D_{R}(M))$. If $S=R\setminus\mathfrak{p}\ $ for some prime ideal $\mathfrak{p}\in\operatorname{Spec}(R)$, we write ${}^{\mathfrak{p}}M$ for $S_{-1}M$. 2. (2) The cosupport of $M$ is defined as follows $\begin{array}[]{ll}\ \operatorname{CoSupp}_{R}M:=\\{\mathfrak{p}\in\operatorname{Spec}(R):\ ^{\mathfrak{p}}M\neq 0\\}.\end{array}$ For brevity we often write $\operatorname{CoSupp}M$ for $\operatorname{CoSupp}_{R}M$ when there is no ambiguity about the ring $R$. Below we recall some properties of cosupport: ###### Lemma 3.2. (cf. [13, Theorem 2.7]) Let $R$ be a ring and $M$ an $R$-module. 1. (1) $\operatorname{CoSupp}M=\operatorname{Supp}D_{R}(M)$. 2. (2) If $M$ is finitely generated, then $\operatorname{CoSupp}M=V(\operatorname{Ann}M)\cap\max(R)$. 3. (3) $\operatorname{CoSupp}M=\emptyset$ if and only if $M=0$. 4. (4) $\operatorname{CoSupp}M\subseteq V(\operatorname{Ann}M)$. 5. (5) If $\ 0\rightarrow M^{{}^{\prime}}\rightarrow M\rightarrow M^{{}^{\prime\prime}}\rightarrow 0$ is exact, then $\operatorname{CoSupp}M=\operatorname{CoSupp}M^{{}^{\prime}}\cup\operatorname{CoSupp}M^{{}^{\prime\prime}}$. 6. (6) If $M$ is representable, then $\operatorname{CoSupp}M=V(\operatorname{Ann}M)$. ###### Proposition 3.3. Let $R$ be a ring and $M$ and $N$ be $R$-modules. Then the following statements are true: 1. (1) $\operatorname{CoSupp}(M)$ is stable under specialization, i.e. $\begin{array}[]{ll}\ \mathfrak{p}\in\operatorname{Cosupp}(M),\mathfrak{p}\subseteq\mathfrak{q}\Rightarrow\mathfrak{q}\in\operatorname{Cosupp}(M).\end{array}$ 2. (2) Let $M$ be a finite module, then $\operatorname{CoSupp}(M\otimes_{R}N)\subseteq\operatorname{Supp}M\cap\operatorname{CoSupp}N$. Proof. 1. (1) Let $\mathfrak{p}\in\operatorname{Cosupp}(M)$, then by definition $D_{R_{\mathfrak{p}}}(D_{R}(M)_{\mathfrak{p}})$ is nonzero and so is $D_{R}(M)_{\mathfrak{p}}$. As $0\neq D_{R}(M)_{\mathfrak{p}}=(D_{R}(M)_{\mathfrak{q}})_{\mathfrak{p}R_{\mathfrak{q}}}$, then $D_{R}(M)_{\mathfrak{q}}\neq 0$. It implies that ${}^{\mathfrak{q}}M\neq 0$. 2. (2) Use [13, 2.5] to prove. $\ \ \ \ \ \ \ \Box$ ###### Lemma 3.4. Let $\mathfrak{a}$ be an ideal of a ring $R$. Let $N$ be an Artinian $R$-module with $\operatorname{Att}_{R}(N)\subseteq V(\mathfrak{a})$. Then $\operatorname{CoSupp}N\subseteq V(\mathfrak{a})$. Proof. Since $N$ is an Artinian module then, the following descending chain $\begin{array}[]{ll}\ \mathfrak{a}N\supseteq\mathfrak{a}^{2}N\supseteq...\supseteq\mathfrak{a}^{n}N\supseteq...\end{array}$ of submodules of $N$ is stable, i.e. there exists an integer $k$ that $\mathfrak{a}^{k}N=\mathfrak{a}^{k+1}N$. As $\operatorname{Att}_{R}(N/\mathfrak{a}^{k}N)=\operatorname{Att}_{R}(N)\cap V(\mathfrak{a})$ (cf. [10, Proposition 5.2]) and $\operatorname{CoSupp}(N/\mathfrak{a}^{k}N)\subseteq V(\mathfrak{a})$ by virtue of Proposition 3.3, hence, by passing to $N/\mathfrak{a}^{k}N$ we may assume that $\mathfrak{a}^{k}N=0$. Let $\mathfrak{p}\in\operatorname{CoSupp}N$ then, ${}^{\mathfrak{p}}N\neq 0$. Thus, for every $s\in S=R\setminus\mathfrak{p}$, $sN\neq 0$ (cf. [13, 2.1]). On the other hand $\bigcap_{n}\mathfrak{a}^{n}N=\mathfrak{a}^{k}N=0$, hence, for every $s\in S$, $sN\not\subseteq\mathfrak{a}^{t}N$. It follows that for all $s\in S$, $s\notin\mathfrak{a}^{t}$ and clearly $\mathfrak{p}\in V(\mathfrak{a})$. $\ \ \ \ \ \ \ \Box$ ###### Corollary 3.5. Let $i\in\mathbb{Z}$. Let $(R,\mathfrak{m})$ be a local ring and $M$ be a finitely generated $R$-module. Assume that $\mathfrak{F}^{i}_{\mathfrak{a}}(M)$ is an Artinian $R$-module, then $\operatorname{CoSupp}\mathfrak{F}^{i}_{\mathfrak{a}}(M)\subseteq V(\mathfrak{a})$. Proof. As $\mathfrak{F}^{i}_{\mathfrak{a}}(M)$ is Artinian and $\mathfrak{a}$-adically complete so, there exists an integer $k$ such that $\bigcap_{n\geq 1}\mathfrak{a}^{n}\mathfrak{F}^{i}_{\mathfrak{a}}(M)=\mathfrak{a}^{k}\mathfrak{F}^{i}_{\mathfrak{a}}(M)=0$. Hence, [2, Proposition 7.2.11] implies that $\operatorname{Att}\mathfrak{F}^{i}_{\mathfrak{a}}(M)\subseteq V(\mathfrak{a})$ and in the light of Lemma 3.4 $\ \operatorname{CoSupp}\mathfrak{F}^{i}_{\mathfrak{a}}(M)\subseteq V(\mathfrak{a})$. $\ \ \ \ \ \ \ \Box$ ###### Remark 3.6. Converse of Corollary 3.5 is not true in general. Let $R=k[\left\|x\right\|]$ denote the formal power series ring over a field $k$. Put $\mathfrak{a}=(x)R$. Then $\begin{array}[]{ll}\ \operatorname{CoSupp}\mathfrak{F}^{0}_{\mathfrak{a}}(R)=\operatorname{Supp}D_{R}(D_{R}(H^{1}_{\mathfrak{a}}(R)))=\operatorname{Supp}H^{1}_{\mathfrak{a}}(R)\subseteq V(\mathfrak{a})\end{array}$ but $\mathfrak{F}^{0}_{\mathfrak{a}}(R)$ is not Artinian . We now turn our attention to prove Theorem 1.2. For this reason we give a preliminary lemma: ###### Lemma 3.7. Let $(R,\mathfrak{m})$ be a $d$-dimensional local ring. Let $M$ be a finitely generated $R$-module. Then $\begin{array}[]{ll}\ \mathfrak{F}^{c}_{\mathfrak{a}}(M)\cong\mathfrak{F}^{c}_{\mathfrak{a}}(R)\otimes_{R}M,\end{array}$ where $c:=\dim R/\mathfrak{a}$. Proof. At first note that by definition of inverse limit, $\mathfrak{F}^{j}_{\mathfrak{a}}(-)$ preserves finite direct sum, for every $j\in\mathbb{Z}$. Furthermore $\mathfrak{F}^{c}_{\mathfrak{a}}(-)$ is a right exact functor (cf. [15, Theorem 4.5]). Hence, by Watts’ Theorem ( [14, Theorem 3.33]) the claim is proved. $\ \ \ \ \ \ \ \Box$ Lemma 3.7 declares that $\mathfrak{F}^{c}_{\mathfrak{a}}(R)=0$ if and only if $\mathfrak{F}^{c}_{\mathfrak{a}}(M)=0$ for all finitely generated $R$-module $M$. In order to prove Theorem 1.2 we utilize the useful consequence of Gruson’s Theorem (see, e.g., [16, Corollary 4.3]) allows us to reduce to the case $M=R$ when considering the cosupport of top formal local cohomology modules: Proof of Theorem 1.2. (1): Since $\mathfrak{F}^{c}_{\mathfrak{a}}(M)\cong\mathfrak{F}^{c}_{\mathfrak{a}(R/J)}(M)$, by Independence Theorem [2, 4.2.1], we may replace $R$ by $R/J$ to assume that $M$ is faithful. Note that for $\dim R/(\mathfrak{a},J)<c$, there is nothing to prove because, $\mathfrak{F}^{c}_{\mathfrak{a}}(M)=0$. In the view of lemma 3.7 and [13, Proposition 2.5], for every $\mathfrak{p}\in\operatorname{Spec}R$ $\begin{array}[]{ll}\ {}^{\mathfrak{p}}\mathfrak{F}^{c}_{\mathfrak{a}}(M)\cong M_{\mathfrak{p}}\otimes_{R_{\mathfrak{p}}}\ ^{\mathfrak{p}}\mathfrak{F}^{c}_{\mathfrak{a}}(R).\end{array}$ As $M_{\mathfrak{p}}$ is a faithful $R_{\mathfrak{p}}$-module, [16, Corollary 4.3] implies that $M_{\mathfrak{p}}\otimes_{R}\ ^{\mathfrak{p}}\mathfrak{F}^{c}_{\mathfrak{a}}(R)=0$ if and only if ${}^{\mathfrak{p}}\mathfrak{F}^{c}_{\mathfrak{a}}(R)=0$, which completes the proof. (2): To prove, we use the localization instead of colocalization in the proof of $(1)$. $\ \ \ \ \ \ \ \Box$ ## 4\. Coassociated primes Let $M$ be an $R$-module. A prime ideal $\mathfrak{p}$ of $R$ is called a coassociated prime of $M$ if there exists a cocyclic homomorphic image $L$ of $M$ such that $\mathfrak{p}=\operatorname{Ann}(L)$. The set of coassociated prime ideals of $M$ is denoted by $\operatorname{Coass}_{R}(M)$ (cf. [17]). When the ambient R is understood, we will often write $\operatorname{Coass}(M)$ instead of $\operatorname{Coass}_{R}(M)$. Note that for an Artinian module the set of coassociated primes is finite. In this section $(R,\mathfrak{m})$ is a local ring and we denote by $D_{R}(M)=\operatorname{Hom}_{R}(M,E(R/\mathfrak{m}))$ the Matlis dual of $R$-module $M$, where $E(R/\mathfrak{m})$ is the injective hull of residue field, so in this case $\operatorname{Coass}(M)=\operatorname{Ass}D_{R}(M)$. Among other results, we will see that under certain assumptions $\operatorname{Cosupp}_{R}(\mathfrak{F}^{i}_{\mathfrak{a}}(M))$ as a subset of $\operatorname{Spec}R$ is closed in the Zariski topology for some $i\in\mathbb{Z}$. ###### Lemma 4.1. Let $(R,\mathfrak{m})$ be a local ring and $M$ an $R$-module. Then the following statements are true: 1. (1) $\operatorname{Coass}(M)\subseteq\operatorname{CoSupp}(M)$. 2. (2) Every minimal element of $\operatorname{CoSupp}(M)$ belongs to $\operatorname{Coass}(M)$. 3. (3) For any Noetherian $\widehat{R}$-module $M$, $\operatorname{Coass}(M)=\operatorname{CoSupp}(M)\subseteq\\{\mathfrak{m}\\}$, where $\widehat{R}$ denotes the $\mathfrak{m}$-adic completion of $R$. Proof. 1. (1) Let $\mathfrak{p}\in\operatorname{Coass}(M)$, then, it implies that $0\neq\operatorname{Hom}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}}/\mathfrak{p}R_{\mathfrak{p}},D_{R}(M)_{\mathfrak{p}})$. Note that it remains nonzero by taking $\operatorname{Hom}_{R_{\mathfrak{p}}}(-,E_{R_{\mathfrak{p}}}(R_{\mathfrak{p}}/\mathfrak{p}R_{\mathfrak{p}}))$ and consequently $\mathfrak{p}\in\operatorname{CoSupp}(M)$. 2. (2) Let $\mathfrak{p}\in\min\operatorname{CoSupp}(M)=\min\operatorname{Supp}D_{R}(M)$, so, $\mathfrak{p}\in\min\operatorname{Ass}D_{R}(M)$. It follows that $\mathfrak{p}\in\min\operatorname{Coass}(M)$. 3. (3) It is clear by (1) and (2).$\ \ \ \ \ \ \ \Box$ It should be noted that $\operatorname{Supp}(\mathfrak{F}^{i}_{\mathfrak{a}}(M))$ is closed when $\operatorname{Ass}(\mathfrak{F}^{i}_{\mathfrak{a}}(M))$ is finite. In fact for a local Gorenstein ring $(R,\mathfrak{m})$, $\ \operatorname{Ass}(\mathfrak{F}^{i}_{\mathfrak{a}}(R))=\operatorname{Ass}D_{R}(H^{\dim R-i}_{\mathfrak{a}}(R))$ see [4] for details. Take into account that it is not finite in general ( see [4] or [1, Remark 2.8(vi)]). ###### Lemma 4.2. Let $(R,\mathfrak{m})$ be a local ring and $M$ be an $R$-module. The set of minimal primes in $\operatorname{CoSupp}(M)$ is finite if and only if $\operatorname{CoSupp}(M)$ is a closed subset of $\operatorname{Spec}R$. Proof. Let $\operatorname{CoSupp}(M)=V(\mathfrak{b})$ for some ideal $\mathfrak{b}$ of $R$. As $R$ is Noetherian then so is $R/\mathfrak{b}$. It turns out that the set of minimal elements of $\operatorname{CoSupp}(M)$ is finite. For the reverse direction, let $\mathfrak{p}_{1},...,\mathfrak{p}_{t}$ be the minimal prime ideals of $\operatorname{CoSupp}(M)$. Put $\mathfrak{q}:=\cap_{i}\mathfrak{p}_{i}$. We claim that $\operatorname{CoSupp}M=V(\mathfrak{q})$. It is clear that $\operatorname{CoSupp}(M)\subseteq V(\mathfrak{q})$. For the opposite direction assume that there is a prime ideal $Q\supset\mathfrak{q}$. Then $Q\supset\mathfrak{p}_{j}$, for some $1\leq j\leq t$ so the proof follows by 3.3(1). $\ \ \ \ \ \ \ \Box$ We deduce from above lemma that the cosupport of formal local cohomology module is closed, whenever its set of coassociated primes is finite. Therefore if one of the situations in Theorem 1.1 is true, the cosupport of formal local cohomology module is closed. Also $\operatorname{CoSupp}\mathfrak{F}^{\dim M}_{\mathfrak{a}}(M)$ is closed, as $\mathfrak{F}^{\dim M}_{\mathfrak{a}}(M)$ is Artinian, whenever $M$ is a finitely generated module over a local ring $(R,\mathfrak{m})$ (cf. [1, Lemma 2.2]). Take into account that when $R$ is a complete local Gorenstein ring and $\mathfrak{F}^{i}_{\mathfrak{a}}(M)$ is assumed to be either Noetherian or Artinian module, then $\begin{array}[]{ll}\ \operatorname{Cosupp}(\mathfrak{F}^{i}_{\mathfrak{a}}(M))=\operatorname{Supp}H^{\dim R-i}_{\mathfrak{a}}(M,R).\end{array}$ By virtue of [1, Theorem 2.7], for a Cohen-Macaulay ring $R$ with $\operatorname{ht}\mathfrak{a}>0$, $\mathfrak{F}^{\dim R/\mathfrak{a}}_{\mathfrak{a}}(R)$ is not Artinian. Moreover $\mathfrak{F}^{\dim M/\mathfrak{a}M}_{\mathfrak{a}}(M)$ is not finitely generated for $\dim M/\mathfrak{a}M>0$ (cf. [1, Theorem 2.6(ii)]). Below we give an alternative proof: ###### Theorem 4.3. Let $\mathfrak{a}$ be an ideal of a local ring $(R,\mathfrak{m})$ and $M$ a finitely generated R-module. Assume that $\dim M/\mathfrak{a}M>0$. Then $\mathfrak{F}^{\dim M/\mathfrak{a}M}_{\mathfrak{a}}(M)$ is not a finitely generated $R$-module. Proof. Put $c:=\dim M/\mathfrak{a}M$. In the contrary assume that $\mathfrak{F}^{c}_{\mathfrak{a}}(M)$ is a finitely generated $R$-module. Let $x\in\mathfrak{m}$ be a parameter element of $M/\mathfrak{a}M$. Hence, [15, Theorem 3.15] implies the following long exact sequence $\begin{array}[]{ll}\ ...\rightarrow\operatorname{Hom}(R_{x},\mathfrak{F}^{c}_{\mathfrak{a}}(M))\rightarrow\mathfrak{F}^{c}_{\mathfrak{a}}(M)\rightarrow\mathfrak{F}^{c}_{(\mathfrak{a},x)}(M)\rightarrow...,\end{array}$ where $i\in\mathbb{Z}$. As $\dim M/(\mathfrak{a},x)M<\dim M/\mathfrak{a}M\ $ then, $\mathfrak{F}^{c}_{(\mathfrak{a},x)}(M)=0$. Now let $f\in\operatorname{Hom}(R_{x},\mathfrak{F}^{c}_{\mathfrak{a}}(M))$. Fix an arbitrary integer $n$, so $\begin{array}[]{ll}\ f(1/x^{n})=x^{m}f(1/x^{m+n})\in x^{m}\mathfrak{F}^{c}_{\mathfrak{a}}(M),\end{array}$ for every integer $m$. It implies that $f(1/x^{n})\in\bigcap_{m}x^{m}\mathfrak{F}^{c}_{\mathfrak{a}}(M)=0$ by Krull’s Theorem and hence, $f=0$. Now it follows that $\mathfrak{F}^{c}_{\mathfrak{a}}(M)=0$, which is a contradiction, see [15, Theorem 4.5]. $\ \ \ \ \ \ \ \Box$ Now we examine the set of coassociated primes of top formal local cohomology to show that by some assumptions on $R$, it could be finite. ###### Proposition 4.4. Let $\mathfrak{a}$ be an ideal of a complete Gorenstein local ring $(R,\mathfrak{m})$ and $c:=\dim R/\mathfrak{a}$. Let $M$ be a finitely generated $R$-module. Then $\begin{array}[]{ll}\ \operatorname{Coass}_{R}(\mathfrak{F}^{c}_{\mathfrak{a}}(M))=\operatorname{Supp}_{R}(M)\cap\operatorname{Ass}(H^{\operatorname{ht}\mathfrak{a}}_{\mathfrak{a}}(R)).\end{array}$ In particular, $\ \operatorname{Coass}_{R}(\mathfrak{F}^{c}_{\mathfrak{a}}(M))$ is finite. Proof. $\begin{array}[]{ll}\ \operatorname{Coass}_{R}(\mathfrak{F}^{c}_{\mathfrak{a}}(M))&=\operatorname{Coass}_{R}(\mathfrak{F}^{c}_{\mathfrak{a}}(R)\otimes_{R}M)\\\ &=\operatorname{Supp}_{R}M\cap\operatorname{Coass}_{R}(\mathfrak{F}^{c}_{\mathfrak{a}}(R))\\\ &=\operatorname{Supp}_{R}M\cap\operatorname{Ass}_{R}(H^{\operatorname{ht}\mathfrak{a}}_{\mathfrak{a}}(R))\end{array}$ where the first equality is clear by Lemma 3.7, the second equality follows by [17, Theorem 1.21]. $\ \ \ \ \ \ \ \Box$ It should be noted that by hypotheses in Proposition 4.4, $\operatorname{ht}\mathfrak{a}=\operatorname{grade}_{R}\mathfrak{a}$ and it is well-known that $\operatorname{Ass}_{R}(H^{\operatorname{grade}_{R}\mathfrak{a}}_{\mathfrak{a}}(R))$ is finite, cf. [3]. ###### Corollary 4.5. Keep the notations and hypotheses in Proposition 4.4, $\begin{array}[]{ll}\ \mathfrak{F}^{c}_{\mathfrak{a}}(M)=0\Longleftrightarrow\operatorname{Supp}_{R}(M)\cap\operatorname{Ass}(H^{\operatorname{ht}\mathfrak{a}}_{\mathfrak{a}}(R))=\emptyset.\end{array}$ ###### Proposition 4.6. Let $i\in\mathbb{Z}$. Let $\mathfrak{a}\subset R$ be an ideal of a ring $R$. If $\operatorname{Coass}_{R}\mathfrak{F}^{i}_{\mathfrak{a}}(R)$ is finite, then so is $\operatorname{Coass}_{R}\mathfrak{F}^{i}_{\mathfrak{a}}(R/H^{0}_{\mathfrak{a}}(R))$. Proof. Consider the exact sequence $\begin{array}[]{ll}\ 0\rightarrow H^{0}_{\mathfrak{a}}(R)\rightarrow R\rightarrow R/H^{0}_{\mathfrak{a}}(R)=\overline{R}\rightarrow 0.\end{array}$ It provides the following long exact sequence $\begin{array}[]{ll}\ ...\rightarrow\mathfrak{F}^{i}_{\mathfrak{a}}(H^{0}_{\mathfrak{a}}(R)){\stackrel{{\scriptstyle\psi}}{{\rightarrow}}}\mathfrak{F}^{i}_{\mathfrak{a}}(R)\stackrel{{\scriptstyle\varphi}}{{\rightarrow}}\mathfrak{F}^{i}_{\mathfrak{a}}(\overline{R})\rightarrow\mathfrak{F}^{i+1}_{\mathfrak{a}}(H^{0}_{\mathfrak{a}}(R))\rightarrow...,\ \ \ \ \ \ \ \ (\ast)\end{array}$ for every $i$. As $\mathfrak{F}^{i}_{\mathfrak{a}}(H^{0}_{\mathfrak{a}}(R))=H^{i}_{\mathfrak{m}}(H^{0}_{\mathfrak{a}}(R))$ is Artinian, it follows that $\operatorname{Coass}(\mathfrak{F}^{i}_{\mathfrak{a}}(H^{0}_{\mathfrak{a}}(R)))$ is finite. By virtue of $(\ast)$, we get the following short exact sequence $\begin{array}[]{ll}\ 0\rightarrow U\rightarrow\mathfrak{F}^{i}_{\mathfrak{a}}(\overline{R})\rightarrow U^{{}^{\prime}}\rightarrow 0,\end{array}$ where $U=\operatorname{coker}\psi$ and $U^{{}^{\prime}}=\operatorname{coker}\varphi$. It implies that $\operatorname{Coass}\mathfrak{F}^{i}_{\mathfrak{a}}(\overline{R})$ is finite. To this end note that $\operatorname{Coass}(U)$ is finite by [17, Theorem 1.10] and $\operatorname{Coass}(U^{{}^{\prime}})$ is finite as $\mathfrak{F}^{i+1}_{\mathfrak{a}}(H^{0}_{\mathfrak{a}}(R))$ is Artinian. $\ \ \ \ \ \ \ \Box$ Now we are going to give more information on the last non-vanishing formal local cohomology module. ###### Theorem 4.7. Let $(R,\mathfrak{m})$ be a local ring of dimension $d>1$. Let $\mathfrak{F}^{d}_{\mathfrak{a}}(R)=0$. Then: 1. (1) If $\mathfrak{p}\in\operatorname{Coass}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)$, then it implies that $\dim(R/(\mathfrak{a},\mathfrak{p}))=d-1$. 2. (2) $\operatorname{Assh}(R)\cap\operatorname{Coass}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)\subseteq\\{\mathfrak{p}\in\operatorname{Spec}R:\dim R/\mathfrak{p}=d,\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})\neq\mathfrak{m}\\}$. 3. (3) If $\operatorname{Coass}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)\subseteq\operatorname{Assh}(R)$, then $\\{\mathfrak{p}\in\operatorname{Spec}R:\dim(R/(\mathfrak{a},\mathfrak{p}))=d-1\\}\subseteq\operatorname{Coass}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)$. Proof. 1. (1) Let $\mathfrak{p}\in\operatorname{Coass}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)$. As $\mathfrak{F}^{d}_{\mathfrak{a}}(R)=0$ then, by [15, Theorem 4.5] we have $\begin{array}[]{ll}\ \dim R/(\mathfrak{a},\mathfrak{p})\leq\dim R/\mathfrak{a}\leq d-1.\end{array}$ On the other hand $\mathfrak{p}\in\operatorname{Coass}(R/\mathfrak{p}\otimes_{R}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R))$, because $\begin{array}[]{ll}\ \operatorname{Coass}(R/\mathfrak{p}\otimes_{R}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R))=\operatorname{Supp}R/\mathfrak{p}\cap\operatorname{Coass}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R).\end{array}$ It yields with the similar argument to lemma 3.7 that $0\neq R/\mathfrak{p}\otimes_{R}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)=\mathfrak{F}^{d-1}_{\mathfrak{a}}(R/\mathfrak{p})$. So, we have $\dim R/(\mathfrak{a},\mathfrak{p})\geq d-1$. It completes the proof. 2. (2) Let $\mathfrak{p}\in\operatorname{Assh}(R)\cap\operatorname{Coass}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)$. Then, similar to $(1)$, $\mathfrak{F}^{d-1}_{\mathfrak{a}}(R/\mathfrak{p})\neq 0$ and moreover $\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})\neq\mathfrak{m}$. To this end note that if $\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})=\mathfrak{m}$ then, $\mathfrak{F}^{d-1}_{\mathfrak{a}}(R/\mathfrak{p})=0$ by Grothendieck’s vanishing Theorem. 3. (3) Let $\mathfrak{p}\in\operatorname{Spec}R$ and $\dim(R/(\mathfrak{a},\mathfrak{p}))=d-1$. Then, it follows that $\emptyset\neq\operatorname{Coass}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R/\mathfrak{p})=\operatorname{Supp}(R/\mathfrak{p})\cap\operatorname{Coass}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)$. Let $\mathfrak{q}\in\operatorname{Coass}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)$ then, $\mathfrak{q}\supseteq\mathfrak{p}$, but by assumption $\mathfrak{q}$ is minimal so we deduce that $\mathfrak{q}=\mathfrak{p}$. $\ \ \ \ \ \ \ \Box$ ###### Remark 4.8. The inclusion in Theorem 4.7(2) is not an equality in general. For example Let $R=k[[x,y,z]]$ denote the formal power series ring in three variables over a field $k$. Let $\mathfrak{a}=(x,y)$ be an ideal of $R$ which is of dimension one and put $\mathfrak{p}=0$. It is clear that $\mathfrak{F}^{3-1}_{\mathfrak{a}}(R)=0=\mathfrak{F}^{3}_{\mathfrak{a}}(R)$, that is $\operatorname{Coass}\mathfrak{F}^{3-1}_{\mathfrak{a}}(R)=\emptyset$. ###### Lemma 4.9. Let $(R,\mathfrak{m})$ be a complete local ring and $\mathfrak{a}$ an ideal of $R$. Let $\mathfrak{p}$ be a minimal prime ideal of $\mathfrak{a}$. Then $\mathfrak{q}\in\operatorname{Coass}_{R}(\widehat{R_{\mathfrak{p}}})$ implies that $\mathfrak{q}\subseteq\mathfrak{p}$, where the functor $\ \widehat{.}\ $ denotes the completion functor. Proof. The proof is straightforward. Let $\mathfrak{q}\in\operatorname{Coass}_{R}(\widehat{R_{\mathfrak{p}}})$, then $\begin{array}[]{ll}\ 0\neq\operatorname{Hom}_{R}(R/\mathfrak{q},\operatorname{Hom}_{R}(\widehat{R_{\mathfrak{p}}},E_{R}(R/\mathfrak{m})))=\operatorname{Hom}_{R}(R/\mathfrak{q}\otimes_{R}\widehat{R_{\mathfrak{p}}},E_{R}(R/\mathfrak{m})).\end{array}$ It yields that $\begin{array}[]{ll}\ 0\neq R/\mathfrak{q}\otimes_{R}\widehat{R_{\mathfrak{p}}}=R/\mathfrak{q}\otimes_{R}R_{\mathfrak{p}}\otimes_{R_{\mathfrak{p}}}\widehat{R_{\mathfrak{p}}}.\end{array}$ It is clear that $R_{\mathfrak{p}}/\mathfrak{q}R_{\mathfrak{p}}\neq 0$ and so $\mathfrak{q}$ must be contained in $\mathfrak{p}$. $\ \ \ \ \ \ \ \Box$ Proof of Theorem 1.4 For $d>2$ and $d=1$, the claim is clear. Let $d=2$. Suppose that $\mathfrak{p}_{1},...,\mathfrak{p}_{r}$ are the minimal prime ideals of $\mathfrak{a}$. Put $S=\bigcap^{r}_{i=1}(R\setminus\mathfrak{p}_{i})$ and choose $y\in\mathfrak{m}\setminus\bigcup^{r}_{i=1}\mathfrak{p}_{i}$. By [2, Theorem 2.2.4], for any $n\in\mathbb{N}$ we have $\begin{array}[]{ll}\ 0\rightarrow H^{0}_{\mathfrak{m}}(R/\mathfrak{a}^{n})\rightarrow R/\mathfrak{a}^{n}\rightarrow D_{(y)}(R/\mathfrak{a}^{n})\rightarrow H^{1}_{\mathfrak{m}}(R/\mathfrak{a}^{n})\rightarrow 0,\end{array}$ where $D_{(y)}(R/\mathfrak{a}^{n})$ is the $(y)$-transform functor. One can see that $D_{(y)}(R/\mathfrak{a}^{n})\cong R_{S}/\mathfrak{a}^{n}R_{S}$, so we get the following exact sequence $\begin{array}[]{ll}\ 0\rightarrow H^{0}_{\mathfrak{m}}(R/\mathfrak{a}^{n})\rightarrow R/\mathfrak{a}^{n}\rightarrow R_{S}/\mathfrak{a}^{n}R_{S}\rightarrow H^{1}_{\mathfrak{m}}(R/\mathfrak{a}^{n})\rightarrow 0.\end{array}$ Furthermore $R_{S}/\mathfrak{a}^{n}R_{S}\cong\oplus^{r}_{i=1}R_{\mathfrak{p}_{i}}/\mathfrak{a}^{n}R_{\mathfrak{p}_{i}}$. All the modules in the above exact sequence satisfy the Mittag-Leffler condition so by applying inverse limits we get $\begin{array}[]{ll}\ 0\rightarrow R/\mathfrak{F}^{0}_{\mathfrak{a}}(R)\rightarrow\oplus^{r}_{i=1}\widehat{R_{\mathfrak{p}_{i}}}\rightarrow\mathfrak{F}^{1}_{\mathfrak{a}}(R)\rightarrow 0.\end{array}$ It yields that $\operatorname{Coass}_{R}(\mathfrak{F}^{1}_{\mathfrak{a}}(R))\subseteq\bigcup^{r}_{i=1}\operatorname{Coass}_{R}(\widehat{R_{\mathfrak{p}_{i}}})\subseteq\operatorname{Coass}_{R}(\mathfrak{F}^{1}_{\mathfrak{a}}(R))\cup\\{\mathfrak{m}\\}$. In the view of lemma 4.9, $\operatorname{Coass}_{R}(\mathfrak{F}^{1}_{\mathfrak{a}}(R))=\bigcup^{r}_{i=1}\operatorname{Coass}_{R}(\widehat{R_{\mathfrak{p}_{i}}})$. Now the claim is proved by [18, Beispiel 2.4]. To this end note that $\operatorname{Coass}_{R}(\widehat{R_{\mathfrak{p}_{i}}})=\operatorname{Coass}_{R_{\mathfrak{p}_{i}}}(\widehat{R_{\mathfrak{p}_{i}}})\cap R$ for every $i\in\\{1,...,r\\}$. $\ \ \ \ \ \ \ \Box$ ###### Remark 4.10. Keep the notations and hypotheses in Theorem 1.4 and let $M$ be a finitely generated $R$-module. As $R$ is complete so by Cohen’s structure Theorem, there exists a Gorenstein local ring $(S,\mathfrak{n})$ where $R$ is a homomorphic image of $S$ and $\dim R=\dim S$. Then by virtue of 3.7 we have $\begin{array}[]{ll}\ \operatorname{Ass}_{R}H^{1}_{\mathfrak{a}S}(M,S)\subseteq\operatorname{Coass}\mathfrak{F}^{d-1}_{\mathfrak{a}}(R)\end{array}$ is finite. ###### Acknowledgement . My thanks are due to my phd. adviser, Professor Peter Schenzel, for his guidance to prepare this paper and useful hints and to the reviewer for suggesting several improvements. Some parts of this paper was written while the author was at Oberwolfach: Representations of Finite Groups, Local Cohomology and Support. Many thanks to the organisers. ## References * [1] M. Asgharzadeh and K. Divaani-Aazar, finiteness properties of formal local cohomology modules and Cohen-Macaulayness, Commun. Alg. 39, no. 3, (2011), 1082-1103(22). * [2] M. Brodmann and R.Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge Univ. Press, 60, Cambridge, (1998). * [3] M. Brodmann and A. Lashgari Faghani, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc. 128, no. 10, 2851-2853; (2000). Press, 60, Cambridge, (1998). * [4] M. Hellus, Local Cohomology and Matlis Duality, Habilitationsschrift, Leipzig, (2006). * [5] M. Hellus and J. Stückrad, On endomorphism rings of local cohomology modules, Proc. Amer. Math. Soc. 136 , no. 7, 2333-2341; (2008). * [6] C. Huneke, Problems on local cohomology: Free resolutions in commutative algebra and algebraic geometry, (Sundance, UT, 1990), 93-108, Jones and Bartlett, (1992). * [7] C. Huneke, D. Katz and T. Marley , On the support of local cohomology, J. Algebra 322 (2009) 3194-3211. * [8] A. Mafi, Results of formal local cohomology modules, To appear in Bull. Malays. Math. Sci. Soc. * [9] L. Melkersson, on asymptotic stability for sets of prime ideals connected with the powers of an ideal, Math. Proc. Camb. Phil. Soc. 107, (1990), 267-271. * [10] L. Melkersson and P. Schenzel, The co-localization of an Artinian module, Proc. Edinburgh Math. Soc. 38, 121-131 (1995). * [11] A. Ogus, Local cohomological dimension of Algebraic Varieties, Ann. of Math. , 98(2), (1973), 327-365. * [12] C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, Publ. Math. I.H.E.S., 42, (1973), 323-395. * [13] A. S. Richardson, Co-localization, co-support and local cohomology, Rocky Mountain J. of Math., 36, 5, (2006), 1679-1703. * [14] J. Rotman, An Introduction to Homological Algebra, Academic Press, Orlando, FL, 1979. * [15] P. Schenzel, On formal local cohomology and connectedness, J. Algebra, 315(2), (2007), 894-923. * [16] W. Vasconcelos, Divisor Theory in Module Categories , North-Holland Math. Stud., vol. 14 , Notas de Matem tica (Notes on Mathematics), vol. 53, North-Holland Publishing Co./American Elsevier Publishing Co., Inc., Amsterdam/Oxford/New York, (1974). * [17] S. Yassemi, Coassociated primes , Comm. Algebra, 23(4), 1473-1498 (1995). * [18] H. Zöschinger, Der Krullsche Durchschnittssatz für kleine Untermoduln, Arch. Math. (Basel), 62(4), (1994), 292-299.	arxiv-papers	2010-11-18T06:30:37	2024-09-04T02:49:14.954570	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Majid Eghbali", "submitter": "Majid Eghbali", "url": "https://arxiv.org/abs/1011.4141" }
1011.4142	General formula for symmetry factors of Feynman diagrams L. T. Hue, H. T. Hung and H. N. Long _Institute of Physics, VAST, 10 Dao Tan, Ba Dinh,10000 Hanoi, Vietnam_ ###### Abstract General formula for symmetry factors (S-factor) of Feynman diagrams containing fields with high spins is derived. We prove that symmetry factors of Feynman diagrams of well-known theories do not depend on spins of fields. In contributions to S-factors, self-conjugate fields and non self-conjugate fields play the same roles as real scalar fields and complex scalar fields, respectively. Thus, the formula of S-factors for scalar theories — theories include only real and complex scalar fields — works on all well-known theories of fields with high spins.Two interesting consequences deduced from our result are : (i) S-factors of all _external connected_ diagrams consisting of only vertices with three different fields, e.g., spinor QED, are equal to unity; (ii) some diagrams with different topologies can contribute the same factor, leading to the result that the inverse S-factor for the total contribution is the sum of inverse S-factors, i.e., $1/S=\sum_{i}(1/S_{i})$. PACS number(s): 11.15.Bt, 12.39.St. Keywords: General properties of perturbation theory, Factorization. ## 1 Introduction In literature, using perturbation theory and Feynman rules, a general Green’s function of an arbitrary theory can be written in terms of sum of Feynman diagrams. Each diagram is associated with a factor known as symmetry factor (S-factor). There are some ways to calculate this factor such as given in [1] (for more details, see [2, 3, 4, 5]) using functional derivative method. A computer program [6] is also written based on this method to find out S-factors of higher-order diagrams from the lower-order ones. Other independent approaches base directly on computer programs such as [7, 8]. S-factors can also be evaluated by using Wick’s theorem, available in many textbooks (see, for example [9, 10, 11, 12, 13]). But the disadvantage of these books as well as the methods mentioned above is none of them give out any general formulas. We can see that Ref.[9] has an expression for connected diagrams of real scalar theories, Ref.[13] has some comments about S-factors for scalar electrodynamics, Ref. [10] for real scalar $\phi^{4}$ and some particular illustrations in Standard Model. Especially, the very detailed investigation into S-factors, which are very close to weights of Feynman diagrams in $\phi^{4}$ theories was presented in [15]. Refs. [16, 17] also contain S-factors of some particular diagrams in QCD. This paper is the development of [14] in which we derived the S-factors for Feynman diagrams of theories with scalar fields. The definition of S-factor can be found in [10]. We can understand this as follows. Using Wick’s theorem for expanding a Green function one often encounters many terms whose contractions are different but contributions are the same. The S-factor is the number of identical terms which are repeatedly counted. In language of Feynman diagram, this factor turns out to be the product of total number of (symmetry) permutations of all vertices and all internal propagators in the diagram, which create new identical diagrams with factors caused by bubbles. In Ref.[14] we have concentrated on two types of fields, namely real and complex scalar fields, and have noted that the distinction between these fields is very important because they contribute different factors to the formula of S-factor[14]: $\displaystyle S=g2^{\beta}2^{d}\prod_{n}(n!)^{\alpha_{n}},$ (1) where $g$ is the number of interchanges of vertices leaving the diagram topologically unchanged, $\beta$ is the number of lines connecting a vertex to itself ($\beta$ is zero if the field is complex), $d$ is the number of double bubbles, and $\alpha_{n}$ is the number of sets of $n$-identical lines connecting the same two vertices. In this paper, by considering some particular cases, we will indicate precisely that in calculating S-factors, we can classify all well-known fields into two classes. The first class comprises self-conjugate fields for which the particle is the same as the antiparticle, such as the real Higgs scalar $\sigma$ in the Standard Model, the photon and the $Z$ boson. We will often refer to this class to be the real scalar-like. The second, all non self- conjugate fields — such as charged particles — will be referred to as complex scalar-like. In analogy with the leptons where $e^{-},\mu^{-}$ and $\tau^{-}$ are called “particles” and $e^{+},\mu^{+}$ and $\tau^{+}$ are called “antiparticles”, hereafter we adopt the convention that the _negative_ electric charged scalar/vector fields (for example $\pi^{-},W^{-}$) will be called particles. Keeping these remarks in mind, we then redefine parameters $g,d,n$ and $\alpha_{n}$ of (1) (detailed in the conclusion), then the formula works on all cases. One more interesting point we would like to mention about this paper is that a simple method of calculating the SF of a particular Feynman diagram emerges directly from the graphical form itself. Especially the $g$-factor, the most complicated factor appearing in our formula (1) as well as [9] and many others textbooks, will be naturally made clear through our calculation. It relates strictly with graphical symmetry properties of the diagram. The outline of our work in this paper is as follows. In the second section, we recall T-product expansions of interaction Lagrangians into N-products and introduce a new definition of vertices and their factors in Feynman diagrams. This helps us simplify our calculation because, for every interaction Lagrangian, we will find factors that really contribute to the SFs and omit other unnecessary factors. The third section is devoted to spinor QED case, the most simple case that contains spinor fields. As mentioned in the abstract, we will show that in the spinor QED, the S-factor of an arbitrary diagram at any order in pertubative expansion is always equal to $1$. This is very useful, for example, in calculating high order QED contributions to lepton Anomalous Magnetic Moment $(g-2)$ [20]. We will also prove that spinor fields behave the same as scalar complex fields. In addition, we will discuss in detail one interesting way of practically determining the $g$ factor from the geometric symmetries of a particular Feynman diagram. In the next three sections some particular cases are illustrated to point out that when calculating S-factor of a diagram, all well-known fields always belong to one of two classes mentioned above. In the last section, we will derive the final formula of the S-factor for general cases. An expression for $g$ factor is also presented in order to determine it from connected diagrams of the total diagram. Examples of the S-factors are illustrated in Appendices A and B. ## 2 Feynman diagrams and symmetry factors Let us start using Wick’s theorem to expand $T$-products of interaction Lagrangians into sums of $N$-products [1, 10]: 1. 1. Real scalar $\phi^{3}$ theory: $\displaystyle\mathcal{L}_{int}^{r}(x)$ $\displaystyle=$ $\displaystyle\frac{\lambda}{3!}\phi^{3}(x),$ $\displaystyle\frac{1}{3!}\phi^{3}(x)\sim\frac{1}{3!}T\left[\phi^{3}(x)\right]$ $\displaystyle=$ $\displaystyle N\left[\frac{1}{3!}\phi^{3}(x)\right]+\frac{3}{3!}\phi(x)\dot{\Delta}(x)$ (2) where each $\dot{\Delta}(x)\equiv\mathop{\vbox{\halign{#\cr\kern 1.72218pt\cr$\hbox{$\hskip 2.97917pt\vrule height=3.65973pt,width=0.29999pt,depth=-0.11153pt\kern-0.29999pt\vrule height=3.65973pt,width=19.45142pt,depth=-3.35974pt\kern-0.29999pt\vrule height=3.65973pt,width=0.29999pt,depth=-0.11153pt$\hss}$\crcr\kern 1.72218pt\nointerlineskip\cr\hbox{$\displaystyle{}{\phi}{(x)}{\phi}(x)$}\crcr}}}\limits$ corresponds to a bubble located at $x$-coordinate in some Feynman diagram. 2. 2. Real scalar $\phi^{4}$ theory: $\displaystyle\mathcal{L}_{int}^{r}(x)$ $\displaystyle=$ $\displaystyle\frac{\lambda}{4!}\phi^{4}(x)$ $\displaystyle\frac{1}{4!}\phi^{4}(x)\sim\frac{1}{4!}T\left[\phi^{4}(x)\right]$ $\displaystyle=$ $\displaystyle N\left[\frac{1}{4!}\phi^{4}(x)\right]+\frac{6}{4!}N\left[\phi^{2}(x)\right]\dot{\Delta}(x)+\frac{3}{4!}\dot{\Delta}(x)\dot{\Delta}(x).$ 3. 3. Complex scalar $\varphi^{4}$ theory: $\displaystyle\mathcal{L}_{int}^{c}(x)$ $\displaystyle=$ $\displaystyle\frac{\rho}{4}[\varphi(x)\varphi^{}(x)]^{2},$ $\displaystyle\frac{1}{4}[\varphi(x)\varphi(x)^{}]^{2}\sim\frac{1}{4}T\left[\varphi(x)\varphi(x)^{}\right]^{2}$ $\displaystyle=$ $\displaystyle N\left[\frac{1}{4}[\varphi(x)\varphi(x)^{}]^{2}\right]+\frac{4}{4}N\left[\varphi(x)\varphi(x)^{}\right]\dot{\Delta}(x)$ (4) $\displaystyle+$ $\displaystyle\frac{2}{4}\dot{\Delta}(x)\dot{\Delta}(x).$ Each term in right hand sides (RHS) of (2), (LABEL:ttichreal4) and (4) changes into one particular kind of vertex in the language of Feynman diagram. They are illustrated in Fig.1, where propagators of real fields are represented as dash lines without directions (arrows), while complex cases are represented as dash lines with _directions_. Vertices are different from each others in numbers of lines and kinds of line they have. This is because terms in RHSs of 1-4 are different in fields and contractions. Now, for a given interaction Lagrangian, we can show exactly all kinds of vertex in the theory. This is very important for us to find out not only $g$ factor relating with vertices but also contributory factors of different kinds of vertex to S-factors. Vertices themselves have well-known factors as _vertex factors_ , which can be ignored due to the fact that S-factors are independent on them. (-120,130)(-120,150)2(-140,110)(-120,130)2(-120,130)(-100,110)2 (-80,140)(10,0,360)2(-80,130)(-80,110)2 (-100,80)[](a) Vertices of $\phi^{3}$(-20,110)(10,140)2(-20,140)(10,110)2 (15,110)(65,110)2(40,120)(10,0,360)2 (80,120)(10,0,360)2(80,140)(10,0,360)2 (40,80)[](b) Vertices of real $\phi^{4}$(140,110)(155,125)2(155,125)(170,140)2 (140,140)(155,125)2(155,125)(170,110)2 (180,110)(200,110)2(200,110)(220,110)2 (200,120)(10,270,630)2 (240,140)(10,270,630)2 (240,120)(10,90,450)2 (190,80)[](c) Vertices of complex $\varphi^{4}$ Figure 1: Vertices of scalar theories Vertex factors, in scalar theories, are simply $i\lambda$ or $i\rho$, while in the others such as in scalar electrodynamics or in quantum chromodynamics, where there exist interactions containing derivatives, are more complicated. For our method, the S-factor determined from (1) depends on values of factors, for example: $1/(3!)$ in $\phi^{3}$, of interacting terms in Lagrangian. These factors are obtained by taking partial derivatives of respective interacting terms. Furthermore, we need to write down each interacting term of the Lagrangian in form of [vertex factor $\times$ $T$-product], then omit this vertex factor in our calculation. The symmetry factor now depends only on $T$-product. A general Feynman diagram derived from the expansion of a general Green’s function consists of many connected pieces (subdiagrams) disconnected with each others. We will call pieces connected vacuum diagrams if they have not any external legs and connected external diagrams if they have at least one external leg [for example, see Fig.A(a.10)]. Every connected subdiagram has its private S-factor. In this work, we concentrate on only aspect of S-factor calculation. Other symmetries, such as the charge conjugation under which diagrams in the QED with odd number of external photon legs give vanishing contributions, are outside the scope. Now we turn to theories of fields with high spins. ## 3 Symmetry factors in spinor QED In spinor Quantum Electrodynamics (QED), the interaction Lagrangian of one fermion field $\psi$ is given by $\displaystyle\mathcal{L}_{int}^{QED}(x)$ $\displaystyle=$ $\displaystyle eq\overline{\psi}(x)\gamma^{\mu}\psi(x)A_{\mu}(x),$ (5) where $e$ is the electromagnetic coupling constant, $q$ is the electric charge of fermion $\psi$ in units of positron charge and $A_{\mu}(x)$ is the electromagnetic field. For the sake of brevity, from now on we will write $\mathcal{L}(x)$ instead of $\mathcal{L}_{int}(x)$. The above Lagrangian has only one interacting term with a vertex factor [$ieq\gamma^{\mu}$]. $T$-product expansion gives: $\displaystyle T\left[\overline{\psi}(x)\gamma^{\mu}\psi(x)A_{\mu}(x)\right]$ $\displaystyle=$ $\displaystyle N\left[\overline{\psi}(x)\gamma^{\mu}\psi(x)A_{\mu}(x)\right]+\mathop{\vbox{\halign{#\cr\kern 1.72218pt\cr$\hbox{$\hskip 2.5pt\vrule height=7.31946pt,width=0.29999pt,depth=1.15515pt\kern-0.29999pt\vrule height=7.31946pt,width=27.80162pt,depth=-7.01947pt\kern-0.29999pt\vrule height=7.31946pt,width=0.29999pt,depth=-0.11153pt$\hss}$\crcr\kern 1.72218pt\nointerlineskip\cr\hbox{$\displaystyle{}{\overline{\psi}}{(x)\gamma^{\mu}}{\psi}(x)$}\crcr}}}\limits A_{\mu}(x)$ (8) $\displaystyle=$ $\displaystyle N\left[\overline{\psi}(x)\gamma^{\mu}\psi(x)A_{\mu}(x)\right]+iS^{\alpha}_{\beta}(x)(\gamma^{\mu})^{\beta}_{\alpha}A_{\mu}(x),$ (9) where $S(x)$ is a fermion bubble. The last expression in (9) has two terms corresponding to two kinds of vertices: The first has one photon leg, one incoming and one outgoing electron leg. The second has one photon leg and one fermion bubble. These vertices are illustrated in Fig.2. (-120,130)(-120,150)1.54(-140,110)(-120,130)(-120,130)(-100,110) (-80,140)(10,0,360)(-80,140)(10,270,90) (-80,130)(-80,110)24 (-120,90)[](a) (-75,90)[](b) Figure 2: Vertices of QED For simplicity in calculating, let us denote two terms in the LHS of (9) as follows: $\displaystyle a_{1}=N\left[\overline{\psi}(x)\gamma^{\mu}\psi(x)A_{\mu}(x)\right],\;\;a_{2}=iS^{\alpha}_{\beta}(x)(\gamma^{\mu})^{\beta}_{\alpha}A_{\mu}(x)\equiv iS(x)\gamma^{\mu}A_{\mu}(x)$ (10) Thus, (9) is rewritten as: $\displaystyle T[\overline{\psi}(x)\gamma^{\mu}\psi(x)A_{\mu}(x)]=a_{1}(x)+a_{2}(x)$ (11) The $n$-point Green’s function $G^{QED}(x_{1},x_{2},...,x_{n})$ of QED is defined as $\displaystyle G^{QED}(x_{1},x_{2},...,x_{n})$ $\displaystyle=$ $\displaystyle\sum_{p=0}^{\infty}\frac{(-i)^{p}}{p!}\int dy_{1}dy_{2}...dy_{p}\langle 0\|T\left[\phi(x_{1})\phi(x_{2})...\phi(x_{n})\right.$ (12) $\displaystyle\times$ $\displaystyle\left.\mathcal{L}(y_{1})\mathcal{L}(y_{2})...\mathcal{L}(y_{p})\right]\|0\rangle,$ where $\phi(x)$ implies a spinor $\overline{\psi}(x),\psi(x)$ or an $A_{\mu}(x)$. The $p$th-order term in this expression is: $\displaystyle G^{QED(p)}(x_{1},x_{2},...,x_{n})$ $\displaystyle=$ $\displaystyle\frac{(-i)^{p}}{p!}\int dy_{1}dy_{2}...dy_{p}\langle 0\|T\left[\phi(x_{1})\phi(x_{2})...\phi(x_{n})\right.$ (13) $\displaystyle\times$ $\displaystyle\left.\mathcal{L}(y_{1})\mathcal{L}(y_{2})...\mathcal{L}(y_{p})\right]\|0\rangle$ Note that QED has some features different from scalar cases. The Lagrangian of QED contains nonvanishing-spin fields, namely half-integer spin fields and spin-1 photon. Spinor fields follow anti-communication relations so when positions of these fields are changed in a product, a minus sign will appear. However, it does not affect the S-factors. Furthermore, every interacting term always has even number of spinor fields so the value of total product in (13) is unchanged regardless positions of these terms. This conclusion is correct for any theories. Then the method used in [14] can again be used as we will discuss next. In a resulting product [$\mathcal{L}(y_{1})\mathcal{L}(y_{2})...\mathcal{L}(y_{p})$] of (13), all terms consisting of the same numbers of $a_{i}$, have the same contributions. Then the sum of these terms is presented as a product of single symbolic term multiplied by a factor deduced from the multinomial formula: $\displaystyle(x_{1}+x_{2}+\cdot\cdot\cdot+x_{r})^{p}$ $\displaystyle=$ $\displaystyle\sum_{p_{1},p_{2},...,p_{r}}\frac{p!}{p_{1}!p_{2}!\cdot\cdot\cdot p_{r}!}x_{1}^{p_{1}}\cdot\cdot\cdot x_{r}^{p_{r}},$ (14) $\displaystyle\textrm{with}\hskip 14.22636ptp_{1}+p_{2}+p_{3}+\cdot\cdot\cdot+p_{r}$ $\displaystyle=$ $\displaystyle p.$ In case of QED, sum of all terms in (13) which have $p_{1}$ factors of $a_{1}$ and $p_{2}$ factors of $a_{2}$ are all presented as product of a single term $a_{1}^{p_{1}}a_{2}^{p_{2}}$ ($p_{1}+p_{2}=p$) multiplied by a factor: $\displaystyle\frac{p!}{p_{1}!p_{2}!}$ (15) In order to do contractions between internal fields of $a_{1}^{p_{1}}a_{2}^{p_{2}}$ and external fields $\phi(x_{1})$, $\phi(x_{2})$, $...,\phi(x_{n})$, we write the T-product as follows: $\displaystyle\langle 0\|T[\phi(x_{1})\phi(x_{2})...\phi(x_{n})a_{1}^{p_{1}}a_{2}^{p_{2}}]\|0\rangle$ $\displaystyle=$ $\displaystyle\langle 0\|T[\phi(x_{1})\phi(x_{2})...\phi(x_{n})N[\overline{\psi}(y_{1})\gamma^{\mu_{1}}\psi(y_{1})A_{\mu_{1}}(y_{1})]...$ $\displaystyle\times$ $\displaystyle N[\overline{\psi}(y_{r})\gamma^{\mu_{r}}\psi(y_{r})A_{\mu_{r}}(y_{r})](-i)S(y_{r+1})\gamma^{\mu_{r+1}}A_{\mu_{r+1}}(x)...(-i)S(y_{r+s})\gamma^{\mu_{r+s}}A_{\mu_{r+s}}(x)]\|0\rangle$ Performing contractions between fields, we rewrite (3) as a sum of terms containing only contractions. The terms with similar contractions lead to the fact that a total contribution of these terms can be presented as a product of a term (now corresponding to a Feynman diagram) multiplied by a new additional factor. This new factor contributes to our S-factor and will be calculated by performing permutations of propagators and vertices. Let us concentrate on symmetries of (3) because this will help us count the number of different terms having the same contribution. There are factors created by two kinds of permutation: ($i$) permutation of propagators (lines) in each vertex and ($ii$) permutation of vertices in one diagram. For propagator permutations, first, there is no permutation in the vertex of type ($a_{2}$) (figure 2.b) having only one leg. Second, according to Wick’s theorem, each vertex of type ($a_{1}$) has three different fields with possibility of contraction with internal fields of other vertices or external fields. These three contractions present as three different lines (see fig.2a). Again, there are no permutations of these lines. Therefore the factor caused by propagator permutations in any vertex is $f_{1}=1$. Next, consider symmetries (or equivalences) between vertices. Vertices in the same kind, which play the same roles in doing contractions, will create distinguishable terms with identical contributions. For QED, there are two types of vertex, number of these terms is given by a factor: $\displaystyle f_{2}=\frac{p_{1}!p_{2}!}{g}$ (17) Let us explain how to obtain this factor. $(p_{1}!p_{2}!)$ is the permutation number of vertices ($p_{1}$ of $a_{1}$ and $p_{2}$ of $a_{2}$) to get new terms, including repeatedly permutations. Factor $g$ cancels the repeat, i.e., permutations are repeatedly counted. It will be more clear when we discuss directly on Feynman diagrams. Combining two factors of (15), (17) and the expanding factor ($1/p!$) in (13) we get a total factor of a particular Feynman diagram appearing in (13): $\displaystyle\frac{1}{S}$ $\displaystyle=$ $\displaystyle\frac{1}{p!}\times\frac{p!}{p_{1}!p_{2}!}\times\frac{p_{1}!p_{2}!}{g}$ (18) $\displaystyle=$ $\displaystyle\frac{1}{g}$ Hence, in QED the symmetry factor is only $g$, the factor belongs to $f_{2}$ factor. It is more easy to understand $g$ in language of Feynman diagram. We will see that $g$ is the number of repeatedly vertex permutations, i.e., the number of vertex permutations that creates identical diagrams. Determining $g$ is rather complicated because we have to make clear relations not only between vertices themselves but also vertices and propagators. Fortunately, we can exploit relations between permutation symmetries and geometrical symmetries of a diagram to solve this problem. Further, $g$ factor of a general diagram can be established from $gs$ of connected pieces. Therefore, to determine $g$ we just consider particular connected subdiagrams based on geometric symmetries of itself. Let us illustrate this by some examples in Fig.3 (-100,150)(20,270,630) (-100,100)[]$(a)\ S=g=1$(-100,130)(-100,115)23 (-130,115)(-100,115)(-100,115)(-70,115) (0,150)(20,0,180) (0,150)(20,180,360) (-20,150)(20,150)26 (0,100)[]$(b)\ S=g=2$(120,150)(20,0,90) (120,150)(20,90,180) (120,150)(20,180,270) (120,150)(20,270,360) (140,150)(100,150)26 (120,170)(120,130)26(120,100)[]$(c)\ S=g=4$(220,150)(20,30,90) (220,150)(20,90,150)(220,150)(20,150,210) (220,150)(20,210,270) (220,150)(20,270,330) (220,150)(20,330,30) (220,170)(220,130)1.56 (202,140)(238,160)1.56 (238,140)(202,160)1.56(220,100)[]$(d)\ S=g=6$ Figure 3: Examples of symmetry factors in QED Let us look at figures 3(b), (c) and (d). Figure (c) has only rotational symmetries of a square, and (d)-a regular hexagon, because fermion lines have directions. With these three diagrams we can rotate (b) an angle $180^{0}$, (c) three angles $90^{0},180^{0},270^{0}$ and (c)-$k\times 60^{0},k=2,...,5$ to get the same diagrams as the origins. Clearly, the number of rotative symmetries (including trivial rotation) is exactly equal to $g$ factor found in [2]. Let us go to other examples in figure 4. Diagram in figure 4a has three identical fermion loops, lying on three vertices of a regular triangle, and three photon propagators (no direction). Then $g$ of the diagram is $3!=6$, is also equal to six symmetries of a regular triangle (two rotation symmetries, three axial and the identical). In Fig.4b, there is no symmetry because of two fixed external propagators. Fig.4c has three connected pieces: two external connected pieces do not contribute any factor while the third (connected vacuum piece) causes a factor $g=2$ by a $180^{0}$ rotation. From the above discussion, all S-factors of diagrams in the QED given in Ref.[2], can be derived. It is worth noting that in the case of spinor QED, all connected pieces relating with external legs, have $S=1$ because external legs cancel their geometrical symmetries. (-100,170)(10,180,360)(-100,170)(10,0,180) (-123,130)(10,290,170)(-123,130)(10,170,290) (-77,130)(10,60,240) (-77,130)(10,240,60) (-110,170)(-130,137)28 (-90,170)(-68,137)28 (-85,123)(-115,123)26 (-100,100)[]$(a)\ S=g=6$(-20,125)(5,125) (5,125)(35,125) (35,125)(60,125) (5,125)(5,140)23 (35,125)(35,140)23 (05,150)(10,-90,90) (5,150)(10,90,-90) (35,150)(10,90,-90) (35,150)(10,-90,90) (25,100)[]$(b)\;S=g=1$(100,140)(125,140)24 (175,140)(195,140)24 (135,140)(10,0,180) (135,140)(10,-180,0) (165,140)(10,-180,0) (165,140)(10,0,180) (150,165)(10,0,180) (150,165)(10,-180,0) (140,165)(160,165)24 (135,100)[]$(c)S=g=2$ Figure 4: Examples in calculation of symmetry factors One more new important conclusion for this section is: in our calculation, fermion fields behave exactly the same as complex scalar fields, except a minus sign for each closed fermion line. For the photon $A_{\mu}$, it has properties of a real scalar field as we will prove in the next section. Thus $S$ in (18) is a special case of formula (1). In appendix A, S-factors of the QED up to fourth order are presented. Ours results are consistent with those in [2]. ## 4 Symmetry factors in scalar Quantum Electrodynamics In scalar Quantum ElectroDynamics (sQED), the interaction Lagrangian consists of both $A_{\mu}$ and a complex scalar field: $\displaystyle\mathcal{L}^{sQED}(x)=ieqA^{\mu}(x)[\varphi^{}(x)\partial_{\mu}\varphi(x)-(\partial_{\mu}\varphi^{}(x))\varphi(x)]+e^{2}q^{2}A_{\mu}(x)A^{\mu}(x)\varphi^{}(x)\varphi(x),$ (19) where $q$ is the electric charge of the complex scalar field $\varphi$. First, we pay attention to the term with derivative. This term should be considered in momentum-space where $\varphi(x)$, $\varphi^{}(x)$ and $A_{\mu}(x)$ have respective momenta $p$, $p^{\prime}$ and $k$. If $\partial^{p}_{\mu}$ denotes that $\partial_{\mu}$ acts only on field having momentum $p$ then we can rewrite: $[\varphi^{}(x)\partial_{\mu}\varphi(x)-(\partial_{\mu}\varphi^{}(x))\varphi(x)]\equiv(\partial^{p}-\partial^{p^{\prime}})_{\mu}[\varphi^{}(x)\varphi(x)]$ (20) This definition helps us easily write down the vertex factor of derivative term in momentum space as $[ieq(p+p^{\prime})^{\mu}]$. $T$-product expansion of the Lagrangian into sum of $N$-products gives: $\displaystyle T\left\\{ieqA^{\mu}(x)[\partial^{p}-\partial^{p^{\prime}}]_{\mu}[\varphi^{}(x)\varphi(x)]+e^{2}q^{2}A_{\mu}(x)A^{\mu}(x)\varphi^{}(x)\varphi(x)\right\\}$ (21) $\displaystyle=$ $\displaystyle ieq(\partial^{p}-\partial^{p^{\prime}})^{\mu}A_{\mu}(x)N[\varphi^{}(x)\varphi(x)]+ieq(\partial^{p}-\partial^{p^{\prime}})^{\mu}A_{\mu}(x)\dot{\Delta}(x)$ $\displaystyle+$ $\displaystyle e^{2}g^{\mu\nu}q^{2}N[A_{\mu}(x)A_{\nu}(x)\varphi^{}(x)\varphi(x)]+(2e^{2}q^{2}g^{\mu\nu})\frac{1}{2}N[A_{\mu}(x)A_{\nu}(x)]\dot{\Delta}(x)$ $\displaystyle+$ $\displaystyle(2e^{2}q^{2}g^{\mu\nu})\frac{1}{2}\dot{\Delta}_{\mu\nu}(x)N[\varphi^{}(x)\varphi(x)]+(2e^{2}q^{2}g^{\mu\nu})\frac{1}{2}\dot{\Delta}_{\mu\nu}(x)\dot{\Delta}(x)$ $\displaystyle=$ $\displaystyle[ieq(\partial^{p}-\partial^{p^{\prime}})^{\mu}]a_{1}+[ieq(\partial^{p}-\partial^{p^{\prime}})^{\mu}]a_{2}+(2e^{2}q^{2}g^{\mu\nu})\frac{1}{2}a_{3}$ $\displaystyle+$ $\displaystyle(2e^{2}q^{2}g^{\mu\nu})\frac{1}{2}a_{4}+(2e^{2}q^{2}g^{\mu\nu})\frac{1}{2}a_{5}+(2e^{2}q^{2}g^{\mu\nu})\frac{1}{2}a_{6},$ where $\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle A_{\mu}(x)N[\varphi^{}(x))\varphi(x)]$ $\displaystyle a_{2}$ $\displaystyle=$ $\displaystyle A_{\nu}(x)\dot{\Delta}(x)$ $\displaystyle a_{3}$ $\displaystyle=$ $\displaystyle N[A_{\nu}(x)A_{\mu}(x)\varphi^{}(x)\varphi(x)],$ $\displaystyle a_{4}$ $\displaystyle=$ $\displaystyle N[A_{\nu}(x)A_{\mu}(x)]\dot{\Delta}(x),$ $\displaystyle a_{5}$ $\displaystyle=$ $\displaystyle\dot{\Delta}_{\mu\nu}(x)N[\varphi^{}(x)\varphi(x)],$ $\displaystyle a_{6}$ $\displaystyle=$ $\displaystyle\dot{\Delta}_{\mu\nu}(x)\dot{\Delta}(x).$ (22) As before, here we have denoted $\dot{\Delta}_{\mu\nu}(x)\equiv\mathop{\vbox{\halign{#\cr\kern 1.72218pt\cr$\hbox{$\hskip 3.75pt\vrule height=3.65973pt,width=0.29999pt,depth=-0.0004pt\kern-0.29999pt\vrule height=3.65973pt,width=24.36736pt,depth=-3.35974pt\kern-0.29999pt\vrule height=3.65973pt,width=0.29999pt,depth=-0.0004pt$\hss}$\crcr\kern 1.72218pt\nointerlineskip\cr\hbox{$\displaystyle{}{A}{{}_{\mu}(x)}{A}_{\nu}(x)$}\crcr}}}\limits$. Let us explain a reason for the factor $\frac{1}{2}$ associated with $a_{i},i=3,4,5,6$. As mentioned in section 2, we have to write: $[e^{2}q^{2}A_{\mu}(x)A^{\mu}(x)\varphi^{}(x)\varphi(x)]=[(2e^{2}g^{\mu\nu})]\times[\frac{1}{2}A_{\mu}(x)A_{\nu}(x)\varphi^{}(x)\varphi(x)]$ because $[(2e^{2}g^{\mu\nu})]$ is well-known vertex factor in literature so the factor $1/2$ (where 2 is derived by taking derivatives of $[A_{\mu}(x)A^{\mu}(x)\varphi^{}(x)\varphi(x)]$ with respect to all fields) is needed for our method. We note that the photon $A_{\mu}$ is real field. Factors $ieq(\partial^{p}-\partial^{p^{\prime}})^{\mu}$ and $(2e^{2}q^{2}g^{\mu\nu}$) are vertex factors, we again ignore them. The vertices in (22) are illustrated in Fig.5. Applying (14), each term of the total Green’s function of scalar QED is product of $[a_{1}^{p_{1}}a_{2}^{p_{2}}a_{3}^{p_{3}}a_{4}^{p_{4}}a_{5}^{p_{5}}a_{6}^{p_{6}}]$ and a factor $f_{1}$: $\displaystyle\frac{p!}{p_{1}!p_{2}!p_{3}!p_{4}!p_{5}!p_{6}!}a_{1}^{p_{1}}a_{2}^{p_{2}}\left[\frac{a_{3}}{2}\right]^{p_{3}}\left[\frac{a_{4}}{2}\right]^{p_{4}}\left[\frac{a_{5}}{2}\right]^{p_{5}}\left[\frac{a_{6}}{2}\right]^{p_{6}}=f_{1}a_{1}^{p_{1}}a_{2}^{p_{2}}a_{3}^{p_{3}}a_{4}^{p_{4}}a_{5}^{p_{5}}a_{6}^{p_{6}},$ $\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle\frac{p!}{2^{p_{3}+p_{4}+p_{5}+p_{6}}p_{1}!p_{2}!p_{3}!p_{4}!p_{5}!p_{6}!}.$ (23) Now, as an example, we investigate contractions of a vertex of type $a_{3}$ with other vertices. This vertex has two identical lines, so that in some case we can change roles of these two lines to create new terms. (-140,150)(-110,150)2(-110,150)(-80,150)2 (-110,120)(-110,150)24(-109,150)2(-110,100)[]$(a_{1})$(-30,120)(-30,150)24(-30,160)(10,270,90)2 (-30,160)(10,90,270)2(-29,150)2(-30,100)[]$(a_{2})$(20,120)(40,140)24(40,140)(60,120)24 (20,160)(40,140)2(40,140)(60,160)2 (40,140)2.3(40,100)[]$(a_{3})$(88,139)(148,139)28 (120,150)(10,90,270)2(120,150)(10,270,90)2 (120,140)2.3(125,100)[]$(a_{4})$(180,150)(10,90,270)2(180,150)(10,270,90)2 (181,129)(10,0,360)1.512 (180,140)2.3(180,100)[]$(a_{5})$(239,132)(10,0,360)112 (210,120)(240,120)2(240,120)(270,120)2 (240,120)2.3(240,100)[]$(a_{6})$ Figure 5: Vertices of scalar QED Each vertex of kind $a_{1}$ or $a_{6}$ has different fields, $a_{5}$ has no relation with other vertices. Vertices $a_{3}$ and $a_{4}$, each has two identical lines. All different contractions of $a_{1,2,3,4,5,6}$ create a new factor: $\displaystyle f_{2}=\frac{2^{p_{3}}2^{p_{4}}}{\prod_{n}(n!)^{\alpha_{n}}},$ (24) where $n$ and $\alpha_{n}$ were mentioned in (1). Next, similar to QED case, a factor caused from making contractions of vertices is given by: $\displaystyle\frac{p_{1}!p_{2}!p_{3}!p_{4}!p_{6}!}{g^{\prime}}=f_{3},$ (25) where $g^{\prime}$ is number of vertex permutations of types $a_{1,2,3,4}$ and $a_{6}$ creating identical diagrams. Then, the total factor is: $\displaystyle f=\frac{1}{S}$ $\displaystyle=$ $\displaystyle\frac{1}{p!}f_{1}f_{2}f_{3}$ (26) $\displaystyle=$ $\displaystyle\frac{1}{p!}\;\frac{p!}{2^{p_{3}+p_{4}+p_{5}+p_{6}}p_{1}!p_{2}!p_{3}!p_{4}!p_{5}!p_{6}!}\;\frac{2^{p_{3}}2^{p_{4}}}{\prod_{n}(n!)^{\alpha_{n}}}\;\frac{p_{1}!p_{2}!p_{3}!p_{4}!p_{6}!}{g^{\prime}}$ $\displaystyle=$ $\displaystyle\frac{1}{g2^{p_{5}+p_{6}}\prod_{n}(n!)^{\alpha_{n}}}\;,$ where $g=g^{\prime}p_{5}!$ now is different from $g^{\prime}$-permutation number of all vertices. We note that $g_{5}!$ related with $a_{5}$ now is included in $g$. It is easy to realize that the quantity $\beta$ that appears in Eq. (1) is given by $p_{5}+p_{6}$ for this case. From (26) we come to following conclusions: 1. 1. $a_{2}$ and $a_{4}$ do not contribute to $\beta$. Remember that this important property of complex scalar fields leads to the discrimination against real ones. Also, non self-conjugate bubble in $a_{5}$ does not create any new factors. $a_{5}$ in $\beta$ therefore comes from bubbles of $A_{\mu}$s. As a consequence, we conclude that $A_{\mu}$s play equivalent roles to real scalar fields. 2. 2. Although this theory contains vertex $a_{5}$ with two different bubbles, the factor $2^{d}$ does not appear. Clearly, $d$ is only the number of vertices with double identical bubbles. This new result is very important for theories with many different fields such as [$\phi^{2}\varphi^{2}$]. . Our formula can be verified by results of Ref.[3]. ## 5 Symmetry factors in QCD The Lagrangian in the QCD is given by $\displaystyle\mathcal{L}^{QCD}=\sum_{i=1}^{3}\overline{\psi}(iD_{\mu}\gamma^{\mu}-m)\psi-\frac{1}{4}F^{a}_{\mu\nu}F^{a\mu\nu},$ (27) where $D_{\mu}=\partial_{\mu}-ig_{S}t_{a}A^{a}_{\mu}$, $t_{a}$s are representation matrices of $SU(3)_{C}$, and $t_{a}=\frac{\lambda_{a}}{2}$ for the basic representation, $F^{a}_{\mu\nu}=\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A^{a}_{\mu}+g_{S}f^{abc}A^{b}_{\mu}A^{c}_{\nu}$. The indices $a,b,c=1,2,...,8$, $\psi$ has three color components: $\psi=(\psi^{R},\ \psi^{G},\ \psi^{B})^{T}$, $A^{a}_{\mu}$s are gauge gluon fields. Expanding the above Lagrangian, we have an interaction lagrangian: $\mathcal{L}_{int}^{QCD}=g_{S}\overline{\psi}\gamma^{\mu}t^{a}\psi A^{a}_{\mu}-g_{S}f^{abc}(\partial_{\mu}A^{a}_{\nu})A^{\mu b}A^{\nu c}-\frac{1}{4}g^{2}_{S}(f^{eab}A^{a}_{\mu}A^{b}_{\nu})(f^{ecd}A^{\mu c}A^{\nu d})$ (28) We emphasize that QCD is different from QED, gluon gauge fields of QCD $A^{a}_{\mu}$s are labeled by a color quantum number $a$ and belong to adjoint representation of $SU(3)_{C}$. But all of $A^{a}_{\mu}$s are real fields, or self-conjugate fields. Since quarks carry colors so gluons must carry them too and physical gauge fields are combinations of $A^{a}_{\mu}$s. However, due to the assumption that all observed particles only are color singlet, we can work with just $A^{a}_{\mu}$s [17]. It is easy to see that the first interacting term identifies with T-product. Hence, we can write this $T$-product in terms of sum of $N$-products[17]: $\displaystyle T\\{\bar{\psi}\gamma^{\mu}t^{a}A^{a}_{\mu}\psi\\}$ $\displaystyle=$ $\displaystyle N[\bar{\psi}\gamma^{\mu}t^{a}A^{a}_{\mu}\psi]+N[\mathop{\vbox{\halign{#\cr\kern 1.72218pt\cr$\hbox{$\hskip 2.5pt\vrule height=7.31946pt,width=0.29999pt,depth=1.15515pt\kern-0.29999pt\vrule height=7.31946pt,width=38.896pt,depth=-7.01947pt\kern-0.29999pt\vrule height=7.31946pt,width=0.29999pt,depth=-0.11153pt$\hss}$\crcr\kern 1.72218pt\nointerlineskip\cr\hbox{$\displaystyle{}{\overline{\psi}}{{}_{(x)}\gamma^{\mu}t^{a}A^{a}_{\mu}}{\psi}_{(x)}$}\crcr}}}\limits]$ (31) $\displaystyle=$ $\displaystyle[\bar{\psi}\gamma^{\mu}t^{a}A^{a}_{\mu}\psi]+iS^{i,\alpha}_{k,\beta}(x)(\gamma^{\mu})^{\beta}_{\alpha}(t^{a})^{k}_{i}A^{a}_{\mu},$ (32) where $\alpha,\beta$ are Dirac indices, and $i,k$ are $SU(3)_{C}$ ones. For the third term of (28), firstly we rewrite it in new form [17]: $\displaystyle\frac{1}{4}(f^{eab}A^{a}_{\mu}A^{b}_{\nu})(f^{ecd}A^{\mu c}A^{\nu d})$ $\displaystyle=$ $\displaystyle\left[f^{eab}f^{ecd}(g^{\mu\alpha}g^{\nu\beta}-g^{\mu\beta}g^{\nu\alpha})\right.$ (33) $\displaystyle+$ $\displaystyle\left.f^{eac}f^{ebd}(g^{\mu\nu}g^{\alpha\beta}-g^{\alpha\nu}g^{\mu\beta})\right.$ $\displaystyle+$ $\displaystyle\left.f^{ead}f^{ecb}(g^{\mu\alpha}g^{\beta\nu}-g^{\beta\alpha}g^{\mu\nu})\right]\times\left[\frac{1}{4!}A^{a}_{\mu}A^{b}_{\nu}A^{c}_{\alpha}A^{d}_{\beta}\right]$ Next we choose T-product of this term as $\frac{1}{4!}T\left[A^{a}_{\mu}A^{b}_{\nu}A^{c}_{\alpha}A^{d}_{\beta}\right]$. The rest is vertex factor. Then we will get: $\displaystyle\frac{1}{4!}T(A^{a}_{\mu}A^{b}_{\nu}A^{c}_{\alpha}A^{d}_{\beta})$ $\displaystyle=$ $\displaystyle\frac{1}{4!}N[A^{a}_{\mu}A^{b}_{\nu}A^{c}_{\alpha}A^{d}_{\beta}]+\frac{6}{4!}\mathop{\vbox{\halign{#\cr\kern 1.72218pt\cr$\hbox{$\hskip 3.75pt\vrule height=3.65973pt,width=0.29999pt,depth=-0.0004pt\kern-0.29999pt\vrule height=3.65973pt,width=10.46011pt,depth=-3.35974pt\kern-0.29999pt\vrule height=3.65973pt,width=0.29999pt,depth=-0.0004pt$\hss}$\crcr\kern 1.72218pt\nointerlineskip\cr\hbox{$\displaystyle{}{A}{{}^{a}_{\mu}}{A}^{b}_{\nu}$}\crcr}}}\limits N[A^{c}_{\alpha}A^{d}_{\beta}]$ (36) $\displaystyle+$ $\displaystyle\frac{3}{4!}\mathop{\vbox{\halign{#\cr\kern 1.72218pt\cr$\hbox{$\hskip 3.75pt\vrule height=3.65973pt,width=0.29999pt,depth=-0.0004pt\kern-0.29999pt\vrule height=3.65973pt,width=10.46011pt,depth=-3.35974pt\kern-0.29999pt\vrule height=3.65973pt,width=0.29999pt,depth=-0.0004pt$\hss}\hbox{$\hskip 24.11346pt\vrule height=7.31946pt,width=0.29999pt,depth=-0.0004pt\kern-0.29999pt\vrule height=7.31946pt,width=9.92345pt,depth=-7.01947pt\kern-0.29999pt\vrule height=7.31946pt,width=0.29999pt,depth=-0.0004pt$\hss}$\crcr\kern 1.72218pt\nointerlineskip\cr\hbox{$\displaystyle{}{A}{{}^{a}_{\mu}}{A}{{}^{b}_{\nu}}{A}{{}^{c}_{\alpha}}{A}^{d}_{\beta}$}\crcr}}}\limits.$ (39) We see that (28) is a part of (27) which brings out S-factors. This part is almost identical with the expansion of real scalar theory (LABEL:ttichreal4), except indices $a$ and $\mu$ of gluon fields. However these indices are quiet. Four field components $A^{a}_{\mu},A^{b}_{\nu},A^{c}_{\alpha}$ and $A^{d}_{\beta}$ are the same after doing contractions to form internal lines without directions. Hence, we can consider them as four identical real scalar fields. Thus, the S-factor formula of this case is also given by the formula (1). The second term in (28) contains derivatives, so it is easier to work in momentum-space. In this space, if we denote momentum of $A^{a}_{\nu},A^{b}_{\mu}$ and $A^{c}_{\sigma}$ correspond to $p,k$ and $q$ then we can write $\partial^{\alpha}A^{a}_{\nu}\equiv(\partial^{\alpha}_{p})A^{a}_{\nu}$, etc…, namely: $f^{abc}(\partial^{\mu}A^{a}_{\nu})A^{b}_{\mu}A^{\nu c}\equiv f^{abc}(\partial^{\mu}_{p})(A^{a}_{\nu}A^{b}_{\mu b}A^{c}_{\sigma})g^{\sigma\nu},$ (40) in sense that $\partial^{\mu}_{p}$ does not operate on any fields except those having momentum $p$. Now (40) can be rewritten in the form [17]: $f^{abc}(\partial^{\mu}A^{a}_{\nu})A^{b}_{\mu}A^{\nu c}=f^{abc}[g^{\mu\nu}(\partial_{p}-\partial_{k})^{\sigma}+g^{\mu\sigma}(\partial_{k}-\partial_{q})^{\nu}+g^{\sigma\nu}(\partial_{q}-\partial_{p})^{\mu}]\times\left[\frac{1}{6}A^{a}_{\nu}A^{b}_{\mu}A^{c}_{\sigma}\right]$ (41) Again, the first factor in (41) is the three-gluon interaction vertex factor in momentum-space and the second, the T-product, is the same as interacting term of $\varphi^{3}$ real scalar theory. In conclusion for calculating S-factor of QCD, all fermion fields can be considered as scalar complex fields, and gluon fields play the roles of real scalar fields. Then, the formula (1) is applicable to the QCD. Some examples are given in Fig.6. These results also agree with those given in Ref. [10] and [16]. (-140,120)(-112,120)34(-70,120)(-30,120)34 (-90,120)(20,0,360)320(-112,120)2.3(-68,121)2.5 (-80,80)[]$a.\;\alpha_{2}=1,g=1$(-80,70)[]$S=2!=2$(0,120)(30,120)34(50,120)(20,0,360)320 (70,120)(110,120)34(30,120)(70,120)34 (28,122)2.3(72,121)2.3 (50,80)[]$b.\;\alpha_{3}=1,g=1$(50,70)[]$S=3!=6$ Figure 6: Examples of S-factors in QCD ## 6 An example of Standard Model Now we turn to the Standard Model. Let us consider a particular coupling between $W$ with charged currents: $\displaystyle\mathcal{L}^{CC}_{f}$ $\displaystyle=$ $\displaystyle\frac{g}{\sqrt{2}}\left[W^{+}_{\mu}J^{+\mu}+W^{-}_{\mu}J^{-\mu}\right]$ (42) $\displaystyle=$ $\displaystyle\sum_{i=1}^{3}\frac{g}{2\sqrt{2}}\left\\{W^{+\mu}\left[\bar{\nu}_{i}\gamma_{\mu}(1-\gamma_{5})e_{i}+\bar{u_{i}}\gamma_{\mu}(1-\gamma_{5})d_{i}\right]\right.$ $\displaystyle+$ $\displaystyle\left.W^{-\mu}\left[\bar{e_{i}}\gamma_{\mu}(1-\gamma_{5})\nu_{i}+\bar{d_{i}}\gamma_{\mu}(1-\gamma_{5})u_{i}\right]\right\\}$ This Lagrangian has twelve terms in the same form as $g/(2\sqrt{2})\bar{\psi}\gamma_{\mu}(1-\gamma_{5})\psi^{\prime}W^{\mu}$ in which all terms have the same vertex factor $[g/(2\sqrt{2})\gamma_{\mu}(1-\gamma_{5})]$. By our choice, T-products have form $\bar{\psi}\psi^{\prime}W^{\mu}$. It is very simple to calculate because $T$-product is equal to $N$-product. In similarity with the case of QED, we easily prove that $W$ field behaves the same way as a complex scalar field. Our analysis leads to a general principle: interactions such as given in (42), Yukawa couplings, etc, are similar to interactions in the spinor QED. Hence we conclude that: S-factors of all _external connected_ (sub-)diagrams containing only vertices with three different fields, are equal to unity. Illustrations can be found in appendix B. We must remember that $W$ boson is complex scalar- like, in S-factor calculation, although in diagrams we do not draw its propagator direction. It is emphasized that Majorana neutrinos belong to real scalar-like. For more details, interested readers can find in Refs.[18, 19]. ## 7 The vacuum diagrams factorization Every Feynman diagram consists of two kinds of well-known connected pieces, namely external connected and vacuum connected subdiagrams. One diagram may include many identical vacuum connected pieces. Conversely, all external connected pieces are different from each others because they connect to different external legs. Each piece has its private S-factor which is independent on the others. It is interesting that the S-factor of a total diagram can be presented as a product of private S-factors of connected pieces, that is well known vacuum diagrams factorization. These private S-factors can be clearly evaluated from our analysis. If there are $i$ different kinds of connected piece (easily classified by geometric properties) then we can label an index $i$ for any thing related with a piece of the $i$th kind, such as $g_{i},\beta_{i},d_{i},n_{i},\alpha_{n_{i}}$ without losing original meanings of $g,\beta,d,\alpha$. Then, each connected piece of kind $ith$ contributes a factor $S_{i}$ to the total S-factor: $S_{i}=g_{i}2^{\beta_{i}}2^{d_{i}}\prod_{n_{i}}(n_{i}!)^{\alpha_{n_{i}}},$ (43) which is the same as (1) except an extra index $i$. Each set consisting of all $k_{i}$ indistinguishable pieces (pieces in kind $ith$) causes a factor $[k_{i}!(S_{i})^{k_{i}}]$. The total S-factor now is presented as a new expression: $\displaystyle S=\prod_{i}[k_{i}!(S_{i})^{k_{i}}]=\prod_{i}[k_{i}!(g_{i})^{i}]\times 2^{\sum_{i}k_{i}\beta_{i}}\times 2^{\sum_{i}k_{i}d_{i}}\times\prod_{i}\prod_{n_{i}}(n_{i}!)^{\alpha_{n_{i}}}$ (44) Comparing with (1) we have $\beta=\sum_{i}k_{i}\beta_{i}$, $d=\sum_{i}k_{i}d_{i}$ and the factor $\prod_{i}\prod_{n_{i}}(n_{i}!)^{\alpha_{n_{i}}}$ can be replaced by $\prod_{n}(n!)^{\alpha_{n}}$, where $\alpha_{n}=\sum_{i}k_{i}\alpha_{n_{i}}$ because $(n,n_{1},n_{2},...=1,2,3,...)$ are running indices. Especially, the relation between $g$ and $g_{i}$s: $g=\prod_{i}[k_{i}!(g_{i})^{k_{i}}],$ (45) can help us practically calculate $g$ from $g_{i}$s. The most convenient property of $g_{i}$ is that it is equal to the number of graphical symmetry transformations of a connected piece in the $i$th kind. In practice, $\alpha_{n},d$ and $\beta$ can directly be deduced from the particular graphical properties of diagram itself. Taking into account of (45), we determine $g$ from $g_{i}$. To see the vacuum factorization, from (44), we group all factors related with connected vacuum pieces (private S-factors, $S_{i}s$, and factors $k_{i}!$-rising from $k_{i}$ identical pieces) in a single factor called vacuum symmetry factor $S_{v}$ and the remaining-external connected ones in another factor $S_{c}$ , then the S-factor of the diagram is divided into two factors $S=S_{v}\times S_{c}$. If we sum all of total diagrams in all orders with their S-factors we will receive results mentioned in Refs.[9, 10]. (-152,118)(-50,120)1.512 (-130,118)(-100,150)1.54 (-100,150)(-70,120)1.54 (-99,150)3 (-130,120)3(-70,120)3 (-115,135)(20,45,225)1.57 (-85,135)(20,-45,135)1.57 (-100,90)[](a) $\;g=1;\alpha_{2}=2$(-100,80)[]$S=4$(-40,130)[]$\longrightarrow$(-30,120)(90,120)1.512 (0,120)(30,150)1.54 (30,150)(60,120)1.54 (15,135)(20,45,225)1.57 (45,135)(20,-45,135)1.57 (31,150)3 (0,120)3(60,120)3 (62,143)(62,148) (14,133)(15,138)(-5,143)(-3,138) (41,140)(43,135) (-18,121)(-13,119)(40,90)[](b) $\;S=1$(32,121)(37,119)(72,121)(77,119) (130,120)(250,120)212 (160,120)(190,150)24 (190,150)(220,120)24 (175,135)(20,45,225)27 (205,135)(20,-45,135)27 (191,150)3 (160,120)3(220,120)3 (222,148)(221,143)(201,141)(201,136) (158,138)(156,143)(175,132)(175,137) (142,122)(147,119)(187,119)(182,122) (232,122)(237,119) (190,90)[](c) $\;g=1;\alpha_{2}=2$(190,80)[]$S=4$ Figure 7: S-factors for diagrams with different propagator directions One more remark we point out here: normally, when drawing a diagram, we just pay attention to directions of momenta while omitting directions of propagators (for example, $W$ boson). This makes us confused in counting $\beta$ and we may lose some diagrams because there are diagrams differing _only in directions_ of propagators. For example, with self-interacting term of $W$ boson we have a diagram without directions of propagators in figure 7.a which has the same S-factor as the one in figure 7.c -the figure including charged transition directions of $W$ (not direction of momentum). But in case of this directional field, there is another different diagram in figure 7.b which does distinguish from the one in figure 7.c by only their directions of lines. Both of them have same contributions but _different_ S-factor values. The S-factor now is related with not only figure 7.a or 7.c but also with both of 7.b and 7.c [14]. For simplicity, we can define a diagram without directions in lines, and call it equivalent diagram. This kind of diagram stands for all diagrams which have the same geometrical shape and contribution but are different in directions of lines. Then the S-factor of a equivalent diagram is different from usual: it is S-factor for the total contribution and the inverse of this factor is the sum of inverse ones of directional diagrams. To determine S-factor of an equivalent diagram (for example, see, Fig.7a), we have to find out S-factors of all possible directional diagrams coming from this non-directional diagram, and denote S-factors $S_{1},S_{2},...$, respectively. Then the S-factor of the equivalent diagram is obtained as follows: $\displaystyle\frac{1}{S}=\sum_{n}\frac{1}{S_{n}}.$ (46) One of our new results that has never been mentioned before: _All well-known formulas for S-factors, including our formula in this paper, only work on directional diagrams where all directions of complex scalar-like fields are pointed out_. For example, in (46) our calculation is only used for $S_{n}$, not for $S$. From now on, our formula implies S-factor of $S_{n}$. ## 8 Conclusion Based on cases illustrated above, we conclude that our calculation does not depend on the spins of fields. It only depends on whether fields presenting a particle and its anti-particle are identical or not. In other words, the class of fields is very important in our calculation of S-factors. We have two classes of field, real scalar-like and complex scalar-like, as mentioned in the second section. For practical calculation, in Table 1 we list some known fields. Table 1: Classification of fields Real scalar-like \| Complex scalar-like ---\|--- Real scalar \| complex scalar Photon $A_{\mu}$ \| spinor Dirac field $Z$ boson \| W boson Gluon \| Ghost Majarona fields \| Now, as our main result, we introduce a general formula of symmetry factor for Feynman diagrams of theories containing many different fields with any spin values. Although it has the same form as the formula for scalar theories (1) $\displaystyle S=g2^{\beta}2^{d}\prod_{n}(n!)^{\alpha_{n}},$ (47) definitions of $\alpha_{n}$, $d$ and $\beta$ are generalized. They are redefined as: • $\alpha_{n}$ is the number of sets of $n$ identical lines connecting the same pairs of vertices(there may be more than such one sets in one vertex pair). * • $d$ is the number of vertices with two identical bubbles. * • $\beta$ is sum of all self-conjugate bubbles coming from self-conjugate fields ($\beta$ vanishes if all fields in the theory belong to non-conjugate fields). * • $g$ is the number of vertex permutations keeping the diagram topologically unchanged. We must emphasize that the most important goal of our work is to find out the general definitions of these parameters in common case. They are more general than [9, 14] and others. The most important thing: formula (47) is applicable to diagrams where all directions of propagators are showed (although they may not be drawn in diagrams). We remind one interesting property of our result: The diagrams with different topologies can contribute the same, and the inverse symmetry factor for the total contribution is therefore the sum of the inverse symmetry ones, i.e., $1/S=\sum_{i}(1/S_{i})$. We have showed that the S-factors of all _external connected_ diagrams containing vertices with three _different_ fields such as interactions in spinor QED, Yukawa couplings, etc, _are equal to unity_ ($S=1$). This conclusion is also correct for all diagrams consisting of only vertices with different legs. We recall that determining the symmetry factor is important because it not only is an important component of modern quantum field theory, but also is used to calculate effective potentials in higher-dimensional theories and cosmological models. 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Kinoshita, M. Nio, Phys. Rev. D 82, 113004 (2010). ## Appendix A Examples of Feynman diagrams in QED up to fourth order. (-150,120)(-110,120)(-110,120)(-70,120) (-110,120)(-110,135)23 (-110,145)(10,90,-90)(-110,145)(10,-90,90) (-100,100)[]$(a1)\ S=g=1$(20,120)(100,120) (60,120)(20,0,180)212 (50,100)[]$(b1)\ S=g=1$(150,120)(230,120) (190,150)(10,0,180) (190,150)(10,-180,0) (180,150)(200,150)24 (195,100)[]$(c1)\ S=g=2$ (-130,140)(-110,140)24 (-90,140)(-70,140)24 (-100,140)(10,0,180) (-100,140)(10,180,360) (-100,100)[]$(a2)\ S=g=1$(-30,120)(50,120) (-10,150)(10,0,180) (-10,150)(10,180,360) (30,150)(10,0,180)(30,150)(10,180,360) (0,150)(20,150)24 (0,100)[]$(b2)\ S=g=2$(80,120)(160,120)212 (100,150)(10,0,180) (100,150)(10,180,360) (140,150)(10,0,180) (140,150)(10,180,360) (110,150)(130,150)24 (125,100)[]$(c2)\ S=g=2$(180,135)(200,135)24 (210,135)(10,0,180) (210,135)(10,180,360) (235,135)(10,0,180) (235,135)(10,180,360)(245,135)(265,135)24 (230,100)[]$(d2)\ S=g=1$ (-130,120)(-100,120) (-100,120)(-70,120) (-100,120)(-100,130)22 (-100,130)(-120,150) (-80,150)(-100,130) (-120,150)(-80,150) (-100,150)(20,0,180)212 (-100,100)[]$(a3)\ S=g=1$(15,120)(35,120) (35,120)(65,120) (65,120)(80,120) (80,120)(110,120) (110,120)(130,120) (50,120)(15,0,180)28 (95,120)(15,0,180)28 (60,100)[]$(b3)\ S=g=1$(150,120)(175,120) (175,120)(205,120) (205,120)(230,120) (190,160)(10,0,180) (190,160)(10,180,360) (180,160)(200,160)24 (190,120)(15,0,180)28 (195,100)[]$(c3)\ S=g=1/2$ (-150,120)(-120,120) (-120,120)(-100,120) (-100,120)(-70,120) (-100,120)(-100,150)26 (-120,120)(-120,150)26 (-110,150)(10,0,180) (-110,150)(10,180,360) (-100,100)[]$(a4)\ S=g=1$(20,120)(100,120) (50,140)(50,170) (50,170)(70,170) (70,170)(70,140) (70,140)(50,140) (50,155)(15,90,270)28 (70,155)(15,270,90)28 (50,100)[]$(b4)\ S=g=2$(150,120)(230,120) (175,160)(10,0,180) (175,160)(10,180,360) (165,160)(185,160)24 (205,160)(10,0,180) (205,160)(10,180,360) (195,160)(215,160)24 (185,100)[]$(c4)\ g_{1}=2;\;S=2!(g_{1})^{2}=8$ (-150,120)(-130,120)24 (-90,120)(-70,120)24 (-130,120)(-120,120) (-100,120)(-90,120) (-120,120)(-100,120) (-110,120)(20,0,180) (-110,120)(10,180,0)26 (-100,100)[]$(a5)\ S=g=1$(20,120)(40,120)24 (80,120)(100,120)24 (40,120)(60,120) (60,120)(80,120) (40,120)(60,160) (60,160)(80,120) (60,120)(60,160)28 (50,100)[]$(b5)\ S=g=1$(150,140)(165,140)23 (215,140)(230,140)23(185,140)(195,140)22 (175,140)(10,0,180) (175,140)(10,180,360) (205,140)(10,0,180) (205,140)(10,180,360) (195,100)[]$(c5)\ S=g=1$ (-150,120)(-70,120)212 (-125,140)(10,0,180) (-125,140)(10,180,360) (-135,140)(-115,140)24 (-95,140)(10,0,180) (-95,140)(10,180,360) (-105,140)(-85,140)24 (-125,100)[]$(a.6)\;g_{1}=2,k_{1}=2$(-125,80)[]$\ S=g=2!(2)^{2}=8$(20,120)(100,120) (45,150)(10,0,180) (45,150)(10,180,360) (35,150)(55,150)24 (45,120)(45,140)23 (75,150)(10,0,180) (75,150)(10,180,360) (65,150)(85,150)24 (50,100)[]$(b.6)\ S=g=2$(150,120)(190,120) (190,120)(230,120) (190,140)(10,90,-90) (190,140)(10,-90,90) (190,120)(190,130)22 (190,150)(190,160)22 (190,170)(10,90,-90) (190,170)(10,-90,90) (195,100)[]$(c.6)\ S=g=1$ (-150,120)(-125,120) (-125,120)(-110,120) (-110,120)(-90,120) (-90,120)(-70,120) (-125,150)(10,90,-90) (-125,150)(10,-90,90) (-125,120)(-125,140)24 (-100,120)(10,0,180)26 (-125,100)[]$(a.7)\ S=g=1$(20,140)(40,140) (40,140)(60,140) (60,140)(80,140) (80,140)(100,140) (60,160)(10,90,-90) (60,160)(10,-90,90) (60,140)(60,150)22 (60,140)(20,180,0)210 (50,100)[]$(b.7)\ S=g=1$(155,140)(175,140)24 (205,140)(225,140)24 (190,140)(15,-180,0) (190,140)(15,90,180) (190,140)(15,0,90) (190,155)(190,165)22 (190,175)(10,90,-90) (190,175)(10,-90,90) (195,100)[]$(c.7)\ S=g=1$ (-150,120)(-70,120)212 (-110,140)(10,0,90) (-110,140)(10,90,180) (-110,140)(10,-180,0) (-120,140)(-100,140)24 (-110,170)(10,90,-90) (-110,170)(10,-90,90) (-110,150)(-110,160)22 (-135,100)[]$(a.8)\ S=g=1$(20,120)(45,120) (45,120)(75,120) (75,120)(100,120) (40,150)(10,0,180) (40,150)(10,-180,0) (50,150)(70,150)23 (80,150)(10,0,180) (80,150)(10,-180,0) (60,120)(15,0,180)28 (50,100)[]$(b.8)\ S=g=2$(150,120)(175,120) (175,120)(205,120) (205,120)(230,120) (175,120)(175,140)24 (205,120)(205,140)24 (175,150)(10,-90,90) (175,150)(10,90,-90) (205,150)(10,90,-90) (205,150)(10,90,-90) (205,150)(10,-90,90) (195,100)[]$(c.8)\ S=g=1$ (-150,120)(-70,120) (-110,150)(10,0,180) (-110,150)(10,-180,0) (-140,150)(10,-180,0) (-140,150)(10,0,180) (-80,150)(10,0,180) (-80,150)(10,-180,0) (-130,150)(-120,150)22 (-100,150)(-90,150)22 (-135,100)[]$(a.9)\ S=g=2$(20,120)(100,120) (30,150)(10,0,180) (30,150)(10,-180,0) (60,150)(10,0,180) (60,150)(10,-180,0) (90,150)(10,0,180) (90,150)(10,-180,0) (80,150)(100,150)24 (40,150)(50,150)22 (50,100)[]$(b.9)\;g_{1}=g_{2}=2;k_{1}=k_{2}=1$(50,80)[]$S=g=g_{1}g_{2}=4$(150,140)(180,140)26 (200,140)(230,140)26 (190,140)(10,0,180) (190,140)(10,-180,0) (170,170)(10,0,180) (170,170)(10,-180,0) (210,170)(10,0,180) (210,170)(10,-180,0) (180,170)(200,170)24 (195,100)[]$(c.9)\ S=g=2$ (-150,140)(-135,140)23 (-85,140)(-70,140)23 (-125,140)(10,0,180) (-125,140)(10,-180,0) (-95,140)(10,-180,0) (-95,140)(10,0,180) (-110,160)(10,0,180) (-110,160)(10,-180,0) (-120,160)(-100,160)24 (-135,100)[]$(a.10)\ S=g=2$(20,150)(30,150)22 (40,150)(10,0,180) (40,150)(10,-180,0) (70,150)(10,0,180) (70,150)(10,-180,0) (100,150)(10,0,180) (100,150)(10,-180,0) (110,150)(120,150)22 (50,150)(60,150)22 (50,100)[]$(b.10)\ S=g=1$(150,120)(230,120)212 (150,140)(170,140)23 (160,140)(10,0,180) (160,140)(10,-180,0) (185,140)(10,0,180) (185,140)(10,-180,0) (220,140)(10,0,180) (220,140)(10,-180,0) (195,140)(210,140)22 (195,100)[]$(c.10)\ S=g=4$ (-150,120)(-70,120)212 (-110,150)(10,0,180) (-110,150)(10,-180,0) (-140,150)(10,-180,0) (-140,150)(10,0,180) (-80,150)(10,0,180) (-80,150)(10,-180,0) (-130,150)(-120,150)22 (-100,150)(-90,150)22 (-135,100)[]$(a.11)\ S=g=2$(20,120)(35,120) (35,120)(100,120) (35,120)(35,140)24 (35,150)(10,90,-90) (35,150)(10,-90,90) (60,150)(10,0,180) (60,150)(10,-180,0) (90,150)(10,0,180) (90,150)(10,-180,0) (70,150)(80,150)22 (50,100)[]$(b.11)\ S=g=2$(150,120)(245,120) (170,140)(180,140)22 (160,140)(10,0,180) (160,140)(10,-180,0) (190,140)(10,0,180) (190,140)(10,-180,0) (215,140)(10,0,180) (215,140)(10,-180,0) (245,140)(10,0,180) (245,140)(10,-180,0) (225,140)(235,140)22 (185,100)[]$(c.11)\ g_{1}=2,k_{1}=2$(185,80)[]$S=g=2!2^{2}=8$ (-150,120)(-130,120) (-90,120)(-70,120) (-130,120)(-120,120) (-120,120)(-100,120) (-100,120)(-90,120) (-110,120)(10,0,180)26 (-110,120)(20,0,180)210 (-130,100)[]$(a.12)\ S=g=1$(15,150)(35,150) (85,150)(105,150) (35,150)(45,150) (45,150)(75,150) (75,150)(85,150) (85,150)(100,150) (65,150)(20,0,180)210 (55,150)(20,-180,0)210 (50,100)[]$(b.12)\ S=g=1$(150,120)(170,120) (170,120)(190,120) (190,120)(210,120) (210,120)(230,120) (180,120)(10,0,180)26 (210,145)(10,-90,90) (210,145)(10,90,-90) (210,120)(210,135)23 (185,100)[]$(c.12)\ S=g=1$ (-150,140)(-130,140)24 (-120,140)(10,0,180) (-120,140)(10,-180,0) (-85,140)(-65,140)24 (-95,140)(10,-180,0) (-95,140)(10,0,180) (-130,170)(10,0,180) (-130,170)(10,-180,0) (-90,170)(10,0,180) (-90,170)(10,-180,0) (-120,170)(-100,170)24 (-135,100)[]$(a.13)\ S=g=2$(-30,120)(55,120)213 (-20,150)(10,0,180) (-20,150)(10,-180,0) (10,150)(10,0,180) (10,150)(10,-180,0) (-10,150)(0,150)22 (35,150)(10,0,180) (35,150)(10,-180,0) (65,150)(10,0,180) (65,150)(10,-180,0) (45,150)(55,150)22 (0,100)[]$(b.13)\ S=g=8$(100,120)(180,120) (140,135)(10,-90,90) (140,135)(10,90,-90) (140,145)(140,155)22 (140,155)(125,175) (125,175)(155,175) (155,175)(140,155) (140,175)(15,0,180)28 (135,100)[]$(c.13)\ S=g=1$(200,150)(215,150)22(215,150)(235,165) (235,165)(235,135) (235,135)(215,150) (235,150)(15,-90,90)28 (275,150)(290,150)22 (265,150)(10,0,180) (265,150)(10,-180,0) (240,100)[]$(d.13)\ S=g=1$ ## Appendix B Examples of Feynman diagrams in SM up to tenth order: $\mu^{-}\rightarrow\nu_{\mu}+e^{-}+\mathaccent 869{\nu_{e}}$. This case we must remember that all W-boson lines are directional, thought we don’t draw. (-150,120)(-130,120) (-130,120)(-110,120) (-110,120)(-90,120) (-90,120)(-50,120) (-90,120)(-90,145)24 (-90,145)(-50,145) (-50,185)(-90,145) (-120,120)(10,0,180)25 (-150,110)[]$\mu$(-40,110)[]$\nu_{\mu}$(-50,190)[]$\mathaccent 869{\nu_{e}}$(-40,145)[]$e$(-100,100)[]$(a.14)\ S=1$(0,120)(40,120) (40,120)(60,120) (60,120)(80,120) (80,120)(100,120) (40,120)(40,145)24 (40,145)(100,145) (100,185)(40,145) (70,120)(10,0,180)25 (0,110)[]$\mu$(110,110)[]$\nu_{\mu}$(90,190)[]$\mathaccent 869{\nu_{e}}$(110,145)[]$e$(50,100)[]$(b.14)\ S=1$(140,120)(190,120) (190,120)(240,120) (190,140)(10,-90,90) (190,140)(10,90,-90) (190,120)(190,130)22 (190,150)(190,160)22 (190,160)(240,160) (230,185)(190,160) (140,110)[]$\mu$(210,140)[]$\mu$(250,110)[]$\nu_{\mu}$(240,190)[]$\mathaccent 869{\nu_{e}}$(250,160)[]$e$(200,100)[]$(c.14)\ S=1$ (-150,120)(-130,120) (-130,120)(-70,120) (-130,140)(-110,140) (-110,140)(-90,140) (-130,120)(-130,140)23 (-90,140)(-70,140) (-70,155)(-130,140) (-100,140)(10,-180,0)26 (-150,110)[]$\mu$(-60,110)[]$\nu_{\mu}$(-60,160)[]$\mathaccent 869{\nu_{e}}$(-60,140)[]$e$(-100,100)[]$(a.15)\ S=1$(0,120)(30,120) (30,120)(70,120) (30,120)(30,140)23 (30,140)(70,140) (70,180)(30,140) (50,160)(10,-135,45)25 (0,110)[]$\mu$(80,110)[]$\nu_{\mu}$(80,170)[]$\mathaccent 869{\nu_{e}}$(80,140)[]$e$(40,100)[]$(b.15)\ S=1$(140,120)(190,120) (190,120)(240,120) (190,140)(10,-90,90) (190,140)(10,90,-90) (190,120)(190,130)22 (190,150)(190,160)22 (190,160)(240,160) (230,185)(190,160) (140,110)[]$\mu$(250,110)[]$\nu_{\mu}$(240,190)[]$\mathaccent 869{\nu_{e}}$(210,140)[]$e$(250,160)[]$e$(200,100)[]$(c.15)\ S=1$ (-150,120)(-135,120) (-135,120)(-115,120) (-115,120)(-100,120) (-100,120)(-85,120) (-85,120)(-65,120) (-65,120)(-50,120) (-100,120)(-100,145)24 (-100,145)(-50,145) (-50,185)(-100,145) (-125,120)(10,0,180)26 (-75,120)(10,0,180)26 (-150,110)[]$\mu$(-40,110)[]$\nu_{\mu}$(-50,190)[]$\mathaccent 869{\nu_{e}}$(-40,145)[]$e$(-100,100)[]$(a.16)\ S=1$(0,120)(40,120) (40,120)(60,120) (60,120)(80,120) (80,120)(100,120) (40,120)(40,130)22 (40,150)(40,160)22 (40,140)(10,-90,90) (40,140)(10,90,-90) (40,160)(100,160) (100,185)(40,160) (70,120)(10,0,180)25 (0,110)[]$\mu$(110,110)[]$\nu_{\mu}$(90,190)[]$\mathaccent 869{\nu_{e}}$(110,160)[]$e$(50,100)[]$(b.16)\ S=1$(57,140)[]$e$(130,120)(150,120) (150,120)(170,120) (160,120)(10,0,180)25 (170,120)(190,120) (190,120)(240,120) (190,140)(10,-90,90) (190,140)(10,90,-90) (190,120)(190,130)22 (190,150)(190,160)22 (190,160)(240,160) (230,185)(190,160) (140,110)[]$\mu$(210,140)[]$\mu$(240,110)[]$\nu_{\mu}$(240,190)[]$\mathaccent 869{\nu_{e}}$(250,160)[]$e$(200,100)[]$(c.16)\ S=1$ (-150,120)(-70,120) (-70,120)(-50,120) (-125,140)(10,0,180) (-125,140)(10,-180,0) (-135,140)(-115,140)23 (-95,140)(10,0,180) (-95,140)(10,-180,0) (-105,140)(-85,140)23 (-70,120)(-70,145)24 (-70,145)(-50,145) (-50,165)(-70,145) (-95,160)[]$\mu$(-125,160)[]$\mu$(-150,110)[]$\mu$(-50,110)[]$\nu_{\mu}$(-50,170)[]$\mathaccent 869{\nu_{e}}$(-40,145)[]$e$(-100,100)[]$(a.17)\ S=2$(0,120)(20,120) (20,120)(40,120) (40,120)(60,120) (60,120)(80,120) (30,120)(10,0,180)25 (60,120)(60,140)23 (30,150)(10,0,180) (30,150)(10,-180,0) (20,150)(40,150)23 (60,140)(80,140) (80,155)(60,140) (0,110)[]$\mu$(45,160)[]$\mu$(80,110)[]$\nu_{\mu}$(90,160)[]$\mathaccent 869{\nu_{e}}$(90,140)[]$e$(40,100)[]$(b.17)\ S=1$(130,120)(180,120) (180,120)(195,120) (205,120)(10,0,180)25 (195,120)(215,120) (215,120)(230,120) (155,140)(10,0,180) (155,140)(10,-180,0) (180,120)(180,140)23 (145,140)(165,140)23 (180,140)(230,140) (230,165)(180,140) (135,110)[]$\mu$(165,160)[]$\mu$(230,110)[]$\nu_{\mu}$(240,170)[]$\mathaccent 869{\nu_{e}}$(240,140)[]$e$(180,100)[]$(c.17)\ S=1$ (-150,120)(-100,120) (-100,120)(-50,120) (-100,145)(-85,145) (-85,145)(-65,145) (-65,145)(-50,145) (-100,120)(-100,145)24 (-100,145)(-50,145) (-50,195)(-100,145) (-75,170)(10,-135,45)25 (-75,145)(10,-180,0)25 (-150,110)[]$\mu$(-50,110)[]$\nu_{\mu}$(-50,180)[]$\mathaccent 869{\nu_{e}}$(-40,145)[]$e$(-100,100)[]$(a.18)\ S=1$(0,120)(40,120) (40,120)(100,120) (40,120)(40,130)22 (40,150)(40,160)22 (40,140)(10,-90,90) (40,140)(10,90,-90) (100,185)(40,160) (40,160)(60,160) (60,160)(80,160) (80,160)(100,160) (70,160)(10,-180,0)25 (0,110)[]$\mu$(100,110)[]$\nu_{\mu}$(90,190)[]$\mathaccent 869{\nu_{e}}$(110,160)[]$e$(50,100)[]$(b.18)\ S=1$(57,140)[]$e$(130,120)(190,120) (190,120)(240,120) (190,140)(10,-90,90) (190,140)(10,90,-90) (190,120)(190,130)22 (190,150)(190,160)22 (190,160)(240,160) (230,200)(190,160) (210,180)(10,-135,45)25 (140,110)[]$\mu$(240,110)[]$\nu_{\mu}$(240,190)[]$\mathaccent 869{\nu_{e}}$(250,160)[]$e$(200,100)[]$(c.18)\ S=1$(210,140)[]$\mu$ (-150,120)(-70,120) (-70,120)(-50,120) (-125,140)(10,0,180) (-125,140)(10,-180,0) (-135,140)(-115,140)23 (-95,140)(10,0,180) (-95,140)(10,-180,0) (-105,140)(-85,140)23 (-70,120)(-70,145)24 (-70,145)(-50,145) (-50,165)(-70,145) (-95,160)[]$e$(-125,160)[]$e$(-150,110)[]$\mu$(-50,110)[]$\nu_{\mu}$(-50,170)[]$\mathaccent 869{\nu_{e}}$(-40,145)[]$e$(-100,90)[]$(a.19)\ g_{1}=1,k_{1}=2$(-100,80)[]$S=2$(0,120)(20,120) (20,120)(40,120) (40,120)(60,120) (60,120)(80,120) (30,120)(10,0,180)25 (60,120)(60,140)23 (30,150)(10,0,180) (30,150)(10,-180,0) (20,150)(40,150)23 (60,140)(80,140) (80,155)(60,140) (0,110)[]$\mu$(45,160)[]$e$(80,110)[]$\nu_{\mu}$(90,160)[]$\mathaccent 869{\nu_{e}}$(90,140)[]$e$(40,90)[]$(b.19)\ S=1$(130,120)(180,120) (180,120)(195,120) (205,120)(10,0,180)25 (195,120)(215,120) (215,120)(230,120) (155,140)(10,0,180) (155,140)(10,-180,0) (180,120)(180,140)23 (145,140)(165,140)23 (180,140)(230,140) (230,165)(180,140) (135,110)[]$\mu$(165,160)[]$e$(230,110)[]$\nu_{\mu}$(240,170)[]$\mathaccent 869{\nu_{e}}$(240,140)[]$e$(180,90)[]$(c.19)\ S=1$ (-150,120)(-70,120) (-70,120)(-50,120) (-125,140)(10,0,180) (-125,140)(10,-180,0) (-135,140)(-115,140)23 (-95,140)(10,0,180) (-95,140)(10,-180,0) (-105,140)(-85,140)23 (-70,120)(-70,145)24 (-70,145)(-50,145) (-50,165)(-70,145) (-95,160)[]$e$(-125,160)[]$\mu$(-150,110)[]$\mu$(-50,110)[]$\nu_{\mu}$(-50,170)[]$\mathaccent 869{\nu_{e}}$(-40,145)[]$e$(-100,90)[]$(a.20)\ g_{1}=g_{2}=1$(-100,70)[]$s=1$(-85,80)[]$k_{1}=k_{2}=1$(-10,120)(70,120) (70,120)(90,120) (15,140)(10,0,180) (15,140)(10,-180,0) (5,140)(25,140)24 (45,140)(10,0,180) (45,140)(10,-180,0) (35,140)(55,140)24 (45,170)(10,0,180) (45,170)(10,-180,0) (55,170)(35,170)24 (70,120)(70,145)24 (70,145)(90,145) (90,165)(70,145) (60,150)[]$\mu$(-5,150)[]$\mu$(30,170)[]$\mu$(-10,110)[]$\mu$(90,110)[]$\nu_{\mu}$(90,170)[]$\mathaccent 869{\nu_{e}}$(100,145)[]$e$(45,80)[]$S=3!=6$(40,90)[]$(b.20)\ g_{1}=1,k_{1}=3$(150,120)(210,120) (210,120)(250,120) (210,120)(210,140)23 (165,140)(10,0,180) (165,140)(10,-180,0) (155,140)(175,140)23 (195,140)(10,0,180) (195,140)(10,-180,0) (185,140)(205,140)23 (195,170)(10,0,180) (195,170)(10,-180,0) (185,170)(205,170)23 (210,140)(250,140) (250,160)(210,140) (150,110)[]$\mu$(160,155)[]$e$(205,155)[]$e$(180,180)[]$e$(250,110)[]$\nu_{\mu}$(260,170)[]$\mathaccent 869{\nu_{e}}$(260,140)[]$e$(190,80)[]$S=6$(190,90)[]$(c.20)\ g_{1}=1,k_{1}=3$ (-150,120)(-70,120) (-70,120)(-50,120) (-125,140)(10,0,180) (-125,140)(10,-180,0) (-135,140)(-115,140)23 (-95,140)(10,0,180) (-95,140)(10,-180,0) (-105,140)(-85,140)23 (-95,170)(10,0,180) (-95,170)(10,-180,0) (-85,170)(-105,170)24 (-70,120)(-70,145)24 (-70,145)(-50,145) (-50,165)(-70,145) (-80,140)[]$\mu$(-145,140)[]$e$(-110,170)[]$e$(-150,110)[]$\mu$(-50,110)[]$\nu_{\mu}$(-50,170)[]$\mathaccent 869{\nu_{e}}$(-40,145)[]$e$(-100,90)[]$(a.21)\ g_{1}=g_{2}=1$(-85,80)[]$k_{1}=2,k_{2}=1$(-100,70)[]$S=2$(0,120)(20,120) (20,120)(40,120) (40,120)(60,120) (60,120)(80,120) (30,120)(10,0,180)25 (60,120)(60,140)23 (15,150)(10,0,180) (15,150)(10,-180,0) (5,150)(25,150)23 (45,150)(10,0,180) (45,150)(10,-180,0) (35,150)(55,150)23 (60,140)(80,140) (80,155)(60,140) (0,110)[]$\mu$(15,170)[]$\mu$(45,170)[]$e$(80,110)[]$\nu_{\mu}$(90,160)[]$\mathaccent 869{\nu_{e}}$(90,140)[]$e$(40,90)[]$(b.21)\ S=1$(130,120)(180,120) (180,120)(195,120) (205,120)(10,0,180)25 (195,120)(215,120) (215,120)(230,120) (140,140)(10,0,180) (140,140)(10,-180,0) (130,140)(150,140)23 (165,140)(10,0,180) (165,140)(10,-180,0) (155,140)(175,140)23 (180,120)(180,140)23 (180,140)(230,140) (230,165)(180,140) (130,110)[]$\mu$(165,160)[]$\mu$(140,160)[]$e$(230,110)[]$\nu_{\mu}$(240,170)[]$\mathaccent 869{\nu_{e}}$(240,140)[]$e$(180,90)[]$(c.21)\ S=1$ (-150,120)(-70,120) (-70,120)(-50,120) (-125,140)(10,0,180) (-125,140)(10,-180,0) (-135,140)(-115,140)23 (-95,140)(10,0,180) (-95,140)(10,-180,0) (-105,140)(-85,140)23 (-95,170)(10,0,180) (-95,170)(10,-180,0) (-85,170)(-105,170)24 (-70,120)(-70,145)24 (-70,145)(-50,145) (-50,165)(-70,145) (-80,140)[]$\mu$(-145,140)[]$\mu$(-110,170)[]$e$(-150,110)[]$\mu$(-50,110)[]$\nu_{\mu}$(-50,170)[]$\mathaccent 869{\nu_{e}}$(-40,145)[]$e$(-100,100)[]$(a.22)\ S=2$(0,120)(60,120) (60,120)(100,120) (60,120)(60,140)23 (15,140)(10,0,180) (15,140)(10,-180,0) (5,140)(25,140)23 (45,140)(10,0,180) (45,140)(10,-180,0) (35,140)(55,140)23 (60,140)(100,140) (100,180)(60,140) (80,160)(10,-135,45)25 (0,110)[]$\mu$(15,160)[]$\mu$(45,160)[]$e$(100,110)[]$\nu_{\mu}$(110,190)[]$\mathaccent 869{\nu_{e}}$(110,140)[]$e$(40,100)[]$(b.22)\ S=1$(130,120)(180,120) (180,120)(230,120) (140,140)(10,0,180) (140,140)(10,-180,0) (130,140)(150,140)23 (165,140)(10,0,180) (165,140)(10,-180,0) (155,140)(175,140)23 (180,120)(180,140)23 (180,140)(195,140) (205,140)(10,-180,0)25 (195,140)(215,140) (215,140)(230,140) (230,165)(180,140) (130,110)[]$\mu$(165,160)[]$\mu$(140,160)[]$e$(230,110)[]$\nu_{\mu}$(240,170)[]$\mathaccent 869{\nu_{e}}$(240,140)[]$e$(180,100)[]$(c.22)\ S=1$ (-150,120)(-70,120) (-70,120)(-50,120) (-125,140)(10,0,180) (-125,140)(10,-180,0) (-135,140)(-115,140)23 (-125,170)(10,0,180) (-125,170)(10,-180,0) (-135,170)(-115,170)23 (-95,140)(10,0,180) (-95,140)(10,-180,0) (-105,140)(-85,140)23 (-95,170)(10,0,180) (-95,170)(10,-180,0) (-85,170)(-105,170)24 (-70,120)(-70,145)24 (-70,145)(-50,145) (-50,165)(-70,145) (-80,140)[]$\mu$(-80,170)[]$e$(-145,140)[]$\mu$(-145,170)[]$e$(-150,110)[]$\mu$(-50,110)[]$\nu_{\mu}$(-50,170)[]$\mathaccent 869{\nu_{e}}$(-40,145)[]$e$(-100,90)[]$(a.23)\ g_{1}=g_{2}=1$(-100,80)[]$k_{1}=k_{2}=2$(-100,70)[]$S=4$(0,120)(60,120) (60,120)(100,120) (60,120)(60,140)23 (15,140)(10,0,180) (15,140)(10,-180,0) (5,140)(25,140)23 (15,170)(10,0,180) (15,170)(10,-180,0) (5,170)(25,170)23 (45,140)(10,0,180) (45,140)(10,-180,0) (35,140)(55,140)23 (45,170)(10,0,180) (45,170)(10,-180,0) (35,170)(55,170)23 (60,140)(100,140) (100,160)(60,140) (0,110)[]$\mu$(0,145)[]$e$(55,150)[]$e$(55,180)[]$e$(0,180)[]$\mu$(100,110)[]$\nu_{\mu}$(110,170)[]$\mathaccent 869{\nu_{e}}$(110,140)[]$e$(40,90)[]$(b.23)\ g_{1}=g_{2}=1$(40,80)[]$k_{1}=3,k_{2}=1$(40,70)[]$S=6$(130,120)(200,120) (200,120)(230,120) (150,140)(10,0,180) (150,140)(10,-180,0) (140,140)(160,140)23 (150,170)(10,0,180) (150,170)(10,-180,0) (140,170)(160,170)23 (180,140)(10,0,180) (180,140)(10,-180,0) (170,140)(190,140)23 (180,170)(10,0,180) (180,170)(10,-180,0) (170,170)(190,170)23 (200,120)(200,140)23 (200,140)(230,140) (230,165)(200,140) (130,110)[]$\mu$(195,150)[]$e$(140,150)[]$e$(195,180)[]$e$(140,180)[]$e$(230,110)[]$\nu_{\mu}$(250,170)[]$\mathaccent 869{\nu_{e}}$(240,140)[]$e$(180,90)[]$(c.23)\ g_{1}=1,k_{1}=4$(180,80)[]$S=24$	arxiv-papers	2010-11-18T06:53:50	2024-09-04T02:49:14.963993	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "L. T. Hue, H. T. Hung and H. N. Long", "submitter": "Long Hoang Ngoc", "url": "https://arxiv.org/abs/1011.4142" }
1011.4239	# Probing GRB environments with time variability: ULTRASPEC fast imaging of GRB 080210 ††thanks: Based on observations collected with the ULTRASPEC visitor instrument built by a consortium from the University of Sheffield, Warwick, the UK Astronomy Technology Centre and ESO, mounted at the ESO/3.6-m telescope on La Silla, Chile, and on target-of-opportunity observations collected in service mode under program ID 080.D-0526, P.I. Vreeswijk, with the FOcal Reducer/low dispersion Spectrograph 2 (FORS2) installed at the Cassegrain focus of the Very Large Telescope (VLT), Unit 1, Antu, operated by the European Southern Observatory (ESO) on Cerro Paranal in Chile. For further information or questions on the content of the paper, please e-mail to annalisa@raunvis.hi.is. A. De Cia1, P. Jakobsson1, G. Björnsson1, P. M. Vreeswijk1,2, V. S. Dhillon3 T. R. Marsh4, R. Chapman1,5, J. P. U. Fynbo2, C. Ledoux6, S. P. Littlefair3 D. Malesani2, S. Schulze1, A. Smette6, T. Zafar2 and E. H. Gudmundsson1. 1 Centre for Astrophysics and Cosmology, Science Institute, University of Iceland, Dunhaga 5, IS-107 Reykjavik, Iceland 2 Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen Ø, Denmark 3 Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK 4 Department of Physics, University of Warwick, Coventry CV4 7AL, UK 5 Centre for Astrophysics Research, University of Hertfordshire, College Lane, Hatfield AL10 9AB, UK 6 European Southern Observatory, Alonso de Córdova 3107, Vitacura, Casilla 19001, Santiago 19, Chile (Accepted 2010 Mm gg. Received 2010 Mm gg; in original form 2010 Mm gg) ###### Abstract We present high time resolution (1.09 s) photometry of GRB 080210 obtained with ULTRASPEC mounted on the ESO/3.6-m telescope, starting 68.22 min after the burst and lasting for 26.45 min. The light curve is smooth on both short (down to 2.18 s) and long time scales, confirmed by a featureless power spectrum. On top of the fireball power-law decay, bumps and wiggles at different time scales can, in principle, be produced by density fluctuations in the circumburst medium, substructures in the jet or by refreshed shocks. Comparing our constraints with variability limits derived from kinematic arguments, we exclude under-density fluctuations producing flux dips larger than 1 per cent with time scales $\Delta t>9.2$ min (2 per cent on $\Delta t>2.3$ min for many fluctuating regions). In addition, we study the afterglow VLT/FORS2 spectrum, the optical-to-X-ray spectral energy distribution (SED) and the time decay. The SED is best fit with a broken power law with slopes $\beta_{\mathrm{opt}}=0.71\pm 0.01$ and $\beta_{X}=1.59\pm 0.07$, in disagreement with the fireball model, suggesting a non-standard afterglow for GRB 080210. We find $A_{V}=0.18\pm 0.03$ mag optical extinction due to SMC- like dust and an excess X-ray absorption of log $(N_{\mathrm{H}}/$cm${}^{-2})=21.58^{+0.18}_{-0.26}$ assuming Solar abundances. The spectral analysis reveals a damped Ly$\alpha$ absorber (log $(N_{\mathrm{H\,{\sc I}}}/$cm${}^{-2})=21.90\pm 0.10$) with a low metallicity ([X/H$]=-1.21\pm 0.16$), likely associated with the interstellar medium of the GRB host galaxy ($z=2.641$). ###### keywords: gamma-rays: bursts - instrumentation: detectors - dust, extinction - ISM: kinematics - ISM: abundances ††pagerange: Probing GRB environments with time variability: ULTRASPEC fast imaging of GRB 080210 ††thanks: Based on observations collected with the ULTRASPEC visitor instrument built by a consortium from the University of Sheffield, Warwick, the UK Astronomy Technology Centre and ESO, mounted at the ESO/3.6-m telescope on La Silla, Chile, and on target-of-opportunity observations collected in service mode under program ID 080.D-0526, P.I. Vreeswijk, with the FOcal Reducer/low dispersion Spectrograph 2 (FORS2) installed at the Cassegrain focus of the Very Large Telescope (VLT), Unit 1, Antu, operated by the European Southern Observatory (ESO) on Cerro Paranal in Chile. For further information or questions on the content of the paper, please e-mail to annalisa@raunvis.hi.is.–Probing GRB environments with time variability: ULTRASPEC fast imaging of GRB 080210 ††thanks: Based on observations collected with the ULTRASPEC visitor instrument built by a consortium from the University of Sheffield, Warwick, the UK Astronomy Technology Centre and ESO, mounted at the ESO/3.6-m telescope on La Silla, Chile, and on target-of-opportunity observations collected in service mode under program ID 080.D-0526, P.I. Vreeswijk, with the FOcal Reducer/low dispersion Spectrograph 2 (FORS2) installed at the Cassegrain focus of the Very Large Telescope (VLT), Unit 1, Antu, operated by the European Southern Observatory (ESO) on Cerro Paranal in Chile. For further information or questions on the content of the paper, please e-mail to annalisa@raunvis.hi.is.††pubyear: 2010 ## 1 Introduction Instrument \| Grism \| Start time \| Exposure time \| $\delta t^{a}$ \| Coverage \| FWHM ---\|---\|---\|---\|---\|---\|--- \| \| (UT hh:mm:ss) \| (s) \| (min) \| (Å) \| (Å) VLT/FORS2 \| — \| 08:26:25 \| 10 \| 36.4 \| $R$ band \| VLT/FORS2 \| 300V \| 08:32:10 \| 600 \| 47.1 \| 3500–9600 \| 13.3 VLT/FORS2 \| 600z+OG590 \| 08:43:13 \| 600 \| 58.1 \| 8000–9000 \| 6.4 VLT/FORS2 \| 1400V \| 08:54:37 \| 600 \| 69.5 \| 4600–5900 \| 2.5 ULTRASPEC \| — \| 08:58:18 \| $1.09\times 1455$ \| 81.4 \| $V$ band \| VLT/FORS2 \| 1200R+GG435 \| 09:05:47 \| 600 \| 80.7 \| 6000–7000 \| 3.0 VLT/FORS2 \| 300V \| 09:17:11 \| 600 \| 92.1 \| 3500–9600 \| 13.3 VLT/FORS2 \| — \| 09:30:53 \| 45 \| 101.2 \| $R$ band \| Table 1: GRB 080210 observation log on date 2008 February 10. a $\delta t$ is the mid-exposure time after the BAT trigger (07:50:06 UT). Long ($>2$ s) soft gamma ray bursts (GRBs) are the most powerful explosions known in the Universe. After the discovery of GRB optical (van Paradijs et al., 1997) and X-ray (Costa et al., 1997) afterglows in 1997, we have learned that they mainly occur in distant galaxies and their connection to core- collapse supernovae is now widely accepted (for a review see Woosley & Bloom, 2006). The diversity amongst individual GRB events, as well as the difficulty in observing such transient sources, challenges theoretical models to explain them, the fireball model providing the best overall agreement (e.g., Rees & Mészáros, 1992; Mészáros & Rees, 1993; Piran, 1999). In this scenario, the GRB afterglow originates from the synchrotron radiation produced by the interaction between the ultra-relativistic ejecta (jet) and the surrounding interstellar medium. Although GRBs can be extremely variable in their prompt phase and X-ray afterglow flares are commonly observed during the first minutes after the burst, the late-time afterglow in general shows a fairly smooth power-law behaviour at different phases (Zhang et al., 2006), from the X-rays to the optical, IR and radio wavelengths. Environmental effects and intrinsic discontinuities can introduce afterglow variability on different time scales, possibly due to (i) ambient density fluctuations (Wang & Loeb, 2000), (ii) substructures in the jets (patchy-jet; Mészáros, Rees & Wijers, 1998), (iii) inhomogeneities on the emitting surface (patchy-shell; Kumar & Piran, 2000), (iv) refreshed shocks (Rees & Mészáros, 1998; Sari & Mészáros, 2000) or (v) late-time central engine activity (Rees & Mészáros, 2000). (i) Density fluctuations can arise from interstellar turbulence or can be generated, before the GRB event, by a variable wind from the progenitor star (see e.g., van Marle et al., 2005). Linear density fluctuations with $dn/n<1$ on a length scale of 1–$10^{3}$ AU could induce fluctuations in the afterglow light curve with a fractional amplitude of up to $\sim 30$ per cent over time scales of tens of minutes in the optical (Wang & Loeb, 2000). (ii) Substructures in the jet can form if the bulk Lorentz factor depends on the angle inside the jet (Mészáros et al., 1998). As the emitting region evolves through this patchy-jet, the flux varies in intensity. (iii) Angular inhomogeneity of the relativistic ejecta can separate the emitting surface into different causally disconnected regions (patchy-shell; Kumar & Piran, 2000; Nakar & Oren, 2004). These wiggles evolve within the emitting surface, which is expanding in time with the blast-wave deceleration, causing variability in the radiation. (iv) If the ejecta have a range of bulk Lorentz factors, the slower shells will catch up with the leading blast wave, once the fireball has been decelerated by the external medium. The refreshed shocks will boost the luminosity of the afterglow (Rees & Mészáros, 1998). (v) The GRB engine could contribute to the variability at late times: as debris accretes onto the black hole in the period following the burst, its extended activity could heat the environment or produce new outflows, giving rise to a detectable component of emission which, like any accretion-powered source, would be variable (Rees & Mészáros, 2000). Thus, the detection (and even the non-detection) of time variability within several minutes to a couple of hours after the burst can provide important constraints on the different proposed scenarios, and therefore on the physics of the evolution of the fireball. Temporal variations in GRB afterglow light curves were first observed, on time scale as short as $\sim 1$ hour, in GRB 011211 (Holland et al., 2002; Jakobsson et al., 2004), induced either by inhomogeneities in the medium surrounding the GRB, or by a patchy jet. Lazzati et al. (2003) have argued that the deviations in the afterglow of GRB 021004 are due to the interaction of the GRB fireball, or jet, with density enhancements in the ambient medium. However, time-resolved polarimetry of the same burst suggested that the variations were produced by a refreshed shock (Björnsson, Gudmundsson & Jóhannesson, 2004). Granot, Nakar & Piran (2003) have also interpreted the variations seen in the light curve of GRB 030329 as due to refreshed shocks. Several limitations challenge the detection of late-time variability in GRB light curves. First, the amplitude of most of the fluctuations that can possibly be expected decays with time. Moreover, different processes (e.g., density fluctuations, patchy shell or refreshed shocks) will physically constrain the variability over only certain time scales at one observation time (Ioka, Kobayashi & Zhang, 2005). In particular, the fastest variability is the hardest to detect, partly because it is intrinsically weaker, but also because the readout noise and dead time of classical CCDs usually limits the time resolution of the observation itself. High-speed photometry can now be achieved thanks to the fast read-out with zero-noise of the frame-transfer electron-multiplying CCDs (EMCCDs). The ULTRASPEC camera (Dhillon et al., 2007) adopts such a CCD to amplify the signal, rendering the read-out noise negligible. In addition, the frame transfer architecture allows the EMCCD to read out a completed exposure whilst the next exposure is being obtained, virtually eliminating the dead time between exposures. We observed GRB 080210 with ULTRASPEC mounted on the ESO 3.6-m telescope in La Silla, allowing 1.09 s time resolution imaging. The highest speed photometry obtained for a GRB afterglow so far is the TORTORA observations of the extremely bright “naked eye” GRB 080319B (Greco et al., 2009; Beskin et al., 2010), with a 0.3 s time bin, from 10 to 100 s after the burst trigger. However, the ULTRASPEC observations of GRB 080210, presented in this paper, probes the afterglow phase, providing the lowest $\Delta t/t$ so far. This opens a new window on the fast-variability study of the afterglow itself. Comparing the variability limits given by Ioka et al. (2005) with the ULTRASPEC observations, we can constrain the properties of the circumburst medium and the shock structure. In addition, we investigate the GRB 080210 host galaxy environment in another way, through ESO-VLT/FORS low- and medium-resolution spectroscopy, as well as optical-to-X-ray spectral energy distribution (SED) modelling. Ly$\alpha$ and metal absorption systems, often observed in GRB lines of sight, can be used to derive physical properties of the absorbing gas clouds, such as kinematics, densities and metallicities (Vreeswijk et al., 2006; Ledoux et al., 2009). On the other hand, the SED provides information on both the dust inside the host galaxy and the spectral properties of the GRB afterglow itself (e.g., Starling et al., 2007). Overall, interpreting the combined optical and X-ray spectra and light curves within the context of the fireball model can provide a probe of the blast-wave physics, as well as the GRB environment (Sari, Piran & Narayan 1998; Zhang et al., 2006). The paper is organized as follows: observations and data reduction are presented in Sec. 2, while their analysis is reported in Sec. 3: first the ULTRASPEC light curve, then the optical spectroscopy, the SED modelling and finally the optical and X-ray afterglow temporal decay. We discuss our results in Sec. 4 and summarize them in the last section. Throughout the paper we use the convention $F_{\nu}\left(t\right)\propto t^{-\alpha}\nu^{-\beta}$ for the flux density, where $\alpha$ is the temporal slope and $\beta$ is the spectral slope. Hereafter we assume a standard $\Lambda$CDM cosmology with $H_{0}$ = 70.4 km s-1 Mpc-1, $\Omega_{M}$ = 0.27 and $\Omega_{\Lambda}$ = 0.73 (Jarosik et al., 2010). Figure 1: Estimated slit throughputs (top panel) of the two 300V spectra due to the wavelength dependence of the PSF and the misalignment of the slit relative to the parallactic angle at the high airmass of these observations (2.3 at epoch 1 and 1.8 at epoch 2). In the bottom panel the ratio between the two slit throughputs (solid line) is overplotted on the ratio between the spectra at the two epochs (dotted line). ## 2 Observations and data reduction ### 2.1 Swift detection The Swift Burst Alert Telescope (BAT) triggered on GRB 080210 on February 10th, 2008 at $T_{0}=$ 07:50:06 UT (Grupe et al., 2008). The duration spanning 90 per cent of the GRB emission (15–350 keV) was $45\pm 11$ s and its 15–150 keV fluence was $1.8\times\,10^{-6}$ erg cm-2. The time integrated BAT spectrum is best fit by a simple power law with photon index $\Gamma=1.77\pm 0.12$ (Ukwatta et al., 2008). An X-ray afterglow was observed with the Swift X-Ray Telescope (XRT), starting 161 s (240 s) after the trigger in windowed timing (photon counting) mode. An optical afterglow was detected by the Swift Ultra-Violet/Optical Telescope at a position R.A. $=16^{\mathrm{h}}45^{\mathrm{m}}04.01^{\mathrm{s}}$ and Decl. $=+13^{\circ}49\arcmin 35.9\arcsec$ (J2000, estimated 90 per cent confidence error radius of 0.6″; Marshall & Grupe, 2008). We retrieved the X-ray light curve and spectra from the Swift repository (Evans et al., 2007, 2009). ### 2.2 ULTRASPEC imaging Figure 2: The fast (1.09 s) sampled ULTRASPEC light curves ($V$ band). The whole observation (4093–5680 s after the trigger) is displayed in the three upper panels, while the first 100 s are shown in the bottom panel (1$\sigma$ errors over-plotted). The raw GRB (solid) and the comparison star (dotted) light curves are shown in the first and second panels, respectively. The GRB light curve, flux calibrated with the comparison star, is shown in panels 3 and 4. The flux decreases as a power law ($F\propto t^{-\alpha}$) with decay index $\alpha=0.74\pm 0.07$ ($\chi^{2}_{\textrm{dof}}=1.03$ for 1453 degrees of freedom). Any short time-scale variation is consistent with statistical fluctuations smaller than $3\sigma$. For 26.45 min, starting at 08:58 UT on February 10th 2008, we observed GRB 080210 with ULTRASPEC at the ESO 3.6-m telescope on La Silla, Chile, mounted on the EFOSC2 spectrograph (D’Odorico, 1988). Because of the frame-transfer capabilities of ULTRASPEC, it is possible to obtain very high time resolution data without sacrificing efficiency. Observations were taken in imaging mode, through the Bessel $V$-band filter. The CCD pixels were binned by $2\times 2$, allowing $\sim$1 s sampling of the light curve, with negligible ($\sim 10$ ms) dead time between exposures. Time-stamping of individual exposures uses a dedicated GPS-based system with a relative accuracy of 50 $\mu$s and an absolute accuracy of a few ms. The data were bias subtracted and subsequently flat-fielded using a median of 100 sky-flat frames. The photometric information about the GRB afterglow was extracted using an implementation of the optimal photometry algorithm of Naylor (1998), which provides significantly better signal-to-noise than aperture photometry for faint sources. A nearby comparison star was used both to estimate the point- spread function and to correct for transparency variations. The position of GRB 080210 was fixed with respect to the position of the comparison star; this ensures that centroiding on the faint GRB does not introduce spurious variability into the light curve. Observations of a flux standard were taken on the following night to place the measurements on a standard photometric system. We were not able to determine the $V$-band extinction coefficient for the night of these observations, so the La Silla average of 0.12 mag/airmass was used to correct for atmospheric extinction. The observation log is presented in Table 1. ### 2.3 VLT/FORS2 observations Starting at 08:32 UT on 2008 February 10th (42 min post-burst), a series of 600 s spectra were obtained with VLT/FORS2 in long-slit spectroscopy mode with a 1$\aas@@fstack{\prime\prime}$0 wide slit, North-South oriented and centred with an $R$-band acquisition image. The sequence of grisms used was 300V, 600z+OG590, 1400V, 1200R+GG435 and finally 300V again. This allowed us to both cover a larger wavelength window with the lower resolution grism (300V) and obtain mid-resolution spectroscopy for different regions of the spectrum. The individual spectra were cleaned of cosmic rays using the Laplacian Cosmic Ray Identification algorithm of van Dokkum (2001). The seeing remained relatively stable during the observations, between 1$\aas@@fstack{\prime\prime}$1 and 1$\aas@@fstack{\prime\prime}$4, yielding the spectral resolutions reported in Table 1. A first analysis of the spectrum revealed an absorption system associated with the host galaxy at redshift $z=2.641$ (Jakobsson et al., 2008a; Fynbo et al., 2009). While the mid-resolution spectra were mainly used for the spectral analysis, we aimed to flux-calibrate the low-resolution 300V spectra for the SED study. However, the two 300V spectra were obtained at high airmass (2.3 and 1.8 respectively) where the difference between the slit position angle and the parallactic angle was 126.6∘ and 133.4∘, respectively. Thus, slit losses influence the continuum level of the spectra, particularly in the blue. In order to correct for this, we computed the slit throughput in the following way. A theoretical model of the point spread function (PSF) as delivered by an 8.2 m diameter Unit Telescope, including the central obscuration caused by the secondary mirror, was built using a piece of IDL code graciously made available by Enrico Fedrigo (private communication). In particular, this model includes the dependence of the diffraction-limited theoretical PSF as a function of wavelength. This model PSF was then convolved with a Gaussian whose wavelength-dependent FWHM follows Roddier’s formula, i.e. $\propto(\lambda/\lambda_{\mathrm{ref}})^{-0.2}$, normalized to the value measured on each observed spectrum at the effective wavelength of the $R$ filter used for the centering of the target ($\lambda_{\mathrm{ref}}=6600$ Å). For each spectrum, the distance of the PSF centre to the slit centre at a given wavelength was assumed to be the differential refraction between this wavelength and $\lambda_{\mathrm{ref}}$ multiplied by the cosine of the difference between the parallactic angle at the time of the observation and the parallactic angle at the time when the object would be at an airmass of 1.41, converted to degrees ($25^{\circ}$ and $18^{\circ}$). Here we assumed that the Longitudinal Atmospheric Dispersion Corrector (LADC) of FORS2 (Avila, Rupprecht & Beckers, 1997) performs optimally up to airmass of 1.41. For the calculation of the differential refraction, we assumed the usual atmospheric conditions at Paranal (temperature $T=12\,^{\circ}\mathrm{C}$, pressure $P=743$ mbar). The integrated value of the flux along the spatial direction entering the spectrograph can then be calculated for each wavelength. The factor representing the slit throughput is then the ratio between this integrated value and the total flux at that wavelength. The top panel of Fig. 1 shows the estimated slit throughputs as a function of wavelength for the two 300V FORS2 spectra. The bottom panel shows the ratio between the throughputs compared with the ratio between the two spectra. The agreement indicates that the slit throughputs have been reasonably well calculated. The response curve correction was performed using observations of the standard star LTT3864. The flux calibration was rescaled using $R$-band VLT/FORS2 images. $R$-band photometry was secured with VLT/FORS2, before and after our spectroscopic observation. Calibration was carried out by observing the Landolt standard fields SA 100 and Rubin 149, which allowed us to obtain a photometric accuracy of 0.02 mag. The GRB observations were carried out at large airmass (1.7–2.5), but so was also one of the two standard fields, which allowed a reliable extinction coefficient to be computed, in agreement with the value tabulated in the ESO web page. The photometric conditions were excellent according to the Paranal night logs. ## 3 Data analysis ### 3.1 ULTRASPEC light curve Figure 3: The power spectrum of the ULTRASPEC light curve in fractional $\textrm{rms}^{2}$ units, after correcting for the decay. The minimum period detectable, given the time resolution, is 2.18 s, while the longest time scale monitored is 1587 s. The peaks in the power spectrum are generated by random noise, see Section 3.1 for more details. The lack of a preferred frequency confirms the smoothness of the light curve. The ULTRASPEC light curve is plotted in Fig. 2. We Fourier transformed the ULTRASPEC light curve in order to investigate the variability and possible periodicities over all time scales. We exclude the flux variation due to the afterglow natural evolution by first correcting the light curve with the power-law fit to the decay. Figure 3 shows the power spectra of the unbinned light curve. Thanks to the fast sampling of ULTRASPEC, we can monitor the power spectrum down to 2.18 s time scales. The power spectrum of the light curve shows a peak at 2.8 s, likely due to random noise. Indeed, the peak height is not significant (3 per cent of the flux, corresponding to log $(\Delta F/F)=-1.57$, (log $\Delta t/t)=-3.2$ at 81.4 min), well below the instrument detection limit, see Sec. 4.2. We performed a Monte Carlo type generation of 100,000 light curves with random variability equal to the measured GRB variability and found peaks of this height and greater to occur in 6 per cent of cases. In addition, the phase-folded light curve does not show any evidence for periodicity. Thus, no frequency is preferred in the power spectrum. The ULTRASPEC light curve is smooth and follows a pure power law within the statistical fluctuations. Line [$\lambda_{\textrm{vac}}$] \| Observed wavelength \| Redshift \| Observed EW \| Notes ---\|---\|---\|---\|--- (Å) \| (Å) \| \| (Å) \| 1400V O i $\lambda 1302$ \| 4740.6 \| 2.6405 \| $4.52\pm 0.42$ \| Si ii $\lambda 1304$ \| 4747.7 \| 2.6398 \| $4.81\pm 0.41$ \| O i* $\lambda 1304$ contribution C ii $\lambda 1334$ \| 4858.4 \| 2.6406 \| $9.71\pm 0.47$ \| C ii* $\lambda 1335$ \| 4864.1 \| 2.6416 \| $1.93\pm 0.22$ \| Si iv $\lambda 1393$ \| 4890.5 \| 2.5088a \| $0.93\pm 0.33$ \| Si iv $\lambda 1402$ not detected Si iv $\lambda 1393$ \| 5073.8 \| 2.6403 \| $5.52\pm 0.28$ \| Si iv $\lambda 1402$ \| 5106.7 \| 2.6404 \| $4.52\pm 0.27$ \| C iv $\lambda 1548$ \| 5431.8 \| 2.5085a \| $2.44\pm 0.21$ \| C iv $\lambda 1550$ \| 5440.8 \| 2.5084a \| $1.93\pm 0.23$ \| Si ii $\lambda 1526$ \| 5556.7 \| 2.6396 \| $5.53\pm 0.34$ \| Two velocity components 5558.7 \| 2.6416 C iv $\lambda 1548$ \| 5636.0 \| 2.6403 \| $9.56\pm 0.30$ \| C iv $\lambda 1550$ \| 5644.9 \| 2.6400 \| $7.75\pm 0.27$ \| Fe ii $\lambda 1608$ \| 5854.4 \| 2.6396 \| $3.54\pm 0.38$ \| Two velocity components 5857.8 \| 2.6416 1200R Al ii $\lambda 1670$ \| 6081.0 \| 2.6396 \| $6.08\pm 0.45$ \| Two velocity components 6083.6 \| 2.6416 Si ii $\lambda 1808$ \| 6580.9 \| 2.6396 \| $2.36\pm 0.39$ \| Two velocity components 6583.9 \| 2.6416 Al iii $\lambda 1854$ \| 6749.8 \| 2.6396 \| $3.36\pm 0.34$ \| Two velocity components 6753.9 \| 2.6416 Al iii $\lambda 1862$ \| 6780.0 \| 2.6396 \| $1.37\pm 0.35$ \| Two velocity components 6783.8 \| 2.6416 600z Fe ii $\lambda 2344$ \| 8532.1 \| 2.6496 \| $5.35\pm 0.30$ \| Two velocity components 8648.5 \| 2.6416 Fe ii $\lambda 2374$ \| 8642.8 \| 2.6396 \| $3.96\pm 0.38$ \| Two velocity components 8648.1 \| 2.6416 Fe ii $\lambda 2382$ \| 8671.5 \| 2.6396 \| $7.23\pm 0.41$ \| Two velocity components 8675.9 \| 2.6416 Table 2: Absorption lines in the medium resolution 1400V, 1200R and 600z spectra. The redshifts for the two-component profiles (short vertical lines) are derived from a Voigt profile fit. Observer frame EWs with $1\sigma$ errors are reported. a Intervening system. Ion [transitions] \| Component a \| Component b \| Total column density \| [X/H] ---\|---\|---\|---\|--- \| log ($N/$cm-2) \| log ($N/$cm-2) \| log ($N/$cm-2) \| Al ii [1670]a \| \| \| $>13.56$ \| $>-2.79$ Al iii [1854,1862] \| $13.77\pm 0.08$ \| $13.91\pm 0.13$ \| $14.14\pm 0.08$ \| $-2.21\pm 0.13$ Fe ii [1608] \| $15.72\pm 0.37$ \| $15.63\pm 0.54$ \| $15.98^{+0.37}_{-0.26}$ \| $-1.42\pm 0.33$ Si ii [1526b, 1808] \| $15.84\pm 0.18$ \| $15.96\pm 0.16$ \| $16.20^{+0.13}_{-0.11}$ \| $-1.21\pm 0.16$ Zn ii [2026]c \| \| \| $13.53\pm 0.14$ \| $-0.93\pm 0.18$ Table 3: The ionic column densities estimated from a simultaneous Voigt profile fit to the lines in the 1400V and 1200R medium resolution grism spectra. aAl ii $\lambda$1670 line is saturated, the lower limit on $N_{\textrm{Al\,{\sc ii}}}$ is derived from the EW (Table 2). bThe Si ii $\lambda$1526 line is only included in a first stage to model the line profiles; the Si abundance is computed using only the weaker and non-saturated Si ii $\lambda$1808 line. cZn abundance estimated from the low resolution 300V spectrum (b${}_{\textrm{turb}}$=39.4, b${}_{\textrm{th}}$=0 km s-1). ### 3.2 VLT/FORS2 spectral analysis Figure 4: A portion of the normalized optical afterglow spectrum, centred on the Ly$\alpha$ absorption line, at the GRB host galaxy redshift. A neutral hydrogen column density fit to the damped Ly$\alpha$ line is shown with a solid line (log $(N_{\mathrm{H\,{\sc I}}}/$cm${}^{-2})=21.90\pm 0.10$), while the $1\sigma$ errors are shown with dashed lines. In the 300V combined spectrum (averaged from the two 300V flux calibrated spectra), we identify a damped Ly$\alpha$ absorber (DLA) in addition to a number of absorption lines at $z=2.641\pm 0.001$, associated with the host galaxy of the burst. We also identify an intervening system at $z_{\textrm{int}}=2.509\pm 0.001$ from Si iv and C iv transitions. A list of the lines detected in the low resolution 300V spectrum and their equivalent widths (EWs) is reported in Fynbo et al. (2009). We measured the EWs for both 300V epochs and find no evidence for spectral variability in the absorption. For the H i DLA fit, we derived log $(N_{\mathrm{H\,{\sc I}}}/$cm${}^{-2})=21.90\pm 0.10$ (Fig. 4). The tentative detection of Ly$\alpha$ emission inside the DLA, as seen from the 1D spectrum (Jakobsson et al., 2008a), is most likely noise, since it is not detected in the 2D frame. A good metallicity estimator is usually the weak transition Zn ii $\lambda$2026, but for GRB 080210 this line is only covered by the low resolution 300V spectrum, allowing only a crude metallicity estimate. From the simultaneous Voigt-profile fit of the Zn ii $\lambda$2026 with the Si ii $\lambda$1526, $\lambda$1808 and Fe ii $\lambda$1608, $\lambda$1611 lines in the low resolution spectrum, performed with the MIDAS/FITLYMAN software (Fontana & Ballester, 1995), we derive a log $(N_{\mathrm{Zn\,{\sc II}}}/$cm${}^{-2})=13.53\pm 0.14$ cm2 (i.e. [Zn/H] $-0.93\pm 0.18$, Doppler thermal broadening $b_{\textrm{th}}=0$ km s-1, and turbulent broadening $b_{\textrm{tur}}=39.4\pm 6.8$ km s-1). The analysis seems to show that the Zn ii $\lambda$2026 line is on the linear part of the curve of growth (i.e. unsaturated), despite the low spectral resolution of the data. The metallicities refer to the solar abundances reported by Asplund et al. (2009). Several lines are also identified in the medium resolution 1400V, 1200R and 600z spectra with the EWs listed in Table 2. Many of the lines associated with the GRB host galaxy system show evidence for a two-component profile (component “a” and “b”). In order to derive reliable column densities and to study the kinematics of the gas, we select the Fe ii, Si ii, Al ii and Al iii lines in the higher resolution grisms 1400V and 1200R that are neither too saturated nor blended with other transitions, and model them simultaneously with a two-component Voigt profile, using the VPFIT111Available at http://www.ast.cam.ac.uk/$\sim$rfc/vpfit.html software. In this way, the line profile of all the species is modelled with the same redshift $z$ and $b_{\textrm{tur}}$, for a given component, resulting in different column densities for different ions. We expect the Fe ii, Si ii, Al ii and possibly Al iii to be co-spatial and therefore to show a similar line profile. This is what we observe in the line-of-sight to GRB 080210. The Si ii $\lambda$1304 transition was excluded from the analysis because it is blended with the possibly dominating O i* $\lambda$1304 line. The 600z spectrum was not included in the Voigt profile fit because of its poorer spectral resolution. The two components that model the line profiles are separated by $148\pm 25$ km s-1. The normalization was determined locally around each line and telluric features were excluded from the fit. Figure 5 shows the two-component Voigt profile fit to the lines in the medium resolution spectra. The abundances and corresponding metallicities are presented in Table 3. Iron is probably depleted onto dust grains and is thus not a good metallicity indicator (Savage & Sembach, 1996). The best metallicity estimate is derived from silicon [Si/H] $=-1.21\pm 0.16$ (1$\sigma$ uncertainties), corresponding to $Z/Z_{\odot}=0.06^{+0.03}_{-0.02}$. Figure 5: The line profiles in the medium resolution 1400V and 1200R spectra are best modelled with two components separated by $148\pm 25$ km s-1 in velocity: $z_{a}=2.6396\pm 0.0003$, $z_{b}=2.6416\pm 0.0003$ (dotted lines), $b_{\textrm{turb, a}}=38\pm 7$ km s-1, $b_{\textrm{turb, b}}=23\pm 6$ km s-1, $b_{\textrm{th}}=0$ km s-1. The dashed curves show the $1\sigma$ errors. ### 3.3 Spectral energy distribution Extinction \| Model \| $\chi^{2}$ [dof] \| $E(B-V)$ \| $N_{\mathrm{H}}$ \| $\beta_{1}$ \| $\beta_{2}$ \| $E_{\textrm{break}}$ ---\|---\|---\|---\|---\|---\|---\|--- type \| \| \| (mag) \| ($10^{22}$ cm-2) \| \| \| (keV) \| PL \| 517.2[39] \| $0.10\pm 0.01$ \| $<0.03$ \| $0.82\pm 0.02$ \| - \| - SMC \| BPL \| 191.6[37] \| $0.06\pm 0.01$ \| $0.38^{+0.19a}_{-0.17}$ \| $0.71\pm 0.02$ \| $1.59\pm 0.11$ \| $1.40^{+0.09}_{-0.13}$ \| TBPL \| 232.0[38] \| $0.07\pm 0.01$ \| $0.27^{+0.18}_{-0.17}$ \| $\Gamma_{2}-0.5$ \| $1.23\pm 0.02$ \| $1.16^{+0.10}_{-0.12}$ \| PL \| 434.9[39] \| $0.17\pm 0.02$ \| $<0.22$ \| $0.91\pm 0.02$ \| - \| - LMC \| BPL \| 194.4[37] \| $0.10\pm 0.02$ \| $0.48^{+0.19}_{-0.18}$ \| $0.77\pm 0.03$ \| $1.59^{+0.11}_{-0.10}$ \| $1.41^{+0.10}_{-0.12}$ \| TBPL \| 238.7[38] \| $0.10\pm 0.02$ \| $0$ \| $\Gamma_{2}-0.5$ \| $1.28\pm 0.03$ \| $1.32\pm 0.11$ \| PL \| 546.9[39] \| $0.21\pm 0.02$ \| $<0.25$ \| $0.91\pm 0.03$ \| - \| - MW \| BPL \| 265.0[37] \| $<0.02$ \| $0.20^{+0.18}_{-0.16}$ \| $0.63^{+0.03}_{-0.01}$ \| $1.59\pm 0.11$ \| $1.39^{+0.09}_{-0.12}$ \| TBPL \| 323.1[38] \| $0.05\pm 0.03$ \| $0$ \| $\Gamma_{2}-0.5$ \| $1.20\pm 0.03$ \| $1.20\pm 0.10$ Table 4: Parameters resulting from a joint fit of the optical-to-X-ray SED at 66 min after the burst, assuming an absorbed PL, BPL or TBPL, see main text. The host galaxy dust extinction is modelled with a SMC, LMC or MW extinction law (Pei 1992) and the excess X-ray absorption is measured assuming solar metallicity. The best fitting model is highlighted. All the errors refer to a 90 per cent confidence level. a $N_{\mathrm{H}}=2.04^{+1.03}_{-0.95}\times 10^{22}$ cm-2 for $Z/Z_{\odot}=0.06$. Figure 6: The SED of the optical 300V spectrum ($\times$) and the X-ray spectrum (+), at 66 min after the burst. The solid line shows the best fitting model, a broken power law with spectral slopes $\beta_{\mathrm{X}}=1.59\pm 0.11$ and $\beta_{\mathrm{opt}}=0.71\pm 0.02$, for an SMC-type dust extinction, see Table 4. The residuals are displayed in the bottom panel. Figure 7: The optical and X-ray afterglow light curves of GRB 080210. The Swift/XRT X-ray light curve (at 1.73 keV) is plotted beneath the optical data. The ULTRASPEC $V$-band light curve is here plotted with a bin factor of 10. The $R$-band decay is derived from the VLT/FORS2 data. Late time points were excluded to avoid a possible break. The solid lines show the fit to the data, while the dotted lines extrapolate the fit to the complete datasets. The vertical line shows the SED time. The errors are 1$\sigma$. In order to fit the optical-to-X-ray SED, we combine the averaged 300V optical with the X-ray spectra. We choose to include the 300V spectrum in the SED to investigate the significant dust reddening reported by Fynbo et al. (2009). Also, the larger spectral window coverage of the 300V as compared to the higher resolution spectra makes it more suitable to investigate the SED. The SED time is chosen at the logarithmic mean between the two 300V observations (3945 s). Using the Swift spectrum repository (Evans et al., 2007, 2009), we extract the X-ray spectrum from a narrow time interval (3690–4200 s, logarithmically centred on the SED time), and use its count rate to scale the X-ray spectrum extracted from a larger time window (3690–106130 s) and with a better signal-to-noise (S/N). This approach assumes no spectral evolution in the X-ray spectrum, as confirmed by extracting spectra for different time windows and by the constant hardness ratio. This allows the optical and the X-ray spectra to be compared in flux. The 300V averaged spectrum was cleaned (absorption lines removed and frequencies bluer than Ly$\alpha$ excluded) and corrected for Galactic extinction. The optical spectrum was then binned into 22 bands (192 Å) in order to obtain the same number of data points as in the X-ray spectrum. The statistical errors were calculated from the variance in the spectrum. A systematic error introduced by the response function was calculated by measuring the amplitude of the spurious wiggles introduced by the flux calibration process. This uncertainty and the error due to the calibration against the VLT photometric measurement were then added in quadrature to the formal error. The total resulting error on the optical spectrum is about 8 per cent of the flux. The optical-to-X-ray SED is shown in Fig. 6. We model the SED from the optical to the X-rays with a single power law (PL), a broken power law (BPL) and a tied broken power law (TBPL), where the spectral slopes are tied to differ by $\Delta\beta=0.5$ to reproduce the spectral break (cooling frequency) expected for a synchrotron spectrum. The fit was performed with the Interactive Spectral Interpretation System (ISIS; Houck & Denicola, 2000) software, which allows all the data to be compared directly in count space. Working in count space has the advantage of not requiring any a priori model for the X-rays, otherwise needed for the conversion of the X-rays into flux (see e.g Starling et al., 2007). The host galaxy dust extinction was modelled with Small Magellanic Cloud (SMC), Large Magellanic Cloud (LMC) or Milky Way (MW) extinction curves (Pei, 1992) and the X-ray absorption (in excess of the Galactic $N_{\textrm{H,Gal}}=(5.59\pm 0.02)\times 10^{20}$ cm2; Kalberla et al., 2005) is measured from a _zphabs_ model in ISIS, assuming Solar metallicity. Table 4 summarizes the fit results. The best fitting model is a BPL with spectral slopes $\beta_{\mathrm{X}}=1.59\pm 0.11$ and $\beta_{\mathrm{opt}}=0.71\pm 0.02$, with the cooling frequency occurring at $E_{\textrm{break}}=1.40^{+0.09}_{-0.13}$ keV in the soft X-ray range. These values are derived for SMC-like extinction (lowest $\chi^{2}=191.6$, for 37 degrees of freedom), while the LMC extinction curve provides a very similar fit, resulting in consistent parameter values. The MW extinction curve and its related 2175 Å bump can be excluded. The optical spectrum is reddened by dust grains at the redshift $z=2.641$ of the host galaxy, with $E(B-V)=0.06\pm 0.01$ mag, or $A_{V}=0.18\pm 0.03$ mag (rest-frame), modelled with an SMC-like extinction curve. We find an excess X-ray absorption of $N_{\mathrm{H}}=0.38^{+0.19}_{-0.17}\times 10^{22}$ cm-2 assuming Solar metallicity, whereas $N_{\mathrm{H}}=2.04^{+1.03}_{-0.95}\times 10^{22}$ cm-2 for $Z/Z_{\odot}=0.06$. The above errors refer to a 90 per cent confidence level. The SED results should be treated with caution, because they are dependent on the slit loss correction of the 300V spectra. In particular, the same SED analysis, but for the optical spectrum that has not been corrected for slit losses, provides a 4 per cent change in the optical slope and 68 per cent in the $E(B-V)$, for the best fit model. ### 3.4 Afterglow evolution Time since GRB (hr) \| Instrument \| Magnitude ---\|---\|--- 0.61 \| VLT/FORS2 \| 18.74$\pm$0.05 1.69 \| VLT/FORS2 \| 19.57$\pm$0.05 55.70 \| Keck-I/LRISa \| 23.97$\pm$0.07 Table 5: The $R$-band photometry, not corrected for Galactic extinction (1$\sigma$ errors). a Perley & Bloom, private communication. Figure 7 shows the afterglow time evolution in the X-ray and optical bands. We converted the X-ray light curve into monochromatic flux at 1.73 keV, the logarithmic average of the XRT band, assuming a spectral slope $\beta_{\mathrm{X}}=1.59$ as derived from the SED (see Section 3.3). Early ($<12$ min) X-ray data were excluded from the fit to avoid the influence of flares. The last XRT data points were also excluded in order to avoid the contribution from a possible break at late time that cannot be constrained. We fit the X-ray light curve with a single power law with temporal slope $\alpha_{\mathrm{X}}=1.24\pm 0.07$ (reduced $\chi^{2}_{\nu}=3.67$ for 23 dof), noting that a broken power law does not improve the fit. The high $\chi^{2}_{\nu}$ could be produced by the wiggles observed in the X-ray light curve, possibly originated by micro-variability. However, the poor sampling of the X-ray light curve does not allow us to investigate this further. The $V$-band temporal decay was derived from a power-law fit to the ULTRASPEC light curve $\alpha_{V}=0.74\pm 0.07$, see Section 3.1. We collected the $R$-band photometric data points from our VLT/FORS2 acquisition images and the Keck-I/LRIS data, reported in Table 5, corrected them for Galactic extinction ($A_{V}=0.276$ mag; Schlegel, Finkbeiner & Davis, 1998), and converted them to flux density. A temporal decay with slope $\alpha_{\textrm{VLT+Keck}}=1.07\pm 0.07$ can be derived by a poor power-law fit to the three $R$-band data points (reduced $\chi^{2}_{\nu}=26.5$), in disagreement with the $V$-band decay. This suggest the presence of a break in the light curves at late times. Thus, the Keck data point was excluded from the temporal decay study to avoid the contribution from the possible break. The $R$-band decay derived from the two VLT data points has a temporal slope $\alpha_{R}=0.75\pm 0.09$, consistent with the $V$ band, where the error was calculated from the minimum and maximum slopes between the two points. ## 4 Discussion ### 4.1 Modelling the afterglow \| $\alpha_{\textrm{obs}}$ \| $\beta_{\textrm{obs}}$ \| Regime \| $\alpha(\beta)_{\textrm{exp}}$ \| $\beta(\alpha)_{\textrm{exp}}$ \| $p(\alpha)$ \| $p(\beta)$ \| $\sigma_{p(\alpha),p(\beta)}$ ---\|---\|---\|---\|---\|---\|---\|---\|--- Optical \| $0.75\pm 0.09$ \| $0.71\pm 0.01$ \| $\nu<\nu_{c}$ \| $1.07\pm 0.02$ \| $0.50\pm 0.06$ \| $2.00\pm 0.12$ \| $2.43\pm 0.02$ \| 3.5 X-rays \| $1.24\pm 0.07$ \| $1.59\pm 0.07$ \| $\nu_{c}<\nu$ \| $1.88\pm 0.11$ \| $1.16\pm 0.05$ \| $2.32\pm 0.09$ \| $3.18\pm 0.14$ \| 5.2 Table 6: The optical and X-ray temporal and spectral indices $\alpha$ and $\beta$ as observed and expected from the fireball model. We assume here an ISM scenario, with no extra energy injection and the slow cooling regime (e.g., Zhang et al., 2006). The errors are $1\sigma$. The electron energy distribution indices $p(\alpha)$, derived from the temporal slope, agree within $2.1\sigma$ between optical and X-rays, while $p(\beta)$ disagree at a $5.3\sigma$ level. The level of agreement between $p(\alpha)$ and $p(\beta)$ is indicated by $\sigma_{p(\alpha),p(\beta)}$. In order to investigate the physics of the GRB 080210 afterglow, we attempt to model it within the synchrotron scenario. In Table 6 the temporal slope $\alpha$ and the spectral slope $\beta$ are collected from both the optical and X-ray analysis (where $F_{\nu}\,{\propto}\,t^{-\alpha}\,\nu^{-\beta}$), as derived above. We first note that the $\beta_{\mathrm{X}}-\beta_{\mathrm{opt}}=0.88\pm 0.07$ disagrees with the $\Delta\beta=0.5$ expected from the fireball model. In particular, this implies that the spectral break is not a cooling break and that the optical and the X-ray emission are not produced by a coherent synchrotron process. Possibly, the optical radiation and the X-rays were emitted in different regions, the overall SED resulting from a composition of two synchrotron spectra. Alternatively, different radiative processes must be invoked to explain the SED. We further test fireball model predictions calculating the electron energy distribution index, $p$, from the temporal and spectral indices, see Table 6. We assuming a simple ISM, slow cooling scenario, with no extra energy injection (Zhang et al., 2006) and the cooling frequency in the soft X-rays, as derived by the SED fit. The electron indices derived from the temporal and spectral slope show a poor agreement ($3.5\sigma$) for the optical band, and disagreement ($5.2\sigma$) for the X-rays. Although the optical and X-ray temporal slopes provide a similar $p(\alpha)$ (within $2.1\sigma$), the electron indices derived from the spectral slopes disagree at a $5.3\sigma$ level between the optical and the X-rays. Thus, the closure relations are not satisfied for the case of GRB 080210. In particular, the X-ray spectral slope seems too steep to be produced by the synchrotron electron cooling expected in the model. One possible reason for this is that the X-ray spectral slope was overestimated due to the degeneracy with the spectral break and the X-ray absorption. However, using the optical data in the SED helps in breaking this degeneracy. This suggests that the the fireball model cannot properly reproduce the GRB 080210 afterglow and therefore cannot be applied to the data. An independent SED study of a larger sample of GRB afterglows shows similar results for GRB 080210 (Zafar et al., in preparation). ### 4.2 Variability The ULTRASPEC capability of observing at 1 s time resolution is a new frontier in the GRB afterglow variability study. But do we expect to see variability on those short time scales? How strong? And what processes can produce such variations? Answering these questions is essential to interpret not only the current light curve but also future observations with ULTRASPEC or any equivalent instrument. In order to address these questions and investigate the ULTRASPEC possibility of detecting fast variability, we analyse here the variability limits, derived by Ioka et al. (2005), based on kinematic arguments, showing that only certain time-scale fluctuations are physically allowed, at a particular observing time. These authors consider: (a) dips in the light curve, (b) bumps produced by density fluctuations, (c) a patchy- shell and (d) a refreshed shock. For the sake of clarity, we report below the limits from Ioka et al. (2005) that are relevant for this paper. (a) The fluctuations that could produce dips in the light curve are limited to $\frac{\|\Delta F_{\nu}\|}{F_{\nu}}\leq\frac{4}{5}\left(\frac{\Delta t}{t}\right)^{2}$ as derived from geometric constraints on the evolving emitting surface, considering causality arguments, relativistic effects and assuming a sudden shut off of the emission to obtain the upper limit on the variability. (b) Regardless of their properties, the density enhancements can decelerate the emitting matter, limiting the variability to $\frac{\|\Delta F_{\nu}\|}{F_{\nu}}\leq\frac{8}{5}\frac{\Delta t}{t}$ assuming the same geometric and causality arguments as above, and that the kinetic energy $E_{\textrm{kin}}$ is uniformly distributed in the variable volume. (c) In case of a patchy shell, the time scale of the fluctuations is initially constrained to grow linearly in time ($\Delta t\sim t$ Nakar & Oren, 2004), limiting the variability time scales to $\frac{\Delta t}{t}\geq 1$ for persistent angular fluctuations. (d) Refreshed shocks can produce bumps with time scales $\frac{\Delta t}{t}\geq\frac{1}{4}$ if the acceleration of the GRB ejecta is hydrodynamic, as a slow shell will expand with its co-moving sound speed and collide with the decelerating leading shock-front. If the emitting region is observed off-axis, i.e. when the line of sight is not aligned with the jet axis, and many regions ($>10^{3}$) contribute to the variability, the dips and density fluctuations, respectively, are limited to (a) $\frac{\|\Delta F_{\nu}\|}{F_{\nu}}\leq\frac{6}{\sqrt{2}}\left(\frac{\Delta t}{t}\right)^{3/2}$ (b) $\frac{\|\Delta F_{\nu}\|}{F_{\nu}}\leq 24\frac{\Delta t}{t}$ as derived by Ioka et al. (2005) from cases (a) and (b) above. Figure 8: Adapted from Ioka et al. (2005). The axes show the relative flux variation amplitude, $\|\Delta F_{\nu}\|/F_{\nu}$, versus the variability time scales over the time of observation, $\Delta t/t$. The solid lines reflect the variability limits derived from dips in the light curves (a), for bumps produced by density fluctuations (b), a patchy shell (c) and a refreshed shock (d). The dotted lines refer to the case of many fluctuation regions and off- axis observations, for dips (a) and for density fluctuations (b). The regions where variability is allowed by each process are indicated by the arrows. The GRB 080210 ULTRASPEC observation time-scale domain ($2.18$ s $<\Delta t<26.45$ min at mid-exposure time after the burst, $t=81.44$ min, in the observer frame), is enclosed by the dashed lines. The variability region that is both physically allowed and detectable by ULTRASPEC is highlighted. The variability limits discussed above are plotted in Fig. 8 (adapted from Ioka et al., 2005), where the regions of allowed variability are indicated by the arrows, for each process. We also indicate the ULTRASPEC observation time domain (outlined by the dashed lines) at mid-exposure time ($81.44$ min), considering the covered time scales above $2.18$ s ($0.03~{}{\textrm{ min}}<\Delta t<26.45$ min) and the instrument detection limit. This limit is calculated from the light curve S/N over a single data point (time bin unit $dt=1.09$ s) and extended to longer time scales ($\Delta F/F({n_{\textrm{bins}}})=\Delta F/F(\textrm{1 bin})/\sqrt{n_{\textrm{bins}}}$), where $n_{\textrm{bins}}$ is the number of time bins for each time scale. It is this detection limit that defines which fluctuations could possibly have been detected in the ULTRASPEC observations. The region where the allowed variability overlaps with the ULTRASPEC monitoring is highlighted in Fig. 8. Given the smoothness of the GRB 080210 light curve, we can exclude any variability in this region, as it is physically allowed but not detected by ULTRASPEC. A number of remarks can be deduced from Fig. 8. The fastest variability, both allowed and observable, can be produced by many density fluctuation regions (b, upper dotted line in Fig. 8). For a single density fluctuation region (b, solid line), the S/N of this ULTRASPEC observation can probe variability only on time scales $\Delta t>72.8$ s. While these limits can provide constraints on the fluctuation amplitudes to be expected in a standard afterglow, they cannot easily be applied to GRB 080210, as this afterglow does not seem to fit the synchrotron model. Nevertheless, the limits on dips in the light curve (a and a lines) do not depend strictly on the fireball model, they only assume a relativistically expanding shell, regardless of the particular model (Fenimore, Madras & Nayakshin, 1996). Thus, given the smoothness of the GRB 080210 light curve, we can limit the possible dips in the light curve to be weaker than 1 per cent in flux, on time scales longer than $\Delta t>9.2$ min. In case of many regions contributing to the dips in the light curve, we can exclude fluctuations stronger than 2 per cent of the flux on time scales $\Delta t>2.3$ min. These constraints are derived from the intersection between the ULTRASPEC detectability limit and the theoretical limits on light curve dips (a and a* lines). These limits on the variation amplitude can be interpreted in terms of under-density of the circumburst region within the fireball model. However, we cannot apply this to the case of GRB 080210 because of its non-standard afterglow physics. Finally, with the current ULTRASPEC dataset, refreshed shocks could in principle have been detected on time scales $20.00$ min $<\Delta t<26.45$ min, but they were not observed. ### 4.3 Host galaxy environment #### 4.3.1 Gas location, metallicity and dust The spectroscopy of the optical afterglow reveals a number of absorption lines due to neutral and low-ionization species, i.e. O i, Si ii, C ii, Fe ii, Al iii and Zn ii, which can be used to investigate the properties of the absorbing region. The ionization potential of O i (13.618 eV) is very close to that of H i (13.598 keV): this already suggests that the two species could be co-spatial. Indeed, in low ionization media O i and H i tend to couple due to charge exchange (Field & Steigman, 1971). The ionization potentials of neutral Si, C, Fe, Al and Zn are well below 13.618 eV. Thus, all the observed species may, in principle, coexist in the same region. However, since the oxygen and hydrogen lines that we detect in the spectrum are saturated, their profiles cannot be used to compare the kinematics. On the other hand, the Voigt profile fit to the Fe ii, Si ii and Al ii transitions indicates that these ions share the same two-component profile. Furthermore, they have comparable ionization potentials. These two pieces of evidence strongly suggest that these species are co-spatial. Al iii is mildly ionized, and therefore belonging to a different gas-phase; however, its double velocity profile indicates that Al iii is still related to the rest of the gas, possibly surrounding the bulk of the H i. Regarding the distance of the burst to the absorber, to first order, we can exclude that the lines are produced in the close vicinity of the GRB, as ionization is expected to occur inside $\sim$10 pc (see eg. Ledoux et al., 2009). The Mg i is a possible distance limit indicator ($>50$ pc; Prochaska, Chen & Bloom, 2006), but none of its transitions were covered by the observations. The actual distance of the bulk of the gas may be much larger. Indeed, absorption systems have been found up to several kpc from the burst (Vreeswijk et al., 2007; D’Elia, 2009; Ledoux et al., 2009), where the distance was computed using a photo-excitation (UV pumping) model of the fine- structure line variability. We also detect fine structure lines (i.e. Si ii, C ii and Fe ii* in the 1400V and 300V grisms, see also Fynbo et al., 2009), but the low resolution of the FORS spectra does not allow any further modelling. From the DLA profile fit, we derived a neutral hydrogen column density, log $(N_{\mathrm{H\,{\sc I}}}/$cm${}^{-2})=21.90\pm 0.10$. This fairly high column density, compared to the low-resolution afterglow sample analysed by Fynbo et al. (2009), causes the neutral hydrogen to screen heavier elements (present in the same gas with much lower abundances) from ionization. Thus, we assume that no ionization effects can significantly influence the metallicity estimate. The best metallicity indicator between the optical absorption lines that we detected is Si ii $\lambda$1808, for which we find [Si/H$]=-1.21\pm 0.16$ ($Z/Z_{\odot}=0.06^{+0.03}_{-0.02}$). This suggests a chemically poor environment, quite common for GRBs with bright optical afterglows, where metallicities fall below $0.3Z_{\odot}$ for most absorbers (Fynbo et al., 2006). From the X-rays, we derive an equivalent hydrogen column density of log $(N_{\mathrm{H}}/$cm${}^{-2})=21.58^{+0.18}_{-0.26}$, assuming Solar abundances, whilst log $(N_{\mathrm{H}}/$cm${}^{-2})=22.31^{+0.18}_{-0.27}$ for $Z/Z_{\odot}=0.06$. The soft X-ray absorption is normally produced by metals in the line-of-sight, e.g., carbon and oxygen (Wilms et al., 2000). The equivalent hydrogen column density measured from the X-ray absorption in GRB 080210 is comparable to the neutral hydrogen column density log $(N_{\mathrm{H}}/$cm${}^{-2})=21.90\pm 0.10$ from the Ly$\alpha$. However, in general the equivalent and neutral hydrogen column densities correlate extremely poorly (Watson et al., 2007). We find a visual extinction, $A_{V}=0.18\pm 0.03$ mag, from the SED fitting and interpret it as due to dust. This value is quite common in GRB afterglows (Kann et al. 2010; Schady et al. 2010; Zafar et al., in preparation.). An SMC extinction law best reproduces the dust extinction that affects the GRB 080210 afterglow spectrum and a MW extinction law can be excluded. Consistent with this, we do not observe the 2175 Å bump (7919 Å in the observer frame), which is a typical signature of the Galactic dust absorption (Pei, 1992). Even though such a structure has been observed in GRB afterglows (Krühler et al., 2008; Elíasdóttir et al., 2009; Perley et al., 2010), it has been shown that in most GRB host galaxies, dust typically displays an SMC extinction curve (e.g., Starling et al., 2007; Kann et al., 2010). While a Galactic extinction law requires roughly the same amount of graphite and silicate grains, the SMC curve can be produced by silicate grains alone (Pei, 1992). Thus, we infer a low graphite dust content for the GRB 080210 host galaxy. Although the presence of dust is expected in DLAs (Pettini et al., 1997), the low metallicity disfavours the production of dust grains, as shown by the relation between dust and metallicity (Vladilo, 1998). ### 4.4 Origin of the intervening system The intervening system at $z=2.508$ would require a relative velocity $v\sim 11,000$ km s-1, if associated with the GRB host galaxy. Velocities up to $3,000$ km s-1 have been observed in GRB afterglow spectra (e.g., Mirabal et al., 2003) or expected by Wolf Rayet wind models (van Marle et al., 2005). However, none of the proposed scenarios seems to be able to reproduce $\sim 10,000$ km s-1. An intriguing possibility is that such an outflow could be accelerated by an active galactic nucleus (AGN). About half of the C iv and Mg ii narrow-line absorbers towards QSOs with apparent outflow velocities of 3,000–12,000 km s-1 are actually intrinsic to the QSO/host (Wild et al., 2008). We cannot exclude the possibility that the host galaxy of GRB 080210 is a low-luminosity AGN, since its emission would fall well below the detectability limit of X-ray telescopes. However, it is unlikely that a $\sim 10,000$ km s-1 fast outflow could remain as narrow in velocity as we observe ($b_{\textrm{turb}}\sim 30$ km s-1). In addition, if the faster outflows occur in the polar direction of an axisymmetric accretion geometry, it might be difficult to locate a star forming region hosting the burst between the nucleus and the accelerated absorbing material without invoking a fine tuned geometry. Moreover, the fastest outflows are typically located very close to the AGN itself and the burst location is unlikely to cross them on the line of sight. Thus, the most favoured origin of the intervening system at $z=2.509$ is an absorber on the line of sight, a cloud or a galaxy $\sim 43$ Mpc from the GRB host galaxy and unrelated to it. Statistically, a significant incidence of intervening systems is expected. Given the number density per unit redshift interval of intervening absorbers, not associated with the host galaxy, the probability of detecting at least one random C iv absorber of rest-frame EW$(\lambda 1548)>0.4$ Å is 34 per cent (Chen et al., 2007). ## 5 Summary We searched for short-term variability, down to 2.18 s, in the ESO 3.6-m/ULTRASPEC observations of the GRB 080210 optical afterglow. The light curve decays as a power law ($\alpha=0.74\pm 0.07$) and appears smooth on all time scales. Nevertheless, the time-monitoring allows us to investigate the circumburst environment and the blast-wave propagation. Comparing our observation with the variability limits derived by Ioka et al. (2005), we can exclude dips in the light curve with amplitude stronger than 1 per cent of the flux on time scales $\Delta t>9.2$ min and stronger than 2 per cent on time scales $\Delta t>2.3$ min, for a single or many under-dense regions respectively. The GRB 080210 optical and X-ray late afterglows decay with temporal slopes $\alpha_{\mathrm{opt}}=0.75\pm 0.09$ and $\alpha_{\mathrm{X}}=1.24\pm 0.07$. The spectral slopes $\beta_{\mathrm{opt}}=0.71\pm 0.01$ and $\beta_{X}=1.59\pm 0.07$ are derived from the joint optical-to-X-ray SED fit with a broken power law ($1\sigma$ errors). We evaluate these observations with the theoretical expectation of the standard model and find no agreement within $5.3\sigma$, suggesting that the GRB 080210 afterglow cannot be produced with the fireball model physics. From the SED analysis, we find that the spectral break is located in the soft X-ray at $E_{\textrm{break}}=1.40^{+0.09}_{-0.13}$ keV, while the X-rays absorption indicates an excess equivalent hydrogen absorption of log $(N_{\mathrm{H}}/$cm${}^{-2})=21.58^{+0.18}_{-0.26}$ assuming Solar abundances, and log $(N_{\mathrm{H}}/$cm${}^{-2})=22.31^{+0.18}_{-0.27}$ for $Z/Z_{\odot}=0.06$ (90 per cent confidence level errors). Optical reddening ($A_{V}=0.18\pm 0.03$ mag) is induced by SMC-like dust (low graphite content). In the optical VLT/FORS2 spectra, we detect several metal absorption lines associated with the GRB host galaxy ($z=2.641$), as well as a DLA (log $(N_{\mathrm{H\,{\sc I}}}/$cm${}^{-2})=21.90\pm 0.10$). We find [Si/H] $=-1.21\pm 0.16$ ($Z/Z_{\odot}=0.06^{+0.03}_{-0.02}$) suggesting a low metallicity environment. A Voigt-profile fit of the medium resolution lines reveals a two-component profile, separated by $148\pm 25$ km s-1, possibly associated with two major clouds along the line of sight within the host galaxy. GRB 080210 represents one of the first attempts to study fast variability in GRB afterglows. Although this particular case must be treated with caution, due to its non-standard afterglow physics, our analysis demonstrated that the expected short-term can be detected by using the high speed read-out of the ULTRASPEC camera, specially for bright afterglows with higher S/N. ## Acknowledgements ADC acknowledges support from the University of Iceland Research Fund. PJ acknowledge support by a Marie Curie European Re-integration Grant within the 7th European Community Framework Program and a Grant of Excellence from the Icelandic Research Fund. The Dark Cosmology Centre is funded by the Danish National Research Foundation. VSD, SPL and TRM are supported by STFC. ULTRASPEC was funded by the EU-OPTICON programme. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. 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(2007) Zhang B., Liang E., Zhang B., 2007, ApJ, 666, 1002	arxiv-papers	2010-11-18T17:22:46	2024-09-04T02:49:15.011404	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "A. De Cia, P. Jakobsson, G. Bj\\\"ornsson, P. M. Vreeswijk, V. S.\n Dhillon, T. R. Marsh, R. Chapman, J. P. U. Fynbo, C. Ledoux, S. P.\n Littlefair, D. Malesani, S. Schulze, A. Smette, T. Zafar and E. H.\n Gudmundsson", "submitter": "Annalisa De Cia", "url": "https://arxiv.org/abs/1011.4239" }
1011.4248	# Physical Conditions in the Broad Line Region of $z\sim 3$ Quasars: A Photoionization Method to Derive $r_{\rm BLR}$ and $M_{\rm BH}$ ♣♣affiliationmark: C. Alenka Negrete11affiliation: anegrete@astroscu.unam.mx, deborah@astroscu.unam.mx and Deborah Dultzin11affiliation: anegrete@astroscu.unam.mx, deborah@astroscu.unam.mx Instituto de Astonomía, Universidad Nacional Autónoma de México, Mexico Paola Marziani22affiliation: paola.marziani@oapd.inaf.it INAF, Astronomical Observatory of Padova, Italy Jack Sulentic33affiliation: sulentic@iaa.es Instituto de Astrofísica de Andalucía, Spain ###### Abstract We present high S/N UV spectra for eight quasars at $z\sim 3$ obtained with VLT/FORS. The spectra enable us to analyze in detail the strongest emission features in the rest-frame range 1400-2000 Å of each source (Ciii]$\lambda$1909, Siiii]$\lambda$1892, Aliii$\lambda$1860, Siii$\lambda$1814, Civ$\lambda$1549 and Siiv$\lambda$1397). Previous work indicates that a component of these lines is emitted in a region with well- defined properties i.e., a high density and low ionization emitting region). Flux ratios Aliii$\lambda$1860/ Siiii]$\lambda$1892, Civ$\lambda$1549/Aliii$\lambda$1860, Siiv$\lambda$1397/ Siiii]$\lambda$1892, Civ$\lambda$1549/Siiv$\lambda$1397 and Siii$\lambda$1814/ Siiii]$\lambda$1892 for this region permit us to strongly constrain electron density, ionization parameter and metallicity through the use of diagnostic maps built from CLOUDY simulations. Reliable estimates of the product density times ionization parameter allow us to derive the radius of the broad line region $r_{\rm BLR}$ from the definition of the ionization parameter. The $r_{\rm BLR}$ estimate and the assumption of virialized motions in the line emitting gas yields an estimate for black hole mass. We compare our results with estimates obtained from the $r_{\rm BLR}$ – luminosity correlation customarily employed to estimate black hole masses of high redshift quasars. galaxies: active — galaxies: high-redshift — quasars: general — quasars: emission lines ††slugcomment: To appear in …00affiliationtext: Based on observations made with ESO Telescopes at Paranal Observatory under programme ID 078.B-0109(A) ## 1 Introduction ### 1.1 Interpreting Quasar Spectra Measuring relevant physical parameters from the observed broad-line spectra of quasars is still an open challenge. Identification and intensity measurements of the strongest emission lines has made possible a rough inference of the typical conditions of the emitting gas since the earliest days of quasar spectroscopy. The very first quasars of intermediate redshift discovered in the 1960s showed a fairly high ionization spectrum, with prominent lines of Civ$\lambda$1549, and Heii$\lambda$1640 in addition to strong Balmer lines of the low-redshift quasars. Photoionization by the central continuum source was considered the preeminent heating mechanism of the emitting gas. Significant Ciii]$\lambda$1909 emission suggested electron densities ($n_{\rm e}$) in the range $10^{9}-10^{10}$ cm-3. The observed intensity ratio Ciii]$\lambda$1909/Civ$\lambda$1549 indicated ionization parameter (defined by Eq. 1 later in this paper) values of the order of $10^{-1}$. This photoionization scenario was successful in explaining at least some quasar optical and UV spectra (see the review by Davidson and Netzer 1979 for a synopsis). More recent work emphasized the existence of several problems with this simple scenario. Low ionization lines (LILs), and especially Feii are too strong to be explained by a photoionized region of moderate density and column density (see for example Dumont and Mathez 1981; Joly 1987; Collin-Souffrin et al. 1988; Dumont and Collin-Souffrin 1990). These authors stressed that the LILs required a denser, low-temperature environment. We unfortunately lack a simple well-defined diagnostic measure of physical conditions in the broad line region. One strategy for estimating electron density in astrophysical sources involves using two emission lines of the same ion, and with similar energies above the ground level, but with different radiative transition probabilities $A_{ki}$. In practice one often chooses two lines from the same term where one forbidden/semi-forbidden transition is associated with another that is semi-formidden/permitted in order to ensure very different values of $A_{ki}$ (for example, [Siiii]$\lambda$1882 and Siiii]$\lambda$1892). This technique is not straightforwardly applicable to the broad lines of quasars, precisely because the lines are broad, suitable candidates are too closely spaced in wavelength, and density is at least an order of magnitudes higher than the critical density for the forbidden transitions used in spectra of planetary nebulæ and Hii regions. In addition the S/N and the resolution of quasar spectra are usually not very high. Feibelman and Aller (1987) used the Ciii]$\lambda$1909/ Siiii]$\lambda$1892 ratio to study the rather high-density environment typical of symbiotic stars ($n_{\rm e}$$\sim 10^{7-10}$ cm-3). In this case we have two semi-forbidden (intercombination) resonance lines with significantly different transition probabilities (see Table References). The lines are emitted by two iso- electronic species with somewhat different ionization potentials (11 eV for C+ vs. 8 eV for Si+). The ionic fraction is dependent on the relative abundance of silicon-to-carbon (to be assumed) as well as on the ionization structure within the emitting region (to be computed). Using lines of different ions introduces additional potentially serious sources of uncertainty. It is perhaps not surprising that most workers believe that $n_{\rm e}$ in the BLR cannot be reliably estimated using quasar spectra. Strong Ciii]$\lambda$1909 emission would imply that $n_{\rm e}$ cannot be very high ($n_{\rm e}$$\sim 10^{11-13}$cm-3). Very high density was invoked to explain the rich low ionization spectrum (especially Feii) seen in the spectra of most quasars. Several lines in the UV spectrum of I Zw 1 point towards high density at least for the LIL emitting zone: prominent Feii, relatively strong Aliii$\lambda$1860, and detection of Ciii$\lambda$1176 (Baldwin et al. 1996; Laor et al. 1997b). The region where these lines are produced cannot emit much Ciii]$\lambda$1909\. But is Ciii]$\lambda$1909 really so strong in most quasars? BLR conditions are certainly complex and a single emitting region is not sufficient to explain both LILs and high ionization lines (HILs). ### 1.2 Quasar Systematics Quasar spectra are not all alike. There are significant differences in line intensity ratios and broad line profiles from object to object (Bachev et al. 2004; Marziani et al. 2010). More importantly, these differences can be organized in a systematic way as has been realized since the early 1990s (Boroson and Green 1992). Since then several authors stressed the importance of the so-called eigenvector 1 (E1) of quasars (e.g. Gaskell et al. 1999). Sulentic et al. (2000, 2007) expanded the E1 trends into a 4-dimensional space involving optical, UV and X-ray measures. They also defined spectral types along a sequence occupied by AGN in an optical plane involving Feii and FWHM H$\beta$ parameters. Objects at extreme ends of the E1 sequence are very different at almost all wavelengths and median spectra computed in spectral bins within this plane emphasize systematic changes in broad line properties (Sulentic et al. 2002, 2007). The differences have motivated the suggestion of a possible dichotomy between Narrow Line Seyfert 1 (NLSy1s) like sources and broader line objects that include almost all radio-loud quasars. The most effective divider of the two quasar types appears to be at FWHM of the H$\beta$ broad component H$\beta_{\mathrm{BC}}\approx$ 4000 km s-1 for low-to- moderate luminosity sources (Marziani et al. 2009). This corresponds to to Eddington ratio $L/L_{\mathrm{Edd}}\sim 0.2\pm$0.1 (Marziani et al. 2003b). The distinction between NLSy1-like objects (hereafter Population A sources with FWHM(H$\beta$) $\leq$ 4000 km s-1) and the rest of quasars (Population B with FWHM(H$\beta$) $\gtrsim$ 4000 km s-1) is of special relevance here. Pop. A sources show relatively low equivalent width lines with e.g. the $\lambda$1900 blend $\sim$30 Å. A close analysis of the Ciii]$\lambda$1909 blend in the prototypical NLSy1 I Zw 1 shows strong Siiii]$\lambda$1892, and Aliii$\lambda$1860 blended with rather weak Ciii]$\lambda$1909 along with prominent Feiii(UV34)($\lambda$1895.5, $\lambda$1914.0, and $\lambda$1926.3) and Feii blends (Laor et al. 1997b; Vestergaard and Wilkes 2001). The Ciii]$\lambda$1909 line is apparently so weak that Feiii$\lambda$1914 may be the most prominent feature at $\lambda\approx$ 1910Å (cf. Hartig & Baldwin 1986). This interpretation is confirmed by detailed deblending of a source (SDSS J120144.36+011611.6) that can be considered a high luminosity analog of I Zw 1 (§7.1). Median composite UV spectra of low-$z$ quasars show that Aliii$\lambda$1860 and Siiii]$\lambda$1892 are more prominent in 4DE1 spectral types A2 and A3. Bins A1, A2, A3 are defined in terms of increasing FeII$\lambda$4570 (see Fig. 1 of Sulentic et al. 2002). I Zw 1, although belonging to the extreme type A3 (Bachev et al. 2004) is not unique as a NLSy1, since the fraction of Pop. A sources in bin A3 and A4 is $\approx$ 20 % of all Pop. A sources in the sample of Zamfir et al. (2010). A significant number of extreme NLSy1 are not classified as quasars in SDSS and must be collected from the galaxy catalog (Hu et al. 2008). ### 1.3 Emission Line Diagnostics and BLR Properties Emission lines and line ratios are used in diagnostic maps to estimate temperatures and electronic densities in galactic and extragalactic photoionized regions. Examples include HII-regions and galaxies with HII- region nuclear spectra where electron densities are less than $n_{\rm e}$$\approx 10^{4}$ cm-3. This method has been successfully applied to the narrow line region (NLR) in AGN. Application to the broad line quasars has yielded results that are difficult to interpret. One recent exception is the work by Maksuoka et al. (2008) who succeeded in analyzing the partly ionized region thought to emit most of the LILs in quasar spectra. They suggest that Oi 8446 and the Caii triplet are emitted by dense, low ionization gas probably located in the periphery of the BLR. If the electron density and ionization conditions are known it is possible to derive, with additional assumptions, the distance of the BLR emitting region from the central continuum source (as stressed earlier also by Baldwin et al. 1996). The physical conditions of photoionized gas can be described by electron density $n_{\rm e}$, hydrogen column density $N_{\rm c}$, metallicity (normalized to solar), shape of the ionizing continuum, and the ionization parameter $U$. The latter represents the dimensionless ratio of the number of ionizing photons and the electron density $n_{\rm e}$ or, equivalently, the total number density of hydrogen $n_{\mathrm{H}}$, ionized and neutral.111In a fully ionized medium $n_{\rm e}$$\approx 1.2$ $n_{\mathrm{H}}$. We prefer to adopt the definition based on $n_{\mathrm{H}}$ because it is the one employed in the CLOUDY computations. Both $U$ and $n_{\mathrm{H}}$ are related through the equation $U=\frac{\int_{\nu_{0}}^{+\infty}\frac{L_{\nu}}{h\nu}d\nu}{4\pi n_{\mathrm{H}}cr^{2}}$ (1) where $L_{\nu}$ is the specific luminosity per unit frequency, $h$ is the Planck constant, $\nu_{0}$ the Rydberg frequency, $c$ the speed of light, and $r$ can be interpreted as the distance between the central source of ionizing radiation and the line emitting region. Note that $Un_{\mathrm{H}}c$ is the ionizing photon flux $\Phi(H)=\frac{Q(H)}{4\pi r^{2}}.$ (2) If we know the product of $n_{\mathrm{H}}$ and $U$, we can estimate the radius $r$ of the BLR from Eq. 1. The dependence of $U$ on $r_{\rm BLR}$ was used by Padovani & Rafanelli (1988) to derive central black hole masses assuming a plausible average value of the product $n_{\mathrm{H}}$$\cdot U$. The typical value of $n_{\rm e}$ was derived at that time from semiforbidden line Ciii]$\lambda$1909 which implied that the density could not be much higher than $n_{\rm e}$$\approx 10^{9.5}$ cm-3 (Osterbrock & Ferland 2006). Padovani (1988) derived an average value $<U\cdot$ $n_{\rm e}$$>\approx 10^{9.8}$ from several sources where $r_{\rm BLR}$ had been determined from reverberation mapping, and for which the number of ionizing photons could be measured from multiwavelength observations. The average value was then used to compute black hole masses for a much larger sample of Seyfert 1 galaxies and low-$z$ quasars (Padovani & Rafanelli 1988; Padovani, Burg & Edelson 1989). Wandel, Peterson & Malkan (1999) compared the results of the photoionization method with the ones obtained through reverberation mapping, found a very good correlation for the masses computed with the two methods, and concluded that “both methods measure the mass of the central black hole.” ### 1.4 Outline of the paper The importance of the product $U\cdot$$n_{\mathrm{H}}$ goes beyond knowledge of the physical conditions in the BLR if it can lead to estimates of BLR radius and black hole mass. This paper identifies suitable emission line ratios that overcome some of the major problems in the analysis of emission lines of quasars. It also defines a photoionization method that can be applied to even the highest redshift quasars making use of high S/N near-IR spectroscopic data. In §2 we present the spectra of 8 pilot sources obtained with the VLT/FORS; in §3 we discuss data reduction; In §4 we describe our method of fitting broad emission line profiles; in §5 we give a phenomenological interpretation of the profile fits; in §6 we describe a method for deriving BLR physical conditions and give the results of our fits; in §7 we discuss two sources not belonging to our sample that show extreme behavior and that are helpful to understand more common quasar spectra; in §8 we give the results for the photoionization method; in §9 we derive the radius of the BLR (its distance from the ionizing source) and the mass of the black hole for each quasar in our sample; in §10 we discuss our results; finally, in §11 we summarize our results and the prospect of a more extended application of our method. All the computations were made considering $H_{0}$ =70 km s-1 Mpc-1 and a relative energy density $\Omega_{\Lambda}=0.70$ and $\Omega_{\mathrm{M}}=0.3$. ## 2 Observations Data were obtained between Nov. 2006 and Jan. 2007 using the VLT2/FORS1 telescope operated in service mode. FORS1 is the visual and near UV focal reducer and low dispersion spectrograph of the Very Large Telescope (VLT) operated by European Southern Observatory (ESO) (Appenzeller et al. 1998). Our VLT sample consists of 8 quasars with $z\sim 3$. In Figure 1 we show the spectra uncorrected for redshift. Tab. 2 provides a log of observations that is organized as follows. Column 1: object name, Col. 2: apparent B magnitude, Col. 3 redshift, Col. 4: line(s) used for redshift estimation: a) Oi$\lambda$1304.8, b) Ciii]$\lambda$1909; Col. 5: absolute B magnitude, Col. 6 flux at 6 cm taken from FIRST (Far InfraRed and Submillimetre Telescope), Col. 7: date (refers to time at start of exposure), Col. 8: Digital Integration Time, Col. 9: number of exposures with integration time equal to DIT, Col. 10: airmass at the beginning of each exposure, Col. 11 $S/N$ in the continuum around 1700Å. The observation of one of our 8 quasars, J00521-1108, yielded only a low S/N spectrum which we retain because observed features in the blend at $\sim 1900$ Å are clear enough to fit the individual lines. Two sources, J01225+1339 and J02287+0002, are BAL quasars. We will keep them separate because Civ$\lambda$1549 is severely affected by absorption. ## 3 Data Reduction Data were reduced using standard iraf tasks. All spectra were wavelength and flux calibrated in the observed frame and then corrected for Galactic extinction. Flux correction was applied using meteorological data provided by ESO. The observed flux was multiplied by the inverse of the light lost computed from the ratio seeing over slit width in arcsec. Correction to rest frame requires estimating the redshift $z$ which is not a trivial task as outlined below. Rest frame correction also involved scaling the specific flux in flux per unit wavelength interval by a factor $(1+z)^{3}$. Measurements were carried out on the rest-frame spectra. It is necessity to describe below two important aspects of the data reduction. ### 3.1 A & B Atmospheric Bands Correction The A or B atmospheric band falls on top of the 1900Å blend in many of the spectra. This is an important region for this study especially because it involves Siiii]$\lambda$1892, Aliii$\lambda$1860, and Siii$\lambda$1814\. In order to remove these absorption features we created an A+ B band template from standard star spectra used as specific flux calibrators. We scaled this template to find a best fit. Fig. 1 shows the A and B absorption correction where we make a line identification to illustrate which lines are affected. In cases where the A or B bands overlap a weak line like Siii$\lambda$1814 the effect is considerable and measures of Siii$\lambda$1814 should not be considered at all or with extreme care. This happens for sources J00103-0037, J03036-0023, and J20497-0554. In cases where one of the bands overlaps a stronger line like Siiii]$\lambda$1892 or Aliii$\lambda$1860, the correction was good enough to permit accurate measures. ### 3.2 Redshift Estimate Normally one uses strong narrow emission lines to set the rest frame in a quasar. In our case no strong narrow lines are available so we consider the peaks of Ly$\alpha$, Civ$\lambda$1549 and Ciii]$\lambda$1909\. The Ly$\alpha$ peak is affected by absorption and Civ$\lambda$1549 is a HIL feature often showing blueshifts and/or asymmetries (Gaskell 1982, Espey et al. 1989, Corbin 1990, Tytler & fan 1992; Marziani et al. 1996; Richards et al. 2002, Baskin & Laor 2005; Sulentic et al. 2007). This is especially true in Pop. A sources. Ciii]$\lambda$1909 is blended with Siiii]$\lambda$1892 and Feiii that is especially prominent in this region and could well affect the peak. Pop. B sources show a rather weak Fe spectrum making the Ciii]$\lambda$1909 peak a more reliable $z$ estimator. Our best option is to use the low ionization line Oi$\lambda$1304 whenever it is strong. However it is blended with low ionization Siii$\lambda$$\lambda$1304,1309 (${}^{2}P^{0}_{3/2,1/2}-^{2}S_{1/2}$). Both Oi$\lambda$1304 and Siii$\lambda$$\lambda$1304,1309 are broad lines and in Pop. B sources might show large redshifts or even significant blueshifts. Simulations in the ($n_{\mathrm{H}}$, $U$) region of interest show Oi$\lambda$1304 $\approx$ 2 Siii$\lambda$1304,1309 and this is confirmed in the spectrum I Zw 1 where Oi$\lambda$1304 and Siii$\lambda$$\lambda$1304,1309 are resolved. The two components of the Siii$\lambda$$\lambda$1304,1309 doublet are set to the same intensity (i.e., we assume an optically thick case). We model the blend Oi$\lambda$1304 + Siii$\lambda$$\lambda$1304,1309 with 5 Gaussians; the three components of the Oi feature are produced by Bowen florescence mechanism, and should show ratios consistent with their transition probabilities. Generating a model spectrum in iraf (lines broadened to 4000 km s-1) yields a rest frame peak wavelength of 1304.8 $\pm$ 0.2 Å (in vacuum) which we use as a reference for our VLT spectra assuming that there is no hint of systematic BC shifts as is the case for all Pop. A sources (the majority in our sample) and many Pop. B sources (Marziani et al. 2003a). Examination of Fig. 2 reveals that the peak of Oi$\lambda$1304 in source J00521-1108 is not observed clearly. We use Ciii]$\lambda$1909 to set the rest frame in this case. There are other sources J00103-0037, J02287+0002, J02390-0038 and J20497-0554 where the redshift estimation using both Oi$\lambda$1304 and Ciii]$\lambda$1909 are not in good agreement. The largest disagreement was found for J02287+0002. Redshifts obtained for the three remaining quasars, J01225+1339, J03036-0023 and J23509-0052, were obtained from Oi$\lambda$1304\. Fig. 3 shows the deredshifted VLT-FORS spectra for our sample of 8 quasars. ## 4 Data Analysis ### 4.1 Methodological Considerations on Multicomponent Fits The specfit IRAF task (Kriss 1994) allows us to fit the continuum, emission and absorption line components, Feii and Feiii blends, etc. We fit two spectral ranges: (1) 1450–1680 Å for analysis of Civ$\lambda$1549 and (2) 1750–2050 Å for analysis of the 1900 Å blend. Significant Feii and Feiii emission are expected close to and underlying the 1900 Å blend. Study of the 1900 Å blend is especially difficult in quasars because the lines are broad and the blending severe. We therefore need to take advantage of several previous results. ### 4.2 Feii and Feiii Emission Our approach is completely empirical and employs an Feii \+ Feiii template taken from templates successfully used in previous works. Our Feii template is based on a CLOUDY simulation and is not very far from the preferred model of Bruhweiler & Verner (2008). Feiii lines are common and strong in the vicinity of Ciii]$\lambda$1909 as is evidenced by their presence in average LBQS (Francis et al. 1991) and SDSS (Vanden Berk et al. 2001) spectra. They appear to be strong when Aliii$\lambda$1860 is also strong (Hartig and Baldwin 1986). Vestergaard & Wilkes (2001) produced an Feiii template based on the UV spectrum of I Zw 1. Since then, Sigut et al. (2004) have modeled the Feiii BLR spectrum. See also Verner et al. (2003) for a plot of emission around 1900 Å. Ly$\alpha$ pumping enhances Feiii (UV 34)$\lambda$1914.0 (Johansson et al. 2000) and this line can be a major contributor to the blend right on the red side of Ciii]$\lambda$1909 (see Fig. 2 of Vestergaard & Wilkes). We reproduced the option B of the empirical Feiii template of Vestergaard & Wilkes (2001), taking advantage of the line identifications from Ekberg (1993). When detected we can use Feii UV 191 to set a rough Feii level while the feature at 2080 Å is helpful for a more precise estimation of the intensity of Feiii. The continuum was fitted using the regions around 1450Å (1750 and 1960Å) that are relatively free of Feii emission (Vanden Berk et al. 2001). We used the same power-law to describe the continuum at both the Civ$\lambda$1549 and 1900Å regions. ### 4.3 Line Components We base our specfit analysis on several previous observational results. The most important ones are as follows: * • Sulentic et al. (2002) gridded the broad component of FWHM H$\beta$BC versus $R_{\rm FeII}$=W(Feii$\lambda$4570blend)/W(H$\beta$BC) parameter plane into bins of fixed $\Delta$ FWHM = 4000 km s-1 and $\Delta$ $R_{\rm FeII}$= 0.5. Quasar spectra in different bins are different in many measures. As mentioned earlier, the largest differences are found between NLSy1-like objects, Pop. A, and broader sources of Pop. B with FWHM(H$\beta$) $\gtrsim$ 4000 km s-1. The gridding of Sulentic et al. (2002) is valid for low $z$ ($<$0.7) quasars. At higher $z$ an adjustment must be made since no sources with FWHM H$\beta_{BC}<3500$ km s-1 exist above luminosity $\sim 10^{48}$ ergs s-1 (Marziani et al. 2009). * • Median spectra were computed for spectral bins from the atlas of Marziani et al. (2003a) who found that H$\beta$ can be described by a Lorentz function in Pop. A sources and by the sum of 2 Gaussians in Pop. B sources (unshifted + broader redshifted components) (Zamfir et al. 2010; Marziani et al. 2010). * • A careful Feii subtraction reveals a blue-shifted H$\beta$ component in some bin A3 sources (i.e. the stongest Feii emitters; Zamfir et al. 2010). * • Civ$\lambda$1549 (HIL) and H$\beta$ (LIL) profiles show significant differences in Pop. A. Large Civ$\lambda$1549 blueshifts ($\lesssim-1000$ km s-1) are observed in Pop. A only (Sulentic et al. 2007). HIL and LIL profiles are more similar in Pop. B sources. * • We do not have H$\beta$ observations for our high-$z$ objects since there are no near IR spectra. We use the results of Marziani et al. (2003b, 2010): they show that the BC of Siiii]$\lambda$1892, Aliii$\lambda$1860 and Civ$\lambda$1549 lines is similar to the one of H$\beta$, including the FWHM and profile shape, either Gaussian or Lorentzian. The similarity helps us to define whether an object is Population A or B in this paper. These observational results point toward three different components in broad line profiles (see Marziani et al. 2010) which can be described as follows: 1. 1. A broad component (BC) showing a roughly symmetric profile with FWHM in the range 1000-5000 km s-1. It is consistent with the component identified by Matsuoka et al. (2008). This broad component dominates LILs in Balmer lines of Pop. A sources while it becomes less prominent in Pop. B. The profile is best modeled by a Lorentzian function in Pop. A sources while Pop. B profiles are better described by a Gaussian (Marziani et al. 2003b). 2. 2. A very broad component (VBC), as seen in LILs and HILs of most pop B sources but is absent from Pop. A profiles. The VBC can be modeled as a Gaussian (FWHM $\sim$ 10000 km s-1) often with a significant shift to the red. It can be called a defining property of Pop. B sources (Marziani et al. 2010). This component is clearly identified in the Civ$\lambda$1549 line of Pop. B objects, and is also appreciable on the red side of Ciii]$\lambda$1909 of Pop. B objects J00103-0037 and J02390-0038. 3. 3. A blueshifted broad component (BBC), defined as the residual emission in the Civ$\lambda$1549 line after subtracting a scaled BC Lorentzian profile (Marziani et al. 2010). This blueshifted component is often prominent in Civ$\lambda$1549 and Ly$\alpha$ of Pop. A sources. It is much less intense in radio-loud Pop. B sources (Marziani et al. 1996; Punsly 2010; Richards et al. 2010). We model this profile as a blueshifted Gaussian. The Gaussian approximation is probably inappropriate especially if the BCC is strong: this component is believed to be produced in a partially-obscured radial flow, not in a virialized emitting system. In the present work we do not even try to fit the blueshifted in the doubly ionized lines. It is quite obvious from the fits that a possible contribution of this component would be negligible. Baldwin et al. (1996) presented a similar analysis. Their Fig. 2 organizes spectra in a sequence that is roughly corresponding to E1, going from Aliii$\lambda$1860-strong sources to objects whose spectra show prominent Ciii]$\lambda$1909 along with weak Aliii$\lambda$1860 (Bachev et al. 2004). Two of the three line components they isolated correspond to the ones we consider in this paper: a blue-shifted feature, and a more symmetric, unshifted and relatively narrow component that we call LIL-BC. Less obvious is the correspondence of a third feature, although it appears to be the redshifted part of what we call the VBC. Several improvements have been introduced since the paper of Baldwin et al. 1996. These improvements are expected to make our analysis easier. First, the definition of a template of Feiii emission (Vestergaard & Wilkes 2001), along with the possibility to model Feii in the 1400–2000 Å spectral region with cloudy (Verner et al. 1999, 2004). The analysis of spectra along the E1 sequence allows one to see trends that make the interpretation of the emission line blends easier (Marziani et al. 2010). ### 4.4 Expected emission from the various components We looked for evidence of three possible components as described above: BC, VBC and BBC only for the most intense HILs: Civ$\lambda$1549 and Siiv$\lambda$1397\. We did not include the contributions of the BBC for the doubly-ionized lines. In the case of Pop. A sources, we indeed assume that the BC contains the vast majority of the light. In the case of Pop. B sources we consider the contribution of the VBC. The Aliii$\lambda$1860 doublet shows no evidence of either a BBC or VBC. There is no obvious BBC for Siiii]$\lambda$1892 in the blend. There is evidence of a VBC of Ciii]$\lambda$1909 extending on the red side of the blend, and this is taken into account. Moreover, we expect that the Aliii$\lambda$1860 doublet is emitted exclusively in the BC, the region where Feii is also emitted. This is empirically confirmed by the aspect of the 1900 blend in many sources, where we do not see any evidence of BBC nor VBC. We remark that the Aliii$\lambda$1860 doublet is relatively unblended, and that a BBC feature as strong as in Civ$\lambda$1549 would not easily escape visual detection.The same is also true for Siiii]$\lambda$1892\. Several fits that included BBC in Ciii]$\lambda$1909 yielded 0 intensity, implying a large Civ$\lambda$1549/Ciii]$\lambda$1909 (Marziani et al. 2010). The Siiv$\lambda$1397 + Oiv]$\lambda$1402 blend closely resembles the shape of Civ$\lambda$1549, suggesting that BBC is relevant, especially for Siiv$\lambda$1397 (Oiv]$\lambda$1402 is expected to give a minority contribution to line emission at the high density derived for the BC; any Oiv]$\lambda$1402 contribution to the BBC is not relevant to our method). BBC is very weak or undetectable in the vast majority of the H$\beta$ profiles analyzed in Marziani et al. 2010 (but see Zamfir et al. 2010 for several cases of H$\beta$ BBC), while prominent in Ly$\alpha$; the Ly$\alpha$/H$\beta$ ratio in this component is high. In summary, BBC is visually strong in Ly$\alpha$, Civ$\lambda$1549, and Siiv$\lambda$1397\. Heii$\lambda$1640 in BBC is needed for a self-consistent fit of the Civ$\lambda$1549+Heii$\lambda$1640 blend. The VBC of Siiii]$\lambda$1892 is poorly constrained, but in the fits where Ciii]$\lambda$1909 VBC is visible, we always find Siiii]$\lambda$1892 $<$ Ciii]$\lambda$1909, consistent with the high ionization level expected in the VBC region. Some Siiv$\lambda$1397 VBC emission is assumed in the Pop. B fits. No VBC emission is expected in Aliii$\lambda$1860 and Feii. These constraints help also to make the fits less ambiguous. The absence of a VBC makes the decomposition of Pop. A spectra easier. As said earlier, the Feii intensity scale of the template (but Feii is in general weak) is anchored to the UV 191 intensity; similarly the Feiii intensity is set by a feature external to the 1900 Å blend (2080 Å). There is no evidence of BBC of Aliii$\lambda$1860 and Siiii]$\lambda$1892; the Aliii$\lambda$1860 doublet profile is mostly unblended and defines the LIL-BC. The additional complication here is the Feiii 1914 line whose intensity is affected by Ly$\alpha$ pumping. Since the shift and FWHM are assumed the same for all lines (and templates) in the 1900 blend, the only free parameters in addition to shift and FWHM are the intensities of 6 components, including the two from the templates, and Siii$\lambda$1814 and Aliii$\lambda$1860 that are not heavily blended. The specfit analysis is especially helpful to measure in a non-subjective way, taking all constraints into account, the two parameters that are most affected by the blend: the intensity of Siiii]$\lambda$1892 and Ciii]$\lambda$1909 (we repeat that any Feiii $\lambda$1914 contribution in excess to the one of the adopted template is included in the estimated Ciii]$\lambda$1909 intensity). In the case of Pop B, the presence of a VBC does not really complicate the fit as a matter of fact. The Ciii]$\lambda$1909 line undeniably shows a VBC protruding on the red side of the 1900 blend. In any case, considering that we can expect the VBC to be assimilable to a shifted Gaussian with FWHM $\sim 10000$km s-1, the unblended part of the Ciii]$\lambda$1909 VBC provides a strong constraint. The Siiii]$\lambda$1892 VBC is certainly the most difficult feature to ascertain, as it buried under Siiii]$\lambda$1892 and Ciii]$\lambda$1909 emission. We rely on the specfit results. that indicate negligible Siiii]$\lambda$1892 VBC. ### 4.5 Errors We identify five sources of error from the conditions for data reduction and methodological considerations described above: 1. 1. A & B atmospheric bands correction (already described in §3.1). 2. 2. Line profile shape, Gaussian or Lorentzian (Pop. A or B). The distinction between Pop. A and B is based on line width with the boundary at FWHM H$\beta\approx$ km s-1 in low luminosity quasars and around $\approx$ 5000 km s-1 at higher luminosity such as the eight sources presented here. Most of our quasars are unambiguously Pop. A or B because of line width and because Pop. B sources show an H$\beta$ VBC and pop A sources a prominent Civ$\lambda$1549 BBC. In these cases only one profile shape (Gaussian or Lorentzian) was fitted. 3. 3. Rest-frame determination using Oi$\lambda$1304 or Ciii]$\lambda$1909\. In some cases the redshift estimates derived from the two lines do not agree, most likely because of absorptions present in Oi$\lambda$1304.8 and because this is not a very intense line. The principal impact of uncertainty in the rest frame placement is estimation of the peak wavelength of Civ$\lambda$1549\. If the line peak differs from $\lambda$1549Å, the BC intensity is diminished and we infer a greater contribution from the blue (BBC) component. Similarly, for the blend 1900Å, the rest frame shift may increase or decrease our estimate for the strength of Ciii]$\lambda$1909 with consequent decrease or increase of the Siiii]$\lambda$1892 contribution. This additional source of uncertainty affects J00103-0037, J02287+0002 and J20497-0554. 4. 4. Feii intensity (continuum placement). Broad Feii emission can produce a pseudo-continuum affecting our estimates of emission line intensities. Siii$\lambda$1814 is especially affected in our spectra because it is weak. Aliii$\lambda$1860 is similarly affected when it is weak. The effect is less noticeable for Civ$\lambda$1549 since expected Feii emission underlying the Civ$\lambda$1549 line is weak also for strong Feii emitters. 5. 5. Broad absorption lines (BALs) in quasars principally affect the blue side of Civ$\lambda$1549\. We also find an absorption feature between FeII$\lambda$1787 and Siii$\lambda$1814 (eg. Fig. 6). In sources J01225+1339 and J02287+0002 we can only set upper limits for line intensities (since we fit unabsorbed components). There are other sources of error such as small BC shifts and FWHM variations. We assume that the BC of the Siiii]$\lambda$1892, Aliii$\lambda$1860, Siii$\lambda$1814 and Civ$\lambda$1549 line has the same FWHM and wavelength shift, although we allow for variations in their relative flux strengths and adopted profile type (Lorentzian for Pop. A and Gaussian for Pop. B). For Feiii(UV34) and Feii we slightly relax this constraint. The Feiii(UV34) and Feii emission is not very strong and the FWHM of individual features is poorly constrained by specfit. In the case of Ciii]$\lambda$1909 we need to consider the possibility that the profile is narrower because there might be a contribution from different regions: indeed, the specfit routine usually converges toward a narrower profile if the Ciii]$\lambda$1909 width is not constrained. The effect depends on the strength of the Feiii $\lambda$ 1914 feature. The goal of this study is to estimate diagnostic line ratios. A posteriori, we can say that the estimated line ratios are rather insensitive to the emission component profile shape: assuming a Gaussian or Lorentzian profile yields the same ratio for the strongest lines (i.e., Civ$\lambda$1549, Siiii]$\lambda$1892, Aliii$\lambda$1860) upon which our analysis is based (with an uncertainty of $\sim$10%). The same conclusion applies to the redshift uncertainty. Only in the case of J02287+0002 the redshift difference produces a significant effect due to a $\Delta z\approx 0.16$. However, we adopt the fit based on Oi$\lambda$1304: the alternative fit produces line ratios that are of not obvious interpretation and at variance with respect to the other sources. The most serious sources of error remain effects of A/B band overlap for Siii$\lambda$1814 and the presence of a BAL for Civ$\lambda$1549. ## 5 Results of Line Component Analysis on Individual Objects In Figures from 4 to 6 we show our best fits for the VLT sample taking into account the considerations described in §4. The fits of Siiv$\lambda$1397 are shown in Fig. 7 and the intensity values and equivalent widths are in Tables 3 and 5. We present here a phenomenological description of the fits. The line profiles, intensities and line ratios usually follow the trends of Pop. A or B sources, although in some objects there are features that are ambiguous (for example the line shape). In these cases however, we have assigned Pop. A or B type on the basis of the FWHM (see Cols. 5 and 6 of Table 7). ### 5.1 Pop. A Objects * • J03036-0023 – We estimate for this source a FWHM(BC)$\sim$3700 km s-1 and we use a Lorentzian function to fit the broad lines. The peak of Civ$\lambda$1549 is blueshifted and requires a strong BBC (Fig. 4(a)). The bump on the red side of Civ$\lambda$1549 can be accounted for by Heii$\lambda$4686 BC and BBC. There is no evidence for a red shifted component in Ciii]$\lambda$1909 (Fig. 4(b)). Aliii$\lambda$1860 is prominent. Unfortunately the blue wing of Siii$\lambda$1814 and the red wing of Feii$\lambda$1787 are affected by A band absorption. * • J20497-0554 – This source shows FWHM(BC)$\sim$3800 km s-1. As for J03036-0023, the Civ$\lambda$1549 line can be accounted for by an unshifted BC (assumed Lorentzian) of a considerably contribution of a BBC (Fig. 4(c)). We see a prominent Aliii$\lambda$1860 line and FeII$\lambda$1787 (Fig. 4(d)). Siiii]$\lambda$1892 is affected by several narrow absorption lines; however, it is obviously strong. The lack of a red wing on Ciii]$\lambda$1909 suggests that no VBC is present. * • J23509-0052 – This source has FWHM(BC)$\sim$3600 km s-1. Civ$\lambda$1549 shows a slight blue asymmetry with a BBC (Fig. 4(e)) required to model it. The contribution of Feii is small and Feii$\lambda$1787 is weak (Fig. 4(f)). Ciii]$\lambda$1909 is very strong. Aliii$\lambda$1860 is affected by A band absorption; the profile we fit is probably an upper limit. This object could well belong to spectral type A1 that includes Pop. A sources with the lowest $R_{\rm FeII}$($\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}0.5$). ### 5.2 Pop. B Objects . * • J00103-0037 – This source has a FWHM(BC)$\sim$4500 kms-1. The red side of Civ$\lambda$1549 is blended with Heii$\lambda$4686\. Fitting a BC with no shift plus a BBC to Civ$\lambda$1549 leaves a very large residual on the red side. A very broad redshifted component (VBC) is needed to model the spectrum (Fig.5 (a)). The faint narrow line under Civ$\lambda$1549 can be explained as the narrow component (NC) of Civ$\lambda$1549(see Sulentic et al. 2007). The presence of a similar NC in Ciii]$\lambda$1909 could possible explain the large residual seen $\sim$1900\. We specifically note the prominent Ciii]$\lambda$1909 emission and weak (but detected) Aliii$\lambda$1860 (Fig. 5(b)). The Feii “bump” at 1787 Å(UV 191) is appreciable. Fainter Feii emission is relatively unimportant because Feii creates a pseudo-continuum. Siii$\lambda$1814 is compromised by A-band absorption. The blend at 1900Å includes a Ciii]$\lambda$1909 VBC and the fit indicates Civ$\lambda$1549/ Ciii]$\lambda$1909 (VBC) $\approx$ 7 which is reasonable. * • J00521-1108 – This source shows the noisiest spectrum in the sample. We fit a FWHM(BC) $\sim$ 5300 km s-1 with Civ$\lambda$1549 requiring a large VBC to account for the red wing. (Fig. 5(c)). Absorption features seriously affect the Civ$\lambda$1549 profile. The profile of Ciii]$\lambda$1909 is strongly asymmetric due to some sort of absorption on the red side. Aliii$\lambda$1860 is weak consistent with pop. B (Fig. 5(d)). * • J02390-0038 – This objects has a somewhat atypical Pop. B spectrum due to a very strong BBC in Civ$\lambda$1549 (Figure 5(e)). This source has a FWHM(BC)$\sim$ 5400 km s-1. Consistent with pop. B we find Feii$\lambda$1787 to be weaker than Siii$\lambda$1814\. Ciii]$\lambda$1909 is flat topped and has a very similar intensity as Siiii]$\lambda$1892\. Both Civ$\lambda$1549 and Ciii]$\lambda$1909 it show red wings indicating VBC emission. The A band absorption lies between Siiii]$\lambda$1892 and Aliii$\lambda$1860 (Fig. 5(f)) producing a possible overestimation of Aliii$\lambda$1860. ### 5.3 BAL QSOs * • J01225+1339 – Civ$\lambda$1549 is highly affected by two broad absorption lines (Fig. 6(a)) with blueshifts of 5200 and 10800 km s-1 at peak absorption with equivalent widths/FWHM -12Å / 3900 km s-1 and -25Å/5200 km s-1, respectively. The blueshift of the Civ$\lambda$1549 peak leads us to suspect a large blueshifted BBC emission component. The 1900 Å blend shows absorptions coincident with Feii$\lambda$1787 and the blue side of Siii$\lambda$1814 which is however unambiguously detected (Fig. 6(b)). Aliii$\lambda$1860 is prominent which implies that this a Pop. A source. The FWHM(BC) $\sim$ 4400 km s-1 is consistent with a high-luminosity Pop. A source. It is also possible that this BAL QSO is an outlier like Mark 231 at low-$z$ (Sulentic et al. 2006), in other words an extreme Pop. A object. The Ciii]$\lambda$1909 is well fitted with a Lorentzian profile. Broad A band atmospheric absorption lies over Siiii]$\lambda$1892. * • J02287+0002 – This object has a very complex spectrum. On one side it has a FWHM(BC) $\approx$ 4700 km s-1 which is typical of low-luminosity Pop. B and the lines profiles are better fitted with Gaussians. On the other side, however, it shows features that are typical of extreme Pop. A sources: prominent Feii$\lambda$1787, strong Aliii$\lambda$1860, no Ciii]$\lambda$1909 VBC (Fig. 6(d)). Considering that the FWHM limit between Pop. A and B is increasing with luminosity, the FWHM(BC) is within the limit of Pop. A. The Ciii]$\lambda$1909 line is not very flat topped but the similar intensities of Ciii]$\lambda$1909 and Siiii]$\lambda$1892 remind the case of J02390-0038. Also because it has a strong blue-shifted component in Civ$\lambda$1549 atypical to Pop. B objects (Fig. 6(c)). The estimated rest frame of this quasar differs by $\sim$ 1400 km s-1 using Oi$\lambda$1304 and Ciii]$\lambda$1909\. This is the largest discrepancy in our sample. In order to evaluate the effect of the $z$ discrepancy we performed two fits using both rest frames. Figs. 6(a) and (b) use the Ciii]$\lambda$1909 restframe. In the 1900 Å blend, we found a contribution of Siiii]$\lambda$1892 similar to Ciii]$\lambda$1909\. If we use the Oi$\lambda$1304 inferred rest frame we show in Figs. 6(e) and (f) that Ciii]$\lambda$1909 becomes stronger with a resultant decrease of Siiii]$\lambda$1892\. A similar effect occurs for Civ$\lambda$1549 broad and blue-shifted components. The Civ$\lambda$1549 BAL shows a blueshift of 9100 km s-1at deepest absorption, a EW of –14 Å and a FWHM of 4600 km s-1. Summing up, we are able to assign an A/B identification to all sources in our sample. The two BAL QSOs appear as objects of extreme Pop. A. We remind that the identification of Ciii]$\lambda$1909 in the BAL QSOs and in sources with strong Aliii$\lambda$1860 is debatable (Hartig and Baldwin 1986): strong Feiii 1914 could take the place of most Ciii]$\lambda$1909 emission. ## 6 Estimation of Physical Conditions in the Emitting regions ### 6.1 CLOUDY Simulations We computed a multidimensional grid of CLOUDY (Ferland et al. 1998) simulations, (see also Korista et al. 1997) to derive $U$ and $n_{\mathrm{H}}$ from our spectral measurements. Simulations span the density range $7.00\leq\log$ $n_{\mathrm{H}}$$\leq 14.00$, and $-4.50\leq\log U\leq 00.00$, in intervals of 0.25. Each simulation was computed for a fixed ionization parameter and density assuming plane parallel geometry. The 2D grid of simulations was repeated twice assuming $N_{\rm c}$$=10^{23}$ and $10^{24}$ cm-2. Several cases were computed also for $N_{\rm c}$= $10^{25}$cm-2. Metallicity was assumed to be either solar or five times solar. Two alternative input continua were used: 1) the standard AGN continuum of CLOUDY which is equivalent to the continuum described by Mathews and Ferland (1987) and 2) the low-$z$ quasar continuum of Laor et al. (1997a). Computed line ratios are almost identical for fixed ($U$, $n_{\mathrm{H}}$). However the ionizing luminosity differs by more than a factor of 2 for a fixed specific continuum luminosity. The contour plots showing the distributions of Ciii]$\lambda$1909/ Siiii]$\lambda$1892, Aliii$\lambda$1860/ Siiii]$\lambda$1892, Siii$\lambda$1814/ Siiii]$\lambda$1892, Siiv$\lambda$1397/ Siiii]$\lambda$1892, Civ$\lambda$1549/Aliii$\lambda$1860and Civ$\lambda$1549/ Siiii]$\lambda$1892 (Fig. 9) are generated from 29 $\times$ 19 = 551 simulations and assume a standard set of simulations using a Mathews and Ferland (1987) continuum, $N_{c}=10^{23}$ cm-2, and solar metallicity. The CLOUDY 08.00 computations included a model of the Fe+ ion with 371 levels. The UV Feii template, described in §4.2 is based on a suitable CLOUDY simulation. Even if a relationship is very likely between the dense low ionization gas producing our diagnostic lines and Feii emission (supported observationally) our diagnostics do not use any Feii computation. The weak line Oi$\lambda$1304 is used only for rest frame estimation and not for diagnostic considerations. Apart from the hypothesis of plane-parallel geometry, no inferences are made about the actual distribution, location, and kinematics of the line emitting gas. The assumption of constant density is crude. If gas is distributed in clouds then magnetic confinement appears to be unnecessary to avoid cloud dispersion from pressure imbalance or cloud shear associated with a hot confining medium. Magnetic confinement could make density uniform within the cloud (Rees 1987; Bottorff and Ferland 2000). ### 6.2 Intermediate Ionization Lines in the Blend at $\lambda$1900 The ratio Ciii]$\lambda$1909/ Siiii]$\lambda$1892 is density dependent because the transition probabilities of the two lines are so different: 114 s-1 vs 12600 s-1 (see Table References). The forbidden lines at 1883Å and 1907Å have such low transition probabilities that they are collisionally quenched at much lower density and will not be considered. Line ratios like Ciii]$\lambda$1909/ Siiii]$\lambda$1892 are useful diagnostics in a rather narrow range of density which depends on the transition probabilities. Above the critical density, emission lines originating from forbidden or semi-forbidden transitions become collisionally quenched, and hence weaker than lines for which collisional effects are still negligible. Ciii]$\lambda$1909 is clearly unsuitable as a diagnostic for $n_{\rm e}$$\gg 10^{11}$ cm-3, as the Ciii]$\lambda$1909/ Siiii]$\lambda$1892 $\rightarrow$ 0\. Feldman et al. (1992) gives critical density of Siiii]$\lambda$1892 $n_{e}\sim 2\cdot 10^{11}$ cm-3. Aliii$\lambda$1860 is a permitted transition with large transition probability ($A\sim 5\cdot 10^{8}$ s-1) and has very-high critical density (i.e., its equivalent width goes to zero toward thermodynamic equilibrium, which occurs at very high density, when all line emission is collisionally quenched). Our 2D array of CLOUDY simulations shows that the ratio Aliii$\lambda$1860/ Siiii]$\lambda$1892 is well suited to sample the density range $10^{10}-10^{12.5}$ cm-3. Within this range the Siiii]$\lambda$1892 intensity decreases smoothly by a factor 10; above the upper limit in density, the predicted intensity of Siiii]$\lambda$1892 decreases. This corresponds to the densest, low ionization emitting regions likely associated to the production of Feii. Intermediate-ionization lines such as Ciii]$\lambda$1909, Aliii$\lambda$1860, Siiii]$\lambda$1892 and Siiv$\lambda$1397avoid the issue of collisional ionization (invoked for the production of low-ionization species by S. Collin and collaborators, as mentioned in the introduction). The widely separated doublet Aliii$\lambda$1860 is expected to be produced entirely in the fully ionized zone (Laor et al. 1997b). CLOUDY simulations confirm this suggestion. Figure 10 (top panel) shows the ionic fraction as a function of the geometrical depth in a cloud (or slab) of fixed column density ($N_{\rm c}$$=10^{23}$ cm-2) and density ($n_{\mathrm{H}}$$=10^{12.5}$cm-2). As it can be seen, Al++, Fe++, and Si++ share a region of dominance deep in the cloud, close to the end of the Strömgren sphere. Beyond, in the partially- ionized zone (PIZ) there is, as by definition, a significant fraction of ionized hydrogen. The dominant ionic stages of Si and Fe become Fe+ and Si+. It is very appropriate to consider the Aliii$\lambda$1860/ Siiii]$\lambda$1892 ratio since the two lines are apparently emitted in the very same zones within the gas slab. Available reverberation mapping results may support this interpretation but are rather difficult to extrapolate since they are limited to an handful of low luminosity objects. The main finding is that Ciii]$\lambda$1909 responds to continuum changes on timescales much longer than Civ$\lambda$1549 and other HILs. This results comes from the analysis of total Ciii]$\lambda$1909 + Siiii]$\lambda$1892 in NGC 3783 (Onken & Peterson 2002), from the Ciii]$\lambda$1909 of NGC 4151 (Metzroth et al. 2006) and of NGC 5548. It is intriguing that the Ciii]$\lambda$1909 cross-correlation delay in NGC 5548 (by far the best studied object) is even larger than that of H$\beta$ (32 ld vs. 20 ld; Peterson et al. 2002; Clavel et al. 1991). For fixed density, lines of higher ionization form at higher photon flux. The C+, Al+, Si+ ionization potential are 24, 18, and 16 eV respectively. These comparable ionization potentials are well below the one of HILs, $X^{i-1}\gtrsim$ 50 eV. However, the much lower Ciii]$\lambda$1909 critical density implies that the Ciii]$\lambda$1909 line should be formed farther out than Siiii]$\lambda$1892 and Aliii$\lambda$1860 if all these lines are produced under similar ionization conditions. ### 6.3 The Siii Contribution to the UV Spectrum of Quasars AGN have a rather rich singly-ionized silicon spectrum in the range 1000-2000 Å, due to several resonant transitions from the ground term $3s^{2}3p^{1}{}^{2}P^{0}$ to terms associated to the $3s^{1}3p^{2}$ ($\lambda$1814 and $\lambda$1304), $3s^{2}4s$ ($\lambda$1531), $3s^{2}3d$ ($\lambda$1263) electronic configurations. A feature on the red side of Ly$\alpha$, at 1263 Å is detected in several Pop. A sources like I Zw 1 (Marziani et al. 2010). The $\lambda$1309 feature is partially blended with the Oi triplet associated with a Bowen fluorescence mechanism from HI Ly$\beta$. The $\lambda$1309 line is well resolved in sources like I Zw 1 with the extension of the broad $\lambda$1309 feature suggesting significant Siii emission around $\lambda$1309\. The $\lambda$1531 feature is blended with Civ$\lambda$1549\. High S/N spectra of low redshift sources from HST (Laor et al. 1994) as well as for quasars at $z\approx$ 4 (Constantin et al. 2002) indicate that the feature is very weak in most sources. The Siii feature at $\lambda$1814 is detected in at least four of the five quasars studied by Laor et al. (1994). In the 6 objects used for the E1 sequence of Marziani et al. (2010), it is detected without reasonable doubt in I Zw 1 only. However this has to do more with the poor S/N of the 1900 Å region than of anything else; good S/N HST spectra show an unambiguous detection in Akn 120 and Mark 509, for example. Early detection of the UV Siii lines from IUE observations suggested collisional excitation and no relevant fluorescence effects (as observed in type Ia supernovæ; Dumont and Mathez 1981). Fluorescence effects and recombination are revealed by optical lines which are emitted through several cascade routes to the ground state. However, a median spectrum of A2 quasars with S/N $\approx$ 200 (Zamfir et al. 2010) barely detects any expected emission feature in the range 4000-6500 Å. The prominence of the $\lambda$1814 feature may then be associated to the relatively low temperature believed to exist in the innermost part of the line emitting “cloud”, $T_{e}\sim 5000-8000^{\circ}$ K. We may consider the equivalent width ($W$) ratio between the doublet lines at 1264 and at 1814 Å both due to a ${}^{2}D_{\frac{5}{2}}\rightarrow^{2}P_{\frac{3}{2}}$ transition; in the simplest case we have: ${W_{1263}}/{W_{1814}}\approx\left({N_{\mathrm{u,1264}}}/{N_{\mathrm{u,1814}}}\right)\left({f_{1264}}/{f_{1816}}\right)\left({\lambda_{1264}}/{\lambda_{1816}}\right)^{2}\approx 0.485\frac{f_{1264}}{f_{1816}}e^{-\frac{\Delta E}{kT}}\approx 0.27$ if $T=5000^{\circ}$, where $f$ is the oscillator strength, $N_{\mathrm{u}}$ are the density of the upper electronic levels giving rise to the two lines, and the energy difference between the transitions is $\Delta E\approx$ 3 eV. ### 6.4 Use of Siiv$\lambda$1397, Civ$\lambda$1549 and Siii$\lambda$1814 The same caveats mentioned for the ratio Ciii]$\lambda$1909/ Siiii]$\lambda$1892 apply to the Aliii$\lambda$1860/ Siiii]$\lambda$1892 ratio as well. The ground term ${}^{1}S_{0}$ has energy 16 eV and 24 eV for Si+2 and C+2 respectively. A dependence on the ionization parameter is expected, as already mentioned. However, given the similarity in the ionization structure of the photo-ionized slab, it is after all not surprising that the ratio Aliii$\lambda$1860/ Siiii]$\lambda$1892 is almost insensitive to the ionization parameter over a wide range of density. We remind that the detection of strong Aliii$\lambda$1860 alone already suggests that we are considering very high density emitting gas even if metallicity is super-solar. Simulations indicate that Aliii$\lambda$1860 should increase smoothly with density and be weakest in canonical BLR if the density is $n_{\mathrm{H}}$$\sim 10^{9}$ cm-3 (cf. Korista et al. 1997). The ratio Aliii$\lambda$1860/ Siiii]$\lambda$1892 is therefore diagnosing high density gas, while the Ciii]$\lambda$1909/ Siiii]$\lambda$1892 ratio covers the domain of $n_{\mathrm{H}}$$\sim 10^{10}$cm-3. The ratio Aliii$\lambda$1860/ Siiii]$\lambda$1892 alone is, generally speaking, insufficient to determine $n_{\mathrm{H}}$. A second diagnostic ratio is needed to constrain $U$ and to unambiguously derive $n_{\mathrm{H}}$. We consider Siii$\lambda$1814/ Siiii]$\lambda$1892, Civ$\lambda$1549/ Siiii]$\lambda$1892, Siiv$\lambda$1397/ Siiii]$\lambda$1892 as three diagnostic ratios suitable for constraining $U$. The Siii$\lambda$1814 doublet is conveniently placed, although weak or undetectable in most sources. The Siii$\lambda$1814/ Siiii]$\lambda$1892 ratio is anti-correlated with $U$ in a regular form, as our cloudy simulations show, especially for $\log U\lesssim-2$, and for density not much above the critical density of Siiii]$\lambda$1892\. The behavior of the ratio Siii$\lambda$1814/ Siiii]$\lambda$1892 resembles the distribution of Ciii]$\lambda$1909/Civ$\lambda$1549 in the plane ($n_{\mathrm{H}}$, $U$) which shows a very regular dependence and a smooth anti-correlation with $U$ until the collisional quenching of Ciii]$\lambda$1909 sets on. However, the IP of Si0 is just 8 eV. The majority of the $\lambda$1814 doublet is emitted in the partially ionized zone (PIZ) near the hydrogen ionization front. Fig 10 (bottom panel) shows the behavior of the volume emissivity $\epsilon$ times the geometrical depth within an emitting slab of gas as a function of the depth itself. Since the actual line emission is proportional to $\epsilon\cdot h$ reported in Fig. 10, the emission of the Aliii$\lambda$1860, and Siiii]$\lambda$1892 lines is negligible in the PIZ, but the one of H$\beta$ certainly is not, and may be well dominating if $N_{\rm c}$$\gg 10^{23}$cm-2. Significant emission in the PIZ is expected also for the Siii$\lambda$1814 doublet. However, for this latter line, total emission should not depend strongly on column density, since the emissivity becomes very low at $N_{\rm c}$$\gtrsim 10^{23}$cm-2. If higher $N_{\rm c}$ are considered (up to 1025cm-2), ratios including Siiii]$\lambda$1892, Aliii$\lambda$1860, and Siii$\lambda$1814 show a very weak dependence on column density, with changes a few percent at worst. We attempted to isolate a Civ$\lambda$1549 and a Siiv$\lambda$1397 component that corresponds to the Aliii$\lambda$1860 and Siiii]$\lambda$1892 lines. This can be a small part of the total Civ$\lambda$1549 and of Siiv$\lambda$1397 emission, but there is no point to consider the whole Civ$\lambda$1549 emission when Civ$\lambda$1549 shows a large blueshift and is much broader than H$\beta$ (Fig. 5 of Sulentic et al. 2000) and Siiii]$\lambda$1892 and Aliii$\lambda$1860 (Fig. 2 of Marziani et al. 2010). Taking into account various sources of ambiguity (mainly uncertainty in the quasar rest frame, S/N, blending with HeII$\lambda$1640), the Civ$\lambda$1549 BC component we measure with our fits is constrained within $\pm$ 50% at worst. Thus we find significant Civ$\lambda$1549 emission from the low-ionization gas, and the basic assumption is that its Civ$\lambda$1549 profile is the same as the Aliii$\lambda$1860, Siiii]$\lambda$1892 lines. If one considers the emissivity behavior, Civ$\lambda$1549 and Siiv$\lambda$1397 are obviously favored within the fully ionized zone. The ratio of Civ$\lambda$1549 and Siiv$\lambda$1397 over Ciii]$\lambda$1909 or over Siiii]$\lambda$1892 increases with ionization parameter in a way that is roughly independent on density until the collisional quenching of the semi- forbidden lines sets on (Fig. 9). The previous considerations are helpful to understand why the Civ$\lambda$1549 and Siiv$\lambda$1397 ratios provide clear diagnostics of the ionization parameter and seem to be in (at least qualitative) agreement with ratios employing lower ionization lines like Siii$\lambda$1814/ Siiii]$\lambda$1892 that is also mainly sensitive to $U$. The striking fact that the Civ$\lambda$1549 and Siiv$\lambda$1397 emission confirms low-ionization supports the hypothesis that the four lines are emitted by the same region. The ratios involving Si only have the obvious advantage that the the determination of the physical parameters is not dependent on metallicity; the drawback of Siiv$\lambda$1397 is that this line is weaker and blended with Oiv]$\lambda$1402\. However, as already pointed out, the Siiv$\lambda$1397 BC intensity should be slightly affected by Oiv]$\lambda$1402\. ## 7 Results on two Extreme, Elucidating Cases We analyze two extreme objects – one extreme Pop. A and one extreme Pop. B – with the same methodology we used previously. The aim of this analysis is to help the interpretation of the line components measured on the spectra of the 8 $z\approx$3 quasars. ### 7.1 SDSS Weak Ciii]$\lambda$1909 source: SDSS J120144.36+011611.6 Our project stems from the careful analysis of the I Zw 1 UV spectrum by Laor et al. (1997b), and the evidence of a well-defined trend in the $\lambda$1900Å blend along the E1 sequence (Aoki & Yoshida 1999, Wills et al. 1999, Bachev et al. 2004, Marziani et al., 2010). The NLSy1 I Zw 1 is known to be a sort of extremum in the E1 sequence: it shows strong Feii and Feiii, prominent Aliii$\lambda$1860 and Siiii]$\lambda$1892 emission. It is an example of the A3 spectral type, whose median 1900Å blend in shown in Fig. 3 of Bachev et al. (2004). Definitely, these sources are present also at intermediate to high redshift. We describe here the analysis of one source, SDSS J120144.36+011611.6 (Fig. 8), which seems to be a high-redshift, high- luminosity replica of I Zw 1, with broader lines (the FWHM limit of NLSy1s and Pop. A sources is luminosity dependent; see Netzer & Trahktenbrot 2007 and Marziani et al. 2009). These sources are Pop. A, and we assume that the profile of the BC is Lorentzian. Hereafter unshifted Lorentzian part of the line is said to be BC. Pop. A sources are also free of any VBC, making the analysis of the blend simpler. Several considerations can be made from Fig. 8. ##### Aliii$\lambda$1860 The lines less ambiguous to measure are the Aliii$\lambda$1860 doublet because the two lines are less blended with other features, and they are remarkably strong. Siiii]$\lambda$1892 is more heavily blended with Ciii]$\lambda$1909 and Feiii. However, the specfit routine allows usually a plausible deconvolution of Siiii]$\lambda$1892, making the Aliii$\lambda$1860/ Siiii]$\lambda$1892 ratio very reliable. Feii and Feiii are obviously strong. We can use Feii UV 191 to set roughly the level of Feii, while the feature at 2080Å is helpful to estimate intensity of Feiii. ##### Feiii Our specfit analysis produces a very weak Ciii]$\lambda$1909 component. A precise measurement of Ciii]$\lambda$1909 is cumbersome, since its intensity depends on the actual Feiii emission. The strong feature at $\lambda$1914 could be dominating, and the template may seriously underestimate it (see Vestergaard and Wilkes 2001 for several alternatives in the empirical Feiii emission of I Zw 1). In any case the residual Ciii]$\lambda$1909 emission is small, suggesting that the $\lambda$1900 blend, once believed to be mostly Ciii]$\lambda$1909, is actually almost void of Ciii]$\lambda$1909 emission in these A3-type sources. ##### Siii$\lambda$1814 The Siii$\lambda$1814 line is well visible in the spectrum of SDSS J120144.36+011611.6 and can be used as a substitute of Civ$\lambda$1549 to measure the ionization parameter. The ratio Siii$\lambda$1814/ Siiii]$\lambda$1892 is mainly sensitive to the ionization parameters, as it is the ratio Civ$\lambda$1549/ Siiii]$\lambda$1892\. The ratio Siii$\lambda$1814/ Siiii]$\lambda$1892 has the considerable advantage of being weekly dependent on metallicity. If metallicity is known (see also §8.1), the ratio Civ$\lambda$1549/Aliii$\lambda$1860 should be in principle preferred since Aliii$\lambda$1860 is emitted through a permitted resonance transition while Siii$\lambda$1814 is emitted in the collisionally excited, partially-ionized zone (PIZ). ##### Civ$\lambda$1549 We fit Civ$\lambda$1549 with a Lorentzian component that is unshifted + a Civ$\lambda$1549 BBC. Note that the lack of shift in Aliii$\lambda$1860 and Siiii]$\lambda$1892 imposes a strong, determinant condition on the strength of the Lorentzian-component in Civ$\lambda$1549\. It is important to stress that some Civ$\lambda$1549 BC emission is expected to be present according to our array of simulations. The ratio Civ$\lambda$1549/Aliii$\lambda$1860 is rather modest, as we can easily see even by eye. We note in passing that the with of BCC is $\approx 10000$ km s-1, with a peak blueshift indicatively of $-6000$km s-1, significantly larger than the one measured on the Civ$\lambda$1549 of I Zw 1 by Marziani et al. (2009). ##### Siiv$\lambda$1397 We fit Siiv$\lambda$1397 with a Lorentzian component that is unshifted + a BBC (that may include Siiv$\lambda$1397 and Oiv]$\lambda$1402 contribution). Results that are consistent with the ones of Civ$\lambda$1549\. The ratio Siiv$\lambda$1397/ Siiii]$\lambda$1892 has the advantage that is almost independent on metallicity. In principle, the crossing point between the Siii$\lambda$1814/ Siiii]$\lambda$1892 and Siii$\lambda$1814/ Siiii]$\lambda$1892 can set a metallicity independent point in the ($n_{\mathrm{H}}$, $U$) plane. If the accuracy of the Siii$\lambda$1814 intensity is good then this point can be used to retrieve information on metallicity (§8.1). We consider the measured line ratios in the plane $U$ vs. $n_{\mathrm{H}}$ (initally assuming metallicity equal to solar), and see where they cross. We find very high density and low ionization (Fig. 11). #### 7.1.1 Along the E1 Sequence: A More Complex Scenario Looking at the fit solution (Fig. 8(b)), it seems that our spectrum has almost no Ciii]$\lambda$1909\. In many ways this is not surprising. The physical solutions in the ($U$, $n_{\mathrm{H}}$) plane of Fig. 11, points toward very low ionization ($U\sim 10^{-3}-10^{-2}$) and high density ($n_{\mathrm{H}}$$\gtrsim 10^{12}$cm-3). At such high values of $n_{\mathrm{H}}$ we expect that Ciii]$\lambda$1909 is collisionally quenched, and no significant emission. The ratio Ciii]$\lambda$1909/Civ$\lambda$1549 is expected to be just $\sim$ 10-2 in the dense LIL-BLR. Therefore we can say that sources like SDSS and I Zw 1 are extreme because all of the $\lambda$1900Å blend is emitted by very low ionization, dense gas. However, as soon as we move away from spectral types A3+ along the E1 sequence, we see that the prominence of Aliii$\lambda$1860 diminishes greatly. The emission disappears altogether at the other end of the E1 sequence, where several lobe-dominated radio-loud sources are found. For most quasars we see that Ciii]$\lambda$1909 is rather strong and unmistakably present. At a first glance this complicates the interpretation of the spectrum. However, one has to consider that Ciii]$\lambda$1909 can be emitted only by gas of much lower $n_{\mathrm{H}}$ than that of the region where the bulk of Siiii]$\lambda$1892 and Aliii$\lambda$1860 is emitted. Our simulations show that Aliii$\lambda$1860 intensity grows smoothly as a function of density in the ionization parameter range of interest $-3\lesssim\log U\lesssim-1$. This said, and considered the smooth trend we see from A3 to B1+, the most reasonable conclusion is that a dense region emits significantly whenever Aliii$\lambda$1860 emission is detected, although the relative prominence of the dense BC changes along the E1 sequence: it accounts for the entire LIL emission only in the most extreme Pop. A sources. The sequence of Fig. 3 of Bachev et al. (2004) seems to be mainly a sequence of prominence of Aliii$\lambda$1860+ Siiii]$\lambda$1892 vs Ciii]$\lambda$1909. So, a very important conclusion is that a very dense, low-ionization region exists in the wide majority of quasars. It is associated with Feii prominence, as such gas is expected to emit a strong low-ionization spectrum. The most extreme Feii emitters are also the most extreme Aliii$\lambda$1860 emitters; in some cases where no Aliii$\lambda$1860 emission is measured, we also fail to detect any Feii emission (Marziani et al. 2010). Where is this region located? Why there are such distinctive line profiles as Lorentzian? The second issue goes beyond the present paper; for the moment our aim is to measure line components of Aliii$\lambda$1860, Siii$\lambda$1814, Siiii]$\lambda$1892 and Civ$\lambda$1549 that come from this region and to estimate its distance from the central black hole (§9). ### 7.2 The Other E1 Extremum: 3C 390.3 3C 390.3 is a lobe-dominated (LD) RL source, with very broad emission lines and no optical Feii within detection limits (see Marziani et al. 2003, for the criterion used). It also shows a large peak shift in its H$\beta$ profile, and prominent narrow lines ([Oiii]$\lambda$5007, H$\beta$), but also narrow components of Civ$\lambda$1549 and Ciii]$\lambda$1909 which are all well separated from the broad H$\beta$ profile. To deconvolve the blends around Ciii]$\lambda$1909 and Civ$\lambda$1549, we assume the same emission line profiles as for H$\beta$. The H$\beta$ broad profile can be described as the sum of a BC and a VBC. An unusual property of the BC is its large peak blueshift (it is unusual because of the shift amplitude: even if median spectra of Pop. B sources show a small BC redshift, there is a pretty large scatter with both red- and blueshifted peaks observed in individual sources; Zamfir et al. 2010). Applying in a self consistent way the BC and VBC to the Ciii]$\lambda$1909 and Civ$\lambda$1549 blends, leads to very interesting results as shown in Figure 8: * $\bullet$ Aliii$\lambda$1860 is very weak or even absent within the S/N limits; * $\bullet$ the Civ$\lambda$1549 profile is very similar to the one of H$\beta$: BC+VBC accounts for more than 90% emission, with a possible, very weak contribution of the blueshifted component which is usually dominating in Pop. A sources; * $\bullet$ the red wing observed in the 1900Å blend is accounted for only if there is a strong Ciii]$\lambda$1909 VBC, Feii and Feiii emission being below the detectability limit in this source; * $\bullet$ there is no evidence of a VBC in Aliii$\lambda$1860; * $\bullet$ the ratio Civ$\lambda$1549/Ciii]$\lambda$1909 BC is $\approx$ 10, a far cry from the ratios observed in extreme Pop. A sources like I Zw 1 and SDSS J120144.36+011611.6; * $\bullet$ even more interesting we find a Civ$\lambda$1549/Ciii]$\lambda$1909 VBC $\approx$10, as for the BC. We conclude that this object is fundamentally different from Pop. A sources. The low ionization, high density BC seems to be absent. The similarity in the Civ$\lambda$1549/Ciii]$\lambda$1909 ratios suggest that we are observing a gas in conditions similar to the one associated to the VBC. We predict that Feii will remain undetected or found to be weak even with S/N $\rightarrow\infty$. In the plane ($U$, $n_{\mathrm{H}}$) of Figure 11, the line ratios converge to a point at $\log U\approx$ -1.5, and log $n_{\mathrm{H}}$$\approx$ 10.1 (the Siii$\lambda$1814 line cannot be measured accurately since S/N is poor). In this case, the Ciii]$\lambda$1909/Civ$\lambda$1549 ratio also converges toward a ($U$, $n_{\mathrm{H}}$) value consistent with the one indicated by the Aliii$\lambda$1860/Siii$\lambda$1814\. In most other cases, this does not happen because the Aliii$\lambda$1860 doublet is too strong to be accounted for by gas of $n_{\mathrm{H}}$$\sim 10^{10}$cm-3 even at super-solar metallicity. Emission occurs at pretty high ionization, in conditions that once upon a time were thought to be standard in quasars (Davidson and Netzer 1979). This Pop. A and B difference at the extrema was already pointed out, in a semi-quantitative way, by Sulentic et al. (2000) and Marziani et al. (2001). It is however important not to generalize the case of 3C 390.3 to the remaining Pop. B sources. In most of them, Aliii$\lambda$1860 is detected, and there is evidence of strong Ciii]$\lambda$1909\. Actually Ciii]$\lambda$1909 emission seems to be appreciable in all of our VLT quasars. This means that we are in a composite situation, where low-ionization, high-density emission is present, along with significant VBC and other emission. The two extreme cases help us however to understand these more complex cases. ### 7.3 The Contribution of Lower Density Gas Once the true intensity of the BC components is retrieved, the presence of significant Ciii]$\lambda$1909 emission complicates the analysis. As pointed out, the photoionization solution for the BC suggests very high density, and in this region no Ciii]$\lambda$1909 emission is expected. Whenever Ciii]$\lambda$1909 is observed, we need to reverse the question: how much does any Ciii]$\lambda$1909 emitting gas contributes to the lines used for diagnostic ratios? Negligible contribution is expected to Aliii$\lambda$1860\. However, this is not true for Civ$\lambda$1549 and Siiii]$\lambda$1892\. Especially among Pop. A2 and A3 objects, it is not so obvious that the profile of Ciii]$\lambda$1909 and Siiii]$\lambda$1892 is the same. It could be well that the Ciii]$\lambda$1909 profile is narrower than the ones of Siiii]$\lambda$1892 and Aliii$\lambda$1860 (as found for SDSS J1201+0116), justifying the idea of Ciii]$\lambda$1909 emission from a disjoint region (§6.2). For the BAL QSOs in our sample and Aliii$\lambda$1860-strong sources most of what we ascribe to Ciii]$\lambda$1909 could be actually Feiii, as suggested by Hartig and Baldwin (1986). However, objects like the typical A1 sources show Siiii]$\lambda$1892 and Ciii]$\lambda$1909 with similar profiles, so in the following we will assume the worst condition, that is all Ciii]$\lambda$1909 emitting gas is contributing to the BC lines. To estimate the contribution of the Ciii]$\lambda$1909 emitting gas to Siiii]$\lambda$1892 and Civ$\lambda$1549, we can consider the trends observed along E1. Siiii]$\lambda$1892 is strong when Aliii$\lambda$1860 is strong, and the Siiii]$\lambda$1892/ Ciii]$\lambda$1909 ratio is lower when Ciii]$\lambda$1909 is strong, as visible in both Baldwin et al. (1996) and Bachev et al. (2004), as well as in the present paper. The observed Siiii]$\lambda$1892/Ciii]$\lambda$1909 ratio in the A1 median bin seems to be as low as in Mark 335, $\approx 0.4$. Pop. B sources that often show prominent Ciii]$\lambda$1909 also show a low Siiii]$\lambda$1892/Ciii]$\lambda$1909 ratio, as also appreciable in the median spectra of Bachev et al. (2004). These trends suggest that most Siiii]$\lambda$1892 is emitted where Aliii$\lambda$1860 is also emitted. As a consequence, any correction due to gas in different physical conditions (lower density) emitting Siiii]$\lambda$1892 is expected not to be dominant unless Ciii]$\lambda$1909 is extremely strong. The Ciii]$\lambda$1909 emitting gas should be at density $\log n\sim\ 9$, or lower. Higher density would imply increasing Siiii]$\lambda$1892/Ciii]$\lambda$1909 to values that would exceed the observed ones even if the LIL-BC is not emitting any significant Siiii]$\lambda$1892 (see Fig. 13). #### 7.3.1 Preliminary analysis of a low-$z$ sample To set these trends on a more quantitative basis we considered the set of pre- costar recalibrated sources, for which Ciii]$\lambda$1909 and Civ$\lambda$1549 data are publicly available (Kuraszkiewicz et al 2002, Evans & Koratkar 2004).We performed measurements (interactively with the task splot of iraf) for about $30$ sources with the highest S/N, holding a Ciii]$\lambda$1909 blend that could be relatively easily deblended, and following the expectations described in §4.4. The rest frame equivalent width of Aliii$\lambda$1860 and Siiii]$\lambda$1892 are found to be highly correlated (Fig. 12). The correlation is due to Siiii]$\lambda$1892 being stronger when Aliii$\lambda$1860 is strong, not necessarily because Ciii]$\lambda$1909 is strong: actually, the Siiii]$\lambda$1892/Ciii]$\lambda$1909 ratio can be low when Ciii]$\lambda$1909 is strong, and the Aliii$\lambda$1860/ Siiii]$\lambda$1892 ratio achieves maximum values when Ciii]$\lambda$1909 is faintest. This is consistent with the bulk of Aliii$\lambda$1860 and Siiii]$\lambda$1892 originating in the same region. At the same time the presence of Ciii]$\lambda$1909 lowers the Aliii$\lambda$1860/ Siiii]$\lambda$1892, Civ$\lambda$1549/ Siiii]$\lambda$1892 and Siiv$\lambda$1397/ Siiii]$\lambda$1892 because of “excess” Siiii]$\lambda$1892 emission associated to the gas emitting Ciii]$\lambda$1909\. #### 7.3.2 The effect of low-density gas on the product ($U$$n_{\mathrm{H}}$) We expect that any correction will increase density (increase Aliii$\lambda$1860/ Siiii]$\lambda$1892 lowering Siiii]$\lambda$1892) and decrease ionization parameter (lowering Civ$\lambda$1549 more than Siiii]$\lambda$1892), but that their product will be less affected. To show the amplitude of the effect we performed an experiment, adding to a pure, high-density solution, contribution from moderate density gas ($\log$ $n_{\mathrm{H}}$$\sim 9-10$). The Fig. 14 shows the displacement in the density – ionization parameter plane for several values of the Ciii]$\lambda$1909/ Siiii]$\lambda$1892 intensity ratio. The values refer to the Ciii]$\lambda$1909 addition over Siiii]$\lambda$1892 in the ideal case, corresponding to observed ratios of 0.34, 0.7, 1.0, 1.1 for an addition of a Ciii]$\lambda$1909 component whose intensity is 0.4, 1.0 ,1.5 ,2.0 the intensity of the Siiii]$\lambda$1892 component associate to the high density gas. As one can see from the figure, even if deviation for $U$ and $n_{\mathrm{H}}$ taken separately are significant, deviations for the product are by far less important. The largest change for the product is found to be $\approx$ 0.15, if we exclude the gray dots corresponding to $\log$ $n_{\mathrm{H}}$= 10 (an unlikley case, since this would imply a correction Siiii]$\lambda$1892/Ciii]$\lambda$1909 $>$ 1, inconsistent with what we observe when Ciii]$\lambda$1909 is strong). Following the expected line ratios of Fig.13 we apply a correction to the BC fluxes that is 0.4 and 1.5 Ciii]$\lambda$1909 for Siiii]$\lambda$1892 and Civ$\lambda$1549 respectively (Figures 15, 16 and 11), corresponding to $\log U=-2$ and $\log$$n_{\mathrm{H}}$= 9. We remark that that if $U$ is lower, the correction will have negligible effect, while assuming a larger $U$ will lead to Civ$\lambda$1549 flux in excess to the one observed. To constrain the ionization parameter we can first consider that, since the Ciii]$\lambda$1909 gas comes from a (relatively) low density region, the contribution to Siii$\lambda$1814 is small: for $\log n\sim\ 9$ and $\log U\approx-2$, the contribution should be $\approx 0.03$ Ciii]$\lambda$1909\. Second, another powerful feature is the Siiv$\lambda$1397 doublet (Baldwin et al. 1996): the Siiv$\lambda$1397 doublet is less affected by the Ciii]$\lambda$1909 correction, the contribution from lower density gas being estimated $\approx$ 0.25 Ciii]$\lambda$1909\. The line ratio Siiv$\lambda$1397/Aliii$\lambda$1860 is also sensitive to ionization and less affected by any lower density correction (provided that the relative abundance of S and Al stays the same, as it seems to be the case). In summary, corrected BC line intensities are computed as follows: $I^{\mathrm{c}}$(Siii$\lambda$1814)BC = $I$(Siii$\lambda$1814)BC – 0.03 $I$(Ciii]$\lambda$1909)BC; $I^{\mathrm{c}}$( Siiii]$\lambda$1892)BC = $I$( Siiii]$\lambda$1892)BC – 0.4 $I$(Ciii]$\lambda$1909)BC; $I^{\mathrm{c}}$(Siiv$\lambda$1397)BC = $I$(Siiv$\lambda$1397)BC – 0.26 $I$(Ciii]$\lambda$1909)BC; $I^{\mathrm{c}}$(Civ$\lambda$1549)BC = $I$(Civ$\lambda$1549)BC – 1.5 $I$(Ciii]$\lambda$1909)BC. ## 8 Results on the $z\approx$ 3 Quasars To estimate $\log$$n_{\mathrm{H}}$ and $\log U$ values, we use the cloudy contour plots of the ratios Aliii$\lambda$1860/ Siiii]$\lambda$1892, Siii$\lambda$1814/ Siiii]$\lambda$1892, Civ$\lambda$1549/ Siiii]$\lambda$1892, Siiv$\lambda$1397/ Siiii]$\lambda$1892 showed in Fig. 9222Note that there are regions where the ratio values are actually undefined: close to the high $U$ limit ($\log U\gtrsim-0.3$), ratios Aliii$\lambda$1860/ Siiii]$\lambda$1892 and Siii$\lambda$1814/ Siiii]$\lambda$1892(with $n_{\mathrm{H}}$$\lesssim 10^{9}$cm-3) should not be considered.. The data points of our objects are in regions were the ratios are well-defined. The ratios Civ$\lambda$1549/ Siiii]$\lambda$1892, Siiv$\lambda$1397/ Siiii]$\lambda$1892, and Siii$\lambda$1814/ Siiii]$\lambda$1892 are mainly sensitive to the ionization parameter $U$, while Aliii$\lambda$1860/ Siiii]$\lambda$1892 and Ciii]$\lambda$1909/ Siiii]$\lambda$1892 are mainly sensitive to the electron density. We know that Ciii]$\lambda$1909 is collisionally quenched at $\log$ $n_{\rm e}$$\gtrsim$ 10 and in the contour plot for Ciii]$\lambda$1909/ Siiii]$\lambda$1892 we see a step around this value. We measure the BC intensity of Siiii]$\lambda$1892, Aliii$\lambda$1860, Siii$\lambda$1814, Siiv$\lambda$1397 and Civ$\lambda$1549; with them we compute the diagnostic ratios (for 3C390.3 we use Ciii]$\lambda$1909 in place of Siii$\lambda$1814; however there is no object similar to 3C390.3 among the $z\approx$ 3 quasars). We present the fluxes of the line components in Tab. 3 and the equivalent width in Tab. 5. Table 4 shows the weak lines around Civ$\lambda$1549\. For Civ$\lambda$1549 line we show the core, blue shifted and the very broad components. Errors are at a $2\sigma$ confidence level, and include the sources of uncertainty described in §4.5. Errors are then quadratically propagated according to standard practice to compute intensity ratios and their logarithm. From Table 3 we can derive the diagnostic ratios. As we see in Fig. 1, Siii$\lambda$1814 is absorbed by telluric B band in J00103-0037, J03036-0023, J20497-0554 (most affected). We will not consider Siii$\lambda$1814 to compute $n_{\mathrm{H}}$ and $U$ on those cases. However, if we take at face value the Siii$\lambda$1814 measure on J00103-0037 and J20497-0554, it will converge close to the point set by the remaining two ratios. We display on a graph a line representing the behavior of each ratio under the assumption of solar metallicity; the ideal point where the lines representing different diagnostic ratios cross determines the values of $\log$$n_{\mathrm{H}}$ and $\log U$. Figs. 15 and 16 shows the contour plots were we can see that the diagnostic ratios converge to rather well defined values. The cross point is very precise for the objects J00103-0037, J00521-1108, J02287+0002 (using $z_{CIII}$), and J20497-0554 ; for the remaining objects J01225+1339, J02287+0002 (using $z_{OI}$), J02390-0038, J03036-0023, and J23509-0052 the cross point is slightly different. We must not forget the errors involving the fits, such as the changing of the shape profile that makes large the peak intensity if is Lorentzian or it could be less intense if it is Gaussian; the Feii pseudo- continuum contribution that affects principally to Siii$\lambda$1814 or the Feiii that in some cases affects Ciii]$\lambda$1909. In principle, the crossing point of the ratios Siiv$\lambda$1397/ Siiii]$\lambda$1892 and Siii$\lambda$1814/ Siiii]$\lambda$1892 is independent on metallicity. Therefore, any significant disagreement between this crossing point and the ratios based on Civ$\lambda$1549 may indicate chemical composition different from the assumed solar one (§8.1). The difficulty here is the large uncertainty of the Siii$\lambda$1814 line. In all contour plots we show the $\pm 2\sigma$ interval as a shaded band. So, it is proper to consider deviations from metallicity only in the case of 4 sources where the crossing point excluding Siii$\lambda$1814 is outside the uncertainty band. For the objects J00103-0037, J03036-0023 and J20497-0554 we exclude the Siii$\lambda$1814 line from the diagnostic ratios because it is affected by absorption. For the remaining objects, we take the average of the crossing contour plots of $\log$(Aliii$\lambda$1860/ Siiii]$\lambda$1892) crossing with $\log$(Civ$\lambda$1549/ Siiii]$\lambda$1892) and $\log$(Aliii$\lambda$1860/ Siiii]$\lambda$1892) crossing with $\log$(Siii$\lambda$1814/ Siiii]$\lambda$1892). For 3C390.3 we use $\log$(Ciii]$\lambda$1909/ Siiii]$\lambda$1892) instead of $\log$(Siii$\lambda$1814/ Siiii]$\lambda$1892). Table 6 summarizes the $\log n_{\mathrm{H}}$ and $\log U$ values including their uncertainty. Since $U$ and $n_{\mathrm{H}}$ are not independent quantities (their correlation coefficient is found to be 0.55), we adopt the appropriate formula for the errors on the product $U$$n_{\mathrm{H}}$ (following Bevington 1969). We present average values of the crossing points for extreme objects of Fig. 11 and 18(d). In the SDSS J12014+0116 case we give full weight to the Siii$\lambda$1814 measurement. The crossing points disagree somewhat for SDSS J12014+0116. This could be due to an underestimate of both Siii$\lambda$1814 and Civ$\lambda$1549 in the fits. On the one hand, Siii$\lambda$1814 is clearly seen and strong but is contaminated by Feii blend; on the other hand, Civ$\lambda$1549 is strongly affected by the blue-shifted component and by the assumption that it is of Gaussian shape. An increase by 30% in the measurement of the intensity of the two line would lead to a better agreement but this is somewhat an ad-hoc speculation. Rather, the significant difference between the crossing point of the Siii$\lambda$1814/ Siiii]$\lambda$1892 and Siiv$\lambda$1397/ Siiii]$\lambda$1892 ratios and the other ones point toward strong metal enrichment. We will show in §8.1 that this is probably the case. At any rate, the convergence is toward a value of $\log U\approx-3$, lower than for the other $z\approx$3 quasars. This is reflected in the Civ$\lambda$1549 EW of this source, also significantly lower. It is intriguing to note that the correction because of lower density drives the other sources toward values of $U$ and $n_{\mathrm{H}}$ that are closer to the ones of SDSS J12014+0116. Table 7 reports the values of the $r_{\rm BLR}$ and the $M_{\rm BH}$ of our 8 objects and the extreme objects in the last two rows. Column 1 identifies the quasar name; Col. 2 gives the quasar proper distance in mega parsecs [Mpc]; Cols. 3 and 4 are the continuum specific flux value at 1350Å and 1700Å respectively, Col. 5 reports the FWHM in km s-1 for the broad components, Col. 6 is the Population designation. Cols. (7) to (10) report the logarithm of the size of the BLR in cm obtained from: a) $1Z_{\odot}$, b) $1Z_{\odot}$ line ratios corrected because of low density emission, c) $5Z_{\odot}$, and d) $5Z_{\odot}$ line ratios corrected because of low density emission. Cols. (11) -(14) list the logarithm of the black hole mass in solar masses in the same order as for $r_{\rm BLR}$. Finally Col. (15) is $M_{\rm BH}$ computed from Vestergaard and Peterson (2006) formula (Equation 10). We will explain in §10 how these quantities are computed. ### 8.1 Effects of Metallicity The strength of Nv$\lambda$1240 relative to Civ$\lambda$1549 and Heii$\lambda$1640 suggests supersolar chemical abundances (Hamann and Ferland 1993; Hamann & Ferland 1999). Chemical abundances may be well 5 to 10 times solar (Dhanda et al. 2007), with Z $\approx$ 5$Z_{\odot}$ reputed typical of high $z$ quasars (Ferland et al. 1996). The E1 sequence seems to be mainly a sequence of ionization in the sense of a steady decrease in prominence of the low-ionization BC toward Population B (Marziani et al. 2001, 2010). However, this is not to neglect that metal-enrichment also plays a role, especially for the most extreme Pop. A sources i.e., those in bin A3 and higher (Sulentic et al. 2001). The lines employed in the present study come from carbon, silicon and aluminium; all these element can be significantly depleted from gas if dust grains are formed (e. g., Mathis 1990). However, the emitting regions where our lines are produced are thought too hot to contain significant amount of dust (a definition of broad line region is right the central engine region below the dust sublimation region: e.g., Elitzur 2009). In addition Si and Al are expected to be produced under similar circumstances in the late stage of evolution of massive stars (Clayton 1983, Ch. 7). We note also that the Civ$\lambda$1549/ Siiii]$\lambda$1892 and Civ$\lambda$1549/Aliii$\lambda$1860 usually give results that are in perfect agreement in the plane ($n_{\mathrm{H}}$,$U$). These findings support our assumption that, if metallicity variations are present, the relative abundance Al to Si remains constant. We considered two cases for enhanced metallicity: (1) constant solar abundance ratio Al:Si:C with $Z=5Z_{\odot}$ (5Z) ; (2) an overabundance of Si with respect to carbon by a factor 3, again with $Z=5Z_{\odot}$ (5ZSA). This condition comes from the yields listed by Woosley and Weaver (1995) from type II Supernovæ. The Si overabundance is also supported by the chemical composition of the gas returned to the interstellar medium by an evolved population with a top-loaded initial mass function simulated using Starburst 99 (Leitherer et al. 1999). The abundance of Al should scale roughly with the one of Si. While some cases with Al scaling with C are possible from the Woosely and Weaver (1995) yields, they are rarer than cases in which Al scales with Si. This latter case is appropriate for the most massive progenitors. Also, the assumption of Al scaling with C with [Si/C] = 0.477 would yield to implausible high density and lack of convergence to a well-defined solution for $\log$$n_{\mathrm{H}}$$\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}14$. We therefore assume in the following that Al scales with Si in the two cases listed above. An array of simulations as a function of ionization parameter and density was computed assuming the conditions (1) and (2) listed in the previous paragraph. As expected, if the solar metallicity is simply scaled by a factor (5Z) we find that the ratio Aliii$\lambda$1860/ Siiii]$\lambda$1892 is not strongly dependent on $Z$: the ratio increases by about 40% passing from $Z=1Z_{\odot}$ to $Z=5Z_{\odot}$, for $\log$$n_{\mathrm{H}}$$\approx 12$ and $\log U\approx$–2. The same is true for the Siii$\lambda$1814/ Siiii]$\lambda$1892 and Siiv$\lambda$1397/ Siiii]$\lambda$1892 ratios. Since the first ratio sets $n_{\mathrm{H}}$, and the last two $U$, the ratios mentioned in this paragraph should be preferred because they provide $n_{\mathrm{H}}$ and $U$ values that are weakly affected by a factor 5 change in metallicity. A posteriori we confirm that the effect on the product $U$$n_{\mathrm{H}}$ derived also with ratios involving Civ$\lambda$1549 is negligible in case 5Z (and it should be even more so if a metallicity increase is $Z_{\odot}\lesssim Z\lesssim 5Z_{\odot}$). The two extreme cases seem to be revealing also as far as metallicity is concerned. Ratios converge to a fairly well defined point in the case of 3C 390.3 (see Fig. 11, left panel, $Z=1Z_{\odot}$ assumed). In this case there is no major evidence of supersolar metallicity. The converse is true in the case of SDSS J120144.36+011611.6. The ratios involving Civ$\lambda$1549 indicate a lower ionization level, with $\log U\sim-3$. This is because C3+ changes ionization state to C2+ for smaller ionization parameters, and so Civ$\lambda$1549 rapidly disappears. The Civ$\lambda$1549 intensity depends weakly on $Z$ while the Aliii$\lambda$1860 and Siiii]$\lambda$1892 lines are more sensitive. The 5Z case yields values closer to the ones obtained from the Si and Al line ratios, but not yet in concordance. If we pass to case 5ZSA with a factor 3 Si overabundance, then the concordance of the line ratios is good, especially if no correction because of low density emission is applied. As a further confirmation we checked that the metallicity-dependent Civ$\lambda$1549/Siiv$\lambda$1397 ratio is in very good agreement with the crossing point of the other lines. Therefore, in this case, we have independent measures of metallicity, $n_{\mathrm{H}}$, and $U$. The case of even higher metallicity, say $Z\sim 10Z_{\odot}$, remains to be explored but may not be appropriate considering the good agreement with 5ZSA. In principle, the discrepancy in the intersection point, with Siiii]$\lambda$1892/Civ$\lambda$1549 and Aliii$\lambda$1860/Civ$\lambda$1549 yielding lower $U$ than the Al-Si ratios, should signal a significant enrichment in Si and Al of the BLR gas. In this case the Siiv$\lambda$1397/Civ$\lambda$1549 ratio should be helpful, as it can be assumed to be dependent mainly on the Si abundance relative to C. Here, more than precisely determining the exact abundance value we are interested in analyzing the effect of large metallicity changes on $U$, $n_{\mathrm{H}}$, and their product. Appreciable discrepancy is visible in the contour plots of the BAL quasar J01225+1339, J03036-0023, J20497-0554 and J23509-0052 if $Z=Z_{\odot}$ is assumed. In the same plots made for $Z=5Z_{\odot}$SA, the agreement becomes better (see Fig. 18). High metallicity yields higher $U$ and smaller $n_{\mathrm{H}}$ if emission line ratios involving Civ$\lambda$1549 are considered. This reflect the increase in abudance of Si and Al with respect to C, and the fact the Siii$\lambda$1814, Siiii]$\lambda$1892, Aliii$\lambda$1860 lines are emitted at lower ionization than Civ$\lambda$1549\. In the case of sources like J02390-0038 and J20497-0554, the discrepancy of the crossing lines might indicate $Z_{\odot}\lesssim Z\lesssim 5Z_{\odot}$, more than the extreme enrichment like the one assumed in 5ZSA, as also suggested by the Civ$\lambda$1549/Siiv$\lambda$1397 ratio. In Fig. 18 and 19 the contour plots are shown for the case 5ZSA. We consider uncorrected and corrected line ratios as two independent cases. The halftone bands also show the importance of accurate Siii$\lambda$1814 measurements to infer unambiguous constrain on metallicity (and, by extension, to estimate $n_{\mathrm{H}}$ and $U$ independently). The product $n_{\mathrm{H}}$$U$ is however much less affected than $n_{\mathrm{H}}$ and $U$ individually (Tab. 6). A first $r_{\rm BLR}$ and $M_{\rm BH}$ estimate can be obtained considering only the Siii$\lambda$1814/ Siiii]$\lambda$1892 and Aliii$\lambda$1860/ Siiii]$\lambda$1892 ratios. We conclude that the effect of scaling the metallicity up to $Z=5Z_{\odot}$ is within the uncertainty of the method, and particularly small if the ratios Siii$\lambda$1814/ Siiii]$\lambda$1892, Aliii$\lambda$1860/ Siiii]$\lambda$1892, and Siiv$\lambda$1397/ Siiii]$\lambda$1892 are considered to compute $n_{\mathrm{H}}$ and $U$. It is significant if strong enrichment of Al and Si over C occurs. A more refined approach could exploit the dependence of Aliii$\lambda$1860/Civ$\lambda$1549 (and Siiii]$\lambda$1892/Civ$\lambda$1549 and Siiv$\lambda$1397/Civ$\lambda$1549) on $Z$ to build a 3D diagram where $n_{\mathrm{H}}$, $U$, $Z$ along with Si-Al enrichment can be determined independently. ### 8.2 Determining the best estimate of $n_{\mathrm{H}}$, $U$ There are two sets of line ratios for each object: one comes from the specfit results, and the other is the one computed after correcting the specfit results because of low density contributions. Solutions with corrected value usually point toward very high density and low ionization, predicting Siii$\lambda$1814 emission even twice as strong as Siiii]$\lambda$1892\. However, increasing the metallicity for corrected ratios leads to better agreement among lines and more reasonable $n_{\mathrm{H}}$, $U$ values, while leaving the product fairly unaffected. Especially the assumption of Silicon - Aluminium enrichment improves the concordance in the intriguing cases of SDSS J12014+0116, J01225+1339, J02287+0002 (using $z_{oi}$), the extreme objects and the two BAL QSOs in our sample. However, apart from the case of SDSS J12014+0116, the enrichment is probably excessive. From the discussion above the independent determination of $n_{\mathrm{H}}$ and $U$ seems possible only if metallicity is at least roughly known. We exclude metallicity cases where we find a sizeable disagreement in the crossing points (with the exceptions of ratios involving Siii$\lambda$1814: the ionization level can be estimated in a $Z$-independent way using the Siiv$\lambda$1397/Siii$\lambda$1814 and the Siiv$\lambda$1397). We consider each individual source with line intensity before and after correction to obtain two independent sets of product $n_{\mathrm{H}}$$U$ values. As mentioned, changing metallicity is not affecting the product $n_{\mathrm{H}}$$U$ as much as $n_{\mathrm{H}}$ and $U$ individually. In case of concordance of the crossing points and of high accuracy in the Siii$\lambda$1814 ratio, $n_{\mathrm{H}}$ $U$, and $Z$ can be considered independently determined. In Table 6 we indicate the values that are deemed most appropriate. ## 9 A Photoionization Method to Compute the Broad Line Region Distance and the Black Hole Mass. The distance of the broad line region ($r_{\rm BLR}$) from the central continuum source and the black hole mass ($M_{\rm BH}$) are key parameters that let us understand the dynamics of the gas in the emitting region and the quasar behavior and evolution. In this work we will use a method based on the determination of $n_{\mathrm{H}}$ and $U$ to compute $r_{\rm BLR}$. Eq. 1 can be rewritten as $r_{BLR}=\left[\frac{\int_{\nu_{0}}^{+\infty}\frac{L_{\nu}}{h\nu}d\nu}{4\pi Un_{\mathrm{H}}c}\right]^{1/2}$ (3) and also as $r_{BLR}=\frac{1}{h^{1/2}c}(Un_{\mathrm{H}})^{-1/2}\left(\int_{0}^{\lambda_{Ly}}f_{\lambda}\lambda d\lambda\right)^{1/2}d_{p}$ (4) where $h$ is the Plank constant, $c$ is the light speed, $d_{\mathrm{p}}$ is the proper distance. The integral is carried out from the Lyman limit to the shortest wavelengths on the rest frame specific flux $f_{\lambda}$. For the integral we will use two Spectral Energy Distributions (SEDs): one described by Mathews & Ferland (1987) and one by Laor et al. (1997a), also reproduced in Fig. 20. Eq. 4 becomes: $r_{\rm BLR}\approx 93\cdot(Un_{\mathrm{e}})_{10}^{-\frac{1}{2}}\cdot f_{\lambda_{0},-15}^{\frac{1}{2}}\tilde{Q}_{H,0.1}^{\frac{1}{2}}\zeta(z,0.3,0.7)~{}~{}\mathrm{ld}$ (5) where $\tilde{Q}_{H}=\int_{0}^{\lambda_{\mathrm{Ly}}}\tilde{n}_{\lambda}\lambda d\lambda$, and $\zeta(z,0.3,0.7)$ is an interpolation function for $d_{\mathrm{p}}$ as a function of redshift. $\tilde{Q}_{H}$ is 0.00963 cmÅ in the case the continuum of Laor et al. (1997) is considered; $\tilde{Q}_{H}\approx$0.02181 cmÅ for Mathews & Ferland (1987). We use their average value, since the derived $U$ and $n_{\mathrm{H}}$ are not sensitive to the two different shapes to a first approximation.333Since the Laor et al. (1997) continuum produces a fewer ionizing photons, the same value of $U$ is obtained at a smaller distance. Knowing $r_{\rm BLR}$ we can calculate the $M_{\mathrm{BH}}$ assuming virial motions of the gas $M_{\mathrm{BH}}=f\frac{\Delta v^{2}r_{\mathrm{BLR}}}{G}.$ (6) or, $M_{\mathrm{BH}}=\frac{3}{4G}f_{0.75}(FWHM)^{2}r_{BLR}$ (7) with the geometry term $f\approx 0.75$, corresponding to $f_{0.75}\approx 1.0$ (Graham et al. 2011, see also Onken et al. 2004 and Woo et al. 2010). Collin et al. (2006) suggest that $f$ is significantly different for Pop.A and B sources; we do not consider here their important result for the sake of comparison with previous work (§10.2). The resulting $r_{\rm BLR}$ and $M_{\rm BH}$ are reported in Columns 7 to 14 of Table 7. Errors are at 2$\sigma$ confidence level and have been computed propagating quadratically the major sources of uncertainty. More precisely, in addition to the error on $\log$($n_{\mathrm{H}}$$U$), the $r_{\rm BLR}$ determination is affected by the uncertainty in the spectrophotometry (specific fluxes of Col. 3 of Tab. 7), and errors on the shape of the ionizing continuum. The two SEDs that we assumed as extreme yield a difference in ionizing photons of a factor 2.2. At a 2$\sigma$ confidence level this corresponds to an uncertainty of $\pm$0.065 in logarithm. An additional source of uncertainty affects $M_{\rm BH}$ due to the FWHM determination. Errors on FWHM are quadratically added to the uncertainty on $r_{\rm BLR}$ in the values reported as $\log$$M_{\rm BH}$ errors. ## 10 Discussion ### 10.1 Previous work There have been several studies aimed at computing $r_{\rm BLR}$ and $M_{\rm BH}$. A direct measure of $r_{\rm BLR}$ through reverberation mapping requires an enormous amount of observational effort and has only been applied to a relatively small number of quasars: slightly less than 50 objects with $z\lesssim 0.4$ (Kaspi et al. 2000, 2005, Peterson et al. 2004; Bentz et al. 2010). A second way to measure $r_{\rm BLR}$ uses a less direct method. Kaspi et al. (2000, 2005) and Bentz et al. (2009) used reverberation mapping results to find, in an empirical way, a relationship between $r_{\rm BLR}$ and the optical continuum luminosity at 5100Å, $r_{\mathrm{BLR}}\propto L^{\alpha}$ (8) with $\alpha\approx 0.52$. Vestergaard & Peterson (2006) obtained a similar result for the optical continuum luminosity with an $\alpha\approx 0.50$ and for the UV continuum at 1350Å, $\alpha\approx 0.53$. These relationships have been used to compute the $r_{\rm BLR}$ not only for nearby objects, but also for high redshift, high luminosity objects. There are other works that use single epoch spectra and the continuum at 3000Å, obtaining an $\alpha\approx 0.47$ (McLure & Jarvis 2002). We can rewrite Eq. 6 as $M_{\mathrm{BH}}\propto f\frac{\mathrm{FWHM}^{2}L^{\alpha}}{G}.$ (9) H$\beta$ is a low ionization strong line whose FWHM has been widely used to determine the $M_{\rm BH}$ for objects mainly up to $z\lesssim 0.9$; above this limit IR spectrometers and large telescopes are needed to cover the redshifted line. For distant objects ($z\sim 2$), an alternative is to use Civ$\lambda$1549, a high ionization line emitted in the UV. However, this line should be used with caution because the line is often blueshifted. This means that at least part of this line is likely emitted in an outflow (Sulentic et al. 2007; Richards et al. 2010). Thus the estimation of $M_{\rm BH}$ using FWHM(Civ$\lambda$1549) tend to be systematically higher than those using FWHM(H$\beta$), especially for objects of Population A. ### 10.2 Comparison with Vestergaard and Peterson (2006) Vestergaard & Peterson (2006) used the relationship $r_{\mathrm{BLR}}\propto L^{0.53}$ to obtain the following formula that relates $M_{\rm BH}$ to the FWHM(Civ$\lambda$1549) and the continuum luminosity at 1350Å: $\log M_{\mathrm{BH}}(\mathrm{C\sc{\i v}})=\log\left\\{\left[\frac{\mathrm{FWHM}(\mathrm{C\sc{\i v}})}{1000\>\mathrm{km}\>\mathrm{s}^{-1}}\right]^{2}\left[\frac{\lambda L_{\lambda}(1350\mathrm{\AA})}{10^{44}\>\mathrm{ergs}\>\mathrm{s}^{-1}}\right]^{0.53}\right\\}+(6.66\pm 0.01)-s_{\mathrm{f}}.$ (10) The scale factor $s_{\mathrm{f}}\approx-0.27$ sets the masses to the $f$ value obtained by Graham et al. (2011). In Cols. 11 and 15 of Table 7 and in Fig. 21 we compare our $M_{\rm BH}$ results with those using Eq. 10. We do not apply corrections for radiation-pressure effects that are likely relevant especially for objects radiating at large Eddington ratio (Netzer 2009; Netzer & Marziani 2010). The difference between this computation and the one reported in Sulentic et al. (2007) is that in the latter work the blueshifted component was not separated from the broad component of Civ$\lambda$1549 just to show how larger values of FWHM(Civ$\lambda$1549) yielded $M_{\rm BH}$ much larger than the ones derived from FWHM(H$\beta$) in Pop. A objects. We compare the masses obtained using our photoionization method with those of Vestergaard & Peterson (2006) in Fig. 21. We use the FWHM of the BC as an estimator of the virial line broadening. From our results we can see that the masses agree within less than 1$\sigma$ uncertainty in the luminosity correlation (0.33). There is a systematic offset of 0.17$\pm$0.10 if uncorrected ratios are used. The mass values obtained after correction for low density gas are systematically lower. This happens because the correction increases the product $U$$n_{\mathrm{H}}$, lowering $r_{\rm BLR}$ and hence $M_{\rm BH}$. The $M_{\rm BH}$ obtained after corrections are within the error bars. The systematic offset is then $0.13\pm 0.12$. It is important to consider that the computed correction is in many ways a maximum correction. Ciii]$\lambda$1909 emission is assumed to have the same FWHM of Siiii]$\lambda$1892 while it could be significantly narrower; in addition part of the Ciii]$\lambda$1909 emission could be due to Feiii $\lambda$ 1914\. In many sources of Pop. A the correction could be ignored altogether. The present results indicate that the Kaspi et al. (2000) relationships can be extended to be used in high redshift objects (or at least until z$\sim$3) if the FWHM of the core broad line region can be well determined and measured. In order to do this, we need: * $\bullet$ Spectra with S/N high enough to see the profile shape that allows decomposition of the Civ$\lambda$1549 line, especially to separate the blue component from the broad core; * $\bullet$ to follow the methodological considerations explained in §4 . Fig. 21 should be looked at with two cautions. First, the correlation is dominated by the luminosity dependence of $r_{\rm BLR}$, used to compute $M_{\rm BH}$ in both cases. Second, the spread of $M_{\rm BH}$ values is small, less than one order of magnitude (and most objects have statistically indistinguishable masses). Our estimated error bars are however smaller compared to the spread expected on the basis of the $r_{\rm BLR}$-$L$ correlation which is, according to Vestergaard & Peterson (2006), $\pm$0.66 at a 2$\sigma$ confidence level. The two shaded bands of Fig. 21 limit the region where we can expect to find data points on the basis of the $r_{\rm BLR}$-$L$ correlation. Clearly, a proper interpretation of the Civ$\lambda$1549 profile may help to reduce the scatter. In any case, our method should provide $M_{\rm BH}$ estimates with somewhat lower uncertainty. It is interesting to note that SDSS J12014+0116 appear at the largest $M_{\rm BH}$, and the agreement with the $r_{\rm BLR}$-$L$ is very good. The $r_{\rm BLR}$ value of 3C 390.3 we obtain is fairly uncertain due to the low S/N. The size computed in the present paper is larger than the reverberation mapping derived $r_{\rm BLR}$ for H$\beta$ and Civ$\lambda$1549, although consistent with the value derived from Ly$\alpha$. 3C 390.3 has the unusual property of having a response time longer in Civ$\lambda$1549 than in H$\beta$, although the large error bars do not exclude that the two lines respond with similar time. This behavior might be related to different physical conditions found for this object. ### 10.3 The LIL-BLR While the existence of high density and low ionization has been invoked since long to explain Feii emission (especially by S. Collin and collaborators, as mentioned in the introduction), we have provided additional evidence that high density and low ionization are indeed diagnosed from emission lines other than Feii blends, and that conditions are the one producing most line emission in extreme Pop. A quasars. Last but not least, both $U$ and $n_{\mathrm{H}}$ can be observationally determined with reasonable accuracy from the diagnostic ratios Siii$\lambda$1814/ Siiii]$\lambda$1892, Siiv$\lambda$1397/ Siiii]$\lambda$1892, Aliii$\lambda$1860/ Siiii]$\lambda$1892, and Civ$\lambda$1549/(Aliii$\lambda$1860 or Siiii]$\lambda$1892). Metallicity can be also constrained from the previous ratios as well as from Civ$\lambda$1549/Siiv$\lambda$1397\. These ratios are pretty well defined, while estimating Ciii]$\lambda$1909 and Feiii$\lambda$1914 relative contribution is not relevant to our method (with the exceptions of sources like 3C 390.3). This low ionization BLR (or LIL BLR) has very similar properties to the Oi and Caii emitting region identified by Matsuoka et al. (2008). The LIL-BLR seems to be present in the vast majority of quasars, probably all the ones with significant Feii emission (Marziani et al. 2010). The low values of $U$ could be a consequence of the high density rather than of a far away location of the emitting region. The assumption of a single well defined value of $U$ and $n_{\mathrm{H}}$ is probably an idealization even for the LIL-BLR taken alone; however the convergence of emission line ratios toward a well defined point, along with the ability to qualitatively explain Feii and most H$\beta$ emission, indicate that the LIL-BLR might be a region with a small range of low $U$ and high $n_{\mathrm{H}}$. Do density and ionization parameter in the LIL BLR converge to a single well defined value with a small dispersion? A tentative answer comes from the values reported in Tab. 6. Excluding 3C390.3, the range $U$ and $n_{\mathrm{H}}$ span is not very large (even considering changes in metallicity): less than one order of magnitude around $\log$ $n_{\mathrm{H}}$$\sim$ 12.5, and ionization parameter $\sim-2.75$, and the spread is not much larger than the uncertainties in the individual measurements. The product $U\cdot$$n_{\mathrm{H}}$ seems to be fairly stable. We have $<\log(U\cdot$$n_{\mathrm{H}}$)$>\approx 9.5$, with a sample dispersion of 0.15 (excluding 3C 390.3). We applied the same method to 14 low-$z$ quasars (Negrete et al. 2010), and we obtain $<\log(U\cdot$$n_{\mathrm{H}}$)$>\approx 9.7$, with a dispersion of 0.3. Assuming $\log(U\cdot$$n_{\mathrm{H}}$) $\approx 9.6$ could be a good approach to estimate $M_{\rm BH}$ if elaborate measurements on Civ$\lambda$1549 and Ciii]$\lambda$1909 are not possible, and only a rough estimate of the Siiii]$\lambda$1892 BC FWHM is available. On the other hand, there is still a major effect of measurement errors on the uncertainty derived for $\log(U\cdot$$n_{\mathrm{H}}$); instrumental improvements may lead to a significant appreciation of object-by-object diversity. The $U\cdot$$n_{\mathrm{H}}$ values reported in Table 6 are not very far from the average value obtained by Padovani and Rafanelli (1988). This is not surprising since the spectra at low and high $z$ seems to show the same diversity, classified through the E1 sequence and the Pop. A/Pop. B distinction. In other words, NLSy1-like sources whose spectrum is similar to I Zw1 appear to be present at high redshift, meaning that the LIL-BLR remains strong and prominent over a wide range of redshifts. A second reason is that Padovani and Rafanelli (1988) considered H$\beta$. Emission of H$\beta$ can be significant under a much wider range of $U$ and $n_{\mathrm{H}}$; however, the Aliii$\lambda$1860, Mgii$\lambda$2800, Feii emitting region should be also a strong producer of H$\beta$. This region with well-defined physical conditions is expected to be the emitter of the core of H$\beta$, i.e., the part of the line responding more strongly to continuum changes. #### 10.3.1 Verification on EW and Line Luminosity Considering that we have very low ionization parameter values, a legitimate question is whether we have a sufficient number of photons to explain the EW and luminosity of the emission lines. We made a preliminary check of consistency for the EW from CLOUDY simulations. A second, a-posteriori test was to consider the predicted line luminosity assuming the actual luminosity of the quasar, the density and the distance $r_{\rm BLR}$ derived from our method, and spherical geometry. A remarkable property common to all spectra is the low EW of the emission lines. The extreme source SDSS J120144.36+011611.6 has total $W$(Civ$\lambda$1549) $\approx$ 19 Å, which becomes $\approx$ 7 Å if only the LIL component is considered. The whole 1900 Å blend has $W\approx$ 20 Å; the individual EWs of Siiii]$\lambda$1892 and Aliii$\lambda$1860 are just a few Å. Similar considerations apply to the other high-$z$ quasars, especially for Pop. A sources. In this case we have $W$(Civ$\lambda$1549)$\approx 20-30$ Å, the equivalent width of the whole 1900 Å $30-40$Å including the uncertain contribution of Feiii and Ciii]$\lambda$1909\. This has the important implication that the ionization parameter cannot be very large. From our simulations, we deduce that the LIL-BLR $\log U$ is always $\lesssim-2.0$. On the other hand, toward the low $U$, high density limit emission lines tend to disappear altogether. The predicted EWs become too low to account for the observed EW with a covering factor $f_{\mathrm{c}}\lesssim 0.5$ if $\log U\lesssim-3$. The values obtained after correction $\log U\approx-3.25$ are still possible within the condition of $f_{\mathrm{c}}\lesssim 0.5$ since the Civ$\lambda$1549 EW is greatly diminished. Cases of very low $U$ are rather indicative of strong metal enhancement than of extremely low ionization level (see §8.1). If we take $\log$ $n_{\mathrm{H}}$$\approx$12.5, $\log U=-2.75$, the predicted equivalent width of is $W$(Civ$\lambda$1549)$\approx$37 Å for a covering factor 0.5, close to the largest values observed in our quasars. However, The Civ$\lambda$1549 EW of SDSS J12014+0116 can be accounted for by the values of $\log U\approx-3.00$ and $\log$ $n_{\mathrm{H}}$$\sim$ 12.75 – 12.50. If we model the BLR with a spherical geometry where the emitting gas covers a fraction $f_{\mathrm{c}}$ of the continuum, CLOUDY computations confirm that the line luminosity can be accounted for. The luminosity at 1700 Å of J00103-0037 is $\log\lambda L_{\lambda}\approx$ 46.6; the predicted luminosity of Civ$\lambda$1549 is $\log L$(Civ$\lambda$1549) $\approx$ 44.5, under the assumption of $Z=Z_{\odot}$, $f_{\mathrm{c}}=0.1$, and Mathews & Ferland continuum shape. The observed Civ$\lambda$1549 line luminosity of J00103-0037, $\approx 9\cdot 10^{44}$ ergs s-1, is obtained with $f_{\mathrm{c}}\approx 0.3$. #### 10.3.2 Analogy with $\eta$ Carinæ The physical conditions we envisage for the BC of quasars find a correspondence in the so-called Weigelt blobs of $\eta$ Carinæ, located in the equatorial plane of the system, perpendicular to the symmetry axis of the bipolar lobes forming the homunculus nebula (cf. Marziani et al. 2010). Unlike the gas of the bipolar lobes, predominantly shock heated, the Weigelt blobs are believed to be dense gas photoionized by the radiation associated to the central, massive star and to a possible companion (e.g., Johansson et al. 2000; Davidson 2005). The spectrum of the Weigelt blobs shows very weak Ciii]$\lambda$1909 along with a prominent line at $\lambda$1914, ascribed to the $z^{7}P_{3}^{0}\rightarrow a^{7}S_{3}$ Feiii transition. The line appears very strong because the upper level is populated by Ly$\alpha$ fluorescence. This very same process is expected to be present also in quasars. Indeed, in I Zw 1, where lines are narrow, and in SDSS J120144.36+011611.6 the peak emission at around 1914 Å is actually visible. The amount of Ly$\alpha$ pumping to the upper level (z${}^{7}P^{\mathrm{o}}_{3}$) of the UV 34 cannot be estimated through the standard edition of CLOUDY (the relevant levels of the UV 34 multiplets of Fe+2 ion are not included). Additional photoionization computations including a suitable Fe+2 model and line transfer should be considered. This is beyond the aim of the present study; we can conclude in a qualitative fashion that the spectrum of the $\eta$ Carinæ blobs supports a view of the low-ionization part of the BLR that is not conventional: very high density gas, at very low ionization. ## 11 Conclusions In this paper we presented new observations of eight high redshift quasars. The spectra were meant to provide high S/N, moderate resolution data on which the Civ$\lambda$1549, Siiii]$\lambda$1892, Aliii$\lambda$1860, and Siii$\lambda$1814 emission line profiles could be accurately analyzed. Line profile fits allowed us to isolate a specific component whose intensity ratios were used to derive consistent values for electron density and ionization parameter. This line component (LIL BC) seems to be emitted predominantly by low ionization, high density gas in the majority of quasars studied thus far by us. These results permitted us to compute the product $n_{\mathrm{H}}$$\cdot U$ and hence the size of the Broad Line Region and the central black hole mass. The method described in this paper rests on the assumption of photoionization as the mechanism of gas heating; on the assumption of isotropic luminosity, and on line ratios predicted by cloudy simulations. The photoionization method explored in this paper offers an estimate of $r_{\rm BLR}$ for each quasar, with some advantages on the $r_{\rm BLR}$ valued derived from the luminosity- size correlation. The luminosity correlation suffers from large scatter and is simply extrapolated to very high luminosity without any support since there are, unfortunately, no conclusive results on reverberation of high luminosity quasars even if heroic efforts are underway (e.g., Trevese et al. 2007, Botti et al. 2010). We found that the black hole masses derived from the computed $r_{\rm BLR}$ and from the virial assumption are in good agreement with the ones derived from the luminosity-size relationship. Actually, Fig. 21 suggests that we might have reduced the errors of the $M_{\rm BH}$ by a factor of two with respect to the expectation from the Vestergaard & Peterson (2006) relationship. We repeat that our $M_{\rm BH}$ and $r_{\rm BLR}$ results are based on the product $n_{\mathrm{H}}$$\cdot U$ and not on values of $n_{\mathrm{H}}$ and of $U$ taken separately. It seems that this product converges to two typical ranges of values, one of them associated to low-ionization, high density gas (the LIL-BLR). For our $n_{\mathrm{H}}$ and $U$ determinations we do not use ratio Ciii]$\lambda$1909/ Siiii]$\lambda$1892 except for 3C 390.3. As we discussed in §7.2, this ratio should not be considered at high density because Ciii]$\lambda$1909 is collisionally quenced if $n_{\mathrm{H}}$ $\gtrsim 10^{10}$ cm-3. Ciii]$\lambda$1909 is produced in conditions that are very different from the ones we found for the LIL-BLR. While the method can be applied to most quasars, the application seems to be especially straightforward to quasars whose spectrum is like SDSS J120144.36+011611.6 (if high metallicity is properly taken into account) or, at the other end, 3C390.3. In the first case we have dominance by the LIL-BLR, in the second case the LIL-BLR seems to be completely absent and physical conditions look radically different: high ionization and moderate density. An inspection of SDSS spectra covering both the 1900 Å blend and Civ$\lambda$1549 (up to $z\approx 3.5$) shows that SDSS J120144.36+011611.6 has many replicas at high redshift, accounting for at least a few percent of all quasars. These high- metallicity objects should be the first candidates to expand black hole mass computations to high redshift without relying on the $r_{\rm BLR}$ \- $L$ correlation. To apply the photoionization method in the most effective way, determining $n_{\mathrm{H}}$ and $U$ with the lowest uncertainty, spectral data should be of moderate resolution ($\lambda/\Delta\lambda\sim 1000$) as well as of high S/N. If the Siii$\lambda$1814 line can be measured in an accurate way, it would be possible to derive independent estimates of $U$, $n_{\mathrm{H}}$, and $Z/Z_{\odot}$ in most quasars. Especially the most extreme (in terms of Aliii$\lambda$1860 strength) objects in bin A3 and A2 hold the promise to make possible an independent estimate of $n_{\mathrm{H}}$, $U$, and metallicity. Clearly, objects in bin A1 resembling their median spectrum are not well suited for an application of the method. Also, any source with Ciii]$\lambda$1909/ Siiii]$\lambda$1892$>$ 1 is subject to a large correction. In light of the many uncertainty, an average value of the product $Un$ (obtained from the objects of the other spectral types) could be considered. Pop. B objects should not avoided entirely, especially whenever Siiii]$\lambda$1892 $\;\buildrel>\over{\sim}\;$ Ciii]$\lambda$1909 after VBC removal. The present exploratory analysis emphasized several sources of uncertainty. However, the parameter needed for $r_{\rm BLR}$ and $M_{\rm BH}$ computation, the product $U$$n_{\mathrm{H}}$, seems to be fairly stable and well-defined. Even with an error of a 0.3 in logarithm, the square root will be subject to a 0.15 uncertainty in logarithm, much lower than the uncertainty associated with the $r_{\rm BLR}$luminosity correlation. The large intrinsic spread of the correlation at low luminosity, its uncertain extrapolation at very high luminosity make preferable a one-by-one determination based on physical properties of an emitting region that remains similar to itself. A. Negrete and D.Dultzin acknowledge support form grant IN111610-3 PAPIIT, DGAPA UNAM. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org. 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Ion \| $\lambda$ \| $X$ \| $E_{l}-E_{u}$ \| Transition \| $A_{ki}$ \| $n_{\mathrm{c}}$ \| Note ---\|---\|---\|---\|---\|---\|---\|--- \| Å \| eV \| eV \| \| s-1 \| cm-3 \| Si II \| 1808.00 \| 8.15 \| 0.000 - 6.857 \| ${}^{2}D^{o}_{3/2}\rightarrow{}^{2}P_{1/2}$ \| $2.54\cdot 10^{6}$ \| $\cdots$ \| 1 Si II \| 1816.92 \| 8.15 \| 0.036 - 6.859 \| ${}^{2}D^{o}_{5/2}\rightarrow{}^{2}P_{3/2}$ \| $2.65\cdot 10^{6}$ \| $\cdots$ \| 1 Al III \| 1854.716 \| 18.83 \| 0.000 - 6.685 \| ${}^{2}P^{o}_{3/2}\rightarrow{}^{2}S_{1/2}$ \| $5.40\cdot 10^{8}$ \| $\cdots$ \| 1 Al III \| 1862.790 \| 18.83 \| 0.000 - 6.656 \| ${}^{2}P^{o}_{1/2}\rightarrow{}^{2}S_{1/2}$ \| $5.33\cdot 10^{8}$ \| $\cdots$ \| 1 \| 1882.7 \| 16.34 \| 0.000 - 6.585 \| ${}^{3}P^{o}_{2}\rightarrow{}^{1}S_{0}$ \| 0.012 \| $6.4\cdot 10^{4}$ \| 1,2,3 Si III] \| 1892.03 \| 16.34 \| 0.000 - 6.553 \| ${}^{3}P^{o}_{1}\rightarrow{}^{1}S_{0}$ \| 16700 \| $2.1\cdot 10^{11}$ \| 1,4,5 \| 1906.7 \| 24.38 \| 0.000 - 6.502 \| ${}^{3}P^{o}_{2}\rightarrow{}^{1}S_{0}$ \| 0.0052 \| $7.7\cdot 10^{4}$ \| 1,2,6 C III] \| 1908.734 \| 24.38 \| 0.000 - 6.495 \| ${}^{3}P^{o}_{1}\rightarrow{}^{1}S_{0}$ \| 114 \| $1.4\cdot 10^{10}$ \| 1,2,4,5 Fe III \| 1914.066 \| 16.18 \| 3.727 - 10.200 \| $z^{7}P^{o}_{3}\rightarrow a^{7}S_{3}$ \| $6.6\cdot 10^{8}$ \| $\cdots$ \| 7 Note. — All wavelengths are in vacuum. (1) Ralchenko, Yu., Kramida, A.E., Reader, J., and NIST ASD Team (2008). NIST Atomic Spectra Database (version 3.1.5). Available at: http://physics.nist.gov/asd3. 2: Feibelman & Aller (1987). 3: $n_{\mathrm{c}}$ computed following Shaw & Dufour (1995). 4: Morton (1991). 5: Feldman (1992). 6: Zheng (1988). 7: Wavelength and $A_{ki}$ from Ekberg (1993), energy levels from Edlén and Swings (1942). Table 2: Basic Properties of Sources and Log of Observations. Object name \| $\rm m_{B}$ \| $z$ \| Line \| $M_{\mathrm{B}}$ \| Flux 6cm (mJy) \| Date \| DIT \| Nexp \| Airmass \| S/N ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- (1) \| (2) \| (3) \| (4) \| (5) \| (6) \| (7) \| (8) \| (9) \| (10) \| (11) J00103-0037 \| 18.39 \| 3.1546 \| 1 \| -25.68 \| 0.40 \| 2006-11-08 \| 1139 \| 3 \| 1.16, 1.13, 1.11 \| 60 J00521-1108 \| 18.70 \| 3.2364 \| 2 \| -25.39 \| 0.43 \| 2007-01-01 \| 1199 \| 3 \| 1.15, 1.21, 1.29 \| 41 J01225+1339 \| 18.24 \| 3.0511 \| 1 \| -25.80 \| * \| 2006-11-08 \| 1259 \| 2 \| 1.36, 1.32 \| 92 J02287+0002 \| 18.20 \| 2.7282 \| 1 \| -25.72 \| 0.35 \| 2006-12-16 \| 1259 \| 2 \| 1.10, 1.12 \| 67 J02390-0038 \| 18.68 \| 3.0675 \| 1 \| -25.36 \| 0.43 \| 2006-11-07 \| 1199 \| 3 \| 1.35, 1.46, 1.60 \| 57 J03036-0023 \| 17.65 \| 3.2319 \| 1 \| -26.44 \| 0.34 \| 2006-12-16 \| 1259 \| 2 \| 1.11, 1.14 \| 88 J20497-0554 \| 18.29 \| 3.1979 \| 1 \| -25.79 \| * \| 2006-11-04 \| 1259 \| 2 \| 1.52, 1.70 \| 54 J23509-0052 \| 18.67 \| 3.0305 \| 1 \| -25.36 \| 0.41 \| 2006-11-07 \| 1199 \| 3 \| 1.10, 1.11, 1.14 \| 62 *footnotetext: Not in FIRST Table 3: Line Fluxes a \| Ciii]$\lambda$1909 \| \| \| \| Civ$\lambda$1549 \| \| Siiv$\lambda$1397 ---\|---\|---\|---\|---\|---\|---\|--- Object \| BC \| VBC \| Siiii]$\lambda$1892 \| Aliii$\lambda$1860 \| Siii$\lambda$1814 \| BC \| blue \| VBC \| \| BC \| blue \| VBC J00103-0037 \| 4.99 $\pm$ 2.11 \| 1.18 $\pm$ 1.38 \| 2.91 $\pm$ 1.03 \| 1.99 $\pm$ 0.81 \| 1.13 $\pm$ 0.82 : \| 14.25 $\pm$ 8.34 \| 6.32 $\pm$ 1.80 \| 8.71 $\pm$ 8.24 \| \| 4.27 $\pm$ 2.38 \| 0.78 $\pm$ 1.52 \| 0.54 $\pm$ 0.89 J00521-1108 \| 3.08 $\pm$ 0.36 \| 0.01 $\pm$ 0.12 * \| 2.77 $\pm$ 0.66 \| 1.55 $\pm$ 0.85 \| 1.06 $\pm$ 0.90 \| 10.89 $\pm$ 2.37 \| 1.40 $\pm$ 2.23 \| 8.27 $\pm$ 2.63 \| \| … \| … \| … J01225+1339 \| 10.71 $\pm$ 1.35 \| … \| 8.26 $\pm$ 1.33 \| 4.35 $\pm$ 2.09 \| 1.01 $\pm$ 0.98 \| 22.73 $\pm$ 6.04 \| 14.21 $\pm$ 1.52 \| … \| \| 11.21 $\pm$ 2.93 \| 6.66 $\pm$ 1.93 \| … J02287+0002 (1) \| 5.28 $\pm$ 1.56 \| 0.01 $\pm$ 0.02 * \| 4.70 $\pm$ 2.01 \| 2.29 $\pm$ 1.13 \| 0.98 $\pm$ 0.79 \| 7.64 $\pm$ 5.00 \| 7.66 $\pm$ 2.32 \| 0.77 $\pm$ 1.63 * \| \| 4.96 $\pm$ 1.04 \| 1.35 $\pm$ 0.80 \| 0.43 $\pm$ 0.56 J02287+0002 (2) \| 6.86 $\pm$ 1.56 \| 0.02 $\pm$ 0.02 * \| 2.78 $\pm$ 2.01 \| 1.77 $\pm$ 1.13 \| 0.94 $\pm$ 0.79 \| 11.48 $\pm$ 5.00 \| 2.99 $\pm$ 2.32 \| 0.64 $\pm$ 1.63 * \| \| 4.88 $\pm$ 1.04 \| 1.24 $\pm$ 0.80 \| 0.57 $\pm$ 0.56 J02390-0038 \| 3.57 $\pm$ 0.78 \| 1.26 $\pm$ 0.88 \| 3.50 $\pm$ 0.40 \| 2.41 $\pm$ 0.64 \| 0.90 $\pm$ 0.83 \| 7.51 $\pm$ 1.25 \| 7.92 $\pm$ 1.37 \| 2.04 $\pm$ 1.52 \| \| 2.64 $\pm$ 0.53 \| 1.49 $\pm$ 0.81 \| 0.23 $\pm$ 0.39 J03036-0023 \| 13.24 $\pm$ 1.09 \| … \| 11.82 $\pm$ 1.21 \| 5.17 $\pm$ 1.48 \| 1.53 $\pm$ 1.16 : \| 29.46 $\pm$ 3.54 \| 20.60 $\pm$ 3.34 \| … \| \| 11.13 $\pm$ 1.86 \| 7.37 $\pm$ 4.12 \| … J20497-0554 \| 8.04 $\pm$ 1.09 \| … \| 7.43 $\pm$ 0.56 \| 3.01 $\pm$ 1.25 \| 1.52 $\pm$ 1.51 : \| 18.39 $\pm$ 2.17 \| 9.23 $\pm$ 2.24 \| … \| \| 6.65 $\pm$ 2.51 \| 1.95 $\pm$ 1.06 \| … J23509-0052 \| 5.24 $\pm$ 1.21 \| … \| 4.61 $\pm$ 1.62 \| 1.50 $\pm$ 0.50 \| 0.40 $\pm$ 0.36 \| 9.24 $\pm$ 1.17 \| 7.80 $\pm$ 2.36 \| … \| \| 3.61 $\pm$ 1.49 \| 2.93 $\pm$ 1.34 \| … Extreme Objects \| \| \| \| \| \| \| \| \| \| \| \| J12014+01161 \| 5.52 $\pm$ 1.07 \| … \| 16.16 $\pm$ 5.90 \| 14.03 $\pm$ 2.54 \| 4.90 $\pm$ 2.95 \| 21.18 $\pm$ 12.48 \| 29.37 $\pm$ 9.98 \| … \| \| 18.45 $\pm$ 6.55 \| 11.81 $\pm$ 9.08 \| … 3C 390.3 \| 3.74 $\pm$ 1.15 \| 4.30 $\pm$ 0.68 \| 2.32 $\pm$ 0.43 \| 0.23 $\pm$ 0.32 * \| 0.40 $\pm$ 0.48 * \| 30.04 $\pm$ 4.99 \| … \| 56.44 $\pm$ 10.66 \| \| … \| … \| … Note. — (a) Units are $10^{-14}$ ergs s-1 cm-2 Å-1. (1) Considering $z_{OI\lambda 1304}$. (2) Considering $z_{CIII]\lambda 1909}$. (:) Siii$\lambda$1814 approximated values due the line is affected by telluric absorptions (see Fig. 1). We do not measure Siiv$\lambda$1397 for J00521-1108 and 3c390.3 because they have low S/N. () Consistent with no emission. Table 4: Weak lines around Civ$\lambda$1549\. a Object \| Niv]$\lambda$1486 \| Siii$\lambda$1533 \| Heii$\lambda$1640 ---\|---\|---\|--- \| \| \| BC \| blue J00103-0037 \| 2.4 $\pm$ 1.9 \| 1.1 $\pm$ 0.8 \| 1.3 $\pm$ 0.4 \| 2.0 $\pm$ 1.8 J00521-1108 \| 0.1 $\pm$ 0.2 \| 1.1 $\pm$ 1.1 \| 1.6 $\pm$ 1.2 \| 0.1 $\pm$ 0.2 J01225+1339 \| … \| 1.0 $\pm$ 1.8 \| 3.2 $\pm$ 3.4 \| 4.3 $\pm$ 1.8 J02287+0002 (1) \| … \| 1.0 $\pm$ 0.8 \| 0.3 $\pm$ 0.1 \| 0.2 $\pm$ 0.2 J02287+0002 (2) \| … \| 0.9 $\pm$ 0.8 \| 0.2 $\pm$ 0.1 \| 0.0 $\pm$ 0.2 J02390-0038 \| 0.5 $\pm$ 0.9 \| 0.9 $\pm$ 0.9 \| 0.2 $\pm$ 0.4 \| 1.6 $\pm$ 1.1 J03036-0023 \| … \| 1.5 $\pm$ 1.2 \| 0.5 $\pm$ 0.5 \| 9.2 $\pm$ 3.4 J20497-0554 \| 0.3 $\pm$ 0.6 \| 1.5 $\pm$ 1.5 \| 1.8 $\pm$ 0.8 \| 4.6 $\pm$ 2.7 J23509-0052 \| … \| 0.4 $\pm$ 0.4 \| 0.2 $\pm$ 0.9 \| 2.5 $\pm$ 1.6 Extreme Objects \| \| \| \| J12014+01161 \| 3.0 $\pm$ 3.5 \| 4.9 $\pm$ 5.8 \| 1.5 $\pm$ 1.7 \| 10.3 $\pm$ 9.2 3C 390.3 \| 5.6 $\pm$ 2.7 \| 1.6 $\pm$ 1.5 \| 2.8 $\pm$ 1.5 \| … Note. — (a) Units are $10^{-14}$ ergs s-1 cm-2 Å-1. (1) Considering $z_{OI\lambda 1304}$. (2) Considering $z_{CIII]\lambda 1909}$. We do not show Heii$\lambda$1640VBC because is very weak, when is considered. Table 5: Equivalent Widths. Object \| Ciii]$\lambda$1909BC \| Ciii]$\lambda$1909Tot \| Siiii]$\lambda$1892 \| Aliii$\lambda$1860 \| Siii$\lambda$1814 \| Civ$\lambda$1549BC \| Civ$\lambda$1549Tot \| Siiv$\lambda$1397BC \| Siiv$\lambda$1397Tot ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- J00103-0037 \| 13.7 $\pm$ 5.8 \| 17.1 $\pm$ 7.2 \| 7.9 $\pm$ 3.0 \| 5.3 $\pm$ 2.8 \| 2.9 $\pm$ 2.5 : \| 29.0 $\pm$ 16.1 \| 59.6 $\pm$ 24.4 \| 7.4 $\pm$ 4.32 \| 9.68 $\pm$ 6.74 J00521-1108 \| 8.2 $\pm$ 1.5 \| 8.2 $\pm$ 2.4 \| 7.3 $\pm$ 2.5 \| 4.0 $\pm$ 2.6 \| 2.6 $\pm$ 3.1 \| 21.0 $\pm$ 18.9 \| 40.1 $\pm$ 26.8 \| … \| … J01225+1339 \| 15.3 $\pm$ 3.5 \| … \| 11.6 $\pm$ 3.1 \| 6.0 $\pm$ 3.6 \| 1.4 $\pm$ 2.3 \| 25.5 $\pm$ 9.1 \| 41.6 $\pm$ 9.7 \| 11.1 $\pm$ 3.95 \| 17.74 $\pm$ 4.71 J02287+0002 (1) \| 15.9 $\pm$ 5.3 \| 15.9 $\pm$ 5.3 \| 14.1 $\pm$ 6.6 \| 6.8 $\pm$ 4.0 \| 2.9 $\pm$ 2.6 \| 20.4 $\pm$ 10.5 \| 42.9 $\pm$ 16.8 \| 12.7 $\pm$ 3.04 \| 17.30 $\pm$ 4.03 J02287+0002 (2) \| 20.5 $\pm$ 5.3 \| 20.5 $\pm$ 5.3 \| 8.3 $\pm$ 6.6 \| 5.2 $\pm$ 4.0 \| 2.7 $\pm$ 2.6 \| 30.8 $\pm$ 10.5 \| 40.6 $\pm$ 16.8 \| 12.7 $\pm$ 3.04 \| 17.85 $\pm$ 4.03 J02390-0038 \| 10.8 $\pm$ 3.3 \| 14.7 $\pm$ 4.5 \| 10.4 $\pm$ 2.0 \| 7.0 $\pm$ 2.4 \| 2.5 $\pm$ 2.5 \| 16.3 $\pm$ 4.1 \| 37.7 $\pm$ 7.0 \| 4.9 $\pm$ 1.32 \| 11.72 $\pm$ 4.15 J03036-0023 \| 12.4 $\pm$ 1.8 \| … \| 10.9 $\pm$ 1.8 \| 4.6 $\pm$ 1.6 \| 1.3 $\pm$ 1.1 : \| 19.7 $\pm$ 3.7 \| 33.3 $\pm$ 4.8 \| 6.4 $\pm$ 1.46 \| 10.71 $\pm$ 2.94 J20497-0554 \| 15.5 $\pm$ 3.5 \| … \| 14.1 $\pm$ 2.3 \| 5.6 $\pm$ 2.9 \| 2.7 $\pm$ 2.9 : \| 25.4 $\pm$ 5.6 \| 38.1 $\pm$ 7.1 \| 8.0 $\pm$ 3.75 \| 10.34 $\pm$ 4.04 J23509-0052 \| 15.8 $\pm$ 4.0 \| … \| 13.7 $\pm$ 5.3 \| 4.4 $\pm$ 1.8 \| 1.1 $\pm$ 0.9 \| 22.0 $\pm$ 4.2 \| 40.5 $\pm$ 10.6 \| 7.7 $\pm$ 3.73 \| 14.02 $\pm$ 5.10 Extreme Objects \| \| \| \| \| \| \| \| \| J12014+0116 \| 2.9 $\pm$ 0.9 \| … \| 8.4 $\pm$ 4.1 \| 7.1 $\pm$ 2.0 \| 2.4 $\pm$ 1.9 \| 8.2 $\pm$ 5.6 \| 19.4 $\pm$ 7.4 \| 6.03 $\pm$ 2.76 \| 9.85 $\pm$ 4.37 3C 390.3 \| 13.1 $\pm$ 4.3 \| 28.4 $\pm$ 6.2 \| 7.9 $\pm$ 3.3 \| 0.7 $\pm$ 0.8 \| 1.2 $\pm$ 1.3 \| 49.1 $\pm$ 8.1 \| 147.2 $\pm$ 19.4 \| … \| … Note. — (1) Considering $z_{OI\lambda 1304}$. (2) Considering $z_{CIII]\lambda 1909}$. (:) Siii$\lambda$1814 approximated values due the line is affected by telluric absorptions. Table 6: Hydrogen Density and Ionization Parameter. \| Log$n_{H}$ \| \| Log$U$ \| \| Log$n_{H}\cdot U$ ---\|---\|---\|---\|---\|--- Object \| $1Z_{\odot}$ \| $1Z_{\odot}$ low dens \| $5Z_{\odot}$ \| $5Z_{\odot}$ low dens \| \| $1Z_{\odot}$ \| $1Z_{\odot}$ low dens \| $5Z_{\odot}$ \| $5Z_{\odot}$ low dens \| \| $1Z_{\odot}$ \| $1Z_{\odot}$ low dens \| $5Z_{\odot}$ \| $5Z_{\odot}$ low dens J00103-0037 \| 12.50 $\pm$ 0.17 \| … \| … \| … \| \| -2.79 $\pm$ 0.19 \| … \| … \| … \| \| 9.71 $\pm$ 0.22 \| … \| \| J00521-1108 \| 12.43 $\pm$ 0.26 \| 12.84 $\pm$ 0.23 \| … \| … \| \| -2.86 $\pm$ 0.15 \| -3.00 $\pm$ 0.11 \| … \| … \| \| 9.58 $\pm$ 0.26 \| 9.85 $\pm$ 0.22 \| \| J00103-0037 * \| 12.50 $\pm$ 0.17 \| … \| … \| … \| \| -2.79 $\pm$ 0.19 \| … \| … \| … \| \| 9.71 $\pm$ 0.22 \| … \| … \| … J00521-1108 \| 12.43 $\pm$ 0.26 \| 12.84 $\pm$ 0.23 \| … \| … \| \| -2.86 $\pm$ 0.15 \| -3.00 $\pm$ 0.11 \| … \| … \| \| 9.58 $\pm$ 0.26 \| 9.85 $\pm$ 0.22 \| … \| … J01225+1339 \| 12.45 $\pm$ 0.22 \| 13.24 $\pm$ 0.20 \| 11.47 $\pm$ 0.28 \| 12.62 $\pm$ 0.25 \| \| -2.96 $\pm$ 0.09 \| -3.42 $\pm$ 0.18 \| -1.83 $\pm$ 0.11 \| -2.79 $\pm$ 0.68 \| \| 9.49 $\pm$ 0.21 \| 9.82 $\pm$ 0.23 \| 9.64 $\pm$ 0.27 \| 9.83 $\pm$ 0.66 J02287+0002 (1) \| 12.35 $\pm$ 0.16 \| … \| 11.64 $\pm$ 0.33 \| 12.28 $\pm$ 0.28 \| \| -2.81 $\pm$ 0.28 \| … \| -2.33 $\pm$ 0.25 \| -2.58 $\pm$ 0.98 \| \| 9.55 $\pm$ 0.28 \| … \| 9.31 $\pm$ 0.36 \| 9.70 $\pm$ 0.94 J02287+0002 (2) \| 12.47 $\pm$ 0.16 \| … \| … \| … \| \| -2.81 $\pm$ 0.28 \| … \| … \| … \| \| 9.67 $\pm$ 0.28 \| … \| … \| … J02390-0038 \| 12.75 $\pm$ 0.12 \| 13.42 $\pm$ 0.21 \| 11.97 $\pm$ 0.15 \| 13.20 $\pm$ 0.13 \| \| -3.18 $\pm$ 0.05 \| -3.60 $\pm$ 0.07 \| -2.19 $\pm$ 0.08 \| -3.29 $\pm$ 0.13 \| \| 9.57 $\pm$ 0.11 \| 9.82 $\pm$ 0.20 \| 9.78 $\pm$ 0.15 \| 9.91 $\pm$ 0.15 J03036-0023 \| 12.32 $\pm$ 0.14 \| 12.92 $\pm$ 0.15 \| … \| … \| \| -2.92 $\pm$ 0.06 \| -3.34 $\pm$ 0.05 \| … \| … \| \| 9.40 $\pm$ 0.14 \| 9.58 $\pm$ 0.15 \| … \| … J20497-0554 \| 12.26 $\pm$ 0.25 \| 12.82 $\pm$ 0.18 \| … \| … \| \| -2.89 $\pm$ 0.12 \| -3.29 $\pm$ 0.13 \| … \| … \| \| 9.37 $\pm$ 0.25 \| 9.53 $\pm$ 0.19 \| … \| … J23509-0052 \| 12.13 $\pm$ 0.24 \| 12.90 $\pm$ 0.17 \| 10.86 $\pm$ 0.23 \| 12.12 $\pm$ 0.58 \| \| -2.89 $\pm$ 0.09 \| -3.51 $\pm$ 0.21 \| -1.93 $\pm$ 0.13 \| -2.99 $\pm$ 0.82 \| \| 9.24 $\pm$ 0.24 \| 9.39 $\pm$ 0.23 \| 8.93 $\pm$ 0.23 \| 9.13 $\pm$ 0.86 Extreme Objects \| \| \| \| \| \| \| \| \| \| \| \| \| \| J12014+01161 \| 12.81 $\pm$ 0.29 \| 12.95 $\pm$ 0.16 \| 12.34 $\pm$ 0.11 \| 12.90 $\pm$ 0.42 \| \| -2.85 $\pm$ 0.18 \| -2.87 $\pm$ 0.08 \| -2.41 $\pm$ 0.25 \| -3.20 $\pm$ 0.58 \| \| 9.97 $\pm$ 0.30 \| 10.09 $\pm$ 0.16 \| 9.93 $\pm$ 0.25 \| 9.70 $\pm$ 0.62 3C 390.3 \| 10.05 $\pm$ 0.34 \| … \| … \| … \| \| -1.48 $\pm$ 0.34 \| … \| … \| … \| \| 8.57 $\pm$ 0.41 \| … \| … \| … Note. — We also show the values considering the correction by the contribution of low density regions (§7.3), $Z=5Z_{\odot}$ (§8.1), and $Z=5Z_{\odot}$ with the correction by the contribution of low density regions. (1) Considering $z_{OI\lambda 1304}$. (2) Considering $z_{CIII]\lambda 1909}$. (*) For J00103-0037 the correction is too large to be reliable. We show in bold numbers the ones that we consider the best. Table 7: The Size of the Broad Line Region and the Black Hole Masses. Object \| $d_{p}[Mpc]$ \| f(1700Å)a \| f(1350Å)a \| FWHMBC \| Pop. \| Log($r_{BLR}$) [cm]b \| \| Log($M_{BH}$) [$M_{\odot}$]b ---\|---\|---\|---\|---\|---\|---\|---\|--- \| $X10^{15}$ \| $X10^{-15}$ \| $X10^{-15}$ \| [km s-1] \| \| original \| low dens \| $5Z_{\odot}$ \| $5Z_{\odot}$low dens \| \| original \| low dens \| $5Z_{\odot}$ \| $5Z_{\odot}$low dens \| V&P (2006)c (1) \| (2) \| (3) \| (4) \| (5) \| (6) \| (7) \| (8) \| (9) \| (10) \| \| (11) \| (12) \| (13) \| (14) \| (15) J00103-0037 \| 6.51 \| 4.8 $\pm$ 1.0 \| 6.8 $\pm$ 1.4 \| 4500 $\pm$ 800 \| B \| 18.10 $\pm$ 0.12 \| … \| … \| … \| \| 9.16 $\pm$ 0.20 \| … \| … \| … \| 9.11 J00521-1108 \| 6.59 \| 6.1 $\pm$ 1.5 \| 8.8 $\pm$ 2.1 \| 5300 $\pm$ 1600 \| B \| 18.23 $\pm$ 0.14 \| 18.09 $\pm$ 0.13 \| … \| … \| \| 9.43 $\pm$ 0.30 \| 9.29 $\pm$ 0.29 \| … \| … \| 9.32 J01225+1339 \| 6.42 \| 8.1 $\pm$ 1.6 \| 10.9 $\pm$ 2.2 \| 4400 $\pm$ 1000 \| A† \| 18.32 $\pm$ 0.12 \| 18.16 $\pm$ 0.13 \| 18.25 $\pm$ 0.15 \| 18.15 $\pm$ 0.33 \| \| 9.36 $\pm$ 0.23 \| 9.20 $\pm$ 0.23 \| 9.29 $\pm$ 0.25 \| 9.19 $\pm$ 0.39 \| 9.19 J02287+0002 (1) \| 6.09 \| 7.4 $\pm$ 2.7 \| 8.4 $\pm$ 3.0 \| 4700 $\pm$ 1000 \| A† \| 18.25 $\pm$ 0.16 \| … \| 18.37 $\pm$ 0.20 \| 18.17 $\pm$ 0.48 \| \| 9.35 $\pm$ 0.25 \| … \| 9.47 $\pm$ 0.27 \| 9.27 $\pm$ 0.51 \| 9.17 J02287+0002 (2) \| \| \| \| \| \| 18.19 $\pm$ 0.16 \| … \| … \| … \| \| 9.29 $\pm$ 0.25 \| … \| … \| … \| J02390-0038 \| 6.45 \| 6.9 $\pm$ 2.1 \| 9.9 $\pm$ 3.0 \| 5400 $\pm$ 1000 \| B \| 18.25 $\pm$ 0.09 \| 18.12 $\pm$ 0.12 \| 18.14 $\pm$ 0.10 \| 18.08 $\pm$ 0.10 \| \| 9.46 $\pm$ 0.18 \| 9.34 $\pm$ 0.20 \| 9.36 $\pm$ 0.19 \| 9.30 $\pm$ 0.19 \| 9.35 J03036-0023 \| 6.58 \| 20.5 $\pm$ 5.7 \| 30.0 $\pm$ 8.4 \| 3700 $\pm$ 600 \| A \| 18.58 $\pm$ 0.10 \| 18.49 $\pm$ 0.10 \| … \| … \| \| 9.47 $\pm$ 0.17 \| 9.38 $\pm$ 0.17 \| … \| … \| 9.30 J20497-0554 \| 6.55 \| 6.6 $\pm$ 1.3 \| 9.5 $\pm$ 1.9 \| 3800 $\pm$ 600 \| A \| 18.34 $\pm$ 0.13 \| 18.26 $\pm$ 0.11 \| … \| … \| \| 9.25 $\pm$ 0.19 \| 9.17 $\pm$ 0.18 \| … \| … \| 9.04 J23509-0052 \| 6.40 \| 4.8 $\pm$ 1.0 \| 6.1 $\pm$ 1.2 \| 3600 $\pm$ 800 \| A \| 18.33 $\pm$ 0.13 \| 18.25 $\pm$ 0.12 \| 18.49 $\pm$ 0.13 \| 18.38 $\pm$ 0.43 \| \| 9.19 $\pm$ 0.23 \| 9.12 $\pm$ 0.23 \| 9.35 $\pm$ 0.23 \| 9.25 $\pm$ 0.47 \| 8.88 Extreme Objects \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| J12014+01161 \| 6.58 \| 21.8 $\pm$ 2.3 \| 31.9 $\pm$ 3.2 \| 4000 $\pm$ 800 \| A \| 18.31 $\pm$ 0.15 \| 18.25 $\pm$ 0.09 \| 18.33 $\pm$ 0.13 \| 18.44 $\pm$ 0.31 \| \| 9.26 $\pm$ 0.23 \| 9.20 $\pm$ 0.19 \| 9.28 $\pm$ 0.22 \| 9.40 $\pm$ 0.43 \| 9.37 3C 390.3 \| 0.24 \| 4.2 $\pm$ 0.7 \| 9.9 $\pm$ 1.6 \| 6400 $\pm$ 2200 \| B \| 17.23 $\pm$ 0.21 \| … \| … \| … \| \| 8.60 $\pm$ 0.37 \| … \| … \| … \| 7.99 Note. — (a) Units of the flux at 1350 and 1700Å are in ergs s-1 cm-2 Å-1. (b) The showed values are the average $\pm$ 0.17 dex of the computation using both SED of Laor (1997) and Mathews & Ferland (1987) (see Fig. 20. (c) We show the comparison between our computations and those using the Vestergaard & Peterson (2006) mehtod. They report an uncertainty of 0.66 dex. (†) Acording to the FWHM it is clasified as pop B, but has other spectral caracteristics of pop. A objects. See §5. (1) Considering $z_{OI\lambda 1304}$. (2) Considering $z_{CIII]\lambda 1909}$ Figure 1: Sample of 8 VLT spectra. Abscissa is obseved wavelength in Å, ordinate is specific flux in units 10-16 ergs s-1 cm-2 Å-1 corrected for Milky Way Galactic extinction. The superimposed dotted line is before atmospheric bands subtraction. We show the positions of the lines of our interest Ciii]$\lambda$1909, Siiii]$\lambda$1892, Aliii$\lambda$1860, Siii$\lambda$1814, Feii$\lambda$1787, Civ$\lambda$1549 and Siiv$\lambda$1397\. J01225+1339 and J02287 are BAL quasars. Figure 2: Oi$\lambda$1304.8 used to place the restframe showed by the mark. Abscissa is rest frame wavelength in Å, ordinate is specific flux in units 10-16 ergs s-1 cm-2 Å-1 corrected for Milky Way Galactic extinction. In J00521-1108 and J02390-0038 the peak of Oi$\lambda$1304.8 is not observed clearly. In J00103-0037, J02287+0002 and J20497-0554, the redshift results, using both Oi$\lambda$1304.8 and Ciii]$\lambda$1909, are ambiguous. In J01225+1339, J03036-0023 and J23509-0052, the redshifts obtained using Oi$\lambda$1304.8 or Ciii]$\lambda$1909 are consistent. Figure 3: Sample of 8 VLT spectra in rest frame wavelength. Abscissa is rest frame in Å, ordinate is specific flux in the rest frame in units 10-13 ergs s-1 cm-2 Å-1. Figure 4: Fits for Pop. A objects: J03036-0023 (a, b), J20497-0554 (c, d), J23509-0052 (e, f). Upper panels show the fits and the lower pannels under the fits show the residuals and also the fitted absorptions lines. Upper abscissa is rest frame wavelength in Å, lower abscissa is in velocity units, ordinate is specific flux in arbitrary units. Vertical dashed line is the restframe for Civ$\lambda$1549 and Ciii]$\lambda$1909\. Long dashed line is the fit, solid dark lines are the broad components: Civ$\lambda$1549 in left panels and Ciii]$\lambda$1909, Siiii]$\lambda$1892, Aliii$\lambda$1860, Siii$\lambda$1814 in right panels. Dotted dark lines under Aliii$\lambda$1860 show the doublet. Short dashed line is Feii. Feiii is shown in dash-triple-dot line in the right panels. Dash-dot line in the left panels is the blue-shifted component of Civ$\lambda$1549 while dotted line is the very broad component, present also in Ciii]$\lambda$1909 for Pop. B objects. In the left panels we show with faint lines the contribution of Niv$\lambda$1486, Siii$\lambda$1533 and Heii$\lambda$1640 core and blue-shifted components. For colors see online figures. Figure 5: Fits for Pop. B objects: up J00103-0037, middle J00521-1108, low J02390-0038. Units and meaning of symbols are the same of Fig. 4. Figure 6: Fits for BAL quasars: up J01225+1339, middle J02287+0002 using $z_{\mathrm{O}{\sc i}\lambda 1304}$ rest frame, low J02287+0002 using $z_{\mathrm{C}{\sc iii}\lambda 1909}$ rest frame. Note in J02287+0002 the line displacement with the consequently line intensity changes, specially in Ciii]$\lambda$1909, Siiii]$\lambda$1892 and Civ$\lambda$1549 broad and blue- shifted components. Units and symbols are the same as in Fig. 4. Figure 7: Fits for Siiv$\lambda$1397: (a) J03036-0023, (b) J20497-0554, (c) 23509-0052, (d) J00103-0037, (e) J02390-0038, (f) J01225+1339, (g) J02287+0002 and (h) J12014+0116. We do not measure Siiv$\lambda$1397 for J00521–1108 and 3C 390.3 because they have low S/N. We asume that the profile of Siiv$\lambda$1397 can be fitted with the same three possible components (core, blue-shifted and red-shifted) as in the case of Civ$\lambda$1549\. Units and symbols are the same as in Fig. 4. Figure 8: Fits for the two extreme objects: up J120144.36+011611.6, low 3C390.3. Units and meaning of symbols are the same of Fig. 4. For 3C390.3 it was needed to fit a narrow unshifted component that we show in dash-dot-dot line. Note for 3C390.3 Siii$\lambda$1814 is almost absent. Figure 9: Isocontours for the ratios (a) $\log$(Ciii]$\lambda$1909/ Siiii]$\lambda$1892), (b) $\log$(Aliii$\lambda$1860/ Siiii]$\lambda$1892), (c) $\log$(Siii$\lambda$1814/ Siiii]$\lambda$1892), (d) $\log$(Siiv$\lambda$1397/ Siiii]$\lambda$1892), (e) $\log$(Civ$\lambda$1549/Aliii$\lambda$1860), and (f) $\log$(Civ$\lambda$1549/ Siiii]$\lambda$1892) derived from CLOUDY simulations. Abscissa is electron density in cm-3, ordinate is the ionization parameter, both in logarithm scale. Figure 10: Top: Ionic Fractions as a function of the geometric depth $h$ in the gas slab. Bottom: Line emissivity per unit volume in units of ergs s-1 cm-3 multiplied by depth. Figure 11: Contour plots for the extreme objects (a) 3C390.9 and (b) J12014+0116. Abscissa is electron density in cm-3, ordinate is the ionization parameter, both in logarithm scale. Solid line is for $\log$(Civ$\lambda$1549/Aliii$\lambda$1860), dot line is for $\log$(Siiv$\lambda$1397/ Siiii]$\lambda$1892), dash line is for $\log$(Aliii$\lambda$1860/ Siiii]$\lambda$1892), long dash line is for $\log$(Aliii$\lambda$1860/Siiv$\lambda$1397), dash dot line is for $\log$(Civ$\lambda$1549/ Siiii]$\lambda$1892) and dash-triple-dot line is for $\log$(Siii$\lambda$1814/ Siiii]$\lambda$1892). The point where the isocontours cross determines the values of Log$n_{e}$ and LogU. The shaded area is the error bands for Siii$\lambda$1814\. In 3C390.3 we add a long dash line for $\log$(Ciii]$\lambda$1909/ Siiii]$\lambda$1892). In J12014+0116 the shaded area is the error bands for Siii$\lambda$1814\. See text for further details. Figure 12: Correlation between rest frame equivalent width of Aliii$\lambda$1860 and Siiii]$\lambda$1892\. Line in unweighted least square best fit. Figure 13: Expected contribution from moderate density emitting gas as a function of density for ionization parameters $\log U=-2$ (black lines) and $\log U=2.5$ (grey lines). Figure 14: Plane $\log U-\log n$ in an expanded scale. The lower-left dot corresponds to a high density solution. Moderate density emitting gas is then added with increasing intensity of a Ciii]$\lambda$1909 component with the four Ciii]$\lambda$1909/ Siiii]$\lambda$1892 ratios reported in the figure. Connected positions corresponds to increasing density of Ciii]$\lambda$1909 emitting gas, $\log n=9,9.5,10$. Note that the case $\log n=10$ (grey symbols) is not appropriate for the observed data. Figure 15: Contour plots for (a) J00103-0037, (b) J00521-1108, (c) J02390-0038, (d) J03036-0023, (e) J20497-0554 and (f) 23509-0052. Units and meaning of symbols are the same of Fig. 11. In the case of the objects shown in panels (c) and (f) the Siii$\lambda$1814 line is very weak and thus unreliable. For the objects in panels (a), (d) and (e) the line is also affected by the telluric absorption. For these objects we rely on the Siiv$\lambda$1397 line. Figure 16: Contour plots for BAL quasars (a) J01225+1339 and (b) J01225+1339 using $z_{OI\lambda 1304}$ and (c) J01225+1339 using $z_{CIII]\lambda 1909}$. Units and meaning of symbols are the same of Fig. 11. Figure 17: Contour plots for corrected values considering the low density emission contribution of Ciii]$\lambda$1909\. Contour plot for J00103-0037 is not shown because the correction is so large to be reliable. For the objects in panels (e) and (f) the Siii$\lambda$1814 line is also affected by the telluric absorption. Abscissa, ordinate and symbols are the same as Fig. 11. Figure 18: Contour plots for (a) J01225+1339, (b) J02287+0002 using $z_{CIII]\lambda 1909}$, (c) J02390-0038, the extreme object (d) J12014+0116, and (e) J23509-0052, from the array of simulations computed for $Z=5Z_{\odot}$. Coordinates and symbols are as for Fig. 11. The intersection point improves in certain cases, but in others is the same. Figure 19: Contour plots for the same objects of the previous figure from the array of simulations computed for $Z=5Z_{\odot}$ with ratios corrected because of low-density emission. Coordinates and symbols are as for Fig. 11. Figure 20: Spectral energy distribution used to compute the number of ionizing photons for Laor et al. (1997) in solid line and Mathews & Ferland (1987) in dotted line. Dashed line shows the Lyman limit. Figure 21: $M_{\rm BH}$ comparison for the high-$z$ sample. Filled symbols refer to uncorrected intensity ratios; open symbols are for intensity ratios corrected because of low-density emission. Circles refer to solar metallicity; squares to 5 times solar metallicity and Si-Al enrichment. Abscissa and ordinate are logarithm of $M_{\rm BH}$ in solar masses. Each point is labeled with the object name in short format. In ordinate we report $M_{\rm BH}$ values obtained with the method of this paper; in abscissa those obtained employing the Vestergaard & Peterson relationship described in the text. The shaded bands limit the $2\sigma$ confidence level spread expected on the basis of the Vestergaard & Peterson relationship.	arxiv-papers	2010-11-18T18:22:28	2024-09-04T02:49:15.044671	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "C. Alenka Negrete, Deborah Dultzin, Paola Marziani and Jack Sulentic", "submitter": "Alenka Negrete", "url": "https://arxiv.org/abs/1011.4248" }
1011.4331		arxiv-papers	2010-11-19T01:08:44	2024-09-04T02:49:15.065010	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Li-Jun Tian, Chang-Qing Zhu, Hong-Biao Zhang, and Li-Guo Qin", "submitter": "Zhu Changqing", "url": "https://arxiv.org/abs/1011.4331" }
1011.4418	# Parameterization of General $Z$-$\gamma$-$Z^{\prime}$ Mixings in an Electroweak Chiral Theory Ying Zhang1111Email address: hepzhy@mail.xjtu.edu.cn, Qing Wang2,3222Corresponding author at: Department of Physics, Tsinghua University, Beijing 100084,P.R.China Email address: wangq@mail.tsinghua.edu.cn 1School of Science, Xi’an Jiaotong University, Xi’an, 710049, P.R.China 2Center for High Energy Physics, Tsinghua University, Beijing 100084, P.R.china 3Department of Physics,Tsinghua University,Beijing 100084,P.R.China (Mar 1, 2011) ###### Abstract A new general parameterization with eight mixing parameters among $Z$, $\gamma$ and an extra neutral gauge boson $Z^{\prime}$ is proposed and subjected to phenomenological analysis. We show that in addition to the conventional Weinberg angle $\theta_{W}$, there are seven other phenomenological parameters $G^{\prime}$, $\xi$, $\eta$, $\theta_{l}$, $\theta_{r}$, $r$, $l$ for the most general $Z$-$\gamma$-$Z^{\prime}$ mixings, in which parameter $G^{\prime}$ arises due to the presence of an extra Stueckelberg-type mass coupling. Combined with the conventional $Z$-$Z^{\prime}$ mass mixing angle $\theta^{\prime}$, the remaining six parameters $\xi$, $\eta$, $\theta_{l}$-$\theta^{\prime}$, $\theta_{r}$-$\theta^{\prime}$, $r$, $l$ are caused by general kinetic mixings. In all the eight phenomenological parameters $\theta_{W}$, $G^{\prime}$, $\xi$, $\eta$, $\theta_{l}$, $\theta_{r}$, $r$, $l$, we can determine the $Z$-$Z^{\prime}$ mass mixing angle $\theta^{\prime}$ and the mass ratio $M_{Z}/M_{Z^{\prime}}$. The $Z$-$\gamma$-$Z^{\prime}$ mixings we discuss are based on the model-independent description of the extended electroweak chiral Lagrangian (EWCL) previous proposed by us. In addition, we show that there are eight corresponding independent theoretical coefficients in our EWCL which are fully fixed by our eight phenomenological mixing parameters. We further find that the experimental measurability of these eight parameters does not rely on the extended neutral current for $Z^{\prime}$, but depends on the $Z-Z^{\prime}$ mass ratio. PACS numbers: 14.70.Pw; 12.60.Cn; 12.15.Mm; 12.15.Lk ## I Introduction One of the simplest and more popular gauge extensions of the standard model (SM) is to add an extra $U(1)$ group associated with the $Z^{\prime}$ gauge boson to the electroweak gauge group $SU(2)_{L}\otimes U(1)_{Y}$, that constitutes one of the ”hot spots” in high energy physics today. The extra gauge boson $Z^{\prime}$ is the carrier of a new gauge force corresponding to the smallest gauge group extensions that plays a crucial role in cosmology, GUT, SUSY and various strong coupling new physics theories associated with new physics beyond SM (for the latest review see LangackerRMP2008 ). As long as there exists a $Z^{\prime}$ particle, it will shift observables from present physics by mixing with the standard electroweak neutral gauge bosons, $\gamma$ and $Z$. The corrections, however, depend on details of the model set-up, and especially on the way the neutral gauge bosons mix. A model-independent way to figure out these mixings is through phenomenological requirements and constraints. Usually, theorists only consider minimal $Z$-$Z^{\prime}$ mass mixing Langacker2009 . A massless photon constrains any possible extension of the mass mixings matrix to be of Stueckelberg-type Zhang2008JHEP . However, theory and phenomenology do not forbid general three-body $Z$-$\gamma$-$Z^{\prime}$ kinetic mixing. In the literature only a few examples have been considered, such as, the special kinetic mixings given in Holdom and Daniel . A general model-independent description of $Z$-$\gamma$-$Z^{\prime}$ mixing is needed to enable data analysis and experimental searches for $Z^{\prime}$ to be more specific and effective, particularly in light of the progress made in the LHC and Tevatron experiments. With this motivation, we are prompted to study the most general gauge boson mixing. In fact, a general description of the $Z^{\prime}$ interaction with SM particles has already been given in our previous work Zhang2008JHEP ; Zhang2009JHEP in which $Z^{\prime}$ is regarded as a gauge boson of a broken $U(1)^{\prime}$ symmetry and the conventional EWCL is extended to include this extra broken $U(1)^{\prime}$ symmetry from original $SU(2)_{L}\otimes U(1)_{Y}\rightarrow U(1)_{em}$ to $SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)^{\prime}\rightarrow U(1)_{em}$. In Ref.Zhang2008JHEP , the bosonic part up to order $p^{4}$ of the most general EWCL involving this $Z^{\prime}$ boson and discovered particles has been proposed that describes the most general $Z$-$\gamma$-$Z^{\prime}$ mixings. In Ref.Zhang2009JHEP , various $Z$-$\gamma$-$Z^{\prime}$ mixings that have appeared in the literature are shown to be included in our EWCL formalism and are further classified into five simple groupings. However, the expressions given in Zhang2008JHEP ; Zhang2009JHEP for these $Z$-$\gamma$-$Z^{\prime}$ mixings are complex and are not suitable for phenomenological investigations. It is the purpose of this paper to improve this shortcoming by setting up a more general parameterization for all $Z$-$\gamma$-$Z^{\prime}$ mixings to facilitate present and future phenomenological analysis in the EWCL given by Zhang2008JHEP . We will discuss the physical meaning, origin and experimental measurability of these parameters within new parameterization. We show that there are eight independent degree of freedoms and all complexities of the mixing can be absorbed into eight phenomenological parameters $\theta_{W}$, $G^{\prime}$, $\xi$, $\eta$, $\theta_{l}$, $\theta_{r}$, $r$, $l$, for which all but the traditional Weinberg mixing angle $\theta_{W}$ and the Stueckelberg-type coupling $G^{\prime}$, combine with the conventional $Z$-$Z^{\prime}$ mass mixing angle $\theta^{\prime}$, and the remaining six parameters $\xi$, $\eta$, $\theta_{l}$-$\theta^{\prime}$, $\theta_{r}$-$\theta^{\prime}$, $r$, $l$ are caused by general kinetic mixings. We will explicitly construct quantitative relations among these mixing parameters and those related to theoretical coefficients appearing in the underlying EWCL. This paper is organized as follows. In Section II, we give a short review of the relevant parts associated with the $Z$-$\gamma$-$Z^{\prime}$ kinetic and mass mixings from the EWCL given in Ref.Zhang2008JHEP , and introduce the mixing matrix. In Section III, we explain the physical meaning and origin of the eight parameters describing the mixing matrix by diagonalizing the mass- squared and kinetic matrices, and construct the relations among the various mixing matrix elements and coefficients in our EWCL. In Section IV, we first discuss the experimental measurability of parameters arising in our new parameterization, then express the EWCL coefficients related to $Z$-$\gamma$-$Z^{\prime}$ mixings in these eight parameters that transfers the measurability from the mixing parameters to the relevant EWCL coefficients. Section V presents a summary. ## II Review of the kinetic and mass mixings from EWCL We begin the discussion by first reviewing the EWCL of $Z^{\prime}$ established in Zhang2008JHEP . The general Lagrangian describing the gauge symmetry breaking $SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)^{\prime}\rightarrow U(1)_{em}$ independent of the details of the symmetry breaking can be constructed in terms of $2\times 2$ non-linear Goldstone field $\hat{U}$ with the following covariant derivative $\displaystyle D_{\mu}\hat{U}=\partial_{\mu}\hat{U}+igW_{\mu}\hat{U}-i\hat{U}(g^{\prime}\frac{\tau_{3}}{2}+\tilde{g}^{\prime})B_{\mu}-i{g^{\prime\prime}}\hat{U}X_{\mu}\;,$ where $W_{\mu}$, $B_{\mu}$ and $X_{\mu}$ are gauge bosons corresponding to $SU(2)_{L}$, $U(1)_{Y}$ and $U(1)^{\prime}$, respectively. Here, carets are used to distinguish extended $U(1)^{\prime}$ breaking quantities from the traditional electroweak breaking quantities in AppelquistPRD1993 . $g$, $g^{\prime}$, $g^{\prime\prime}$ and $\tilde{g}^{\prime}$ are $SU(2)_{L}$ coupling, conventional $U(1)_{Y}$ coupling, $U(1)^{\prime}$ coupling and special Stueckelberg-type gauge coupling, respectively. In paper Zhang2008JHEP , the bosonic part of the Lagrangian up to order $p^{4}$ has been presented. Because of our interests here in $Z^{\prime}$ mixing effects, we focus only on the neutral gauge boson mixing parts, which can be divided into a mass part $\mathcal{L}_{M}$ $\displaystyle\mathcal{L}_{M}$ $\displaystyle=$ $\displaystyle-\frac{1}{4}f^{2}\mathrm{tr}[\hat{V}_{\mu}^{2}]+\frac{1}{4}\beta_{1}f^{2}\left(\mathrm{tr}[T\hat{V}_{\mu}]\right)^{2}+\frac{1}{4}\beta_{2}f^{2}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}^{\mu}]+\frac{1}{4}\beta_{3}f^{2}\left(\mathrm{tr}[\hat{V}_{\mu}]\right)^{2}$ (1) $\displaystyle\stackrel{{\scriptstyle\mbox{\tiny unitary gauge}}}{{======}}\frac{f^{2}}{8}(1\\!-\\!2\beta_{1})(gW^{3}_{\mu}\\!-g^{\prime}B_{\mu})^{2}+\frac{f^{2}}{2}(1\\!-\\!2\beta_{3})(g^{\prime\prime}X_{\mu}\\!+\tilde{g}^{\prime}B_{\mu})^{2}$ $\displaystyle+\frac{f^{2}}{2}\beta_{2}(g^{\prime\prime}X_{\mu}+\tilde{g}^{\prime}B_{\mu})(gW^{3,\mu}-g^{\prime}B^{\mu})=\frac{1}{2}\mathcal{V}^{T}_{\mu}\mathcal{M}^{2}_{0}\mathcal{V}_{\mu}$ and kinetic part $\mathcal{L}_{K}$ $\displaystyle\mathcal{L}_{K}$ $\displaystyle=$ $\displaystyle-\frac{1}{4}B_{\mu\nu}^{2}-\frac{1}{2}\mathrm{tr}[W_{\mu\nu}^{2}]-\frac{1}{4}X_{\mu\nu}^{2}+\frac{1}{2}\alpha_{1}gg^{\prime}B_{\mu\nu}\mathrm{tr}[TW^{\mu\nu}]+\frac{1}{4}\alpha_{8}g^{2}\left(\mathrm{tr}[TW_{\mu\nu}]\right)^{2}$ (2) $\displaystyle+g{g^{\prime\prime}}\alpha_{24}X_{\mu\nu}\mathrm{tr}[TW^{\mu\nu}]+g^{\prime}{g^{\prime\prime}}\alpha_{25}B_{\mu\nu}X^{\mu\nu}$ $\displaystyle\stackrel{{\scriptstyle\mbox{\tiny unitary gauge}}}{{======}}-\frac{1}{4}B_{\mu\nu}B_{\mu\nu}-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}-\frac{1}{4}(1\\!-\\!\alpha_{8}g^{2})(\partial_{\mu}W^{3}_{\nu}\\!-\partial_{\nu}W^{3}_{\mu})^{2}$ $\displaystyle+\frac{1}{2}\alpha_{1}gg^{\prime}B_{\mu\nu}(\partial_{\mu}W^{3}_{\nu}\\!-\partial_{\nu}W^{3}_{\mu})+gg^{\prime\prime}\alpha_{24}X^{\mu\nu}(\partial_{\mu}W^{3}_{\nu}-\partial_{\nu}W^{3}_{\mu})+g^{\prime}g^{\prime\prime}\alpha_{25}B_{\mu\nu}X^{\mu\nu}$ $\displaystyle=$ $\displaystyle-\frac{1}{4}\mathcal{V}^{T}_{\mu\nu}\mathcal{K}_{0}\mathcal{V}^{\mu\nu}.$ Here, $T\equiv\hat{U}^{\dagger}\tau_{3}\hat{U}$ and $\hat{V}_{\mu}\equiv(\hat{D}_{\mu}\hat{U})\hat{U}^{\dagger}$ are $SU(2)_{L}$ covariant operators. In $\mathcal{L}_{M}$, the first term is the conventional non-linear $\sigma$ model term and the fourth term is a new non-linear $\sigma$ model term due to the presence of the $U(1)^{\prime}$ Goldstone boson. The second term is the conventional custodial symmetry breaking term. The third term is the mixing of the second and fourth terms. For $\mathcal{L}_{K}$, with the exception of the standard kinetic terms for the $U(1)_{Y}$, $SU(2)_{L}$ and $U(1)^{\prime}$ gauge bosons, the terms with coefficients $\alpha_{1}$, $\alpha_{24}$ and $\alpha_{25}$ are the kinetic mixing terms between $U(1)$ and the diagonal part of the $SU(2)_{L}$ gauge fields, between $U(1)^{\prime}$ and the diagonal part of the $SU(2)_{L}$ gauge fields, and between $U(1)$ and $U(1)^{\prime}$ gauge fields, respectively. The term with coefficients $\alpha_{8}$ is the correction term for the diagonal part of $SU(2)_{L}$ gauge field. These coefficients parameterize the most general kinetic mixing among the $Z$-$\gamma$-$Z^{\prime}$ bosons. For convenience, all these terms have been abbreviated into matrix forms in the unitary gauge $\hat{U}=1$ in the gauge boson vector $\mathcal{V}^{T}_{\mu}=(W^{3}_{\mu},B_{\mu},X_{\mu})$, the field strength tensor $\mathcal{V}_{\mu\nu}\equiv\partial_{\mu}\mathcal{V}_{\nu}-\partial_{\nu}\mathcal{V}_{\mu}$, the mass-squared matrix $\mathcal{M}_{0}^{2}$ and the kinetic matrix $\mathcal{K}_{0}$. The mass-squared and kinetic matrices are $\displaystyle\mathcal{M}_{0}^{2}=f^{2}\left(\begin{array}[]{ccc}\frac{g^{2}}{4}(1\\!-\\!2\beta_{1})&-\frac{gg^{\prime}}{4}(1\\!-\\!2\beta_{1})+\frac{g\tilde{g}^{\prime}}{2}\beta_{2}&\frac{gg^{\prime\prime}}{2}\beta_{2}\\\ -\frac{gg^{\prime}}{4}(1\\!-\\!2\beta_{1})+\frac{g\tilde{g}^{\prime}}{2}\beta_{2}&\frac{g^{\prime 2}}{4}(1\\!-\\!2\beta_{1})+\tilde{g}^{\prime 2}(1\\!-\\!2\beta_{3})-g\tilde{g}^{\prime}\beta_{2}&-\frac{g^{\prime}g^{\prime\prime}}{2}\beta_{2}+g^{\prime\prime}\tilde{g}^{\prime}(1\\!-\\!2\beta_{3})\\\ \frac{gg^{\prime\prime}}{2}\beta_{2}&-\frac{g^{\prime}g^{\prime\prime}}{2}\beta_{2}+g^{\prime\prime}\tilde{g}^{\prime}(1\\!-\\!2\beta_{3})&g^{\prime\prime 2}(1\\!-\\!2\beta_{3})\end{array}\right)\;,~{}~{}~{}$ (6) $\displaystyle\hskip 113.81102pt\mathcal{K}_{0}=-\frac{1}{4}\left(\begin{array}[]{ccc}1-g^{2}\alpha_{8}&-gg^{\prime}\alpha_{1}&-2gg^{\prime\prime}\alpha_{24}\\\ -gg^{\prime}\alpha_{1}&1&-2g^{\prime}g^{\prime\prime}\alpha_{25}\\\ -2gg^{\prime\prime}\alpha_{24}&-2g^{\prime}g^{\prime\prime}\alpha_{25}&1\end{array}\right)\;.$ (10) From $\mathcal{M}_{0}^{2}$ and $\mathcal{K}_{0}$, we see that three body $Z$-$\gamma$-$Z^{\prime}$ mixing is controlled by 11 dimensionless coefficients: 4 gauge couplings $g,g^{\prime},\tilde{g}^{\prime},g^{\prime\prime}$, 3 mass-mixing low-energy constants $\beta_{1}$, $\beta_{2}$, $\beta_{3}$ and 4 kinetic-mixing low- energy constants $\alpha_{1}$, $\alpha_{8}$, $\alpha_{24}$, $\alpha_{25}$. Among these, only nine play roles in the sense that we can redefine nine new coefficients by absorbing $\beta_{1}$ and $\beta_{3}$ as follows $\displaystyle g^{\prime}=\frac{\bar{g}^{\prime}}{\sqrt{1-2\beta_{1}}}\hskip 42.67912ptg=\frac{\bar{g}}{\sqrt{1-2\beta_{1}}}\hskip 42.67912ptg^{\prime\prime}=\frac{\bar{g}^{\prime\prime}}{\sqrt{1-2\beta_{3}}}\hskip 42.67912pt{\tilde{g}^{\prime}}=\frac{\bar{\tilde{g}}^{\prime}}{\sqrt{1-2\beta_{3}}}\;,$ (11) $\displaystyle\beta_{2}=\bar{\beta}_{2}\sqrt{1\\!-\\!2\beta_{1}}\sqrt{1\\!-\\!2\beta_{3}}\hskip 17.07182pt\alpha_{a}=gg^{\prime}\bar{\alpha}_{1}\hskip 17.07182pt\alpha_{b}=g^{2}\bar{\alpha}_{8}\hskip 17.07182pt\alpha_{c}=gg^{\prime\prime}\bar{\alpha}_{24}\hskip 17.07182pt\alpha_{d}=g^{\prime}g^{\prime\prime}\bar{\alpha}_{25}\;.~{}~{}$ (12) Then $\mathcal{M}_{0}^{2}$ and $\mathcal{K}_{0}$ of these redefined nine coefficients become $\displaystyle\mathcal{M}_{0}^{2}$ $\displaystyle=$ $\displaystyle f^{2}\left(\begin{array}[]{ccc}\frac{\bar{g}^{2}}{4}&-\frac{\bar{g}\bar{g}^{\prime}}{4}+\frac{\bar{g}\bar{\tilde{g}}^{\prime}}{2}\bar{\beta}_{2}&\frac{\bar{g}\bar{g}^{\prime\prime}}{2}\bar{\beta}_{2}\\\ -\frac{\bar{g}\bar{g}^{\prime}}{4}+\frac{\bar{g}\bar{\tilde{g}}^{\prime}}{2}\bar{\beta}_{2}&\frac{\bar{g}^{\prime 2}}{4}+\bar{\tilde{g}}^{\prime 2}-\bar{g}\bar{\tilde{g}}^{\prime}\bar{\beta}_{2}&-\frac{\bar{g}^{\prime}\bar{g}^{\prime\prime}}{2}\bar{\beta}_{2}+\bar{g}^{\prime\prime}\bar{\tilde{g}}^{\prime}\\\ \frac{\bar{g}\bar{g}^{\prime\prime}}{2}\bar{\beta}_{2}&-\frac{\bar{g}^{\prime}\bar{g}^{\prime\prime}}{2}\bar{\beta}_{2}+\bar{g}^{\prime\prime}\bar{\tilde{g}}^{\prime}&\bar{g}^{\prime\prime 2}\end{array}\right)\;,$ (16) $\displaystyle\mathcal{K}_{0}$ $\displaystyle=$ $\displaystyle-\frac{1}{4}\left(\begin{array}[]{ccc}1-\alpha_{b}&-\alpha_{a}&-2\alpha_{c}\\\ -\alpha_{a}&1&-2\alpha_{d}\\\ -2\alpha_{c}&-2\alpha_{d}&1\end{array}\right)\;.$ (20) Furthermore there exists a scale symmetry for $\mathcal{M}_{0}^{2}$ and $\mathcal{K}_{0}$, i.e., these are invariant under the following transformation determined by an arbitrary parameter $\zeta$, $\displaystyle\bar{g}\rightarrow\zeta\bar{g}\hskip 28.45274pt\bar{g}^{\prime}\rightarrow\zeta\bar{g}^{\prime}\hskip 28.45274pt\bar{g}^{\prime\prime}\rightarrow\zeta\bar{g}^{\prime\prime}\hskip 28.45274pt\bar{\tilde{g}}^{\prime}\rightarrow\zeta\bar{\tilde{g}}^{\prime}\hskip 28.45274ptf\rightarrow\frac{1}{\zeta}f$ (21) with $\bar{\beta}_{2},\alpha_{a},\alpha_{b},\alpha_{c},\alpha_{d}$ unchanged. Since the dimensional coefficient $f$ does not enter into the final mixing matrix, the above scale symmetry implies that among the nine redefined theoretical coefficients, only eight of these are independent, and span the largest mixing space for an extra neutral gauge boson $Z^{\prime}$. We take these eight theoretical coefficients as $\bar{g}/\bar{g}^{\prime}$, $\bar{\tilde{g}}^{\prime}/\bar{g}^{\prime\prime}$, $\bar{g}_{Z}/\bar{g}^{\prime\prime}$, $\bar{\beta}_{2}$, $\alpha_{a}$, $\alpha_{b}$, $\alpha_{c}$, $\alpha_{d}$ with $\displaystyle\bar{g}_{Z}\equiv\sqrt{\bar{g}^{2}+\bar{g}^{\prime 2}}\;.$ (22) These will provide all combinations of extra neutral vector boson corrections to low-energy EW physics via through mixings. As discussed in Zhang2009JHEP , then inputting different set of values for these coefficients, the effective theory can recuperate the various $Z^{\prime}$ models that have been presented in the literature. The mixings can be disentangled by diagonalizing the mass- squared matrix $\mathcal{M}_{0}^{2}$ and kinetic matrix $\mathcal{K}_{0}$ simultaneously, i.e. through introducing in a $3\times 3$ real matrix $U$ which relates the interaction eigenstate $(W^{3}_{\mu},B_{\mu},X_{\mu})$ to the mass eigenstate $(Z_{\mu},A_{\mu},Z^{\prime}_{\mu})$ in the following manner $\displaystyle\left(\begin{array}[]{c}W^{3}_{\mu}\\\ B_{\mu}\\\ X_{\mu}\end{array}\right)=U\left(\begin{array}[]{c}Z_{\mu}\\\ A_{\mu}\\\ Z^{\prime}_{\mu}\end{array}\right)\;.$ (29) The $U$ matrix has to fulfill conditions $\displaystyle U^{T}\mathcal{M}_{0}^{2}U=\mathrm{diag}(M_{Z}^{2},0,M_{Z^{\prime}}^{2})\hskip 113.81102ptU^{T}\mathcal{K}_{0}U=-\frac{1}{4}\mathrm{diag}(1,1,1)\;.$ (30) In Refs.Zhang2008JHEP ; Zhang2009JHEP , we have already discussed the exact form of $U$, although in practice its physical meaning tends to get lost due to its complex form, and is not suitable in presenting phenomenological arguments. Here, we simplify its expression by re-parameterizing it as follows, $\displaystyle U\equiv\left(\begin{array}[]{ccc}s_{W}\xi+c_{W}c_{l}l{}{}&~{}~{}s_{W}a{}{}&~{}~{}s_{W}\eta+c_{W}s_{r}r\\\ c_{W}\xi-s_{W}c_{l}l{}{}&~{}~{}c_{W}a{}{}&~{}~{}c_{W}\eta-s_{W}s_{r}r\\\ -c_{W}\xi G^{\prime}+s_{W}c_{l}lG^{\prime}-s_{l}l{}{}&~{}~{}-c_{W}aG^{\prime}{}{}&~{}~{}-c_{W}\eta G^{\prime}+s_{W}s_{r}rG^{\prime}+c_{r}r\end{array}\right)=U_{0}U_{1}\;,~{}~{}$ (34) $\displaystyle U_{0}\equiv\left(\begin{array}[]{ccc}c_{W}&s_{W}&0\\\ -s_{W}&c_{W}&0\\\ s_{W}G^{\prime}&-c_{W}G^{\prime}&1\end{array}\right)\hskip 85.35826ptU_{1}\equiv\left(\begin{array}[]{ccc}lc_{l}&0&rs_{r}\\\ \xi&a&\eta\\\ -ls_{l}&0&rc_{r}\end{array}\right)\;,$ (41) in which there are three angle parameters $\theta_{W},\theta_{r},\theta_{l}$ establishing the trigonometric values $c_{i}\equiv\cos\theta_{i}$, $s_{i}\equiv\sin\theta_{i}$ for $i=W,l,r$ and six other mixing parameters $G^{\prime},a,\xi,\eta,r,l$, totally nine in all. Among these nine parameters, $a=a(\theta_{W},\theta_{r},\theta_{l},G^{\prime},\xi,\eta,r,l)$ is a single relation determining one of the other eight parameters, the detailed dependence will be given later in (IV). Thus only eight of nine parameters in (34) are independent, the degree of freedoms just matches the number of independent theoretical coefficients for electroweak gauge boson mixings that we counted before. In fact, because of the massless photon, parameter $a$ is a normalization constant and plays the role of rescaling the photon field, which does not cause observable effects in the two-point vertices involving electroweak gauge bosons. Note that in the SM tree diagram limit, $U_{0}$ is a pure Weinberg rotation with $G^{\prime}=0$ and $U_{1}$ is a unit matrix with $\theta_{l}=\theta_{r}=\xi=\eta=0$ and $l=r=a=1$. ## III Phenomenological parameters in terms of diagonalization and EWCL coefficients Next, we explain the physical meaning and origin of the eight parameters $\theta_{W}$, $G^{\prime}$, $\xi$, $\eta$, $\theta_{r}$, $\theta_{l}$, $r$, $l$ by diagonalizing the mass-squared matrix $\mathcal{M}_{0}^{2}$ and kinetic matrix $\mathcal{K}_{0}$. First, $G^{\prime}$ is defined in such a way that it relates to the Stueckelberg-type coupling $\bar{\tilde{g}}^{\prime}$ as $\displaystyle G^{\prime}\equiv\frac{\bar{\tilde{g}}^{\prime}}{\bar{g}^{\prime\prime}}=\frac{\tilde{g}^{\prime}}{g^{\prime\prime}}\;.$ (42) i.e., $G^{\prime}$ is derived from the Stueckelberg coupling as the ratio of the Stueckelberg coupling and conventional $U(1)^{\prime}$ coupling. In our EWCL formalism, the deviation from SM has two sources: a Stueckelberg-type interaction for $B_{\mu}$ and the extra $U(1)^{\prime}$ interaction from gauge boson $X_{\mu}$, with $G^{\prime}$ the relative ratio of the interaction strengths between these two types of sources. Theoretically $G^{\prime}$ can take arbitrary real numbers, in particular $G^{\prime}=\infty$ and $G^{\prime}=0$ correspond to $g^{\prime\prime}=0,~{}\tilde{g}^{\prime}$ finite and $\tilde{g}^{\prime}=0,~{}g^{\prime\prime}$ finite, respectively. However, phenomenological analysis shows that a very large $G^{\prime}$ is not physically realistic as Ref.StSM gives $G^{\prime}=\tilde{g}^{\prime}/g^{\prime\prime}=1.9/149\approx 0.013$. If we ignore $G^{\prime}$, the rotation matrix $U_{0}$ then reverts to the standard Weinberg rotation with Weinberg angle $\theta_{W}$ defined as $\displaystyle c_{W}\equiv\frac{\bar{g}}{\bar{g}_{Z}}\hskip 28.45274pts_{W}\equiv\frac{\bar{g}^{\prime}}{\bar{g}_{Z}}\hskip 56.9055pt\mbox{or}\hskip 56.9055pt\theta_{W}=\arctan\frac{\bar{g}^{\prime}}{\bar{g}}=\arctan\frac{g^{\prime}}{g}\;.~{}~{}~{}~{}$ (43) The Weinberg angle originates from mixing of field $W^{3,\mu}$ and $B^{\mu}$ and the Weinberg rotation enables the part of the mass matrix associated with $\gamma$ and $Z$ to be diagonalized if the $Z^{\prime}$ particle and the Stueckelberg coupling are neglected. Once the Stueckelberg coupling $\bar{\tilde{g}}^{\prime}$ shows up, there will be off diagonal matrix elements involving $\gamma$-$Z$ and $\gamma$-$Z^{\prime}$ mixings. To disentangle these mixings, we add $G^{\prime}$ terms to the $U_{0}$ matrix and after the $U_{0}$ rotation, we find $\displaystyle U_{0}^{T}\mathcal{M}_{0}^{2}U_{0}=f^{2}\left(\begin{array}[]{ccc}\frac{1}{4}\bar{g}_{Z}^{2}&0&\frac{1}{2}\bar{g}_{Z}\bar{g}^{\prime\prime}\bar{\beta}_{2}\\\ 0&0&0\\\ \frac{1}{2}\bar{g}_{Z}\bar{g}^{\prime\prime}\bar{\beta}_{2}&0&\bar{g}^{\prime\prime 2}\end{array}\right)\;.$ (47) This is a typical $Z$-$Z^{\prime}$ mixing matrix. We apply a further matrix $\tilde{U}_{0}$ with rotation angle $\theta^{\prime}$ to diagonalize (47), i.e. $\displaystyle\tilde{U}_{0}=\left(\begin{array}[]{ccc}c^{\prime}&0&s^{\prime}\\\ 0&1&0\\\ -s^{\prime}&0&c^{\prime}\end{array}\right)\hskip 56.9055pt\tilde{U}_{0}^{T}U_{0}^{T}\mathcal{M}_{0}^{2}U_{0}\tilde{U}_{0}=\mathrm{diag}(M^{2},0,{M^{\prime}}^{2})$ (51) with $c^{\prime}=\cos\theta^{\prime},s^{\prime}=\sin\theta^{\prime}$. We find that it fixes the rotation angle $\theta^{\prime}$ as follows $\displaystyle\theta^{\prime}=\arctan\frac{\Delta_{g}-\sqrt{\Delta_{g}^{2}+16\bar{g}_{Z}^{2}\bar{g}^{\prime\prime 2}\bar{\beta}_{2}^{2}}}{4\bar{\beta}_{2}\bar{g}^{\prime\prime}\bar{g}_{Z}}\hskip 56.9055pt\Delta_{g}=\bar{g}_{Z}^{2}-4\bar{g}^{\prime\prime 2}\;.$ (52) Hence $\theta^{\prime}$ originates from the $Z$-$Z^{\prime}$ mass mixing ,its role being to disentangle this mixing, and appears in most of the new physics models involving the $Z^{\prime}$ boson. With the zero eigenvalue in (51) corresponding to the massless photon, the two other nonzero eigenvalues in (51) are $\displaystyle\frac{M^{2}}{f^{2}}=\frac{1}{4}\bar{g}_{Z}^{2}c^{\prime 2}+\bar{g}^{\prime\prime 2}s^{\prime 2}-s^{\prime}c^{\prime}\bar{g}_{Z}\bar{g}^{\prime\prime}\bar{\beta}_{2}\hskip 28.45274pt\frac{M^{\prime 2}}{f^{2}}=\bar{g}^{\prime\prime 2}c^{\prime 2}+\frac{1}{4}\bar{g}_{Z}^{2}s^{\prime 2}+s^{\prime}c^{\prime}\bar{g}_{Z}\bar{g}^{\prime\prime}\bar{\beta}_{2}\;.$ (53) Here $M$ and $M^{\prime}$ are just the $Z$ and $Z^{\prime}$ masses if there are no Stueckelberg and kinetic mixings. For $g^{\prime\prime}=\tilde{g}^{\prime}=0$, (47) is already diagonal with eigenvalues $\frac{1}{4}f^{2}\bar{g}_{Z}^{2},0,0$, and there is no need to apply further rotation; clearly, $\theta^{\prime}=0$ is given by (52) resulting in a unit matrix $\tilde{U}_{0}$. This further simplifies the eigenvalues of (53) to $M^{2}/f^{2}=\frac{1}{4}\bar{g}_{Z}^{2}$ and $M^{\prime 2}/f^{2}=0$. Here $M^{\prime}=0$ implies the mass of $Z^{\prime}$ is zero and $Z^{\prime}$ decouples from $Z$ and $\gamma$. After diagonalizing the mass-squared matrix $\mathcal{M}_{0}^{2}$, the next logical step is to further diagonalize the kinetic matrix $\mathcal{K}_{0}$. Considering that after the rotation $U_{0}\tilde{U}_{0}$ which diagonalizes $\mathcal{M}_{0}^{2}$, the kinetic matrix $\mathcal{K}_{0}$ is already transformed to symmetric form $\displaystyle\tilde{U}_{0}^{T}U_{0}^{T}\mathcal{K}_{0}U_{0}\tilde{U}_{0}=\left(\begin{array}[]{ccc}k_{1}&k_{2}&k_{3}\\\ k_{2}&k_{4}&k_{5}\\\ k_{3}&k_{5}&k_{6}\end{array}\right)$ (57) with $\displaystyle k_{1}$ $\displaystyle=$ $\displaystyle 1-2s_{W}s^{\prime}c^{\prime}G^{\prime}+s_{W}^{2}c^{\prime 2}G^{\prime 2}+2c_{W}s_{W}c^{\prime 2}\alpha_{a}-c_{W}^{2}c^{\prime 2}\alpha_{b}$ (58) $\displaystyle+(4c_{W}s^{\prime}c^{\prime}-4c_{W}s_{W}c^{\prime 2}G^{\prime})\alpha_{c}+(-4s_{W}s^{\prime}c^{\prime}+4s_{W}^{2}c^{\prime 2}G^{\prime})\alpha_{d}\;,$ $\displaystyle k_{2}$ $\displaystyle=$ $\displaystyle c_{W}s^{\prime}G^{\prime}-c_{W}s_{W}c^{\prime}G^{\prime 2}+(s_{W}^{2}-c_{W}^{2})c^{\prime}\alpha_{a}-c_{W}s_{W}c^{\prime}\alpha_{b}$ (59) $\displaystyle+[2s_{W}s^{\prime}+2(c_{W}^{2}-s_{W}^{2})c^{\prime}G^{\prime}]\alpha_{c}+(2c_{W}s^{\prime}-4c_{W}s_{W}c^{\prime}G^{\prime})\alpha_{d}\;,$ $\displaystyle k_{3}$ $\displaystyle=$ $\displaystyle-s_{W}(s^{\prime 2}-c^{\prime 2})G^{\prime}+s_{W}^{2}s^{\prime}c^{\prime}G^{\prime 2}+2c_{W}s_{W}s^{\prime}c^{\prime}\alpha_{a}-c_{W}^{2}c^{\prime}s^{\prime}\alpha_{b}$ (60) $\displaystyle+[2c_{W}(s^{\prime 2}-c^{\prime 2})-4c_{W}s_{W}s^{\prime}c^{\prime}G^{\prime}]\alpha_{c}+[2s_{W}(c^{\prime 2}-s^{\prime 2})+4s_{W}^{2}s^{\prime}c^{\prime}G^{\prime}]\alpha_{d}\;,$ $\displaystyle k_{4}$ $\displaystyle=$ $\displaystyle 1+c_{W}^{2}G^{\prime 2}-2c_{W}s_{W}\alpha_{a}-s_{W}^{2}\alpha_{b}+4c_{W}s_{W}G^{\prime}\alpha_{c}+4c_{W}^{2}G^{\prime}\alpha_{d}\;,$ (61) $\displaystyle k_{5}$ $\displaystyle=$ $\displaystyle- c_{W}c^{\prime}G^{\prime}-c_{W}s_{W}s^{\prime}G^{\prime 2}+(s_{W}^{2}-c_{W}^{2})s^{\prime}\alpha_{a}-c_{W}s_{W}s^{\prime}\alpha_{b}$ (62) $\displaystyle-[2s_{W}c^{\prime}-2(c_{W}^{2}-s_{W}^{2})s^{\prime}G^{\prime}]\alpha_{c}-(2c_{W}c^{\prime}+4c_{W}s_{W}s^{\prime}G^{\prime})\alpha_{d}\;,$ $\displaystyle k_{6}$ $\displaystyle=$ $\displaystyle 1+2s_{W}s^{\prime}c^{\prime}G^{\prime}+s^{\prime 2}s_{W}^{2}G^{\prime 2}+2c_{W}s_{W}s^{\prime 2}\alpha_{a}-c_{W}^{2}s^{\prime 2}\alpha_{b}$ (63) $\displaystyle-(4c_{W}s^{\prime}c^{\prime}+4c_{W}s_{W}s^{\prime 2}G^{\prime})\alpha_{c}+(4s_{W}s^{\prime}c^{\prime}+4s^{\prime 2}s_{W}^{2}G^{\prime})\alpha_{d}\;.$ Note that as long as we have a nonzero Stueckelberg coupling $G^{\prime}$, the rotated kinetic matrix $\tilde{U}_{0}^{T}U_{0}^{T}\mathcal{K}_{0}U_{0}\tilde{U}_{0}$ is not diagonal, even if the kinetic mixing coefficients $\alpha_{a}$,$\alpha_{b}$,$\alpha_{c}$,$\alpha_{d}$ all vanish. For the special case $g^{\prime\prime}=\tilde{g}^{\prime}=0$, the matrix elements reduce to $k_{3}=k_{5}=0$ and $k_{6}=1$. With these results, we introduce the matrix $\tilde{U}_{1}$ to further diagonalize the rotated kinetic matrix $\tilde{U}_{0}^{T}U_{0}^{T}\mathcal{K}_{0}U_{0}\tilde{U}_{0}$ $\displaystyle\tilde{U}_{1}\\!\equiv\tilde{U}_{0}^{-1}U_{1}\\!=\left(\begin{array}[]{ccc}l\cos(\theta_{l}\\!-\\!\theta^{\prime})&0&r\sin(\theta_{r}\\!-\\!\theta^{\prime})\\\ \xi&a&\eta\\\ -l\sin(\theta_{l}\\!-\\!\theta^{\prime})&0&r\cos(\theta_{r}\\!-\\!\theta^{\prime})\end{array}\right)\;,~{}~{}~{}~{}~{}$ (67) which changes the diagonal matrix $\mathrm{diag}(M^{2},0,{M^{\prime}}^{2})$ to $\mathrm{diag}(M_{Z}^{2},0,M_{Z^{\prime}}^{2})$ with $\displaystyle M_{Z}^{2}$ $\displaystyle=$ $\displaystyle M^{2}l^{2}\left[\cos^{2}(\theta_{l}-\theta^{\prime})+\frac{\cos(\theta_{l}-\theta^{\prime})\sin(\theta_{r}-\theta^{\prime})\sin(\theta_{l}-\theta^{\prime})}{\cos(\theta_{r}-\theta^{\prime})}\right]\;,$ (68) $\displaystyle M_{Z^{\prime}}^{2}$ $\displaystyle=$ $\displaystyle{M^{\prime}}^{2}r^{2}\left[\cos^{2}(\theta_{r}-\theta^{\prime})+\frac{\cos(\theta_{r}-\theta^{\prime})\sin(\theta_{r}-\theta^{\prime})\sin(\theta_{l}-\theta^{\prime})}{\cos(\theta_{l}-\theta^{\prime})}\right]\;,$ (69) as long as we take $\displaystyle\frac{\tan(\theta_{l}-\theta^{\prime})}{\tan(\theta_{r}-\theta^{\prime})}=\frac{M^{2}}{M^{\prime 2}}\;.$ (70) i.e. $\displaystyle\tilde{U}_{1}^{T}\tilde{U}_{0}^{T}U_{0}^{T}\mathcal{K}_{0}U_{0}\tilde{U}_{0}\tilde{U}_{1}=U_{1}^{T}U_{0}^{T}\mathcal{K}_{0}U_{0}U_{1}=U^{T}\mathcal{K}_{0}U=-\frac{1}{4}\mathrm{diag}(1,1,1)\;.$ (71) We see that the parameters in (67) play the role of generating most general kinetic mixings. In particular, $\xi$ and $\eta$ originate from $Z-\gamma$ and $Z^{\prime}-\gamma$ mixings respectively, while $l$, $r$, $\theta_{l}-\theta^{\prime}$ and $\theta_{r}-\theta^{\prime}$ originate from the most general $Z$ and $Z^{\prime}$ redefinition and mixing which need four independent parameters (two from redefinition and the other two from kinetic mixings). The $\theta^{\prime}$ appearing in (67) in the combinations of $\theta_{l}-\theta^{\prime}$ and $\theta_{r}-\theta^{\prime}$ is needed to subtract out $Z$-$Z^{\prime}$ mass mixing from general $Z$-$\gamma$-$Z^{\prime}$ mixings, leaving only the pure kinetic mixings. If there are no kinetic mixings, then $\displaystyle a=l=r=1\hskip 56.9055ptG^{\prime}=\xi=\eta=0\hskip 56.9055pt\theta_{l}=\theta_{r}=\theta^{\prime}\;.$ (72) By further requiring no $Z$-$Z^{\prime}$ mass mixing by taking $\theta^{\prime}=0$ in above result, we recover the SM tree diagram limit mentioned previously. Using (71), we then find $\displaystyle\frac{1}{a^{2}}=k_{4}$ (73) which only rescales the photon field to normalized kinetic form. Equation (70) gives one relation between the angle combinations $\theta_{l}-\theta^{\prime}$ and $\theta_{r}-\theta^{\prime}$, (71) further fixes $\tan(\theta_{l}-\theta^{\prime})$ through the following quadratic equation $\displaystyle\Big{\\{}\frac{k_{2}k_{5}}{k_{4}}-k_{3}\Big{\\}}M^{2}M^{\prime 2}\frac{\tan^{2}(\theta_{l}-\theta^{\prime})}{M^{4}}$ $\displaystyle+\Big{\\{}(k_{1}-\frac{k_{2}^{2}}{k_{4}})M^{\prime 2}+(\frac{k_{5}^{2}}{k_{4}}-k_{6})M^{2}\Big{\\}}\frac{\tan(\theta_{l}-\theta^{\prime})}{M^{2}}+\Big{\\{}k_{3}-\frac{k_{2}k_{5}}{k_{4}}\Big{\\}}=0$ (74) There are two solutions from the above equation: one of these we choose so that it vanishes in the limit $k_{1}=k_{4}=k_{6}=1,k_{2}=k_{3}=k_{5}=0$ for fixed $M^{2}$ and $M^{\prime 2}$, the other nonzero solution corresponds to having the$Z$ mass vanish and $\gamma$ receiving a nonzero mass. Combining the solution of (74) with equation (70), we obtain $\theta_{l}-\theta^{\prime}$ and $\theta_{r}-\theta^{\prime}$. $r$ and $l$ can be determined by $\displaystyle\frac{1}{l^{2}}=\cos^{2}(\theta_{l}-\theta^{\prime})\Big{\\{}(k_{6}-\frac{k_{5}^{2}}{k_{4}})\tan^{2}(\theta_{l}-\theta^{\prime})+2(\frac{k_{2}k_{5}}{k_{4}}-k_{3})\tan(\theta_{l}-\theta^{\prime})+k_{1}-\frac{k_{2}^{2}}{k_{4}}\Big{\\}}~{}~{}$ (75) $\displaystyle\frac{1}{r^{2}}=\cos^{2}(\theta_{r}\\!-\\!\theta^{\prime})\Big{\\{}(k_{1}-\frac{k_{2}^{2}}{k_{4}})\tan^{2}(\theta_{r}-\theta^{\prime})+2(k_{3}-\frac{k_{2}k_{5}}{k_{4}})\tan(\theta_{r}-\theta^{\prime})+k_{6}-\frac{k_{5}^{2}}{k_{4}}\Big{\\}}$ (76) With $l$, $r$, $\theta_{l}-\theta^{\prime}$ and $\theta_{r}-\theta^{\prime}$, $\xi$ known, and $\eta$ are re-expressible $\displaystyle\frac{\xi}{l}$ $\displaystyle=$ $\displaystyle\frac{k_{5}\sin(\theta_{l}-\theta^{\prime})-k_{2}\cos(\theta_{l}-\theta^{\prime})}{k_{4}}$ (77) $\displaystyle\frac{\eta}{r}$ $\displaystyle=$ $\displaystyle-\frac{k_{2}\sin(\theta_{r}-\theta^{\prime})+k_{5}\cos(\theta_{r}-\theta^{\prime})}{k_{4}}\;.$ (78) As an example, we give the explicit result for the special case $g^{\prime\prime}=\tilde{g}=0$, (present situation is $0/0$ case, here in the limiting procedure we let $\tilde{g}$ approach zero first, then take $g^{\prime\prime}$ to zero, because as we mentioned before $G^{\prime}$ is small from purely phenomenological estimations.) where the above considerations program gives result: $\displaystyle\theta_{l}=\theta_{r}=\theta^{\prime}=G^{\prime}=\eta=0\hskip 28.45274pt\frac{1}{a^{2}}=k_{4}\hskip 28.45274pt\frac{1}{l^{2}}=k_{1}\\!-\frac{k_{2}^{2}}{k_{4}}\hskip 28.45274ptr=1\hskip 28.45274pt\xi=-\frac{k_{2}l}{k_{4}}$ (79) $\displaystyle M^{2}_{Z}\\!=M^{2}l^{2}\hskip 99.58464ptM^{2}=\frac{1}{4}\bar{g}_{Z}^{2}f^{2}\hskip 91.04872ptM_{Z^{\prime}}^{2}\\!=M^{\prime 2}\\!=0$ (80) Up to this stage, once we know the coefficients in mass-squared matrix $\mathcal{M}_{0}^{2}$ and kinetic matrix $\mathcal{K}_{0}$, i.e. $f$ and eight theoretical coefficients of EWCL $\bar{g}/\bar{g}^{\prime},\bar{\tilde{g}}^{\prime}/\bar{g}^{\prime\prime},\bar{g}_{Z}/\bar{g}^{\prime\prime},\bar{\beta_{2}},\alpha_{a},\alpha_{b},\alpha_{c},\alpha_{d}$, we can obtain the final phenomenological mixing parameters $\theta_{W},\theta_{r},\theta_{l},G^{\prime},\xi,\eta,l,r$, and intermediate mixing angle $\theta^{\prime}$, photon normalization factor $a$. In particular, the intermediate mass-squared ratio $M^{2}/M^{\prime 2}$ is determined from (70) and the physical mass ratio $M_{Z}/M_{Z^{\prime}}$ can be expressed as $\displaystyle\frac{M_{Z}}{M_{Z^{\prime}}}=\frac{l}{r}\frac{\sin^{1/2}(2\theta_{l}-2\theta^{\prime})}{\sin^{1/2}(2\theta_{r}-2\theta^{\prime})}\;.$ (81) This result offers a hope in predicting the $Z^{\prime}$ mass in mixing parameters. Unfortunately, the mixing parameters themselves are not easy to test. In the next section, we will discuss the experimental measurability of the mixing parameters. Here we would rather treat the above relation as an additional constraint used in determining parameters for a given $Z-Z^{\prime}$ mass ratio. Phenomenologically, a more important question is, once we know the eight phenomenological mixing parameter $\theta_{W},\theta_{r},\theta_{l},G^{\prime},\xi,\eta,l,r$ from fitting the experiment data, how can we obtain the corresponding eight theoretical coefficients $\bar{g}/\bar{g}^{\prime},\bar{\tilde{g}}^{\prime}/\bar{g}^{\prime\prime},\bar{g_{Z}}/\bar{g}^{\prime\prime},\bar{\beta_{2}},\alpha_{a},\alpha_{b},\alpha_{c},\alpha_{d}$. Considering that the mixing parameter $G^{\prime}=\bar{\tilde{g}}^{\prime}/g^{\prime\prime}$ has already appeared in $\mathcal{M}_{0}^{2}$, i.e. it is both a theoretical coefficient and a phenomenological parameter, the problem remaining is to fix the other seven coefficients $\bar{g}/\bar{g}^{\prime},\bar{g}_{Z}/\bar{g}^{\prime\prime},\bar{\beta_{2}},\alpha_{a},\alpha_{b},\alpha_{c},\alpha_{d}$ in eight phenomenological parameters $\theta_{W},\theta_{r},\theta_{l},G^{\prime},\xi,\eta,l,r$. Since the details of computation are very complex, here we only outline the calculations: We choose seven equations (43), (70), (74), (75) , (76), (77), and (78) for which the auxiliary quantity $\theta^{\prime}$ is further determined by (52), $M^{2}/M^{\prime 2}$ by (53), and $k_{1},k_{2},k_{3},k_{4},k_{5},k_{6}$ by (61) to (63). Solving these equations, we can in principle express these theoretical coefficients in phenomenological parameters. With the expressions of the EWCL coefficients of the phenomenological parameters, and with help of (52) and (73), the conventional $Z$-$Z^{\prime}$ mass mixing angle $\theta^{\prime}$, the ratio $M_{Z}/M_{Z^{\prime}}$ and $a$ can all be expressed in the eight phenomenological mixing parameters. The above procedure yields completely general results. To terms of order $p^{4}$, we give explicit expressions for six phenomenological parameters $\theta_{r},\theta_{l},\xi,\eta,l,r$ in terms of theoretical coefficients $\bar{g}^{\prime}/\bar{g},\bar{\tilde{g}}^{\prime}/\bar{g}^{\prime\prime},\bar{g}_{Z}/\bar{g}^{\prime\prime},\bar{\beta}_{2},\alpha_{a},\alpha_{b},\alpha_{c},\alpha_{d}$: $\displaystyle\theta_{r}$ $\displaystyle\simeq$ $\displaystyle\theta^{\prime}+\frac{4s_{W}\bar{g}^{\prime\prime 2}}{\Delta_{g}}G^{\prime}+\frac{s_{W}(5\bar{g}_{Z}^{2}+12\bar{g}^{\prime\prime 2})}{\Delta_{g}}G^{\prime}\theta^{\prime 2}-\frac{4c_{W}(-2\bar{g}_{Z}^{2}s_{W}^{2}+\Delta_{g})\bar{g}^{\prime\prime 2}}{\Delta_{g}^{2}}G^{\prime}\alpha_{a}$ (82) $\displaystyle-\frac{4s_{W}c_{W}^{2}\bar{g}_{Z}^{2}\bar{g}^{\prime\prime 2}}{\Delta_{g}^{2}}G^{\prime}\alpha_{b}-\frac{8c_{W}\bar{g}^{\prime\prime 2}}{\Delta_{g}}\alpha_{c}+\frac{8s_{W}\bar{g}^{\prime\prime 2}}{\Delta_{g}}\alpha_{d}$ $\displaystyle\theta_{l}$ $\displaystyle\simeq$ $\displaystyle\theta^{\prime}+\frac{s_{W}\bar{g}_{Z}^{2}}{\Delta_{g}}G^{\prime}+\frac{s_{W}(3\bar{g}_{Z}^{2}+20\bar{g}^{\prime\prime 2})}{\Delta_{g}}G^{\prime}\theta^{\prime 2}-\frac{c_{W}\bar{g}_{Z}^{2}(-2\bar{g}_{Z}^{2}s_{W}^{2}+\Delta_{g})}{\Delta_{g}^{2}}G^{\prime}\alpha_{a}$ (83) $\displaystyle-\frac{s_{W}c_{W}^{2}\bar{g}_{Z}^{4}}{\Delta_{g}^{2}}G^{\prime}\alpha_{b}-\frac{2c_{W}\bar{g}_{Z}^{2}}{\Delta_{g}}\alpha_{c}+\frac{2s_{W}\bar{g}_{Z}^{2}}{\Delta_{g}}\alpha_{d}$ $\displaystyle r$ $\displaystyle\simeq$ $\displaystyle 1-s_{W}G^{\prime}\theta^{\prime}+\frac{2c_{W}s_{W}(\bar{g}_{Z}^{2}+4\bar{g}^{\prime\prime 2})}{\Delta_{g}}G^{\prime}\alpha_{c}+\frac{2(c_{W}^{2}\bar{g}_{Z}^{2}+4(c_{W}^{2}-2)\bar{g}^{\prime\prime 2})}{\Delta_{g}}G^{\prime}\alpha_{d}\;,$ (84) $\displaystyle l$ $\displaystyle\simeq$ $\displaystyle 1+s_{W}G^{\prime}\theta^{\prime}-s_{W}c_{W}\alpha_{a}+\frac{c_{W}^{2}}{2}\alpha_{b}-\frac{2c_{W}s_{W}(\bar{g}_{Z}^{2}+4\bar{g}^{\prime\prime 2})}{\Delta_{g}}G^{\prime}\alpha_{c}+\frac{2s_{W}^{2}(\bar{g}_{Z}^{2}+4\bar{g}^{\prime\prime 2})}{\Delta_{g}}G^{\prime}\alpha_{d}\;,~{}~{}~{}~{}$ (85) $\displaystyle\xi$ $\displaystyle\simeq$ $\displaystyle- c_{W}G^{\prime}\theta^{\prime}+(2c_{W}^{2}-1)\alpha_{a}+c_{W}s_{W}\alpha_{b}+\frac{8(2c_{W}^{2}-1)\bar{g}^{\prime\prime 2}}{\Delta_{g}}G_{3}\alpha_{c}-\frac{16c_{W}s_{W}\bar{g}^{\prime\prime 2}}{\Delta_{g}}G^{\prime}\alpha_{d}\;,$ (86) $\displaystyle\eta$ $\displaystyle\simeq$ $\displaystyle c_{W}G^{\prime}-\frac{c_{W}}{2}G^{\prime}\theta^{\prime 2}+\frac{2s_{W}(c_{W}^{2}\bar{g}_{Z}^{2}-2\bar{g}^{\prime\prime 2})}{\Delta_{g}}G^{\prime}\alpha_{a}+\frac{\bar{g}_{Z}^{2}c_{W}s_{W}^{2}}{\Delta_{g}}G^{\prime}\alpha_{b}+2s_{W}\alpha_{c}+2c_{W}\alpha_{d}\;.$ (87) Here, $\theta^{\prime}\simeq-2\bar{g}_{Z}\bar{g}^{\prime\prime}\bar{\beta}_{2}/\Delta_{g}$, $\theta_{W}=\arctan\bar{g}^{\prime}/\bar{g}$ and $G^{\prime}=\bar{\tilde{g}}^{\prime}/\bar{g}^{\prime\prime}$. Moreover, we obtain, $\displaystyle a\simeq 1+c_{W}s_{W}\alpha_{a}+\frac{s_{W}^{2}}{2}\alpha_{b}-2c_{W}s_{W}G^{\prime}\alpha_{c}-2c_{W}^{2}G^{\prime}\alpha_{d}\;,$ (88) $\displaystyle\theta^{\prime}\simeq\frac{\bar{g}_{Z}^{2}\theta_{r}-4\bar{g}^{\prime\prime 2}\theta_{l}}{\Delta_{g}}\;.$ (89) Note that since (72) tells us that if there is no kinetic mixings, $\theta_{l}=\theta_{r}=\theta^{\prime}$, then the differences $\theta_{l}-\theta^{\prime}$ and $\theta_{r}-\theta^{\prime}$ reflect the effects caused by kinetic mixings. Substituting (82) and (83) into (70), we find the result for $M^{2}/M^{\prime 2}$ which just matches the results that we obtained from (53). Although our result here already includes in all possible mixings cases, pure $Z$-$Z^{\prime}$ mass mixing is worth a special discussion: we find that the limit $G^{\prime}=\alpha_{c}=\alpha_{d}=0$ can not be taken at the very beginning, since this will lead to $\theta_{r}=\theta_{l}=\theta^{\prime}$ from (82) to (83) and then limit problems $0/0$ in (70) for $M^{2}/M^{\prime 2}$. To obtain the correct result, we need first to maintain $G^{\prime}$ and $\alpha_{c}$, $\alpha_{d}$ with nonzero values through to completion of the computation of the ratio $M^{2}/M^{\prime 2}$, then taking its vanishing limit. This is an interesting new phenomena, i.e. nonzero $G^{\prime}$ and $\alpha_{c}$, $\alpha_{d}$ extensions make that $M^{2}/M^{\prime 2}$ can be expressed in mixing parameters. In contrast with the pure $Z$-$Z^{\prime}$ mass mixing case that from (53) we find that just the mixing angle $\theta^{\prime}$ can not fully fix the value of $M^{2}/M^{\prime 2}$ as we are left with $\bar{\beta}_{2}$ degrees of freedom remaining. ## IV Measurability of the parameters and relevant EWCL coefficients Compared with the coefficients in EWCL, our eight parameters $\theta_{W}$, $G^{\prime}$, $\xi$, $\eta$, $\theta_{l}$, $\theta_{r}$, $r$, $l$ are more close to experimental data and more easily determined by experiment. Once these are known, the relevant EWCL coefficients can be further determined by establishing relations between these parameters and the EWCL coefficients. In this section, we begin by discussing how these parameter values can be fixed in principle from experiment, and then construct the relations among the EWCL coefficients and parameters. Experimentally, with the exception of $SU(2)_{L}$ coupling $g$ which can be determined from charged currents, the main means to determine the mixing parameters is by testing the structure of the electro-magnetic and neutral currents. The corresponding Lagrangian is $gW^{3}_{\mu}J^{3,\mu}+g^{\prime}B_{\mu}J_{Y}^{\mu}+g^{\prime\prime}X_{\mu}J_{X}^{\mu}$, where $J^{3,\mu}$ is the third component of the conventional weak isospin current, $J_{Y}^{\mu}$ is the hypercharge current, and $J_{X}^{\mu}$ is the current coupled to the extra $X_{\mu}$ boson. The Lagrangian of the electro- magnetic and neutral currents that couple to the physical bosons $\gamma,Z,Z^{\prime}$ becomes $eJ_{\mathrm{em}}^{\mu}A_{\mu}+g_{Z}J_{Z}^{\mu}Z_{\mu}+g^{\prime\prime}J_{Z^{\prime}}^{\mu}Z^{\prime}_{\mu}$. With the help of (29), we can read off $\displaystyle eJ_{\mathrm{em}}^{\mu}=gU_{1,2}J^{3,\mu}+g^{\prime}U_{2,2}J_{Y}^{\mu}+g^{\prime\prime}U_{3,2}J_{X}^{\mu}=gs_{W}a[J^{3,\mu}+J_{Y}^{\mu}]+g^{\prime\prime}U_{3,2}J_{X}^{\mu}$ (90) $\displaystyle g_{Z}J_{Z}^{\mu}=gU_{1,1}J^{3,\mu}+g^{\prime}U_{2,1}J_{Y}^{\mu}+g^{\prime\prime}U_{3,1}J_{X}^{\mu}$ $\displaystyle\hskip 14.22636pt=g[(s_{W}\xi+c_{W}c_{l}l)J^{3,\mu}+(s_{W}\xi- s_{W}c_{l}l\tan\theta_{W})J_{Y}^{\mu}]+g^{\prime\prime}U_{3,1}J_{X}^{\mu}$ (91) $\displaystyle g^{\prime\prime}J_{Z^{\prime}}^{\mu}=gU_{1,3}J^{3,\mu}+g^{\prime}U_{2,3}J_{Y}^{\mu}+g^{\prime\prime}U_{3,3}J_{X}^{\mu}\;,$ (92) with $U_{i,j}$ a general matrix element of mixing matrix $U$, and we have used the result $gU_{1,2}=g^{\prime}U_{2,2}$ combined (34) and (43). In principle, once experiments finally fix the coefficients $U_{i,j}$, then from (34), we can determine all eight parameters $\theta_{W}$, $G^{\prime}$, $\xi$, $\eta$, $\theta_{l}$, $\theta_{r}$, $r$, $l$. Considering that $Z^{\prime}$ has not been discovered as yet in current experiments, we divide the present experimental measurability of the parameters into two stages: 1. 1. Suppose we can measure $eJ_{\mathrm{em}}^{\mu}$ and $g_{Z}J_{Z}^{\mu}$ experimentally but not know what $J_{Z^{\prime}}^{\mu}$ and $J_{X}^{\mu}$ are. This is the present SM situation as it stands and is independent of details of the $Z^{\prime}$ model. Then (90) implies that we can determine $gs_{W}a$ and the electro-magnetic coupling $e$ now must be identified as $e=gs_{W}a$. Compared with conventional relation in SM, we find that an extra correction factor $a$ appears in the relation. Considering that $e$ and $g$ can be measured from electro-magnetic and charge currents respectively, we can then derive $s_{W}a$. Further, from (91), we find $g(s_{W}\xi+c_{W}c_{l}l)$ and $g(s_{W}\xi-s_{W}c_{l}l\tan\theta_{W})$. Then, in this first stage, combined with known $g$, we can obtain four combinations of the eight parameters: $g$, $s_{W}a$, $s_{W}\xi+c_{W}c_{l}l$ and $s_{W}\xi-s_{W}c_{l}l\tan\theta_{W}$. 2. 2. Suppose in addition to $eJ_{\mathrm{em}}^{\mu}$ and $g_{Z}J_{Z}^{\mu}$, we also know $J_{X}^{\mu}$. This can be realized if we have a prior $U(1)^{\prime}$ charges for the SM fermions which is $Z^{\prime}$ model- dependent. Then from (90) and (34), $g^{\prime\prime}U_{3,2}=g^{\prime\prime}(s_{W}\eta+c_{W}s_{r}r)$ is obtainable; from (91) and (34), $g^{\prime\prime}U_{3,1}=g^{\prime\prime}(c_{W}\eta-s_{W}s_{r}r)$ is calculable. We find at this second stage, we can obtain a further two combinations of the eight parameters. Therefore, before needing to measure $g^{\prime\prime}J_{Z^{\prime}}^{\mu}$, the above two stages already enable us evaluate seven of the eight parameters. Using (81), the remaining unknown parameter can be determined once we assume a $Z-Z^{\prime}$ mass ratio. Thus, even without the knowledge of $g^{\prime\prime}J_{Z^{\prime}}^{\mu}$, and as long as the $Z-Z^{\prime}$ mass ratio is fixed, we can now measure all eight phenomenological parameters. In consequence, we can express the EWCL coefficients in these parameters. Up to order $p^{4}$, the theoretical coefficients $\bar{g}_{Z}/\bar{g}^{\prime\prime},\bar{\beta}_{2},\alpha_{a},\alpha_{b},\alpha_{c},\alpha_{d}$ in phenomenological parameters $\theta_{W},\theta_{r},\theta_{l},G^{\prime},\xi,\eta,l,r$ can be written as $\displaystyle\frac{\bar{g}_{Z}}{\bar{g}^{\prime\prime}}$ $\displaystyle\simeq$ $\displaystyle\frac{2(\theta_{l}-\theta^{\prime})}{\theta_{r}-\theta^{\prime}}\;,$ (93) $\displaystyle\bar{\beta}_{2}$ $\displaystyle\simeq$ $\displaystyle-\frac{\bar{g}_{Z}^{2}\theta_{r}-4\bar{g}^{\prime\prime 2}\theta_{l}}{2\bar{g}_{Z}\bar{g}^{\prime\prime}}\;,$ (94) $\displaystyle\alpha_{a}$ $\displaystyle=$ $\displaystyle-\frac{1}{4s_{W}c_{W}\bar{g}^{\prime\prime 2}\Delta_{g}}\Big{\\{}s_{W}(\bar{g}_{Z}^{2}s_{W}^{2}+2(c_{W}^{2}-2)\bar{g}^{\prime\prime 2})\Delta_{g}G^{\prime}\theta^{\prime}$ (95) $\displaystyle+8s_{W}^{2}\bar{g}^{\prime\prime 2}\Delta_{g}(l-1)+(\bar{g}_{Z}^{2}s_{W}^{2}+(4-2c_{W}^{2})\bar{g}^{\prime\prime 2})\Delta_{g}(r-1)$ $\displaystyle-4c_{W}s_{W}\bar{g}^{\prime\prime 2}\Delta_{g}\xi+c_{W}(-\bar{g}_{Z}^{4}s_{W}^{2}-2\bar{g}^{\prime\prime 2}c_{W}^{2}\Delta_{g})G^{\prime}\eta\Big{\\}}$ $\displaystyle\alpha_{b}$ $\displaystyle=$ $\displaystyle-\frac{1}{4c_{W}^{2}\bar{g}^{\prime\prime 2}\Delta_{g}}\Big{\\{}s_{W}(\Delta_{g}-2c_{W}^{2}\bar{g}_{Z}^{2}+4c_{W}^{2}\bar{g}^{\prime\prime 2})\Delta_{g}G^{\prime}\theta^{\prime}$ (96) $\displaystyle+8\bar{g}^{\prime\prime 2}(1-2c_{W}^{2})\Delta_{g}(l-1)+((1-2c_{W}^{2})\bar{g}_{Z}^{2}+4\bar{g}^{\prime\prime 2}s_{W}^{2})\Delta_{g}(r-1)$ $\displaystyle-8s_{W}c_{W}\bar{g}^{\prime\prime 2}\Delta_{g}\xi+c_{W}(-\bar{g}_{Z}^{4}s_{W}^{2}-16\bar{g}^{\prime\prime 4}s_{W}^{2}+\bar{g}_{Z}^{2}c_{W}^{2}\Delta_{g})G^{\prime}\eta\Big{\\}}$ $\displaystyle\alpha_{c}$ $\displaystyle=$ $\displaystyle\frac{1}{8s_{W}c_{W}\bar{g}^{\prime\prime 2}\Delta_{g}}\Big{\\{}s_{W}c^{2}_{W}\Delta_{g}^{2}\theta^{\prime}+s^{2}_{W}c^{2}_{W}(5\bar{g}_{Z}^{2}+14s_{W}\bar{g}^{\prime\prime 2})\Delta_{g}G^{\prime}\theta^{\prime 2}$ (97) $\displaystyle- c_{W}^{2}s_{W}\Delta_{g}^{2}\theta_{r}-8s_{W}^{2}\bar{g}^{\prime\prime 2}(-\bar{g}_{Z}^{2}s_{W}^{2}+4\bar{g}^{\prime\prime 2})G^{\prime}(l-1)$ $\displaystyle+(s_{W}^{4}\bar{g}_{Z}^{4}-16\bar{g}^{\prime\prime 4}+2\bar{g}^{\prime\prime 2}c_{W}^{4}\bar{g}_{Z}^{2}+8\bar{g}^{\prime\prime 4}c_{W}^{2})G^{\prime}(r-1)$ $\displaystyle+4s_{W}c_{W}\bar{g}^{\prime\prime 2}(\bar{g}_{Z}^{2}(c_{W}^{2}-2)+4\bar{g}^{\prime\prime 2})G^{\prime}\xi+4s_{W}^{2}c_{W}\bar{g}^{\prime\prime 2}\Delta_{g}\eta\Big{\\}}$ $\displaystyle\alpha_{d}$ $\displaystyle=$ $\displaystyle-\frac{1}{8\bar{g}^{\prime\prime 2}\Delta_{g}}\Big{\\{}s_{W}\Delta_{g}^{2}\theta^{\prime}+4\bar{g}^{\prime\prime 2}\Delta_{g}G^{\prime}+(5\bar{g}_{Z}^{2}s_{W}^{2}+(12-14c_{W}^{2})\bar{g}^{\prime\prime 2})\Delta_{g}G^{\prime}\theta^{\prime 2}$ (99) $\displaystyle- s_{W}\Delta_{g}^{2}\theta_{r}-8s_{W}^{2}\bar{g}_{Z}^{2}\bar{g}^{\prime\prime 2}G^{\prime}(l-1)+\bar{g}_{Z}^{2}(-\bar{g}_{Z}^{2}s_{W}^{2}+(-4+2c_{W}^{2})\bar{g}^{\prime\prime 2})G^{\prime}(r-1)$ $\displaystyle+4s_{W}c_{W}\bar{g}_{Z}^{2}\bar{g}^{\prime\prime 2}G^{\prime}\xi-4c_{W}\bar{g}^{\prime\prime 2}\Delta_{g}\eta\Big{\\}}$ Where $\theta^{\prime}$ is given by (89) and $\bar{g}_{Z}/\bar{g}^{\prime\prime}$ is given by (93). The remaining two theoretical coefficients $\bar{g}^{\prime}/\bar{g}$ and $\bar{\tilde{g}}^{\prime}/\bar{g}^{\prime\prime}$, which are already determined in (43) and (42) respectively, are not displayed with the above formulae. Substituting the results back into (73) and combining with (61), we further obtain $\displaystyle a$ $\displaystyle=$ $\displaystyle 1-\frac{1}{8c_{W}^{2}\bar{g}^{\prime\prime 2}\Delta_{g}}\Big{\\{}s_{W}(s_{W}^{2}\bar{g}_{Z}^{2}-4\bar{g}^{\prime\prime 2})\Delta_{g}G^{\prime}\theta^{\prime}+8s_{W}^{2}\bar{g}^{\prime\prime 2}\Delta_{g}(l-1)$ $\displaystyle+(\bar{g}_{Z}^{2}s_{W}^{2}+4\bar{g}^{\prime\prime 2})\Delta_{g}(r-1)-8s_{W}c_{W}\bar{g}^{\prime\prime 2}\Delta_{g}\xi- c_{W}(\bar{g}_{Z}^{4}s_{W}^{2}-4\bar{g}_{Z}^{2}\bar{g}^{\prime\prime 2}c_{W}^{2}+16\bar{g}^{\prime\prime 4})G^{\prime}\eta\Big{\\}}$ The results (89) to (IV) indicate that once we known the eight phenomenological parameters $\theta_{W},G^{\prime},\xi,\eta,\theta_{l},\theta_{r},r,l$, the conventional $Z$-$Z^{\prime}$ mixing angle $\theta^{\prime}$, then the general $Z$-$\gamma$-$Z^{\prime}$ mixing coefficients $\bar{g}/\bar{g}^{\prime},\bar{g}_{Z}/\bar{g}^{\prime\prime},G^{\prime},\bar{\beta}_{2},\alpha_{a},\alpha_{b},\alpha_{c},\alpha_{d}$, and $a$ are fixed, where the $a$ parameter although appears in phenomenological role, as discussed earlier, it is derivable from the other eight parameters through (IV). ## V Summary To summarize our results, based on the extended electroweak chiral Lagrangian previously proposed by us, we have found that there are eight independent degrees of freedoms to describe the most general $Z$-$\gamma$-$Z^{\prime}$ mixings that correspond to the eight independent theoretical coefficients $\bar{g}/\bar{g}^{\prime},\bar{\tilde{g}}^{\prime}/\bar{g}^{\prime\prime},\bar{g}_{Z}/\bar{g}^{\prime\prime}$, $\bar{\beta}_{2}$, $\alpha_{a}$, $\alpha_{b}$, $\alpha_{c}$, $\alpha_{d}$ in our electroweak chiral Lagrangian. For convenience in phenomenological analysis, we have proposed a new general parameterization involving these eight parameters that describe the $Z$-$\gamma$-$Z^{\prime}$ mixings, which include the conventional Weinberg angle $\theta_{W}$ and a Stueckelberg-type coupling $G^{\prime}$. Combined with the conventional $Z$-$Z^{\prime}$ mass mixing parameter $\theta^{\prime}$, we find that parameters $\xi$, $\eta$, $\theta_{l}$-$\theta^{\prime}$, $\theta_{r}$-$\theta^{\prime}$, $r$, $l$ reflect the general kinetic mixings among the $Z$-$\gamma$-$Z^{\prime}$. With this parameterization, $\theta_{W}$, $G^{\prime}$, $\xi$, $\eta$, $\theta_{l}$, $\theta_{r}$, $r$, $l$, we can fully determine the $Z$-$Z^{\prime}$ mass mixing angle $\theta^{\prime}$ and the mass ratio $M_{Z}/M_{Z^{\prime}}$. Experimentally, with the knowledge of charge currents, neutral currents and the current for extra gauge boson $X_{\mu}$, combined with mass ratio $M_{Z}/M_{Z^{\prime}}$, we can in principle measure all eight parameters. ## Acknowledgments This work was supported by National Science Foundation of China (NSFC) under Grant No. 10875065 and No.10947152. ## References * (1) P. Langacker, Rev. Mod. Phys. 81, 1199(2008) and references therein. * (2) P. Langacker, arXiv:0911.4294. * (3) Y. Zhang, S.-Z. Wang and Q. Wang, JHEP 03, 047(2008). * (4) B. Holdom, Phys. Lett. B166, 196(1986); B259, 329(1991). * (5) D.Feldman, Z.-W. Lu, P.Nath, Phys. Rev. D75, 115001(2007); D.Feldman, B.Kors, P.Nath, Phys. Rev. D75, 023503(2007); D.Feldman, Z.-W. Lu, P.Nath, G.Peim, Phys. Rev. D81, 095017(2010). * (6) Y. Zhang, and Q. Wang, JHEP 07, 012(2009). * (7) T. Appelquist, G.-H. Wu, Phys. Rev. D 48, 3235(1993). * (8) B. Körs and P. Nath, Phys. Lett. B 586, 366 (2004).	arxiv-papers	2010-11-19T13:05:27	2024-09-04T02:49:15.079408	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ying Zhang, Qing Wang", "submitter": "Wang Qing", "url": "https://arxiv.org/abs/1011.4418" }
1011.4453	# Metallic nanograins: spatially nonuniform pairing induced by quantum confinement M. D. Croitoru1,3 A. A. Shanenko2 C. C. Kaun3 F. M. Peeters2 1Institut für Theoretische Physik III, Universität Bayreuth, 95440 Bayreuth, Germany 2Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium 3Research Center for Applied Sciences, Academia Sinica, 11529 Taipei, Taiwan ###### Abstract It is well-known that the formation of discrete electron levels strongly influences the pairing in metallic nanograins. Here we focus on another effect of quantum confinement in superconducting grains that was not studied previously, i.e., spatially nonuniform pairing. This effect is very significant when single-electron levels form bunches and/or a kind of shell structure: in highly symmetric grains the order parameter can exhibit variations with position by an order of magnitude. Nonuniform pairing is closely related to a quantum-confinement induced modification of the pairing- interaction matrix elements and size-dependent pinning of the chemical potential to groups of degenerate or nearly degenerate levels. For illustration we consider spherical metallic nanograins. We show that the relevant matrix elements are as a rule enhanced in the presence of quantum confinement, which favors spatial variations of the order parameter, compensating the corresponding energy cost. The size-dependent pinning of the chemical potential further increases the spatial variation of the pair condensate. The role of nonuniform pairing is smaller in less symmetric confining geometries and/or in the presence of disorder. However, it always remains of importance when the energy spacing between discrete electron levels $\delta$ is approaching the scale of the bulk gap $\Delta_{B}$, i.e., $\delta>0.1$-$0.2\,\Delta_{B}$. metallic nanograins, nonuniform pairing, superconducting correlations, matrix elements ###### pacs: 74.20.Fg, 74.78.Na ## I Introduction Quantum confinement plays a fundamental role in superconductors with nanoscale dimensions. Interplay of quantum confinement and pairing correlations results in important qualitative changes in the superconductor characteristics. parm ; blat ; mush ; smit ; brau ; perali ; sier ; glad ; yuzb ; kres01 ; kres02 ; sh01 ; sh02 ; sh03 ; sh04 ; sh05 ; gar Because of technological reasons quasi-0D superconducting structures (i.e., ensembles of small grains) were the first where this interplay was investigated experimentally. Initial attempts by Giaever and Zeller at the end of 60s used tunneling studies on large ensembles of superconducting particles. gaiv01 Since that time most of the studies on superconducting correlations in grains were performed with grain powders li01 ; li02 or on films made of crystalline granules separated by amorphous inter-granular space. bose01 ; bose02 In the pioneering work of Ralph et al. ralph01 ; ralph02 the discrete electron spectrum was measured for a single grain. Their technique (single-electron tunneling spectroscopy) enabled them for the first time to probe superconducting correlations in an individual ${\rm Al}$ grain. Very recently, STM was used to detect the superconducting gap of an isolated ultra-small lead grain deposited onto a silicon substrate (see e.g. Refs. bose03, ; bose04, ). These advances opened new prospects to examine superconductivity in individual metallic nanograins with unprecedented detail, e.g., to investigate the influence of the confinement on the superconducting correlations. The main feature of a superconducting nanograin that makes them different from a bulk superconductor is the formation of discrete electron levels with average energy spacing $\delta\approx 2\pi^{2}\hbar^{2}/(mk_{F}V)$, with $k_{F}$ the bulk Fermi wave number and $V$ the system volume. It can be of the same order as the bulk gap $\Delta_{B}$, or even larger in the case of ultra- small nanograins. Therefore, size-quantization of the electron spectrum can have a substantial impact on the basic superconducting characteristics of such quasi-0D superconducting systems. The understanding of the fundamental properties of superconducting correlations in low-dimensional structures, in particular in isolated metallic grains, has experienced a remarkable development in the last two decades. Theoretical aspects, which have attracted the most attention are the following. The problem of the breakdown of BCS superconductivity in ultra- small metallic grains was addressed in several papers. delft ; smit ; mat The effect of the shell structure in the single-electron spectrum on the superconducting correlations was pointed out for nanograins glad ; gar and ultrasmall metallic clusters. kres01 ; kres02 The ground state properties of the BCS pairing Hamiltonian of ultra-small grains were considered beyond the mean-field approximation using the Richardson exact solution. sier ; glad ; yuzb A spatially uniform pairing was assumed in these and other works and, as a consequence, the matrix elements of the pairing interaction were taken independent of the relevant single-electron quantum numbers, i.e., they were set to $-g/V$, with $g>0$ the coupling constant and $V$ the volume. note1 This is, say, a bulk-like approximation recovered when the single-electron wave functions are taken as plane waves. However, the translational invariance is broken in nanograins, which leads to a position-dependent order parameter. As a result, the pairing gap becomes strongly dependent on the relevant quantum numbers, which is directly related to a confinement-induced modification (as compared to $-g/V$) of the matrix elements controlling the scattering of the time reversed states. Another important issue is that single-electron levels can form bunches and even a kind of shell structure in symmetric confining geometries. In this case the chemical potential $\mu$ can be pinned to a group of nearly degenerate or degenerate levels. This is of importance because the density of states in the vicinity of $\mu$ strongly influences the superconducting correlations. In other words, such a pinning plays the role of a filter that selects the contribution of a particular single-electron shell (or of a group of close levels) to the superconducting order parameter. Such a contribution is, as a rule, spatially nonuniform. The aim of the present paper is to investigate effects related to a spatially nonuniform pairing in metallic nanograins, which was not studied in previous publications. For illustrative purposes we consider metallic spherical nanograins, where the spatial dependence of the superconducting condensate is pronounced (the order parameter can vary with position by an order of magnitude). In less symmetric confining geometries and/or in the presence of disorder spatial variations of the order parameter are reduced. However, our study implies that nonuniform pairing remains of importance when the interlevel spacing $\delta$ is approaching the scale of the order of the bulk gap $\Delta_{B}$. Any remaining grouping of single-electron levels, that is always present in real samples, even strengthens the effect of interest. We work in the mean-field approximation and, thus, stay in the regime $\delta\lesssim\Delta_{B}$. Below we consider ${\rm Sn}$ and ${\rm Al}$ with $\Delta_{B}=0.616$ and $0.25\,{\rm meV}$, respectively (for the parameters used below). Using the above values of $\Delta_{B}$, we find that the mean- field approach is valid for $D>6$-$8\,{\rm nm}$, with $D$ the sphere diameter. Our paper is organized as follows. In Sec. II, we outline the formalism how to obtain a self-consistent solution to the problem. In Sec. III, we present our numerical results. In particular, in Sec. III.1 we investigate the effects of quantum confinement on pairing correlations through the modifications of the matrix elements of the pairing interaction and the size-dependent pinning of $\mu$ to single-electron shells. Sec. III.2 is focused on a spatial distribution of the pair condensate and its relation to modifications of the matrix elements and the size-dependent pinning of $\mu$. In Sec. III.3 we discuss the interplay of Andreev reflection with quantum confinement, resulting in the formation of Andreev-type states and significant dependence of the pairing gaps on the relevant quantum numbers. A short summary and discussion are given in Sec. IV. ## II Formalism The reduction of the system to the nanometer scale leads to the formation of a discrete electron spectrum. Moreover, in the presence of quantum confinement, the translational invariance of the system is broken, and the superconducting order parameter is position dependent, i.e., $\Delta=\Delta({\bf r})$. For the mean-field treatment of such a situation, it is appropriate to use the Bogoliubov-de Gennes (BdG) equations, bogl ; degen which can be written as $\displaystyle E_{i}\|u_{i}\rangle=\widehat{H}_{e}\|u_{i}\rangle+{\widehat{\Delta}}\|v_{i}\rangle,$ (1a) $\displaystyle E_{i}\|v_{i}\rangle={\widehat{\Delta}}^{\ast}\|u_{i}\rangle-\widehat{H}_{e}^{\ast}\|v_{i}\rangle,$ (1b) where $E_{i}$ stands for the Bogoliubov-quasiparticle (bogolon) energy, $\widehat{\Delta}=\Delta(\widehat{\bf r})$ (with $\widehat{\bf r}$ the position operator) and the single-electron Hamiltonian is referred to the chemical potential $\mu$, i.e., $\widehat{H}_{e}({\bf r})=\frac{\widehat{\bf p}^{2}}{2m_{e}}+V(\widehat{\bf r})-\mu.$ (2) We remark that any magnetic effects are beyond the scope of the present paper. For simplicity, the confining interaction $V({\bf r})$ is taken as zero inside the specimen and infinite outside: $V({\bf r})=V_{B}\;\vartheta(R-\rho)$ with the barrier potential $V_{B}\to\infty$ ($R=D/2$ and $\rho$ is the radial coordinate for the spherical confining geometry). As a mean-field approach, the BdG equations should be solved in a self- consistent manner $\Delta({\bf r})=g\sum_{i}\langle{\bf r}\|u_{i}\rangle\langle v_{i}\|{\bf r}\rangle\tanh(\frac{\beta E_{i}}{2}),$ (3) where $g>0$ is the coupling constant for the effective electron-electron interaction approximated by the delta-function potential, i.e., $\langle{\bf r},{\bf r}^{\prime}\|\Phi\|{\bf r},{\bf r}^{\prime}\rangle=-g\delta({\bf r}-{\bf r}^{\prime})$. The sum in Eq. (3) runs over the states with the single- electron energy $\xi_{i}=\bigl{[}\langle u_{i}\|\widehat{H}_{e}\|u_{i}\rangle+\langle v_{i}\|\widehat{H}_{e}\|v_{i}\rangle\bigr{]}\in[-\hbar\omega_{D},\hbar\omega_{D}],$ (4) with $\omega_{D}$ the Debye frequency. As is known, the solution of the BdG equations has two branches: $(i,+)$ and $(i,-)$ (see Ref. swid, ) for which we have $E_{i,+}>0$ and $E_{i,-}<0$. The sum in Eq. (3) should be taken over the physical states [the $(i,+)$ branch], i.e., $E_{i}=E_{i,+}$. For a given mean electron density $n_{e}$ the chemical potential $\mu$ is determined from $n_{e}=\frac{2}{V}\sum_{i}\bigl{[}f_{i}\langle u_{i}\|u_{i}\rangle+(1-f_{i})\langle v_{i}\|v_{i}\rangle\bigr{]},$ (5) with $V=\frac{4}{3}\pi R^{3}$ the volume of the spherical grain. For conventional superconductors the energy gap is typically much smaller than the chemical potential. As a result, $\mu$ stays nearly the same when passing from the normal state to the superconducting one. degen Therefore, one can solve Eq. (5) in the absence of superconducting order ($\Delta({\bf r})=0$). In a spherical nanograin, because of symmetry reasons, the order parameter depends only on the radial coordinate, i.e., $\Delta=\Delta(\rho)$. Therefore the pseudospinor in the particle-hole space can be characterized by the quantum numbers of the angular momentum, i.e., ($l,m$). The angular part of the pseudospinor $\Psi_{i}$ is given by the spherical harmonics $Y_{lm}(\theta,\varphi)$ in polar coordinates $(\rho,\theta,\varphi)$, i.e., $\langle{\bf r}\|\Psi_{i}\rangle=Y_{lm}(\theta,\varphi)\left(\begin{array}[]{c}u_{jl}(\rho)\\\ v_{jl}(\rho)\end{array}\right),$ (6) where $i=\\{j,l,m\\}$, with $j$ the radial quantum number associated with the quantum-confinement boundary conditions $u_{jl}(\rho)\|_{\rho=R}=v_{jl}(\rho)\|_{\rho=R}=0.$ (7) To solve the BdG equations (1a) and (1b) numerically, $u_{jl}(\rho)$ and $v_{jl}(\rho)$ are expanded in the eigenfunctions of the single-electron Hamiltonian $\widehat{H}_{e}$ [see Eq. (2)]. In addition, iterations should be invoked, to account for the self-consistency relation given by Eq. (3). This program is significantly simplified by keeping only the pairing of the time- reversed states, and which is a standard approximation for the problem of superconducting correlations in nanograins. In the framework of the BdG equations this can be done through the so-called Anderson approximate solution for which the particle- and hole-like wave functions are assumed to be proportional to the single-electron wave function. It means that $u_{jl}(\rho)={\cal U}_{jl}\,\chi_{jl}(\rho),\;u_{jl}(\rho)={\cal V}_{jl}\,\chi_{jl}(\rho),$ (8) with the radial part of the single-electron wave function given by $\chi_{jl}(\rho)=\frac{\sqrt{2}}{R^{3/2}j_{l+1}(\alpha_{jl})}j_{l}(\alpha_{jl}\frac{\rho}{R}),$ (9) with $j_{l}(x)$ the $l$-order spherical Bessel function of the first kind and $\alpha_{jl}$ its $j$-node. The coefficients ${\cal U}_{jl}$ and ${\cal V}_{jl}$ (taken as real) obey the standard constraint (see, e.g., Refs. ket, ) ${\cal U}^{2}_{jl}+{\cal V}^{2}_{jl}=1.$ (10) Then, inserting Eq. (8) into Eqs. (1a) and (1b) we find the following set of coupled equations (here $E_{jlm}=E_{jl}$ and $\xi_{jlm}=\xi_{jl}$): $\displaystyle[E_{jl}-\xi_{jl}]~{}{\cal U}_{jl}=\Delta_{jl}\,{\cal V}_{jl},$ (11a) $\displaystyle[E_{jl}+\xi_{jl}]~{}{\cal V}_{jl}=\Delta_{jl}\,{\cal U}_{jl},$ (11b) with $\Delta_{jl}=\int\limits_{0}^{R}{\rm d}\rho\,\rho^{2}\,\chi^{2}_{jl}(\rho)\Delta(\rho)$ (12) and $\xi_{jl}=\frac{\hbar^{2}}{2m_{e}}\frac{\alpha_{jl}^{2}}{R^{2}}-\mu.$ (13) A nontrivial physical solution of Eqs. (11a) and (11b) exists only when $E_{jl}=\sqrt{\xi_{jl}^{2}+\Delta^{2}_{jl}}.$ (14) The Anderson prescription about the pairing of the time-reversed states allows one to rephrase the self-consistency relation [see Eq. (3)] as follows: $\Delta_{j^{\prime}l^{\prime}}=-\sum\limits_{jl}(2l+1)\;\frac{M_{j^{\prime}l^{\prime},jl}\;\Delta_{jl}}{2\sqrt{\xi^{2}_{jl}+\Delta^{2}_{jl}}}\tanh(\frac{\beta E_{jl}}{2}),$ (15) where $\displaystyle M_{j^{\prime}l^{\prime},jl}=-\frac{g}{4\pi}\int\limits_{0}^{R}\\!\\!{\rm d}\rho\,\rho^{2}~{}\chi_{j^{\prime}l^{\prime}}^{2}(\rho)~{}\chi_{jl}^{2}(\rho).$ To derive Eq. (15), one should keep in mind the property of the spherical harmonics $\sum_{m=-l}^{l}\;\|Y_{lm}(\theta,\varphi)\|^{2}=\frac{2l+1}{4\pi}$. We remark that $M_{j^{\prime}l^{\prime},jl}$ is nothing else but the pairing- interaction matrix element $\langle i^{\prime},\bar{i^{\prime}}\|\Phi\|i,\bar{i}\rangle$ (with $\bar{i}=\\{j,l,-m\\}$) averaged over the states with $m=-l,\ldots l$ and $m^{\prime}=-l^{\prime},\ldots l^{\prime}$, i.e., $M_{j^{\prime}l^{\prime},jl}=\frac{1}{(2l^{\prime}+1)(2l+1)}\sum\limits_{m^{\prime}=-l^{\prime}}^{l^{\prime}}\sum\limits_{m=-l}^{l}\langle i^{\prime},\bar{i^{\prime}}\|\Phi\|i,\bar{i}\rangle.$ As seen from Eq. (12), a spatially uniform order parameter means that the pairing gaps $\Delta_{jl}$ do not depend on the quantum numbers $j$ and $l$. This is compatible with Eq. (15) only when $M_{j^{\prime}l^{\prime},jl}$ does not depend on $j^{\prime}$ and $l^{\prime}$. According to the definition given by Eq. (LABEL:M_mat), we have $M_{j^{\prime}l^{\prime},jl}=M_{jl,j^{\prime}l^{\prime}}$ and, so, if $M_{j^{\prime}l^{\prime},jl}$ does not depend on $j^{\prime},l^{\prime}$, it does not depend on $j,l$ either. So, we arrive at the standard simplified approach of investigating the pairing correlations in metallic grains (see the discussion in the Introduction). Below we show that the spatial dependence of the order parameter can not be ignored in superconducting nanograins, which implies significant variations of the matrix elements and pairing gaps with the relevant quantum numbers. After a numerical solution of Eq. (15), the position-dependent order parameter can be calculated from $\Delta(\rho)=\sum\limits_{jl}\Delta^{(jl)}(\rho),$ (16) with the shell-dependent contribution $\Delta^{(jl)}(\rho)$ given by $\Delta^{(jl)}(\rho)=\frac{g}{8\pi}\,(2l+1)\,\frac{\chi_{jl}^{2}(\rho)\,\Delta_{jl}}{\sqrt{\xi^{2}_{jl}+\Delta^{2}_{jl}}}\tanh(\frac{\beta E_{jl}}{2}).$ (17) ## III Discussion of results ### III.1 Enhanced intrashell matrix elements and quantum-size pinning of the chemical potential Figure 1: Critical temperature versus the grain diameter as calculated for: (a) $M_{j^{\prime}l^{\prime},jl}\neq-g/V$ and $\mu\neq\mu_{B}$; (b) $M_{j^{\prime}l^{\prime},jl}=-g/V$ and $\mu\neq\mu_{B}$; and (c) $M_{j^{\prime}l^{\prime},jl}\neq-g/V$ and $\mu=\mu_{B}$. The dashed curves in (a) show approximate lower and upper boundaries for the quantum-size oscillations of $T_{c}$, both curves represent the same dependence $T_{c}/T_{c,B}=1+a(D/D_{0})^{3/2}$, with $D_{0}=50\,{\rm nm}$ and $a=1$ (the lower boundary) and $a=3.5$ (the upper one). The same curves are also given in (b) and (c), for comparison. Numerical calculations were performed with the set of parameters typical for tin degen ; fett : $\hbar\omega_{D}/k_{B}=195~{}{\rm K}$, $gN(0)=0.25$, with $N(0)$ the bulk density of states at the Fermi level (we use the bulk electron density $n_{e}=148\,{\rm nm}^{-3}$, see, e.g., Ref. ash, ). Figure 1(a) shows the critical temperature (in units of the bulk critical temperature $T_{c,B}$) versus the nanograin diameter $D$ as calculated from Eq. (15) when the matrix elements of the electron-electron interaction and the size-dependent variation of the chemical potential have been fully taken into account. Results in Fig. 1 are presented for a step $\Delta R=0.01~{}{\rm nm}$. For each radius the critical temperature was defined as the temperature above which the spatially-averaged order parameter $\langle\Delta(\rho)\rangle$ becomes smaller than $0.01$ of its value at $T=0$. Our numerical results exhibit two features typical of the size- dependent pairing characteristics in high-quality superconducting nanograins and nuclei. First, we observe an overall increase of $T_{c}$ with decreasing $D$ (it is very pronounced due to the highly-symmetric confining geometry). Second, $T_{c}$ oscillates wildly with $D$. This oscillatory behavior can be understood in the following way. The pair correlations are nonzero only for the states within a finite range (the Debye window) around the chemical potential $\mu$. Moreover, the main contribution to the sum in Eq. (15) comes from the states in the very vicinity of the Fermi level, because in this case the expression $\Delta_{jl}/\sqrt{\xi^{2}_{jl}+\Delta^{2}_{jl}}\simeq 1$ $(\xi_{jl}\simeq 0)$. When varying the nanograin size, the number of states in the Debye window changes. The smaller the diameter, the smaller the number of relevant states contributing to the pairing characteristics and, as a result, the more significant is such a change. This change is not monotonous but rather oscillating due to a permanent competition between incoming and outcoming states. As a consequence, all basic pairing characteristics, e.g., $T_{c}$ and pairing gaps $\Delta_{jl}$, exhibit quantum-size oscillations. It is not only typical of nanograins with superconducting correlations (see, e.g., the recent paper bose04 ) but it is also present in superconducting nanowires sh01 ; sh02 ; sh03 ; sh04 ; sh05 and nanofilms. guo ; eom Such oscillations are pronounced for small diameters/thicknesses but decay with increasing the characteristic size so that $T_{c}$ approaches the bulk critical temperature $T_{c,B}$ (for our parameter $T_{c,B}=4.01\,{\rm K}$). It is interesting to note that the overall increase of $T_{c}$ with decreasing $D$ in Fig. 1(a) is similar to a size-dependent enhancement of the pairing gap in nuclei, where it is proportional to $1/\sqrt{A}$ (see, e.g., Ref. sat, ), with $A$ the number of nucleons. In particular, the two dashed curves in Fig. 1(a) show approximate upper and lower boundaries for $T_{c}$, highlighting the magnitude of the quantum-size oscillations: both curves represent the same dependence, i.e., $T_{c}/T_{c,B}=1+a\,(D_{0}/D)^{3/2}$, with $D_{0}=50\,{\rm nm}$ and $a=1$ and $3.5$ for the lower and upper boundaries, respectively [$(D_{0}/D)^{3/2}\propto N^{-1/2}_{e}$, with $N_{e}=n_{e}V$ the number of electrons]. We remark that real samples exhibit inevitable shape and size fluctuations that affect the high-degeneracy of single-electron levels. Hence, measurements on an ensemble of nanograins will significantly smooth the quantum-size oscillations in the critical temperature and reduce its overall enhancement with decreasing nanograin size (see, also, Sec. IV). For instance, in experimentally fabricated tin nanograins of a semi-spherical shape the observed enhancement of the excitation gap over its bulk value is about bose04 $60\%$ for the particle heights $\approx 10$-$20\,{\rm nm}$. This is significantly smaller than the enhancement of $T_{c}$ shown in Fig. 1(a). However, detailed investigations of the enhancement of $T_{c}$ in superconducting nanograins is beyond the scope of our present paper. Here we are interested in a spatially nonuniform distribution of the pair condensate which is of importance even in the presence of shape and size fluctuations and disorder (see the discussion in Sec. IV). In order to outline the role of the matrix elements $M_{j^{\prime}l^{\prime},jl}$ [see Eq. (LABEL:M_mat)] of the electron-electron interaction we also show what happens when the true matrix elements are simply replaced by those of the bulk-like form: $M_{j^{\prime}l^{\prime},jl}=-g/V$, which is what is usually done when investigating the superconducting correlations in nanograins. The results are displayed in Fig. 1(b) and, as seen, the difference with respect to Fig. 1(a) is significant. To simplify the comparison, we show also in Fig. 1(b) two solid curves that represent the radius-dependent upper and lower values of $T_{c}$ from Fig. 1(a). Table 1: Matrix elements $M_{j^{\prime}l^{\prime},jl}=M_{jl,j^{\prime}l^{\prime}}$ in units of $-g/V$ calculated at $D=7.1\,{\rm nm}$ for quantum numbers such that $\xi_{j^{\prime}l^{\prime}},\xi_{jl}<\hbar\omega_{D}$: $M_{j^{\prime}l^{\prime},jl}$ \| $j^{\prime}$ \| $l^{\prime}$ \| $j$ \| $l$ ---\|---\|---\|---\|--- 10.62 \| 31 \| 11 \| 31 \| 11 1.9 \| 31 \| 11 \| 23 \| 29 1.33 \| 31 \| 11 \| 19 \| 39 0.64 \| 31 \| 11 \| 8 \| 71 0.41 \| 31 \| 11 \| 1 \| 101 4.71 \| 23 \| 29 \| 23 \| 29 1.7 \| 23 \| 29 \| 19 \| 39 0.7 \| 23 \| 29 \| 8 \| 71 0.43 \| 23 \| 29 \| 1 \| 101 3.72 \| 19 \| 39 \| 19 \| 39 0.77 \| 19 \| 39 \| 8 \| 71 0.46 \| 19 \| 39 \| 1 \| 101 2.69 \| 8 \| 71 \| 8 \| 71 0.69 \| 8 \| 71 \| 1 \| 101 3.61 \| 1 \| 101 \| 1 \| 101 To clarify the physical reason why using the true matrix elements leads to significant deviations from the results found for $M_{j^{\prime}l^{\prime},jl}=-g/V$, we show in Table I the numerical values of $M_{j^{\prime}l^{\prime},jl}$ (calculated in units of $-g/V$) for $D=14.2\,{\rm nm}$ (only the states within the Debye window are given). As seen, the diagonal (intrashell) matrix elements $M_{jl,jl}$ are strongly enhanced as compared to $-g/V$. However, the matrix elements controlling the scattering of the time reversed states between different shells (intershell) are often decreased in absolute value with respect to $-g/V$. So, the question arises why the superconducting correlations are enhanced for the true matrix elements? The point is that the intershell interactions are of less importance due to a size-dependent pinning of the chemical potential to the groups of degenerate or nearly degenerate levels (shells can be often close to each other in energy), see the next paragraph. When $\mu$ is pinned to a particular shell, then the single-electron energy measured from $\mu$ is zero for the states from this shell. These states make a major contribution to superconducting correlations unless diameters are not large enough $D<20$-$30\,{\rm nm}$, in other words, the number of contributing shells is less than $10$-$15$. In this case the superconducting correlations are nearly determined by the pairing gap $\Delta_{jl}$ associated with the shell pinned to $\mu$. From Eq. (15) it is seen that $\Delta_{jl}$ for the states with $\xi_{jl}=0$ is mainly governed by the intrashell matrix element $M_{jl,jl}$. For instance, when ignoring the contribution of all other states one simply obtains (at $T=0$) $\Delta_{jl}\approx-(l+\frac{1}{2})\,M_{jl,jl}.$ When the diameter increases beyond $20$-$30\,{\rm nm}$, then the intershell matrix elements approach $-g/V$ while the intrashell matrix elements are still significantly different from the bulk-like behavior. However, the role of the states with $\xi_{jl}=0$ is becoming less and less important for larger diameters due to the presence of larger and larger number of shells making a contribution to the pairing correlations. As a consequence, the difference between the data in Figs. 1(a) and (b) decreases when approaching $D=35$-$40\,{\rm nm}$, together with the amplitude of the quantum-size oscillations of $T_{c}$. Figure 2: (a) Size variations of the chemical potential, accompanied by an overall shift of $\mu$ to upper values with decreasing $D$. (b) Details of the quantum-size pinning of $\mu$ (filled squares) to the single-electron levels (solid curves), small diameters are shown for simplicity. Figure 3: Spatial distribution of the pair condensate in spherical nanograins: $\Delta(\rho)$ (calculated at $T=0$) versus $\rho$ for diameters $D=12\,{\rm nm}$ (a), $13.52\,{\rm nm}$ (b), $14.2\,{\rm nm}$ (c), $16\,{\rm nm}$ (d), $16.4\,{\rm nm}$ (e) and $17.54\,{\rm nm}$ (f). In the fully self-consistent scheme the chemical potential is determined in such a way that the mean electron density $n_{e}$ is constant [see Eq. (5)]. However, size-dependent variations of $\mu$ are of importance not only because they simply prevent the mean electron density from deviations. In fact, such deviations are almost insignificant: our calculations for $\mu=\mu_{B}$ show that $n_{e}$ decreases by a few percent when $D$ reduces to $10$-$20\,{\rm nm}$. A more interesting thing is that the size-dependent variations of $\mu$ have a pronounced effect on the superconducting correlations. In particular, this can be seen from Fig. 2(c), where $T_{c}$ is calculated for the true matrix elements and $\mu=\mu_{B}$. What is the reason for this suppression of $T_{c}$? In the presence of the formation of strongly degenerate electron levels or bunches of electron levels with almost negligible spacing between them, the chemical potential lies mostly at the highest partly-filled degenerate level (see, e.g., Refs. kres01, ; kres02, ). Pairing correlations are significant only within the Debye window around the chemical potential $\mu$ and are strongest delft exactly at $\mu$. Hence, when $\mu$ is pinned to a shell level, this favors the pairing correlations at this level and, in turn, through the self-consistency relation, favors the pairing correlations at neighboring shells. In other words, if the level to which the chemical potential is pinned is highly degenerate than the phase space for the strongest pair scattering is enlarged and, consequently, the system gains in interaction energy and, as a result, superconducting correlations are strongly enhanced. It is different when $\mu$ is not pinned to a shell, which is mostly the case for a constant chemical potential, e.g., for $\mu=\mu_{B}$. Here the relevant shells entering the Debye window are as a rule specified by $\xi_{jl}\not=0$ and, so, their contributions are diminished. The above discussion is further illustrated by our numerical results for $\mu$ in Fig. 2. As seen from panel (a), when keeping the electron density of the system constant, $\mu$ slightly shifts systematically up with decreasing $D$ and exhibits size-dependent oscillations, as seen from Fig. 2(a). These oscillations are a signature of the size-dependent pinning of $\mu$ to groups of degenerate or nearly degenerate single-electron levels. This is clearly seen from Fig. 2(b), where variations of $\mu$ (filled squares) are plotted versus $D$ together with the single-electron energies measure from the band bottom, i.e., $\frac{\hbar^{2}}{2m_{e}}\frac{\alpha^{2}_{jl}}{R^{2}}$ (solid curves). For the sake of simple illustration, panel (b) shows the data for extremely small diameters, where the energy spacing between the shell levels is pronounced and, as a result, the size-dependent oscillations of $\mu$ are not so wild as it happens for higher diameters. As follows from Fig. 2(b) $\mu$ is pinned to a shell level in most cases, which, as mentioned above, represents incomplete shells. Sometimes $\mu$ can be found between two neighboring shell levels, which corresponds to the case of a fully occupied lower shell. ### III.2 Spatially nonuniform pair condensate In the previous paragraph we considered the effect of quantum confinement on pairing correlations through the matrix elements and quantum-size pinning of $\mu$. As discussed at the end of Sec. II, a framework which incorporates both issues appears to be only consistent when the position-dependent superconducting order parameter is taken into consideration. Thus, our results discussed in the previous section suggest that the spatial variations of $\Delta(\rho)$ will be pronounced even in nanograins with diameters up to $D=20$-$30\,{\rm nm}$. However, it is usually argued that spatial variations of $\Delta(\rho)$ cost significant extra energy and, so, they are strongly suppressed when $D\ll\xi$, with $\xi$ the bulk coherence length (see, for instance, Ref. mat, ). In addition, $D$ should be larger than $\lambda_{F}$: in practice, $k_{F}D\sim 10$ is assumed to be sufficient to ignore any spatial dependence of the order parameter. kres01 ; kres02 For typical metallic parameters $k_{F}D\sim 200$-$400$ for $D=10$-$20\,{\rm nm}$ and this is the reason why the spatial dependence of the order parameter was ignored in most papers on superconducting correlations in nanograins. To go in a more detail on this point, we below discuss our numerical results on $\Delta(\rho)$. In Fig. 3 the radial dependence of the superconducting order parameter is shown as calculated from Eq. (16) for $D=12\,{\rm nm}$ (a), $13.52\,{\rm nm}$ (b), $14.2\,{\rm nm}$ (c), $16\,{\rm nm}$ (d), $16.4\,{\rm nm}$ (e) and $17.54\,{\rm nm}$ (f). The shells making a contribution to the superconducting correlations are also displayed in each panel, and the quantum numbers of the shell level pinned to $\mu$ are underlined. As seen, we in general have a nonuniform distribution of the pair condensate for diameters $D=10$-$20\,{\rm nm}$, which is in agreement with our expectations. For example, let us consider the results plotted in panel (c). Here $\mu$ is pinned to the shell level $(l,j)=(101,1)$ and, so, single-electron states with $j=1$ and $l=101$ make a major contribution to $\Delta(\rho)$, which results in a significant enhancement of the order parameter next to the edge, i.e., for $\rho/R=0.9$-$1.0$. The profile of this enhancement is determined by the radial wave function $\chi^{2}_{1,101}(\rho)$ with two pronounced local maxima ($\Delta/\Delta_{B}=14.3$ and $7.2$ at $\rho/R=0.9$ and $0.97$, respectively) and one node (recall that $j$ is the number of the nodes of the radial wave function). All the other shells displayed in Fig. 3(c) are specified by $\xi_{jl}\not=0$ and, as a result, their contributions is much less significant. The local maximum $\Delta(\rho)/\Delta_{B}=2.3$ at $\rho/R=0.1$ is due to states $(j,l)=(31,11)$. The shells with $(j,l)=(23,29)$ and $(19,29)$ are responsible for local enhancements of the order parameter up to $1.5$-$2.0\Delta_{B}$ at $\rho/R=0.27$ and $0.36$, respectively. At last, the shell $(8,71)$ produces the local maximum at $\rho/R=0.64$. In general, the larger the angular momentum, the larger the values of $\rho/R$ at which the corresponding single-electron states have an effect on the profile of $\Delta(\rho)$. It is worth noting that typically, the order parameter is strongly suppressed in the center ($\rho=0$) except of rare cases when states with zero angular momentum contribute to the pairing correlations. One such example is given in Fig. 3(d), where a narrow pick can be seen at $\rho=0$ due to the contribution of the shell with $(j,l)=(41,0)$. Figure 4: The order parameter $\Delta(\rho)$ for sufficiently large diameters $D=33.6\,{\rm nm}$ (a), $D=35\,{\rm nm}$ (b) and $D=36\,{\rm nm}$ (c). From Fig. 3 it follows that the radial distribution of the pair condensate remains strongly nonuniform even for $D\approx 20\,{\rm nm}$. We would like to note that when selecting concrete values of $D$ for Fig. 3, we did not even take diameters for which $T_{c}$ is close to the upper dashed curve in Fig. 1(a). In the case of a strong enhancement of $T_{c}$ the radial distribution of the pair condensate is as a rule strongly nonuniform. The points selected for Fig. 3 are mainly in a vicinity of the lower dashed curve in panel (a) of Fig. 1: for $D=13.52,\,14.2,\,16$ and $16.4\,{\rm nm}$ we have $T_{c}/T_{c,B}=6.41,\,6.48,\,4.082$ and $3.78$, respectively. However, even in this case the order parameter can vary with position by an order of magnitude. Spatial variations of $\Delta(\rho)$ are significantly relaxed only when $D$ approaches $30$-$40\,{\rm nm}$, as seen from Fig. 4. For our parameters $k_{F}=16.4\,{\rm nm}$ and, so, we obtain $k_{F}D\approx 300$ for $D\approx 20\,{\rm nm}$. Hence, the criterion $k_{F}D\gg 1$ is not very useful in order to estimate the effect of spatial variations of the pair condensate. Based on our numerical study, we would like to suggest another criterion related to a more sensitive energy scale, which in the superconducting state is governed by the bulk pairing gap $\Delta_{B}$. The spatial distribution of the order parameter is always strongly inhomogeneous when $\delta\sim\Delta_{B}$ (here it is even better to replace $\Delta_{B}$ by the size-dependent pairing gap). The spatial variations decay with a decrease in the ratio of the mean interlevel spacing to the bulk order parameter, i.e., $\delta/\Delta_{B}$, and our numerical results suggest that such variations are significantly reduced only when $\delta/\Delta_{B}<0.05$-$0.1$ (recall that effects of a magnetic field are beyond the scope of our paper). For ${\rm Sn}$ spherical superconducting grains this regime is achieved when $D>40$-$50\,{\rm nm}$ (note that $\delta\approx 2\pi^{2}\hbar^{2}/(mk_{F}V)$ underestimates the intershell spacing for spherical confining potential). Despite that our results are for a highly symmetric confining geometry, we can expect that the order parameter will be always spatially nonuniform for $\delta/\Delta_{B}>0.1$, even when shape imperfections and disorder dissolve a shell structure. The reason is that the number of contributing states (i.e., the states in the energy interval $\approx[\mu-\Delta_{B},\mu+\Delta_{B}$) is not very large for $\delta/\Delta_{B}>0.1$. In this case the states pinned to $\mu$ always make a major contribution to the order parameter and, so, the profile of the squared absolute value of the corresponding wave function will mainly determine the spatial distribution of the condensate. Thus, the domain $\delta/\Delta_{B}=0.1$-$1.0$ is in general characterized by strong effects due to the spatially nonuniform pairing. We remark that our conclusions do not contradict the usual argument that spatial variations of the order parameter cost extra energy. Let us compare a bulk superconductor with a superconducting nanograin. In bulk the relevant matrix elements controlling the scattering of the time-reversed states are $-g/V$ and the order parameter is spatially uniform (in the absence of a magnetic field). As opposed to bulk, the pair condensate significantly varies with position in nanograins, which results, of course, in an increase of the kinetic energy. However, the intrashell matrix elements are now enhanced in absolute value as compared to $-g/V$ due to quantum confinement. This compensates energy costs of spatial variations of the order parameter. The discussion in the previous paragraph is also related to arguments that invoke the conventional Ginzburg-Landau theory. According to this arguments the order parameter is uniform in samples with size smaller than the bulk coherence length. When applying this to nanograins, one can conclude that the pair condensate should not vary with its position. However, this is not true. It is well-known that one should be careful when applying the conventional Ginzburg-Landau theory to superconductors with characteristic size smaller than the zero-temperature (BCS) coherence length $\xi_{0}$. Strictly speaking, Gor’kov’s derivation of the conventional Ginzburg-Landau formalism from the BCS approach is not applicable on a scale smaller than $\xi_{0}$ (see,.e.g., Ref. fett, ). For ${\rm Sn}$ we have $\xi_{0}\approx 230\,{\rm nm}$ (see, e.g., Ref. degen, ). Thus, in the case of interest $D\ll\xi_{0}$, and one can hardly invoke the conventional Ginzburg-Landau formalism to check whether or not $\Delta(\rho)$ varies with $\rho$. ### III.3 Confinement-induced Andreev-type states Here we would like to discuss one more issue related to a spatially nonuniform pairing in nanograins. This is the formation of Andreev-type states induced by quantum confinement sh03 ; sh06 (see also a similar paper torm discussing Andreev-type states in an ultracold trapped superfluid Fermi gas). Since the 60s (see Refs. saint, ; car, ; andr, ) it is known that quasiparticles can “feel” a spatial variation of the superconducting order parameter as a kind of potential barrier. This physical mechanism (referred to as Andreev mechanism below) is the basis for Andreev quantization investigated previously for the core of a single vortex for the mixed state of a type-II superconductor car and for an isolated normal region of the intermediate state of a type-I superconductor andr (or for a similar case of SNS contacts saint ). Based on our consideration of Sec. III.2, one can expect that Andreev-type states can play a remarkable role in superconducting nanograins due to significant spatial variations of the superconducting order parameter. This is very similar to recently investigated Andreev-type states in superconducting nanowires/nanofilms sh03 ; sh06 , where the pair condensate is position dependent in the direction perpendicular to the nanowire/nanofilm due to the quantization of the perpendicular electron motion. In Ref. sh06, it was shown that $\Delta_{i}=\int\\!{\rm d}^{3}r\,\Delta({\bf r})\,\Bigl{[}\|u_{i}({\bf r})\|^{2}+\|v_{i}({\bf r})\|^{2}\Bigr{]},$ (18) which means that the pairing energy gap $\Delta_{i}$ is the averaged value of the order parameter ”watched” by the quasiparticles with quantum numbers $i$. Note that $\|u_{i}({\bf r})\|^{2}+\|v_{i}({\bf r})\|^{2}$ can be interpreted as the spatial distribution of quasiparticles according to the well-known constraint $\int{\rm d}^{3}r(\|u_{i}({\bf r})\|^{2}+\|v_{i}({\bf r})\|^{2})=1$ [see, e.g., Ref. ket, and Eq. (10)]. When inserting Eqs. (6) into Eq. (18), one can easily obtain Eq. (12) with $\Delta_{i}=\Delta_{jl}$. If quasiparticles avoid the domains of enhanced pair condensate, the corresponding integral in the right-hand-side of Eq. (18) becomes smaller and, hence, such quasiparticles have smaller pairing gaps $\Delta_{jl}$. They can be referred to as Andreev-type states. Our numerical study of quantum-number dependent pairing gaps $\Delta_{jl}$ for metallic nanograins reveals a significant role of Andreev mechanism. Let us consider $D=13.52\,{\rm nm}$, the corresponding spatial distribution of the pair condensate is given in Fig. 3(b). To show how different species of quasiparticles are distributed in the radial direction in this case, the radial-dependent shell contributions (at $T=0$) $\Delta^{(jl)}(\rho)$ [see Eqs. (16) and (17)] are plotted in Fig. 5(a). We remark that such a representation is more informative than simply a plot of $\|u_{jl}(\rho)\|^{2}+\|v_{jl}(\rho)\|^{2}$. First, the radial dependence of $\Delta^{(jl)}\propto\chi^{2}_{jl}(\rho)$ is the same as that of $\|u_{jl}(\rho)\|^{2}+\|v_{jl}(\rho)\|^{2}\propto\chi^{2}_{jl}(\rho)$ [see Eq. (8)]. Second, a plot of $\Delta^{(jl)}(\rho)$ gives also information how the corresponding states contribute to $\Delta(\rho)$. From Fig. 3(c) we can see that a significant enhancement of the order parameter occurs at $\rho/R=0.45$-$0.7$. From Fig. 5(a) it is clear that this enhancement is due to the states with $(j,l)=(14,48)$ and $(9,63)$. Other shells, i.e., $(27,16)$ and $(23,25)$ contribute less, and the corresponding quasiparticles, representing Andreev-type states, are mainly located beyond the domain $\rho=0.45-1.0$. As a result, they have smaller pairing gaps, i.e., $\Delta_{27,16}=2.65\,\Delta_{B}$ and $\Delta_{23,25}=2.81\,\Delta_{B}$, as compared to $\Delta_{14,48}=4.098\,\Delta_{B}$ and $\Delta_{9,63}=5.77\,\Delta_{B}$. As seen, the quasiparticles with $(j,l)=(27,16)$ are most successful in avoiding the local enhancement of $\Delta(\rho)$ at $\rho/R=0.45$-$0.7$ and, so, $\Delta_{27,16}$ is the smallest pairing gap. Such a manifestation of Andreev mechanism is not a particular feature of $D=13.52\,{\rm nm}$. In general, $\Delta_{jl}$ strongly varies with $j$ and $l$ for diameters $<30-40\,{\rm nm}$, i.e., where spatial variations of the order parameter are still pronounced. Quite often such variations can be an order of magnitude, as, e.g., for $D=14.2\,{\rm nm}$ (see $\Delta(\rho)$ given in Fig. 2(c)). At this diameter a great enhancement of $\Delta(\rho)$ takes place at $\rho/R=0.9$. This is due to the contribution of the shell with $(j,l)=(1,101)$ [see Fig. 5(b)]. Other shells make much less important inputs and the corresponding quasiparticles are mainly distributed beyond the domain $\rho/R=0.9$-$1.0$. So, as compared to $\Delta_{1,101}=9.32\,\Delta_{B}$, they have significantly smaller pairing gaps, i.e., $\Delta_{31,11}=1.6\Delta_{B}$, $\Delta_{23,29}=1.62\,\Delta_{B}$, $\Delta_{19,39}=1.72\,\Delta_{B}$ and $\Delta_{8,71}=2.35\,\Delta_{B}$. Thus, the interplay of Andreev mechanism and quantum confinement is responsible for variations of $\Delta_{jl}$ with the relevant quantum numbers. Figure 5: Shell-dependent contributions to the order parameter $\Delta_{jl}(\rho)$ for relevant shells: (a), $D=13.52\,{\rm nm}$, $(j,l)=(27,16),\,(23,25),\,(14,48)$ and $(9,63)$; (b) $D=14.2\,{\rm nm}$, $(j,l)=(31,11),\,(23,29),\,(19,39),\,(8,71)$ and $(1,101)$. One could expect that such a serious difference in pairing gaps of different quasiparticle species can result in a pronounced drop of the ratio of $\Delta_{E}$ (the minimal energy gap) to the critical temperature $k_{B}T_{c}$, similar to the case for quantum superconducting nanowires. sh03 The main idea here is that $\Delta_{E}$ is governed by Andreev-type states and, hence, is decreased. Unlike $\Delta_{E}$, $T_{c}$ is controlled by the quasiparticles making a major contribution to $\Delta(\rho)$ and, so, $T_{c}$ is coupled to their higher pairing gaps. As a result, $\Delta_{E}/k_{B}T_{c}$ can be significantly smaller than in bulk. For instance, one can expect that $\Delta_{E}=\Delta_{31,11}=1.6\,\Delta_{B}$ at $D=14.2\,{\rm nm}$ while $T_{c}$ is governed by $\Delta_{1,101}=9.32\,\Delta_{B}$. However, this is not correct for nanograins. The point is that $\Delta_{E}$ is a spectroscopical gap which is probed by STM. It is defined as $\Delta_{E}=\min_{jl}E_{jl}$. For nanowires the subband-dependent pairing gap is always the minimal quasiparticle energy due to a quasi-free spectrum in the direction parallel to the nanowire. For nanograins this is different. In particular, for $D=14.2\,{\rm nm}$ we have the following single-electron energies (absorbing $\mu$) of the relevant shells: $\xi_{31,11}=-18.6\,\Delta_{B}$, $\xi_{23,29}=-26.03\,\Delta_{B}$, $\xi_{19,39}=20.9\,\Delta_{B}$, $\xi_{8,71}=22.6\,\Delta_{B}$ and $\xi_{1,101}=0$. Hence, one can calculate that $\Delta_{E}=E_{1,101}=\Delta_{1,101}$ in spite of the fact that $\Delta_{1,101}$ is the largest pairing gap. Thus, although Andreev mechanism plays a significant role in superconducting nanograins, it can hardly be probed by STM-measurements due to the nonzero interlevel spacing, unlike quantum superconducting nanowires. ## IV Conclusions and discussion In conclusion, we have shown that the spatial distribution of the pair condensate is essentially nonuniform in metallic nanograins. In particular, the spatially nonuniform pairing can proliferate in nanograins even when $k_{F}D\sim 300$ and, so, the usual criterion to neglect variations of the superconducting condensate with position, i.e., $k_{F}D\gg 1$, is not very useful and can result in wrong conclusions. This is the reason why effects due to spatially nonuniform pairing in superconducting grains were previously overlooked. Our study suggests that a new criterion should be based on a more delicate energy scale (as compared to the Fermi energy), which, in the superconducting state, is given by the bulk order parameter $\Delta_{B}$. It turns out that the pairing becomes spatially nonuniform when the interlevel spacing $\delta$ exceeds $0.1$-$0.2\,\Delta_{B}$. Variations of the order parameter with position exhibit a pronounced enhancement with an increase of $\delta/\Delta_{B}$. When $\delta\sim\Delta_{B}$, such variations can be almost an order of magnitude in highly symmetric grains. At first sight, this seems impossible because it costs extra energy for such spatial variations. However, a nonuniform distribution of the pair condensate is accompanied by enhanced pairing interaction matrix elements, which compensates the energy cost for an inhomogeneous distribution of the condensate. Another point is the size-dependent pinning of the chemical potential to groups of degenerate or nearly degenerate energy levels. Such a pinning plays the role of a filter that increases the contribution of the single-electron levels in the vicinity of the chemical potential and suppresses contributions of other states. This results in an additional mechanism favoring spatially nonuniform pairing in metallic nanograins. Figure 6: (a) Single-electron energies $\xi_{i}$ (given in units of the Debye energy $\hbar\omega_{D}$) ordered in ascending manner versus the ordering number $N$ for the rectangular-shaped aluminum nanograin with $L_{x}=7.06\,{\rm nm},\,L_{y}=L_{x}/1.1,\,L_{z}=1.1\,L_{x}$ (squares) and for a cubic nanograin with $L_{x}=7.06\,{\rm nm}$ (triangles). (b) The size- dependent excitation gap $\Delta_{E}/\Delta_{B}$ versus $L_{x}$ (in steps of $\delta L_{x}=0.01\,{\rm nm}$) for an aluminum nanograin of the rectangular shape with the dimensions $L_{x},L_{y}=L_{x}/1.1,L_{z}=1.1\,L_{x}$: squares represent the results calculated with the modified matrix elements and with proper variations of $\mu$; stars are the data obtained for the bulk-like matrix elements $-g/V$ and $\mu=\mu_{B}$. (c) The spatial distribution of the pair condensate in the rectangular grain with $L_{x}=7.06\,{\rm nm}$ and $L_{y}=L_{x}/1.1,\,L_{z}=1.1\,L_{x}$, $\Delta(x)=\Delta(x,y,z)\|_{y=x/1.1,z=1.1x}$. In this paper we investigated a highly symmetric confining geometry. Due to this feature the problem becomes effectively one-dimensional (the order parameter depends only on the radial coordinate) and, so, sufficiently large diameters up to $D\approx 40\,{\rm nm}$ can be investigated. This size is almost impossible to reach theoretically for grains with the order parameter depending on three relevant coordinates due to time consuming numerical calculations. Such an effectively one-dimensional problem has large degeneration factors for the corresponding shell structure, resulting in a significant enhancement of the pairing correlations. In reality there can be several issues that may lead to a splitting of the shell levels. It will decrease the degeneration factors and, so, reduce the pairing correlations, since the main contribution to the sum in the gap equation comes from the transitions within the same shell pinned to the chemical potential. Among such issues is the Jahn-Teller deformation, i.e., the transformation of a spherical nanograin with incompletely filled shells to an ellipsoidal shape. In addition, the surface imperfections and impurities can significantly change the distribution of single-electron levels. However, our qualitative results are quite generic and do not depend on a particular shape of a nanograin and the presence of possible imperfections. For instance, when $\delta\sim\Delta_{B}$ the pair condensate will always be spatially nonuniform because only a few single-electron levels enters the energy interval $\approx[\mu-\Delta_{B},\mu+\Delta_{B}]$. Due to the dominant contribution of such levels to $\Delta({\bf r})$, one can expect that the pair condensate acquires a profile governed by the squared absolute value of the wave function for the single-electron state closest in energy to $\mu$. This is significantly strengthened by an increase (in absolute value) of the diagonal matrix elements $\langle i,\bar{i}\|\Phi\|i,\bar{i}\rangle$ and, in addition, by the pinning of the chemical potential to the single-particle levels. We remark that the diagonal matrix elements, i.e., $\langle i,\bar{i}\|\Phi\|i,\bar{i}\rangle$ [see the definition for $\Phi$ below Eq. (3)] are always enhanced as compared to $-g/V$ in the presence of quantum confinement, whatever disorder and shape imperfections. This can be seen from the following simple arguments. Introducing $\varphi_{i}({\bf r})$, the wave function associated with state $i$, one can write $\langle i,\bar{i}\|\Phi\|i,\bar{i}\rangle=-g\int\\!\\!{\rm d}^{3}r\;\|\varphi_{i}({\bf r})\|^{4}.$ Due to the normalization condition we have $\|\varphi_{i}({\bf r})\|^{2}=\frac{1}{V}+d_{i}({\bf r})$, where $\int{\rm d}^{3}rd_{i}({\bf r})=0$. Then, the above matrix element can be rearranged as $\langle i,\bar{i}\|\Phi\|i,\bar{i}\rangle=-\frac{g}{V}\Bigl{[}1+V\\!\int\\!\\!{\rm d}^{3}r\,d^{\,2}_{i}({\bf r})\Bigr{]}.$ The second term in the brackets is always positive in the presence of quantum confinement, i.e., when $d_{i}({\bf r})\not=0$. It is zero only when $\varphi_{i}({\bf r})$’s are chosen in the form of plane waves, which results in $\langle i,\bar{i}\|\Phi\|i,\bar{i}\rangle=-g/V$. The above discussion can be supplemented by our numerical results calculated from the BCS-like equation similar to Eq. (15) but now for aluminum nanograins of rectangular shape with dimensions $L_{x},\,L_{y}=L_{x}/1.1,\,L_{z}=1.1L_{x}$. For aluminum we have degen ; fett ; ash : $\hbar\omega_{D}/k_{B}=375~{}{\rm K}$, $gN(0)=0.18$, and $\mu_{B}=11.67\,{\rm eV}$, which corresponds to the electron density $n_{e}=181\,{\rm nm}^{-3}$. In Fig. 6(a) single-electron levels arranged in the ascending order are shown within the Debye window for the rectangular nanograin with $L_{x}=7.06\,{\rm nm}$ (squares). The same is also given here for a cubic aluminum nanograin with $L_{x}=L_{y}=L_{z}=7.06\,{\rm nm}$ (triangles). As seen, single-electron levels for the rectangular shape are distributed in a nearly equidistant manner (with $\delta\approx 0.2$-$0.3\,{\rm meV}\sim\Delta_{B}$) contrary to the states corresponding to the cubic geometry. It is well-known that an almost equidistant distribution smit of single-electron levels near $\mu$ is also expected in the presence of significant imperfections such as the surface roughness and/or impurities. So, our results in Fig. 6 give a feeling about the role of the spatially nonuniform pairing in disordered metallic grains. The excitation energy gap $\Delta_{E}$ for the rectangular nanograin is shown in units of $\Delta_{B}$ in Fig. 6 as a function of $L_{x}$ in the interval $L_{x}=7$-$8\,{\rm nm}$. Here squares represent our results calculated with the modification of the matrix elements and with $\mu$ varying with $L_{x}$; stars are the results found for the bulk-like matrix elements $-g/V$ and $\mu=\mu_{B}$. As seen, $\Delta_{E}$ ($\propto T_{c}$) is now two-times enhanced as compared to $\Delta_{B}$ (on average), which is much less significant than for highly symmetric grains (compare with Fig. 1) due to a splitting of the shell levels. However, the effect of interest is still pronounced: $\Delta_{E}$ calculated for the modified matrix elements and with account of size variations of $\mu$ is generally larger by a factor of $1.5$-$2.0$. The spatial profile of the order parameter is nonuniform with local enhancements over its average value by about $100\%$, see, e.g., Fig. 6(c). For rectangular grains with $L_{x}=7$-$8\,{\rm nm}$ we have $\delta\sim\Delta_{B}$. However, as we checked, the spatially nonuniform pairing and the related effects of the modification of the relevant matrix elements and the size-dependent pinning of $\mu$ are of significance even for smaller $\delta$’s, i.e., when $\delta>0.1$-$0.2\,\Delta_{B}$ ($L_{x}<14$-$15\,{\rm nm}$). For instance, at $L_{x}=11\,{\rm nm}$ the order parameter exhibits variations of about $30$-$40\%$ of its averaged value. These results are in agreement with our expectations based on the investigation of the highly symmetric spherical grains. ###### Acknowledgements. 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Törma, Phys. Rev. Lett. 95, 170407 (2005). * (42) P. G. de Gennes, and D. Saint-James, Phys. Lett. 4, 151 (1963). * (43) C. Caroli, P. G. de Gennes, and J. Matricon, Phys. Lett. 9, 307 (1964). * (44) A. F. Andreev, Sov. Phys. JETP 22, 455 (1966). * (45) A. A. Shanenko, M. D. Croitoru, and F. M. Peeters, Phys. Rev. B 78, 054505 (2008).	arxiv-papers	2010-11-19T15:47:25	2024-09-04T02:49:15.088688	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "M. D. Croitoru, A. A. Shanenko, C. C. Kaun and F. M. Peeters", "submitter": "Mihail Croitoru", "url": "https://arxiv.org/abs/1011.4453" }
1011.4524	# $K\to\pi\pi$ matrix elements from mixed action lattice QCD Jack Laiho Department of Physics and Astronomy, University of Glasgow, Glasgow, Scotland, UK E-mail jlaiho@fnal.gov Ruth S. Van de Water Physics Department, Brookhaven National Laboratory, Upton, New York, USA E-mail ruthv@bnl.gov ###### Abstract: We present a new method for determining $K\to\pi\pi$ matrix elements from lattice simulations that is less costly than direct simulations of $K\to\pi\pi$ at physical kinematics. It improves, however, upon the traditional “indirect” approach of constructing the $K\to\pi\pi$ matrix elements using NLO $SU(3)$ $\chi$PT, which can lead to large higher-order chiral corrections. Using the explicit example of the $\Delta I=3/2$ $(27,1)$ operator to illustrate the method, we obtain a value for Re($A_{2}$) that agrees with experiment and has a total uncertainty of $\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 316\relax$}}$ 20%. Although our simulations use domain- wall valence quarks on the MILC asqtad-improved gauge configurations, this method is more general and can be applied to calculations with any fermion formulation. ## 1 Motivation Lattice calculations of $K\to\pi\pi$ matrix elements are important for understanding the Standard Model and in constraining physics beyond the Standard Model. For example, they are needed to explain the origin of the $\Delta I=1/2$ rule and to compute the long-distance contributions to neutral kaon mixing [1]. Because the lowest-order Standard Model contributions to $\epsilon^{\prime}/\epsilon$ are from 1-loop electroweak penguin diagrams, $K\to\pi\pi$ decay is sensitive to physics at very high scales. Many extensions of the Standard Model lead to new particles that enter the loops, and these contributions to $K\to\pi\pi$ may be sufficiently large that they can be observed once the hadronic uncertainties in the weak matrix elements are small enough. A standard way of searching for new physics in the flavor sector is by overconstraining the angles and sides of the CKM unitarity triangle [2]. This requires precise experimental measurements and equally well-controlled theoretical calculations of hadronic weak matrix elements using lattice QCD. For many years improved measurements and calculations have simply confirmed the Standard Model CKM framework at the few-percent level, but recent $N_{f}=2+1$ flavor lattice calculations of $B_{K}$ with $\sim$ 4% precision [3] have revealed a 2-3$\sigma$ tension in the CKM unitarity triangle [1, 4, 5]. This tension, which may be due to kaon or $B_{d}$-meson mixing, is illustrated in Fig. 1. Almost all constraints on the CKM unitarity triangle, however, come from the $B$-meson sector. Thus it is essential to place other constraints on the unitarity triangle from the kaon sector in order to test whether the amount of observed $CP$-violation in the $B$-meson sector is the same as in the kaon sector. Once lattice QCD calculations of $K\to\pi\pi$ matrix elements are sufficiently precise, they can be combined with the experimental measurement of $\epsilon_{K}^{\prime}/\epsilon_{K}$ to impose an additional constraint on the apex of the CKM unitarity triangle. Figure 1: Global fit of the CKM unitarity triangle [5]. ## 2 New approach to $K\to\pi\pi$ matrix elements The Maiani-Testa no-go theorem states that physical matrix elements cannot be extracted from Euclidean correlation functions with multi-hadron states [6]. Therefore two general approaches have been developed to evade the Maiani-Testa theorem and allow the determination of $K\to\pi\pi$ matrix elements using lattice QCD. The “direct” Lellouch-Lüscher finite-volume method [7] is the most straightforward to implement, but it is computationally demanding because it requires a large ($\sim$ 6 fm) box and physical light-quark masses. The “indirect” method constructs $K\to\pi\pi$ matrix elements using the low-energy constants (LEC’s) of $SU(3)$ $\chi$PT obtained from calculating simpler lattice quantities such as $K\to 0$ and $K\to\pi$. Although it was shown that all LEC’s through next-to-leading order can be obtained from such “simple” lattice quantities [8], this approach relies on the use of $SU(3)$ $\chi$PT at the kaon mass, where the convergence of the chiral expansion is quite slow. Li and Christ studied the extraction of $K\to\pi\pi$ matrix elements with $N_{f}=2+1$ dynamical domain-wall lattice simulations using the “indirect” method in Ref. [9]. They concluded that large uncertainties in the LO and NLO $SU(3)$ LEC’s and the slow convergence of $SU(3)$ $\chi$PT at the scale of the kaon mass lead to large errors that make the extraction of $K\to\pi\pi$ matrix elements using the “indirect” method unreliable. Our procedure therefore addresses these drawbacks of the traditional approach and improves upon it in several ways. In the combined chiral-continuum extrapolation, we use the physical pseudoscalar meson masses and decay constant. This leads to better fits as measured by the correlated $\chi^{2}$/d.o.f.; nontrivial agreement between the NLO mixed-action $\chi$PT prediction for the isovector scalar correlator and lattice simulation data lends support to this approach [10]. When the fixed-order (NLO) fit is bad, we approximate higher order terms in the chiral expansion by polynomials. This leads to larger leading-order terms and hence suggests better convergence than was found in Ref. [9]. For example, Fig. 2 shows the $SU(3)$ $\chi$PT fit of $B_{K}$ along with the sizes of the various contributions [11]. The NLO corrections are only approximately 1/3 of the LO terms even at $m_{s}/2$. Figure 2: Chiral and continuum extrapolation of $B_{K}$ [11] (left plot) along with convergence of the $SU(3)$ $\chi$PT fit (right plot). Circles (squares) denote $a\sim 0.12$ fm ($a\sim 0.09$ fm) data. Only degenerate points are shown, but the fit also includes non-degenerate data. The cyan band is the degenerate quark mass full QCD curve ($m_{x}=m_{y}=m_{l}=m_{h}$) in the continuum limit. The y-intercept of the band gives the LEC $B_{0}$, the value of $B_{K}$ in the $SU(3)$ chiral limit. The right-most point on both plots corresponds to $\sim m_{s}/2$. Despite these findings, however, NLO $\chi$PT corrections can still be 50% or more for some quantities. Therefore, to achieve the precision needed for $K\to\pi\pi$ we do not rely on the “indirect” method alone. Rather, we combine indirect and direct methods in a cost-effective way. We bypass the Maiani- Testa theorem by simulating with both pions at rest. We fit the numerical data to NLO mixed-action $\chi$PT plus higher-order analytic terms, extrapolate to the continuum, and interpolate to the point at which $m_{K}=m_{K}^{\textrm{phys.}}$ and $m_{\pi}=m_{K}^{\textrm{phys.}}/2$. Thus we avoid relying upon $SU(3)$ $\chi$PT to extrapolate to the physical kaon mass, where we expect higher-order corrections to be significant. We then correct this unphysical kinematics point using fixed-order $SU(3)$ $\chi$PT. The low energy constants needed for this correction can be obtained from simpler quantities such as $f_{K}$, $K^{0}$-$\bar{K^{0}}$, and $K\to\pi$. Since the kaon is tuned to its physical value, terms involving only kaons are correct to all orders in the $SU(3)$ chiral expansion; we therefore expect higher-order corrections to be small. We can test this approach using the known quantities $f_{K}$ and $f_{\pi}$; the results are shown in Fig. 3. The size of the NLO corrections are quite small (below 10% for $f_{\pi}$ and below 5% for $f_{K}$), indicating that the systematic uncertainty due to truncating the chiral expansion is under control. Figure 3: Demonstration of the method for $f_{\pi}$ (left plot) and $f_{K}$ (right plot). Errors on the circular points only show the statistical errors in the numerator. Vertical error bands denote the total (statistical plus systematic) uncertainties in $f_{\pi}$ and $f_{K}$. After the interpolation to the unphysical kinematics point $m_{K}=m_{K}^{\textrm{phys.}}$ and $m_{\pi}=m_{K}^{\textrm{phys.}}/2$, the size of the NLO $\chi$PT correction is below 10% for $f_{\pi}$ and below 5% for $f_{K}$. ## 3 Preliminary determination of Re($A_{2}$) We now use our approach to determine the $(27,1)$ $\Delta I=3/2$ $K\to\pi\pi$ matrix element, which can be combined with continuum Wilson coefficients [12] to obtain Re($A_{2}$). We compute the matrix element in unquenched lattice QCD using asqtad-improved staggered sea quarks and domain-wall valence quarks. This mixed-action approach shares the primary advantages of both staggered and domain-wall lattice simulations. We use the publicly-available 2+1 flavor MILC gauge configurations [13], and simulate with several valence and sea quark masses. Although our preliminary analysis only uses two lattice spacings, we have generated data at three lattice spacings ($a\sim 0.06$, 0.09 and 0.12 fm) and will include all of it in a future publication. Our lightest taste- pseudoscalar sea-sea pion has $m_{\pi,5}=240$ MeV while our lightest valence- valence pion has $m_{\pi}=210$ MeV. On the $a\sim 0.06$ fm ensembles, the heaviest (taste-singlet) sea-sea pion is also quite light $m_{\pi,I}=270$ MeV. This gives us good control over our combined chiral-continuum extrapolation using mixed-action $\chi$PT. The approximate chiral symmetry of the valence sector ($m_{\textrm{res}}<3$ MeV on all lattice spacings) makes analysis of the mixed-action simulation data simpler than the purely staggered case. Only two additional parameters appear at 1-loop in the mixed action $\chi$PT expressions for $m_{PS}$, $f_{PS}$, and $B_{K}$ as compared to the purely domain-wall case [14], and they can both be obtained from spectrum calculations. Furthermore, nonperturbative renormalization using the method of Rome-Southampton [15] can be carried out in a straightforward manner. Finally, the success of our earlier mixed-action lattice calculation of $B_{K}$ [11] indicates that the mixed-action method is also a good way to determine $K\to\pi\pi$ matrix elements. Table 1: Data used for the preliminary determination of Re($A_{2}$). The columns show the (i) approximate lattice spacings, (ii) lattice volumes, (iii), nominal up/down ($m_{l}$) and strange quark ($m_{h}$) masses in the sea, (iv) corresponding pseudoscalar taste pion mass, (v) partially quenched valence quark masses ($m_{x}$), (vi) lightest available domain-wall pion mass, and (vii) number of configurations analyzed on each ensemble. \| sea sector \| valence sector \| ---\|---\|---\|--- $a$(fm) \| $\left(\frac{L}{a}\right)^{3}\times\frac{T}{a}$ \| $am_{l}/am_{h}$ \| $am_{\pi,5}$ \| $am_{x}$ \| $am_{\pi}$ \| $N_{\rm conf.}$ 0.06 \| $64^{3}\times 144$ \| 0.0018/0.018 \| 0.06678(3) \| 0.0026, 0.0108, 0.033 \| 0.06376(96) \| 96 0.06 \| $48^{3}\times 144$ \| 0.0036/0.018 \| 0.09353(7) \| 0.0036, 0.0072, 0.0108, 0.033 \| 0.07458(76) \| 129 0.12 \| $24^{3}\times 64$ \| 0.005/0.05 \| 0.15970(13) \| 0.007, 0.02, 0.03, 0.05, 0.065 \| 0.1718(11) \| 218 0.12 \| $20^{3}\times 64$ \| 0.007/0.05 \| 0.18887(8) \| 0.01, 0.02, 0.03, 0.04, 0.05, 0.065 \| 0.1968(08) \| 279 Figure 4 shows the interpolation to the unphysical kinematics point $m_{K}=m_{K}^{\textrm{phys.}}$ and $m_{\pi}=m_{K}^{\textrm{phys.}}/2$. Before the interpolation, we adjust the data points by the known 1-loop finite volume corrections, which only depend upon the valence quark masses [16]. The interpolation currently uses LO $\chi$PT supplemented by NLO analytic terms, including a term proportional to $a^{2}$ so that we can take the continuum limit. We have finished calculating the mixed-action 1-loop chiral logs, however, and are now working on incorporating them into the fit. We then correct the unphysical kinematics “2$\pi$” point to physical kinematics using $SU(3)$ $\chi$PT: $\displaystyle\langle\pi^{+}\pi^{-}\|{\mathcal{O}}_{i}\|K^{0}\rangle_{\textrm{phys.}}$ $\displaystyle=$ $\displaystyle\langle\pi^{+}\pi^{-}\|{\mathcal{O}}_{i}\|K^{0}\rangle_{\textrm{$2\pi$}}\times(1+\delta_{\textrm{$\chi$PT}})\,,$ (1) where the correction factor is given by $\displaystyle\delta_{\textrm{$\chi$PT}}$ $\displaystyle=$ $\displaystyle\left(\langle\pi^{+}\pi^{-}\|{\mathcal{O}}_{i}\|K^{0}\rangle_{\textrm{phys.}}-\langle\pi^{+}\pi^{-}\|{\mathcal{O}}_{i}\|K^{0}\rangle_{\textrm{$2\pi$}}\right)/\langle\pi^{+}\pi^{-}\|{\mathcal{O}}_{i}\|K^{0}\rangle_{\textrm{$2\pi$}}\,.$ (2) Because we have already interpolated to the physical kaon mass, terms in the $\chi$PT expression that are only functions of the kaon mass cancel in the numerator, and the correction is needed only for the short extrapolation from $m_{K}/2$ to $m_{\pi}$. We therefore expect the convergence properties to be better and the truncation errors to be smaller than if we were to extrapolate up to the kaon mass. At leading order, the expression for the $\Delta I=3/2$ $(27,1)$ $K\to\pi\pi$ matrix element is $\displaystyle\langle\pi^{+}\pi^{-}\|{\mathcal{O}}^{\Delta I=3/2}_{(27,1)}\|K^{0}\rangle_{\textrm{LO}}$ $\displaystyle=$ $\displaystyle 4iB_{0}f_{0}(m^{2}_{K}-m^{2}_{\pi})/3\,,$ (3) where $f_{0}$ and $B_{0}$ are the pion decay constant and $B_{K}$ in the chiral limits, respectively. The $\chi$PT correction factor is then $\displaystyle\delta_{\textrm{$\chi$PT}}^{\textrm{LO}}=\left[(m_{K}/2)^{2}-m_{\pi}^{2}\right]/\left[m_{K}^{2}-(m_{K}/2)^{2}\right]$ (4) and is only 23%, which is much smaller than the chiral correction factors observed by Li and Christ using the standard direct approach [9]. Because the low-energy constants $f_{0}$ and $B_{0}$ cancel in the ratio, there is no ambiguity regarding the choice of $SU(3)$ LEC’s. Figure 4: Left plot: interpolation of Re($A_{2}$) to the unphysical kinematics point $m_{K}=m_{K}^{\textrm{phys.}}$ and $m_{\pi}=m_{K}^{\textrm{phys.}}/2$. Circles (squares) denote $a\sim 0.12$ fm ($a\sim 0.06$ fm) data. Fit lines correspond to the degenerate mass case and should pass through the filled symbols. Right plot: correction of Re($A_{2}$) to physical kinematics using LO $SU(3)$ $\chi$PT. The renormalization factor for the $(27,1)$ $\Delta I=3/2$ operator is the same as $B_{K}$, so we can use the result for $Z_{B_{K}}$ from Ref. [11]. We obtain $\displaystyle\textrm{Re}(A_{2})=1.568(86)\times 10^{-8},$ (5) where the error is statistical only. This agrees with the experimental measurement, $\textrm{Re}(A_{2})_{\textrm{exp}}=1.50\times 10^{-8}\textrm{GeV}$ [17]. Table 2 presents an estimated error budget for $\textrm{Re}(A_{2})$. We assume that the truncation error due to leaving out NLO corrections is half the size of the LO correction, and estimate that total uncertainty in our preliminary result is below 20%. This should improve further with the use of our full data set. The $\chi$PT truncation error and error from the uncertainty in the LEC’s will likely also decrease with the use of the NLO $\chi$PT correction factor. We restrict our lightest valence quark mass to maintain $m_{\pi}L\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 318\relax$}}3.5$, and estimate that our finite volume errors are a few percent using 1-loop FV$\chi$PT. We will perform an explicit finite-volume study, however, before publication. Our preliminary result is renormalized using lattice perturbation theory because we have not yet completed the nonperturbative renormalization on the $a\sim 0.06$ fm ensembles. We expect these $Z$-factors to be reliable within systematic uncertainties, however, because we find good agreement between $Z_{B_{K}}$computed nonperturbatively and using lattice perturbation theory on the $a\sim 0.12$ and $a\sim 0.09$ ensembles. Table 2: Estimated total error budget for $\textrm{Re}(A_{2})$. Each source of uncertainty is given as a percentage. uncertainty \| $\qquad\textrm{Re}(A_{2})$ ---\|--- statistics \| 4.7% $\chi$PT truncation error \| 12% uncertainty in leading-order LECs \| 4% discretization errors \| 4% finite volume errors \| few percent renormalization factor \| 3.4% scale and quark-mass uncertainties \| 3% Wilson coefficients \| few percent total \| less than 20% ## References * [1] A. J. Buras and D. Guadagnoli, Phys. Rev. D 78, 033005 (2008). * [2] M. Antonelli, et al., Phys. Rept. 494, 197-414 (2010). * [3] See V. Lubicz, PoS LAT2009, 013 (2009) and Refs. therein. * [4] E. Lunghi and A. Soni, Phys. Lett. B 666, 162 (2008). * [5] J. Laiho, E. Lunghi and R. S. Van de Water, Phys. Rev. D 81, 034503 (2010); updates at www.latticeaverages.org. * [6] L. Maiani and M. Testa, Phys. Lett. B 245, 585 (1990). * [7] L. Lellouch and M. Lüscher, Commun. Math. Phys. 219, 31 (2001). * [8] J. Laiho and A. Soni, Phys. Rev. D 71, 014021 (2005). * [9] S. Li and N. H. Christ, PoS LATTICE2008, 272 (2008). * [10] C. Aubin, J. Laiho and R. S. Van de Water, Phys. Rev. D 77, 114501 (2008). * [11] C. Aubin, J. Laiho and R. S. Van de Water, Phys. Rev. D 81, 014507 (2010). * [12] M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Z. Phys. C 68, 239 (1995). * [13] C. Aubin et al. [MILC], Phys. Rev. D 70, 114501 (2004). * [14] O. Bar, C. Bernard, G. Rupak and N. Shoresh, Phys. Rev. D 72, 054502 (2005); C. Aubin, J. Laiho and R. S. Van de Water, Phys. Rev. D 75, 034502 (2007). * [15] G. Martinell, C. Pittori, C. T. Sachrajda, M. Testa and A. Vladikas, Nucl. Phys. B 445, 81 (1995). * [16] C. J. D. Lin, G. Martinelli, C. T. Sachrajda et al., Nucl. Phys. B619, 467-498 (2001). * [17] A. J. Buras, arXiv:hep-ph/9806471.	arxiv-papers	2010-11-19T21:41:05	2024-09-04T02:49:15.099836	{ "license": "Public Domain", "authors": "Jack Laiho and Ruth S. Van de Water", "submitter": "Ruth Van de Water", "url": "https://arxiv.org/abs/1011.4524" }
1011.4595	# The signature of LLAGNs in the nearby universe D. M. Neri-Larios, J. P. Torres-Papaqui, R. Coziol, J. M. Islas-Islas, and R. A. Ortega-Minakata Departamento de Astronomía, Universidad de Guanajuato, Apartado Postal 144, 36000 Guanajuato, Gto, México (daniel@astro.ugto.mx) ###### Abstract We have used the diagnostic diagram that compares the ratio of emission lines [NII]$\lambda$6584/H$\alpha$ with the equivalent width of [NII]$\lambda$6584, as proposed by Coziol et al. (1998), to determine the source of ionization of SDSS NELGs that cannot be classified by standard diagnostic diagrams, because the emission line [OIII]$\lambda$5007, H$\beta$, or both, are missing. We find these galaxies to be consistent with low luminosity AGNs, suggesting that this characteristic is the signature of the LLAGNs in the nearby Universe. Galaxies: Emission lines — Galaxies: Active — Galaxies: Low-Luminosity Active Galatic Nuclei ## 1 Resumen Hemos utilizado el diagrama diagnóstico que compara la razón de las líneas de emisión [NII]$\lambda$6584/H$\alpha$ con el ancho equivalente de [NII]$\lambda$6584, propuesto por Coziol et al. (1998), para determinar la fuente de ionización de galaxias con líneas de emisión agostas tomadas del SDSS donde la emisión en [OIII]$\lambda$5007, en H$\beta$, o ambas, no están presentes y por esta razón fueron excluidas de análisis previos. Estas galaxias se encuentran generalmente poblando la región del digarama consistente con AGNs de baja luminosidad, sugiriendo que esta característica es la firma de los LLAGNs en el universo cercano. ## 2 Introduction and Discussion The most common (standard) diagnostic diagrams devised to identify the source of ionization of Narrow Emission Line Galaxies (NELGs) in the nearby universe are those that compare the line ratios [OIII]$\lambda$5007/H$\beta$ with [NII]$\lambda$6584/H$\alpha$ [OI]$\lambda 6300/H\alpha$, or [SII]$\lambda\lambda 6717,6731/H\alpha$. Applying different separation sequences Kewley at al. (2001) and Kauffmann et al. (2003) proposed to discriminate between NELGs ionized by thermal photo-ionization, consistent with Star Forming Galaxies (SFGs), and NELGs ionized by non thermal photo- ionization source, generally called Active Galactic Nuclei (AGNs). Galaxies with intermediate line ratios are classified as Transitory Objects (TOs). The standard diagnostic diagrams are unfortunately useless in cases where some of the emission lines listed above are missing. In the SDSS survey this leaves an important fraction of galaxies unclassifiable (Cid Fernandes et al. 2010). The main interest of NELGs with emission lines missing is that they are particularly frequent in dense galactic structures, like clusters of galaxies (Phillips et al. 1986) and compact groups (Coziol et al. 1998; Martínez et al. 2010). In Coziol et al. (1998) it was shown that even after subtracting different templates from the spectra the missing lines do not appear. This is why these authors have used an alternative ”diagnostic diagram” to classify these galaxies. The NII diagram compares the equivalent width (EW) of [NII]$\lambda$6584, which is unaffected by the template subtraction, with the corrected by template ratio [NII]$\lambda 6584/H\alpha$. From the SDSS DR5 we have downloaded a sample of 476931 NELGs spectra, using the Virtual Observatory service111 http://www.starlight.ufsc.br. Keeping only galaxies that have emission lines S/N$\geq 3$ (S/N$>$10 in the continuum) reduces our sample to 224846 galaxies. In this last sample we count 34307 galaxies without H$\beta$, 12455 without [OIII] and 2840 without both lines, which represents 22% of the sample. These galaxies were systematically discarded by Cid Fernandes et al. (2010) in their recent study about the “forgotten” population of weak line galaxies (WLGs). Note also that since our galaxies have S/N$\geq 3$ they do not classify as WLGs. In Figure 1 we show the NII diagram for the NELGs with the emission lines missing. 62.3% of the galaxies without [OIII] falls on the AGN side of the NII diagram (to the right of separation line at -0.3 in [NII]/H$\alpha$). This fraction increases to 91% for the galaxies without both lines and 93% for the galaxies without H$\beta$. Therefore, most of the NELGs with emission lines missing are AGNs. These results differ significantly from what Cid Fernandes et al. (2010) have obtained. They found very few WLG are AGNs. This suggests that the absence of emission lines points to a different nature for the galaxies. Most of the NELGs with emission lines missing have EW below Log(EW[NII])$=0.6$: 77% without [OIII], 94% without H$\beta$ and 98% without both lines. By definition, the EW is the ratio between the flux in the emission line and the flux in the adjacent continuum. This parameter is consequently sensitive to the luminosity of the emission line and to the underlying stellar population. A small value of EW suggests the galaxies have an early-type morphology (see the article by Torres-Papaqui et al. in this proceeding), and/or that the emission line has a low luminosity. This is confirmed in Figure 1 where it is seen that the H$\alpha$ luminosity decreases with the EW. The median H$\alpha$ luminosity below Log(EW[NII])$=0.6$ is that of the galaxies without H$\beta$, which is 5.6$\times$1039 erg s-1. This property categorizes these galaxies as Low Luminosity AGNs (LLAGNs). Figure2 shows that the NELGs with emission lines missing have broad FWHM, similar to luminous AGNs. After correction for the resolution of the instrument, the FWHM values fall between 170 up to 700 km s-1. A higher fraction of the galaxies without [OIII] have lower FWHM, which is consistent with the high number of TOs and SFGs in this sample. The FWHM seems like another good criterion to separate AGNs from SFGs. We have found the NELGs with emission lines missing to be in majority LLAGNs. Therefore, this characteristics–the absence of emission lines–could be taken as the signature of LLAGNs in the nearby universe (Coziol et al. 1998). Summing all the luminous AGNs with the LLAGNs and TOs in our sample suggests that almost half of the NELGs in the nearby universe are AGNs (Miller et al. 2003). This result seems now much more consistent with the high number of QSOs observed at high redshifts. This observation may support the standard interpretation of AGNs (independent of their luminosity) as galaxies with active black holes in their nucleus. We also acknowledge support from PROMEP (Grant No. 103.5-10-4684). Figure 1: Contours of luminosity in H$\alpha$. The vertical line at log([NII]/H$\alpha$)=-0.3 separates SFGs from AGNs. The two lines at -0.4 and -0.1 define the buffer zone inhabited by TOs. Figure 2: Distributions of the FWHM of H$\alpha$: a) NELGs without[OIII]; b) NELGs without H$\beta$; c) NELGs without both lines. The point-dashed line corresponds to the distribution of luminous AGNs. ## References * Adelman-McCarthy et al. (2007) Adelman-McCarthy J.K. et al. 2007, ApJS, 172, 634 * Baldwin et al. (1981) Baldwin, J.A., Phillips, M.M., & Terlevich, R., 1981, PASP, 93, 5 * Cid Fernandes et al. (2005) Cid Fernandes, R., Mateus, A., Sodré, L., Stasińska, G., & Gomes, J.M., 2005, MNRAS, 348, 363 * Coziol et al. (1998) Coziol, R., Ribeiro, A.L.B., de Carvalho, R.R., & Capelato, H.V., 1998, ApJ, 493, 563 * Greene & Ho (2006) Greene, J.E., Ho, L.C., 2006, ApJ, 641, 117 * Martínez et al. (2008) Martínez, M.A., del Olmo, A., Coziol, R., & Focardi, P., 2008, ApJ, 678, L9 * Martínez et al. (2010) Martínez, M.A., del Olmo, A., Coziol, R., & Perea, J., 2010, AJ, 139, 1199 * Phillips et al. (1986) Phillips, M.M., Jenkins, C.R., Dopita, M.A., Sadler E.M., & Binette, L., 1986, AJ, 91, 1062 * Veilleux & Osterbrock (1987) Veilleux, S., & Osterbrock, D.E., 1987, ApJS, 63, 295	arxiv-papers	2010-11-20T18:24:19	2024-09-04T02:49:15.106480	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "D.M. Neri-Larios, J.P. Torres-Papaqui, R. Coziol, J.M. Islas-Islas,\n R.A. Ortega-Minakata", "submitter": "Juan Pablo Torres-Papaqui PhD", "url": "https://arxiv.org/abs/1011.4595" }
1011.4651	# Diplomarbeit Karim Adiprasito Karim ADIPRASITO111The final preparation of this paper was supported by the Deutsche Forschungsgemeinschaft within the research training group ‘Methods for Discrete Structures’ (GRK1408). Institute of Mathematics, Discrete Geometry Group, Free University Berlin, Arnimallee 2, 14195 Berlin e-mail: karim.adiprasito@fu-berlin.de # Characterization of polytopes via tilings with similar pieces Karim Adiprasito Karim ADIPRASITO111The final preparation of this paper was supported by the Deutsche Forschungsgemeinschaft within the research training group ‘Methods for Discrete Structures’ (GRK1408). Institute of Mathematics, Discrete Geometry Group, Free University Berlin, Arnimallee 2, 14195 Berlin e-mail: karim.adiprasito@fu-berlin.de ###### Abstract Generalizing results by Valette, Zamfirescu and Laczkovich, we will prove that a convex body $K$ is a polytope if there are sufficiently many tilings which contain a tile similar to $K$. Furthermore, we give an example that this can not be improved. Consider a convex body (a compact convex set with nonempty interior) $K$ in $\mathbb{R}^{n}$, which is tiled into a finite number of convex bodies. A tiling $T$, as a formal object, will here be the set consisting of all its tiles, and all tiles will be assumed to be convex bodies. Also, this investigation will only consider proper tilings, that is, tilings which are not trivial (meaning they consist of more than just one tile). We consider the following problem: Let there be tiles similar to $K$. Does it follow that $K$ is a polytope? For dimension 2, M. Laczkovich ([1]) could show that if one tile is similar to $K$, and if the tiling is proper, $K$ is in fact a polytope. He generalized a remark by G. Valette and T. Zamfirescu in [2]. Now the question arises: is this extendable to dimension 3? Already the original paper by Laczkovich contained a remark that the immediate generalization must be wrong and left the problem open for higher dimensions. Indeed, a circular cone can easily be tiled in such a way that one tile is similar to the cone by just cutting near the apex. T. Zamfirescu asked whether this was optimal, i.e.: Consider a convex body $K$ in 3-space which is tiled in such a way such that 2 tiles are similar to $K$. Then is $K$ a polytope? This will turn out to be true, and is a very special case of the general theorem. In dimensions higher than 3, a condition only on the number of similar tiles will never be sufficient (see the example at the end of this paper). The additional condition will encode some information on how the tiles are located relative to the convex body itself. In consistency with our observations, the condition will degenerate in dimensions 2 and 3. We thus are able to answer Zamfirescu’s question as well as solve the problem. Let us state the theorem: ###### Theorem 1. Let $K$ be a $n$-dimensional convex body, and let $T_{i},\ \\{1,2,3,...,n-1\\}$ be $n-1$ proper tilings of $K$, each of which contains a tile $L_{i}$ similar to $K$. If the convex hull of the fixed points $x_{L_{i}}$ forms a nondegenerate $n-2$-dimensional simplex, then $K$ is a polytope. Here, the point $x_{L}$ is the fixed point of the similarity $f_{L}$ from $K$ to $L$. If there is more than one similarity from $K$ to $L$, choose one. The simplex referred to in theorem 1 will be called the tip simplex of $K$ with respect to the tilings involved. Often, one is in the situation to have a single tiling, and several tiles are similar to $K$. This is a special case of above theorem, which is a bit harder to prove, since then one has to deal with interdependencies between the tiles similar to $K$ when deforming tilings. ###### Corollary 2. Let $K$ be a $n$-dimensional convex body, which is properly tiled into a finite number of convex bodies, $n-1$ of which are similar to $K$. Denote these particular tiles by $L_{i}$. If the convex hull of the fixed points $x_{L_{i}}$ forms a nondegenerate $(n-2)$-dimensional simplex, then $K$ is a polytope. ###### Corollary 3. Let $K$ be a convex body in $\mathbb{R}^{n},\ n\leq 3$, which is properly tiled into a finite number of convex bodies, $n-1$ of which are similar to $K$. Then $K$ is a polygon/ a polyhedron. Even more, in dimension 3, symmetry also plays a role in this calculation: ###### Corollary 4. Let $K$ be a convex body in $\mathbb{R}^{3}$, which is properly tiled into a finite number of convex bodies, $1$ of which is similar to $K$. If there are two similarities $f_{L},f_{L}^{\prime}$ from $K$ to the tile $L$ with different fixpoints respectively, then $K$ is a polytope. ###### Proof. We can regard this situation in the following way: $K$ has two tilings $T$ and $T^{\prime}$ which are the same, but in $T$ $f_{L}$ is the similarity from $K$ to $L$, and in $T^{\prime}$ $f_{L}^{\prime}$ is considered to be the similarity from $K$ to $L$. Theorem 1 then concludes the proof. ∎ Denote by $\textsc{bd}(M),\ \textsc{int}(M),\ \mathscr{E}(M),\ \textsc{conv}(M)$ the the boundary/ the interior/ set of extremal points/ the convex hull of a set $M$. Also, $B_{\varepsilon}(x)$ denotes the set of all points in $\mathbb{R}^{n}$ with distance less than $\varepsilon$ to $x$. Consider the general situation of a convex body $K$ tiled into convex bodies $K\supset P_{j}\in T,\ j\in[s]:=\\{1,2,3,...s\\}$. For every 2 tiles $P_{i},\ P_{j}$ there is a hyperplane $H_{ij}$ seperating the two. For a fixed $i\in[s]$, let $F_{ij}$ be the halfspace with boundary $H_{ij}$ containing $P_{i}$. Then, $P_{i}=K\cap\bigcap_{i\neq j}F_{ij}$, and every extremal point $x\in\textsc{int}(K)$ of $P_{i}$ is a vertex of $\bigcap_{i\neq j}F_{ij}$. In particular, $\mathscr{E}(T):=\cup{\mathscr{E}(P_{j})}$ has finitely many elements in common with $\textsc{int}(K)$, i.e. $\mathscr{E}(T)\cap\textsc{int}(K)$ is finite. Suppose $L=P_{1}$ is similar to $K$, then the similarity transformation $f_{L}$ from $K$ to $L$ maps $\mathscr{E}(K)$ to $\mathscr{E}(L)$ bijectively. If the fixed point $x_{L}$ of $f_{L}$ lies in $\textsc{int}(K)$, then for some iteration of $f_{L}$, $f_{L}^{n}(K)$ lies in the interior of $K$. Since $\mathscr{E}(f_{L}^{n}(K))=\mathscr{E}(f_{L}^{n}(K))\cap\textsc{int}(K)$ is finite, so is $\mathscr{E}(K)$, and $K$ is a polytope. Therefore, we can assume that $x_{L}\in\textsc{bd}(K)$. ## Adapting a tiling In this section, we focus on adapting tilings until they have features that come handy in the later proof. First, we want $L$ to be homothetic to $K$. Then we want to move the fixed point to an arbitrary point of the tip simplex. Before we begin, let us state our methods to deform a tiling: Let $K$ be a convex body, and let $T$ be a tiling of $K$, containing a tile $L$ similar to $K$. Let $f_{L}$ be a similarity mapping from $K$ to $L$. Define $f_{L}(T)$ to be the function $f_{L}$ applied to all the tiles of $T$, and thus a tiling of $L$. The idea is that we can refine the tiling $T$ to $T^{\prime}$ by defining $T^{\prime}:=f_{L}(T)\cup T\setminus\\{L\\}$. In this situation, we will also write $T^{\prime}=f_{L}(T)+T$. The similarity mappings of newly created tiles will be defined by the composition of the similarities involved. We call this procedure iterating a tiling. In a related situation, suppose we are given two tilings $T_{1}$ and $T_{2}$, we could form the tiling $T_{1}T_{2}:=\\{P\cap Q\|\ P\in T_{1},\ Q\in T_{2}\\}$. If $T_{1}$ tiles a set $K_{1}$, and $T_{2}$ tiles $K_{2}$, then $T_{1}T_{2}$ is a tiling of the intersection of $K_{1}$ and $K_{2}$. In particular, if $K_{1}=K_{2}$, then $T_{1}T_{2}$ is a (possibly refined) tiling of $K_{1}$. Let us apply the above methods to concrete tilings to show that simplifications are indeed possible. ###### Lemma 5. Let $K$ be a convex body as above, $T$ a tiling which contains a tile $L$ similar to $K$. Then there is a tiling $T^{\prime}$ of $K$ which contains a homothetic copy of $K$ whose fixed point coincides with $x_{L}$. ###### Proof. Denote by $f_{L}$ the similarity transformation from $K$ to $L$. We may assume $x_{L}=0$. Let $O_{n}$ denote the set of orthogonal transformations of $R^{n}$. Let $\lambda$ denote the similarity ratio of $f_{L}$, and define $M_{L}:=\lambda^{-1}f_{L}$. Then $M_{L}\in O_{n}$. Since $O_{n}$ forms a compact subset of the space of n-dimensional matrices, it follows that the powers of $M_{L}$ can get as close to the identity matrix as we please. Write $B_{\varepsilon}:=B_{\varepsilon}(0)$ for the open ball of radius $\varepsilon$ around $0$ and $TC(K):=TC_{0}(K)=TC_{x_{L}}(K)$ for the solid tangent cone of $K$ at 0. Obviously, the solid tangent cones of later iterations of $f_{L}(K)$ are included in earlier: $TC(K)\supset TC(L)\supset TC(f^{2}_{L}(K))\supset TC(f^{3}_{L}(K))...$ But since we can get $f^{i}_{L}$ as close to a homothety as we want, we can get $B_{\varepsilon}\cap TC(f^{i}_{L}(K))$ as close to $B_{\varepsilon}\cap TC(K)$ (with respect to, for example, the Pompeiu-Hausdorff metric and a fixed $\varepsilon>0$) as we want by choosing $i\in\mathbb{N}$. Thus, the solid tangent cones of $K$ and $f^{i}_{L}(K),\ i\in\mathbb{N}$ at $x_{L}$ coincide. In particular, because all tiles are compact, convex and have nonempty interior, their tangent cones are never degenerate, and it follows that $L$ is the only tile of $T$ which contains $0=x_{L}$. Then there is an $\varepsilon>0$ such that $K\supset B_{\varepsilon}\cap K=B_{\varepsilon}\cap L=B_{\varepsilon}\cap TC(K).$ In particular, $x_{L}$ is not an accumulation point of extremal points of $K$. Choose $R$ such that $B_{R}\supset K$, set $t:=R/\varepsilon$ and $d$ such that $\lambda^{d}<\varepsilon$ and $t\lambda^{d}<1$. There is a tiling $T$ of $K$ such that $f_{L}^{d}(K)$ is a tile of $K$. $tM_{L}^{-d}(T)$ gives a tiling of $tM_{L}^{-d}(K)$, and by the choice of $t$, $tM_{L}^{-d}(K)\supset K$. Thus, $tM_{L}^{-d}(T)\\{K\\}$ is a tiling of $K$ which contains $L^{\prime}:=tM_{L}^{-d}(f_{L}^{d}(K))=t\lambda^{d}K\subsetneq K$, which is a homothetic copy of $K$. ∎ Before we go on, let us consider the above constructed tiling. Recall the notion of hyperplanes $H_{1j}$ seperating the tiles $P_{1}=L^{\prime}$ and $P_{j}$ of $K$, then $L^{\prime}$ is the convex hull of $\\{x_{L}\\}\cup(H_{1j}\cap TC(K))=\\{x_{L}\\}\cup(H_{1j}\cap K).$ Thus, we have the following Lemma: ###### Lemma 6. $K$ can be written as $\textsc{conv}(\\{x_{L}\\}\bigcup_{i}B_{i}),$ where the $B_{i}\subset\textsc{bd}(K)$ are finitely many convex compact $n-1$-dimensional sets with boundary (which can be chosen to be disjoint from $\\{x_{L}\\}$). We will call these sets a base of $K$. Now that the tiles are directed nicely, we want to turn to adjusting their position. ###### Lemma 7. Let $K$ be a convex body as above, $T_{i}$ tilings fulfilling the conditions of theorem 1, $S$ the tip simplex, $x_{0}$ a point in the tip simplex. Then there is a tiling $T^{\prime}$ with a tile $L$ similar to $K$ such that $x_{L}=x_{0}$. ###### Proof. If the tip simplex is just a point, the Lemma is trivial. Suppose therefore that there are at least two tilings with $T_{1}$ and $T_{2}$ with tiles $L_{1}$ respectively $L_{2}$ which are similar to $K$ and have distinct fixed points, and suppose the corresponding similarities are chosen to be homotheties. Then the tiling $T_{1}+f_{L_{1}}(T_{2})$ will contain a homothetic copy of $K$: $L_{3}=f_{L_{1}}(L_{2})$. $L_{3}$ is new to us, because the corresponding fixed point will not coincide with $x_{L_{1}}$ or $x_{L_{2}}$, but will lie in their convex hull. By iterating this procedure, we see that there is a dense subset $M$ of $\textsc{conv}(\\{x_{L_{1}},x_{L_{2}}\\})$ we can make the fixed points of similarities lie in. Next, suppose $x_{0}\in\textsc{conv}(\\{x_{L_{1}},x_{L_{2}}\\})\setminus M$. Find a tiling $T$ with a similar copy $L$ of $K$ such that $x_{L}\in\textsc{conv}(\\{x_{L_{1}},x_{0}\\})$. (Again, we assume the similarities to be homotheties.) Define $H(x)$ to be the homothety which fixes $x_{L_{1}}$ and maps $f_{L}(x_{0})$ to $x_{0}$, and let $T^{\prime}:=H(T)$, which covers $K$. $T^{\prime}\\{K\\}$ is a tiling of $K$. If $L$ is chosen small enough, $H(L)$ is a proper subset of $K$ and thus a tile of $T^{\prime}\\{K\\}$ with fixed point $x_{0}$. With higher dimensional tip simplices, this construction works just in the same way. ∎ ## Conclusion of proof We will prove theorem 1 by induction. In dimension 1, it is trivial, in dimension 2, it was proven by Laczkovich. Let us assume theorem 1 is proven in dimension $n-1$. Let $K$ be a convex body fulfilling the conditions of the theorem 1 in Dimension $n$, let $x$ be some point in the relative interior of the tip simplex $S$, and let $H$ be some $n-1$-dimensional hyperplane of $\mathbb{R}^{n}$ containing $x$. $H\cap K$ is a $n-1$-dimensional convex set. It could even be of smaller dimensions, so let us just assume $H\cap K$ is not a point. Let $T^{\prime}_{i}$ be proper tilings of $K$ whose similarities are homotheties and whose fixed points are also the extremal points of the $n-3$-dimensional simplex $S\cap H$. $H\cap K$ inherits a tiling structure from $K$ by means of intersection: $T^{\prime}_{i}$ induces the tiling $\\{H\\}T^{\prime}_{i}$ on $H\cap K$. This tiling is proper if $H$ does not coincide with any of the hyperplanes separating an $L_{i}$ from an adjacent tile, which is true for all but finitely many choices of $H$. Note that because $L^{\prime}_{i}$ is a homothetic copy of $K$ whose fixed point lies in $H$, $H\cap L^{\prime}_{i}$ will be an element of $T^{\prime}_{i}$ which is a homothetic copy of $H\cap K$. The fixed point of $H\cap L^{\prime}_{i}$ in $H\cap K$ coincides with the fixed point of $L^{\prime}_{i}$ in $K$ and is therefore an extremal point of $S\cap H$. Since $x$ is a relative interior point of $S$, $S\cap H$ is a nondegenerate $n-3$-dimensional simplex spanned by the fixed points of $H\cap L^{\prime}_{i}$, which in turn are elements of the proper tilings $\\{H\\}T^{\prime}_{i}$ of $H\cap K$. Using the induction hypothesis, we see that $H\cap K$ must be a polytope. ###### Proposition 8. Let $K$ be a convex body in $\mathbb{R}^{n}$, which is tiled as in the description of theorem 1. Let $S$ be the tip simplex, $x$ a point in the relative interior of this simplex, and let $H$ be some hyperplane of $\mathbb{R}^{n}$ containing $x$. Then $H\cap K$ is a polytope in all but possibly a finite number of cases. ###### Proof of theorem 1. Assume theorem 1 is proven in dimension $n-1$. As already stated, $K$ is the convex hull of the union of a finite number of $n-1$ dimensional compact convex sets and a point $x$ in the relative interior of the tip simplex. Since we supposed $K$ is not a polytope, one of these sets has infinitely many extremal points which are also extremal points of $K$. Call this set $B$, and recall that $B\subset\textsc{bd}(K)$. Pick another point $y$ in the relative interior of the tip simplex of $K$. Choose a $n-1$-dimensional affine subspace $H$ parallel to $B$ and containing $y$. If $H\cap\textsc{conv}(\\{x\\}\cup B)=\emptyset$, interchange the roles of $y$ and $x$. If $H$ contains $x$, tilt $H$ just a little such that it intersects $\textsc{conv}(\\{x\\}\cup B)$ but neither intersects $\\{x\\}$ nor the base $B$. Also, assume $H$ is not one of the finite hyperplanes excluded in proposition 8. The intersection of $H$ and $K$ is, as we know from proposition 8, a polytope. But $H\cap B$ will have infinitely many extremal points, as it is a (possibly dilated, if we tilted $H$) homothetic copy of $B$. Thus, $K$ and $B$ can only share a finite number of extremal points, in contradiction with the assumption. ∎ ### Sharpness of results We could now ask if the above results are optimal, and indeed, they are. First note that as soon as we have constructed (in any dimension) a tiling $T$ of a convex body $K$ which contains (at least) 2 similar copies $L,L^{\prime}$ of $K$, then we could create a tiling with more similar copies by regarding the tiling $f_{L}(T)+T$. Note however that we can never make a degenerate tip simplex nondegenerate using this method. The dimension of the tip simplex is thus the real condition to make $K$ a polytope, a property not visible in dimensions 2 or 3. Using induction on dimensions, we can construct a convex body which even allows us to see that the condition on the tip simplex is optimal, in particular, $n-3$-dimensional tip simplices are not enough to conclude that $K$ is a polytope. Figuratively speaking, we take a circular cone, which shows theorem 1 to be sharp in dimension 3, and take it as a base for a 4-dimensional cone, which in turn forms a base of a five-dimensional cone etc. This example is just an extension of the example known for dimension 3. To make a concrete example with $n-3$ dimensional tip simplex in dimension $n>2$, use the following construction: Regard the convex body $K$ which is the set of all points $\sqrt{x_{1}^{2}+x_{2}^{2}}+\sum_{i\in\\{3,4,5,...,n\\}}x_{i}\leq 1;\ x_{i}\geq 0\ \forall i\in\\{3,4,5,...,n\\}$ where the $x_{i}$ are coordinates with respect to some base $\\{e_{1},e_{2},e_{3},...,e_{n}\\}$ of $\mathbb{R}^{n}$. We will show that the tip simplex $S$ is $\textsc{conv}(\bigcup_{i\in\\{3,4,5,...,n\\}}\\{e_{i}\\})$. We regard the homotheties $f_{i}(x)=\frac{1}{2}(x-e_{i})+e_{i}$ for $i\in\\{3,4,5,...,n\\}$. Their fixed points span the said tip simplex, and the interior of the images of $K$ does not intersect. Thus, it remains to show that the remaining tile is convex. But this is simple, since it can be written as intersection of the convex sets $K$ and $X_{i},\ i\in\\{3,4,5,...,n\\}$, $X_{i}:=\\{x\in\mathbb{R}^{n}\|x_{i}\leq\frac{1}{2}\\}.$ ## References * [1] M. Laczkovich; Decomposition of convex figures into similar pieces, Discrete and Comp. Geometry Vol. 13, 143-148, 1995 * [2] G. Valette, T. Zamfirescu; Les partages d’un polygone convexe en 4 polygones semblambes au premier, J. Combin. Theory Ser. B, Vol 16, 1-16, 1974	arxiv-papers	2010-11-21T11:42:26	2024-09-04T02:49:15.112477	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Karim Adiprasito", "submitter": "Karim Alexander Adiprasito", "url": "https://arxiv.org/abs/1011.4651" }
1011.4667	††thanks: Corresponding author # Quench dynamics of the topological quantum phase transition in the Wen- plaquette model Long Zhang Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Su-Peng Kou spkou@bnu.edu.cn Department of Physics, Beijing Normal University, Beijing 100875, China Youjin Deng Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China ###### Abstract We study the quench dynamics of the topological quantum phase transition in the two-dimensional transverse Wen-plaquette model, which has a phase transition from a $Z_{2}$ topologically ordered to a spin-polarized state. By mapping the Wen-plaquette model onto a one-dimensional quantum Ising model, we calculate the expectation value of the plaquette operator $F_{i}$ during a slowly quenching process from a topologically ordered state. A logarithmic scaling law of quench dynamics near the quantum phase transition is found, which is analogous to the well-known static critical behavior of the specific heat in the one-dimensional quantum Ising model. ## I Introduction Ultracold atoms provide an ideal platform for experimental studies of the time evolution of quantum systems, and make it desirable for related theoretical explorations on dynamics of quantum phase transitions in various models. These explorations mainly focus on nonequilibrium dynamics in quantum systems which undergo a quantum phase transition when a system parameter is varied (_quantum quench_)Pol_colloquium . These experimental and theoretical studies can potentially help to pave the way for future technologies and provide a deeper understanding of quantum many-body physics, particularly the universal scaling behavior in the quench dynamics. Recently, a new type of phase transition, the so-called topological quantum phase transition (TQPT) has attracted considerable research attentionWenbook ; wen3 ; wen4 ; wen1 ; Kitaev ; nayak ; lidar ; chamon ; xiang ; yujing ; vid1 ; vid2 . TQPT is a kind of phase transition between two quantum states with the _same_ symmetry. It is fundamentally different from the usual symmetry- breaking phase transition, and involves a new type of order — _topological order_ , as introduced by WenWenTo1 . In such an ordered state, there is no local order parameter, and the state is robust against arbitrary local perturbations. On this basis, quantum systems with topological order have been proposed to build robust quantum memoriesKitaev and topological quantum computer (TQC)kou1 ; kou2 . In Refs.kou1 ; kou2 , it was shown that the pure state of topological order can be obtained via an adiabatical and continuous evolution process from a non-topologically ordered state and further be used as the initial state for TQC. Nevertheless, the detailed dynamics of such a quench process, particularly the universal scaling behavior near TQPT, has not been studied yet. In the last decade, several exactly solvable spin models with topological order were found, such as the toric-code modelKitaev , the Wen-plaquette modelWen and the Kitaev model on a hexagonal latticeKitaev2 . These spin models provide a framework to study the TQPT and its quench dynamics. Thus, in recent years, some research groups have studied the quench dynamics of TQPT in the Kitaev model (e.g. Mondal _et al._ quh-Kiv ) or the toric-code model (e.g. Tsomokos _et al._ quh-tc ). Mondal _et al._ found a relationship between the quench rate and the defect density in the one-dimensional (1D) and two- dimensional (2D) Kitaev model in the limit of slow quench rate and generalized the result to the defect density of a $d$ dimensional quantum model. Tsomokos _et al._ investigated how a topologically ordered ground state in the toric- code model reacts to rapid quenches. They tested several cases and showed which kind of quench can preserve or suppress the topological order. In this work, we study the quench dynamics of TQPT from a topologically ordered state to a non-topological order in the Wen-plaquette model. To characterize the phase transition, we calculate the expectation value of plaquette operator $F_{i}$ during the quenching process, which is related to the number of quasiparticles in the topologically ordered state. Our results provide helpful information about the whole quenching process. The remaining of this paper is organized as follows. Section II describes an exact mapping from the 2D transverse Wen-plaquette model onto the 1D Ising chain. In Sec. III, we study the TQPT of the 2D transverse Wen-plaquette model and give some results about its order parameters. In Sec. IV, we give the solutions to the dynamics of TQPT in the transverse Wen-plaquette model. A brief discussion is given in Sec. V. ## II Mapping the transverse Wen-plaquette onto Ising model Figure 1: (Color online) The lattice where Wen-plaquette model is located. A plaquette is defined by $F_{i}=\tau_{i}^{y}\tau_{i+\hat{x}}^{x}\tau_{i+\hat{x}+\hat{y}}^{y}\tau_{i+\hat{y}}^{x}$. We start with the Hamiltonian of the Wen-plaquette model on a square lattice with periodic boundary conditions in both directions: $H_{W}=-g\sum_{i}F_{i},\quad F_{i}=\tau_{i}^{y}\tau_{i+\hat{x}}^{x}\tau_{i+\hat{x}+\hat{y}}^{y}\tau_{i+\hat{y}}^{x},$ (1) where $\tau_{i}^{x}$ and $\tau_{i}^{y}$ are Pauli operators on site $i$, $\hat{x}$ and $\hat{y}$ are the unit vectors in x-axis and y-axis, respectively (see Fig. 1). Because of the commutativity of $H$ and $F_{i}$, the energy eigenstates can be labeled by the eigenstates of $F_{i}$. We can easily find $F_{i}^{2}=1$, so the eigenvalues of $F_{i}$ are $F_{i}=1$ and $F_{i}=-1$, which gives the exact ground state energy. In the case of $g>0$, the ground state is $F_{i}=1$ for every plaquette, and the elementary excitation is $F_{i}=-1$ on one plaquette (denoted by $i$) with an energy gap $E_{g}-E_{0}=2g$. On an even-by-even lattice, there are two types of plaquettes — the even plaquettes and the odd plaquettes respectively. As a result, one may define two kinds of bosonic quasiparticles: $Z_{2}$ charge and $Z_{2}$ vortex (see detailed calculations in Ref. Wen or Ref. Wenbook ). A $Z_{2}$ charge is defined by $F_{i}=-1$ on an even sub-plaquette while a $Z_{2}$ vortex defined by $F_{i}=-1$ on an odd. Thus a fermion can be regarded as the bound state of a $Z_{2}$ charge and a $Z_{2}$ vortex. Now we consider the Wen-plaquette model in a transverse field, which is defined by: $H_{W}^{\prime}=-g\sum_{i}F_{i}-J\sum_{i}\tau_{i}^{x}.$ (2) This model on a square lattice can be mapped onto the 1D quantum Ising model with the Hamiltonian yujing $H_{I}=-\sum_{n=1}^{N}(g_{I}{\sigma}_{n}^{x}+{\sigma}_{n}^{z}{\sigma}_{n}^{z}),$ (3) where $\sigma_{n}^{x}$ and $\sigma_{n}^{z}$ are Pauli operators. To derive the mapping, one can calculate the commutation relations (see detailed calculations in Appendix): $\displaystyle[F_{i},\tau_{j}^{x}]=2F_{i}\tau_{j}^{x}(\delta_{i,j-\hat{x}}+\delta_{i,j-\hat{y}}),$ $\displaystyle[F_{i},F_{j}]=0,$ $\displaystyle[\tau_{i}^{x},\tau_{j}^{x}]=0.$ (4) These relations correspond to those in Ising model: $\displaystyle[\sigma_{i}^{x},\sigma_{j}^{z}\sigma_{j+1}^{z}]=2\sigma_{i}^{x}\sigma_{j}^{z}\sigma_{j+1}^{z}(\delta_{i,j}+\delta_{i,j+1}),$ $\displaystyle[\sigma_{i}^{x},\sigma_{j}^{x}]=0,$ $\displaystyle[\sigma_{i}^{z}\sigma_{i+1}^{z},\sigma_{j}^{z}\sigma_{j+1}^{z}]=0.$ (5) Then we obtain the mapping $F_{i}\leftrightarrow\sigma_{i}^{x},\quad\tau_{i}^{x}\leftrightarrow\sigma_{i}^{z}\sigma_{i+1}^{z}.$ (6) Accordingly, the Hamiltonian (2) can be mapped onto the 1D quantum Ising model like following: $H_{W}^{\prime}\rightarrow-\sum_{\alpha}\sum_{i}(g\sigma_{\alpha i}^{x}+J\sigma_{\alpha i}^{z}\sigma_{\alpha i+1}^{z}),$ (7) where the subscript $\alpha$ implies there is one or more Ising chains. The number of Ising chains is determined by the size of the square lattice for the Wen-plaquette model (see Ref.yujing ). Since the Ising chains decouple from each other, we can consider only one Ising chain without loss of generality, and reduce the Hamiltonian (7) to $H_{W}^{\prime}=-J\sum_{i}(g_{W}\sigma_{i}^{x}+\sigma_{i}^{z}\sigma_{i+1}^{z}),\quad g_{W}=\frac{g}{J}.$ (8) Then we may explore the quantum properties of the original Wen-plaquette model by studying the corresponding 1D Ising model. ## III String order parameters in Wen-plaquette model For the 1D transverse Ising model (3), there are two phases: in the limit of $g_{I}\gg 1$, the ground state is a paramagnet with all spins polarized along x-axis, $\left\langle\sigma_{i}^{x}\right\rangle\rightarrow 1$; in the limit of $g_{I}\ll 1$, there are two degenerate ferromagnetic ground states with all spins along positive or negative z-axis and $\left\langle\sigma_{i}^{x}\right\rangle\rightarrow 0$. Consequently, in Hamiltonian (8), there is a quantum critical point atBaxter $g_{W}=\frac{g}{J}=1$ (9) that divides the two phases. Accordingly, the original transverse Wen- plaquette model also has two phases separated by this quantum critical point — in the region of $g_{W}>1$, the system is a topologically ordered state; in the region of $g_{W}<1,$ it’s a spin-polarized state. Noting that the local order parameters cannot be used to learn the nature of TQPT any more, we introduce two non-local order parameters $\psi_{1}$ and $\psi_{2}$ as string order parameters (SOP’s) in the transverse Wen-plaquette model. They are defined by the expectations of string operators $\prod_{i}F_{i}$ and $\prod_{i}\tau_{i}^{x}$ with $i$ as the site index along a string in the diagonal directionyujing , i.e. $\psi_{1}\equiv\left\langle\prod_{i}F_{i}\right\rangle$ and $\psi_{2}\equiv\left\langle\prod_{i}\tau_{i}^{x}\right\rangle$, respectively. We then calculate these two SOP’s by using the mapping in Eq.(6). For $\psi_{2}$, one has $\psi_{2}=\left\langle\prod_{i}\tau_{i}^{x}\right\rangle=\left\langle\sigma_{1}^{z}\sigma_{2}^{z}\sigma_{2}^{z}\sigma_{3}^{z}\cdots\sigma_{n-1}^{z}\sigma_{n}^{z}\right\rangle=\left\langle\sigma_{1}^{z}\sigma_{n}^{z}\right\rangle,$ (10) which becomes the correlation of two spins in the Ising chain of length $n$. By employing the Jordan-Wigner transformation $\sigma_{n}^{x}=1-2c_{n}^{\dagger}c_{n},\quad\sigma_{n}^{z}=-(c_{n}+c_{n}^{\dagger})\nu,$ (11) where $\nu\equiv\prod_{m<n}(1-2c_{m}^{\dagger}c_{m})=\prod_{m<n}(c_{m}c_{m}^{\dagger}-c_{m}^{\dagger}c_{m})=\prod_{m<n}A_{m}B_{m},$ (12) where $A_{m}=c_{m}^{\dagger}+c_{m}$, $B_{m}=c_{m}^{\dagger}-c_{m}$, $c^{\dagger}_{m}$ and $c_{m}$ are the creation and annihilation operators for fermions, we obtain $\displaystyle\psi_{2}$ $\displaystyle=\left\langle\sigma_{1}^{z}\sigma_{n}^{z}\right\rangle=\left\langle(c_{1}+c_{1}^{\dagger})\nu(c_{n}+c_{n}^{\dagger})\right\rangle$ $\displaystyle=\left\langle B_{1}A_{2}B_{2}\cdots B_{n-1}A_{n}\right\rangle.$ (13) Following the Wick’s theorem, we can transform Eq.(III) into a Toeplitz determinant asbook20 $\left\|\begin{array}[c]{cccc}G_{12}&G_{13}&\cdots&G_{1n}\\\ G_{22}&G_{23}&\cdots&G_{2n}\\\ \vdots&\vdots&\ddots&\vdots\\\ G_{n-1,2}&G_{n-1,3}&\cdots&G_{n-1,n}\end{array}\right\|,$ (14) where $G_{ij}=-\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathrm{d}k\frac{g_{W}-\cos k-i\sin k}{\sqrt{(g_{W}-\cos k)^{2}+\sin^{2}k}}e^{ik(i-j)}.$ (15) In the thermodynamic limit $N\rightarrow\infty$, we have (see Ref. McCoy1 or Ref. McCoy2 ) $\psi_{2}\sim\left\\{\begin{array}[c]{ll}(1-g_{W}^{2})^{1/4}&\text{when $g_{W}<1$}\\\ 0&\text{when $g_{W}\geq 1$}\end{array}\right..$ (16) The exponent $1/4$ here agrees with the critical exponent $2\beta/\nu=1/4$ (see Ref.Baxter ). By the same method we can obtain the result of $\psi_{1}$, $\psi_{1}=\left\langle\prod_{i}F_{i}\right\rangle=\left\langle\prod_{i}\sigma_{i}^{x}\right\rangle.$ (17) In addition, we can take advantage of the duality of 1D Ising model (see Ref. book20 ) : $s_{i}^{x}=\sigma_{i}^{z}\sigma_{i+1}^{z}\quad\text{and}\quad s_{i}^{z}=\prod_{k<i}\sigma_{k}^{x}.$ (18) Thus the result in Eq.(17) is turned into $\psi_{1}=m$, where $m\equiv\left\langle\sigma_{i}^{z}\right\rangle$ is the spontaneous magnetization. We can find that when $N\rightarrow\infty$, the spin correlation (10) is the square of $m$. Consequently, from Eq.(16), we have $\psi_{1}\sim\left\\{\begin{array}[c]{ll}0&\text{when $g_{W}\leq 1$}\\\ (1-g_{W}^{-2})^{1/8}&\text{when $g_{W}>1$}\end{array}\right..$ (19) From the calculations above, one may notice that the non-local SOPs in a 2D transverse Wen-plaquette model are transformed to the local order parameters in the dual 1D Ising model. ## IV Quench dynamics of TQPT In this section we study the dynamics of TQPT in the transverse Wen-plaquette model. The Kibble-Zurek mechanism (KZM)Kibble ; Zurek is a general theory to explore the dynamics of second order phase transitions including the quantum case. According to KZM, during a quench-induced phase transition, the system undergoes three stages of evolution: adiabatic–pulse–adiabatic. It predicts that the density of topological defects, which are generated by the pulse evolution, is a function of quench time. We can solve the dynamic problems in the Wen-plaquette model by taking a sequence of transformations as in Ref. quench . First, through the Jordan- Wigner transformation (11), the spin operators $\sigma_{i}^{x,z}$ are represented by fermionic operators $c_{n}$. Second, the operators $c_{n}$ are Fourier transformed into the momentum space: $c_{n}=\frac{e^{-i\pi/4}}{\sqrt{N}}\sum_{k}c_{k}e^{ikn}.$ (20) Third, after Bogoliubov transformation, one have $c_{k}=u_{k}\eta_{k}+v_{-k}^{\ast}\eta_{-k}^{\dagger},$ (21) where $\eta_{k}$ and $\eta_{k}^{\dagger}$ are fermionic operators. For dynamic problems, the expression (21) becomes $c_{k}(t)=u_{k}(t)\widetilde{\eta}_{k}+v_{-k}^{\ast}(t)\widetilde{\eta}_{-k}^{\dagger}.$ (22) Now, a dynamic solution can be written in the form of Bogoliubov mode $(u_{k}(t),$ $v_{k}(t))$ after this whole transformation procedure. By this method, the dynamics of the quantum Ising model can be expressed as the time evolution of the Bogoliubov mode via so-called Bogoliubov-de Gennes dynamic equations (see Ref. quench ): $\left\\{\begin{array}[c]{l}i\hbar\mathrm{d}u_{k}/\mathrm{d}t=+2(g_{I}(t)-\cos k)u_{k}+2\sin kv_{k}\\\ i\hbar\mathrm{d}v_{k}/\mathrm{d}t=-2(g_{I}(t)-\cos k)v_{k}+2\sin ku_{k}\end{array}\right..$ (23) Consider a linear quench, i.e. $g_{I}(t<0)=-\frac{t}{{\tau}_{Q}},$ (24) where $t$ varies from $-\infty$ to $0$ and the quench time $\tau_{Q}$ characterizes the quench rate (defined by $1/\tau_{Q}$). Equations (23) can be transformed into the form of Landau-Zener (LZ) modelLZmodel (the connection between the KZM and the LZ model can be found in Ref. Dam ; Dam-Zuk ) : $\left\\{\begin{array}[c]{l}i\hbar\mathrm{d}u_{k}/\mathrm{d}\tau=-\frac{1}{2}(\tau\Delta_{k})u_{k}+\frac{1}{2}v_{k}\\\ i\hbar\mathrm{d}v_{k}/\mathrm{d}\tau=+\frac{1}{2}(\tau\Delta_{k})v_{k}+\frac{1}{2}u_{k}\end{array}\right.,$ (25) where $\tau=4\tau_{Q}\sin k(\frac{t}{\tau_{Q}}+\cos k),$ and $\Delta_{k}^{-1}=4\tau_{Q}\sin^{2}k.$ From equations (25), we derive a second order differential equation for $v_{k}$: $\frac{\mathrm{d}^{2}v_{k}}{\mathrm{d}\tau^{2}}+(\frac{1}{4}\tau^{2}\Delta^{2}_{k}+\frac{i\Delta_{k}}{2}+\frac{1}{4})v_{k}=0.$ (26) After the substitutions $s=\frac{1}{4i\Delta_{k}}\quad\text{and}\quad z=\sqrt{\Delta_{k}}\tau e^{i\pi/4},$ (27) we have $\frac{\mathrm{d}^{2}v_{k}(z)}{\mathrm{d}z^{2}}+(s+\frac{1}{2}-\frac{1}{4}z^{2})v_{k}(z)=0.$ (28) This kind of differential equation has a general solution $\begin{array}[c]{l}v_{k}(\tau)=-[aD_{-s-1}(-iz)+bD_{-s-1}(iz)],\\\ u_{k}(\tau)=(-\Delta_{k}\tau+2i\frac{\partial}{\partial\tau})v_{k}(\tau),\end{array}$ (29) where $D_{m}(x)$ is the so-called parabolic cylinder function (PCF) or Weber- Hermite functionWhi-Wat . According to the boundary conditions and the characters of the PCF, one may derive the approximative solutions to equations (28) at the end of linear quench for $t=0$ (See Ref. quench ): $\begin{array}[c]{l}{\|u_{k}\|}^{2}=\frac{1-\cos k}{2}+e^{-2\pi\tau_{Q}\sin^{2}k},\\\ {\|v_{k}\|}^{2}=1-{\|u_{k}\|}^{2},\\\ u_{k}v_{k}^{\ast}=\frac{1}{2}\sin k+sgn(k)e^{-\pi\tau_{Q}\sin^{2}k}\sqrt{1-e^{-\pi\tau_{Q}\sin^{2}k}}e^{i\varphi_{k}},\end{array}$ (30) with the condition $\tau_{Q}\gg 1$. Figure 2: (Color online) The expectation value of $F_{i}$ at $t=0$ varying with the quench time $\tau_{Q}$, where $f_{1}=1-\frac{1}{\pi}\int_{-\pi}^{\pi}\mathrm{d}k(\frac{1+\cos k}{2}-e^{-2\pi\tau_{Q}\sin^{2}k})$ and the approximative result $f_{2}=\frac{1}{\pi\sqrt{2\tau_{Q}}}$. Now we calculate the dynamic solutions in the Wen-plaquette model via the same transformation procedure plus the mapping (6). We shall also consider $g_{W}(t<0)=-\frac{t}{{\tau}_{Q}}.$ (31) The quenching process can be set as tuning the strength of the transverse field from zero to very large compared with the coupling constant $g$. The phase transition is thus from a topologically ordered state to a spin- polarized state during the time-evolution from $t\rightarrow-\infty$ to $t=0$. First, we calculate the expectation value of $F_{i}$ after the quenching process. From the Jordan-Wigner transformation, $\sigma_{n}^{x}=1-2c_{n}^{\dagger}c_{n}$, we have $\left\langle F_{i}\right\rangle\rightarrow\left\langle\sigma_{n}^{x}\right\rangle=\left\langle\left(1-2c_{n}^{\dagger}c_{n}\right)\right\rangle.$ Through the Fourier transformation, we express $c_{m}^{\dagger}c_{n}$ in momentum space $c_{m}^{\dagger}c_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathrm{d}k\int_{-\pi}^{\pi}\mathrm{d}k^{\prime}c_{k}^{\dagger}c_{k^{\prime}}e^{i(k^{\prime}n-km)}.$ (32) After the Bogoliubov transformation (21), we obtainintro1 $\displaystyle\langle c_{m}^{\dagger}c_{n}\rangle$ $\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathrm{d}k\int_{-\pi}^{\pi}\mathrm{d}k^{\prime}e^{i(k^{\prime}n-km)}v_{-k}v_{-k}^{\ast}\delta_{k,k^{\prime}}$ $\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathrm{d}k\|v_{k}\|^{2}e^{ik(n-m)}.$ (33) From the solutions (30), we derive $\displaystyle\left\langle F_{i}\right\rangle$ $\displaystyle\rightarrow\left\langle\sigma_{n}^{x}\right\rangle=1-2\cdot\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathrm{d}k(\frac{1+\cos k}{2}$ $\displaystyle-e^{-2\pi\tau_{Q}\sin^{2}k})\overset{\tau_{Q}\gg 1}{\simeq}\frac{1}{\pi\sqrt{2\tau_{Q}}}.$ (34) We can see the original state in the limit of $g\gg 1$, which is a topological ground state with $\left\langle F_{i}\right\rangle\rightarrow\left\langle\sigma_{n}^{x}\right\rangle\rightarrow 1$, will finally evolve into the trivial spin-polarized state with $\left\langle F_{i}\right\rangle\rightarrow\left\langle\sigma_{n}^{x}\right\rangle\rightarrow 0$. Figure 2 shows how the expectation value of $F_{i}$ at the end of quenching process ($t=0$) varies with the quench rate. We can obtain the the numbers of $Z_{2}$ charges and $Z_{2}$ vortices by defining $\mathcal{N}_{c}\equiv\frac{1}{2}\sum_{i\in\mathrm{even}}(1-F_{i})=\sum_{i\in\mathrm{even}}c_{i}^{\dagger}c_{i}$ (35) and $\quad\mathcal{N}_{v}\equiv\frac{1}{2}\sum_{i\in\mathrm{odd}}(1-F_{i})=\sum_{i\in\mathrm{odd}}c_{i}^{\dagger}c_{i},$ respectively. Before the quench, the system is in a topological ground state. There is no quasiparticles ($N_{c}\equiv\langle\mathcal{N}_{c}\rangle=0$, $N_{v}\equiv\langle\mathcal{N}_{v}\rangle=0$), so we have $F_{i}=1$ for all plaquettes. At the end of the quench, we can also calculate the density of plaquettes $F_{i}=-1$intro2 by $\displaystyle n=$ $\displaystyle\frac{1}{2N}\langle\sum_{i}(1-F_{i})\rangle=\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathrm{d}k(\frac{1+\cos k}{2}$ $\displaystyle-e^{-2\pi\tau_{Q}\sin^{2}k})\overset{\tau_{Q}\gg 1}{\simeq}\frac{1}{2}-\frac{1}{2\pi\sqrt{2\tau_{Q}}}.$ (36) It is obvious that when $\tau_{Q}\rightarrow\infty$, half of the plaquettes will be turned into $F_{i}=-1$, which implies $\left\langle F_{i}\right\rangle\rightarrow 0.$ However, this result cannot give us any information about the quenching process or the critical behaviors. In order to obtain such information, we find that, for $\tau_{Q}\gg 1$, the expression $\|v_{k}(t)\|^{2}=\frac{1}{2}(1+\frac{\cos k+t/\tau_{Q}}{\sqrt{1+2t/\tau\cos k+(t/\tau_{Q})^{2}}})-e^{-2\pi\tau_{Q}\sin^{2}k}$ (37) is a time-dependent approximate function for the general solution $\|v_{k}(\tau)\|^{2}$ in Eq.(29), if we cut off the negative part of the curve (see Fig. 3). Figure 3: (Color online) The colored curves represent $\|v_{k}(\tau)\|^{2}$ in the original form of the PCF varying with momenta k from $-\pi$ to $\pi$, where the red one is at $t=-0.5\tau_{Q}$, the blue at $t=-\tau_{Q}$, the green at $t=-2\tau_{Q}$ with $\tau_{Q}=50$ in all the three cases. The dashed, dotted and dot-dashed curves, respectively, represent the approximate function for $\|v_{k}(\tau)\|^{2}$ with corresponding parameters, with the negative parts under the x-axis being cut off. They match with the colored curves very well By using this approximate function, we calculate the value $\left\langle F_{i}\right\rangle$ during the quenching process as a function of the time $t$ near the quantum critical point. The result is shown in Fig. 4. Figure 4: (Color online) The expectation value of $F_{i}$ during the quenching process varying from $t=-2\tau_{Q}$ to $t=0$ with $\tau_{Q}=50$ and the absolute value of the slope ($\|\frac{\mathrm{d}\langle F_{i}\rangle}{\mathrm{d}t}\|$) corresponding to the curve above. The small graph shows $\|\frac{\mathrm{d}\langle F_{i}\rangle}{\mathrm{d}t}\|\sim\ln\|t/\tau_{Q}+1\|$ from both sides approaching the critical point $t=-\tau_{Q}$. Further calculations show that the derivative of the expectation-value $\left\langle F_{i}\right\rangle$ diverges at point $t=t_{c}=-\tau_{Q}$ as $\frac{\mathrm{d}\langle F_{i}\rangle}{\mathrm{d}t}\rightarrow-\infty$, which characterizes the critical point. In particular, we obtain a logarithmic scaling law of quench dynamics near the quantum phase transition as $\frac{\mathrm{d}\langle F_{i}\rangle}{\mathrm{d}t}\sim\ln\|t-t_{c}\|.$ (38) Such a dynamics is analogous to the static scaling behavior of the specific heat near the critical point for the 1D quantum Ising model. The physical picture can be described as follows. At beginning, the external field is weak and can be treated as a perturbation, which creates vortices or charges and drives them to move around and annihilate each other. As a result, the density of quasiparticles remains small. As the strength of the field continuous to grow, the density of plaquettes $F_{i}=-1$ increases rapidly near the point $g_{W}=1$ (that is $t=-\tau_{Q}$). The rate of density-changing diverges with a logarithmic scaling law. Finally, at the end of the quench, the field becomes so strong that the vortices and charges are all confined and cannot be treated as quasiparticles. If the quenching process is infinitely slow, half of the plaquttes overturns. In addition we shall notice that only the linear quench is considered here. However, for other cases, our method is not reliable. Inspired by other papers on quantum quench in the toric-code model (e.g. Rahmani _et al._ Rah-Cha , where a sudden quench is studied), we can study the quench problems more generally through the time evolution of the entanglement entropy in the Wen- plaquette model. Nevertheless, we will not consider this in this work. ## V Conclusion In summary, we study the dynamics of TQPT caused by a linear quench in the transverse Wen-plaquette model. We first show how to derive the mapping from the 2D Wen-plaquette onto the 1D Ising model by comparing their commutation relations. Based on this mapping, we point out the quantum critical point in the transverse Wen-plaquette model and calculate its non-local order parameters. We then calculate the expectation value of $F_{i}$ at the end of the quench, and further show how this value varies during the whole process. In particular, we find a logarithmic scaling law of quenching process near the TQPT. Finally we address the realization of the Wen-plaquette model in an optical lattice of cold atoms. Because the Wen-plaquette model can be regarded as an effective model of the Kitaev model on a two dimensional hexagonal lattice, one may first realize the Kitaev model. The Hamiltonian of the Kitaev model isKitaev2 $\mathrm{H}=\sum_{j+l=\text{even}}(\mathrm{J}_{x}\sigma_{j,l}^{x}\sigma_{j+1,l}^{x}+\mathrm{J}_{y}\sigma_{j-1,l}^{y}\sigma_{j,l}^{y}+\mathrm{J}_{z}\sigma_{j,l}^{z}\sigma_{j,l+1}^{z})$ (39) where $j$ and $l$ denote the column and row indices of the lattice. In the limit of $\mathrm{J}_{x}\mathrm{\gg J}_{z}\sim\mathrm{J}_{y}$ in this model, the effective Hamiltonian of Kitaev model is simplified into that of the Wen- plaquette model as $H_{0}=-\frac{\mathrm{J}_{z}^{2}\mathrm{J}_{y}^{2}}{16\|\mathrm{J}_{x}\|^{3}}\sum_{i}\sigma_{\text{left}(i)}^{x}\sigma_{\text{right}(i)}^{x}\sigma_{\text{up}(i)}^{y}\sigma_{\text{down}(i)}^{y}.$ (40) Then one can use the Kitaev model in the limit $\mathrm{J}_{x}\mathrm{\gg J}_{z}\sim\mathrm{J}_{y}$ on a torus to do the TQC. The realization of the Kitaev model on the 2D hexagonal lattice has been proposed in Ref.du ; zo1 . The essential idea realizing the Kitaev model is to induce and control virtual spin-dependent tunneling between neighboring atoms in the lattice that results in a controllable Heisenberg exchange interaction. The authors acknowledge that this research is supported by NFSC Grant No. 10874017, 10975127, National Basic Research Program of China (973 Program) under the grant No. 2011CB92180, the Anhui Provincial Natural Science Foundation under Grant No. 090416224, and the Chinese Academy of Sciences. ## Appendix In this part, we give detailed calculations about commutation relations (II), which is related to the consistency between Wen-plaquette model and Ising model. The key point is to calculate $[F_{i},F_{j}]$, $\displaystyle[F_{i},F_{j}]$ $\displaystyle=[\tau_{i}^{x}\tau_{i+\hat{x}}^{y}\tau_{i+\hat{y}+\hat{x}}^{x}\tau_{i+\hat{y}}^{y},\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}]$ $\displaystyle=[\tau_{i}^{x},\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}]\tau_{i+\hat{x}}^{y}\tau_{i+\hat{y}+\hat{x}}^{x}\tau_{i+\hat{y}}^{y}+\tau_{i}^{x}$ $\displaystyle[\tau_{i+\hat{x}}^{y},\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}]\tau_{i+\hat{y}+\hat{x}}^{x}\tau_{i+\hat{y}}^{y}+\tau_{i}^{x}\tau_{i+\hat{x}}^{y}$ $\displaystyle[\tau_{i+\hat{y}+\hat{x}}^{x},\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}]\tau_{i+\hat{y}}^{y}+\tau_{i}^{x}\tau_{i+\hat{x}}^{y}\tau_{i+\hat{y}+\hat{x}}^{x}$ $\displaystyle[\tau_{i+\hat{y}}^{y},\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}].$ (41) According to the commutation relations of Pauli operators, $\displaystyle[\tau_{i}^{x},\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}]$ $\displaystyle=[\tau_{i}^{x},\tau_{j}^{x}]\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}+\tau_{j}^{x}[\tau_{i}^{x},\tau_{j+\hat{x}}^{y}]\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}$ $\displaystyle+\tau_{j}^{x}\tau_{j+\hat{x}}^{y}[\tau_{i}^{x},\tau_{j+\hat{y}+\hat{x}}^{x}]\tau_{j+\hat{y}}^{y}+\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}[\tau_{i}^{x},\tau_{j+\hat{y}}^{y}]$ $\displaystyle=2i\delta_{i,j+\hat{x}}\tau_{j}^{x}\tau_{i}^{z}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}+2i\delta_{i,j+\hat{y}}\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{i}^{z};$ $\displaystyle[\tau_{i+\hat{x}}^{y},\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}]$ $\displaystyle=[\tau_{i+\hat{x}}^{y},\tau_{j}^{x}]\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}+\tau_{j}^{x}[\tau_{i+\hat{x}}^{y},\tau_{j+\hat{x}}^{y}]\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}$ $\displaystyle+\tau_{j}^{x}\tau_{j+\hat{x}}^{y}[\tau_{i+\hat{x}}^{y},\tau_{j+\hat{y}+\hat{x}}^{x}]\tau_{j+\hat{y}}^{y}+\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}[\tau_{i+\hat{x}}^{y},\tau_{j+\hat{y}}^{y}]$ $\displaystyle=-2i\delta_{i+\hat{x},j}\tau_{j}^{z}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}-2i\delta_{i,j+\hat{y}}\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{z}$ $\displaystyle\tau_{j+\hat{y}}^{y};$ (43) $\displaystyle[\tau_{i+\hat{x}+\hat{x}}^{x},\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}]$ $\displaystyle=[\tau_{i+\hat{x}+\hat{x}}^{x},\tau_{j}^{x}]\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}+\tau_{j}^{x}[\tau_{i+\hat{x}+\hat{x}}^{x},\tau_{j+\hat{x}}^{y}]\tau_{j+\hat{y}+\hat{x}}^{x}$ $\displaystyle\tau_{j+\hat{y}}^{y}+\tau_{j}^{x}\tau_{j+\hat{x}}^{y}[\tau_{i+\hat{x}+\hat{x}}^{x},\tau_{j+\hat{y}+\hat{x}}^{x}]\tau_{j+\hat{y}}^{y}+\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}$ $\displaystyle[\tau_{i+\hat{x}+\hat{x}}^{x},\tau_{j+\hat{y}}^{y}]$ $\displaystyle=2i\delta_{i+\hat{y},j}\tau_{j}^{x}\tau_{j+\hat{x}}^{z}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}+2i\delta_{i+\hat{x},j}\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}$ $\displaystyle\tau_{j+\hat{y}}^{z};$ (44) $\displaystyle[\tau_{i+\hat{y}}^{y},\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}]$ $\displaystyle=[\tau_{i+\hat{y}}^{y},\tau_{j}^{x}]\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}+\tau_{j}^{x}[\tau_{i+\hat{y}}^{y},\tau_{j+\hat{x}}^{y}]\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}$ $\displaystyle+\tau_{j}^{x}\tau_{j+\hat{x}}^{y}[\tau_{i+\hat{y}}^{y},\tau_{j+\hat{y}+\hat{x}}^{x}]\tau_{j+\hat{y}}^{y}+\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}[\tau_{i+\hat{y}}^{y},\tau_{j+\hat{y}}^{y}]$ $\displaystyle=-2i\delta_{i+\hat{y},j}\tau_{j}^{z}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}-2i\delta_{i,j+\hat{x}}\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{z}$ $\displaystyle\tau_{j+\hat{y}}^{y},$ (45) we can calculate (Appendix) in following four cases: 1. 1. when $i=j+\hat{x}$, $\displaystyle[F_{i},F_{j}]$ $\displaystyle=2i\tau_{j}^{x}\tau_{j+\hat{x}}^{z}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}\tau_{i+\hat{x}}^{y}\tau_{i+\hat{y}+\hat{x}}^{x}\tau_{i+\hat{y}}^{y}-2i\tau_{i}^{x}\tau_{i+\hat{x}}^{y}\tau_{i+\hat{y}+\hat{x}}^{x}$ $\displaystyle\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{z}\tau_{j+\hat{y}}^{y}$ $\displaystyle=-2\tau_{j}^{x}\tau_{j+\hat{x}}^{z}\tau_{j+\hat{y}+\hat{x}}^{z}\tau_{j+\hat{y}}^{y}\tau_{i+\hat{x}}^{y}\tau_{i+\hat{y}+\hat{x}}^{x}+2\tau_{i+\hat{x}}^{y}\tau_{i+\hat{y}+\hat{x}}^{x}\tau_{j}^{x}\tau_{j+\hat{x}}^{z}$ $\displaystyle\tau_{j+\hat{y}+\hat{x}}^{z}\tau_{j+\hat{y}}^{y}$ $\displaystyle=0;$ (46) 2. 2. when $i=j+\hat{y}$, $\displaystyle[F_{i},F_{j}]$ $\displaystyle=2i\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{z}\tau_{i+\hat{x}}^{y}\tau_{i+\hat{y}+\hat{x}}^{x}\tau_{i+\hat{y}}^{y}-2i\tau_{i}^{x}\tau_{j}^{x}\tau_{j+\hat{x}}^{y}$ $\displaystyle\tau_{j+\hat{y}+\hat{x}}^{z}\tau_{j+\hat{y}}^{y}\tau_{i+\hat{y}+\hat{x}}^{x}\tau_{i+\hat{y}}^{y}$ $\displaystyle=-2\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{z}\tau_{j+\hat{y}}^{z}\tau_{i+\hat{y}+\hat{x}}^{x}\tau_{i+\hat{y}}^{y}+2\tau_{j+\hat{y}}^{z}\tau_{j}^{x}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{z}$ $\displaystyle\tau_{i+\hat{y}+\hat{x}}^{x}\tau_{i+\hat{y}}^{y}$ $\displaystyle=0;$ (47) 3. 3. when $i+\hat{x}=j$ $\displaystyle[F_{i},F_{j}]$ $\displaystyle=-2i\tau_{i}^{x}\tau_{j}^{z}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}\tau_{i+\hat{y}+\hat{x}}^{x}\tau_{i+\hat{y}}^{y}+2i\tau_{i}^{x}\tau_{i+\hat{x}}^{y}\tau_{j}^{x}\tau_{j+\hat{x}}^{y}$ $\displaystyle\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{i+\hat{y}}^{z}\tau_{i+\hat{y}}^{y}$ $\displaystyle=-2\tau_{i}^{x}\tau_{j}^{z}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{i+\hat{y}+\hat{x}}^{z}\tau_{i+\hat{y}}^{y}+2\tau_{i}^{x}\tau_{i+\hat{x}}^{z}\tau_{j+\hat{x}}^{y}\tau_{j+\hat{y}+\hat{x}}^{x}$ $\displaystyle\tau_{j+\hat{y}}^{z}\tau_{i+\hat{y}}^{y}$ $\displaystyle=0;$ (48) 4. 4. when $i+\hat{y}=j$ $\displaystyle[F_{i},F_{j}]$ $\displaystyle=2i\tau_{i}^{x}\tau_{i+\hat{x}}^{y}\tau_{j}^{x}\tau_{j+\hat{x}}^{z}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}\tau_{i+\hat{y}}^{y}-2i\tau_{i}^{x}\tau_{i+\hat{x}}^{y}\tau_{i+\hat{y}+\hat{x}}^{x}\tau_{j}^{z}$ $\displaystyle\tau_{j+\hat{x}}^{y}\tau_{i+\hat{y}+\hat{x}}^{x}\tau_{i+\hat{y}}^{y}$ $\displaystyle=-2\tau_{i}^{x}\tau_{i+\hat{x}}^{y}\tau_{j}^{z}\tau_{j+\hat{x}}^{z}\tau_{j+\hat{y}+\hat{x}}^{x}\tau_{j+\hat{y}}^{y}+2\tau_{i}^{x}\tau_{i+\hat{x}}^{y}\tau_{i+\hat{y}+\hat{x}}^{z}\tau_{j}^{z}\tau_{j+\hat{y}+\hat{x}}^{x}$ $\displaystyle\tau_{i+\hat{y}}^{y}$ $\displaystyle=0.$ (49) It is clear that we also have $[F_{i},F_{j}]=0$ in other cases. Then, we only need to verify $[F_{i},\tau_{j}^{x}]=2F_{i}\tau_{j}^{x}(\delta_{i,j-\hat{x}}+\delta_{i,j-\hat{y}})$, while others can be easily obtained from the commutation relations of Pauli operators. The verification is shown in the following: $\displaystyle[F_{i},\tau_{j}^{x}]$ $\displaystyle=-2i\delta_{j,i+\hat{x}}\tau_{i}^{x}\tau_{i+\hat{x}}^{z}\tau_{i+\hat{y}+\hat{x}}^{x}\tau_{i+\hat{y}}^{y}-2i\delta_{j,i+\hat{y}}\tau_{i}^{x}\tau_{i+\hat{x}}^{y}\tau_{i+\hat{y}+\hat{x}}^{x}\tau_{i+\hat{y}}^{z}$ $\displaystyle=-2\delta_{j,i+\hat{x}}\tau_{i}^{x}\tau_{i+\hat{x}}^{x}\tau_{i+\hat{x}}^{y}\tau_{i+\hat{y}+\hat{x}}^{x}\tau_{i+\hat{y}}^{y}-2\delta_{j,i+\hat{y}}\tau_{i}^{x}\tau_{i+\hat{x}}^{y}\tau_{i+\hat{y}+\hat{x}}^{x}$ $\displaystyle\tau_{i+\hat{y}}^{x}\tau_{i+\hat{y}}^{y}$ $\displaystyle=-2\delta_{j,i+\hat{x}}\tau_{j}^{x}F_{i}-2\delta_{j,i+\hat{y}}\tau_{j}^{x}F_{i}$ $\displaystyle=2F_{i}\tau_{j}^{x}(\delta_{j,i+\hat{x}}+\delta_{j,i+\hat{y}}).$ (50) ## References * (1) A. Polkovnikov, K. 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Zurek, Phys. Rev. A 75, 052321 (2007). * (28) L. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory, 2nd Ed. (Pergamon Press, Oxford,1965); C. Zener, Proc. Roy. Soc. Lond. A 137, 696 (1932). * (29) B. Damski, Phys. Rev. Lett. 95, 035701 (2005). * (30) B. Damski and W. H. Zurek, Phys. Rev. A 73, 063405 (2006). * (31) E. T. Whittaker, and G. N. Watson, A Course of Modern Analysis, (Cambridge University Press, Cambridge, England, 1958). * (32) The expression (15) is also calculated by this whole procedure. * (33) It should be noticed that $Z_{2}$ charges and $Z_{2}$ vortices are on longer the elementary excitations or quasipartices of the post-quench system. * (34) A. Rahmani, and C. Chamon, Phys. Rev. B 82. 134303 (2010). * (35) L.-M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 91, 090402 (2003). * (36) A. Micheli, G. K. Brennen, P. Zoller, Nature Physics, 2, 341 (2006).	arxiv-papers	2010-11-21T15:16:59	2024-09-04T02:49:15.118655	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Long Zhang, Su-Peng Kou, Youjin Deng", "submitter": "Long Zhang", "url": "https://arxiv.org/abs/1011.4667" }
1011.4679	# Top quark pair production at the Tevatron Purdue University E-mail on behalf of the CDF and D0 collaborations ###### Abstract: The top quark has been discovered in 1995 by the CDF and D0 experiments located at the two beam-crossing points at the Tevatron $p\bar{p}$ collider. The top quark is the most massive of the known elementary particles. At hadron-hadron colliders, top quarks are mostly produced in pairs through strong interactions. The measurement of the cross section for top quark pair production ($\sigma_{t\bar{t}}$) is a test of our understanding of QCD prediction at next-to-leading order (NLO). This measurement requires a deep understanding of the backgrounds to top quark production, thus setting the ground for the measurements of top quark intrinsic properties. Finally, measuring $\sigma_{t\bar{t}}$ allows to set limits on new physics models that predict new particles coupling to the most recently discovered quark. The top quark discovery needed only few tens of pb-1. The Tevatron integrated luminosity is now about 9 fb-1, more than two orders of magnitude larger. In this document I will present some of the latest measurements of total and differential cross sections for top quark pair production performed by the CDF and D0 collaborations, that analyze up to about 6 fb-1 to provide a refined understanding of this extremely interesting particle. The identification of hadronic jets with substructure is thought to be a crucial tool towards the search for new phenomena at the Large Hadron Collider (LHC) proton-proton collisions at $\geq 7$ TeV using the CMS and ATLAS detectors. The first search for highly boosted top quarks has been performed by CDF and is presented here. ## 1 Introduction The top quark has been intriguing physicists ever since its discovery in 1995 at the CDF and D0 experiments [1]. In fact, its mass was found to be surprisingly large, some thirty times larger than the one of the bottom quark. Having a Yukawa coupling very close to one, speculation arose on whether the top quark might have a special role in the electroweak symmetry breaking mechanism. The Standard Model (SM) predicts that it is produced mostly in pairs at hadron colliders through QCD processes, with its cross section at the Tevatron being calculated up to the (N)NLO [2]. Theoretical and experimental cross sections are quoted in this paper for an assumed top quark pole mass of 172.5 GeV/c2, compatible with its current best experimental determination [3]. Several authors agree on the cross section at $p\bar{p}$ collisions at 1.96 TeV to be $\sim 7.5$ pb. With an integrated luminosity soon reaching 10 fb-1, the number of top pair events produced in the two collisions points where CDF and D0 are located is about 150 000. The top quark also has a ”special” phenomenology in that it is the sole quark that decays before hadronizing. It decays almost 100% of the times into $Wb$, with subsequent $W$ decays to leptons or quarks. Top quark pair decay modes are labeled as lepton+jets, dileptonic, or all hadronic depending on the combination of hadronic or leptonic decays of the two $W$ bosons. The golden mode for top quarks is the lepton+jets signature, where the charged lepton is an electron or muon; the dilepton and all-hadronic channels provide unique challenges but also complementary information. Summing up the event selection acceptance in the three signatures, the CDF and D0 experiments analyze about 9% of the produced top quark pair events. Leptonic tau decays are identified through their electron or muon decays. Events with hadronically decaying taus are collected by CDF and D0 with different strategies: D0 adopts explicit tau identification to suppress the otherwise large QCD backgrounds where jets mimic both the tau and the neutrino signature, while CDF uses the $\not$$E_{T}$ plus $b$-jets signature (vetoing identified electrons and muons). This brings the total acceptance up to about 13%, i.e. the ability to analyze $\sim$1000 top pair events per integrated fb-1. ## 2 Measurement of total cross sections The most precise measurements come from the lepton+jets signature: both CDF and D0 require the presence of a high $P_{T}$ electron or muon, large missing transverse energy ($\not$$E_{T}$) and at least three high $P_{T}$ jets. The main background is $W$ plus jets production, where its cross section is poorly known. CDF and D0 [4] use machine-learning techniques using as inputs several kinematical and topological distributions to discriminate between the top quark signal and the $W$ plus jets background. A maximum likelihood fit of the discriminant output distribution allows the extraction of the number of top quark events, and thus of $\sigma_{t\bar{t}}$. An alternative analysis uses $b$-jet identification algorithms ($b$-tagging) to achieve large signal-to- background ratio (S/B), thus enabling to perform a simpler counting experiment to measure $\sigma_{t\bar{t}}$. The event yields as a function of the number of $b$-tagged jets can be seen in Fig. 1 for the D0 analysis. CDF measures the ratio of $t\bar{t}$ to $Z\to\ell\ell$, thus effectively trading the dominant luminosity uncertainty for the smaller theoretical uncertainty on the $Z$ production cross section. With a precision below 7%, this is the most precise determination of $\sigma_{t\bar{t}}$ as of today. A different CDF analysis uses the very loose signature of $\geq 1$jet, $\not$$E_{T}$and one electron/muon to measure simultaneously the top quark pair signal and the backgrounds contribution [5]. CDF and D0 measure $\sigma_{t\bar{t}}$ with lower precision in the dilepton [6] and all-hadronic sample [7], where the cause for the former is the limited statistics, and for the latter the poor knowledge of the QCD background. SM extensions such as supersymmetry (SUSY) predicts the presence of charged Higgs bosons, in addition to the SM neutral one. In this scenario the $t\to H^{-}b$ decay can compete with the SM decay. Depending on the model parameters, the $H^{\pm}\to\tau^{\pm}\nu$ or $H^{\pm}\to c\bar{s}$ decay dominates, thus altering the lepton ratio in top pair decays. D0 measures this ratio for a Higgs mass range $M_{W}<M_{H}^{\pm}<M_{top}$, and excludes $BR(t\to Hb)$ of 0.1(0.45) at 95% confidence level (C.L.) for leptonic (hadronic) decays [8]. Top quark pair production is a large background to the low mass SM Higgs, SUSY and technicolor signatures, in the $\not$$E_{T}$plus two $b$-jets signature. CDF uses neural networks first to suppress the dominant QCD background in the sample, then to isolate the signal and measure $\sigma_{t\bar{t}}$ [9]. All the CDF and D0 measurements of total $t\bar{t}$ cross sections are in good agreement with the SM expectations. The two collaborations plan to combine these measurements to further increase precision. A further stringent test of QCD consists in measuring the $t\bar{t}+1$ jet cross section. This is important also as at the LHC top quark pairs are produced mostly together with extra jet radiation. CDF measures simultaneously $\sigma_{t\bar{t}+0\mathrm{jet}}$ and $\sigma_{t\bar{t}+1\mathrm{jet}}$ in the lepton+jet decay mode, the latter with a precision better than 15% [10]. Figure 1: The left plot shows the lepton plus jets sample composition, as a function of the number of $b$-tagged jets. The right plot shows the distribution of the two leading jet mass in events selected with one jets with $P_{T}$¿400 GeV/c. Events with two top quarks decaying hadronically would appear in the boxed region with both jets with mass $\sim M_{top}$. ## 3 Measurement of differential cross sections The measurement of the differential cross section is potentially sensitive to broad enhancements of the spectrum and interference effects. It has also been suggested that the measurement of $d\sigma_{t\bar{t}}/dm(t\bar{t})$ would allow a measurement of the top quark pole mass $M_{top}$, without the ambiguity of the definition of the top quark mass intrinsic in the Monte Carlo generators. The cross section $d\sigma_{t\bar{t}}/d(X)$ where $X$ is either $m(t\bar{t})$ or $P_{T}^{top}$ has been computed at NLO in perturbative QCD. CDF measured $d\sigma_{t\bar{t}}/dm(t\bar{t})$ while D0 measured $d\sigma_{t\bar{t}}/dP_{T}^{top}$, finding good agreement with the SM predictions [11]. ## 4 Search for boosted top quarks Top quarks decaying hadronically would appear as a jet with substructure when the quark is very highly boosted. At the Tevatron, this condition is satisfied when the top quark $P_{T}$ is $>400$ GeV/c, with $\sigma_{t\bar{t}}^{\mathrm{boost}}$ of the order of few fb. Unfortunately, QCD dijet production still dominates the sample. CDF studied very high $P_{T}$ jets and isolates the region where at least one top would appear as a jet with mass $\simeq M_{top}$ (see Fig. 1) and sets limits on $\sigma_{t\bar{t}}^{\mathrm{boost}}<55$ fb at 95% C.L. ## 5 Conclusions Top quark physics is a crucial part of the Tevatron program. The measurement of the pair production cross section allows the understanding of the sample composition, fundamental to perform top properties measurements such as its mass, spin, charge, and more. It also allows precision tests of pQCD, and establishes the $t\bar{t}$ background to both SM and new physics searches. The first measurements of differential $t\bar{t}$ cross sections have been performed with 1 fb-1 of data. Searching boosted top quarks allows studies of jets substructure and the development of tools for searching the Higgs boson and new physics at the LHC. So far, the study of top quark production and decay confirms the SM nature of the top quark. The additional collisions that could be obtained with the proposed Tevatron run extension, and the ones already coming from the LHC collider, will provide by 2014 to the high energy physics community a sample of $O(10^{6})$ top quarks, thus further expanding our understanding of the last discovered quark. ## References * [1] F. Abe et al. [CDF Collaboration], Phys. Rev. Lett. 74 2626; S. Abachi et al. [D0 Collaboration], Phys. Rev. Lett. 74, 2632. * [2] V. Ahrens, A. Ferroglia, M. Neubert, B. D. Pecjak and L. L. Yang, JHEP 1009 097; M. Cacciari, S. Frixione, M. L. Mangano, P. Nason and G. Ridolfi, JHEP 0809 127; N. Kidonakis and R. Vogt, Phys. Rev. D 78, 074005; S. Moch and P. Uwer, Nucl. Phys. Proc. Suppl. 183, 75. * [3] TEVWG and CDF and D0 Collaborations, arXiv:1007.3178. * [4] T. Aaltonen et al. [CDF Collaboration], Phys. Rev. Lett. 105 012001; A. Abazov et al. [D0 Collaboration], D0 Conference Note 6037. * [5] T. Aaltonen et al. [CDF Collaboration], CDF Conference Note 10137. * [6] T. Aaltonen et al. [CDF Collaboration], CDF Conference Note 10163; A. Abazov et al. [D0 Collaboration], D0 Conference 6038. * [7] T. Aaltonen et al. [CDF Collaboration], Phys. Rev. D 81, 052011; V. M. Abazov et al. [D0 Collaboration], Phys. Rev. D 82, 032002. * [8] V. M. Abazov et al. [D0 Collaboration], Phys. Rev. D 80, 071102. * [9] T. Aaltonen et al. [CDF Collaboration], CDF Conference Note 10237. * [10] T. Aaltonen et al. [CDF Collaboration], CDF Conference Note 9850. * [11] T. Aaltonen et al. [CDF Collaboration], Phys. Rev. Lett. 102, 222003; V. M. Abazov et al. [D0 Collaboration], Phys. Lett. B 693, 515. * [12] T. Aaltonen et al. [CDF Collaboration], CDF Conference Note 10234.	arxiv-papers	2010-11-21T16:52:02	2024-09-04T02:49:15.125873	{ "license": "Public Domain", "authors": "Fabrizio Margaroli (for the CDF and D0 collaborations)", "submitter": "Fabrizio Margaroli", "url": "https://arxiv.org/abs/1011.4679" }
1011.4767	# Instantaneous interquark potential in generalized Landau gauge in SU(3) lattice QCD: a possible gauge for the quark potential model Takumi Iritani Department of Physics, Graduate School of Science, Kyoto University, Kitashirakawa-oiwake, Sakyo, Kyoto 606-8502, Japan Hideo Suganuma Department of Physics, Graduate School of Science, Kyoto University, Kitashirakawa-oiwake, Sakyo, Kyoto 606-8502, Japan ###### Abstract We investigate “instantaneous interquark potential”, an interesting gauge- dependent quantity defined from the spatial correlator $\langle\mathrm{Tr}[U_{4}^{\dagger}(s)U_{4}(s^{\prime})]\rangle$ of the temporal link-variable $U_{4}$, in detail in generalized Landau gauge using SU(3) quenched lattice QCD. While the instantaneous potential has no linear part in the Landau gauge, in the Coulomb gauge, it is expressed by the Coulomb plus linear potential, where the slope is 2-3 times larger than the physical string tension, and the lowest energy state is considered to be a gluon-chain state. Using the generalized Landau gauge, we find that the instantaneous potential can be continuously described between the Landau and the Coulomb gauges, and it approximately reproduces the physical interquark potential in a specific intermediate gauge, which we call “$\lambda_{C}$-gauge”. This $\lambda_{C}$-gauge is expected to provide a quark-potential-model picture, where dynamical gluons do not appear. We also investigate $T$-length terminated Polyakov-line correlator and its corresponding “finite-time potential” in generalized Landau gauge. ###### pacs: 12.38.Gc,14.70.Dj,12.39.Jh,12.39.Pn ## I Introduction Nowadays, quantum chromodynamics (QCD) is established as the fundamental theory of the strong interaction, and perturbative QCD gives a standard framework to describe high-energy reactions of hadrons. QCD is a nonabelian gauge theory constructed from quarks and gluons, and color SU(3) gauge symmetry is one of the guiding principles in formulating QCD Nambu66 ; GWP73 . For actual perturbative calculations of QCD, the gauge has to be fixed to remove gauge degrees of freedom. In the low-energy region, however, QCD exhibits a strong-coupling nature and the resulting nonperturbative QCD is a very difficult and complicated theory. On the other hand, the quark potential model has been phenomenologically used for the description of low-energy properties of hadrons in terms of their underlying structure. The quark potential model is a successful nonrelativistic or semi-relativistic framework with a potential instantaneously acting among quarks, and describes many hadron properties in terms of quark degrees of freedom. In this model, there are no dynamical gluons, and gluonic effects indirectly appear as the instantaneous interquark potential. (Here, the concept of “instantaneous” is to be considered at the scale of the effective model.) In spite of a great success of the quark potential model, its relation to QCD is not yet so clear, and to link the quark potential model from QCD is one of the important subjects in hadron physics Isgur8586 ; BPSV0005 ; TS0304 . In principle, the quark model may be obtained from QCD after integrating out the gluon degrees of freedom in the path integral formalism. Or, from the viewpoint of “gauge” in QCD, the quark model without dynamical gluons may be regarded as a gauge-fixed effective theory of QCD. In fact, the quark potential model does not have local color SU(3) symmetry but has global color SU(3) symmetry Isgur8586 ; HanNambu , since each quark has a color and dynamical gluons are absent in this model framework. If so, what gauge of QCD corresponds to the quark model? Since the quark model has global color SU(3) symmetry and spatial-rotation symmetry, one may first consider that the Landau gauge and the Coulomb gauge would be candidates of such a gauge for the quark model. Actually, the Coulomb gauge is defined by minimizing “spatial gluon-field fluctuations” in total, as will be shown in Sec.II, and therefore one may expect only a small gluon-field fluctuation appearing in the Coulomb gauge. Similarly, the Landau gauge is defined by minimizing “total gluon-field fluctuations” in Euclidean QCD, so that only a small gluon-field fluctuation may be expected in the Landau gauge. Such a minimal gluonic content in these gauges seems to be preferable for the modeling only with quarks. In this paper, we investigate the instantaneous interquark potential in the Landau, the Coulomb, and their intermediate gauges, i.e., “generalized Landau gauge” (or “$\lambda$-gauge”), in SU(3) lattice QCD, from the viewpoint of the quark potential model Iritani10 . Here, the generalized Landau gauge is a natural general gauge to connect the Landau, the Coulomb, and the temporal gauges, by one real parameter $\lambda$. Besides the quark-model arguments, it is meaningful to investigate the connection between the Landau and the Coulomb gauges, using the generalized Landau gauge. Actually, these gauges have been often used as the typical gauge in QCD, but their physical pictures seem to be rather different for several important arguments in QCD. As the typical example, the color confinement, which is an important gauge- invariant QCD phenomenon, can be explained from various viewpoints in various gauges. In the Landau gauge, the color confinement is mathematically investigated by the Kugo-Ojima criterion, in terms of the BRST charge and the inverse Higgs theorem KugoOjima . In the Coulomb gauge, the color confinement is argued from the viewpoint of a large instantaneous Coulomb energy Gribov ; Zwanziger98 ; Greensite03 ; Greensite04 , and its resulting gluon-chain picture GreensiteThorn ; tHooft . Taking the maximally Abelian (MA) gauge, the quark confinement has been discussed in terms of the dual superconductor picture NambutHooftMandelstam . Of course, in gauge theories, the physical quantities never depend on the gauge choice. However, according to the gauge choice, the physical picture can be changed, and the role of the gauge field, which is fundamental field of the gauge theory, can be also changed. Then, it is important to link the different gauges, and investigate the role of gluons in each gauge. The role and the properties of gluons are expected to be clarified by the overview on the structure of gauge dependence. One of the aim in this paper is to investigate the gluonic properties through a continuous view from the Landau gauge to the Coulomb gauge, using generalized Landau gauge. In particular, we clarify the behavior of the instantaneous potential, as an interesting gluonic correlation. The organization of this paper is as follows. In Sec.II, we briefly review the properties of the Landau gauge and the Coulomb gauge. In Sec.III, we give the formalism of generalized Landau gauge ($\lambda$ gauge). In Sec.IV, we formulate the instantaneous potential in lattice QCD. In Sec.V, we show the lattice QCD results. In Sec.VI, we investigate Polyakov-line correlators and its relation to the potential. Sec.VII will be devoted to Summary and Discussions. ## II Landau gauge and Coulomb gauge In this section, we briefly review the properties of the Landau gauge and the Coulomb gauge. ### II.1 Landau gauge The Landau gauge is one of the most popular gauges in QCD, and its gauge fixing is given by $\partial_{\mu}A_{\mu}=0,$ (1) where $A_{\mu}(x)\equiv A_{\mu}^{a}(x)T^{a}\in\mathfrak{su}(N_{c})$ are gluon fields, with $\mathfrak{su}(N_{c})$ generator $T^{a}(a=1,2,\dots N_{c}^{2}-1)$. The Landau gauge keeps the Lorentz covariance and the global SU($N_{c}$) color symmetry. These symmetries simplify the tensor structure of various quantities in QCD. For example, the gluon propagator $D_{\mu\nu}^{ab}(p)$ is simply expressed as $D_{\mu\nu}^{ab}(p)=D(p^{2})\delta^{ab}\left(g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right),$ (2) due to the symmetries and the transverse property. Owing to this advanced feature, the Landau gauge is often used both in the Schwinger-Dyson formalism HigashimaMiransky ; Alkofer01 and in lattice QCD studies for quarks and gluons Mandula99 ; Iritani09 . In Euclidean QCD, the Landau gauge has a global definition to minimize the global quantity, $R_{\rm L}=\int d^{4}x\mathrm{Tr}\left\\{A_{\mu}(x)A_{\mu}(x)\right\\}=\frac{1}{2}\int d^{4}xA_{\mu}^{a}(x)A_{\mu}^{a}(x),$ (3) by the gauge transformation. This global definition is more strict, and the local form in Eq.(1) can be obtained from the minimization of $R_{\rm L}$. Since the quantity $R_{\rm L}$ physically means the total amount of gauge- field fluctuations, and therefore the Landau gauge maximally suppresses artificial gauge-field fluctuations originated from the gauge degrees of freedom. Here, we comment on non-locality of the gauge fields. Through the gauge fixing procedure, gauge fields have non-locality stemming from the Faddeev-Popov determinant. In the Landau gauge, this non-locality of gauge fields is Lorentz covariant. Using the Landau gauge, or a covariant and globally symmetric gauge, the color confinement has been mathematically investigated in terms of the BRST charge and the inverse Higgs theorem, which is known as the “Kugo-Ojima criterion” KugoOjima . ### II.2 Coulomb gauge The Coulomb gauge is also a popular gauge in QCD, and its gauge fixing is given by $\partial_{i}A_{i}=0.$ (4) This condition resembles the Landau gauge condition (Eq.(1)), but there are no constraints on $A_{0}$. In the Coulomb gauge, the Lorentz covariance is partially broken, and gauge field components are completely decoupled into two parts, $\vec{A}$ and $A_{0}$: $\vec{A}$ behave as canonical variables and $A_{0}$ becomes an instantaneous potential. Similarly in the Landau gauge, the Coulomb gauge has a global definition to minimize the global quantity $R_{\rm Coul}\equiv\int d^{4}x\mathrm{Tr}\left\\{A_{i}(x)A_{i}(x)\right\\}=\frac{1}{2}\int d^{4}xA_{i}^{a}(x)A_{i}^{a}(x)$ (5) by the gauge transformation. Note here that the Euclidean metric is not necessary for the global definition of the Coulomb gauge. Note also that there appears no nonlocality in the temporal direction in the Coulomb gauge. Due to this feature, a hadron mass measurement can be safely performed using a spatially-extended quark source in the Coulomb gauge in lattice QCD calculations CGsource . In the Coulomb gauge, one of the advantages is the compatibility with the canonical quantization ItzyksonZuber . The QCD Hamiltonian is expressed as $\displaystyle H$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int d^{3}x\left(\vec{E}^{a}\cdot\vec{E}^{a}+\vec{B}^{a}\cdot\vec{B}^{a}\right)$ (6) $\displaystyle+\frac{1}{2}\int d^{3}xd^{3}y\rho^{a}(x)K^{ab}(x,y)\rho^{b}(y),$ where $\rho^{a}$ is the color charge density, $\vec{E}^{a}$ and $\vec{B}^{a}$ are the color electric and magnetic field, respectively. Here, $K^{ab}(x,y)$ is the instantaneous Coulomb propagator Greensite03 defined as $K^{ab}(x,y)=[M^{-1}(-\nabla^{2})M^{-1}]_{xy}^{ab},$ (7) with the Faddeev-Popov operator $M^{ac}=-\partial^{2}\delta^{ac}-\varepsilon^{abc}A_{i}^{b}\partial_{i}.$ (8) The confinement picture in the Coulomb gauge focuses on the Coulomb energy including the inverse of $M$. Here, the Coulomb energy is the non-local second term of the QCD Hamiltonian (6), and is regarded as the instantaneous potential. Near the Gribov horizon, where the Faddeev-Popov operator $M$ has zero eigenvalues Gribov , the Coulomb energy at large quark distance is expected to be largely enhanced and leads to a confining interquark potential, which is called as the “Gribov-Zwanziger scenario” Gribov ; Zwanziger98 . As Zwanziger showed, the Coulomb energy (instantaneous potential) $V_{\rm Coul}(R)$ in the Coulomb gauge gives an upper bound on the static interquark potential $V_{\rm phys}(R)$ Zwanziger03 , i.e., $V_{\rm phys}(R)\leq V_{\rm Coul}(R).$ (9) This inequality indicates that if the physical interquark potential is confining then the Coulomb energy $V_{\rm Coul}$ is also confining. Actually, lattice QCD calculations Greensite03 show that the Coulomb energy (the instantaneous potential) between a quark and an antiquark leads to a linear potential, which characterizes the confinement. However, the slope of the instantaneous potential is too large, i.e., $2\sim 3$ times larger than the physical string tension, and this Coulomb system turns out to be an excited state. As for the ground-state of the quark-antiquark system, Thorn and Greensite proposed the “gluon-chain picture” in the Coulomb gauge GreensiteThorn . In fact, to screen the large Coulomb energy between the quark and the antiquark, chain-like gluons are dynamically generated between them. This gluon-chain is expected to give the linear potential between quarks. In other words, QCD string can be regarded as a “chain” of gluons in the Coulomb gauge. ## III Generalized Landau gauge In this section, we investigate the “generalized Landau gauge”, or “$\lambda$ gauge” Bernard90 , which continuously connects the Landau and the Coulomb gauges. Since the Landau gauge and the Coulomb gauge are useful gauges and give different interesting pictures in QCD, it is meaningful to show the linkage of these gauges. To link these gauges, we generalize the gauge fixing condition (1) as $\partial_{i}A_{i}+\lambda\partial_{4}A_{4}=0,$ (10) by introducing one real parameter $\lambda$ Bernard90 . The case of $\lambda=1$ corresponds to the Landau gauge fixing condition, the Coulomb gauge is achieved at $\lambda=0$, and also temporal gauge for $\lambda\rightarrow\infty$. Therefore, we can analyze gauge dependence of various properties from the Landau gauge toward the Coulomb gauge by varying $\lambda$-parameter from 1 to 0. $\lambda$-gauge keeps the global SU($N_{c}$) color symmetry, but partially breaks the Lorentz symmetry like the Coulomb gauge, except for $\lambda$=1. In Euclidean QCD, the global definition of $\lambda$-gauge is expressed by the minimization of $R^{\lambda}\equiv\int d^{4}x\left[\mathrm{Tr}\left\\{A_{i}(x)A_{i}(x)\right\\}+\lambda\mathrm{Tr}\left\\{A_{4}(x)A_{4}(x)\right\\}\right]$ (11) by the gauge transformation. Here, the $\lambda$-parameter controls the ratio of the gauge-field fluctuations of $\vec{A}$ and $A_{4}$. Lattice QCD is formulated on the discretized Euclidean space-time, and the theory is described with the link-variable $U_{\mu}(x)\equiv e^{iagA_{\mu}(x)}\in{\rm SU}(N_{c})$, with the lattice spacing $a$ and the gauge coupling constant $g$, instead of gauge fields $A_{\mu}(x)\in\mathfrak{su}(N_{c})$. $\lambda$-gauge fixing condition is expressed in terms of the link-variable as the maximization of a quantity $R_{\rm latt}^{\lambda}[U]\equiv\sum_{x}\left\\{\sum_{i}\mathrm{Re}\mathrm{Tr}U_{i}(x)+\lambda\mathrm{Re}\mathrm{Tr}U_{4}(x)\right\\}$ (12) by the gauge transformation of the link-variables, $U_{\mu}(x)\rightarrow\Omega(x)U_{\mu}(x)\Omega^{\dagger}(x+\hat{\mu}),$ (13) with the gauge function $\Omega\in\mathrm{SU}(3)$. In the continuum limit of $a\rightarrow 0$, this condition results in the minimization of $R^{\lambda}$ in Eq.(11), and satisfies the local $\lambda$-gauge fixing condition of Eq.(10). Note here that the gluon-field fluctuation is strongly suppressed in the generalized Landau gauge with $\lambda\neq 0$, so that one can use the expansion of the link-variable $U_{\mu}(x)\equiv e^{iagA_{\mu}(x)}\simeq 1+iagA_{\mu}(x)+\mathcal{O}(a^{2})$ for small lattice spacing $a$, and the gluon field $A_{\mu}(x)$ can be defined by $A_{\mu}(x)\equiv\frac{1}{2iag}\left[U_{\mu}(x)-U_{\mu}^{\dagger}(x)\right]_{\rm traceless}\in\mathfrak{su}(N_{c})$ (14) without suffering from large gluon fluctuations stemming from the gauge degrees of freedom. ## IV Polyakov-line correlators and instantaneous potential In this section, we formulate Polyakov-line correlators and instantaneous potential in lattice QCD. First, we consider the Euclidean continuum theory to make clear the physical interpretation of the terminated Polyakov-line correlator. Considering the source field $J_{\mu}(x)$ which couples to the gauge field $A_{\mu}(x)$, the generating functional is given by $Z[J]\equiv\langle\ e^{i\int d^{4}x\mathcal{L}_{\rm int}(J)}\rangle=\langle\exp\\{ig\int d^{4}xA_{\mu}^{a}(x)J^{\mu}_{a}(x)\\}\rangle.$ (15) We consider a closed-loop current such as the Wilson loop $W(R,T)$, which is constructed in a gauge-invariant manner. From the relation $Z[J]\propto e^{-E_{J}T}$, the ground-state energy between static quark and antiquark pair is expressed as $V_{\rm phys}(R)=-\lim_{T\rightarrow\infty}\frac{1}{T}\ln\langle W(R,T)\rangle.$ (16) Next, we consider the color current $J_{\mu}(x)$ of a quark located at $\vec{x}=\vec{a}$ and an antiquark at $\vec{x}=\vec{b}$. For the case that these sources are generated at $t=0$ and annihilated at $t=T$, the color current $J_{\mu}(x)$ is expressed as $J_{\mu}(x)=\delta_{\mu 4}\left[\delta(\vec{x}-\vec{a})-\delta(\vec{x}-\vec{b})\right]\theta(T-t)\theta(t),$ (17) in the generalized Landau gauge. In this case, the current $J_{\mu}$ is not conserved, and it breaks the gauge invariance. In a similar manner to the Wilson loop $W(R,T)$, we define the “energy” of two sources in the presence of $J_{\mu}(x)$ as $V(R,T)=-\frac{1}{T}\ln\langle\mathrm{Tr}[L(\vec{a},T)L^{\dagger}(\vec{b},T)]\rangle$ (18) with $R=\|\vec{a}-\vec{b}\|$, using the terminated Polyakov-line $L(\vec{x},T)\equiv P\exp\left(ig\int_{0}^{T}dx_{4}A_{4}(\vec{x},x_{4})\right),$ (19) with $P$ being the path-ordered product. As a caution, for $\lambda\neq$ 0, $V(R,T)$ is gauge-dependent and does not mean the energy of a physical state, due to the temporal nonlocality of the Faddeev-Popov determinant. Next, we consider $N_{x}\times N_{y}\times N_{z}\times N_{t}$ lattice with the lattice spacing $a$. Using temporal link-variables, the terminated Polyakov- line with length $T$ is defined as $L(\vec{x},T)=U_{4}(\vec{x},a)U_{4}(\vec{x},2a)\cdots U_{4}(\vec{x},T).$ (20) The terminated Polyakov-line is generally gauge variant, and its expectation value depends on the choice of gauge. Only for $T=N_{t}$, the trace of the Polyakov-line coincides the Polyakov loop, which is gauge invariant. In the Coulomb gauge, the expectation value of the terminated Polyakov-line is zero due to the remnant symmetry. (See Appendix A.) In this paper, we consider the Polyakov-line correlator in generalized Landau gauge denoted by $G_{\lambda}(R,T)=\langle\mathrm{Tr}[L^{\dagger}(\vec{x},T)L(\vec{y},T)]\rangle$ (21) with $R=\|\vec{x}-\vec{y}\|$. From this correlator, we define “finite-time potential”, $V_{\lambda}(R,T)\equiv-\frac{1}{T}\ln G_{\lambda}(R,T).$ (22) Here, for the simple expression, a normalization factor 1/3 is dropped off for $G_{\lambda}(R,T)$, since it only gives a constant term in the potential $V_{\lambda}(R,T)$. Particularly for $T=a$, we call $V_{\lambda}(R)\equiv V_{\lambda}(R,a)=-\frac{1}{a}\ln\langle\mathrm{Tr}\big{[}U_{4}^{\dagger}(\vec{x},a)U_{4}(\vec{y},a)\big{]}\rangle$ (23) as “instantaneous potential”. Here, these quantities depend on $\lambda$-parameter. In the Coulomb gauge, the instantaneous potential $V_{\lambda=0}(R)$ (or the Coulomb energy) gives a linear potential, but its slope is about $2\sim 3$ times larger than the physical string tension Greensite03 . In the Landau gauge, the instantaneous potential $V_{\lambda=1}(R)$ has no linear part Nakamura06 , which is also expected from the exponential reduction of the gluon propagator Mandula99 ; Iritani09 and the Lorentz symmetry. In Sec.VII, we will discuss the relation between the gluon propagator and the instantaneous potential in the Landau gauge. ## V Lattice QCD results We perform SU(3) lattice QCD Monte Carlo calculations at the quenched level. We use the standard plaquette action with the lattice parameter $\beta\equiv\frac{2N_{c}}{g^{2}}=5.8$ on a $16^{4}$-size lattice. The lattice spacing $a$ is 0.152 fm, which is determined so as to reproduce the string tension as $\sqrt{\sigma}=427$ MeV STI . We use the gauge configurations, which are picked up every 1000 sweeps after a thermalization of 20000 sweeps. After the generation of gauge configurations, we perform gauge fixing by maximizing $R^{\lambda}_{\rm latt}[U]$. In this paper, we use the Landau gauge ($\lambda$=1), the Coulomb gauge ($\lambda$=0), and their intermediate gauges with $\lambda=0.75,0.50,0.25,0.10,0.05,0.04,0.03,0.02,0.01$. We investigate in detail the region near the Coulomb gauge ($\lambda$=0), since the behavior of the instantaneous potential largely changes for $\lambda\sim$0, as will be shown later. The number of gauge configurations is 50 for each $\lambda$. We adopt the jackknife method to estimate the statistical error. Here, we comment on $\lambda$-gauge fixing convergence. We fix the gauge by maximizing the quantity $R^{\lambda}_{\rm latt}[U]$ in Eq.(12), which corresponds to $\partial_{i}A_{i}+\lambda\partial_{4}A_{4}=0$. Therefore, to check the convergence of gauge fixing, we evaluate $\epsilon_{\lambda}$ defined by $\displaystyle\epsilon_{\lambda}$ $\displaystyle\equiv$ $\displaystyle\langle\left(\partial_{i}A_{i}^{a}+\lambda\partial_{4}A_{4}^{a}\right)^{2}\rangle$ (24) $\displaystyle\equiv$ $\displaystyle\frac{1}{(N_{c}^{2}-1){N_{\rm site}}}\sum_{x=1}^{N_{\rm site}}\sum_{a=1}^{N_{c}^{2}-1}\Big{\\{}\sum_{i=1}^{3}\left[A_{i}^{a}(x)-A_{i}^{a}(x-\hat{i})\right]$ $\displaystyle+$ $\displaystyle\lambda\left[A_{4}^{a}(x)-A_{4}^{a}(x-\hat{4})\right]\Big{\\}}^{2},$ with the gluon field $A_{\mu}(x)=A_{\mu}^{a}(x)T^{a}$ given in Eq.(14). We iterate the gauge transformation to satisfy $\epsilon_{\lambda}<{10}^{-12}$ finally. As for the instantaneous potential $V_{\lambda}(R)$, this convergence condition is very strict. Actually, we can obtain stable the lattice data of $V_{\lambda}(R)$ even with $\epsilon_{\lambda}<{10}^{-4}$. Also, we comment that the calculation cost of the gauge fixing is rapidly increasing as $\lambda$ approaches to zero, while the Coulomb gauge ($\lambda=0$) itself can be easily achieved. Considering this critical slowing down of the gauge fixing Bernard90 ; Cucchieri07 , we adopt a relatively small-size lattice of $16^{4}$ with $\beta=5.8$, although its physical volume of about $(2.4{\rm fm})^{4}$ is large enough to extract the relevant region for the interquark potential. ### V.1 “Instantaneous inter-quark potential” in generalized Landau gauge We investigate the instantaneous potential $V_{\lambda}(R)$ defined by Eq.(23) in generalized Landau gauge. Figure 1 shows lattice QCD results of $V_{\lambda}(R)$ for typical values of $\lambda$. In this figure, the statistic error is small and the error bars are hidden in the symbols. In the Coulomb gauge ($\lambda=0$), the instantaneous potential shows linear behavior, while there is no linear part at all in the Landau gauge ($\lambda=1$). Thus, there is a large gap between these gauges in terms of the instantaneous potential. In our framework, however, these two gauges are connected continuously. By varying the $\lambda$-parameter from 1 to 0 in the generalized Landau gauge, we find that the instantaneous potential $V_{\lambda}(R)$ changes continuously, and the infrared slope of the potential $V_{\lambda}(R)$ at $R\simeq 0.8{\rm fm}$ grows monotonically, from the Landau gauge to the Coulomb gauge, as shown in Fig. 1. Note here that the growing of the infrared slope of $V_{\lambda}(R)$ is quite rapid for $\lambda\lesssim 0.1$, near the Coulomb gauge, while the infrared slope is rather small and almost unchanged for $\lambda=0.1\sim 1$. Figure 1: “Instantaneous potential” $V_{\lambda}(R)$ in generalized Landau gauge for typical values of $\lambda$. The symbols denote lattice QCD results, and the curves fit-results using Coulomb plus linear form, $V_{\lambda}(R)=-A_{\lambda}/R+\sigma_{\lambda}R+C_{\lambda}$, in the region of $R\lesssim 0.8$ fm. For $\lambda=1\sim 0.1$, the potential has almost no linear part. For $\lambda\lesssim 0.1$, the linear potential grows rapidly, and $\sigma_{\lambda}\simeq 2.6\sigma_{\rm phys}$ at $\lambda=0$. To analyze the instantaneous potential $V_{\lambda}(R)$ quantitatively, we fit the lattice QCD results using the Coulomb plus linear Ansatz as $V_{\lambda}(R)=-\frac{A_{\lambda}}{R}+\sigma_{\lambda}R+C_{\lambda},$ (25) where $A_{\lambda}$ is the Coulomb coefficient and $C_{\lambda}$ a constant. Here, $\sigma_{\lambda}$ is the infrared slope of the potential, which we call as “instantaneous string tension”. Besides the Coulomb plus linear Ansatz, we try several candidates of the functional form, $-A/R+\sigma(1-e^{-\varepsilon R})/\varepsilon$, $-A\exp(-mR)/R$, $-A/R+\sigma R^{d}$, and $-A/R^{d}$ apart from an irrelevant constant, but they are less workable. The curves in Fig. 1 are the best-fit results using Eq.(25). The Coulomb plus linear Ansatz works well at least for $R\lesssim 0.8$fm, which is relevant region for hadron physics. We note that the Yukawa form of $-Ae^{-mR}/R$ also works well near the Landau gauge, which will be discussed in relation to the gluon-propagator behavior in Sec.VI-C. We summarize the best-fit parameters and the fit-range in Table 1. While the Coulomb coefficient $A_{\lambda}$ has a relatively weak $\lambda$-dependence, the instantaneous string tension $\sigma_{\lambda}$ shows a strong $\lambda$-dependence near the Coulomb gauge, i.e., $\lambda\lesssim 0.1$. We comment on the asymptotic value of the instantaneous potential $\sigma_{\lambda}$. In the deep IR limit, $R\rightarrow\infty$, $V_{\lambda}(R)$ goes to a saturated value, except for $\lambda=0$. This feature is closely related to the property of the temporal link-variable correlator $\langle\mathrm{Tr}\\{U_{4}^{\dagger}(x)U_{4}(y)\\}\rangle$, as will be discussed in Sec.VI. Table 1: Best-fit parameters on the instantaneous potential using $V_{\lambda}(R)=-A_{\lambda}/R+\sigma_{\lambda}R+C_{\lambda}$, and the ratio of the slope $\sigma_{\lambda}$ to the physical string tension $\sigma_{\rm phys}$. The standard parameters of the physical interquark potential are $A_{\rm phys}\simeq 0.27$ and $\sigma_{\rm phys}\simeq 0.89$GeV/fm STI . The string tension $\sigma_{\lambda}$ is rather small for $\lambda=0.1\sim 1$. $\lambda$ \| fit-range [fm] \| $A_{\lambda}$ \| $\sigma_{\lambda}$ [GeV/fm] \| $C_{\lambda}$ [GeV] \| $\sigma_{\lambda}/\sigma_{\rm phys}$ ---\|---\|---\|---\|---\|--- 0.00 \| 0.1-1.0 \| 0.167(11) \| 2.283(35) \| -0.881(20) \| 2.57(4) 0.01 \| 0.1-0.8 \| 0.287(27) \| 1.476(78) \| -0.617(46) \| 1.66(9) 0.02 \| 0.1-0.8 \| 0.346(32) \| 1.005(90) \| -0.481(54) \| 1.13(10) 0.03 \| 0.1-0.8 \| 0.372(32) \| 0.728(86) \| -0.416(53) \| 0.82(10) 0.04 \| 0.1-0.8 \| 0.382(30) \| 0.557(79) \| -0.386(50) \| 0.63(9) 0.05 \| 0.1-0.8 \| 0.386(29) \| 0.441(73) \| -0.372(47) \| 0.50(8) 0.10 \| 0.1-0.8 \| 0.365(20) \| 0.169(46) \| -0.390(31) \| 0.19(5) 0.25 \| 0.1-0.8 \| 0.281(6) \| -0.005(13) \| -0.544(9) \| -0.01(1) 0.50 \| 0.1-0.8 \| 0.198(0) \| -0.042(1) \| -0.724(1) \| -0.05(0) 0.75 \| 0.1-0.8 \| 0.152(1) \| -0.043(3) \| -0.839(2) \| -0.05(0) 1.00 \| 0.1-0.8 \| 0.123(2) \| -0.040(3) \| -0.917(3) \| -0.04(0) Now, we focus on the $\lambda$-dependence of instantaneous string tension $\sigma_{\lambda}$ in Fig. 2. For $0.1\lesssim\lambda\leq 1$, including the Landau gauge ($\lambda$=1), $\sigma_{\lambda}$ is almost zero, so that this region can be regarded as “Landau-like.” For $\lambda\lesssim 0.1$, $V_{\lambda}(R)$ is drastically changed near the Coulomb gauge, and $\sigma_{\lambda}$ grows rapidly in this small region. Finally, in the Coulomb gauge ($\lambda$=0), one finds $\sigma_{\lambda}\simeq 2.6\sigma_{\rm phys}$, with $\sigma_{\rm phys}\simeq 0.89$GeV/fm. Figure 2: Instantaneous string tension $\sigma_{\lambda}$, the slope of the linear part in the instantaneous potential $V_{\lambda}(R)$ in the generalized Landau gauge. The right upper figure is a close-up near the Coulomb gauge ($\lambda$=0). For $0\leq\lambda\lesssim 0.1$, $\sigma_{\lambda}$ changes rapidly from $2.6\sigma_{\rm phys}$ to 0, while $\sigma_{\lambda}$ is rather small for $0.1\lesssim\lambda\leq 1$. The instantaneous string tension $\sigma_{\lambda}$ coincides with the physical value around $\lambda_{C}\simeq 0.02$. Note that the instantaneous string tension $\sigma_{\lambda}$ continuously changes from $0$ to $(2\sim 3)\sigma_{\rm phys}$, according to the change from the Landau gauge to the Coulomb gauge, and therefore there exists some specific $\lambda$-parameter of $\lambda_{C}\in[0,1]$ where the slope of the instantaneous potential $V_{\lambda}(R)$ coincides with the physical string tension $\sigma_{\rm phys}$. Since the instantaneous potential generally depends on the lattice parameter $\beta$, i.e., the lattice spacing $a$ Greensite03 ; Iritani10 , the value of $\lambda_{C}$ is $\beta$-dependence, although its dependence would be rather weak, as will be discussed in Sec.VI. However, from the continuity between the overconfining potential in the Coulomb gauge and the saturated potential in the Landau gauge, there must exist $\lambda_{C}\in[0,1]$ where the instantaneous string tension $\sigma_{\lambda}$ coincides with $\sigma_{\rm phys}$ for each lattice spacing. We call this specific gauge as “$\lambda_{C}$-gauge”. From Fig.2, the value of $\lambda_{C}$ is estimated to be about 0.02 at $\beta$=5.8. Note here that $\lambda_{C}\simeq 0.02\ll 1$ is very small, and then the $\lambda_{C}$-gauge is close to the Coulomb gauge, which indicates the small temporal non-locality. Figure 3 shows the instantaneous potential $V_{\lambda}(R)$ at $\lambda=0.02\simeq\lambda_{C}$ and the physical static interquark potential $V_{\rm phys}(R)$. In this $\lambda_{C}$-gauge, $V_{\rm phys}(R)$ is found to be approximately reproduced by $V_{\lambda_{C}}(R)$ for $R\lesssim$ 0.8fm. While the physical static potential $V_{\rm phys}(R)$ is derived from the large-T behavior of the Wilson loop $W(R,T)$ Rothe as $V_{\rm phys}(R)=-\lim_{T\rightarrow\infty}\frac{1}{T}\ln\langle W(R,T)\rangle,$ (26) only instantaneous corerlation of the temporal link-variable $U_{4}$ approximately reproduces $V_{\rm phys}(R)$ in the $\lambda_{C}$-gauge, i.e., $\displaystyle V_{\rm phys}(R)$ $\displaystyle\simeq$ $\displaystyle V_{\lambda_{C}}(R)$ (27) $\displaystyle=$ $\displaystyle-\frac{1}{a}\ln\langle\mathrm{Tr}U_{4}^{\dagger}(\vec{x},a)U_{4}(\vec{y},a)\rangle_{\lambda_{C}},$ as is schematically illustrated in Fig.4. Figure 3: The comparison between the instantaneous potential $V_{\lambda}(R)$ at $\lambda=0.02(\simeq\lambda_{C})$ (black dots), and the physical interquark potential $V_{\rm phys}(R)$ (solid line). In $\lambda_{C}$-gauge, $V_{\rm phys}(R)=-\lim_{T\rightarrow\infty}\frac{1}{T}\ln\langle W(R,T)\rangle\simeq- A_{\rm phys}/R+\sigma_{\rm phys}R$ ($A_{\rm phys}\simeq$ 0.27, $\sigma_{\rm phys}\simeq$ 0.89GeV/fm) is approximately reproduced by the instantaneous potential $V_{\lambda_{C}}(R)=-\frac{1}{a}\ln\langle\mathrm{Tr}U_{4}^{\dagger}(\vec{x},a)U_{4}(\vec{y},a)\rangle_{\lambda_{C}}$. Figure 4: The schematic illustration of the physical interquark potential $V_{\rm phys}(R)$ and the instantaneous potential $V_{\rm inst}(R)$. In $\lambda_{C}$-gauge, the instantaneous potential $V_{\lambda_{C}}(R)$ approximately reproduces the physical potential $V_{\rm phys}(R)$. On the relation to the confinement, which is a gauge independent phenomenon, the role of gluons generally depends on the choice of gauges, and the physical picture of the confinement would be changed according to gauges. For example, in the Coulomb gauge, the instantaneous Coulomb energy gives an overconfining potential, and the ground-state of the quark-antiquark system is described as the gluon-chain state. On the other hand, in the Landau gauge, the instantaneous potential has no linear part, and the ghost behavior in the deep-infrared region would be more important for the confinement. In the $\lambda_{C}$-gauge, the physical interquark potential $V_{\rm phys}(R)$ is approximately reproduced by the instantaneous potential $V_{\lambda_{C}}(R)$. This physically means that all other complicated effects including dynamical gluons and ghosts are approximately cancelled in the $\lambda_{C}$-gauge, and therefore we do not need to introduce any redundant gluonic degrees of freedom. The absence of dynamical gluon degrees of freedom would be a desired property for the quark model picture. ### V.2 “Finite-time potential” and “finite-time string tension” In the previous subsection, we investigated the instantaneous potential $V_{\lambda}(R)$, which is defined by the Polyakov-line correlator with a minimum length on the lattice. For the quark-potential model, it is desired that the interquark potential does not have large dependence on the temporal length $T$ of the typical reaction scale. From this viewpoint, we investigate the “finite-time potential” $V_{\lambda}(R,T)$ defined by Eq.(18) in Sec.IV, and its temporal-length dependence. Here, $V_{\lambda}(R,T)$ is expressed by $T$-length terminated Polyakov-line $L(\vec{x},T)$ in Eq.(19), and a generalization of the instantaneous potential $V_{\lambda}(R)$. First, we consider the Coulomb gauge Greensite03 ; Iritani10 . Figure 5 shows the lattice QCD result for $V_{\lambda}(R,T)$ in the Coulomb gauge. Similar to the instantaneous potential, $V_{\lambda}(R,T)$ is well reproduced by the Coulomb plus linear form. However, the parameter values are changed according to $T$-length. In particular, the slope of the potential becomes smaller as $T$ becomes larger, which shows an “instability” of $V_{\lambda}(R,T)$ in terms of $T$ in the Coulomb gauge. For general $\lambda$, finite-time potential $V_{\lambda}(R,T)$ is found to be reproduced by the Coulomb plus linear form as $V_{\lambda}(R,T)=-\frac{A_{\lambda}(T)}{R}+\sigma_{\lambda}(T)R+C_{\lambda}(T),$ (28) at least for $R\lesssim 0.8$fm, similarly for the instantaneous potential. Figure 5: “Finite-time potential” $V_{\lambda}(R,T)$ in the Coulomb gauge ($\lambda=0$) for $T$=1,2,3,4,5. Here, for the comparison, an irrelevant constant is shifted for each $T$. The curves denote the fit-results using the Coulomb plus linear form. The slope of $V_{\lambda}(R,T)$ is clearly changed according to $T$. Since our main interest is linear part of the potential, we focus on “finite- time string tension” $\sigma_{\lambda}(T)$, the slope of $V_{\lambda}(R,T)$. Figure 6 shows $\sigma_{\lambda}(T)$ in generalized Landau gauge for typical values of $\lambda$. In Table 2, we summarize the best-fit parameters of $\sigma_{\lambda}(T)$ at $T=1,2,\dots,6$, and the ratio of $\sigma_{\lambda}(1)/\sigma_{\lambda}(6)$, $\sigma_{\lambda}(1)/\sigma_{\rm phys}$, and $\sigma_{\lambda}(6)/\sigma_{\rm phys}$, respectively. Figure 6: $T$-length dependence of “Finite-time string tension” $\sigma_{\lambda}(T)$, the infrared slope of finite-time potential $V_{\lambda}(R,T)$, in generalized Landau gauge for several typical $\lambda$-values. Near the Coulomb gauge, e.g., for $\lambda\lesssim 0.03$, $\sigma_{\lambda}(T)$ goes to the same value for large $T\sim$ 1fm. For $\lambda\gtrsim 0.1$, $\sigma_{\lambda}(T)$ is an increasing function of $T$. In fact, even though the instantaneous potential has no linear part, the linear part of $V_{\lambda}(R,T)$ appears and gradually grows, as the Polyakov-line grows. Table 2: Finite-time string tension $\sigma_{\lambda}(T)$ in generalized Landau gauge for $T=1,2,\dots,6$, together with the ratio, $\sigma_{\lambda}(1)/\sigma_{\lambda}(6)$, $\sigma_{\lambda}(1)/\sigma_{\rm phys}$, and $\sigma_{\lambda}(6)/\sigma_{\rm phys}$. The fit-range is the same as that listed in Table 1. \| $\sigma_{\lambda}(T)$ [GeV/fm] \| \| \| ---\|---\|---\|---\|--- $\lambda$ \| $T=1$ \| $T=2$ \| $T=3$ \| $T=4$ \| $T=5$ \| $T=6$ \| $\sigma_{\lambda}(1)/\sigma_{\lambda}(6)$ \| $\sigma_{\lambda}(1)/\sigma_{\rm phys}$ \| $\sigma_{\lambda}(6)/\sigma_{\rm phys}$ 0.00 \| 2.283(35) \| 1.704(11) \| 1.463(8) \| 1.322(16) \| 1.244(24) \| 1.147(75) \| 1.99(13) \| 2.57(4) \| 1.29(8) 0.01 \| 1.476(78) \| 1.466(30) \| 1.348(17) \| 1.252(16) \| 1.191(27) \| 1.181(43) \| 1.25(8) \| 1.66(9) \| 1.33(5) 0.02 \| 1.005(90) \| 1.225(55) \| 1.225(31) \| 1.176(20) \| 1.135(25) \| 1.119(53) \| 0.90(9) \| 1.13(10) \| 1.26(6) 0.03 \| 0.728(86) \| 1.034(67) \| 1.119(43) \| 1.113(30) \| 1.104(30) \| 1.086(40) \| 0.67(8) \| 0.82(10) \| 1.22(5) 0.04 \| 0.557(79) \| 0.896(72) \| 1.030(51) \| 1.057(35) \| 1.053(30) \| 1.019(53) \| 0.55(8) \| 0.63(9) \| 1.15(6) 0.05 \| 0.441(73) \| 0.785(76) \| 0.947(59) \| 1.002(43) \| 1.021(35) \| 1.020(44) \| 0.43(7) \| 0.50(8) \| 1.15(5) 0.10 \| 0.169(46) \| 0.467(71) \| 0.684(70) \| 0.811(58) \| 0.872(50) \| 0.875(56) \| 0.19(5) \| 0.19(5) \| 0.98(6) 0.25 \| -0.005(13) \| 0.152(42) \| 0.324(60) \| 0.474(65) \| 0.586(62) \| 0.690(48) \| -0.01(2) \| -0.01(1) \| 0.78(5) 0.50 \| -0.042(1) \| 0.015(21) \| 0.111(41) \| 0.218(57) \| 0.320(69) \| 0.423(79) \| -0.10(2) \| -0.05(0) \| 0.48(9) 0.75 \| -0.043(3) \| -0.025(11) \| 0.028(27) \| 0.100(44) \| 0.181(58) \| 0.275(69) \| -0.16(4) \| -0.05(0) \| 0.31(8) 1.00 \| -0.040(3) \| -0.041(5) \| -0.012(18) \| 0.039(32) \| 0.104(45) \| 0.180(56) \| -0.22(7) \| -0.04(0) \| 0.20(6) On the $T$-dependence of the finite-time string tension $\sigma_{\lambda}(T)$, there are three groups of the $\lambda$-parameter region: (i) Coulomb-like region ($0\leq\lambda\ll\lambda_{C}$), (ii) Landau-like region ($\lambda_{C}\ll\lambda\leq 1$), and (iii) $\lambda_{C}$-like region ($\lambda\sim\lambda_{C}$). (i) The first category is the Coulomb-like region of $0\leq\lambda\ll\lambda_{C}$, i.e., $0\leq\lambda\lesssim 0.01$. In this region, the instantaneous string tension $\sigma_{\lambda}\equiv\sigma_{\lambda}(T=1)$ is larger than the physical string tension $\sigma_{\rm phys}$, and the instantaneous potential $V_{\lambda}(R)\equiv V_{\lambda}(R,T=1)$ gives an overconfining potential. The ground-state of the quark-antiquark system is considered as the gluon- chain state. As the temporal length $T$ of the Polyakov-line increases, finite-time string tension $\sigma_{\lambda}(T)$ decreases and approaches to the physical string tension $\sigma_{\rm phys}$. This decreasing behavior is interpreted that the component of the ground-state, i.e., the gluon-chain state, becomes dominant as $T$-length becomes large. Thus, this region would not be compatible with the quark potential picture, because of the large $T$-dependence of the confining force $\sigma_{\lambda}(T)$ in addition to the dynamical generation of the gluon-chain. (ii) The second category is the Landau-like region of $\lambda_{C}\ll\lambda\leq 1$, i.e. $0.1\lesssim\lambda\leq 1$. In this region, the instantaneous string tension is almost zero, i.e., $\sigma_{\lambda}\simeq 0$, and finite-time string tension $\sigma_{\lambda}(T)$ is an increasing function of $T$. Although its asymptotic value is unclear for $T\sim 0.8$ fm, $\sigma_{\lambda}(T)$ seems to approach to the physical string tension $\sigma_{\rm phys}$, which will be discussed in Sec.VI. (iii) The third category is the $\lambda_{C}$-like region of $\lambda\sim\lambda_{C}$, i.e., $0.01\lesssim\lambda\lesssim 0.1$. In this region, the instantaneous string tension $\sigma_{\lambda}$ is approximately equal to the physical string tension, i.e., $\sigma_{\lambda}\simeq\sigma_{\rm phys}(\simeq$ 0.89GeV/fm), and the instantaneous potential $V_{\lambda}(R)$ approximately reproduces the physical static potential $V_{\rm phys}(R)$, as shown in Fig. 3. As the temporal length $T$ increases, finite-time string tension $\sigma_{\lambda}(T)$ is slightly changed and takes a little larger value ($\simeq$ 1.1GeV/fm) around $T\simeq$ 0.8fm. In particular, near $\lambda_{C}\simeq 0.02$, $\sigma_{\lambda}(T)$ shows only a weak $T$-dependence, while $\sigma_{\lambda}(T)$ largely changes as $T$ in the Coulomb gauge. As a whole, finite-time potential $V_{\lambda}(R,T)$ has small $T$-dependence, as shown in Fig. 7. This is also a desired feature for the linkage to the quark potential model. Figure 7: Finite-time potential $V_{\lambda}(R,T)$ at $\lambda=0.02(\simeq\lambda_{C})$. For the comparison, an irrelevant constant is shifted for each $T$. The slope of $V_{\lambda}(R,T)$ is almost the same for $T=1,2,\cdots 5$, and thus the shape of $V_{\lambda}(R,T)$ is rather stable against the temporal length $T$. When the length $T$ of the Polyakov-line increases, $\lambda$-dependence of $\sigma_{\lambda}(T)$ is weakened, and $\sigma_{\lambda}(T)$ seems to converge to the physical string tension $\sigma_{\rm phys}$ for enough large $T$, as indicated in Fig.6. There are two ingredients on the above gauge-dependence ($\lambda$-dependence): one is the large excess of the Coulomb energy in the Coulomb gauge, the other is the non-locality from the Faddeev-Popov determinant. From the fixing condition of generalized Landau gauge in Eq.(10), one finds that the $\lambda$-parameter controls the non-locality in the temporal direction. In the Landau gauge, the non-locality appears equally in spatial and temporal directions, while temporal non-locality disappears in the Coulomb gauge. If the Coulomb-energy excess can be neglected, e.g., for large $\lambda$, $V_{\lambda}(R,T)$ is expected to reproduce the static potential $V_{\rm phys}(R)$, when $T$-length and $R$ are large enough to neglect the non-locality scale. Owing to the $\lambda$-dependence of the non-locality, such a $T$-length exceeding the non-locality is to be larger for larger $\lambda$ in the Landau-like region. Near the $\lambda_{C}$-gauge, finite-time potential $V_{\lambda}(R,T)$ has only weak $T$-length dependence. In other words, this can be regarded as an approximate “fixed point” against $T$ around $\lambda_{C}\simeq 0.02$. We conjecture that this is due to the approximate cancellation between the Coulomb-energy excess and the non-locality. Actually, in contrast to the large $T$-dependence of $V_{\lambda}(R,T)$ in the Coulomb gauge as shown in Fig.5, $V_{\lambda_{C}}(R,T)$ is rather stable against $T$-length in the $\lambda_{C}$-gauge, as shown in Fig.7. ## VI Terminated Polyakov-line correlator and potentials In the previous section, we investigated instantaneous potential $V_{\lambda}(R)$ and finite-time potential $V_{\lambda}(R,T)$, which are derived from the correlation of terminated Polyakov-line $L(\vec{x},T)$. In this section, we investigate properties of the Polyakov-line correlator, and clarify its relation to $V_{\lambda}(R)$ and $V_{\lambda}(R,T)$. ### VI.1 Asymptotic behavior of link-variable correlator and instantaneous potential First, we investigate the spatial correlator $G_{\lambda}(R)$ of the temporal link-variable $U_{4}$, and the relation to the instantaneous potential $V_{\lambda}(R)\equiv-\frac{1}{a}{\rm ln}G_{\lambda}(R)$. For the large spatial separation of $R\equiv\|\vec{x}-\vec{y}\|\rightarrow\infty$, $G_{\lambda}(R)$ behaves asymptotically as $\displaystyle G_{\lambda}(R)$ $\displaystyle\equiv$ $\displaystyle\langle\mathrm{Tr}\ [U_{4}^{\dagger}(\vec{x},t)U_{4}(\vec{y},t)]\rangle$ (29) $\displaystyle\rightarrow$ $\displaystyle\langle(U_{4})_{ij}^{}\rangle\langle(U_{4})_{ij}\rangle=\frac{1}{3}\langle\mathrm{Tr}\ U_{4}\rangle^{2},$ where $\langle(U_{4})_{ij}\rangle=\frac{1}{3}\langle\mathrm{Tr}\ U_{4}\rangle\delta_{ij}\in\mathbf{R}$ from the global color symmetry. Here, $\frac{1}{3}\langle\mathrm{Tr}\ U_{4}\rangle^{2}$ is found to give the lower bound of $G_{\lambda}(R)$. If $\langle\mathrm{Tr}\ U_{4}\rangle$ takes some finite value, $V_{\lambda}(R)$ inevitably saturates for large $R$. Then, $\langle\mathrm{Tr}\ U_{4}\rangle=0$ is a necessary condition for the deep- infrared confinement feature of $V_{\lambda}(R=\infty)=\infty$. In the Coulomb gauge, $\langle\mathrm{Tr}\ U_{4}\rangle$ is zero due to the remnant symmetry, as is shown in Appendix. Therefore, as $R\rightarrow\infty$, the correlator $G_{\lambda}(R)$ converges to zero, and $V_{\lambda}(R)\equiv-\frac{1}{a}{\rm ln}G_{\lambda}(R)\rightarrow+\infty$, which corresponds to the deep-infrared confinement. For the general case of $\lambda\neq 0$, however, $\langle\mathrm{Tr}\ U_{4}\rangle$ has a non-zero value, and $G_{\lambda}(R)$ approaches to some finite constant. The finiteness of $\langle\mathrm{Tr}\ U_{4}\rangle$ gives a saturation of $V_{\lambda}(R)$, which leads to the absence of its linear part in the case of rapid convergence. Figure 8 shows $G_{\lambda}(R)$ and its asymptotic value $\frac{1}{3}\langle\mathrm{Tr}\ U_{4}\rangle^{2}$ in the Landau and the Coulomb gauges. In the Landau gauge, $\langle\mathrm{Tr}\ U_{4}\rangle$ has a large expectation value, according to the maximization of $\sum_{x}\mathrm{Re}\ \mathrm{Tr}\ U_{\mu}(x)$, and $G_{\lambda}(R)$ rapidly converges to a finite constant for $R\gtrsim$ 0.4fm, which leads to a rapid saturation of the instantaneous potential $V_{\lambda}(R)$. In the Coulomb gauge, we find $\langle\mathrm{Tr}\ U_{4}\rangle=0$, and $G_{\lambda}(R)$ decreases monotonically to zero as $R$, which leads to $V_{\lambda}(R)=+\infty$ for $R=\infty$. Figure 8: The spatial correlator $G_{\lambda}(R)\equiv\langle\mathrm{Tr}\ U_{4}^{\dagger}(\vec{x},a)U_{4}(\vec{y},a)\rangle$ $(R=\|\vec{x}-\vec{y}\|)$ in the Landau gauge (open circles) and the Coulomb gauge (open triangles), together with its asymptotic value of $\frac{1}{3}\langle\mathrm{Tr}\ U_{4}\rangle^{2}$ (solid lines and cross symbols). In the Landau gauge, $\langle\mathrm{Tr}\ U_{4}\rangle\neq 0$, and $G_{\lambda}(R)$ rapidly converges to a constant for $R\gtrsim$ 0.4fm, which leads to a rapid saturation of the instantaneous potential $V_{\lambda}(R)\equiv-\frac{1}{a}{\rm ln}G_{\lambda}(R)$. In the Coulomb gauge, $\langle\mathrm{Tr}\ U_{4}\rangle=0$, and $G_{\lambda}(R)$ decreases monotonically to zero as $R$, which leads to $V_{\lambda}(R)=+\infty$ for $R=\infty$. Figure 9 shows $\lambda$-dependence of $\frac{1}{3}\langle\mathrm{Tr}\ U_{4}\rangle$ in generalized Landau gauge. For $\lambda\neq 0$, $\langle\mathrm{Tr}\ U_{4}\rangle$ takes a non-zero real value, and it approaches to zero continuously as $\lambda\rightarrow 0$. Here, it largely changes in the small region of $0\leq\lambda\lesssim 0.1$. The finiteness of $\langle\mathrm{Tr}\ U_{4}\rangle$ is directly related to the infrared damping of the correlator $G_{\lambda}(R)$ and the infrared form of the instantaneous potential $V_{\lambda}(R)$. Figure 9: The expectation value of $\frac{1}{3}\langle\mathrm{Tr}\ U_{4}\rangle$ in generalized Landau gauge. For $\lambda\neq 0$, $\langle\mathrm{Tr}\ U_{4}\rangle$ takes a non-zero value, and it approaches to zero continuously as $\lambda\rightarrow 0$. The value of $\langle\mathrm{Tr}\ U_{4}\rangle$ relates to the infrared behavior of the correlator $G_{\lambda}(R)$ and the instantaneous potential $V_{\lambda}(R)$. ### VI.2 Asymptotic behavior of Polyakov-line correlator and $T$-length potential Next, we consider $T$-length terminated Polyakov-line correlator $G_{\lambda}(R,T)$, which behaves asymptotically as $\displaystyle G_{\lambda}(R,T)$ $\displaystyle\equiv$ $\displaystyle\langle\mathrm{Tr}L^{\dagger}(\vec{x},T)L(\vec{y},T)\rangle$ (30) $\displaystyle\rightarrow$ $\displaystyle\frac{1}{3}\langle\mathrm{Tr}\ L(T)\rangle^{2}$ for large separation of $R=\|\vec{x}-\vec{y}\|$. As well as the instantaneous potential, $\frac{1}{3}\langle\mathrm{Tr}\ L(T)\rangle^{2}$ is found to give the lower bound of the correlator $G_{\lambda}(R,T)$, and the finiteness of $\langle\mathrm{Tr}\ L(T)\rangle$ is responsible for the infrared saturation of the finite-time potential $V_{\lambda}(R,T)$. Figure 10 shows $T$-dependence of the terminated Polyakov-line $\frac{1}{3}\langle\mathrm{Tr}\ L(T)\rangle$ in generalized Landau gauge for typical values of $\lambda$. In the Coulomb gauge ($\lambda$=0), $\langle\mathrm{Tr}\ L(T)\rangle$ is always zero as well as $\langle\mathrm{Tr}\ U_{4}\rangle$, which means that $V_{\lambda}(R=\infty,T)=-\frac{1}{T}{\rm ln}G_{\lambda}(R=\infty,T)=+\infty$ for any values of $T$. Actually, finite-time potential $V_{\lambda}(R,T)$ always has a linear part in the Coulomb gauge, as shown in Fig.5. For $\lambda\neq 0$, $\langle\mathrm{Tr}\ L(T)\rangle$ is a decreasing function of $T$, and it converges to zero in large-$T$ limit. At $T=N_{t}$, the $T$-length terminated Polyakov-line $\langle\mathrm{Tr}\ L(T)\rangle$ results in the Polyakov loop, and $\langle\mathrm{Tr}\ L(N_{t})\rangle=0$ in the confinement phase. Therefore, $\langle\mathrm{Tr}\ L(T)\rangle$ converges to zero as $T\rightarrow N_{t}$, and then one finds $\displaystyle G_{\lambda}(R=\infty,N_{t})$ $\displaystyle=$ $\displaystyle 0,$ (31) $\displaystyle V_{\lambda}(R=\infty,N_{t})$ $\displaystyle=$ $\displaystyle-\frac{1}{T}{\rm ln}G_{\lambda}(\infty,N_{t})=+\infty,$ (32) which gives a confinement potential. From Fig.10, this convergence is found to be fast for smaller $\lambda$-value, and such a convergence is closely related to the growing of finite-time string tension $\sigma_{\lambda}(T)$. Figure 10: $T$-dependence of $\frac{1}{3}\langle\mathrm{Tr}\ L(T)\rangle$ in generalized Landau gauge. In the Coulomb gauge, $\langle\mathrm{Tr}\ L(T)\rangle$ is always zero. For $\lambda\neq 0$, $\langle\mathrm{Tr}\ L(T)\rangle$ is a decreasing function of $T$, and it converges to zero in large-$T$ limit. ### VI.3 Gluon propagator and instantaneous potential in the Landau gauge In this subsection, we discuss the relation between the gluon propagator and the instantaneous potential in the Landau gauge. The gluon propagator is a two-point function of the gauge field $A_{\mu}(x)$, and is defined in Euclidean QCD as $D_{\mu\nu}(x,y)\equiv\langle\mathrm{Tr}A_{\mu}(x)A_{\nu}(y)\rangle,$ (33) where the trivial color structure is dropped off by taking the trace. In the Landau gauge, we use the expression of $A_{\mu}$ in terms of $U_{\mu}$ as $A_{4}(x)=\frac{1}{2iag}[U_{4}(x)-U_{4}^{\dagger}(x)]+\mathcal{O}(a^{2}).$ (34) Note that this expression is only justified in the Landau gauge, or more generally in large-$\lambda$ gauges, where the fluctuation of $A_{4}$ is highly suppressed. Then, the gluon propagator $D_{\mu\nu}$ is expressed using link-variables, e.g., $\displaystyle a^{2}g^{2}D_{44}(x,y)=a^{2}g^{2}\langle\mathrm{Tr}A_{4}(x)A_{4}(y)\rangle$ (35) $\displaystyle\simeq$ $\displaystyle-\frac{1}{4}\langle\mathrm{Tr}[U_{4}(x)-U_{4}^{\dagger}(x)][U_{4}(y)-U_{4}^{\dagger}(y)]\rangle$ $\displaystyle=$ $\displaystyle\langle\mathrm{Tr}[U_{4}(x)U_{4}^{\dagger}(y)]\rangle$ $\displaystyle\quad-\frac{1}{4}\langle\mathrm{Tr}[U_{4}(x)+U_{4}^{\dagger}(x)][U_{4}(y)+U_{4}^{\dagger}(y)]\rangle.$ The last term in Eq.(35) has only $O(a^{4})$-order $(x-y)$-dependence, and actually it changes only a few $\%$ in the Landau gauge at $\beta$=5.8, so that we here approximate this term as a constant $C$. Thus, Eq.(35) reduces $a^{2}g^{2}D_{44}(x,y)\simeq\langle\mathrm{Tr}[U_{4}(x)U_{4}^{\dagger}(y)]\rangle-C,$ (36) and we calculate the instantaneous potential $V_{\rm inst}(R)$ as $\displaystyle V_{\rm inst}(R)$ $\displaystyle=$ $\displaystyle-\frac{1}{a}\ln\langle\mathrm{Tr}[U_{4}(\vec{x},a)U_{4}^{\dagger}(\vec{y},a)]\rangle$ (37) $\displaystyle=$ $\displaystyle-\frac{1}{a}\ln\left[C+a^{2}g^{2}D_{44}(R)\right]$ $\displaystyle\simeq$ $\displaystyle-\frac{ag^{2}}{C}D_{44}(R)+\mathrm{const.}$ In this way, the instantaneous potential $V_{\rm inst}(R)$ is expressed by using the $44$-component of the gluon propagator in the Landau and large-$\lambda$ gauges. In the previous work Iritani09 , we have found that the Landau-gauge gluon propagator is well reproduced by the four-dimensional Yukawa-function as $D(r)\equiv D_{\mu\mu}(r)\propto\frac{1}{r}e^{-mr},$ (38) with the Yukawa mass-parameter $m\simeq$ 0.6GeV, in the region of $r=0.1\sim 1$ fm. Apart from a pre-factor from the tensor factor, $D_{44}(R)$ approximately behaves as the Yukawa-function, and therefore the instantaneous potential is expressed as $V_{\rm inst}(R)\simeq-\frac{A}{R}e^{-mR}+\mathrm{const.}$ (39) in the Landau gauge. Figure 11: Fit-result of the instantaneous potential $V_{\rm inst}(R)$ in the Landau gauge using Coulomb plus linear form (solid line) and Yukawa-function form (dashed line). Both forms well reproduce lattice QCD result. The best-fit parameter of the Yukawa mass is $m$ = 0.634(3)GeV, which coincides with the infrared effective gluon mass obtained from the Landau-gauge gluon propagator Iritani09 . Figure 11 shows the two fit-results of instantaneous potential in the Landau gauge, using the Coulomb plus linear form $V_{\rm Coul.+lin.}(R)=-A/R+\sigma R$ and the Yukawa-function form $V_{\rm Yukawa}(R)=-A\exp(-mR)/R$. Both functions well reproduce the lattice QCD result. The best-fit Yukawa mass- parameter is $m$=0.634(3)GeV, and this value coincides with the infrared effective gluon mass obtained from the Landau-gauge gluon propagator Iritani09 . We note again that this relation is only valid in the Landau and large-$\lambda$ gauges, where the temporal link-variable $U_{4}$ can be expanded in terms of lattice spacing $a$, and the last-term in Eq.(35) is almost constant. ## VII Summary and Discussion In this paper, aiming to grasp the gauge dependence of gluon properties, we have investigated generalized Landau gauge and applied it to instantaneous interquark potential in SU(3) quenched lattice QCD at $\beta$=5.8. In the Coulomb gauge, the instantaneous potential is expressed by the sum of Coulomb potential and linear potential with 2-3 times larger string tension. In contrast, the instantaneous potential has no linear part in the Landau gauge. Thus, there is a large gap between these two gauges. Using generalized Landau gauge, we have found that the instantaneous potential $V_{\lambda}(R)$ is connected continuously from the Landau gauge towards the Coulomb gauge, and the linear part in $V_{\lambda}(R)$ grows rapidly in the neighborhood of the Coulomb gauge. Since the slope $\sigma_{\lambda}$ of the instantaneous potential $V_{\lambda}(R)$ grows continuously from 0 to 2-3$\sigma_{\rm phys}$, there must exist some specific intermediate gauge where the slope $\sigma_{\lambda}$ coincides with the physical string tension $\sigma_{\rm phys}$. From the lattice QCD calculation, the specific $\lambda$-parameter, $\lambda_{C}$, is estimated to be about $0.02$. In this $\lambda_{C}$-gauge, the physical static interquark potential $V_{\rm phys}(R)$ is approximately reproduced by the instantaneous potential $V_{\lambda}(R)$. We have also investigated $T$-length terminated Polyakov-line correlator, and its corresponding finite-time potential $V_{\lambda}(R,T)$, which is a generalization of the instantaneous potential $V_{\lambda}(R)$, in generalized Landau gauge. The behavior of the slope $\sigma_{\lambda}(T)$ of the finite- time potential is classified into three groups: the Coulomb-like gauge ($0\leq\lambda\lesssim 0.01$), the Landau-like gauge ($0.1\lesssim\lambda\leq 1$), and neighborhood of $\lambda_{C}$-gauge ($\lambda\sim\lambda_{C}\simeq 0.02$). In the Coulomb-like gauge, the slope $\sigma_{\lambda}(T)$ is a decreasing function of $T$, and seems to approach to physical string tension $\sigma_{\rm phys}$ for large $T$. In the Landau-like gauge, $\sigma_{\lambda}(T)$ is an increasing function. Around the $\lambda_{C}$-gauge, $\sigma_{\lambda}(T)$ has a weak $T$-length dependence. We have also investigated $T$-length terminated Polyakov-line correlator and its relation to the finite-time potential. Finally, we consider a possible gauge of QCD to describe the quark potential model from the viewpoint of instantaneous potential. The quark potential model is a successful nonrelativistic framework with a potential instantaneously acting among quarks, and describes many hadron properties in terms of quark degrees of freedom. In this model, there are no dynamical gluons, and gluonic effects indirectly appear as the instantaneous interquark potential. As for the Coulomb gauge, the instantaneous potential has too large linear part, which gives an upper bound on the static potential Zwanziger03 . It has been suggested by Greensite et al. that the energy of the overconfining state is lowered by inserting dynamical gluons between (anti-)quarks, which is called “gluon-chain picture”. This gluon-chain state is considered as the ground-state in the Coulomb gauge Greensite03 ; Greensite04 ; GreensiteThorn . Therefore, dynamical gluon degrees of freedom must be also important to describe hadron states in the Coulomb gauge. For $\lambda_{C}$-gauge, the physical interquark potential $V_{\rm phys}(R)$ is approximately reproduced by the instantaneous potential $V_{\lambda_{C}}(R)$ unlike the Coulomb gauge, as schematically shown in Fig.12. This physically means that all other complicated effects including dynamical gluons and ghosts are approximately cancelled in the $\lambda_{C}$-gauge, and therefore we do not need to introduce any redundant gluonic degrees of freedom. The absence of dynamical gluon degrees of freedom would be a desired property for the quark model picture. The weak $T$-length dependence of $\sigma_{\lambda}(T)$ around the $\lambda_{C}$-gauge ($T$-length stability) is also a suitable feature for the potential model. Figure 12: The schematic illustration of the Coulomb gauge and the $\lambda_{C}$-gauge. In the Coulomb gauge, the instantaneous potential gives an “overconfining state” as an excited-state for the quark-antiquark system. The ground-state is considered as the gluon-chain state, which contains dynamical gluons between static sources. In the $\lambda_{C}$-gauge, the instantaneous potential gives the physical interquark potential approximately, and dynamical gluons need not appear. In this way, as an interesting possibility, the $\lambda_{C}$-gauge is expected to be a useful gauge in considering the linkage from QCD to the quark potential model. In addition, since $\lambda_{C}\simeq 0.02\ll 1$ is a very small parameter in this framework, it is interesting to apply the perturbative technique in terms of $\lambda_{C}$ for the calculation of the Faddeev-Popov determinant and so on. ###### Acknowledgements. The authors thank Dr. H. Iida for his useful arguments. H.S. is supported in part by the Grant for Scientific Research [(C) No.19540287, Priority Areas “New Hadrons” (E01:21105006)] from the Ministry of Education, Culture, Science and Technology (MEXT) of Japan. This work is supported by the Global COE Program, “The Next Generation of Physics, Spun from Universality and Emergence”. The lattice QCD calculations have been done on NEC-SX8 at Osaka University. ## Appendix A On the Coulomb gauge In this appendix, we briefly discuss the difference between the Coulomb gauge and the $\lambda\rightarrow 0$ limit of generalized Landau gauge. ### A.1 Residual gauge degrees of freedom in the Coulomb gauge We comment on the residual gauge degrees of freedom in the Coulomb gauge. In lattice QCD, the Coulomb gauge fixing condition is expressed by the maximization of the quantity $R_{\rm Coul}[U]\equiv\sum_{\vec{x},t}\sum_{i=1}^{3}\mathrm{Re}\ \mathrm{Tr}\ U_{i}(\vec{x},t)$ (40) by the gauge transformation. Now, we consider the spatially-global gauge transformation as $\displaystyle U_{i}(\vec{x},t)$ $\displaystyle\rightarrow$ $\displaystyle\Omega(t)U_{i}(\vec{x},t)\Omega^{\dagger}(t),$ (41) $\displaystyle U_{4}(\vec{x},t)$ $\displaystyle\rightarrow$ $\displaystyle\Omega(t)U_{4}(\vec{x},t)\Omega^{\dagger}(t+1),$ (42) with the gauge function $\Omega(t)\in{\rm SU}(N_{c})$. $R_{\rm Coul}[U]$ is invariant under this transformation, $\displaystyle R_{\rm Coul}[U]$ $\displaystyle\equiv$ $\displaystyle\sum_{\vec{x},t}\sum_{i=1}^{3}\mathrm{Re}\ \mathrm{Tr}\ U_{i}(\vec{x},t)$ (43) $\displaystyle\rightarrow$ $\displaystyle\sum_{\vec{x},t}\sum_{i=1}^{3}\mathrm{Re}\ \mathrm{Tr}\ \Omega(t)U_{i}(\vec{x},t)\Omega^{\dagger}(t)$ $\displaystyle=$ $\displaystyle\sum_{\vec{x},t}\sum_{i=1}^{3}\mathrm{Re}\ \mathrm{Tr}\ U_{i}(\vec{x},t).$ Therefore, the Coulomb gauge has the corresponding residual gauge degrees of freedom. Under this residual symmetry, however, $\mathrm{Tr}\ U_{4}$ is gauge-variant as $\displaystyle\mathrm{Tr}\ U_{4}(\vec{x},t)$ $\displaystyle\rightarrow$ $\displaystyle\mathrm{Tr}\ \Omega(t)U_{4}(\vec{x},t)\Omega^{\dagger}(t+1)$ (44) so that the expectation value of $\mathrm{Tr}\ U_{4}$ is to be zero in the Coulomb gauge. In the generalized Landau gauge with non-zero $\lambda$-parameter, this residual symmetry does not exist, and hence $\mathrm{Tr}\ U_{4}$ has a finite expectation value. (See Fig.9.) ### A.2 Convergence into Coulomb gauge We here investigate the convergence of the generalized Landau gauge into the Coulomb gauge in the limit of $\lambda\rightarrow 0$. To check the convergence, we evaluate the quantity $\langle(\partial_{i}A_{i}^{a})^{2}\rangle$, Figure 13: The lattice QCD result of $\langle(\partial_{i}A_{i}^{a})^{2}\rangle$ in the generalized Landau gauge. As $\lambda\rightarrow 0$, this quantity goes to zero monotonically. $\displaystyle\langle\left(\partial_{i}A_{i}^{a}\right)^{2}\rangle$ $\displaystyle\equiv$ $\displaystyle\frac{1}{(N_{c}^{2}-1){N_{\rm site}}}$ (45) $\displaystyle\times$ $\displaystyle\sum_{x=1}^{N_{\rm site}}\sum_{a=1}^{N_{c}^{2}-1}\Big{\\{}\sum_{i=1}^{3}\left[A_{i}^{a}(x)-A_{i}^{a}(x-\hat{i})\right]\Big{\\}}^{2}.~{}~{}~{}~{}$ In the Coulomb gauge, $\langle(\partial_{i}A_{i}^{a})^{2}\rangle$ is equal to zero. Figure 13 shows the lattice QCD result of $\langle(\partial_{i}A_{i}^{a})^{2}\rangle$, which is monotonically decreasing toward zero by varying $\lambda$ from 1 to 0. This result supports that the generalized Landau gauge approaches to the Coulomb gauge in the $\lambda\rightarrow 0$ limit. ## References (1) Y. Nambu, Proceedings of Preludes Theoretical Physics, in honor of V.F.Weisskopf (North-Holland, Amsterdam, 1966). * (2) D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973); H.D. Politzer, Phys. Rev. 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Itzykson and J. Zuber, “Quantum Field Theory”, (McGraw-Hill, New York, 1980). * (22) D. Zwanziger, Phys. Rev. Lett. 90, 102001 (2003). * (23) C. Bernard, D. Murphy, A. Soni, and K. Yee, Nucl. Phys. B. (Proc. Suppl.) 17, 593 (1990); C. Bernard, D. Murphy, A. Soni, Nucl. Phys. B (Proc. Suppl.) 20, 410 (1991). * (24) A. Nakamura and T. Saito, Prog. Theor. Phys. 115, 189 (2006). * (25) H. Suganuma, T.T. Takahashi, and H. Ichie, “Color Confinement and Hadrons in Quantum Chromodynamics” (World Scientific, Singapore, 2004), p. 249; T.T. Takahashi et al., Phys. Rev. D 65, 114509 (2002); Phys. Rev. Lett. 86, 18 (2001). * (26) A. Cucchieri, A. Maas, and T. Mendes, Mod. Phys. Lett. A22, 2429 (2007). * (27) H.J. Rothe, “Lattice Gauge Theories: An Introduction” 3rd ed. (World Scientific, Singapore, 2005).	arxiv-papers	2010-11-22T10:42:34	2024-09-04T02:49:15.133161	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Takumi Iritani, Hideo Suganuma", "submitter": "Takumi Iritani", "url": "https://arxiv.org/abs/1011.4767" }
1011.4962	# Can IBEX Identify Variations in the Galactic Environment of the Sun using Energetic Neutral Atom (ENAs)? Priscilla C. Frisch Dept. Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637 frisch@oddjob.uchicago.edu Jacob Heerikhuisen CSPAR, University of Alabama, Huntsville, AL jacobh@ucr.edu Nikolai V Pogorelov CSPAR, University of Alabama, Huntsville, AL nikolaip@ucr.edu Bob DeMajistre Applied Physics Laboratory, Johns Hopkins University, Laurel, MD Bob.DeMajistre@jhuapl.edu Geoffrey B Crew Massachusetts Institute of Technology, Cambridge, MA gbc@haystack.mit.edu Herbert O Funsten Los Alamos National Laboratory, Los Alamos, NM hfunsten@lanl.gov Paul Janzen Dept. Physics Astronomy, University of Montana, Missoula, MT paul.janzen@umontana.edu David J McComas11affiliation: University of Texas, San Antonio, TX Southwest Research Institute, San Antonio, TX DMcComas@swri.edu Eberhard Moebius Space Science Center, University of New Hampshire, Durham, NH eberhard.moebius@unh.edu Hans-Reinhard Mueller Dept. of Physics and Astronomy, Dartmouth College, Hanover, NH Hans- Reinhard.Mueller@Dartmouth.edu Daniel Brett Reisenfeld Dept. Physics Astronomy, University of Montana, Missoula, MT dan.reisenfeld@umontana.edu Nathan A Schwadron Boston University, Boston, MA nathanas@bu.edu Jonathan D. Slavin Harvard-Smithsonian Center for Astrophysics, Harvard, Cambridge, MA jslavin@cfa.harvard.edu Gary Paul Zank CSPAR, University of Alabama, Huntsville, AL zank@email.cspar.uah.edu ###### Abstract The Interstellar Boundary Explorer (IBEX) spacecraft is providing the first all-sky maps of the energetic neutral atoms (ENAs) produced by charge-exchange between interstellar neutral Ho atoms and heliospheric solar wind and pickup ions in the heliosphere boundary regions. The ’edge’ of the interstellar cloud presently surrounding the heliosphere extends less than 0.1 pc in the upwind direction, terminating at an unknown distance, indicating that the outer boundary conditions of the heliosphere could change during the lifetime of the IBEX satellite. Using reasonable values for future outer heliosphere boundary conditions, ENA fluxes are predicted for one possible source of ENAs coming from outside of the heliopause. The ENA production simulations use three- dimensional MHD plasma models of the heliosphere that include a kinetic description of neutrals and a Lorentzian distribution for ions. Based on this ENA production model, it is then shown that the sensitivities of the IBEX 1.1 keV skymaps are sufficient to detect the variations in ENA fluxes that are expected to accompany the solar transition into the next upwind cloud. Approximately 20% of the IBEX 1.1 keV pixels appear capable of detecting the predicted model differences at the $3\sigma$ level, with these pixels concentrated in the Ribbon region. Regardless of the detailed ENA production model, the success of the modeled ${\bf B}\cdot{\bf R}\sim 0$ directions in reproducing the Ribbon locus, together with our results, indicate that the Ribbon phenomenon traces the variations in the heliosphere distortion caused by the relative pressures of the interstellar magnetic and gaseous components. ISM: magnetic fields, clouds, HI — solar system: general — stars: winds, outflows ††slugcomment: published ApJ, 2010 ## 1 Introduction The dynamic heliosphere varies with the properties of the surrounding interstellar cloud and the solar wind. A theoretical study of the expected heliosphere response to different types of interstellar clouds show that both the overall dimensions and hydrogen filtration should vary substantially with variations in the physical properties of circumheliospheric interstellar material (Müller et al., 2006, 2008). The presence of an interstellar magnetic field causes heliosphere asymmetries that can diagnose the properties of the surrounding interstellar material, but which are partially offset by the charge-exchange coupling of interstellar Ho and protons upstream of the heliopause (Pogorelov et al., 2009c; Opher et al., 2009; Ratkiewicz et al., 2008; Izmodenov, 2009). The velocity discontinuity observed between interstellar gas inside of the heliosphere (e.g. Witte, 2004) and interstellar material (ISM) towards the nearest stars in the upwind direction (36 Oph and $\alpha$ Cen, Landsman et al., 1984; Lallement et al., 1995; Wood et al., 2000; Linsky & Wood, 1996) is usually interpreted to indicate that the Sun is immersed in one interstellar cloud today, but will enter a separate cloud sometime in the next $\sim 4000$ years. The Interstellar Boundary Explorer (IBEX) is for the first time mapping heliospheric energetic neutral atoms, formed by charge-exchange between solar wind ions and pickup ions with neutral interstellar atoms (McComas et al., 2009b, a; Funsten et al., 2009b; Fuselier et al., 2009; Schwadron et al., 2009). In this paper we show that, depending on the source of ENAs observed, IBEX is capable of detecting the variable heliosphere boundary conditions that might accompany the expected (someday, it could be soon) solar transition into a new interstellar environment in the upwind direction. The discovery of an unexpected ’Ribbon’ of ENA emission, in directions where the interstellar magnetic field draping over the heliosphere is thought to be perpendicular to the sightline, showed that IBEX may be detecting plasma- neutral interactions beyond the heliopause. The similar spectra of ENAs in the Ribbon and adjacent sightlines suggest that the Ribbon represents a selection effect rather than an ENA population with an fully independent origin. McComas et al. (2009a) and Schwadron et al. (2009) noted that the ribbon is organized by the most likely direction of the external interstellar magnetic field (ISMF), and proposed several different potential sources of the Ribbon including the possibility that the Ribbon might be created from a population of anisotropic suprathermal ions gyrating around the interstellar magnetic field just outside the heliopause. These ions could be indigenous to the outer heliosheath (beyond the heliopause) or more likely would arise from ENAs that propagated out from the supersonic solar wind and/or inner heliosheath (between the termination shock and the heliopause); these authors noted that the problem with this idea is that the relatively confined pitch angle distributions would need to be maintained long enough to create ”secondary ENAs”, which likely takes several years on average. Subsequently, Heerikhuisen et al. (2010) incorporated this idea, where outward propagating ENAs create an anisotropic population of pickup ions (PUI) in the outer heliosheath that seed ”secondary ENA” production several hundred AU upwind of the heliopause, into a quantitative model. While it is still uncertain how long the ion ring beam takes to scatter into a shell distribution (Florinski, Zank, Heerikhuisen, Hu and Khazanov, submitted), with marginally stable ring distributions predicted by some models for the distribution of the pitch angles of pickup ions in the outer heliosheath (Gamayhunov, Zhang, Rassoul, submitted to ApJ), the Heerikhuisen et al. simulation assumes that the re-neutralization time is essentially instantaneous compared to the scattering time so that the new secondary ENA will have a preferred direction that is perpendicular to the local ISMF direction. In this model, IBEX then sees these secondary ENAs where the gyration plane of the ion is aligned with the sightline to IBEX, i.e. where the sightline is perpendicular to the ISMF direction. IBEX data are uniquely qualified to simultaneously test both the direction of the ISMF at the heliosphere and the density of interstellar neutrals. The ISMF drapes around the heliosphere, rotating by $\sim 30^{\circ}$ between ’infinity’ and the heliopause near the nose. The Ribbon is $\sim 20^{\circ}$ wide and at least $270^{\circ}$ long, possibly forming a complete circle in the sky. IBEX only sees ENAs with momentum vectors directed back towards the inner heliosphere. The long mean free paths of ENAs, e.g. $\sim 200$ AU for 1.1 keV ENAs in $n\mathrm{(p^{+})}$$\sim 0.1$ cm-3 plasma, allow detection in the inner heliosphere of secondary ENAs formed in regions beyond the heliopause with elevated interstellar densities and relatively isotropic ENA velocities (compared to the outwards radial flows for primary ENAs produced in the supersonic solar wind, although not compared to the inner heliosheath ion populations). The solar wind contributing to ENA production includes both core ions from the expanding solar corona, and pickup ions formed inside of the heliosphere by charge exchange between interstellar neutrals and the core solar wind. The predictive capabilities of global heliosphere models have improved significantly to accommodate observational constraints placed by the 10 AU difference in the termination shock distances found by the Voyager 1 and Voyager 2 spacecraft (Stone, 2007; Opher et al., 2009), the $\sim 5^{\circ}$ offset between the upwind directions of interstellar Ho and Heo flowing into the heliosphere determined from SOHO/SWAN and Ulysses/GAS data (Lallement et al., 2005; Witte, 2004; Frisch, 2008, where the Heo upwind direction must first be converted to J2000 coordinates for this comparison), the properties of the ISM surrounding the heliosphere (Slavin & Frisch, 2008), and now the IBEX data on ENA fluxes and the Ribbon. Although the IBEX Ribbon itself was not predicted by models of ENA formation in the heliosphere, the global heliosphere models provided the ISMF orientation that matches well with the configuration of the Ribbon in the sky (Schwadron et al., 2009; Pogorelov et al., 2009c). In the discussions below we rely on the ENA production models quantified by Heerikhuisen et al. (2010); Pogorelov et al. (2009b) to predict the ENA fluxes for a heliosphere immersed in the next interstellar cloud versus the present- day surrounding interstellar cloud. The ENA production simulations use three- dimensional MHD plasma models of the heliosphere that include a kinetic description of neutrals and a Lorentzian distribution for solar wind protons to approximate the suprathermal population of pick-up ions in the heliosheath region. The interstellar neutrals, that are coupled self-consistently to the plasma component in the MHD heliosphere models, act to symmetrize the heliosphere. Any asymmetry in the quiescent plasma distribution (e.g. created by ISMF) results in variations in the number of charge-exchange events, creating new ions that are decelerated by the heliopause and therefore that mitigate the asymmetry (Pogorelov et al., 2008a, 2009c, 2009b). The LISM flow, for the assumed ISMF strength, is subfast magnetosonic (Table 1). This results in the absence of a bow shock in front of the heliopause, and increases the width of the region where the ISMF deviates from its unperturbed orientation. Since the local ISMF direction varies as interstellar protons approach the heliopause, the mean-free-path for the charge exchange interaction must also be included self-consistently in any Ribbon production models. The Ribbon ENAs are formed upwind of the heliopause in the Heerikhuisen et al. (2010) model, so the outer boundary conditions set by the ISM have a direct impact on predicted ENA fluxes and provide a means of estimating ENA variations from a cloud transition, regardless of whether the model is correct in detail. Any model that reproduces the observed ${\bf B}\cdot{\bf R}\sim 0$ of the Ribbon, which is seen where the ISMF (${\bf B}$) is perpendicular to the sightline (${\bf R}$), should provide useful insights into the deformation of the heliosphere due to altered boundary conditions from the variable interstellar wind. Our conclusions here rely explicitly on the assumption that the Heerikhuisen et al. (2010) model provides a viable description of ENA production, both for the cloud we are in today and for the nearby cloud observed towards $\alpha$ Cen and 36 Oph in the upwind direction. We focus on the 1.1 keV data, because the contrast between Ribbon ENA fluxes and diffuse ENA fluxes is stronger at this energy, and partly because of the enhanced outwards flow of ENAs from core solar wind ions that have typical energies near 1 keV. The ENA spectra are explicitly predicted by the heliosphere models, however this spectral information is not used here. The observed fluxes of $\sim 4.5$ keV ENAs are an order of magnitude lower than the 1.1 keV fluxes and the mean-free-paths are $\sim 50$% larger. Since the outflowing ion fluxes will decrease as $\sim R^{-2}$ with distance from the Sun $R$, both parent ion densities and the resulting 1-AU ENA densities are lower when production regions are further from the Sun. ## 2 Properties of the Upwind ISM The boundary conditions of the heliosphere are set by the ISM, and vary over geologically short time-scales. Interstellar clouds within $\sim 50$ pc flow past the Sun with heliocentric velocities that cluster around $\sim 28$ km s-1 (after projection effects are removed), or $\sim 1$ parsec per 35,000 yrs (Frisch & Slavin, 2006). If nearby ISM is in pressure equilibrium, then models of the interstellar cloud around the heliosphere (Slavin & Frisch, 2008) combined with ISM data (Redfield & Linsky, 2004) yield an estimate for the typical cloud length of $\sim 1$ pc. Cloud column densities for 23 cloud components towards stars within 10 pc yield a range of cloud lengths 0.06–3 pc, giving typical cloud crossing times for the Sun of $\sim 1,450-2.8\times 10^{5}$ years.111These estimates assume an equilibrium thermal pressure of $\sim 3\times 10^{-13}$ dynes cm-2 for the present-day cloud, uniform magnetic and cosmic ray pressures, D/H$\sim 1.6\times 10^{-5}$, and a uniform proton density of 0.08 cm-3. The velocity of the ISM at the heliosphere, which we term here the heliospheric ISM but which is also known as either the Local Interstellar Cloud (LIC) or the circumheliospheric ISM, is best set by the velocity of interstellar Heo observed inside of the heliosphere by the Ulysses satellite (Witte, 2004). Interstellar Heo experiences minimal filtration in the heliosheath regions ($<2$%, Müller & Zank, 2004) and is not subject to radiation pressure, so that variation in the interstellar Heo velocity as it traverses the heliosphere is minimal. The first comparisons between the velocities of interstellar gas in the heliosphere and outside of the heliosphere, towards stars in the upwind direction, showed differences of over 3 km s-1 (Adams & Frisch, 1977). When the 26.3 km s-1 Heo velocity is projected towards the nearest star in the upwind direction $\alpha$ CMa, 1.3 pc away and 50∘ from the heliosphere nose, the projected heliospheric ISM velocity is –17.0 km s-1, in contrast to the observed cloud velocity from the unsaturated Fe+ and Mg+ lines of $-18.0\pm 0.2$ km s-1 (Linsky & Wood, 1996). The heliosphere nose direction is given by the upwind direction of Heo flowing through the heliosphere, or $\lambda,\beta=255.4^{\circ}\pm 0.5^{\circ},+5.1\pm 0.2^{\circ}$ (epoch J2000, Witte, 2004, and private communication). The star 36 Oph is 6 pc away and 10∘ from the heliospheric nose. The projected heliospheric ISM velocity in this direction is –25.9 km s-1, versus the observed cloud velocity from the Fe+ and Mg+ lines of $-28.1\pm 0.2$ km s-1 (Wood et al., 2000). The limit on a cloud component at the heliospheric ISM velocity towards 36 Oph is $\sim 6\times 10^{16}$ cm-2, giving an upper limit to the heliospheric ISM edge222We use $n\mathrm{(H^{\circ})}$=0.19 cm-2 for the heliospheric ISM Ho density, based on Model 26 in the radiative transfer models of Slavin & Frisch (2008). in this direction of 0.1 pc which will be traversed in less than 4000 years. The interstellar cloud observed towards 36 Oph and $\alpha$ Oph is known as the G-cloud (Lallement & Bertin, 1992; Frisch et al., 2002; Redfield & Linsky, 2008), since it is close to the Sun in the galactic hemisphere. Although other possible clouds have been suggested as the next cloud to be encountered by the heliosphere (Frisch, 2003), the G-cloud remains the most likely future heliospheric environment. The scenario examined here assumes that the Sun transitions directly from the heliospheric ISM to the G-cloud seen towards 36 Oph and $\alpha$ Cen. We determine the properties of the next cloud by assuming that the thermal and magnetic pressures in the heliospheric ISM are equal, and that the heliospheric ISM and G-clouds are in pressure equilibrium. The G-cloud temperature is found from the mass-dependent Doppler broadened widths of interstellar absorption lines, and is $5400\pm 500$ K towards $\alpha$ Cen, and $5900\pm 500$ towards 36 Oph (where the cloud column density is also larger by 70%). The cooler G-cloud temperatures are thus compensated by neutral densities that are 20% larger than the heliospheric ISM. The 9% higher velocity of the G-cloud, in comparison to the velocity of the ISM now surrounding the Sun, will increase the interstellar ram pressure even if thermal pressure and ionization levels remain constant. We test the sensitivity of ENA emission to the ISMF direction using two separate ISMF directions for the G-cloud. The first assumption is that the directions of the ISMF in the heliospheric ISM and G-clouds are the same (Models 2 and 3 in Table 1). For Model 4 we make an arbitrary333Caveat: This direction, which is arbitrary from an interstellar viewpoint, was selected to lie on the great circle that divides the hot and cold poles of the cosmic microwave background dipole moment, and which passes through the heliosphere nose region (Frisch, 2010). assumption for the G-cloud ISMF direction, which is $28^{\circ}$ different from today’s field but less than the $\sim 30^{\circ}$ rotation of the ISMF between the ISM and the heliopause for the upwind direction. The detailed heliosphere boundary conditions used here for the next-cloud are listed in Table 1. We show below that the IBEX Ribbon is highly sensitive to even small variations in the ISMF direction, even when increased Ho densities mitigate the influence of the ISMF on heliospheric asymmetries. ## 3 ENA fluxes from encounter with Next Interstellar Cloud The heliosphere model has been run for the interstellar boundary conditions listed for Models 1-4 in Table 1. The 1.1 keV ENA fluxes predicted by the resulting models are displayed in Fig. 1. Model 1 corresponds to the heliosphere model displayed in Schwadron et al. (2009). Model 2 (from Heerikhuisen et al., 2010) is the updated model we use for the heliosphere today. Models 3 and 4 correspond to the anticipated heliosphere environment in the next interstellar cloud, with Model 3 having a similar ISMF configuration as Model 2, and Model 4 showing a ISMF field with a different direction. The increased Ho density in Model 3 produces a brighter ribbon, while the ram and thermal pressures slightly increase and the magnetic pressure slightly decreases. The magnetic field in Model 2 plays a bigger role in deforming the heliosphere than in Model 3. Model 4 has the Ribbon shifted significantly because of the different ISMF direction. The differences between the predicted ENA fluxes for Model 2 (the ’today- cloud’) and Model 3 (the G-cloud assuming the same ISMF direction as today- cloud) are directly displayed in Fig. 2. The ENA flux differences are obtained by subtracting the predicted fluxes of Model 2 ($F_{\mathrm{Mod2}}$) from the predicted fluxes of Model 3 ($F_{\mathrm{Mod3}}$, left). The most significant difference between the two models is the ram pressure of the neutrals, which is a factor of 1.8 larger for Model 3 versus Model 2. Over most of the sky, the higher flux of interstellar Ho into the heliosphere for Model 3 generates larger ENA fluxes, with the differences approaching the brightest observed fluxes. However the red pixels in Fig. 2, left, show regions where the today- cloud has higher fluxes than the next-cloud, and represent the small shift in the Ribbon position due to the increase in the ratio of thermal to magnetic pressures, $P_{\mathrm{ther}}$/$P_{\mathrm{mag}}$, in Model 3 compared to Model 2. The effect of increased Ho densities and thermal pressures are also seen in the increased ENA emissivity of the eastern flank of the heliosphere in the nose direction, where there is a bulge in total pressure (magnetic and thermal, see Fig. 1 in Heerikhuisen et al., 2010). Fig. 2, right, shows the flux differences between Model 2 and Model 4, where the ISMF direction in the G-cloud has been allowed to vary by 20∘. This modest variation in the ISMF direction, while retaining the same field strength, leads to an obvious shift in the Ribbon location, and shows that variations in the direction of the ISMF draping over the heliosphere should be apparent in the ENA data. The ENA-production model used here predicts that the ribbon and non-ribbon regions respond differently to variations in the interstellar density, because the ribbon also directly traces heliosphere asymmetries created by the ISMF. Fig. 3 shows the model differences modified by the ratio of $n\mathrm{(H^{\circ})}$ in Models 2 and 3 (0.65). The left figure shows 0.65 $F_{\mathrm{Mod3}}$– $F_{\mathrm{Mod2}}$, and the right figure shows 0.65$F_{\mathrm{Mod4}}$– $F_{\mathrm{Mod2}}$. The background blue regions, where difference counts are $\sim 0$, indicate regions where the ENA fluxes are linearly related to the neutral interstellar density. The variations in the ribbon colors show that the ribbon ENAs trace the distortion of the heliosphere, which depends on the asymmetries introduced by the relative interstellar, ram, and thermal pressures, and which changes with different ISM conditions. The differences in the ENA fluxes predicted by the two next-cloud models are quite obvious in the high-flux regions of the Ribbon, but less obvious (for this color scale) for the directions towards the tail. In order to emphasize the differences in the weaker diffuse ENA emission originating in the low-flux tail region, Fig. 4 shows the percentage differences between Models 3 and 2, ($F_{\mathrm{Mod3}}$–$F_{\mathrm{Mod2}}$)/$F_{\mathrm{Mod2}}$, and Models 4 and 2, ($F_{\mathrm{Mod4}}$–$F_{\mathrm{Mod2}}$)/$F_{\mathrm{Mod2}}$, with enhanced color scales. The percentage differences in the tail region for Model 3 are larger than for Model 4. The outer heliosheath region around the tail is the most disturbed part of the model results, since the flows are subsonic. Hence larger relative variations in ENA fluxes are possible. The capability of IBEX to detect the ENA variations shown in Figs. 2–4 rests on the predicted differences in the modeled ENA fluxes compared to the flux uncertainties for the IBEX data. For this comparison, we use the first 1.1 keV ENA flux maps in the third energy passband (”ESA 3”) of the six energy passbands in the IBEX-HI neutral atom imager (Funsten et al., 2009a), where fluxes are an order of magnitude larger than at 4.5 keV (ESA 6). Fig. 5, left, shows ESA3 fluxes, after correction for the Compton-Getting (CG) shift in the energy and spatial distribution of high velocity particles due to the 30 km s-1 orbital motion of the Earth (e.g. Gleeson & Axford, 1968).444We have used the IBEX Compton-Getting corrected data set ”flxset_hd60-id- base-0071-2010-04-09.sav”, that is available at the IBEX Science Operations Center (ISOC). IBEX pixels are $\sim 7^{\circ}$. The CG corrections are based on a power-law energy spectra that are derived from adjacent energy passbands, typically $\sim E^{-1.6}$, which is evaluated over the look-direction and convolved with the energy response of the detector (see the Appendix in McComas et al., 2010, submitted to GRL, for details on the CG correction to the ENA energies measured by IBEX). The $1\sigma$ uncertainties ($dF_{\mathrm{1\sigma}}$) on these fluxes have been determined from the Poisson statistics propagated from the measurement uncertainties. The IBEX data are built from data processed by the IBEX Science Operations Center (ISOC) for the first all-sky IBEX map (orbits 11-34). The uncertainties in the ESA3 fluxes are shown in Fig. 5, right. The measurement uncertainties for 1.1 keV fluxes can be compared to the predicted ENA flux differences for the next-cloud versus the present cloud, e.g. $\|\Delta F_{\mathrm{model}}\|$=$\|$$F_{\mathrm{Mod3}}$–$F_{\mathrm{Mod2}}$$\|$, based on the Heerikhuisen et al. (2010) model. For the comparison we preselect data points with signal-to-noise S/N$>3$. In Fig. 6, individual pixels in the Ribbon region for both models have values of $\|\Delta F_{\mathrm{model}}\|$/$dF_{\mathrm{1\sigma}}$$>>5$. The same difference map is plotted in Fig. 7, but now color-coded to enhance the differences in the tail region. Lower ENA fluxes towards the tail yield $\|\Delta F_{\mathrm{model}}\|$/$dF_{\mathrm{1\sigma}}$$\sim 1-2$ for individual pixels, which is somewhat larger for Model 3 than for Model 4. Groups of 25 pixels would yield a factor of 5 improvement in the S/N of the difference maps, while effectively smoothing the data over $\sim 125$ square-degrees, and still should provide a significant test. In order to use ENAs from the tail for identifying the next-cloud, either pixels in the tail must be grouped to improve statistics, or the comparison should wait for the better statistics of future skymaps. The predicted ENA flux differences between the today-cloud and next cloud are testable with IBEX data. Twenty percent of the ESA 3 (1.1 keV) pixels with signal-to-noise S/N$>3$ test the flux differences between Model 3 and Model 2 at the $3\sigma$ level, or $\|\Delta F_{\mathrm{model}}\|$/$dF_{\mathrm{1\sigma}}$$>3$. In addition, 49% of the pixels test these flux differences at the $1\sigma$ level, with $\|\Delta F_{\mathrm{model}}\|$/$dF_{\mathrm{1\sigma}}$$>1$ (Fig. 8). Similar values are found for comparisons between the predicted flux differences between Model 4 and Model 2, where 18% of the pixels show model differences that are larger than the $3\sigma$ ESA3 flux uncertainties. We have also evaluated the variations in the 4.5 keV ENA fluxes for the environment of the next cloud (Fig. 9). Although the count rates in IBEX-HI ESA 6, at 4.5 keV, are lower by an order of magnitude than at 1.1 keV (Fig. 10, left), flux variations are predicted to occur when the effect of the increased interstellar density and velocity are included (Model 3 vrs. Model 2), as well as when the ISMF direction varies (Model 4 vrs. Model 2). For example, Model 2 has fluxes 5–6 counts cm-2 sec-1 sr-1 keV-1 at the locations $\lambda$,$\beta$=$253^{\circ},-33^{\circ}$ and $\lambda$,$\beta$=$268^{\circ},8^{\circ}$. For the same locations, Model 3 has fluxes a factor of $\sim 2$ higher. Further study of the energy spectrum of heliosheath ions is in progress, however, since the relatively large tail brightness predicted at 4.5 keV by Model 2 is difficult to distinguish in the data. ## 4 Discussion If the Ribbon ENAs are produced as secondary ENAs beyond the heliopause as suggested by McComas et al. (2009) and Schwadron et al. (2009), and quantified by Heerikhuisen et al. (2010), then the position and intensity of the Ribbon provides a robust diagnostic of the interstellar magnetic field direction, neutral densities, and the cloud ram pressure at the heliosphere. Differences between the velocity of ISM inside of the heliosphere and towards the nearest stars in the upwind direction indicate that the Sun is near or at the edge of an interstellar cloud. Based on observations of the upwind ISM, and assuming that the upwind cloud is in pressure equilibrium with the heliospheric ISM, the next cloud is modeled with densities that are $\sim 50$% larger, and a heliocentric velocity 9% larger, than the cloud today. We predict the flux of ENAs from the next interstellar cloud to surround the heliosphere, and compare those predictions with the measurement uncertainties of the 1.1 keV ENA fluxes detected by IBEX in its first 6 month skymap. These results rely explicitly on the assumption that the Heerikhuisen et al. (2010) model is a viable description of ENA production both for the cloud we are in today and for the nearby cloud observed towards $\alpha$ Cen and 36 Oph in the upwind direction. Although our detailed conclusions rely on the accuracy of the Heerikhuisen et al. (2010) model, this study is a useful gedanken experiment that will help us understand the Ribbon sensitivity to variations in the properties of the ISM around the heliosphere. The variations in ENA fluxes predicted for entry into the next cloud significantly exceed the measurement uncertainties for 20% of the ESA 3 pixels, which tend to be concentrated in the upwind hemisphere, and the variations are larger by a factor of two in some regions. The variations occur because the relative contributions of magnetic pressure and thermal ram pressure that deform the heliosphere are sensitive functions of the boundary conditions imposed by the ISM. If, in addition, the direction of the interstellar magnetic field shifts by as much as $28^{\circ}$, which is slightly larger than the Ribbon width, then significant differences in the ENA fluxes should be observed in individual pixels near the Ribbon. The heliosphere regions with very low ENA fluxes, such as the tail, provide a test of the cloud properties only if pixels are combined for better statistical significance before comparison with model predictions. The long term capability of IBEX to realistically detect such variations, which will also be superimposed on possible solar cycle variations, requires that the efficiencies in the IBEX sensors (conversion, scattering, sputtering in the conversion subsystem, secondary electron emission at the detector foils, microchannel plate efficiencies, for instance) either remain stable over years, or alternatively that the instrument performances are tightly monitored. IBEX-HI detector sensitivity is continuously monitored in a number of ways, such as comparison of coincident and non-coincident count rates (Funsten et al., 2005), and periodic gain tests. Variations in the energy dependence and fluxes of ENAs will occur because of the variation of solar wind properties over the solar cycle. These solar cycle contributions fortunately can be modeled in detail using past and present data on the solar wind, and models of the heliosphere response to these variations. Every IBEX skymap is a historical map of the solar cycle because of the energy dependence of ENA travel times and cross sections (McComas et al. submitted, 2010), so that unraveling the solar cycle dependence will simultaneously constrain the ENA production models and improve future predictions of the ENA variations due to the next interstellar cloud. In this discussion we considered the scenario where the next cloud is faster, slightly cooler, and more dense than the heliospheric ISM gas, as expected from pressure equilibrium and observational data. This study is a proof-of- concept, since the properties of the cloud edges are not established. The more extreme possibilities for the next galactic environment to be encountered by the Sun include a hot plasma without neutrals, and a cloud interface that is either evaporative or mixed with hot plasma by shear flows. If interstellar clouds within 10 pc are in pressure equilibrium, they will typically fill 20% of the sightline to the stars. The intervening voids evidently will be filled with the low density hot gas that creates the Local Bubble soft X-ray emission, although the emissivity of this plasma is somewhat uncertain because of solar wind contamination (Koutroumpa et al., 2008). Should the heliosphere enter the diffuse plasma attributed to the Local Bubble interior, both interstellar neutrals and exo-heliospheric ENAs will vanish. IBEX and other spacecraft will readily detect this condition. Another possibility for the next solar environment would be an evaporative interface that would form upwind between the heliospheric ISM and hot plasma. Such an interface will show steep increases in the cloud velocity and pressure, and decreases in density, over spatial scales that are determined by the angle between the ISMF and cloud surface (Slavin & Frisch, 2008). The present study considers the ENA fluxes for two separate models of the circumheliospheric cloud, but ignores possible variations due to changes in the heliosphere configuration during the transitions between the two clouds. The predicted thickness of the conductive boundary on the cloud around the heliosphere, defined as where the temperature falls to 50% of the asymptotic temperature, is 0.32–0.34 pc for an ISMF direction that makes an angle of 30o with the cloud surface (Slavin & Frisch, 2008, Models 26 and 27, also see Fig. 2, where the cloud edge starts at 3 pc). In the upwind direction the outflow speeds in the conductive boundary are 20–30 km s-1, and opposite to the cloud motion, so that the Sun could traverse the conductive boundary in approximately $\sim 12,000$ years for these models. Based on the above models, we expect the change in heliosphere properties for such an environment to be clearly observable in the resulting ENA flux detected by IBEX. A turbulent mixing layer will also produce strong gradients in the temperature and ionization of the surrounding ISM (Slavin et al., 1993). The ENA emissions for a conductive boundary on the surrounding cloud are discussed in detail in Grzedzielski et al. (2010), where the Sun is estimated to emerge from the interface within $\sim 500$ years. An alternative possibility is that the G-cloud may be denser than has been assumed in this study. If the interstellar $\mathrm{N(Ca^{+}})$ absorption formed at the G-cloud velocity is entirely within a few parsecs of the Sun, then comparisons between the clouds in the $\alpha$ Cen and $\alpha$ Oph sightlines suggest a tiny cold cloud in addition to the warmer gas (Frisch, 2003). The comparisons in this paper are made without consideration of the solar cycle, although the outer heliosheath regions respond to the variations in the solar wind dynamic pressure and magnetic field that characterize the solar activity cycle (Washimi & Tanaka, 1999; Scherer & Fahr, 2003; Zank & Müller, 2003; Pogorelov et al., 2009a). Although the solar cycle will cause the heliosphere to expand and contract as the solar wind dynamic pressure changes, these pulses travel only a relatively short distance upstream of the heliopause ($\sim 100$ AU). The influence of the solar cycle on ENA production and the Ribbon phenomenon is not yet understood. The Ribbon intensity may vary over latitudes due to the ion energy differences and travel times. The extremely low levels of solar activity during the first year of IBEX observations suggests the solar activity cycle variations must first be understood before reaching a conclusion that we have entered a new interstellar cloud (Pogorelov et al., 2008b; Sternal et al., 2008). As the theoretical models of ENA production become increasingly robust, we expect that studies such as this will yield definitive information on both the heliosphere boundary conditions and the physical properties of the interstellar cloud around the Sun. 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Quantity \| LICbbThese values for the ISM forming the heliosphere boundary conditions are based on Model 26 in Slavin & Frisch (2008) and Witte (2004). \| Model 1ccSchwadron et al. (2009) used this model (Pogorelov et al., 2008a) in the initial analysis of IBEX data. \| Model 2ddThis model reproduces the IBEX ribbon (Heerikhuisen et al., 2010). \| Model 3eeThe next-cloud model, assuming the same ISMF direction as the today-model, Model 2. \| Model 4ffThe next-cloud model, assuming an ISMF direction that differs from Model 2. ---\|---\|---\|---\|---\|--- \| \| Original \| Today \| Next (same ISMF) \| Next (new ISMF) $n\mathrm{(H^{\circ})}$ (cm-3) \| 0.19 \| 0.15 \| 0.15 \| 0.23 \| 0.23 $n\mathrm{(p^{+})}$ (cm-3) \| 0.06 \| 0.06 \| 0.06 \| 0.08 \| 0.08 $V$ (km s-1) \| –26.3 \| –26.4 \| –26.4 \| –28.8 \| –28.8 $T$ (K) \| 6300 \| 6530 \| 6530 \| 5400 \| 5400 $\|$${B}$$\|$ ($\mu$G) \| [2.7]ggDetermined by assuming that thermal and magnetic pressures are equal. \| 3. \| 3. \| 2.8 \| 2.8 ${B}$ direction, $\lambda$,$\beta$ \| \| $237.^{\circ},30^{\circ}$ \| $224^{\circ},41^{\circ}$ \| $224^{\circ},41^{\circ}$ \| $252^{\circ},42^{\circ}$ ${B}$ direction, $\ell$,$b$ \| \| $22.^{\circ},41^{\circ}$ \| $36^{\circ},53^{\circ}$ \| $36^{\circ},53^{\circ}$ \| $40.^{\circ},32^{\circ}$ Magnetosonic Mach \| 1.0 \| 1.1 \| 1.1 \| 0.8 \| 0.8 number \| \| \| \| \| Figure 1: Four models of ENA fluxes at 1.1 keV, based on different values for the ISM that constrains the heliosphere (see Table 1). Model 1 is the initial model used to evaluate the IBEX results (Schwadron et al., 2009). Model 2 is the assumed benchmark model for the production of ENAs observed by IBEX today (Heerikhuisen et al. 2010). Models 3 and 4 utilize the same model, but with different heliosphere boundary conditions appropriate for the next upwind cloud. Model 3 represents 1.1 keV ENAs for a heliosphere constrained by the physical parameters of the next cloud in the upwind direction, where it is assumed that cloud is in pressure equilibrium with the circumheliospheric gas. The same ISMF direction of $\lambda,\beta=224^{\circ},41^{\circ}$ is assumed for Models 2 and 3. Model 4 is the same as Model 3, except that the direction of the ISMF differs by $28^{\circ}$, and is directed towards $\lambda,\beta=252^{\circ},42^{\circ}$. The dashed lines, in this and subsequent figures, intersect at the longitude of the heliosphere nose ($\lambda=255.4^{\circ}$) and the ecliptic plane. Figure 2: The differences between the ENA fluxes produced by the nominal G-cloud and the nominal LIC. Left: Model 3–Model 2 ($F_{\mathrm{Mod3}}$– $F_{\mathrm{Mod2}}$). Right: Model 4–Model 2 ($F_{\mathrm{Mod4}}$– $F_{\mathrm{Mod2}}$). The different distortions of the heliosphere caused by variations in the relative magnetic and thermal ram pressures of the ISM are apparent on the northeast flank of the heliopause in the left figure. The right figure shows that the ribbon configuration is highly sensitive to variations in the ISMF direction. The differences in the distortion of the heliosphere towards the tail region are marginally visible. Fluxes are given in the units of counts cm-2 sec-1 sr-1 keV-1. Figure 3: The differences between the ENA fluxes produced by the nominal G-cloud and the nominal present-day cloud, after normalizing the fluxes in Models 3 and 4 by the ratio of the neutral densities (0.65). Left: ($n$(H∘)Mod2/$n$(H∘)Mod3)$F_{\mathrm{Mod3}}$– $F_{\mathrm{Mod2}}$. Right: ($n$(H∘)Mod2/$n$(H∘)Mod4)$F_{\mathrm{Mod4}}$– $F_{\mathrm{Mod2}}$. Zero values of the difference (blue) along the ribbon imply that the ribbons in the two models overlay each other. The background blue regions, where difference counts are $\sim 0$, indicate the regions where ENA fluxes are linearly related to the neutral interstellar density in these models. Fluxes are given in the units of counts cm-2 sec-1 sr-1 keV-1. Figure 4: The percentage differences between the 1.1 keV ENA fluxes produced by the nominal G-cloud and the nominal present-day cloud. Left: 100($F_{\mathrm{Mod3}}$–$F_{\mathrm{Mod2}}$)/$F_{\mathrm{Mod2}}$. Right: 100($F_{\mathrm{Mod4}}$–$F_{\mathrm{Mod2}}$)/$F_{\mathrm{Mod2}}$. The right figure shows that the ribbon configuration is highly sensitive to any variation in the ISMF direction. Figure 5: Left: The first 1.1 keV IBEX ENA skymap, corrected for the Compton- Getting effect (see text). Right: The $1\sigma$ statistical uncertainties on these data (see text). Figure 6: The differences between ENA fluxes predicted by models constrained by different boundary conditions, and compared to the $1\sigma$ measurement uncertainties in the ESA 3 energy passband (1.1 keV, see text). Left: The ratio of the absolute value of Model 3 minus Model 2 fluxes, to $1\sigma$ uncertainties ($\|$$F_{\mathrm{Mod3}}$–$F_{\mathrm{Mod2}}$$\|$)/$dF_{\mathrm{1\sigma}}$). Right: Same as left figure, but showing the absolute value of Model 4 minus Model 2 fluxes ($\|$$F_{\mathrm{Mod4}}$–$F_{\mathrm{Mod2}}$$\|$)/$dF_{\mathrm{1\sigma}}$). Pixels are left blank where fluxes are insignificant (S/N$<3$). Figure 7: Same plot as Fig. 6, except that the color scale is coded to enhance the low flux areas in the tail region. Figure 8: The ESA 3 fluxes (ordinate) are compared to the significance of the model differences (abscissa) at each pixel on the sky for Models 3 and 2 (left) and Models 4 and 2 (right). The abscissa shows the differences in the fluxes of the two models, divided by the $1\sigma$ measurement uncertainties on the ESA 3 data. Only ”good” ESA 3 pixels with S/N$>3$ are plotted. The differences between Models 3 and 2 are tested at the $3\sigma$ level by 20% of all ESA 3 pixels, while 49% of all pixels test these differences at the $1\sigma$ level, for example. Figure 9: The predicted ENA fluxes at 4.5 keV (ESA 6 energy passband) are shown for Model 2 (left) and Model 3 (right). Figure 10: The observed ESA 6 ENA fluxes at 4.5 keV are shown (left), together with the $1\sigma$ uncertainties on those fluxes (right).	arxiv-papers	2010-11-22T22:05:49	2024-09-04T02:49:15.149823	{ "license": "Public Domain", "authors": "P. C. Frisch, J. Heerikhuisen, N. V. Pogorelov, B. DeMajistre, G. B.\n Crew, H. O. Funsten, P. Janzen, D. J. McComas, E. Moebius, H.-R. Mueller, D.\n B. Reisenfeld, N. A. Schwadron, J. D. Slavin, G. P.Zank", "submitter": "Priscilla Chapman Frisch", "url": "https://arxiv.org/abs/1011.4962" }
1011.4968	# Model Dependence of the 2H Electric Dipole Moment Iraj R. Afnan School of Chemical and Physical Sciences Flinders University, GPO Box 2100, Adelaide 5001, Australia Iraj.Afnan@Flinders.edu.au Benjamin F. Gibson Theoretical Division, Los Alamos National Laboratory Los Alamos, NM 87545, USA bfgibson@lanl.gov ###### Abstract Background: Direct measurement of the electric dipole moment (EDM) of the neutron lies in the future; measurement of a nuclear EDM may well come first. The deuteron is one nucleus for which exact model calculations are feasible. Purpose: We explore the model dependence of deuteron EDM calculations. Methods: Using a separable potential formulation of the Hamiltonian, we examine the sensitivity of the deuteron EDM to variation in the nucleon- nucleon interaction. We write the EDM as the sum of two terms, the first depending on the target wave function with plane-wave intermediate states, and the second depending on intermediate multiple scattering in the 3P1 channel, the latter being sensitive to the off-shell behavior of the 3P1 amplitude. Results: We compare the full calculation with the plane-wave approximation result, examine the tensor force contribution to the model results, and explore the effect of short range repulsion found in realistic, contemporary potential models of the deuteron. Conclusions: Because one-pion exchange dominates the EDM calculation, separable potential model calculations will provide an adequate description of the 2H EDM until such time as a better than 10% measurement is obtained. ###### pacs: 11.30.Er,21.10.Ky,21.45.Bc,24.80.+y ## I Introduction With the discovery of parity ($P$) violation, which was suggested by Lee and Yang LY56 , Landau La57 deduced that charge conjugation and parity ($CP$) invariance implies that the electric dipole moment (EDM) of particles, e.g. the neutron, should be zero. If the $CPT$ theorem is valid, which is the case for gauge theories, then any $CP$ violation would also imply a corresponding time reversal ($T$) invariance violation. Predating the discovery of parity violation in the weak interaction, Purcell and Ramsey PR50 had pointed out that there was lacking any experimental test of parity conservation in the strong interaction. With their student Smith SPR57 they set limits on the EDM of the neutron of the order of $d_{n}<5\times 10^{-20}\,e\,cm$. The Standard Model of fundamental interactions predicts values for EDMs (due to second order W boson exchange) which are significantly smaller than contemporary experiments can detect, of the order of $10^{-31}\,e\,cm$. Therefore, an unambiguous observation of a nonzero EDM at current capabilities would imply a yet to be discovered source of $CP$ violation He95 ; KL97 . The new physics could arise in the strong interaction sector (e.g., the $\theta$ term), or in the weak interaction sector [e.g., Super Symmetric models or Left/Right (boson mass) symmetry breaking]. Current limits on the nucleon EDM are of the order of $10^{-26}\,e\,cm$. Even were one to establish a nonzero neutron and proton EDM, those two results would at best determine the isoscalar and isotensor components but would not isolate any isovector component. Thus, one would need a third measurement, such as the deuteron EDM, to fully elucidate the isospin nature of the EDM operator. Both $PT$ violating and $P$ conserving, $T$ violating potentials may give rise to an EDM He95 , but one-pion exchange contributes only to the former. We concentrate here upon the effects due to $PT$ invariance violation in the nuclear potential. The deuteron is attractive as the focus of an EDM investigation, both theoretically and experimentally, because a method has been proposed to directly measure the EDM of charged ions in a storage ring Kh98 ; Fa04 ; Se04 ; Or06 . A permanent EDM can arise because a $PT$ violating interaction can induce a small P-state admixture in the deuteron wave function, one which produces a non vanishing matrix element of the charge dipole operator $\tau^{z}_{-}e\vec{r}$. Although this two-body EDM contribution must be disentangled from the one-body contributions of the neutron and proton, the neutron and proton EDMs tend to cancel in the case of the isospin zero 2H. (If the nucleon EDM were a pure isoscalar as is the case in the $\theta$ model, then this cancellation would be exact.) Therefore, the $PT$ violating nucleon- nucleon (NN) interaction can contribute significantly to the deuteron EDM. Because the deuteron is reasonably understood and has been accurately modeled, reliable calculations are possible. Our purpose is to address the sensitivity of the deuteron EDM to the nuclear physics in the modeling of the nucleon- nucleon interaction. Beyond understanding the model dependence of the 2H EDM, our goal is to determine an appropriate model approximation with which one might reliably calculate the nuclear physics contribution to the 3He and 3H EDMs. Therefore, we examine the uncertainties in the deuteron EDM calculation arising from the short range repulsion in the ground state wave function, the dependence on the size of the deuteron D-state, and the properties of the 3P1 continuum in intermediate states. For the purpose of completeness and to place our work in context, we note that Avishai Av85 first estimated the two-body deuteron EDM [ see Eq. (2) ] $d_{D}^{(2)}$ using a separable potential model due to Mongan Mo69 . He reported a value of $-0.91\ A\,e\,fm$ when he utilized the physical pion mass for the exchanged meson. [Note: To exclude the $PT$ violating and strong coupling constants in the one pion exchange nucleon-nucleon interaction for the quoted values of the EDM, we have introduced $A=\bar{g}^{(1)}_{\pi NN}\,g_{\pi NN}/(16\pi)$.] However, there is an ambiguity in Avishai’s results, in that he states his final result in terms of $A/2$. Because the particular separable potentials used by Avishai were not specified, we were unable to fully confirm his reported numbers. Khriplovich and Korkin KK00 later estimated $d_{D}^{(2)}$ using a zero-range approximation in the chiral limit ($m_{\pi}\rightarrow 0$) and obtained a value of $-0.92\ A\,e\,fm$. This result does not depend upon the 3P1 interaction and should, therefore, be directly comparable to our ‘plane wave’ result. Finally, using the Argonne and Nijmegen contemporary realistic potential models A$v_{18}$, Reid93, and Nijm II SKTS94 Liu and Timmermans LT04 obtained for the polarization component of the two-body contribution to the deuteron EDM $d_{D}^{(2)}$ values of $-0.72\ A\,e\,fm$, $-0.73\ A\,e\,fm$, and $-0.74\ A\,e\,fm$, respectively. These relatively model-independent results suggest that pion exchange is indeed the essential aspect of the model. The differing degree of softness of the three potentials at intermediate range correlates with the values for $d_{D}^{(2)}$, the Nijm II potential being the softest and producing the largest EDM. The important conclusion for our purpose is that all three models yield essentially the same result; within the range of uncertainty defined by the three models utilized, the value of the polarization component of $d_{D}^{(2)}$ can be said to be $\approx-0.73\pm.01\ A\,e\,fm$. Moreover, Liu and Timmermans estimated that the meson exchange current contribution was substantially smaller, calculated to be less than 5% of the potential model contribution. In any case, our goal is to determine an appropriately simple model with which one can calculate reliably the 2H, 3He, and 3H EDMs, so that our numerical comparisons will be made with the $-0.73\pm.01\ A\,e\,fm$ value. ## II Nucleon contributions The total one-body contribution $d_{D}^{(1)}$ to the deuteron EDM due to the neutron and proton is the sum of the individual nucleon EDMs: $d_{D}^{(1)}=d_{n}+d_{p}\;,$ (1) whereas the total deuteron EDM is the sum of this one-body contribution and the two-body contribution $d_{D}^{(2)}$, $d_{D}=d_{D}^{(1)}+d_{D}^{(2)}=(d_{n}+d_{p})+d_{D}^{(2)}\;.$ (2) As has been noted, the neutron and proton EDMs can arise from a variety of sources. Because we have nothing new to add to prior analyses of the nucleon EDM, we adopt the approach advanced by Liu and Timmermans LT04 : $d_{D}^{(1)}\simeq 0.22\times 10^{-2}\bar{G}_{\pi}^{(1)}+O(\bar{G}_{\pi}^{(0,2)},\bar{G}_{\rho,\omega,\eta})\;,$ (3) which is expressed in terms of $\bar{G}_{X}^{(i)}$, the product of the strong coupling constant $g_{XNN}$ and the associated $PT$ violating meson-nucleon coupling constant $\bar{g}_{X}^{(i)}$. (For example, $\bar{G}_{\pi}^{(1)}=\bar{g}_{\pi NN}^{(1)}\,g_{\pi NN}$.) As noted in Ref. LT04 , the contributions from the neutron and proton EDMs have a sizable theoretical uncertainty, but the significant cancellation between $d_{n}$ and $d_{p}$ is clear. For the two-body contribution to $d_{D}^{(2)}$ the mean value obtained by Liu and Timmermans can be expressed as $d_{D}^{(pol)}=1.45\times 10^{-2}\bar{G}_{\pi}^{(1)}\;;$ (4) this corresponds to the EDM value of $-0.73\ A\,e\,fm$. Hence, for the deuteron there can be little doubt that the nuclear physics contribution to $d_{D}^{(2)}$ dominates. Even an uncertainty of 50% in $d_{D}^{(1)}$ contributes only in a minor way. It is the nuclear model aspects of the $d_{D}^{(2)}$ dominant term in the 2H EDM that we investigate below in detail. ## III Two-body Contributions The interaction Hamiltonian for the ground state of the system consists of two components: (i) The strong interaction component $v$ based on nucleon-nucleon potentials with parameters adjusted to fit the experimental phase shifts. (ii) The $PT$ violating component $V$ which we parametrize in terms of one pion exchange (OPE) with one strong interaction vertex $g_{\pi NN}$ and a $PT$ violating vertex $\bar{g}^{(1)}_{\pi NN}$. As a result our Hamiltonian takes the form $H=H^{S}+H^{PT}\quad\mbox{where}\quad H^{S}=H_{0}+v\quad\mbox{and}\quad H^{PT}=V\ .$ (5) Because $H^{PT}$ will mix different parity states, i.e., for the deuteron we get coupling between the 3S1-3D1 large component and the 3P1 small component, we can write the Schrödinger equation for the Hamiltonian in Eq. (5) $H\,\|\Psi\rangle=E\,\|\Psi\rangle$ (6) as a set of coupled equations of the form $\displaystyle(E-H_{0})\,\|\Psi_{L}\rangle$ $\displaystyle=$ $\displaystyle v\,\|\Psi_{L}\rangle+V\,\|\Psi_{S}\rangle\ $ (7) $\displaystyle(E-H_{0})\,\|\Psi_{S}\rangle$ $\displaystyle=$ $\displaystyle v\,\|\Psi_{S}\rangle+V\,\|\Psi_{L}\rangle\ ,$ (8) where the total wave function is the sum of the large and small components: $\|\Psi\rangle=\|\Psi_{L}\rangle+\|\Psi_{S}\rangle$. Because $V\ll v$, we have that $V\,\|\Psi_{S}\rangle\ll v\,\|\Psi_{L}\rangle$, and we can, to a good approximation, write Eq. (7) as $(E-H_{0})\,\|\Psi_{L}\rangle=v\,\|\Psi_{L}\rangle\ ,$ (9) which is the Scrödinger equation for the ground state of the system in the absence of the $PT$ violating interaction. On the other hand, the small component of the wave function $\|\Psi_{S}\rangle$ is given by the solution of Eq. (8) in terms of the amplitude $t(E)$ for the strong potential $v$ as $\|\Psi_{S}\rangle=G(E)\,V\,\|\Psi_{L}\rangle\quad\mbox{with}\quad G(E)=G_{0}(E)+G_{0}(E)\,t(E)\,G_{0}(E)\ ,$ (10) where $G_{0}(E)=(E-H_{0})^{-1}$ is the free Green’s function, and $t(E)$ is the amplitude in the partial wave of the small component of the wave function, e.g., for the deuteron $t(E)$ is the amplitude in the 3P1 partial wave at the ground state energy. Because the dipole operator $O_{d}=\frac{e}{2}\ \sum_{i}\vec{r}_{i}\ \tau_{z}(i)$ (11) is odd under parity, we can write the two-body deuteron EDM ($d_{D}^{(2)}$) in terms of the total ground state wave function $\|\Psi\rangle=\|\Psi_{L}\rangle+\|\Psi_{S}\rangle$ as $d_{D}^{(2)}=\langle\Psi\|\,O_{d}\,\|\Psi\rangle=\langle\Psi_{L}\|\,O_{d}\,\|\Psi_{S}\rangle+\langle\Psi_{S}\|\,O_{d}\,\|\Psi_{L}\rangle\ ,$ (12) where the matrix element of the dipole operator between the small and large component of the wave function can be written in terms of the charge $e$ and the constant $A$ as $\displaystyle\langle\Psi_{L}\|\,O_{d}\,\|\Psi_{S}\rangle$ $\displaystyle=$ $\displaystyle\langle\Psi_{L}\|\,O_{d}\,G_{0}(E)\,V\,\|\Psi_{L}\rangle+\langle\Psi_{L}\|\,O_{d}\,G_{0}(E)\,t(E)\,G_{0}(E)\,V\,\|\Psi_{L}\rangle\ $ (13) $\displaystyle\equiv$ $\displaystyle\frac{e}{2}\,\left[d_{PW}+d_{MS}\right]\,A\quad\mbox{with}\quad A\equiv\frac{\bar{g}^{(1)}_{\pi NN}\,g_{\pi NN}}{16\pi}\ .$ (14) In Eq.(13) the first term on the right hand side (rhs) involves a complete set of intermediate plane wave states and is, up to a constant, the ‘plane wave’ contribution $d_{PW}$. The second term on the rhs of Eq. (13) involves multiple scattering via the amplitude $t(E)$ and is the ‘multiple scattering’ contribution $d_{MS}$. One should note that $E<0$ is the ground state energy, and as a result we need the amplitude $t(E)$ at an unphysical point corresponding to the 2H bound state energy. ## IV Numerical results The primary motivation for the present investigation is: (i) to determine the sensitivity of $d_{D}^{(2)}$ to properties of the deuteron, e.g. the $D$-state probability and the short range behavior of the deuteron wave function. (ii) to determine the relative importance of $d_{PW}$ and $d_{MS}$. This will suggest the significance of multiple scattering terms as one proceeds to heavier nuclei. (iii) The role of the 3P1 interaction in determining the magnitude of $d_{MS}$ and therefore the appropriateness of the $d_{PW}$ approximation in heavier nuclei. Before we proceed to illustrate the sensitivity of the deuteron EDM to nuclear structure effects due to the nuclear interaction, we should detail our choice of nucleon-nucleon interactions and their fit to those aspects of the two-body data relevant to the determination of the EDM. ### IV.1 Two-body potentials The input two-body interactions consists of: (i) The $PT$ violating one pion exchange potential. (ii) The deuteron wave function in the absence of the $PT$ violating interaction. (iii) The 3P1 interaction that couples to the deuteron 3S1-3D1 potential as a result of the introduction of the $PT$ violating potential. The choice of these interactions is motivated by the questions raised regarding the sensitivity of the EDM to nuclear structure effects and the hope of extending the analysis to 3H and 3He using the $d_{PW}$ approximation. For the $PT$ violating interaction we have chosen the standard isovector one- pion exchange given by PH92 $V=-A\left[(\vec{\sigma}^{(-)}\cdot\hat{r})\,\tau_{z}^{(+)}+(\vec{\sigma}^{(+)}\cdot\tau_{z}^{(-)}\right]\,f(r)\ ,$ (15) where the radial dependence is given by $f(r)=-\frac{1}{m_{\pi}}\,\frac{d}{dr}\left(\frac{e^{-m_{\pi}r}}{r}\right)\ ,$ (16) with $m_{\pi}$ being the pion mass. Here we have combined the strength of the strong and $PT$ violating vertices in the constant $A$ given in Eq. (14). This allows us to express the numerical value of the EDM in terms of $A\,e$ with $e$ the charge on the proton. Finally, the spin and isospin operators in Eq. (15) are given by $\vec{\sigma}^{(\pm)}=(\vec{\sigma}^{(1)}\pm\vec{\sigma}^{(2)})$ and $\tau_{z}^{(\pm)}=(\tau_{z}^{(1)}\pm\tau_{z}^{(2)})$. The strong 3S1-3D1 interaction basically defines the deuteron wave function. Here we resort to a separable representation of the interaction to simplify the computation when we proceed to the EDM for the three-nucleon system. As a result the partial wave expansion of the strong interaction in momentum space is written as $\langle\vec{k}\|\,v\,\|\vec{k}^{\prime}\rangle=\sum_{Sjtm}\sum_{\ell\ell^{\prime}}\ \langle\hat{k}\|{\cal Y}^{t}_{(\ell S)jm}\rangle\ v^{Sjt}_{\ell\ell^{\prime}}(k,k^{\prime})\ \langle{\cal Y}^{t}_{(\ell^{\prime}S)jm}\|\hat{k}^{\prime}\rangle\ ,$ (17) with $\|{\cal Y}^{t}_{(\ell S)jm}\rangle$ eigenstates of the orbital angular momentum $\ell$, spin $S$, total angular momentum $j$ and isospin $t$. The separability of the potential is defined by the requirement that $v^{\alpha}_{\ell\ell^{\prime}}(k,k^{\prime})=g^{\alpha}_{\ell}(k)\ \lambda^{\alpha}_{\ell\ell^{\prime}}\ g^{\alpha}_{\ell^{\prime}}(k^{\prime})\ ,$ (18) where $\alpha=(Sjt)$. Here we wish to examine the role of the $D$-state probability and short range nature of the nucleon-nucleon interaction. For that we consider two classes of interactions: (i) The Yamaguchi and Yamaguchi (YY) YY54 separable potential with 4% and 7% $D$-state probability. Each has a different $D$-state probability and no short range repulsion. (ii) The Unitary Pole Approximation (UPA) AR73 ; AR75 to the original Reid soft core potential (Reid68) Re68 and the Nijmegen modified Reid potential (Reid93) SKTS94 . The UPA potential by definition generates the same deuteron wave function as the original potential AR75 that provided the optimum fit to the available data at the time the potentials were constructed and includes short range repulsion. In addition the models have different $D$-state probabilities for the deuteron. For the Yamaguchi and Yamaguchi potentials YY54 the form factor $g_{\ell}^{\alpha}(k)$ is given by $g_{\ell}(k)=\frac{k^{\ell}}{(k^{2}+\beta_{\ell}^{2})^{(\ell+2)/2}}\ ,$ (19) where the parameters $\beta_{\ell}$ and $\lambda_{\ell\ell^{\prime}}$ are detailed in Table 1. Also included in this table are the binding energy $\epsilon_{D}$ and the quadrupole moment $Q_{D}$ for these two potentials. Table 1: Parameters for the Yamaguchi-Yamaguchi potentials YY54 with 4% and 7% $D$-state probability for the deuteron. Also included are the binding energy and quadrupole moments. $D$-state \| $\beta_{0}$ \| $\beta_{2}$ \| $\lambda_{00}$ \| $\lambda_{02}$ \| $\lambda_{22}$ \| $\epsilon_{D}$ (MeV) \| $Q_{D}$ ---\|---\|---\|---\|---\|---\|---\|--- 4% \| 1.3134 \| 1.5283 \| -0.6419 \| 1.0849 \| -1.8320 \| 2.2234 \| 0.2821 7% \| 1.2410 \| 1.9480 \| -0.3776 \| 1.6975 \| -7.6301 \| 2.2265 \| 0.2826 Table 2: The strength $\lambda_{\ell\ell^{\prime}}$ for the UPA approximation to the Reid68 Re68 . and Reid93 SKTS94 potentials potential \| $\lambda_{00}$ \| $\lambda_{02}$ \| $\lambda_{22}$ ---\|---\|---\|--- Reid68 \| -5.2896725E-02 \| -2.4385786E+00 \| 1.1850926E+00 Reid93 \| -4.7704789E-01 \| -1.8111764E+00 \| 2.5825467E-01 In constructing the UPA to the Reid68 Re68 and Reid93 SKTS94 we have used the method of moments AR75 to solve the Schrödinger equation for the deuteron wave function in coordinate space using the original potentials. This was achieved by taking the form factors such that the resultant deuteron wave functions for the Reid68 and Reid93 are linear combinations of the Yamaguchi- Yamaguchi type wave functions with different range parameters $\beta_{i}$, and therefore of the form $g_{\ell}(k)=\sum_{i=1}^{12}\ \frac{c_{\ell}^{i}\ k^{\ell}}{(k^{2}+\beta_{i}^{2})^{(\ell+2)/2}}\ .$ (20) The strengths of the UPA potential ($\lambda_{\ell\ell^{\prime}}$), adjusted to reproduce the matrix elements of the original Reid68 and Reid93 potentials, are given in Table 2, while the parameters of the UPA form factors $\beta_{i}$ and $c^{i}_{\ell}$ for $\ell=0$ and $2$ are given in Table 3. Here, we have chosen the range parameters $\beta_{i}$ to be multiples of the pion mass with the hope of reproducing some of the analytic structure of the one pion tail in the original Reid potentials. Table 3: The form factor parameters of the UPA approximation to the Reid68 Re68 and Reid93 SKTS94 potentials . \| \| Reid68 \| Reid93 ---\|---\|---\|--- $i$ \| $\beta_{i}$ (fm-1) \| $c^{i}_{0}$ \| $c^{i}_{2}$ \| $c^{i}_{0}$ \| $c^{i}_{2}$ 1 \| 0.7 \| 7.21186419E-03 \| -2.24457073E-03 \| 6.30646724E-03 \| -3.08893140E-03 2 \| 1.4 \| 1.78826642E-01 \| -3.31063031E-01 \| 2.12846533E-01 \| -3.01564884E-01 3 \| 2.1 \| 1.31260692E+00 \| -1.04745293E+00 \| 6.05450638E+00 \| -1.78185516E+00 4 \| 2.8 \| 2.13430424E+00 \| -1.43628043E+00 \| -2.57777824E+01 \| 7.87042755E-01 5 \| 4.2 \| 1.46578861E+02 \| -1.95695256E+01 \| 3.20079733E+02 \| -2.53483826E+01 6 \| 5.6 \| -8.10387728E+02 \| 3.12782173E+00 \| -1.49174373E+03 \| 4.67387261E+01 7 \| 7.0 \| 1.12934549E+03 \| 1.51126963E+02 \| 2.32746050E+03 \| 3.37908596E+01 8 \| 9.8 \| -5.87779728E+02 \| -4.26701986E+02 \| -2.57402658E+03 \| -2.10353562E+02 9 \| 12.6 \| -2.27638508E+02 \| 5.92398037E+02 \| 2.53223423E+03 \| 3.41412020E+02 10 \| 15.4 \| 5.33784864E+02 \| -3.73533199E+02 \| -1.31246553E+03 \| -2.42126156E+02 11 \| 21.0 \| -2.53746105E+02 \| 9.68400708E+01 \| 2.66329930E+02 \| 7.60609941E+01 12 \| 26.6 \| 6.63870056E+01 \| -2.09513706E+01 \| -4.84106437E+01 \| -1.89979113E+01 To establish the quality of the UPA deuteron wave function generated using the method of moments we present in Table 4 the deuteron properties for the original potential and the UPA for both Reid68 and Reid93. Also included are the effective range parameters to illustrate the domain of agreement in the scattering amplitude between the original and the UPA potential. It is clear from these results that the method of moments gives a very good representation of the original deuteron wave function and can reproduce the effective range parameters. Table 4: Comparison of the deuteron properties for the original potential and the UPA potential for both Reid68 and Reid93. Tabulated are the binding energy $\epsilon_{D}$, the asymptotic $S$-wave normalization $A_{S}$, the ratio of the asymptotic $D$-wave to $S$-wave $\eta$, the quadrupole moment $Q_{D}$, and the $D$-state probability $P_{D}$. Also included are the scattering length $a_{t}$ and effective range $r_{t}$. \| Reid68 \| Reid93 ---\|---\|--- \| UPA \| Original \| UPA \| Original $\epsilon_{D}$ \| 2.2246 \| 2.2246 \| 2.2246 \| 2.2246 $A_{S}$ \| 0.87893 \| 0.87758 \| 0.8863 \| 0.8853 $\eta=A_{D}/A_{S}$ \| 0.026556 \| 0.026223 \| 0.02565 \| 0.0251 $Q_{D}$ \| 0.2800 \| 0.27964 \| 0.2709 \| 0.2703 $P_{D}$ \| 6.4691 \| 6.4696 \| 5.699 \| 5.699 $a_{t}$ \| 5.408 \| 5.390 \| 5.445 \| 5.422 $r_{t}$ \| 1.752 \| 1.720 \| 1.799 \| 1.755 Finally, to examine the importance of multiple scattering in determining the deuteron EDM, we need to introduce a 3P1 interaction to calculate $d_{MS}$. Here we need to know how important is the fit to the data and the role of the off-shell amplitude in determining the magnitude of $d_{MS}$. To simplify the evaluation of $d_{MS}$, we have chosen to use separable potentials with different form factors. The Mongan Mo69 potentials used by Avishai Av85 come with different form factors, and therefore different off-shell properties. They are either rank one or rank two to optimize the fit to the data; i.e., the potentials are of the form $v_{{}^{3}P_{1}}(k,k^{\prime})=\sum_{i=1}^{n}\ g_{i}(k)\,\lambda_{i}\,g_{i}(k^{\prime})\ ,$ (21) where $n=1$ for rank-one potentials and $n=2$ for rank-two potentials. For the form factors $g_{i}(k)$ we will use the four different forms chosen by Mongan (see Table 5). Considering the fact that Mongan adjusted the parameters of his potentials to fit the Livermore data of the 1960’s, we need first compare the phase shifts predicted by the Mongan potentials and those that we constructed to fit the latest Nijmegen Nij93 $np$ data. In Fig. 1 we compare the 3P1 phase shifts for rank-one and rank-two Case I form factors for Mongan’s potentials with those refitted to the Nijmegen data. Also included are the Nijmegen Nij93 $np$ phase shifts. It is clear from the the results in Fig. 1 that the original Mongan potentials give a poor fit to the current data, while the new fits reproduce the data to a much better degree. Since the 3P1 amplitude required for the determination of $d_{MS}$ is evaluated at the deuteron binding energy, i.e., below the elastic threshold, it is essential that we fit well the low energy phase shifts. Because these are small, we have chosen the criteria for a good fit $\chi^{2}$ defined as $\chi^{2}=\sum_{i=1}^{n}\ \frac{\|\delta_{i}^{\rm th}-\delta_{i}^{\rm exp}\|^{2}}{\|\delta_{i}^{\rm exp}\|^{2}}\ ,$ (22) where $n=11$ is the number of data points below 300 MeV. In Table 5 we present new fits to the Nijmegen $np$ data for the different form factors used by Mongan 111The Case III form factor was motivated by the observation that the on-shell Born amplitude for a rank-one separable potential is identical to the on-shell Born amplitude resulting from meson exchange potential with a meson mass $\beta_{1}$.. Included are rank-one and rank-two potentials and the $\chi^{2}$ for each potential. It is clear from the $\chi^{2}$ that the rank two potentials give a better fit. This is especially true for the Case I form factor. In the following discussion of the deuteron EDM we will consider these different 3P1 potentials to establish the importance of fitting the data and the role of the off-shell behavior of the amplitude. Figure 1: Comparison of the 3P1 phase shifts for the Mongan potentials (Old) with Case I form factor and rank one (R=1) and rank two (R=2) with the new fit (New) and the experimental ($Exp.$) Nijmegen Nij93 $np$ data. Table 5: The parameters of the ‘New’ rank-one and rank-two potentials with the different Mongan form factors. The parameters are adjusted by minimizing the $\chi^{2}$ defined in Eq. (22) taking the experimental phases from the latest Nijmegen Nij93 $np$ phase shift analysis. The form factor for Case III is written in terms of $Q_{1}(\xi)$ the Legendre function of the second kind. Potential \| form factor $g_{i}(k)$ \| Rank \| $\beta_{1}$ \| $\lambda_{1}$ \| $\beta_{2}$ \| $\lambda_{2}$ \| $\chi^{2}$ ---\|---\|---\|---\|---\|---\|---\|--- Case I \| $k/(k^{2}+\beta_{i}^{2})$ \| 1 \| 1.725 \| 0.95 \| - \| - \| 0.62 \| \| 2 \| 0.90 \| 0.059 \| 3.58 \| -2.0 \| 0.02 Case II \| $k/(k^{2}+\beta_{i}^{2})^{3/2}$ \| 1 \| 2.38 \| 9.35 \| - \| - \| 0.81 Case III \| $\left[\frac{1}{k^{2}\pi}Q_{1}(1+\frac{\beta_{i}^{2}}{2k^{2}})\right]^{1/2}~{}$ \| 1 \| 1.68 \| 60.0 \| - \| - \| 0.19 \| \| 2 \| 1.20 \| 120.0 \| 4.4 \| -2.3 \| 0.12 Case IV \| $k/(k^{2}+\beta_{i}^{2})^{2}$ \| 1 \| 2.715 \| 147.0 \| - \| - \| 0.78 ### IV.2 The deuteron EDM We now turn to the study of the sensitivity of the deuteron EDM to the nuclear structure effects as defined by the strong nucleon-nucleon interactions detailed above. We first consider the sensitivity of the two-body deuteron EDM $d_{D}^{(2)}$ to the $D$-state probability ($P_{D}$). In Table 6 we summarize the contributions to the deuteron EDM for the four different deuteron wave functions being considered. For the 3P1 interaction we use a rank-two Mongan Case I potential (fitted to the latest Nijmegen phase shifts Nij93 ). Also included are the results of Khriplovich and Korkin KK00 . We observe that in the plane wave approximation ($d_{PW}$) there is little variation with $P_{D}$, and the short range repulsion incorporated in the two Reid potential wave functions provides no more than a 10% reduction in $d_{PW}$. Moreover, the results are effectively consistent with the zero range (chiral limit) approximation of Khriplovich and Korkin. In particular, the plane wave results for the two YY models suggest that the dependence upon the deuteron $D$-state probability is such that an S-state deuteron result would approach that of Ref. KK00 . In contrast, the multiple scattering contribution ($d_{MS}$), which is of the opposite sign to the plane wave term, varies considerably depending upon the short range character of the deuteron wave function. In particular, the two Reid potentials with different $P_{D}$ values yield quite similar values of $d_{MS}$, but these are only half those generated by the YY potentials. The difference between the YY and Reid potential models can be understood in light of our knowledge that there is no explicit short range repulsion in the YY potentials. We will return to this difference when we address the role of the off-shell behavior of the 3P1 amplitude in determining the magnitude of the multiple scattering contribution $d_{MS}$. From these results we may conclude that the strong repulsion at short distance in realistic nucleon-nucleon potentials reduces the effects of multiple scattering in the matrix element to such an extent that the multiple scattering contribution $d_{MS}$ is only about 20% of the plane wave contribution $d_{PW}$. Furthermore, as noted above, the final results are not particularly sensitive to $P_{D}$. Table 6: The variation of the two-body EDM with $D$-state probability of the deuteron. For the 3P1 interaction we use the ‘New’ fit Case I rank-two potential. Also included are the results of Khriplovich and Korkin KK00 . 3S1-3D1 \| $P_{d}$ \| $d_{PW}(A\,e\,fm)$ \| $d_{MS}(A\,e\,fm)$ \| $d_{D}^{(2)}(A\,e\,fm)$ ---\|---\|---\|---\|--- YY 4% \| 4% \| -1.035 \| 0.4115 \| -0.6234 Reid93 \| 5.7% \| -0.9715 \| 0.2009 \| -0.7706 Reid68 \| 6.5% \| -0.9620 \| 0.1718 \| -0.7902 YY 7% \| 7% \| -1.083 \| 0.4271 \| -0.6564 Khriplovich et al. \| \| -0.92 \| \| To establish the importance of the multiple scattering contribution ($d_{MS}$) to the total two-body deuteron EDM, we turn to the dependence of $d_{D}^{(2)}$ on the choice of the 3P1 interaction. But first we need to examine the sensitivity of the multiple scattering contribution to the 3P1 phase shifts. This can be achieved by comparing the results for the EDM using the ‘Old’ Mongan fit to the 1960’s Livermore phase shift analysis and the ‘New’ fit with the same separable potential form factors to the latest Nijmegen Nij93 $np$ data. We have in Table 7 the EDM results for the rank-one separable potentials with Case I and III form factors. For the deuteron wave function we have used either the UPA to the Reid68 or the YY 4% potentials. It is clear from these results that the multiple scattering contribution ($d_{MS}$) is reduced as a result of the fit to the more recent phase shift analysis (compare rows four and five or rows six and seven in Table 7). This reduction in $d_{MS}$ is consistent with the observation that the ‘New’ 3P1 potentials provide less repulsion (i.e. smaller phase shifts, see Fig. 1) and, therefore, substantially smaller multiple scattering contributions than the old fits due to Mongan. This observation is encouraging for extending the above analysis based on $d_{PW}$ to the three-nucleon EDM, as the new $np$ data suggest a reduced contribution from the multiple scattering term. We now return to the role of the short range repulsion in the deuteron wave function on the magnitude of the multiple scattering term $d_{MS}$ as illustrated in Table 6. In comparing the results for the Reid68 and 4% YY deuterons (column three and five in Table 7) for the Case I and Case III 3P1 potentials, we find that the multiple scattering term is suppressed for both 3P1 potentials. This suggests that the effect tabulated in Table 6 might be valid in general. which implies that the inclusion of multiple scattering will require a more realistic treatment of the deuteron wave function than is the case for the zero range approximation employed by Khriplovich and Korkin KK00 . In fact for some combination of deuteron wave function and 3P1 interaction ( 4% YY and Case III Old) the multiple scattering contribution ($d_{MS}$) is about the same size as the plane-wave approximation ($d_{PW}$) and as a result the deuteron EDM $d_{D}^{(2)}$ is suppressed by an order of magnitude compared to the combination Reid68 and Case I New. Table 7: Variation in the deuteron EDM with changes in the $np$ phase shifts for two rank-one separable potentials having different form factors as defined by Mongan Mo69 . Here ‘New’ refers to the fit to the latest Nijmegen Nij93 $np$ phase shifts while ‘Old’ refers to the original Mongan fit. 3S1-3D1 \| Reid68 \| YY 4% ---\|---\|--- \| $d_{PW}=-0.96$ \| $d_{PW}=-1.04$ Case \| $\chi^{2}$ \| $d_{MS}$ \| $d_{D}^{(2)}$ \| $d_{MS}$ \| $d_{D}^{(2)}$ I (New) \| 0.62 \| 0.21 \| -0.75 \| 0.57 \| -0.47 I (Old) \| 1.90 \| 0.31 \| -0.66 \| 0.78 \| -0.26 III (New) \| 0.19 \| 0.25 \| -0.71 \| 0.77 \| -0.27 III (Old) \| 6.67 \| 0.42 \| -0.54 \| 1.16 \| 0.12 Table 8: The dependence of $d_{MS}$ on the 3P1 separable potential form factor as defined by Mongan Mo69 and that are fit to the latest $np$ phase shifts. The Reid93 or the 4% YY deuteron wave function is used in all cases as indicated. \| \| \| Reid93 \| YY 4% ---\|---\|---\|---\|--- \| \| \| $d_{PW}=-0.9715$ \| $d_{PW}=-1.035$ Case \| Rank \| $\chi^{2}$ \| $~{}d_{MS}(A\,e\,fm)$ \| $d_{D}^{(2)}(A\,e\,fm)$ \| $~{}d_{MS}(A\,e\,fm)$ \| $d_{D}^{(2)}(A\,e\,fm)$ I \| 1 \| 0.62 \| 0.2583 \| -0.7132 \| 0.5665 \| -0.4684 I \| 2 \| 0.02 \| 0.2009 \| -0.7706 \| 0.4115 \| -0.6234 II \| 1 \| 0.81 \| 0.2229 \| -0.7486 \| 0.3807 \| -0.6542 III \| 1 \| 0.19 \| 0.3075 \| -0.6640 \| 0.7654 \| -0.2696 III \| 2 \| 0.12 \| 0.3805 \| -0.5910 \| 1.108 \| 0.0734 IV \| 1 \| 0.78 \| 0.2153 \| -0.7562 \| 0.3277 \| -0.7072 We now turn to the role of the off-shell behavior of the 3P1 amplitude in the deuteron EDM. Here again we make use of the different separable potentials with the different form factors used by Mongan after readjusting the parameters of the potential to fit the latest Nijmegen Nij93 $np$ phase shifts. The parameters of these ‘New’ potentials are given in Table 5. In Table 8 we report the multiple scattering contribution $d_{MS}$ and the two- body EDM $d_{D}^{(2)}$ for these separable potentials. In each case we have made use of either the Reid93 or 4% YY deuteron wave functions in the calculations. Here we observe that for the Reid93 deuteron there is a smaller variation in $d_{D}^{(2)}$ than is the case for the 4% YY deuteron. This is due to the fact the the multiple scattering contributions, $d_{MS}$, for the 4% YY deuteron have a substantially larger variation for the different fits to the $np$ data. This is consistent with the results in Table 7 and is due to the absence of short range repulsion in the YY potentials. Here we can raise a number of questions regarding the role of the 3P1 amplitude in determining the magnitude of the multiple scattering contribution $d_{MS}$. These are: * • Why is $d_{MS}$ almost a factor of two smaller for the Reid93 when compared to that for the 4% YY potential? * • Why, for the Reid93 deuteron, is $d_{MS}$ about the same for all form factors with the possible exception of Case III which gives the largest contribution? * • Why is it that for the 4% YY deuteron $d_{MS}$ has a much larger variation than is the case for Reid93? To address these questions and to try to correlate the results in Table 8 with the off-shell behavior of the 3P1 amplitude, we need to examine the analytic continuation of the $P$-wave scattering wave function to the deuteron pole. This is defined in momentum space in terms of the half off-shell $t$-matrix as $\displaystyle\psi_{\alpha}(k)$ $\displaystyle=$ $\displaystyle G_{0}(-\epsilon_{D},k)\ t_{\alpha}(k,i\kappa;-\epsilon_{D})$ (23) $\displaystyle=$ $\displaystyle G_{0}(-\epsilon_{D},k)\ \mathbf{g}_{\alpha}(k)\,\bm{\tau}_{\alpha}(-\epsilon_{D})\,\mathbf{g}_{\alpha}^{\dagger}(i\kappa)$ $\displaystyle\equiv$ $\displaystyle\Psi_{\alpha}(k)\,\bm{\tau}_{\alpha}(-\epsilon_{D})\,\mathbf{g}_{\alpha}^{\dagger}(i\kappa)\ ,$ where $\alpha$ labels the 3P1 channel, $\epsilon_{D}=\frac{\kappa^{2}}{2\mu}$ is the binding energy of the deuteron, and $\mu$ is the $np$ reduced mass. Here the free Green’s function at the deuteron energy is given by $G_{0}(-\epsilon_{D},k)=-(2\mu)(\kappa^{2}+k^{2})^{-1}$, while the amplitude $t_{\alpha}(k,i\kappa;-\epsilon_{D})$ is the half off-shell 3P1 $t$-matrix evaluated at the deuteron pole. In the second line of Eq. (23) we have written the off-shell $t$-matrix in its separable form with $\bm{\tau}_{\alpha}(-\epsilon_{D})=\left[\,\bm{\lambda}_{\alpha}+2\mu\int\limits_{0}^{\infty}dk\,k^{2}\ \frac{\mathbf{g}_{\alpha}^{\dagger}(k)\,\mathbf{g}_{\alpha}(k)}{\kappa^{2}+k^{2}}\,\right]^{-1}\ .$ (24) For rank-one potentials $\bm{\tau}_{\alpha}(-\epsilon_{D})$ is positive definite since the potential is repulsive (i.e., $\lambda_{\alpha}>0$). As a result, the scattering wave function can be written as $\psi_{\alpha}(k)=\chi_{\alpha}(k)\ \sqrt{\tau_{\alpha}(-\epsilon_{D})}\ g_{\alpha}(i\kappa)\ .$ (25) This definition of the function $\chi_{\alpha}(k)$ is motivated in the following discussion of the matrix elements of the dipole operator $O_{d}$ and the $PT$ violating one pion exchange potential $V$ that go into the evaluation of $d_{MS}$. Figure 2: Comparison of the 3P1 ‘scattered function’ $\chi_{\alpha}(k)$ defined in Eq. (25) for the rank-one separable potentials that fit the latest Nijmegen Nij93 $np$ phase shifts. In Fig. 2 we plot the function $\chi_{\alpha}(k)$ for all the rank-one potentials used in Table 8. A careful inspection of this figure reveals that: (i) The ‘scattered function’ $\chi_{\alpha}(k)$ for the Case III form factor is substantially larger for $k<1.0$ fm-1 than that of the other three form factors. (ii) For $k>3$ fm-1 the Case III ‘scattered function’ has the longest range followed by Case I and then Case II and finally Case IV. This is clear from the choice of form factors as given in Table 5. To establish how this momentum dependence of the 3P1 ‘scattered function’ effects the multiple scattering contribution $d_{MS}$ to the deuteron EDM, we recall from Eq. (14) that $d_{MS}$ can be written as $d_{MS}=-2\left[\,\mathbf{O}_{sp}+\mathbf{O}_{dp}\,\right]\,\bm{\tau}(-\epsilon_{D})\,\left[\,\mathbf{V}_{ps}+\mathbf{V}_{pd}\,\right]\ ,$ (26) where $\mathbf{O}_{D\alpha}=\langle\Psi_{D}\|\,O_{d}\,\|\Psi_{\alpha}\rangle\quad\mbox{and}\quad\mathbf{V}_{\alpha D}=\langle\Psi_{\alpha}\|\,V\,\|\psi_{D}\rangle\ ,$ (27) with $D=\ ^{3}$S1 or 3D1 and $\alpha=^{3}$P1. For rank-one separable potentials we can absorb a factor of $\sqrt{\tau(-\epsilon_{D})}$ into the matrix elements, i.e. ${\cal O}_{\alpha p}\equiv O_{\alpha\beta}\sqrt{\tau(-\epsilon_{D})}$ and ${\cal V}_{p\alpha}\equiv\sqrt{\tau(-\epsilon_{D})}\,V_{p\alpha}$ and therefore for rank-one potentials we have $d_{MS}=-2\left[\,{\cal O}_{sp}+{\cal O}_{dp}\,\right]\ \left[\,{\cal V}_{ps}+{\cal V}_{pd}\,\right]\ .$ (28) The values of ${\cal O}_{\alpha p}$ and ${\cal V}_{p\alpha}$ for the four different form factors and with a deuteron wave function given by either the 4% YY or the Reid93 are presented in Table 9. It is clear from these results that the matrix elements of the dipole operator $O_{d}$, which is long range in coordinate space, are to a good approximation independent of the deuteron wave function and to within 20% independent of the 3P1 potential. On the other hand the matrix elements of the $PT$ violating one pion exchange potential, which probes the short range behavior of both the 3P1 and the deuteron wave function, are clearly model dependent. In particular, for the Reid93 deuteron with short range repulsion, the variation in ${\cal V}_{p\alpha}$ is small with the Case III form factor giving the largest contribution and Case IV yielding the smallest contribution followed by Case II and Case I. This is consistent with the observation made in the Fig. 2 insert regarding the asymptotic behavior of the function $\chi_{\alpha}(k)$. This is also consistent with the observation in Table 8 for rank-one potentials. On the other hand, for the 4% YY deuteron, with no short range repulsion, the matrix elements are almost a factor of two larger with the Case III form factor giving the largest contribution and Case IV the smallest. From the results in Table 9 we may conclude that it is the matrix element of the $PT$ violating one pion exchange potential that probes the short range behavior of the 3P1 and deuteron wave functions and, as a result, determines the magnitude of $d_{MS}$. To that extent it is essential that one generate those two wave functions in a consistent frame work. On the other hand, when the deuteron includes the short range behavior dictated by modern nucleon-nucleon interactions, the contribution of the multiple scattering term $d_{MS}$ is suppressed ($\approx 20$%) in comparison to the plane wave contribution $d_{PW}$. This suggests that one may be able to evaluate the EDM for the three-nucleon system in the plane wave approximation in such a model with an error of the order of 20%. Table 9: The matrix elements of the dipole operator $O_{d}$ and the $PT$ violating one pion exchange potential $V$ for the four different form factors and two different deuteron wave functions. deuteron \| Case \| ${\cal O}_{sp}$ \| ${\cal O}_{dp}$ \| ${\cal V}_{ps}$ \| ${\cal V}_{pd}$ ---\|---\|---\|---\|---\|--- 4% YY \| I \| -0.4197 \| -0.05599 \| 0.5533 \| 0.04211 \| II \| -0.4039 \| -0.05422 \| 0.3794 \| 0.03618 \| III \| -0.4819 \| -0.06300 \| 0.6578 \| 0.04444 \| IV \| -0.4185 \| -0.05622 \| 0.3124 \| 0.03276 Reid93 \| I \| -0.4221 \| -0.06069 \| 0.2169 \| 0.09031 \| II \| -0.4068 \| -0.05906 \| 0.1928 \| 0.04641 \| III \| -0.4852 \| -0.06712 \| 0.2269 \| 0.05154 \| IV \| -0.4224 \| -0.06105 \| 0.1793 \| 0.04338 Finally, the results in Table 9 for ${\cal O}_{\alpha p}$ and ${\cal V}_{p\alpha}$ indicate that the contribution from the $D$-wave component of the deuteron wave function are an order of magnitude smaller than the $S$-wave component. This may suggest that one could neglect the $D$-wave component in the calculating $d_{MS}$ and a simplification of the calculation of the multiple scattering term in heavier nuclei. This observation is consistent with the results in Table 6 where the changes in the multiple scattering contribution has a variation of about 10% with $D$-state probability. ## V Conclusions From our analysis we offer the following conclusions: (i) In the absence of multiple scattering ($d_{MS}=0$) the variation in $d^{(2)}_{D}$ due to differences in the deuteron wave functions is less than 5%, and the value of $d_{PW}$ is consistent with the zero range (chiral limit) results of Khriplovich and Korkin KK00 . (ii) The contribution from multiple scattering $d_{MS}$ is sensitive to the short range behavior of the deuteron wave function, and the $d_{MS}$ contribution is about 20% for realistic parametrizations of the deuteron such as those represented by the Reid93 potential model. This suggests that we can extend the analysis to heavier nuclei in the plane wave approximation with an estimated error of $\approx 20$%. (iii) As suggested by Liu and Timmermans, one pion exchange dominates the deuteron EDM calculation. (iv) The contribution from the 3P1 interaction via $d_{MS}$ depends on the phase shifts in this channel as well as the off- shell behavior of the amplitude. (v) A comparison of our Reid93 results with those of Liu and Timmermans LT04 indicates that one can use a separable potential approximation in heavier nuclei, e.g., 3He and 3H, with minimal loss in accuracy. Moreover, until deuteron EDM experiments attain an uncertainty of less than 10%, simple separable potential model calculations should provide an adequate description. ## VI Acknowledgement The work of BFG was performed under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No DE-AC52-06NA25396. ## References * (1) T. D. Lee and C. N. Yang, Phys. Rev. 104, 254 (1956). * (2) L. Landau, Nucl. Phys. 3. 127 (1957). * (3) E. M. Purcell and N. F. Ramsey, Phys. Rev. 78, 807 (1950). * (4) J. H. Smith, E. M. Purcell, and N. F. Ramsey, Phys. Rev. 108, 120 (1957). * (5) P. Herczeg, Symmetries and Fundamental Interactions in Nuclei edited by W. C. Haxton and E. M. Henley (World Scientific, Singapore), 1995, pp. 89-125. * (6) B. Khriplovich and S. K. Lamoreaux, CP Violation without Strangeness: Electric Dipole Moments of Partllces, Atoms, and Molecules (Springer-Verlag, Berlin, 1997). * (7) I. B. Khriplovich, Phys. Lett. B 444, 98 (1998). * (8) F. J. M. Farley, K. Jungmann, J. P. Miller, W. M. Morse, Y. F. Orlov, B. L. Roberts, Y. K. Semertzidis, A. Silenko, and E. J. Stephenson, Phys. Rev. Lett. 93, 052001 (2004). * (9) Y. K. Semertzidis, M. Aoki, M. Auzinsh, V. Balankin, A. Bazhan, G. W. Bennett, R. M. Carey, P. Cushman, P. T. Debevec, A. Dudnikov, F. J. M. Farley, D. W. Hertzog, et al., in Intersections of Particle and Nuclear Physics: 8th Conference, CIPANP2003, edited by Z. Parsa, AIP Conf. Proc. 698, 200 (2004). * (10) Y. F. Orlov, W. M. Morse, and Y. K. Semertzidis, Phys. Rev. Lett. 96, 214802 (2006). * (11) Y. Avishai, Phys. Rev. D 32, 314 (1985). * (12) T. R. Mongan, Phys. Rev. 175, 1260 (1968); Phys. Rev. 178, 1597 (1969). * (13) I. B. Khriplovich and R. V. Korkin, Nucl. Phys. A 665, 365 (2000). * (14) V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen, and J. J. de Swart, Phys. Rev. C 49, 2950 (1994). * (15) C.-P. Liu and R. G. E. Timmermans, Phys. Rev. C 70, 055501 (2004). * (16) P. Herczeg, Hyperfine Interaction, 75, 127 (1992). * (17) Y. Yamaguchi and Y. Yamaguchi, Phys. Rev. 95, 1635 (1954). * (18) I. R. Afnan and J. M. Read, Australian J. Phys. 26, 725 (1973); * (19) I. R. Afnan and J. M. Read, Phys. Rev 12, 293 (1975). * (20) R. V. Reid, Ann. Phys. (N.Y.) 50, 411 (1968). * (21) V. G. J. Stoks, R. A. M. Klomp, M. C. M. Rentmeester, and J. J. de Swart, Phys. Rev. C 48, 792 (1993).	arxiv-papers	2010-11-22T22:39:14	2024-09-04T02:49:15.158889	{ "license": "Public Domain", "authors": "Iraj R. Afnan and Benjamin F. Gibson", "submitter": "Benjamin Gibson", "url": "https://arxiv.org/abs/1011.4968" }
1011.5068	# GLASMA EVOLUTION IN PARTONIC MEDIUM A.V. Nazarenko nazarenko@bitp.kiev.ua Bogolyubov Institute for Theoretical Physics, 14-b, Metrologichna Str., Kiev 03680, Ukraine ###### Abstract We examine a scenario of the abelianized Glasma evolution with accounting for back-reaction of partonic medium in ultrarelativistic heavy-ion collisions. We announce that such a generalization leads to the instabilities and the presence of negative color conductivity in the system. heavy-ion collisions; negative color conductivity ###### pacs: 24.85.+p, 41.20.Jb ## I INTRODUCTION Phenomenological analyses of experimental data indicate that the quark-gluon plasma (QGP) can be formed in ultrarelativistic A+A collisions exp . Its local thermalization and isotropization should be mainly related to the fast processes stimulated by instabilities at small times after collision Mrow ; Mrow2 . In the present theoretical picture of ultrarelativistic heavy-ion collisions ETS , the early stage is preferably characterized by the large number of partons with “small” momenta of the order of the so-called saturation momentum $\Lambda_{s}$, which are better viewed as a classical Yang-Mills field in vacuum GV , sometimes named as “Glasma”. The initial conditions for Glasma evolution are determined by the Color Glass Condensate (CGC) concept by McLerran-Venugopalan (MV) McLerran , where the field sources before collision can be presented by the randomly distributed valent quarks of colliding hadrons and are located (due to Lorentz contraction) on infinitesimally thin sheets running along the light-cone. These sources are also treated as the hard partons with “large” momenta, which escape quickly from the system after collision. Thus, the original MV model neglects the interaction between the field and the hard partonic medium. The space-time dynamics of the Yang-Mills fields in vacuum (“the melting of CGC”) in assumption of boost invariance was investigated numerically, and the energy and the number distributions of the classically produced gluons were computed (see, for example, review Leon ). Moreover, it was shown that the violations of boost invariance cause a non-Abelian Weibel instability Wei leading the field (soft) modes to grow with proper time RV . However, the effect of isotropization is out of this model. On the other hand, the hard partons (produced after moment of collision) with large transverse momentum $p_{T}$ can be studied within the framework of transport theory, and if the presence of the soft classical field is neglected, the time evolution of the these partons is described by Boltzmann equation with a collision kernel GM ; BMS ; GPZ ; MG ; DG ; NVC (for comment, see 111This list of references contains also some works on approaches based on the relaxation time approximation to the Boltzmann equation.). However, it has been argued that the collective processes caused by the soft gauge field should be dominant in equilibration of QGP instabilities developed due to anisotropic distributions of released hard partons DN ; XG ; BMSS ; Akk . The third regime where the back-reaction of the field on the hard partons (treated as particles) is still weak but where the self-interaction of the former may be strongly nonlinear is governed by a “hard-loop” effective action which has been derived in Ref. MRS for arbitrary momentum-space anisotropies. It is interesting to note that the numerical studies of anisotropic hard partonic modes coupled to unstable soft modes revealed the tendency of the non-Abelian gauge fields to “abelianize” during the stage of instabilities AL ; DN . It means that the field commutators become much smaller than the fields themselves. Moreover, the dynamics of the Abelian and non-Abelian fields is qualitatively the same, if these fields are not strong enough. In this paper, we examine a scenario of Glasma evolution with CGC-like initial conditions, when the presence of the (momentum-)anisotropic medium of hard partons is also taken into account. Our goal is to evaluate analytically the behavior of such a system in short-time interval in weak-interaction regime, when the application of the abelianized version of the field dynamics is possible. Although the last condition demands to consider the system at relatively large times after A+A collision (as follows from numerical investigations) but simplifies the problem considerably. However, it is already pointed out in Refs. GKNS ; SNK ; SKN that the early equilibration of QGP is not necessary to describe pion and kaon spectra observed experimentally at RHIC in Brookhaven. Since the momentum-space anisotropy of the system can be estimated by means of transport coefficients, we attempt to calculate a conductivity tensor and to determine an effect of instabilities on it. It is expected that the back- reaction can lead to a negative color conductivity in the boost-invariant case. ## II THE MODEL FORMULATION As was mentioned in Introduction, the classical Yang-Mills theory in space- time with pseudo-cylindrical metric $\displaystyle ds^{2}=d\tau^{2}-\tau^{2}d\eta^{2}-dr_{T}^{2}-r^{2}_{T}d\varphi^{2},$ (1) $\displaystyle\tau=\sqrt{t^{2}-z^{2}},\quad\eta=\frac{1}{2}\ln{\frac{t+z}{t-z}},$ (2) ($\tau$ and $\eta$ are proper time and space-time rapidity, respectively) has been abelianized since $\tau_{0}\approx 3/\Lambda_{s}$, where $\Lambda_{s}\approx 2$ GeV GV . It means that we actually come to the Maxwell theory with 4-potential $A_{\mu}$ (hereafter, we neglect the normalization constant $1/\sqrt{N_{c}}$, where $N_{c}$ is the number of colors). The free-field theory in mid-rapidity region in the case of central collisions, when the potentials are parametrized as $A_{\tau}=0$ (CGC-like gauge fixing), $A_{\eta}\equiv\Phi(\tau,r_{T})$, $A_{r_{T}}=0$, $A_{\varphi}\equiv\Psi(\tau,r_{T})$, has been already examined (see Ref. SNK ) in order to describe the space-time evolution of the field flow (collective velocity) at pre-thermal stage of collisions. It turns out that the results obtained are qualitatively the same like in the case of non-Abelian model from Ref. KF . Here we generalize the abelianized Glasma equations by inclusion of sources in the right-hand side: $\displaystyle\partial^{2}_{\tau}\Phi-\frac{1}{\tau}\partial_{\tau}\Phi-\partial^{2}_{r_{T}}\Phi-\frac{1}{r_{T}}\partial_{r_{T}}\Phi=J_{\eta}(\tau,r_{T}),$ (3) $\displaystyle\partial^{2}_{\tau}\Psi+\frac{1}{\tau}\partial_{\tau}\Psi-\partial^{2}_{r_{T}}\Psi+\frac{1}{r_{T}}\partial_{r_{T}}\Psi=J_{\varphi}(\tau,r_{T}).$ (4) Note that $J_{\tau}=0$, $J_{r_{T}}=0$ and the current conservation is satisfied automatically. In the context of A+A collisions, the presence of the sources corresponds to the existence of the medium. Accounting for hard partonic component (viewed as particle subsystem), we aim to investigate the field and particle dynamics. The components of the current in Minkowskian space-time are $J^{\mu}=g\int p^{\mu}(f_{+}-f_{-})\frac{d^{3}p}{p^{0}},$ (5) where $p^{0}\equiv\|{\bf p}\|$ (the case of massless partons), $f_{\pm}$ are distribution functions of (scalar) partons. The distribution of partons is supposed to be anisotropic in the momentum space and inhomogeneous in configuration one. Space-time development of functions $f_{\pm}$ is determined by Vlasov equations which we will formulate below. Note that “$-g$” corresponds to the charge of electron in the context of electrodynamics. A toy field model with non-trivial right-hand side has been already investigated in Ref. ALMY . It is useful to parametrize momenta as $(p^{\mu})=(p_{T}\cosh{y},p_{T}\cos{\phi},p_{T}\sin{\phi},p_{T}\sinh{y})$, where $y$ is momentum rapidity. In the terms of our variables, one has: $\displaystyle J_{\tau}=g\int p^{2}_{T}\cosh{\theta}(f_{+}-f_{-})dp_{T}dyd\phi,$ (6) $\displaystyle J_{\eta}=-\tau g\int p^{2}_{T}\sinh{\theta}(f_{+}-f_{-})dp_{T}dyd\phi,$ (7) $\displaystyle J_{r_{T}}=-g\int p^{2}_{T}\cos{\xi}(f_{+}-f_{-})dp_{T}dyd\phi,$ (8) $\displaystyle J_{\varphi}=-r_{T}g\int p^{2}_{T}\sin{\xi}(f_{+}-f_{-})dp_{T}dyd\phi,$ (9) where $\theta=y-\eta$, $\xi=\phi-\varphi$. Taking conditions $J_{\tau}=0$, $J_{r_{T}}=0$ into account, the difference $f_{+}-f_{-}$ should be odd function of $\theta$ and $\xi$ during evolution. The evolution of $f_{\pm}$ is generated by Vlasov equations: $(\hat{L}\pm g\hat{F})f_{\pm}=0,$ (10) where $\hat{L}\equiv p^{\mu}\partial_{\mu}$, $\hat{F}\equiv p^{\mu}F_{\mu\nu}\partial^{\nu}_{p}$, $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.$ Since the sources (partons) are randomly distributed at the initial moment $\tau_{0}$, we put $f^{0}_{+}=f^{0}_{-}=f^{0}$, where $f^{0}$ is defined as $(dN_{h}/d^{3}xd^{3}p)\|_{\tau_{0}}$, and $\hat{L}f^{0}=0$ in our investigations. It means that the system is neutral and the currents are absent at $\tau_{0}$. Using the curved coordinates, we obtain $\hat{L}=p_{T}\left(\cosh{\theta}\partial_{\tau}+\frac{\sinh{\theta}}{\tau}\partial_{\eta}+\cos{\xi}\partial_{r_{T}}+\frac{\sin{\xi}}{r_{T}}\partial_{\varphi}\right),$ (11) $\displaystyle\hat{F}=-\frac{\partial_{\tau}\Phi}{\tau}\partial_{y}+\frac{\partial_{r_{T}}\Phi}{\tau}(\sinh{\theta}\sin{\xi}\partial_{\phi}-\cosh{\theta}\cos{\xi}\partial_{y})$ $\displaystyle+\frac{\partial_{\tau}\Psi}{r_{T}}(\sinh{\theta}\sin{\xi}\partial_{y}+\cosh{\theta}\cos{\xi}\partial_{\phi})+\frac{\partial_{r_{T}}\Psi}{r_{T}}\partial_{\phi}.$ (12) We can see that the operator of Lorentz force $\hat{F}$ is simply the operator of rotation in momentum space and, therefore, conserves the absolute value of transverse momentum $p_{T}$. It is expected that such structure of the Lorentz force should lead to the momentum transmission between different directions and, consequently, to the instabilities in this system. ## III SOLUTION OF EQUATIONS In this Section, we concentrate on finding the solution of the set of the coupled Maxwell–Vlasov equations. Since the Glasma field is essential at early stage of nuclear collisions (in contrast with the hard partons or quarks), the study of the field dynamics actually dominates. By this way, it is necessary to express the particle currents through the fields. In general, the Vlasov equations are complicated. For this reason, we are forced to use a method for approximate solving this set. Fluctuation of distribution function, which arises during fairly small time interval $\Delta\tau$, can be found in the linear approximation in $g$: $f_{\pm}=f^{0}\mp g\delta f.$ (13) In this approximation, the space-time evolution of correction $\delta f$, determining difference $f_{+}-f_{-}=-2g\delta f$ and the current components, results from the following equation: $\hat{L}\delta f=\hat{F}f^{0}.$ (14) Note that this approximation does not permit us to investigate an isotropization of the particle (hard parton) kinematic part of the energy- momentum tensor, proportional to the sum $f_{+}+f_{-}$. It is expected that such isotropization effect can be observed if the correction of the order $g^{2}$ is included. Nevertheless, the approximation under consideration allows one to study the instabilities in the system. If $\tau\to\tau_{0}$, there exists an approximate solution, which is short- living in time and localized in space, $\delta f\approx\frac{\tau-\tau_{0}}{p_{T}\cosh{\theta}}\hat{F}f^{0}.$ (15) It is easy to verify that the action of evolution operator $\hat{L}$ on this expression gives us $\hat{L}\frac{\tau-\tau_{0}}{p_{T}\cosh{\theta}}\hat{F}f^{0}=\hat{F}f^{0}+(\tau-\tau_{0})W(\tau),$ (16) where $\displaystyle W(\tau)$ $\displaystyle=$ $\displaystyle\left[\partial_{\tau}+\frac{\tanh{\theta}}{\tau}(\partial_{\eta}+\tanh{\theta})\right.$ (17) $\displaystyle+$ $\displaystyle\left.\frac{\cos{\xi}}{\cosh{\theta}}\partial_{r_{T}}+\frac{\sin{\xi}}{r_{T}\cosh{\theta}}\partial_{\varphi}\right](\hat{F}f^{0}).$ Thus, if $\tau\to\tau_{0}$, the last term in right-hand side of (16) vanishes. Also note that this is the simplified proof of Eq. (15). In order to understand this approximation in details, see Appendix A. Often the model initial boost-invariant distributions in central heavy-ion collisions take the form $f^{0}=f^{0}(p_{T},\theta)=f^{0}(p_{T},-\theta)$ (note that for the sake of correctness $f^{0}$ has to be also a function of $r_{T}$). In this case, we obtain $J_{\eta}=\sigma_{\eta}(\tau)\partial_{\tau}\Phi,\qquad J_{\varphi}=\sigma_{\varphi}(\tau)\partial_{\tau}\Psi,$ (18) where $\sigma_{\eta}(\tau)=2(\tau-\tau_{0})\sigma_{0},\qquad\sigma_{\varphi}(\tau)=-(\tau-\tau_{0})\sigma_{0}$ (19) are conductivities. The common multiplier dependent on the initial distribution of partons is $\sigma_{0}\equiv-2\pi g^{2}\int\limits_{0}^{\infty}dp_{T}\int\limits_{-\infty}^{\infty}dy\partial_{y}f^{0}p_{T}\tanh{\theta}.$ (20) We can immediately see that $\sigma_{\eta}(\tau)$ and $\sigma_{\varphi}(\tau)$ have the different signs. It says about the presence of negative color conductivity driving to instability in the system. The mechanism of this instability looks simple: we deal with situation when the particles (partons) give the energy to the field. Now let us analyze the properties of $\sigma_{0}$. Firstly, we assume that the initial distribution $f^{0}$ is the product of the function of $(p_{T},\theta)$ and the spatial distribution $(dN_{h}/d^{3}x)\|_{\tau_{0}}(r_{T})$ in transverse plane. Taking into account that the initial distribution is even function of $\theta$, one gets $\sigma_{0}=A\left.\frac{dN_{h}}{d^{3}x}\right\|_{\tau_{0}}>0,$ (21) where $A$ is a positive constant arising after integration over momentum variables. Thus, $\sigma_{\varphi}<0$, while $\sigma_{\eta}>0$. It means that the color negative conductivity takes place in the transverse plane. At this stage the natural question arises: how the negative conductivity does look in the laboratory reference frame. Eqs. (18) are actually the Ohm law, where $E_{\eta}\equiv F_{\tau\eta}=\partial_{\tau}\Phi$, $E_{\varphi}\equiv F_{\tau\varphi}=\partial_{\tau}\Psi$ are the (chromo)electric field strengths. Introducing $E_{i}=F_{ti}$ in the Minkowskian space-time, we find that $E_{\eta}=\tau E_{z}$, $E_{\varphi}=r_{T}\cosh{\eta}(-\sin{\varphi}E_{x}+\cos{\varphi}E_{y})+r_{T}\sinh{\eta}(-\sin{\varphi}F_{zx}+\cos{\varphi}F_{zy})$. In these terms the current components are $\displaystyle J_{t}$ $\displaystyle=$ $\displaystyle-\sinh{\eta}\sigma_{\eta}E_{z},\quad J_{z}=\cosh{\eta}\sigma_{\eta}E_{z},$ (22) $\displaystyle J_{x}$ $\displaystyle=$ $\displaystyle-\frac{\sin{\varphi}}{r_{T}}\sigma_{\varphi}E_{\varphi},\quad J_{y}=\frac{\cos{\varphi}}{r_{T}}\sigma_{\varphi}E_{\varphi}.$ (23) If $\eta=0$ and $\varphi=0$, one has that $J_{t}=J_{x}=0$, $J_{y}=\sigma_{\varphi}E_{y}$, $J_{z}=\sigma_{\eta}E_{z}$. Thus the color negative conductivity takes place under some conditions (related with the value of angles) in the laboratory reference frame. This effect is associated with filamentation in the plasma Mrow2 . Since it is hard to find the general solution of field equations for arbitrary distribution $(dN_{h}/d^{3}x)\|_{\tau_{0}}$, we try to study the particular case, when $(dN_{h}/d^{3}x)\|_{\tau_{0}}={\rm const}$. This assumption simplifies the problem significantly. When $\sigma_{0}$ is a constant, the spatial dependence of the field potentials is immediately derived by using the Bessel-Fourier transform: $\displaystyle\Phi(\tau,r_{T})$ $\displaystyle=$ $\displaystyle\int\limits_{0}^{\infty}\Phi_{0}(k_{T})g_{\eta}(\tau,k_{T})J_{0}(k_{T}r_{T})dk_{T},$ (24) $\displaystyle\Psi(\tau,r_{T})$ $\displaystyle=$ $\displaystyle r_{T}\int\limits_{0}^{\infty}\Psi_{0}(k_{T})g_{\varphi}(\tau,k_{T})J_{1}(k_{T}r_{T})dk_{T},$ (25) where the initial conditions resulting from CGC concept are applied: $\displaystyle g_{\eta}\|_{\tau_{0}}=0,\quad\left.\frac{\partial_{\tau}g_{\eta}}{\tau}\right\|_{\tau_{0}}=k_{T},$ (26) $\displaystyle g_{\varphi}\|_{\tau_{0}}=1,\quad\tau\partial_{\tau}g_{\varphi}\|_{\tau_{0}}=0.$ (27) In principal, functions $g_{\eta}(\tau,k_{T})$, $g_{\varphi}(\tau,k_{T})$ can be expressed for arbitrary $\tau_{0}\not=0$ in terms of Heun functions. However, these expressions are complicated for heuristic analysis of our model and its applications. For this reason, we write down the functions $g_{\eta,\varphi}$ at $\tau_{0}\to 0$: $g_{\eta}=-\frac{k_{T}}{2\sigma_{0}}\exp{\left(\frac{1}{2}\sigma_{0}\tau^{2}\right)}M\left(-\frac{k^{2}_{T}}{4\sigma_{0}},\frac{1}{2};-\sigma_{0}\tau^{2}\right),\qquad g_{\varphi}=\frac{1}{\tau}\sqrt{\frac{2}{\sigma_{0}}}\exp{\left(-\frac{1}{4}\sigma_{0}\tau^{2}\right)}M\left(\frac{k^{2}_{T}-\sigma_{0}}{2\sigma_{0}},0;\frac{1}{2}\sigma_{0}\tau^{2}\right),$ (28) where $M(a,b;z)$ is the Whittaker function. It is easy to verify that the occurrence of negative color conductivity $\sigma_{\varphi}$ leads to a growth of the some components of magnetic and electric fields in comparison with the case of the theory without partonic medium. It is important that the Abelian magnetic field exhibits a growth, draining some energy from the particle reservoir. The instabilities are related with the presence of exponents in functions $g_{\eta,\varphi}$; Whittaker functions change actually a phase of oscillations only in comparison with the free theory. If $\sigma_{0}\to 0$ and $\tau_{0}\to 0$, we come to the well-known expressions (see, for example, Ref. Lappi and references therein): $g_{\eta}(\tau,k_{T})=\tau J_{1}(k_{T}\tau),\quad g_{\varphi}(\tau,k_{T})=J_{0}(k_{T}\tau),$ (29) where $J_{n}(z)$ is the Bessel function. These expressions correspond to the perturbative (lowest order in the source charge densities) solution. Now it is necessary to determine the functions $\Phi_{0}(k_{T})$ and $\Psi_{0}(k_{T})$. They originate from the initial conditions for the field equations. Note that $\Psi_{0}(k_{T})$ and $\Phi_{0}(k_{T})$ are fluctuating quantities within the CGC concept, and the pair correlator of the (Yang–Mills) potentials is observable only. However the field potentials in our approach are not stochastic quantities in the contrast with CGC ideology because we goal to constitute the initial conditions on the base of the statistically averaged components of the energy-momentum tensor accounting for the spatial inhomogeneity. In order to derive $\Psi_{0}(k_{T})$ and $\Phi_{0}(k_{T})$, let us use the energy density distribution and the requirement of the absence of field flow at the initial moment. In mid-rapidity region ($\eta=0$) and $\varphi=0$ (note that transverse directions are equal in the system with cylindrical symmetry), when $T_{tt}=T_{\tau\tau}$, $T_{tx}=T_{\tau x}$, we have that $\displaystyle T_{tt}\|_{\tau_{0}}$ $\displaystyle\equiv$ $\displaystyle\varepsilon(r_{T})$ (30) $\displaystyle=$ $\displaystyle\frac{1}{2}\left(\left.\frac{\partial_{r_{T}}\Psi}{r_{T}}\right\|_{\tau_{0}}\right)^{2}+\frac{1}{2}\left(\left.\frac{\partial_{\tau}\Phi}{\tau}\right\|_{\tau_{0}}\right)^{2},$ $\displaystyle T_{tx}\|_{\tau_{0}}$ $\displaystyle=$ $\displaystyle 0,$ (31) where $\varepsilon(r_{T})$ is assumed to be the known function from numerical calculations or physical point of view. Our trick consists in division of the energy density between different field components: $\partial_{r_{T}}\Psi\|_{\tau_{0}}=\sqrt{\alpha}r_{T}f(r_{T}),\quad\left.\frac{\partial_{\tau}\Phi}{\tau}\right\|_{\tau_{0}}=\sqrt{1-\alpha}f(r_{T}),$ (32) where $f(r_{T})\equiv\sqrt{2\varepsilon(r_{T})}$ and $\alpha$ is a some separation constant (in general, $\alpha$ should be function of $r_{T}$). Since the potentials $\Psi$, $\Phi$ are real, one has that $0\leq\alpha\leq 1$. In principal, $\alpha$ is arbitrary constant. Practically, it turns out that $\alpha\approx 1/2$ (it follows from comparison of electric and magnetic strengths within the numerical approach). Note that the observables of the source-free theory are independent on $\alpha$. Thus, one finds $\Psi_{0}(k_{T})=\sqrt{\alpha}\tilde{f}(k_{T}),\quad\Phi_{0}(k_{T})=\sqrt{1-\alpha}\tilde{f}(k_{T}),$ (33) here $\tilde{f}(k_{T})=\int\limits_{0}^{\infty}f(r_{T})J_{0}(k_{T}r_{T})r_{T}dr_{T}.$ (34) These expressions finally determines the fields in our model. ## IV APPLICATIONS In the previous Sections we have formulated the model of Glasma in hard partonic medium. Since the classical field modes are usually interpreted as soft partons, their spatial dependence at early stage of A+A collision may be done within the framework of Glauber model. However, the explanations of experimental data can be efficiently done with application of another distributions too. As it was demonstrated in Ref. SKN , the Gaussian distribution of soft partons leads to the adequate pion spectra produced after collision at RHIC. To formulate the field initial conditions, here we would like to choose the same approximation for the energy density at the initial moment, $\varepsilon(r_{T})=E\exp{\left(-\frac{r^{2}_{T}}{2R^{2}}\right)},$ (35) where $E=45$ GeV/fm3, $R=3.768$ fm. Then we find that $\tilde{f}(k_{T})=2^{3/2}\sqrt{E}R^{2}\exp{(-k_{T}^{2}R^{2})}.$ (36) The boost-invariant distribution function $f^{0}$ (defining conductivities) is completely arbitrary at this point, so in order to proceed one needs to assume a specific form for it. In what follows we will require that $f^{0}$ is obtained from isotropic function, $N_{0}\exp{\left(-\frac{p^{0}}{p_{h}}\right)},$ (37) by the replacement $y\to\theta$ in $p^{0}=p_{T}\cosh{y}$ and by the rescaling of one dimension in momentum space, $f^{0}=N(\zeta)\exp{\left(-\frac{p_{T}}{p_{h}}\sqrt{\cosh^{2}{\theta}+\zeta\sinh^{2}{\theta}}\right)},$ (38) where $p_{h}$ takes the role of saturation moment, $\zeta>-1$ is a parameter reflecting the strength of the partonic medium anisotropy and $N(\zeta)$ is a normalization constant. Note that $\zeta>0$ corresponds to a contraction of the distribution in the $z$-direction whereas $-1<\zeta<0$ corresponds to a stretching of the distribution in the $z$-direction. Constant $N(\zeta)$ is simply determined by requiring the number density to be the same both for isotropic and anisotropic systems and can be evaluated (by integration over momentum variables) to be $N(\xi)=N_{0}\sqrt{1+\zeta}.$ (39) Integrating over momentum, the multiplier defining the conductivities is $\sigma_{0}=4\pi g^{2}N_{0}p^{2}_{h}C(\zeta),$ (40) where $C(\zeta)=\frac{2}{3}(1+\zeta)^{3/2}F\left(\left[2,\frac{3}{2}\right],\left[\frac{5}{2}\right],-\zeta\right).$ (41) The coefficient $C(\zeta)$ in the region $\zeta\in(-1,\infty)$ is determined by hypergeometric function $F$ and is such that $C(-1)=0$ (the case of source- free theory), $C(0)=2/3$ (for isotropic medium), $C(\infty)=\pi/2$. Figure 1: Time evolution of the field energy density split into longitudinal and transverse electric ($E$) and magnetic ($B$) components at $r_{T}=0.1$ [fm], $\eta=0$, $\varphi=0$. The top panel corresponds to the free theory with $\sigma_{0}=0$. The bottom panel demonstrates the growth of $E_{L}$ and $B_{T}$ at $\sigma_{0}=0.25$ [fm-2]. Fig. 1 shows how the exponentially growing energy transferred from hard to soft partons is distributed among magnetic and electric fields at $\sigma_{0}\not=0$ in comparison with the case of free field theory. The dominant contribution is still in longitudinal electric field (in accordance with CGC-like initial conditions). Nevertheless, we see that the transverse magnetic field demonstrates unstable behavior too while this effect is absent in the free theory. Since the particle subsystem gives the energy to the field, the total field energy density (as the sum of components) tends to grow with time. Note that a similar model with expanding Abelian field coupled to the hard partons, when the strict boost invariance of fields is relaxed, has been already developed in Ref. RR in the context of the quark-gluon plasma. ## V CONCLUSIONS Generalizing the space-time evolution of the expanding Glasma with CGC-like initial conditions by inclusion of small density of the hard partons anisotropically distributed in momentum space, we observe the instabilities (due to transferring energy from hard to soft partons) in the case of the abelianized boost-invariant model. Here we propose to measure an anisotropy by means of transport coefficients like the conductivity tensor in the contrast with the usual approach based on the energy-momentum tensor. As the result, the instabilities in the system under consideration lead to a conclusion of the presence of negative color conductivity in A+A collisions. The sign of conductivity in transverse plane depends on the angle what says about filamentation inherent to Weibel instabilities in plasma. Unfortunately, the approximate solution derived to the transport equations does not permit us to achieve the isotropization here. This problem should be investigated in details and will be published elsewhere. ## ACKNOWLEDGEMENTS The research of author was partially supported by the Foundation of Department of Physics and Astronomy of NAS of Ukraine. ## APPENDIX A. PROPAGATOR OF TRANSPORT EQUATION Here, we would like to discuss in details the approximation which we have applied previously. To find $\delta f$, we have to determine operator $\hat{L}^{-1}$, inverse to the first-order evolution operator $\hat{L}$. The inversion procedure of the evolution operator of transport equation was elaborated by Landau and, generally speaking, results in emergence of the Landau damping in plasma. Let $G$ is the solution of the following equation: $\hat{L}G(\tau,\theta,{\bf r}_{T}\|\tau^{\prime},\theta^{\prime},{\bf r}_{T}^{\prime})=\frac{\delta(\tau-\tau^{\prime})}{\tau}\delta(\theta-\theta^{\prime})\delta^{2}({\bf r}_{T}-{\bf r}_{T}^{\prime}),$ (42) where right hand side is the 4-dimensional $\delta$-function with respect to the pseudo-cylindric measure $\tau d\tau d\theta d^{2}r_{T}$. Without loosing generality, we limit ourselves by the case, when $\tau_{0}\to 0$, and by using the transverse coordinates ${\bf r}_{T}$ instead of cylindrical $(r_{T},\varphi)$. Using the formulas from Appendix B, $G$ is represented as $G(\tau,\theta,{\bf r}_{T}\|\tau^{\prime},\theta^{\prime},{\bf r}_{T}^{\prime})=\lim_{\varepsilon\to+0}\frac{i}{(2\pi)^{4}p_{T}}\int\frac{{\rm e}^{i\omega[\tau\cosh{(\xi-\theta)}-\tau^{\prime}\cosh{(\xi-\theta^{\prime})}]-i{\bf k}_{T}({\bf r}_{T}-{\bf r}^{\prime}_{T})}}{{\bf k}_{T}{\bf v}_{T}-\omega\cosh{\xi}-i\varepsilon}\omega d\omega d\xi d^{2}k_{T},$ (43) where ${\bf v}_{T}\equiv{\bf p}_{T}/p_{T}$ and the Landau damping is already taken into account due to auxiliary formula: $\lim_{\varepsilon\to+0}\frac{1}{x-i\varepsilon}={\cal P}\frac{1}{x}+i\pi\delta(x).$ (44) Discarding the spatial dispersion, we have to assume that $k_{T}\ll\omega\cosh\xi$. It leads to simplification: $\displaystyle G(\tau,\theta,{\bf r}_{T}\|\tau^{\prime},\theta^{\prime},{\bf r}_{T}^{\prime})\approx\frac{1}{p_{T}}\Theta(\tau\cosh{\theta}-\tau^{\prime}\cosh{\theta^{\prime}})$ $\displaystyle\times\delta(\tau\sinh{\theta}-\tau^{\prime}\sinh{\theta^{\prime}})\delta^{2}({\bf r}_{T}-{\bf r}^{\prime}_{T}).$ (45) where $\Theta$ is Heaviside function defined as $\Theta(x<0)=0$, $\Theta(x=0)=1/2$, $\Theta(x>0)=1$. This approximation says that the system is homogeneous at significantly large times and the fluctuations are localized in space. Furthermore, let the Björken scaling flow, when $\theta\approx 0$, take place. It means that $\theta$ and $\theta^{\prime}$ should be equal and gives us that $G(\tau,\theta,{\bf r}_{T}\|\tau^{\prime},\theta^{\prime},{\bf r}_{T}^{\prime})\approx\frac{\Theta(\tau-\tau^{\prime})}{p_{T}\tau^{\prime}\cosh{\theta^{\prime}}}\delta(\theta-\theta^{\prime})\delta^{2}({\bf r}_{T}-{\bf r}^{\prime}_{T}).$ (46) More precisely, it can be derived from condition, $\tau\sinh\theta={\rm const}$, resulting in $\sinh\theta d\tau+\tau\cosh d\theta=0,$ (47) where $d\tau=\tau-\tau^{\prime}$, $d\theta=\theta-\theta^{\prime}$. Assuming that $\tau$ is small and the expression under integration is not essentially changed at this time range, we can do the following replacement: $\int\limits_{0}^{\infty}d\tau^{\prime}\Theta(\tau-\tau^{\prime})F(\tau^{\prime})\to\tau F(\tau).$ (48) Then, one obtains that $\displaystyle\int G(\tau,\theta,{\bf r}_{T}\|\tau^{\prime},\theta^{\prime},{\bf r}_{T}^{\prime})F(\tau^{\prime},\theta^{\prime},{\bf r}_{T}^{\prime})\tau^{\prime}d\tau^{\prime}d\theta^{\prime}d^{2}r^{\prime}_{T}\approx$ $\displaystyle\approx\frac{\tau}{p_{T}\cosh{\theta}}F(\tau,\theta,{\bf r}_{T}).$ (49) This formula determines the solution of inhomogeneous transport equation with a source $F$. ## APPENDIX B. INTEGRAL TRANSFORMATION The Fourier transformation reads $f(t,z)=\int\limits_{-\infty}^{\infty}\frac{d\mu d\mu^{\prime}}{(2\pi)^{2}}\int\limits_{-\infty}^{\infty}f(p,s){\rm e}^{i\mu(t-p)-i\mu^{\prime}(z-s)}dpds.$ Let us introduce new variables: $\displaystyle t=\tau\cosh{\theta},\quad z=\tau\sinh{\theta},$ $\displaystyle p=\rho\cosh{\psi},\quad s=\rho\sinh{\psi},$ $\displaystyle\mu=\lambda\cosh{\phi},\quad\mu^{\prime}=\lambda\sinh{\phi}.$ If $f(t,z)=F(\tau,\theta)$, one gets the following transformation rule: $\displaystyle\tilde{F}(\lambda,\phi)=\int\limits_{-\infty}^{\infty}F(\rho,\psi){\rm e}^{-i\rho\lambda\cosh{(\phi-\psi)}}\rho d\rho d\psi,$ $\displaystyle F(\tau,\theta)=\frac{1}{(2\pi)^{2}}\int\limits_{-\infty}^{\infty}\tilde{F}(\lambda,\phi){\rm e}^{i\tau\lambda\cosh{(\phi-\theta)}}\lambda d\lambda d\phi.$ For example, we find that $\frac{1}{\tau}\delta(\Delta\tau)\delta(\Delta\theta)=\frac{1}{(2\pi)^{2}}\int\limits_{-\infty}^{\infty}{\rm e}^{i\lambda[\tau\cosh{(\phi-\Delta\theta)}-\tau_{0}\cosh{\phi}]}\lambda d\lambda d\phi,$ where $\Delta\tau=\tau-\tau_{0}$, $\Delta\theta=\theta-\theta_{0}$. ## References * (1) I. 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Phys. G: Nucl. Part. Phys. 35 (2008) 104071. * (25) R.J. Fries, J.I. Kapusta and Y. Li, Nucl. Phys. A774, 861 (2006). * (26) P. Arnold, J. Lenaghan, G.D. Moore and L.G. Yaffe, Phys. Rev. Lett. 94, 072302 (2005). * (27) T. Lappi, Phys. Lett. B643, 11 (2006); arXiv: hep-ph/0606207 (2006). * (28) P. Romatschke, A. Rebhan, Phys. Rev. Lett. 97, 252301 (2006); hep-ph/0605064 (2006).	arxiv-papers	2010-11-23T11:49:50	2024-09-04T02:49:15.169837	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "A.V. Nazarenko", "submitter": "Andrey Nazarenko", "url": "https://arxiv.org/abs/1011.5068" }
1011.5082	# The parity of specular Andreev reflection under mirror operation in zigzag graphene ribbon Yanxia Xing1,3, Jian Wang1,∗, and Qing-feng Sun2,† 1Department of Physics and the Center of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China. 2Beijing National Lab for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China. 3Department of Physics, Beijing Institute of Technology, Beijing 100081, China. ###### Abstract It is known that the parity of reflection amplitude can either be even or odd under the mirror operation. Up to now, all the parities of reflection amplitude in the one-mode energy region are even under the mirror operation. In this paper, we give an example of odd parity for Andreev reflection (AR) in a three-terminal graphene-supercondutor hybrid systems. We found that the parity is even for the Andreev retroreflection (ARR) and odd for specular Andreev reflection (SAR). We attribute this remarkable phenomenon to the distinct topology of the band structure of graphene and the specular Andreev reflection involving two energy bands with different parity symmetry. As a result of odd parity of SAR, the SAR probability of a four-terminal system with two superconducting leads (two reflection interfaces) can be zero even when the system is asymmetric due to the quantum interference of two ARs. ###### pacs: 72.80.Vp, 74.45.+c, 73.40.-c, 74.25.F- Since the experimental realization of grapheneref1 , it has become an exciting arena for theoretical and technological investigations.ref2 A number of new phenomena have been predicted and verified experimentally. For instance, in the presence of magnetic field, it exhibits a distinctive half-integer quantum Hall effect.ref1 Its quasi-particles obey the Dirac-like equation and have relativistic-like behaviors.ref2 Due to the relativistic effect, the Klein tunneling occurs where an incident electron in graphene can pass through a potential barrier with probability one.Klein Then a graphene p-n junction can be used to focus Dirac electron current with a negative refractive index.altshuler ; lens Since good contacts between superconducting leads and graphene have been realized experimentally,ref16 the transport study through graphene based normal-metal-superconductor (GNS) heterojunction becomes feasible. In the presence of a normal metal (graphene)-superconducting interface, an incoming electron converts into a hole and a cooper pair is formed that enters the superconductor. Due to the relativistic nature of the electron in graphene, the electron-hole conversion can either be intraband (within conduction or valence band) or interband (between conduction and valence bands). When the electron-hole conversion is intraband, it corresponds to the usual Andreev reflection (AR)ref15 or Andreev retroreflection (ARR) because the reflected hole is along the incident direction. This ARR occurs for both relativistic and non-relativistic electrons. When the electron-hole conversion is interband, the reflected hole is along specular direction and a specular Andreev reflection (SAR) takes place,Beenakker which can lead to novel phenomena as we will discuss below. It is known that the parity is a fundamental quantity in physics and reflection is a general physical phenomenon in nature. In this paper, we discuss the parity of reflection amplitude for graphene in contact with superconductor leads. In general, the parity of a reflection amplitude can be either even or odd when the system is under mirror operation. However, for all previous known reflection events, the reflection amplitudes in the one-mode energy region have even parity under the mirror operation. It is yet to find an odd-parity reflection event. In this paper, we found, for the first time, that the SAR amplitude has an odd parity under the mirror operation for zigzag graphene ribbons with even number of chains. This means that the phases of SAR amplitude for a graphene-superconductor hybrid system and its mirror system differ by $\pi$. We attribute this phenomenon to the unique band structure of the graphene. Obviously this phase difference does not affect any observable quantities for each system. When two systems couple together, however, this $\pi$ phase manifest through quantum interference between two SARs. So this $\pi$ phase shift has important consequences for a four terminal device with two superconducting leads (see Fig.2(a)). When two superconducting leads are symmetrically attached to the device, the quantum interference of the left and right SAR leads to a destructive or constructive interference depending on whether the phase difference of superconducting leads is zero or $\pi$. Importantly, when two superconducting leads are asymmetrically attached to the device, the same interference pattern occurs provided that the Dirac point $E_{0}$ is in line with the condensate of superconducting lead. The quantum interference between pairs of the AR can be tuned by shifting the Dirac point, the asymmetry of the two superconducting leads, as well as the phase between two superconducting leads. Due to the odd parity of SAR, the interference pattern for SAR is phase contrasted to that of ARR where the parity is even. Before doing numerical calculation, we first prove that the phases of SAR amplitude of two systems (i) and (ii) in Fig.1(a) differ by $\pi$, i.e., the parity of SAR is odd under mirror operation. Note that for graphene systems electrons in valence and conduction band are usually referred as electrons and holes, respectively. In the presence of superconducting lead the reference point of electrons and holes is the Fermi level in the superconducting lead. In the following, we will refer electrons (holes) as electrons above (below) Fermi level in superconducting lead. Denote $\psi^{+}_{c}$ ($\psi_{v}^{+}$) the wavefunction of electrons in conduction (valence) band moving in +y direction and $\psi^{-}_{c}$ ($\psi_{v}^{-}$) in -y direction in the zigzag graphene nanoribbon lead. It was known that under reflection ${\hat{P}}:x\rightarrow-x$, $\psi_{c}^{\pm}$ is symmetric while $\psi_{v}^{\pm}$ is anti-symmetric if the energy of electron is in the first transmission channelselect [see Fig.1(a)], i.e., $\displaystyle{\hat{P}}\psi_{c}^{\pm}(x,y)$ $\displaystyle=$ $\displaystyle\psi_{c}^{\pm}(-x,y)$ $\displaystyle{\hat{P}}\psi_{v}^{\pm}(x,y)$ $\displaystyle=$ $\displaystyle-\psi_{v}^{\pm}(-x,y).$ (1) which is one of the unique features of zigzag edge nanoribbons with even number of chains. Assuming the incident electron from the terminal-1, the wavefunctions for SAR $\psi_{1,3}$ in zigzag nanoribbon lead 1 or 3 of the system (i) can be written as $\displaystyle\psi_{1}^{(i)}$ $\displaystyle=$ $\displaystyle\psi_{e}^{+}+r_{11}\psi_{e}^{-}+r_{11A}\psi_{h}^{-}$ $\displaystyle\psi_{3}^{(i)}$ $\displaystyle=$ $\displaystyle t_{13}\psi_{e}^{+}+r_{13A}\psi_{h}^{+}$ (2) where $r_{11}$ is the normal reflection amplitude, $t_{13}$ is the transmission amplitude, $r_{11A}$ and $r_{13A}$ are the Andreev reflection amplitudes with the reflected hole to the terminal-1 and 3, respectively. Similarly the wavefunctions for the system (ii) are given by $\displaystyle\psi_{1}^{(ii)}$ $\displaystyle=$ $\displaystyle\psi_{e}^{+}+{\bar{r}}_{11}\psi_{e}^{-}+{\bar{r}}_{11A}\psi_{h}^{-}$ $\displaystyle\psi_{3}^{(ii)}$ $\displaystyle=$ $\displaystyle{\bar{t}}_{13}\psi_{e}^{+}+{\bar{r}}_{13A}\psi_{h}^{+}$ (3) Since the system (i) is related to (ii) by the reflection operator ${\hat{P}}$, we have $\psi_{\alpha}^{(i)}={\hat{P}}\psi_{\alpha}^{(ii)}$ with $\alpha=1,3$. Note that for SAR, the electron is in the conduction band while the hole is in the valence band, i.e., $\psi_{e}=\psi_{c}$ and $\psi_{h}=\psi_{v}$. From this relation together with Eqs.(1), (2), and (3), we obtain $\displaystyle r_{11A}=-{\bar{r}}_{11A},~{}~{}~{}r_{13A}=-{\bar{r}}_{13A}$ $\displaystyle r_{11}={\bar{r}}_{11},~{}~{}~{}t_{13}={\bar{t}}_{13}$ (4) Note that the origin of this $\pi$ phase shift (odd parity) is the interband conversion from the electron to the hole. Therefore the $\pi$ phase shift does not occur for ARR since it involves only intraband conversion. Now we verify this statement numerically using a tight-binding model (see below for detailed description of the model and numerical procedure). The numerical results of AR probability $R_{11A(13A)}=\|r_{11A(13A)}\|^{2}$ for two systems are shown in Fig.1(b). As expected the AR probability are exactly the same for two systems. However, the phase of AR amplitudes $r_{11A(13A)}$ denoted as $\Phi^{i,ii}_{11(13)}$ are different. It is shown in Fig.1(c) and Fig.1(d) that ARR amplitudes ($\|E_{0}\|>\|E_{F}\|$, with $\|E_{F}\|=0.5$) are the same for two systems in Fig.1(a) while the SAR amplitudes ($\|E_{0}\|<\|E_{F}\|$) have a $\pi$ phase shift. It confirms the odd parity for interband electron-hole conversion, which comes from the distinct topology of the band structure of graphene. To see the consequence of the odd parity of SAR, we examine a symmetric four- terminal device with two superconducting leads depicted in Fig.2(a) (by setting asymmetry $\delta N=0$ and phase difference $\delta\phi=0$). For this system, two beams from terminal-1 has a $\pi$ phase shift due to odd parity of SAR and interferes destructively at terminal-3 giving rise to a vanishing SAR coefficient. However, we can arrive the same conclusion using symmetry argument as follows. Since the system is symmetric with respect to $x=0$, we must have $r_{13A}={\bar{r}}_{13A}$ when the reflection operation along x-direction is applied. While from Eq.(4), $r_{13A}=-{\bar{r}}_{13A}$. So the AR probability $R_{13A}=\|r_{13A}\|^{2}$ for SAR can also be zero from symmetry point of view.cheng Therefore we conclude that the symmetric device can not be used to test the odd parity of SAR. In the following, we demonstrate that due to the $\pi$ phase shift the destructive interference still occurs in a four-probe devices with two superconducting leads attached asymmetrically and hence can be used to test the odd parity of SAR. For this purpose, we consider an asymmetric four-terminal device consisting of a zigzag graphene ribbon with two superconducting leads as shown in Fig.2(a). The Hamiltonian of the graphene isref19 $H_{0}=\sum_{\bf i}\epsilon_{\bf i}a^{\dagger}_{\bf i}a_{\bf i}-\sum_{<{\bf ij}>}ta_{\bf i}^{\dagger}a_{\bf j}$. Here $a_{\bf i}$ and $a_{\bf i}^{\dagger}$ are the annihilation and creation operators at site ${\bf i}$, $\epsilon_{\bf i}$ is the on-site energy which can be controlled experimentally by the gate voltageref1 , and the hopping constant $t=2.75eV$ represents the nearest carbon bond energy. The pair potential (energy gap) of superconducting terminal-$\beta$ with $\beta=2,4$ is $\tilde{\Delta}_{\beta}=\Delta_{\beta}e^{i\varphi_{\beta}}$ with $\Delta_{2}=\Delta_{4}=\Delta\simeq 1meV$. In numerical calculations,cheng we fix Fermi energy $E_{F}$ and tune the Dirac point $E_{0}$. We have used $\Delta$ as the energy unit. Now we study the interference between two ARs from GNS junctions as shown in Fig.2(a) in which two superconducting leads 2 and 4 are asymmetrically attached to the zigzag nanoribbon. The horizontal distance $\delta N$ between two GNS junctions measures the asymmetry of two GNS junctions. The scattering process can be qualitatively understood as follows. For simplicity, we assume $\phi_{2}=\phi_{4}$ for the moment. As shown schematically in Fig.2(a), for SAR the particle-like electrons in terminal-1 split into two beams and are scattered separately by two GNS junctions (green horizontal lines) as holes that finally recombine at terminal-3. We examine the total phase accumulated for each beam that involves the following three processes. Before reaching the first GNS junction (denoted by the left vertical green line) two beams of electrons propagate with the same momentum $k_{x}$. After reaching the second GNS junction (denoted by the right vertical green line) two beams of holes also propagate with the same momentum $k^{\prime}_{x}$. Obviously phases accumulated in the above two processes for both beams are the same. Between them two beams propagate with different momenta $k_{x}$ and $k^{\prime}_{x}$. Hence the phase difference between two beams is $\phi=(k_{x}-k^{\prime}_{x})\delta x$ with $\delta x=b\delta N$, where $b=\sqrt{3}a$ and $a$ the lattice constant. This phase difference can be tuned by varying the Dirac point $E_{0}$ or the asymmetry $\delta N$ giving rise to a complicated interference pattern (see Fig.2). In particular, this phase difference can be zero if $(k_{x}-k^{\prime}_{x})=0$ (i.e.,$E_{0}=0$) or $\delta N=0$. In general, the total phase difference is $\phi=(k_{x}-k^{\prime}_{x})\delta x+\phi_{2}-\phi_{4}$. Interference pattern of AR probability $R_{13A}$ for system depicted in Fig.2(a) with pair potential phase difference of two superconductors $\delta\varphi=0$ and $\pi$ ($\delta\varphi\equiv\varphi_{2}-\varphi_{4}$) are then plotted in Fig.2(b) and (c), respectively. For Fig.2(b) following observations are in order: (1) For the geometrically symmetric system ($\delta N=0$), the interference is always destructive with zero $R_{13A}$ as long as $\|E_{0}\|<\|E_{F}\|$.cheng Clearly this is due to the $\pi$ phase shift depicted in Fig.1(d) and is consistent with the band selection rule.select (2) When Dirac point $E_{0}$ is in line with the condensate energy of the superconductor, i.e., when $E_{0}=0$, $R_{13A}$ is again zero no matter what value $\delta N$ assumes. This means that there is a completely destructive interference between two beams scattered by two GNS junctions attached asymmetrically to the graphene nano-ribbon. This behavior can be understood as follows. When $E_{0}=0$ the incoming electron and reflected hole have the same propagating momentum $k_{x}$ and thus path 1 and 2 in Fig.2(a) experience the same quantum phase $k_{x}\delta x$ except at the superconducting leads. Hence the total phase difference is only due to the $\pi$ phase shift between two SARs. (3) $R_{13A}$ is an even function of Dirac point $E_{0}$ because of the electron-hole symmetry in graphene. Due to the geometric symmetry, $R_{13A}$ is also an even function of asymmetry $\delta N$. (4) For nonzero $E_{F}$, the closer the Dirac point $E_{0}$ to $E_{F}$, the more rapidly $R_{13A}$ oscillates as we vary $\delta N$. This is because the difference of propagating momentum $k_{x}-k^{\prime}_{x}$ increases monotonically as $E_{0}$ approaches to $E_{F}$. (5) When $E_{0}$ is in the vicinity of $E_{F}$, $R_{13A}$ can reach 0.9 which is much larger than that when $\|E_{0}\|>\|E_{F}\|$. This is because when $E_{F}$ is very close to $E_{0}$, the edge states of zigzag ribbon begin to contribute, then electron is easier to be scattered by two GNS junctions located also at edges of zigzag ribbon. Considering the pseudo-spin conservation, large $R_{13A}$ is always found in the region of $\|E_{0}\|<\|E_{F}\|$, i.e., the SAR region. (6) There is an overall fine oscillation with a period of $\delta N=3b$. Similar behavior was also found in zigzag ribbons with a p-n junction where the conductance is determined by the relative displacement $\delta$ along the p-n junction.Beenakker1 In Fig.2(c) with the superconducting phase difference $\delta\varphi=\pi$, we see that the interference pattern is contrary to $\delta\varphi=0$ [Fig.2(b)] where the constructive interference becomes destructive and vice versa. To further analyze the interference pattern, we plot in Fig.3(a) the total $R_{13A}$ vs Dirac point $E_{0}$ for different asymmetry $\delta N$ with the phase difference between two superconducting leads $\delta\varphi=0$ [main panel of Fig.3(a)] or $\delta\phi=\pi$ [inset of Fig.3(a)]. Clearly the interference (oscillatory) pattern occurs only for asymmetric systems ($\delta N\neq 0$) with oscillation frequency proportional to $\delta N$. When pair potential phase difference $\delta\varphi=\pi$ is introduced, the interference pattern reverses, and $R_{13A}$ with $\delta N=0$ becomes the envelop function of $R_{13A}$ for all nonzero $\delta N$. In Fig.3(b) we plot $R_{13A}$ vs $\delta N$ for different widths $W$ of nanoribbon. It is shown clearly that $R_{13A}$ is a periodic function of $\delta N$ with larger periodicity for larger $W$. In the inset of Fig.3(b) we plot this period versus the width for different $E_{0}$. The period $P$ is obtained in two ways: (1). from the expression $P=2\pi/(k_{x}-k^{\prime}_{x})$ where the momenta $k_{x}$ and $k_{x}^{\prime}$ can be obtained from the band structure for a given $E_{0}$ (black symbols). (2). directly from main panel of Fig.3(b) (red solid circle). From the inset, it clearly shows that two periods are exactly the same giving strong evidence that the interference pattern of AR probability are indeed from two reflected hole beams. Finally, the interference pattern of AR probability $R_{11A}$ is also studied (not shown). We found that only ARR probability $R_{11A}$ ($\|E_{0}\|>E_{F}=0.2\Delta$) exhibits interference pattern. We note that since there is no $\pi$ phase shift involved in ARR, when $\delta N=0$ reflected electrons through two GNS junctions interfere constructively when $\delta\varphi=0$ and destructively when $\delta\varphi=\pi$ which is in contrast to SAR in Fig.2. In fact, interference patterns of SAR and ARR are always phase contrast not only for $\delta N=0$ but also for all other $\delta N$. To test the odd parity of SAR experimentally, it relies on the fabrication of high quality zigzag graphene nanoribbons. It has been achieved by several laboratories using different methods last year including the method to unzip the multi-walled carbon nanotube (CNT),ref13 the anisotropic etching by thermally activated nickel nanoparticles,ref14 and use reconstruction of the edge to make zigzag graphene nanoribbons.experiment In view of the above experimental breakthrough, we expect that the setup to test our predicted phenomenon can be realized experimentally. To reduce the experimental challenge, we have considered an unzipped CNT device, i.e., (n,n) CNT-zigzag graphene-(n,n) CNT, obtained by unzipping a few unit cells in the central part of an armchair CNT which has been achieved experimentally.ref13 For this system, the wavefunction in the armchair CNT has the same symmetry as that of the zigzag graphene ribbon. Following the same procedure leading to Eq.(4), we have shown that the unzipped CNT in contact with a superconducting lead has the odd parity under mirror operation. Similar conclusions drawn from GNS can be obtained for unzipped CNT with two superconducting leads. In conclusion, up to now, the parity of reflection amplitude was found to be even under the mirror operation. Here we have provided an example of odd parity for the reflection amplitude, the SAR amplitude in the zigzag graphene- superconductor hybrid system. This odd parity is due to the combination of unique band structure of the graphene and the electron-hole conversion involving two energy bands with different parity symmetry. The signature of odd parity of SAR can be found from the quantum constructive interference in a four terminal system with two superconducting leads attached asymmetrically. Furthermore, the interference pattern due to odd parity of SAR is phase contrasted to that of ARR where the parity is even. Acknowledgement We gratefully acknowledge the financial support from a RGC grant (HKU 705409P) from the Government of HKSAR and from NSF-China under Grant Nos.10974236 and 10821403. ## References * (1) ∗e-mail: jianwang@hkusua.hku.hk. * (2) †e-mail: sunqf@aphy.iphy.ac.cn. * (3) K. S. Novoselov et al, Science 306, 666 (2004); K. S. Novoselov et al, Nature (London) 438, 197 (2005); Y. Zhang et al, Nature (London) 438, 201 (2005). * (4) C. W. J. Beenakker, Rev. Mod. Phys. 80, 1337 (2008); A.H. Castro Neto et al, Rev. Mod. Phys. 81, 109 (2009); A. Rycerz et al, Nature Phys. 3, 172 (2007). * (5) M. I. Katsnelson et al, Nature Phys. 2, 620 (2006); A.-F. Young et al, Nature Phys. 5, 222 (2009); A. V. Shytov et al, Phys. Rev. Lett. 101, 156804 (2008). * (6) V. V. Cheianov et al, Science 315, 1252 (2007). * (7) Y. Xing, J. Wang and Q.-F. Sun, Phys. Rev. B 81, 165425 (2010). * (8) H. B. Heersche et al, Nature 446, 56 (2007); F. Miao et al, Science 317, 1530 (2007). * (9) A. F. Andreev, Sov. Phys. JETP 19, 1228 (1964). * (10) C. W. J. Beenakker, Phys. Rev. Lett. bf 97, 067007 (2006). * (11) J. Nakabayashi et al, Phys. Rev. Lett 102, 066803 (2009). * (12) S. Cheng et al, Phys. Rev. Lett. 103, 167003 (2009); Q.-F. Sun and X.C. Xie, J. Phys.: Condens. Matter 21, 344204 (2009). * (13) D. N. Sheng et al, Phys. Rev. B 73, 233406 (2006); Z. Qiao and J. Wang, Nanotechnology 18, 435402 (2007). W. Long et al, Phys. Rev. Lett. 101, 166806 (2008); J. Li and S.-Q. Shen, Phys. Rev. B 78, 205308 (2008). * (14) A. R. Akhmerov et al, Phys. Rev. B 77, 205416 (2008). * (15) L. Jiao et al, Nature (London), 458, 877 (2009). * (16) L. C. Campos et al, Nano. Lett. 9, 2600 (2009). * (17) Ç. Ö. Girit et al, ibid, 323 1705 (2009). Figure 1: (Color online) Panel (a): zigzag ribbons with even number of chains (gray honeycomb) attached by a superconducting lead on the left and right (orange honey comb), respectively. For SAR the incoming electrons (red arrow) are scattered by the GNS junction (green solid line) as holes (blue arrow). The corresponding wave functions at sublattice “A” (solid circle) and “B” (hollow circle) for the lowest subband in conduction band (bottom) and the highest subband in valence band (top) are shown schematically. Panel (b): AR probability from terminal-1 to terminal-1 $R_{11A}$ and to terminal-3 $R_{13A}$ vs. Dirac point $E_{0}$. Panels (c) and (d): AR phase $\Phi^{i,ii}_{11}$ (c) and $\Phi^{i,ii}_{13}$ (d) of two systems in panel (a) and their phase different $\Phi^{i}_{11(13)}-\Phi^{ii}_{11(13)}$ vs. $E_{0}$. Figure 2: (Color online) Panel (a): sketch of AR interferometer in which the zigzag ribbon is asymmetrically attached by two superconductor lead-2 and 4. Electrons in terminal-1 can be Andreev reflected into terminal-3 by either top or bottom GNS junction (horizontal green lines). Panel(b) and (c): the contour plot of $R_{13A}$ vs Dirac point $E_{0}$ and asymmetry $\delta N$. The phase difference of two superconductor leads $\delta\varphi$ is zero in panel (b) and $\pi$ in panel (c). The other parameters: Fermi energy $E_{f}=0.8$, number of chains in zigzag ribbon $N=40$ corresponding to width $60a$, the width of superconductor lead $W_{S}=10b$, where $b=\sqrt{3}a$. Figure 3: (Color online) Panel (a): With fixed Fermi level $E_{F}=0.8$, total AR probability $R_{13A}$ vs Dirac point $E_{0}$ for different asymmetry $\delta N$. In the main panel $\delta\varphi=0$, and while $\delta\varphi=\pi$ in the inset. Panel (b): $R_{13A}$ vs asymmetry $\delta N$ with $E_{0}=0.3t$ for different width $W$ from $10\times 3a$ to $38\times 3a$ with the interval $2\times 3a$ along the black arrow. Inset panel: the hollow signs are the period $P$ obtained from the main panel and the solid red circles are the period $P$ from the energy band with the expression $P=2\pi/(k_{x}-k_{x}^{\prime})$. The other parameters: $\delta\varphi=0$, $E_{F}=0.8$.	arxiv-papers	2010-11-23T13:03:00	2024-09-04T02:49:15.177399	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Yanxia Xing, Jian Wang, and Qing-feng Sun", "submitter": "Xing Yanxia", "url": "https://arxiv.org/abs/1011.5082" }
1011.5226	# Fractal Geometry of Angular Momentum Evolution in Near-Keplerian Systems M. Atakan Gürkan Astronomical Institute “Anton Pannekoek”, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands Leiden University, Leiden Observatory, P.O. Box 9513, 2300 RA Leiden, The Netherlands E-mail: ato.gurkan@gmail.com (Accepted 201x XXX xx. Received 201x XXX xx; in original form 201x XXX xx) ###### Abstract In this paper, we propose a method to study the nature of resonant relaxation in near-Keplerian systems. Our technique is based on measuring the fractal dimension of the angular momentum trails and we use it to analyze the outcome of $N$-body simulations. With our method, we can reliably determine the timescale for resonant relaxation, as well as the rate of change of angular momentum in this regime. We find that growth of angular momentum is more rapid than random walk, but slower than linear growth. We also determine the presence of long term correlations, arising from the bounds on angular momentum growth. We develop a toy model that reproduces all essential properties of angular momentum evolution. ###### keywords: methods: statistical — methods: N-body simulations — stellar dynamics — Galaxy: centre ††pagerange: Fractal Geometry of Angular Momentum Evolution in Near-Keplerian Systems–References††pubyear: 201x ## 1 Introduction Dynamical systems where the gravitational potential is dominated by a single mass are called near-Keplerian (Tremaine, 2005), since the orbits of smaller masses in such systems are very close to Keplerian conic sections. A characteristic feature of these systems is the existence of various distinct timescales over which the dynamical variables change. For positions and velocities, this is the dynamical time $t_{\mathrm{dyn}}\sim(a^{3}/GM)^{1/2}$, where $M$ is the mass of the central object and $a$ is a given orbit’s semimajor axis. The bound orbits (ellipses) precess over the precession time $t_{\mathrm{prec}}\sim(M/Nm_{\star})t_{\mathrm{dyn}}$, where $m_{\star}$ is the mass of smaller objects and $N$ is the number of these objects within $a$. The timescale for the evolution of energy is the relaxation time, $t_{\mathrm{rlx}}\sim(M^{2}/m_{\star}^{2}N\ln\Lambda)t_{\rm dyn}$. These timescales form an hierarchy $t_{\mathrm{rlx}}\gg t_{\mathrm{prec}}\gg t_{\mathrm{dyn}}$ and lead to three regimes for angular momentum evolution (Tremaine, 1999). For short timescales ($t\lesssim t_{\mathrm{dyn}}$), the changes in angular momentum has no correlation since the torques felt at different parts of the orbit vary rapidly. For intermediate timescales ($t_{\mathrm{dyn}}\lesssim t\lesssim t_{\mathrm{prec}}$), where we are effectively averaging over orbits, the changes in angular momentum are correlated, since the configuration of the orbits change slowly. For long timescales ($t\gtrsim t_{\mathrm{prec}}$), the correlation is lost again, since the orbits are randomized as they precess in different directions at different rates. The presence of an intermediate regime with enhanced angular momentum evolution was first recognized in the literature by Rauch & Tremaine (1996). They called this process resonant relaxation, since it is the result of a near resonance between angular and radial frequencies of the orbit. It plays a central role in various scenarios that are proposed to take place in the vicinity of supermassive black holes (see, e.g., Magorrian & Tremaine, 1999; Alexander, 2008). There are two major uncertainties regarding resonant relaxation. The first one is the boundaries, especially the upper limit, for the intermediate regime. The timescale over which the torques are coherent is evidently related to the precession time $t_{\mathrm{prec}}$, but the detailed nature of this relationship is unknown. Order of magnitude estimates for the timescales are not sufficient, since in some cases the lifetimes of the stars in the systems under question are comparable to the timescales of the dynamical processes (for an example at the Galactic centre, see Madigan et al., 2009). The nature of evolution of angular momentum over intermediate timescales is also uncertain. In this regime, since the torques are correlated, the evolution of angular momentum is not going to be diffusive ($\Delta L\propto\Delta t^{1/2}$). Rauch & Tremaine (1996) seem to suggest that the evolution in this regime is ballistic ($\Delta L\propto\Delta t$, see their Eq.6 and Fig.1) and this is also adopted by Eilon et al. (2009); but their simulations contain too few stars to draw conclusions in this regard. Indeed, in this work we find that the nature of the angular momentum evolution in the intermediate regime is between diffusive and ballistic. We expose this behaviour and determine the timescales by carrying out simplified simulations of near-Keplerian systems and a fractal analysis of the evolution of angular momentum. In section 2, we give a brief review of fractals and present a new method for determining fractal dimension, suitable for trails obtained in numerical simulations. In section 3 we describe our simulations and the fractal analysis of the angular momentum evolution. In section 4, we demonstrate that such an evolution can be mimicked by a simple random walk that retains a limited term memory. We discuss the limitations and implications of our findings in section 5. ## 2 Fractal Dimension Fractals (Mandelbrot, 1982) are geometric objects that exhibit a number of properties that make them suitable for modelling physical processes and natural structures. The defining property of fractals is that their Hausdorff- Besicovitch dimension (hereafter fractal dimension, $D$) exceeds their topological dimension. A number of methods for calculating the fractal dimension is given by Mandelbrot (1982), here we give a brief sketch and develop a new one that is suitable for our purposes. A fractal is a self similar object; that is, part of it exhibits similar properties to the whole, sometimes only in a statistical sense. If a self similar object is made up of $n$ copies of itself, each of which is smaller (in length) by $1/m$, then the object has fractal dimension $D=\ln n/\ln m$. It is trivial to see that this leads to correct numbers for self similar Euclidean objects. For example, any line segment can be thought to be composed of $n=3$ identical copies of itself, each of which is smaller by $1/3$ ($m=3$), giving $D=1$; or a rectangular prism can be cut into $n=8$ identical copies of itself, each of which is smaller (in length) by $1/2$ ($m=2$), giving $D=3$. We can use this technique to calculate the dimension of a well known fractal, the Koch curve (Fig. 1). This curve consists of $n=4$ identical copies of itself111Koch curve can also be seen as consisting of two copies of itself, each of which is smaller by $1/\sqrt{3}$., each of which is smaller by $1/3$, leading to a fractal dimension $D=\ln 4/\ln 3\sim 1.262$. Figure 1: The Koch curve, obtained by repeatedly transforming a line segment. The result of the first transformation is shown as a broken line in red, to demonstrate the self similar structure more clearly. Each part of the Koch curve resting on the straight parts of this broken line is a smaller but an otherwise identical copy of itself. Another way to obtain the fractal dimension is to measure the length of a curve in increasing detail. Since as we use smaller and smaller rulers, we will be resolving more and more detail; the measured length of the curve $L$, is a function of the ruler length $\varepsilon$ (Mandelbrot, 1967): $L(\varepsilon)\propto\varepsilon^{1-D}\,.$ (1) As an example, let us assume that we measure the length of the Koch curve by a given ruler and obtain $L_{0}$. When we reduce the ruler length by $1/3$, we shall be traversing the smaller copies in exactly the same manner we traversed the whole curve. Since there are four smaller copies, each of which will be measured to have a length $1/3$ times the whole curve, we have $L^{\prime}=4/3\times L$. To state in more general terms, if we modify our ruler length by $\varepsilon^{\prime}\leftarrow(1/m)\varepsilon$, the measured length changes by $L^{\prime}\leftarrow(n/m)L$. It is easy to see that this scaling is satisfied when $L\propto\varepsilon^{1-(\ln n/\ln m)}$. This method of determining the (fractal) dimension of a curve readily generalizes to real life curves, which are self similar only in a statistical sense (Mandelbrot, 1967). For our purposes, it is more convenient to interpret a curve as the motion trail of an object. As we check the position of the object on this trail at decreasing time intervals, we will be resolving more details and the total trail length we calculate is going to increase. In other words, the trail length is going to be a function of the sampling interval $\chi$. For example, if we sample the motion of an object on the Koch curve $4$ times, we will be measuring the trail shown as the broken line in Figure 1, leading to an increase in length by $4/3$, with respect to going from the beginning to the end in one step. In more general terms, when the sampling interval is modified by $\chi^{\prime}\leftarrow(1/n)\chi$, the measured length becomes $L^{\prime}\leftarrow(n/m)L$. In this approach, the fractal dimension is determined through the relation $L(\chi)\propto\chi^{\frac{1}{D}-1}\,.$ (2) This technique of measuring fractal dimension of a trail can be easily applied to the results from dynamical simulations. ## 3 Simulations ### 3.1 Simulation Method and Parameters The system we simulated has three components. At the centre lies a stationary supermassive black hole (SMBH) of mass $M_{\rm SMBH}=4\times 10^{6}M_{\sun}$. Around the SMBH, we have $N_{\rm field}=1200$ field stars each with mass $m_{\rm field}=2.5M_{\sun}$, semi-major axes distributed in a powerlaw cusp $M_{\rm cusp}(r)\propto r^{3/2}$ from $a_{\rm min}=0.0001$ to $a_{\rm max}=0.03\,{\rm pc}$, with eccentricities between $e_{\rm min}=0$ and $e_{\rm max}=0.95$, distributed following the distribution function $g(e)\propto e$. The final component is the massless test stars all with semimajor axis $a_{\rm test}=0.01\,{\rm pc}$, and eccentricities $e_{\rm test}=0.2$, $0.75$, and $0.85$ (12 stars each). Other orbital elements (inclination, longitude of the ascending node, argument of pericentre and mean anomaly) of all stars are picked at random. The exact values chosen do not matter too much, since over the course of the simulation the eccentricities are randomized. We chose these parameters to have a system that somewhat resembles the environment of S-stars observed at our Galactic centre. There are large uncertainties regarding the star distributions at this environment (Merritt, 2010, and references therein), and $t_{\rm dyn}\ll t_{\rm prec}\ll t_{\rm rlx}$ hierarchy may not even exist there. However, this condition would hold for a region around a SMBH that developed a Bahcall-Wolf cusp (Bahcall & Wolf, 1976, 1977), so we expect our method would be applicable to such systems. The details of the code that is used for the simulations is going to be explained in detail elsewhere, here we point out only the key features. Both the field stars and the test stars feel the potential of the SMBH, including the general relativistic (GR) correction that leads to the prograde precession of the orbits222We use the treatment of Saha & Tremaine (1992, their eq. 30). The test stars also feel the individual potential of the field stars, which leads to retrograde precession of their orbits and changes in angular momentum and energy. The field stars do not feel the individual potential of each other, but instead see the potential of a smooth cusp that is consistent with their distribution. This approximation decreases the time required for force computation significantly and was already employed by Rauch & Tremaine (1996) in their $N$-body simulations. For the interactions between the test stars and field stars we use a softening kernel ($\mathrm{K}_{2}$) of Dehnen (2001), with a softening length $10^{-4}$ pc. The units we adopted are $G=M_{\mathrm{field}}=3000M_{\sun}=1\,\mathrm{pc}=1$. The extended mass distribution in the background cluster precesses the orbits in a retrograde fashion, and the GR effects lead to prograde precession. By our choice of parameters, these two effects cancel each other for a star with $a\sim 0.01\,\mathrm{pc}$ and $e\sim 0.6$. Mass precession becomes more effective as semi-major axes get larger and eccentricities get smaller, while the GR precession has the opposite behaviour. We carry out the integration of the orbits using a high order Runge-Kutta- Nyström method, discovered by Blanes & Moan (2000, their SRKN${}^{b}_{11}$). We split the Hamiltonian into Keplerian and perturbation parts (Kinoshita et al., 1991) to increase efficiency and avoid spurious precession. This scheme advances the Keplerian orbital elements correctly, except for the truncation and the roundoff errors (the largest accumulated error per star amounts to $\sim 10^{-5}$ over the course of the whole simulation). In particular, the evolution of the Runge-Lenz vector does not exhibit a linear drift as in the case of potential energy-kinetic energy splitting (Hut et al., 1995). Our treatment of the GR perturbation leads to a small error in mean motion, but since we are interested in changes that take place over many orbits, this error is not important. We use shared adaptive timesteps, but time-symmetrize the integration with the method of Hut et al. (1995). Our simulations last 30 code units which corresponds to a few precession times of the slowest precessing test stars. Figure 2: The evolution of angular momentum (black) and energy (red) for a test star with initial eccentricity $e_{0}=0.75$. For comparison, a Brownian motion (blue) and a fractional Brownian motion (yellow; $c=18$, $n=4000$) data as described in sec. 4 are also plotted. All curves are sampled at the same abscissae, and the ordinates are adjusted to have unit variance. We record the energy and angular momentum of each test star throughout the simulation. In Figure 2 we show the evolution of these quantities for a star with initial eccentricity $e_{0}=0.75$. Even by eye, it is possible to tell that these quantities show a different behaviour. For comparison, in this figure we also plot the curves generated by Brownian and fractional Brownian motion, as described in Section 4. ### 3.2 Analysis of Simulation Results Figure 3: Measured trail lengths as a function of duration of the sampling intervals for angular momentum trails and the twice-broken power-law fits made to them. Sets are artificially offset from each other to avoid confusion. The gray lines in the background are to lead the eye and have slopes $-1/2$. We measure the length of the angular momentum trail of each test star by sampling it at intervals ranging from the full length of the simulation down to a fraction of the orbital period. The dependence of the total measured trail length on the sampling interval is shown in Figure 3 for a few stars with different initial eccentricities. Starting from long sampling intervals and moving towards shorter ones, a number of features can be observed on this figure: * • For long timescales, the curves do not have the slope $-1/2$ (corresponding to dimension $D=2$, the value for Gaussian random walks (Mandelbrot, 1982)), but are somewhat steeper. This is to be expected since the energy of an orbit changes through relaxation and this process is much slower; hence the angular momentum evolution is bounded unlike a true random walk, and cannot be described by simple diffusion, even for long timescales. * • There is a marked transition to a more coherent motion as indicated by a decrease in the slope. The point of this transition can be determined with reasonable accuracy for a given star, but it is not common for all stars. * • Even though the slope decreases, it never becomes zero; hence the evolution of angular momentum is never ballistic ($\Delta L\not\propto\Delta t$). * • This slope can also be determined with reasonable accuracy for a given star, but varies from star to star. * • Looking at the eccentricity evolution of the stars reveals that the transition point is later for the stars that moved into $e\sim 0.6$ region where precession is slow. This is in harmony with the expectation that for more slowly precessing stars, the coherent torques last longer. * • The slope increases again for short timescales, but much before the period of the stars is reached. This randomization of the torques is a result of the stochastic nature of the processes that develop the torques and dominates the shorter timescale randomization that would result from orbital motion, at least down to the timescales we resolve. Our results verify the presence of an intermediate regime, where the angular momentum evolution is enhanced. The evolution of angular momentum in this regime is not as rapid as ballistic growth $\Delta L\propto\Delta t$, but more rapid than diffusive growth $\Delta L\propto\Delta t^{1/2}$. This manner of evolution can be seen as the generalization of Brownian motion called fractional Brownian motion. Mandelbrot & van Ness (1968) describes various properties of this motion, along with applications. Mandelbrot (1982) gives further generalizations and methods to produce such curves. Even though these approaches are mathematically elegant and complete, in the next section we propose a simpler model that is easier to attach a physical interpretation. ## 4 A Simple Toy Model for Angular Momentum Evolution One dimensional Brownian motion for a particle’s position $P(t)$ can be generated as follows. Let the motion over $\Delta t$ consist of $N$ steps with equal duration $\delta t=\Delta t/N$. We choose an initial value $P_{0}$ and at each step either increase or decrease the value of $P$ by $\delta P$. We decide which action to take by generating a sequence of random numbers $X_{i}$ uniformly sampled from the interval $[-0.5,0.5]$ and choosing a threshold value $q=0$. At each step, if $X_{i}<q$ we increase the value of $P$ and otherwise decrease it. To make the variance of the motion independent of the number of steps we choose $\delta P\propto 1/\sqrt{N}$. This scheme leads to Brownian motion (Gaussian random walk) for small $\delta t$, i.e, a large number of steps (Falconer, 2003, Chap. 16). It can be extended to a vector variable by letting each component perform independent Brownian motions. We can introduce correlations between the increments of the variable $P$ by using a “repository”. For this, we keep track of the last $n$ values of $X_{i}$ and let our threshold value be proportional to their average $q=c\left<X_{i}\right>_{n}$, where $c$ is some constant. Here, $n$ determines the length of the correlations and $c$ determines their strength. We generated a few sets of data this way and measured the lengths of the resulting trails with differing sampling durations. Figure 4: Measured trail lengths as a function of duration of the sampling intervals for various generated random walk data sets and the twice-broken power-law fits made to them. The set in the middle (green) has $c=18$, $n=4000$. The sets above it have varying $c=12,6,0$ and the sets below it have varying $n=2000,1000,500$ (see the text for explanation of these parameters). All sets have $N=200\,000$ points. To avoid confusion, sets are artificially offset from each other; but to make comparison easier, top four and bottom four sets are redrawn with dashed black lines at the top and bottom of the figure with identical offsets. The gray lines in the background are to lead the eye and have slopes $-1/2$. The curves generated this way (Fig. 4) show very similar characteristics to angular momentum evolution. They exhibit an intermediate regime with lowered slope, whereas for long and short timescales the motion is more randomized. The upper bound of the intermediate regime is determined by the parameter $n$: the break occurs when the sampling interval matches $n\times\delta t$. The other parameter $c$ determines the slope in the intermediate and short timescale regime. Larger $c$ leads to a more coherent motion and a slope closer to $0$. Figure 5: Same as Fig. 4 but for bounded random walks. The slopes for long sampling intervals are steeper than $-1/2$. We also generated similar data with bounded random walks. For those, we started with a (vector) variable of unit magnitude $\|\mathbf{P}(t=0)\|=1$, and whenever a step led to $\|\mathbf{P}\|>1.5$, we took that step in the opposite direction. This limit roughly corresponds to the limit experienced by an orbit starting with initial eccentricity $e_{0}=0.75$, $J_{\mathrm{circular}}/J_{0}\sim 1.5$. The results from this bounded random walks are shown in Figure 5, exhibiting the long term correlations similar to angular momentum evolution. ## 5 Discussion In this work, we studied the evolution of angular momentum in near-Keplerian systems by analyzing the outcome of $N$-body simulations. The simulation method we use incorporates certain approximations. Our test stars are massless, so they do not cause a back-reaction in the surrounding cluster. Furthermore, the field stars all have the same mass. A mass spectrum would change the granularity of the background potential, affecting the applied torques and possibly the coherence timescale. Finally, since the field stars do not see the granularity of their own potential, their angular momenta do not evolve. This decreases the rate of randomization of the background cluster, since vector resonant relaxation can change the orientation of the orbits on timescales comparable to mass and GR precession for some systems333We thank an anonymous referee for pointing out this shortcoming. (for a comparison of these timescales for the Galactic centre, see Kocsis & Tremaine (2010)). If the torques on a star mainly change because of the rearrangement of the background cluster, rather than the reorientation of the orbit, this decrease in randomization would alter the rate of angular momentum evolution. All these approximations limit the domain of applicability of our $N$-body simulation approach; however, none of them alter the essential mechanism by which the angular momentum evolves. We analyzed this evolution by calculating the fractal dimension of the angular momentum trail. With this method it is possible to reliably determine the onset of and the rate of evolution in different regimes. A key result of our analysis is that the evolution of angular momentum is neither diffusive nor ballistic in any regime, as was previously assumed in other studies (e.g. Hopman & Alexander, 2006; Eilon et al., 2009). This seems to contradict the results of the numerical experiments done by Rauch & Tremaine (1996). The reason for this discrepancy is not clear, but we speculate that it arises from the low number of stars used in that work. We also developed a toy model that reproduces the features of angular momentum evolution. This model has adjustable parameters with clear physical interpretations. The relation between the appropriate values of these parameters and the physical variables requires more detailed and extensive analysis, which is currently underway. Apart from studying different initial conditions, we also plan to analyze the components of the torque parallel and perpendicular to the angular momentum separately. The nature of these torques can be very different (Rauch & Tremaine, 1996; Gürkan & Hopman, 2007), so a separate analysis should lead to a better understanding. Koutsoyiannis (2002) developed fractional Gaussian noise generators (FGNGs) similar to our repository model. In that work he compares autoregressive moving average (ARMA) models to FGNGs, and finds that ARMA models are inferior for describing long-term correlations. These models need to be modified to have long term memory (Shumway & Stoffer, 2000), to describe angular momentum evolution in near-Keplerian systems. Alternatively, they can be used to describe short term correlations for torques, and long term correlations can be introduced by taking physical bounds into account, as is done here. After this paper was submitted, Madigan et al. (2010) also submitted a paper containing their analysis of this problem with ARMA models. They use ARMA(1,1) model, which fixes the value of the autocorrelation function for very large timescales to zero (see their figure 4) and hence does not lead to any correlations beyond a given time. The computer programs used to generate and analyze the data are available from the author upon request. ## Acknowledgments This work is supported by a Netherlands Organization for Scientific Research (NWO) Veni Fellowship. Most of the simulations were done on Lisa cluster, maintained by SARA, the Dutch National High Performance Computing and e-Science Support Center. I thank all SARA staff for doing an exceptional job for maintaining this cluster and in particular to Walter Lioen for his help. Usage of Lisa was possible through a grant (client number 10450) to Simon Portegies Zwart. I am grateful to Clovis Hopman, Yuri Levin, Ann-Marie Madigan and İnanç Adagideli for fruitful discussions on this topic, and an anonymous referee for comments that improved this paper. ## References * Alexander (2008) Alexander T., 2008, in S. K. Chakrabarti & A. S. 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1011.5380	# Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean and Hyperbolic spaces Vicent Gimeno# Departament de Matemàtiques-INIT, Universitat Jaume I, Castelló, Spain. gimenov@guest.uji.es and Vicente Palmer Departament de Matemàtiques-INIT, Universitat Jaume I, Castelló, Spain. palmer@mat.uji.es ###### Abstract. We study the topology of (properly) immersed complete minimal surfaces $P^{2}$ in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces, (see [12]). We present an alternative and partially unified proof of the Chern-Osserman inequality satisfied by these minimal surfaces, (in $\mathbb{R}^{n}$ and in $\mathbb{H}^{n}(b)$), based in the isoperimetric analysis above alluded. Finally, we show a Chern-Osserman type equality attained by complete minimal surfaces in the Hyperbolic space with finite total extrinsic curvature. ###### Key words and phrases: Area growth, minimal surfaces, Chern-Osserman inequality, finite topological type, compactification, Euler characteristic. ###### 2000 Mathematics Subject Classification: Primary 53C20 ; Secondary 53C42, 49Q05 # Supported by the Fundació Caixa Castelló-Bancaixa Grants P1.1B2006-34 and P1.1B2009-14 * Supported by MICINN grant No. MTM2010-21206-C02-02. ## 1\. Introduction Let us consider $P^{2}$ be a complete and minimal surface immersed in $\mathbb{R}^{n}$ and with finite total curvature $\int_{P}K^{P}d\sigma<\infty$, being $K^{P}$ the Gauss curvature of the surface. Then we have the following equality (resp. inequality), known as the Chern-Osserman formula, (see [1], [3] and [8]): (1.1) $-\chi(P)=\frac{1}{4\pi}\int_{P}\\|B^{P}\\|^{2}d\sigma-\operatorname{Sup}_{r}\frac{\operatorname{Vol}(P^{2}\cap B^{0,n}_{r})}{\operatorname{Vol}(B^{0,2}_{r})}\leq\frac{1}{4\pi}\int_{P}\\|B^{P}\\|^{2}d\sigma-k(P)$ where $\chi(P)$ is the Euler characterisitic of $P$, $k$ is its number of ends, $B^{P}$ is the second fundamental foorm of $P$ in $\mathbb{R}^{n}$ and $B^{b,n}_{r}$ denotes the geodesic $r$-ball in the simply connected real space form $\mathbb{K}^{n}(b)$. To have finite total scalar (extrinsic) curvature $\int_{P}\\|B^{P}\\|^{2}d\sigma<\infty$ is equivalent to the finiteness of the total Gaussian curvature (the original assumption in [3]) when the surface is minimal and immersed in $\mathbb{R}^{n}$. From this point of view, it is natural to wonder if it is possible to stablish a Chern-Osserman inequality (or equality) for complete minimal surfaces with finite total extrinsic curvature (properly) immersed in the hyperbolic space. This question has been addressed by Q. Chen and Y. Cheng in the papers [4] and [5]. They proved, for a complete minimal surface $P^{2}$ (properly) immersed in $\mathbb{H}^{n}(b)$ and such that $\int_{P}\\|B^{P}\\|d\sigma<\infty$, that $\operatorname{Sup}_{r}\frac{\operatorname{Vol}(P^{2}\cap B^{-1,n}_{r})}{\operatorname{Vol}(B^{-1,2}_{r})}<\infty$ and the following version of the Chern-Osserman Inequality, in terms of the volume growth of the extrinsic balls: (1.2) $-\chi(P)\leq\frac{1}{4\pi}\int_{P}\\|B^{P}\\|^{2}d\sigma-\operatorname{Sup}_{r}\frac{\operatorname{Vol}(P^{2}\cap B^{-1,n}_{r})}{\operatorname{Vol}(B^{-1,2}_{r})}$ The proofs given by these authors are different for those for the Euclidean case, and rely heavily on the properties of the hyperbolic functions. We present in this paper a partial unification of the proof of the Chern- Osserman inequality (in terms of the volume growth) for complete minimal surfaces with finite total extrinsic curvature immersed in Euclidean or Hyperbolic spaces. This partial unification is based in obtaining estimates for the Euler characteristic of the extrinsic balls (given in Lemma 3.1, and Proposition 3.2) and in the isoperimetric inequality for the extrinsic balls given in Theorem 1.1 in [12]. These results are based, in its turn, on the divergence Theorem and the Hessian and Laplacian comparison theory of restricted distance function, (see [6], [7] and [13]) which involves bounds on the mean curvature of the submanifold. We have proved the following Chern-Osserman inequality, which encompasses inequalities (1.1) and (1.2): ###### Theorem A. Let $P^{2}$ be an complete minimal surface immersed in a simply connected real space form with constant sectional curvature $b\leq 0$, $\mathbb{K}^{n}(b)$. Let us suppose that $\int_{P}\\|B^{P}\\|^{2}d\sigma<\infty$. Then 1. (1) $P$ has finite topological type. 2. (2) $\operatorname{Sup}_{t>0}(\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B^{b,2}_{t})})<\infty$ 3. (3) $-\chi(P)\leq\frac{\int_{P}\\|B^{P}\\|^{2}}{4\pi}-\operatorname{Sup}_{t>0}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}$ where $\chi(P)$ is the Euler characteristic of $P$. Although with this approach we are not able to state equality (1.1) in the Euclidean setting, we shall prove in Theorem B the following Chern-Osserman type equality for cmi surfaces in the Hyperbolic space: ###### Theorem B. Let $P^{2}$ be a complete immersed minimal surface in $\mathbb{H}^{n}(b)$. Let us suppose that $\int_{P}\\|B^{P}\\|^{2}d\sigma<\infty$. Then (1.3) $-\chi(P)=\frac{1}{4\pi}\int_{P}\\|B^{P}\\|^{2}d\sigma-\operatorname{Sup}_{t>0}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}-\frac{1}{2\pi}G_{b}(P)$ where $G_{b}(P)$ is a nonnegative and finite quantity which do not depends on the exhaustion by extrinsic balls $\\{D_{t}\\}_{t>0}$ of $P$ and is given by (1.4) $\displaystyle G_{b}(P)$ $\displaystyle:=\lim_{t\to\infty}\left(h_{b}(t)\operatorname{Vol}(B^{b,2}_{t})(\frac{(\operatorname{Vol}(D_{t}))}{\operatorname{Vol}(B^{b,2}_{t})})^{\prime}\right.$ $\displaystyle\left.+\int_{\partial D_{t}}\langle B^{P}(e,e),\frac{\operatorname{\nabla}^{\perp}r}{\\|\operatorname{\nabla}^{P}r\\|}\rangle d\sigma_{t}\right)$ ### 1.1. Outline The outline of the paper is following. In Section §.2 we present the basic facts about the Hessian comparison theory of restricted distance function we are going to use, obtaining as a corollary the compactification of cmi surfaces in $\mathbb{K}^{n}(b)$ with finite total extrinsic curvature, (Corollary 2.3). Section §.3 is devoted to the unified proof of the Chern- Osserman inequality for complete minimal surfaces with finite total extrinsic curvature immersed in Euclidean and Hyperbolic spaces (Theorem A), and in Section §.4 it is proved a Chern-Osserman type equality satisfied by the cmi surfaces in $\mathbb{H}^{n}(b)$ (Theorem B). ## 2\. Preliminaires ### 2.1. The extrinsic distance We assume throughout the paper that $P^{2}$ is a complete, non-compact, immersed, $2$-dimensional submanifold in a simply connected real space form of non-positive constant sectional curvature $\mathbb{K}^{n}(b)$, ($\mathbb{K}^{n}(b)=\mathbb{R}^{n}$ when $b=0$ and $\mathbb{K}^{n}(b)=\mathbb{H}^{n}(b)$ when $b<0$) . All the points in these manifolds are poles. Recall that a pole is a point $o$ such that the exponential map $\exp_{o}\colon T_{o}N^{n}\to N^{n}$ is a diffeomorphism. For every $x\in N^{n}\setminus\\{o\\}$ we define $r_{o}(x)=\operatorname{dist}_{N}(o,x)$, and this distance is realized by the length of a unique geodesic from $o$ to $x$, which is the radial geodesic from $o$. We also denote by $r$ the restriction $r\|_{P}:P\to\mathbb{R}_{+}\cup\\{0\\}$. This restriction is called the extrinsic distance function from $o$ in $P^{m}$. The gradients of $r$ in $N$ and $P$ are denoted by $\operatorname{\nabla}^{N}r$ and $\operatorname{\nabla}^{P}r$, respectively. Let us remark that $\operatorname{\nabla}^{P}r(x)$ is just the tangential component in $P$ of $\operatorname{\nabla}^{N}r(x)$, for all $x\in S$. Then we have the following basic relation: (2.1) $\nabla^{N}r=\operatorname{\nabla}^{P}r+(\operatorname{\nabla}^{N}r)^{\bot},$ where $(\operatorname{\nabla}^{N}r)^{\bot}(x)=\nabla^{\bot}r(x)$ is perpendicular to $T_{x}P$ for all $x\in P$. On the other hand, we should recall that all immersed surfaces $P$ in the real space forms of non-positive constant sectional curvature $N^{n}=\mathbb{K}^{n}(b)$ which satisfies $\int_{P}\\|B^{P}\\|^{2}d\sigma<\infty$ are properly immersed (see [1], [10] and [11]). Therefore, we can omit the hypothesis about the properness of the immersion when we assume that $\int_{P}\\|B^{P}\\|^{2}d\sigma<\infty$. ###### Definition 2.1. Given a connected and complete surface $P^{2}$ properly immersed in a manifold $N^{n}$ with a pole $o\in N$, we denote the extrinsic metric balls of radius $t>0$ and center $o\in N$ by $D_{t}(o)$. They are defined as the intersection $D_{t}(o)=B^{N}_{t}(o)\cap P=\\{x\in P\colon r(x)<t\\},$ where $B^{N}_{t}(o)$ denotes the open geodesic ball of radius $R$ centered at the pole $o$ in $N^{n}$. ###### Remark a. We want to point out that the extrinsic domains $D_{t}(o)$ are precompact sets, (because we assume in the definition above that the submanifold $P$ is properly immersed), with boundary $\partial D_{t}(o)$ being a immersed curve in $P$. The generical smoothness of $\partial D_{t}(o)$ follows from the following considerations: the distance function $r$ is smooth in $\mathbb{K}^{n}(b)\setminus\\{o\\}$ since $\mathbb{K}^{n}(b)$ to possess a pole $o\in\mathbb{K}^{n}(b)$, ($b\leq 0$). Hence the restriction $r\|_{P}$ is smooth in $P$ and consequently the radii $t$ that produce smooth boundaries $\partial D_{t}(o)$ are dense in $\mathbb{R}$ by Sard’s theorem and the Regular Level Set Theorem. ###### Remark b. When the submanifold considered is totally geodesic, namely, when $P$ is a Hyperbolic or an Euclidean subespace of the ambient real space form, the extrinsic balls become geodesic balls, and its boundary is the distance sphere. We recall here that the mean curvature of the geodesic sphere in the real space form $\mathbb{K}^{n}(b)$, ’pointed inward’ is (see [12]): $h_{b}(t)=\left\\{\begin{array}[]{l}\sqrt{b}\cot\sqrt{b}t\,\,\text{ if }\,\,b>0\\\ 1/t\,\,\text{ if }\,\,b=0\\\ \sqrt{-b}\coth\sqrt{-b}t\,\,\text{ if }\,\,b<0\end{array}\right.$ ### 2.2. Hessian comparison analysis of the extrinsic distance Let us consider now $D_{t}$ an extrinsic ball in a complete and properly immersed minimal surface $P$ in the real space form $\mathbb{K}^{n}(b)$ with $b\leq 0$. We are going to apply Gauss-Bonnet formula to the curve $\partial D_{t}$. To do that, we need to compute its geodesic curvature in the following ###### Proposition 2.2. Given $\partial D_{t}$ the smooth closed curves in $P$, (2.2) $k_{g}^{\partial D_{t}}=\frac{h_{b}(t)}{\\|\operatorname{\nabla}^{P}r\\|}+\langle B^{P}(e,e),\frac{\operatorname{\nabla}^{\bot}r}{\\|\operatorname{\nabla}^{P}r\\|}\rangle$ ###### Proof. Let $\\{e,\nu\\}\subset TP$ be an orthonormal frame along the curve $\partial D_{t}$, where $e$ is the unit tangent vector to $\partial D_{t}$ and $\nu=\frac{\operatorname{\nabla}^{P}r}{\\|\operatorname{\nabla}^{P}\\|}$ is the unit normal to $\partial D_{t}$ in $P$, pointed outward. From the definition of geodesic curvature of the extrinsic boundaries $\partial D_{t}$, we have (2.3) $k_{g}^{t}=-\langle\nabla_{e}^{P}e,\frac{\nabla^{P}r}{\\|\nabla^{P}r\\|}\rangle$ Then, having on account the definition of Hessian $Hess^{P}r(e,e)=\langle\nabla^{P}\nabla^{P}r,e\rangle$ and the fact that $\nabla^{P}r$ and $e$ are orthogonal, (2.4) $k_{g}^{t}=\frac{1}{\\|\nabla^{P}r\\|}Hess^{P}r(e,e)$ But, given $X\in T_{q}P$ unitary, (see [7] and [13] for detailed computations): (2.5) $\operatorname{Hess}^{P}(r)(X,X)\,=\,h_{b}(r)\left(\,1-\langle\,X,\nabla^{\mathbb{K}^{n}(b)}r\,\rangle^{2}\,\right)+\langle\,\nabla^{\mathbb{K}^{n}(b)}r,\,B^{P}(X,X)\,\rangle\,$ where $B^{P}$ is the second fundamental form of $P$ in $N$. Applying at this point equation (2.5): (2.6) $k_{g}^{t}=\frac{1}{\\|\nabla^{P}r\\|}\\{h_{b}(r)+\langle\nabla^{\perp}r,B^{P}(e,e)\rangle\\}$ ∎ Now, we consider $\\{D_{t}\\}_{t>0}$ an exhaustion of $P$ by extrinsic balls. Recall than an exhaustion of the submanifold $P$ is a sequence of subsets $\\{D_{t}\subseteq P\\}_{t>0}$ such that: * • $D_{t}\subseteq D_{s}$ when $s\geq t$ * • $\cup_{t>0}D_{t}=P$ Using the equality (2.2) for the geodesic curvature of the extrinsic curves we have the following result ###### Theorem 2.3. Let $P^{2}$ be an complete minimal surface immersed in a simply connected real space form with constant sectional curvature $b\leq 0$, $\mathbb{K}^{n}(b)$. Let us suppose that $\int_{P}\\|B^{P}\\|^{2}d\sigma<\infty$. Then (i) $P$ is diffeomorphic to a compact surface $P^{}$ punctured at a finite number of points. (ii) For all sufficiently large $t>R_{0}>0$, $\chi(P)=\chi(D_{t})$ and hence, given $\\{D_{t}\\}_{t>0}$ an exhaustion of $P$ by extrinsic balls, $\chi(P)=\lim_{t\to\infty}\chi(D_{t})$ ###### Proof. Let us consider $\\{D_{t}\\}_{t>0}$ an exhaustion of $P$ by extrinsic balls, centered at the pole $o\in\mathbb{K}^{n}(b)$. We apply Lemma 2.2 to the smooth curves $\partial D_{t}$: As $-\\|B^{P}\\|\leq\langle B^{P}(e,e),\operatorname{\nabla}^{\bot}r\rangle\leq\\|B^{P}\\|$ we have, on the points of the curve $q\in\partial D_{t}$, (2.7) $\displaystyle\\|\operatorname{\nabla}^{P}r\\|(q)\cdot k_{g}^{\partial D_{t}}(q)$ $\displaystyle=h_{b}(r_{o}(q))+\langle B^{P}(e,e),\operatorname{\nabla}^{\bot}r\rangle(q)\,$ $\displaystyle\geq h_{b}(r_{o}(q))-\\|B^{P}\\|(q)$ Using now Proposition 2.2 in [1], when $P^{2}$ is a cmi in $\mathbb{R}^{n}$ or Lemma 3.1 in [11], when $P^{2}$ is a cmi in $\mathbb{H}^{n}(b)$, we know that $\\|B^{P}\\|(q)$ goes uniformly to $0$ as $t=r_{o}(q)\to\infty$. Hence, for all the points $q\in\partial D_{t}$ and for sufficiently large $t$, (2.8) $\\|\operatorname{\nabla}^{P}r\\|(q)\cdot k_{g}^{\partial D_{t}}(q)\ >0$ Hence, $\\|\operatorname{\nabla}^{P}r\\|>0$ in $\partial D_{t}$, for all sufficiently large $t$. Fixing a sufficienty large radius $R_{0}$, we can conclude that the extrinsic distance $r_{o}$ has no critical points in $P\setminus\operatorname{D}_{R_{0}}$. The above inequality implies that for this sufficienty large fixed radius $R_{0}$, there is a diffeomorphism $\Phi:P\setminus\operatorname{D}_{R_{0}}\to\partial D_{R_{0}}\times[0,\infty[$ In particular, $P$ has only finitely many ends, each of finite topological type. To proof this we apply Theorem 3.1 in [9], concluding that, as the extrinsic annuli $A_{R_{0},R}(o)=D_{R}(o)\setminus D_{R_{0}}(o)$ contains no critical points of the extrinsic distance function $r_{o}:P\longrightarrow\mathbb{R}^{+}$ because inequality (2.8), then $D_{R}(o)$ is diffeomorphic to $D_{R_{0}}(o)$ for all $R\geq R_{0}$. The above diffeomorfism implies that we can construct $P$ from $D_{R_{0}}$ ($R_{0}$ big enough) attaching annulis and that $\chi(P\setminus D_{t})=0$ when $t\geq R_{0}$. Then, for all $t>R_{0}$, $\chi(P)=\chi(D_{t}\cup(P\setminus D_{t}))=\chi(D_{t})$ ∎ ## 3\. Proof of Theorem A We begin with the following results which are the common ingredient of the proof, both for the Euclidean and Hyperbolic cases : ###### Lemma 3.1. Let $P^{2}\subset\mathbb{K}^{n}(b)$ be a surface properly immersed in a real space form with curvature $b\leq 0$, let $D_{t}$ be an extrinsic disc in $P$ of radius $t>0$ and let $\partial D_{t}$ the extrinsic circle. Then: (3.1) $\int_{\partial D_{t}}\frac{\|\|\nabla^{\bot}r\|\|^{2}}{\|\|\nabla^{P}r\|\|}d\sigma_{t}\leq\int_{\partial D_{t}}\frac{1}{\|\|\nabla^{P}r\|\|}-h_{b}(t)\operatorname{Vol}(D_{t})d\sigma_{t}$ ###### Proof. Tracing equality (2.5) we obtain the following expression for the Laplacian of the extrinsic distance in this context: (3.2) $\Delta^{P}(r)\,=\,(m-\\|\nabla^{P}r\\|^{2})h_{b}(r)+m\langle\,\nabla^{N}r,\,H_{P}\,\rangle\quad,$ where $H_{P}$ denotes the mean curvature vector of $P$ in $N$ and $h_{b}(r)$ is the mean curvature of the geodesic $r$-spheres in $\mathbb{K}^{n}(b)$. Applying divergence theorem we have (3.3) $\displaystyle\int_{\partial D_{t}}\frac{\|\|\nabla^{\bot}r\|\|^{2}}{\|\|\nabla^{P}r\|\|}d\sigma_{t}=\int_{\partial D_{t}}\frac{1}{\|\|\nabla^{P}r\|\|}d\sigma_{t}-\int_{\partial D_{t}}\|\|\nabla^{P}r\|\|d\sigma_{t}=\int_{\partial D_{t}}\frac{1}{\|\|\nabla^{P}r\|\|}d\sigma_{t}$ $\displaystyle-\int_{D_{t}}\Delta^{P}rd\sigma=\int_{\partial D_{t}}\frac{1}{\|\|\nabla^{P}r\|\|}d\sigma_{t}-\int_{D_{t}}(2-\|\|\nabla^{P}r\|\|^{2})h_{b}(r)d\sigma$ $\displaystyle\leq\int_{\partial D_{t}}\frac{1}{\|\|\nabla^{P}r\|\|}d\sigma_{t}-\int_{D_{t}}h_{b}(r)d\sigma\leq\int_{\partial D_{t}}\frac{1}{\|\|\nabla^{P}r\|\|}d\sigma_{t}-h_{b}(t)\operatorname{Vol}(D_{t})$ ∎ ###### Proposition 3.2. Let $P^{2}\subset\mathbb{K}^{n}(b)$ be a complete minimal surface properly immersed in a real space form with curvature $b\leq 0$, let $D_{t}$ be an extrinsic disc in $P$ of radius $t>0$ and let $\partial D_{t}$ be its boundary. Then: (3.4) $\displaystyle-2\pi\chi$ $\displaystyle(D_{t})+(b+\frac{f_{b,\alpha}^{2}(t)h_{b}(t)}{2})\operatorname{Vol}(D_{t})$ $\displaystyle+(h_{b}(t)-\frac{f_{b,\alpha}^{2}(t)}{2})\int_{\partial D_{t}}\frac{1}{\\|\nabla^{P}r\\|}d\sigma_{t}\leq\frac{1}{2}R(t)+\frac{1}{2f_{b,\alpha}^{2}(t)}R^{\prime}(t)$ where $R(t)=\int_{D_{t}}\\|B^{P}\\|^{2}d\sigma$, $\\|B^{P}\\|$ is the norm of the second fundamental form of $P$ in $\mathbb{K}^{n}(b)$, $\chi(D_{t})$ is the Euler’s characterisc of $D_{t}$ and, given $\alpha\in]0,2[$ , $f_{b,\alpha}^{2}(t)=\alpha h_{b}(t)$ ###### Proof. Integrating along $\partial D_{t}$ equation (2.2) and using Gauss-Bonnet theorem and co-area formula, (see [14]), we obtain (3.5) $\displaystyle 2$ $\displaystyle\pi\chi(D_{t})-\int_{D_{t}}K^{P}d\sigma=$ $\displaystyle h_{b}(t)\int_{\partial D_{t}}\frac{1}{\\|\operatorname{\nabla}^{P}r\\|}d\sigma_{t}+\int_{\partial D_{t}}\langle B^{P}(e,e),\frac{\operatorname{\nabla}^{\bot}r}{\\|\operatorname{\nabla}^{P}r\\|}\rangle d\sigma_{t}$ where we denote as $K^{P}$ the Gauss curvature of $P$. But , on $\partial D_{t}$, $-\\|B^{P}\\|\frac{\\|\operatorname{\nabla}^{\bot}r\\|}{\\|\operatorname{\nabla}^{P}r\\|}\leq\langle B^{P}(e,e),\frac{\operatorname{\nabla}^{\bot}r}{\\|\operatorname{\nabla}^{P}r\\|}\rangle\leq\\|B^{P}\\|\frac{\\|\operatorname{\nabla}^{\bot}r\\|}{\\|\operatorname{\nabla}^{P}r\\|}$ so, as $f_{b,\alpha}(t)\geq 0\,\forall t>0$, having into account the inequality among the arithmetic and geometric mean and applying co-area formula: (3.6) $\displaystyle 2$ $\displaystyle\pi\chi(D_{t})-\int_{D_{t}}K^{P}d\sigma=h_{b}(t)\int_{\partial D_{t}}\frac{1}{\\|\operatorname{\nabla}^{P}r\\|}d\sigma_{t}$ $\displaystyle+\int_{\partial D_{t}}\langle B^{P}(e,e),\frac{\operatorname{\nabla}^{\bot}r}{\\|\operatorname{\nabla}^{P}r\\|}\rangle d\sigma_{t}\,\geq\,h_{b}(t)\int_{\partial D_{t}}\frac{1}{\\|\operatorname{\nabla}^{P}r\\|}d\sigma_{t}$ $\displaystyle-\frac{1}{2}\int_{\partial D_{t}}\frac{\\|B^{P}\\|^{2}}{f_{b,\alpha}^{2}(r)\\|\operatorname{\nabla}^{P}r\\|}d\sigma_{t}-\frac{1}{2}\int_{\partial D_{t}}\frac{f_{b,\alpha}^{2}(r)\\|\operatorname{\nabla}^{\bot}r\\|^{2}}{\\|\operatorname{\nabla}^{P}r\\|}d\sigma_{t}$ $\displaystyle\geq h_{b}(t)\int_{\partial D_{t}}\frac{1}{\\|\operatorname{\nabla}^{P}r\\|}d\sigma_{t}-\frac{1}{2f_{b,\alpha}^{2}(t)}R^{\prime}(t)-\frac{f_{b,\alpha}^{2}(t)}{2}\int_{\partial D_{t}}\frac{\\|\operatorname{\nabla}^{\bot}r\\|^{2}}{\\|\operatorname{\nabla}^{P}r\\|}d\sigma_{t}$ Then, using inequality (3.1) of Lemma 3.1 in the last member of the inequalities (3.6) and applying Gauss equation for minimal surfaces in the real space forms $\mathbb{K}^{n}(b)$, we have (3.7) $\displaystyle 2\pi\chi$ $\displaystyle(D_{t})-b\operatorname{Vol}(D_{t})+\frac{1}{2}R(t)\geq(h_{b}(t)-\frac{f_{b,\alpha}^{2}(t)}{2})\int_{\partial D_{t}}\frac{1}{\\|\operatorname{\nabla}^{P}r\\|}d\sigma_{t}$ $\displaystyle-\frac{1}{2f_{b,\alpha}^{2}(t)}R^{\prime}(t)+\frac{f_{b,\alpha}^{2}(t)h_{b}(t)}{2}\operatorname{Vol}(D_{t})$ and hence (3.8) $\displaystyle-2\pi\chi$ $\displaystyle(D_{t})+(b+\frac{f_{b,\alpha}^{2}(t)h_{b}(t)}{2})\operatorname{Vol}(D_{t})$ $\displaystyle+(h_{b}(t)-\frac{f_{b,\alpha}^{2}(t)}{2})\int_{\partial D_{t}}\frac{1}{\\|\nabla^{P}r\\|}\leq\frac{1}{2}R(t)+\frac{1}{2f_{b,\alpha}^{2}(t)}R^{\prime}(t)$ ∎ We are going to divide the proof in two cases: the Case I, where the ambient space is the Hyperbolic space $\mathbb{H}^{n}(b)$, and the Case II where the ambient space is the Euclidean space $\mathbb{R}^{n}$. ### Case I Let us consider $P$ (properly) immersed in $\mathbb{H}^{n}(b)$. Let $\\{D_{t}\\}_{t>0}$ be an exhaustion of $P$ by extrinsic balls. Using co-area formula, we know that (3.9) $\frac{d}{dt}\operatorname{Vol}(D_{t})=\int_{\partial D_{t}}\frac{1}{\\|\nabla^{P}r\\|}d\sigma_{t}$ Hence, applying Proposition 3.2 we have (3.10) $\displaystyle-2$ $\displaystyle\pi\chi(D_{t})+(b+\frac{f_{b,\alpha}^{2}(t)h_{b}(t)}{2})\operatorname{Vol}(D_{t})$ $\displaystyle+(h_{b}(t)-\frac{f_{b,\alpha}^{2}(t)}{2})\frac{d}{dt}\operatorname{Vol}(D_{t})\leq\frac{1}{2}R(t)+\frac{1}{2f_{b,\alpha}^{2}(t)}R^{\prime}(t)$ On the other hand, from 3.9, $\frac{d}{dt}\operatorname{Vol}(D_{t})\geq\operatorname{Vol}(\partial D_{t})$. Therefore, using inequality (3.10) we obtain (3.11) $\displaystyle-2\pi\chi(D_{t})$ $\displaystyle+\operatorname{Vol}(D_{t})\left[(b+\frac{f_{b,\alpha}^{2}(t)h_{b}(t)}{2})+(h_{b}(t)-\frac{f_{b,\alpha}^{2}(t)}{2})\frac{\operatorname{Vol}(\partial D_{t})}{\operatorname{Vol}(D_{t})}\right]$ $\displaystyle\leq\frac{1}{2}R(t)+\frac{1}{2f_{b,\alpha}^{2}(t)}R^{\prime}(t)$ Applying isoperimetric inequality in [12], (Theorem 1.1), we have (3.12) $\displaystyle-2\pi\chi(D_{t})$ $\displaystyle+\operatorname{Vol}(D_{t})\left[(b+\frac{f_{b,\alpha}^{2}(t)h_{b}(t)}{2})+(h_{b}(t)-\frac{f_{b,\alpha}^{2}(t)}{2})\frac{\operatorname{Vol}(S_{t}^{b,1})}{\operatorname{Vol}(B_{t}^{b,2})}\right]$ $\displaystyle\leq\frac{1}{2}R(t)+\frac{1}{2f_{b,\alpha}^{2}(t)}R^{\prime}(t)$ Hence, using the fact that $b\operatorname{Vol}(B^{b,2}_{t})+h_{b}(t)\operatorname{Vol}(S^{b,1}_{t}=2\pi\,\,\,\forall t>0$ we obtain, with some computations (3.13) $\begin{split}-2\pi\chi(D_{t})+\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}&\left[2\pi-2\pi\frac{f_{b,\alpha}^{2}(t)}{2}\frac{\operatorname{Vol}(B^{b,2}_{t})}{\operatorname{Vol}(S_{t}^{b,1})}\right]\\\ \leq\frac{1}{2}R(t)+\frac{1}{2f_{b,\alpha}^{2}(t)}R^{\prime}(t)\end{split}$ Therefore, for all $t>0$, (3.14) $\begin{split}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}\left(1-\frac{\alpha h_{b}(t)}{2}\frac{\operatorname{Vol}(B_{t}^{b,2})}{\operatorname{Vol}(S_{t}^{b,1})}\right)&-\chi(D_{t})\\\ \leq\frac{R(t)}{4\pi}+\frac{R^{\prime}(t)}{4\pi\alpha h_{b}(t)}\end{split}$ As $\frac{\|\|B^{P}\|\|^{2}}{h_{b}(t)}\leq\frac{1}{\sqrt{-b}}\|\|B^{P}\|\|^{2}$, then $\int_{P}\|\|B^{P}\|\|^{2}d\sigma<\infty$ implies $\int_{P}\frac{\|\|B^{P}\|\|^{2}}{h_{b}(t)}d\sigma<\infty$. Hence, by co-area formula: (3.15) $\int_{0}^{\infty}\left(\int_{\partial D_{t}}\frac{\|\|B^{P}\|\|^{2}}{\|\|\operatorname{\nabla}^{P}r\|\|h_{b}(r)}\right)dt=\int_{0}^{\infty}\left(\frac{R^{\prime}(t)}{h_{b}(t)}\right)dt<\infty$ Therefore, there is a monotone increasing (sub)sequence $\\{t_{i}\\}_{i=1}^{\infty}$ tending to infinity, (namely, $t_{i}\to\infty$ when $i\to\infty$), such that $\frac{R^{\prime}(t_{i})}{h_{b}(t_{i})}\rightarrow 0$ when $i\to\infty$. Let us consider the exhaustion of $P$ by these extrinsic balls, namely, $\\{D_{t_{i}}\\}_{i=1}^{\infty}$. Then we have, replacing $t$ for $t_{i}$ and taking limits when $i\to\infty$ in inequality (3.14) and applying Theorem 2.3 (ii), (3.16) $\begin{split}\operatorname{Sup}_{i}&\frac{\operatorname{Vol}(D_{t_{i}})}{\operatorname{Vol}(B_{t_{i}}^{b,2})}\left(1-\frac{\alpha}{2}\right)-\chi(P)\\\ &\leq\lim_{i\to\infty}\frac{R(t_{i})}{4\pi}=\frac{1}{4\pi}\int_{P}\\|B^{P}\\|^{2}d\sigma<\infty\end{split}$ for all $\alpha$ such that $0<\alpha<2$. Hence, as $\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}$ is a continuous non decreasing function of $t$, we can conclude that $\operatorname{Sup}_{t>0}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}<\infty$ and $-\chi(P)<\infty$. Then, letting $\alpha$ tend to $0$ in (3.16), we get, for all $t>0$: (3.17) $\displaystyle\operatorname{Sup}_{t>0}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}-\chi(P)\leq\frac{\int_{P}\\|B^{P}\\|^{2}}{4\pi}$ ### Case II Let us consider $P$ immersed in $\mathbb{R}^{n}$. We consider, as in the proof above, an exhaustion of $P$ by extrinsic balls, $\\{D_{t}\\}_{t>0}$, but now, and following [1], these extrinsic balls will be centered at the origin $0\in\mathbb{R}^{n}$, which we assume, without loss of generality, that belongs to the surface $P$. Applying Proposition 3.2 we have (3.18) $\displaystyle-2\pi\chi$ $\displaystyle(D_{t})+(\frac{\alpha}{2t^{2}})\operatorname{Vol}(D_{t})$ $\displaystyle+(\frac{1}{t}-\frac{\alpha}{2t})\int_{\partial D_{t}}\frac{1}{\\|\nabla^{P}r\\|}\leq\frac{1}{2}R(t)+\frac{t}{2\alpha}R^{\prime}(t)$ Now, as $\int_{P}\|\|B^{P}\|\|^{2}d\sigma<\infty$, we can apply Proposition 2.2 in [1], so we have, for $\alpha\in]0,2[$, (3.19) $\frac{t}{2\alpha}R^{\prime}(t)=\frac{t}{2\alpha}\int_{\partial D_{t}}\frac{\\|B^{P}\\|^{2}}{\\|\nabla^{P}r\\|}d\sigma\leq\frac{\mu(t)}{2\alpha t}\int_{\partial D_{t}}\frac{1}{\\|\nabla^{P}r\\|}d\sigma$ being $\mu(t)$ such that $\lim_{t\to\infty}\mu(t)=0$ and therefore, from (3.18), (3.20) $\displaystyle-2\pi\chi(D_{t})+\operatorname{Vol}(D_{t})(\frac{\alpha}{2t^{2}})$ $\displaystyle+(\frac{1}{t}-\frac{\alpha}{2t}-\frac{\mu(t)}{2\alpha t})\int_{\partial D_{t}}\frac{1}{\\|\nabla^{P}r\\|}d\sigma_{t}\,\leq\,\frac{1}{2}R(t)$ On the other hand, $\frac{1}{t}-\frac{\alpha}{2t}-\frac{\mu(t)}{2\alpha t}\geq 0$ if and only if $\mu(t)\leq\alpha(2-\alpha)$, which it is true for $t$ big enough, namely, for $t>t_{\alpha}$ because $\lim_{t\to\infty}\mu(t)=0$. Hence, as $\operatorname{Vol}(\partial D_{t})\leq\int_{\partial D_{t}}\frac{1}{\\|\nabla^{P}r\\|}d\sigma_{t}$, and applying Theorem 1.1 in [12], we have that inequality (3.20) becomes, for all $t>t_{\alpha}$ (3.21) $\displaystyle-2\pi\chi(D_{t})$ $\displaystyle+\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{0,2})}\left[2\pi(1-\frac{\alpha}{2}-\frac{\mu(t)}{2\alpha})+\frac{\pi\alpha}{2}\right]\leq\,\frac{1}{2}R(t)$ Then, taking limits when $t\to\infty$ in inequality (3.21) and applying Theorem 2.3, we have that $\lim_{t\to\infty}\mu(t)=0$ and $\chi(P)=\lim_{t\to\infty}\chi(D_{t})$, so we obtain, for all $\alpha$ such that $0<\alpha<2$: (3.22) $\displaystyle 2\pi\operatorname{Sup}_{t}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{0,2})}$ $\displaystyle\left(1-\frac{\alpha}{2}+\frac{\pi\alpha}{2}\right)$ $\displaystyle-2\pi\chi(P)\leq\frac{\int_{P}\\|B^{P}\\|^{2}}{2}<\infty$ Therefore we obtain $\operatorname{Sup}_{t>0}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{0,2})}<\infty$ and $-\chi(P)<\infty$. Then, letting $\alpha$ tend to $0$ we obtain, for all $t>0$: (3.23) $\displaystyle\operatorname{Sup}_{t>0}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{0,2})}-\chi(P)\leq\frac{\int_{P}\\|B^{P}\\|^{2}}{4\pi}$ ## 4\. Proof of Theorem B In Corollary 2.3, it was obtained a sufficienty large radius $R_{0}$, such that the extrinsic distance $r_{p}$ has no critical points in $P\setminus\operatorname{D}_{R_{0}}$. Hence for this sufficienty large fixed radius $R_{0}$, there is a diffeomorphism $\Phi:P\setminus\operatorname{D}_{R_{0}}\to\partial D_{R_{0}}\times[0,\infty[$ so, in particular, $P$ has only finitely many ends, each of finite topological type. The above diffeomorfism implied that we could construct $P$ from $D_{R_{0}}$ ($R_{0}$ big enough) attaching annulis and that $\chi(P\setminus D_{t})=0$ when $t\geq R_{0}$, and hence for all $t>R_{0}$, $\chi(P)=\chi(D_{t})$. Let us consider now an exhaustion by extrinsic balls $\\{D_{t}\\}_{t>0}$ of $P$ such that the extrinsic distance $r_{o}$ has no critical points in $P\setminus\operatorname{D}_{R_{0}}$. Applying now Gauss-Bonnet Theorem to the extrinsic balls $D_{t}$ (4.1) $2\pi\chi(P)=\int_{D_{t}}K^{P}d\sigma+\int_{\partial D_{t}}k_{g}d\sigma_{t}$ Having in to account equation (2.2) and the Gauss formula, we have, for all sufficiently large radius $t>R_{0}$ (4.2) $\displaystyle 2\pi\chi(P)$ $\displaystyle=-\frac{1}{2}\int_{D_{t}}\\|B^{P}\\|^{2}+b\operatorname{Vol}(D_{t})+h_{b}(t)\left(\operatorname{Vol}(D_{t})\right)^{\prime}$ $\displaystyle+\int_{\partial D_{t}}\langle B^{P}(e,e),\frac{\operatorname{\nabla}^{\perp}r}{\\|\operatorname{\nabla}^{P}r\\|}\rangle d\sigma_{t}=-\frac{1}{2}\int_{D_{t}}\\|B^{P}\\|^{2}d\sigma$ $\displaystyle+\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}\left(b\cdot\operatorname{Vol}(B_{t}^{b,2})+h_{b}(t)(\operatorname{Vol}(D_{t}))^{\prime}\frac{\operatorname{Vol}(B_{t}^{b,2})}{\operatorname{Vol}(D_{t})}\right.$ $\displaystyle\left.+\frac{\operatorname{Vol}(B_{t}^{b,2})}{\operatorname{Vol}(D_{t})}\int_{\partial D_{t}}\langle B^{P}(e,e),\frac{\operatorname{\nabla}^{\perp}r}{\\|\operatorname{\nabla}^{P}r\\|}\rangle d\sigma_{t}\right)$ But $2\pi=b\cdot\operatorname{Vol}(B_{t}^{b,2})+h_{b}(t)\operatorname{Vol}(S^{b,1}_{t})\,\,\,\forall t>0$, so, for all sufficiently large radius $t>R_{0}$ and after some computations: (4.3) $\displaystyle 2\pi\chi(P)=-\frac{1}{2}\int_{D_{t}}\\|B^{P}\\|^{2}d\sigma$ $\displaystyle+2\pi\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}+h_{b}(t)\operatorname{Vol}(B^{b,2}_{t})(\frac{(\operatorname{Vol}(D_{t}))}{\operatorname{Vol}(B^{b,2}_{t})})^{\prime}$ $\displaystyle+\int_{\partial D_{t}}<B^{P}(e,e),\frac{\operatorname{\nabla}^{\perp}r}{\\|\operatorname{\nabla}^{P}r\\|}>d\sigma_{t}$ The above equation is valid for all $t>R_{0}$, so, taking limits when $t\to\infty$, we can define (4.4) $\displaystyle G_{b}(P)$ $\displaystyle:=\lim_{t\to\infty}\left(h_{b}(t)\operatorname{Vol}(B^{b,2}_{t})(\frac{(\operatorname{Vol}(D_{t}))}{\operatorname{Vol}(B^{b,2}_{t})})^{\prime}\right.$ $\displaystyle\left.+\int_{\partial D_{t}}<B^{P}(e,e),\frac{\operatorname{\nabla}^{\perp}r}{\\|\operatorname{\nabla}^{P}r\\|}>d\sigma_{t})\right)$ Using equalities (4.3), we have that (4.5) $G_{b}(P)=2\pi\chi(P)+\frac{1}{2}\int_{D_{t}}\\|B^{P}\\|^{2}d\sigma-2\pi\operatorname{Sup}_{t}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{b,2})}<\infty$ and hence, $G_{b}(P)$ do not depends on the exhaustion $\\{D_{t}\\}_{t>0}$. ## References [1] M. T. Anderson, The compactification of a minimal submanifold in Euclidean space by the Gauss map , I.H.E.S. Preprint, 1984 * [2] S.S. Chern and R. Osserman Complete minimal surfaces in euclidean space (Academic Press, New York, 1984). * [3] S.S. Chern and R. Osserman, Complete minimal surface in $E^{n}$, J. d’Analyse Math. 19 (1967), 15-34. * [4] Chen Qing, On the area growth of minimal surfaces in $\mathbb{H}^{n}$, Geometriae Dedicata, 75 (1999), 263–273. * [5] Chen Qing and Cheng Yi, Chern-osserman inequality for minimal surfaces in $\mathbb{H}^{n}$, Proc. Amer. Math Soc., Vol. 128, 8, (1999), 2445-2450. * [6] R. Greene and S. Wu Function theory on manifolds which posses a pole, Lecture Notes in Math.,699, (1979), Springer Verlag, Berlin. * [7] A. Hurtado and V. Palmer, A note on the p-parabolicity of submanifolds, Pot. Analysis 34, (2), (2011),101–118. * [8] L.P. Jorge and W. H. Meeks, The topology of minimal surfaces of finite total Gaussian curvature, Topology, 122, (1983), 203-221. * [9] J. Milnor, Morse theory, Princeton University Press, New Jersey, 1969. * [10] S. Muller and V. Sverak, On surfaces of finite total curvature, J. Differential Geometry, 42, 2, (1995), 229-257. * [11] G. De Oliveira, Compactification of minimal submanifolds of hyperbolic space, Comm. An. and Geom., 1 (1993), 1-29. * [12] V. Palmer, Isoperimetric Inequalities for extrinsic balls in minimal submanifolds and their applications, J. London Math. Soc. (2) 60 (1999), 607-616. * [13] V. Palmer, On deciding whether a submanifold is parabolic of hyperbolic using its mean curvature Simon Stevin Transactions on Geometry, vol 1. 131-159, Simon Stevin Institute for Geometry, Tilburg, The netherlands, 2010. * [14] T. Sakai, Riemannian Geometry, Translations of Mathematical Monographs, vol. 149, A.M.S.1996.	arxiv-papers	2010-11-24T14:34:34	2024-09-04T02:49:15.196689	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Vicent Gimeno and Vicente Palmer", "submitter": "Vicent Gimeno", "url": "https://arxiv.org/abs/1011.5380" }
1011.5408	]Corresponding author. E-mail: myqiang@nju.edu.cn. # Deterministic endless collective evolvement in active nematics Xia-qing Shi National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China Center for Soft Condensed Matter Physics and Interdisciplinary Research, Soochow University, Suzhou 215006, China Yu- qiang Ma [ National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China Center for Soft Condensed Matter Physics and Interdisciplinary Research, Soochow University, Suzhou 215006, China ###### Abstract We propose a simple deterministic dynamic equation and reveal the mechanism of large-scale endless evolvement of spatial density inhomogeneity in active nematic. We determine the phase regions analytically. The interplay of density, magnitude of nematic order, and nematic director is crucial for the long-wave-length instability and the emergence of seemingly fluctuated collective motions. Ordered nematic domains can absorb particles, grow and divide endlessly. The present finding extends our understanding of the large- scale and seemingly fluctuated organization in active fluids. Assemblies of active particles, which may be a physical abstraction of running animalsbird1 , flying birdsbird2 , swimming bacteriabact1 , migrating cellscell1 or even cytoskeletoncytoskeleton2 , have been served as a new building block for physicists over the last decade or so to understand the common collective behavior in these non-equilibrium systemsvicsek1 . Active nematic, a recently proposed concept for a kind of apolar active particles, is formed by driven rod-like particles with head-tail symmetry through the randomly driving force along the rod orientation axis at the single-particle level[7(a)-7(f)]. The symmetry of the system is not broken by applying such micro-driven forces until spontaneous symmetry breaking occurs. Recently, simulations and experiments in active nematic system show well-organized collective motions with system-sized fluctuation[7(b)-7(e)]. For example, splitting and merging of large-scale self-organized structures are exemplified in the simulation on active nematics[7(c)]. Experiments also show large-scale collective swarming and swirling in driven granular rods monolayer[7(d),7(e)]. These observations lead us to think about the nature of such seemingly fluctuated collective motions. It is currently unclear whether these large- scale collective motions arise in a deterministic manner or as a result of noises applied upon the system. Moreover, are they genuinely restless on large scale and evolving without end? These important aspects are still not well addressed in previous studies. In the present study, we start from a deterministic equation to study the mechanism of restless collective evolution in active nematics. We reveal a new phase that is characterized by the unattainability of stable steady state. We first identify that, if the steady state can be reached, the system investigated here favors spatially homogeneous state. On the other hand, by taking account of the interplay between particle density and local nematic order (i.e., magnitude and orientation), the linear stability analysis shows that homogeneous nematic state can be unstable to fluctuations of small wave number. Therefore, the system enters into a chaotic phase region with no stable steady state. Large-scale spatial inhomogeneity of density and nematic order is developed as a result of long-wavelength instability. The spatial inhomogeneity in turn changes the direction of the nematic director, leading to a non-ending evolvement of the system. Numerical flux analysis shows that the particle-rich nematic domains are surrounded by particles fluxes, and evolve via absorbing particles from low-density isotropic medium, growing, and extending itself and breaking into small pieces. More importantly, all these seemingly fluctuated collective motions giving rise to giant number fluctuations are governed by a deterministic equation which is essentially free of noises. We notice that one salient feature of simulation rules for active nematics by Chate et al.[7(c)] is that particle rotations are governed through inter- particle nematic interaction while spatial translational movements are free of such interactions. Experimentally particles are driven along their long axis, inducing strong longitudinal diffusion, and they can thrust into the media with the supply of kinetic energy[7(e)]. A simple diffusion equation which follows these observations can be written as(see supp ): $\displaystyle\partial_{t}f(\textbf{r},\textbf{u},t)=\nabla[D_{\\|}{\bf u}{\bf u}\nabla+D_{\perp}(\textbf{I}-{\bf u}{\bf u})\nabla]f({\bf r},{\bf u})$ (1) $\displaystyle+\mathscr{R}[D_{r}\mathscr{R}f({\bf r},{\bf u})+D_{r}\mathscr{R}w({\bf r},{\bf u})f({\bf r},{\bf u})],$ where $D_{\\|}$ and $D_{\perp}$ are the parallel and perpendicular components of the translational diffusion constants. $D_{r}$ is the rotational diffusion constant, and the rotational operator $\mathscr{R}$ is defined by $\mathscr{R}={\bf u}\times\partial_{\bf u}$Shimada . $f(\textbf{r},\textbf{u},t)$ is the particle number distribution function where the spatial coordinate ${\bf r}$ and the unit vector ${\bf u}$ denote the center-of-mass position and long-axis direction of particles, respectively. $w(\textbf{r},\textbf{u})$ is a self-consistent interacting potential which has $\pm{\bf u}$-symmetry. In two-dimensional(2-D) case, the most common form of such interacting potential is the excluded-volume-like interaction $w(\textbf{r},\textbf{u})=l^{2}\int{\rm d}{\bf u}^{\prime}\|{\bf u}\times{\bf u}^{\prime}\|f({\bf r},{\bf u}^{\prime})$, where $l$ is the particle length. Figure 1: Three phases are separated by the blue curves. Polar plots (a)-(f) indicate the instability regime for $\kappa$ and $\theta$. The locations ($D_{0}$, $\delta$) of their origins are used as the parameters to produce these plots. (a) $\|D_{0}\|=0$, $\delta=0.8$, (b) $\|D_{0}\|=2/3$, $\delta=0.4$, (c) $\|D_{0}\|=0$, $\delta=0.4$, (d) $\|D_{0}\|=1/3$, $\delta=0.01$, (e) $\|D_{0}\|=2/3$, $\delta=0.01$, and (f) $\|D_{0}\|=0$, $\delta=0.01$. The black and red branches in polar plots represent the cases of positive and minus $D_{0}$, respectively. The inset shows the corresponding instability modes $D_{r}^{-1}\lambda^{+}_{\tilde{\rho},S_{\rho}}\|_{\theta=0}$ for these polar plots. The diffusion equation Eq.(LABEL:dyadd11) for active nematics satisfies particle number conservation with the spatial translational current ${\bf J}^{s}({\bf r},{\bf u})=-D_{\\|}{\bf u}{\bf u}\nabla f({\bf r},{\bf u})-D_{\perp}(\textbf{I}-{\bf u}{\bf u})\nabla f({\bf r},{\bf u})$ and the local rotational current ${\rm J}^{r}({\bf r},{\bf u})=-D_{r}\mathscr{R}f({\bf r},{\bf u})-D_{r}\mathscr{R}w({\bf r},{\bf u})f({\bf r},{\bf u})$, which are independent of each other. The translational current is purely diffusive. In the Fourier space as defined by $f({\bf r},{\bf u})=\int{\rm d}{\bf r}\tilde{f}({\bf k},{\bf u})e^{-i{\bf k}\cdot{\bf r}}$, the spatial fluctuation modes are governed by diffusive decaying term $-k^{2}(D_{\\|}{\rm cos}^{2}\varphi+D_{\perp}{\rm sin}^{2}\varphi)\tilde{f}({\bf k},{\bf u})$, where $\varphi$ is the angle ${\bf k}$ makes with ${\bf u}$. For positive $D_{\\|}$ and $D_{\perp}$, all the spatial fluctuation modes decay except for $k=0$ which indicates total particle number conservation. Therefore, as governed by these decaying diffusive modes, the spatial term suggests that only the spatially homogeneous state will probably be the stable steady state if there are no particle sources and sinks in the system and at the boundaries. However, when $D_{\\|}\neq D_{\perp}$, we will show that the spatially homogeneous state may become unstable to fluctuations in the nematic state. Consequently, the system is deprived of all possible stable steady state and becomes restless and evolves endlessly, similar to the deterministic nonperiodic flow found in turbulenceLorenz . We first examine the spatially homogeneous dynamic equation derived from Eq.(LABEL:dyadd11) by neglecting the spatial derivatives: $\partial_{t}f(\textbf{u},t)=\mathscr{R}[D_{r}\mathscr{R}f({\bf u})+D_{r}\mathscr{R}w({\bf u})f({\bf u})].$ Determined by the integration kernal of the self-consistent interacting potential $w(\textbf{u})$, $f(\textbf{u},t)$ has $\pm{\bf u}$-symmetry. In spatially homogeneous nematic state, we assume that the nematic director ${\bf n}_{0}$ is in the $x$-axis, and thus the distribution function can be expanded as $f(\textbf{u},t)=(2\pi)^{-1}\rho\sum_{n=0}^{\infty}a_{n}(t){\rm cos}2n\phi$, where $\rho$ is the particle number density, $\phi$ is the angle of the unit vector ${\bf u}$, and $n=0,1,2\cdots\infty$. The dynamic equation of the coefficient $a_{n}(t)$ is given by ${(4D_{r})}^{-1}{\partial_{t}a_{n}(t)}=-n^{2}a_{n}+\frac{\rho l^{2}n^{2}a_{n}}{(4n^{2}-1)\pi}+\rho l^{2}\sum_{m=1}^{\infty}\frac{nma_{m}(a_{\|n-m\|}-a_{\|n+m\|})}{(4m^{2}-1)\pi}$, where $a_{0}(t)=1$ and $a_{1}(t)=2S(t)$, since the number density $\rho=\int{\rm d}{\bf u}f({\bf u},t)$ and the nematic order parameter $S(t)=\int{\rm d}{\bf u}{\rm cos}2\phi f({\bf u},t)/\rho$. By setting $a_{n}=0$ for $n\geq 3$, the truncated dynamic equation for the nematic order parameter can be written as: $\frac{\partial_{t}S(t)}{4D_{r}}=(\tilde{\rho}-1)S-\frac{3\tilde{\rho}^{2}S^{3}}{4(5-\tilde{\rho})}$, where $\tilde{\rho}=\rho/\rho^{}$ is the rescaled number density, and the critical density $\rho^{}=3\pi/2l^{2}$, beyond which the system enters into a spatially homogenous nematic state. Next, we examine the linear stability of such spatially homogeneous nematic state above $\rho^{}$, by expanding the number distribution function $f({\bf r},{\bf u})={(2\pi)^{-1}}\rho({\bf r})[1+4({u}_{\alpha}{u}_{\beta}-\delta_{\alpha\beta}/2)Q_{\alpha\beta}({\bf r})]$ with the inclusion of the alignment tensor $Q_{\alpha\beta}({\bf r})=S({\bf r})(\hat{n}_{\alpha}\hat{n}_{\beta}({\bf r})-\delta_{\alpha\beta}/2)$ where $\hat{\bf n}(\bf r)$ is the unit vector of the nematic director. We assume that nematic director is along the $x$ axis of the system. Small fluctuations of density and nematic director near the ordered nematic state are given by $\delta\tilde{\rho}({\bf r})=\tilde{\rho}({\bf r})-\tilde{\rho}_{0}$ and $S\delta n_{y}({\bf r})=Q_{xy}({\bf r})$, respectively. Here, $\tilde{\rho}_{0}=\rho_{0}/\rho^{}$ is the reduced bulk particle density, and $\delta n_{y}({\bf r})$ is the $y$-component of the deviated nematic director. Noting that ${\bf n}_{0}({\bf r})=\hat{{\bf x}}$ and $\|{\bf n}\|=1$, $\delta n_{y}({\bf r})$ is the only possible small fluctuation of nematic director ${\bf n}({\bf r})$. The resulting hydrodynamic equations can be obtained from Eq. (LABEL:dyadd11), yielding $\displaystyle\partial_{t}\delta\tilde{\rho}\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\frac{D_{p}}{2}\partial_{\alpha}^{2}\delta\tilde{\rho}+\frac{D_{n}}{2}(\partial_{x}^{2}-\partial_{y}^{2})(\tilde{\rho}S)$ (2) $\displaystyle+2D_{n}\tilde{\rho}_{0}S\partial_{x}\partial_{y}\delta n_{y},$ $\displaystyle\tilde{\rho}_{0}S\partial_{t}\delta n_{y}\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\frac{D_{n}}{4}\partial_{x}\partial_{y}\delta\rho+\frac{D_{p}}{2}\tilde{\rho_{0}}S\partial_{\alpha}^{2}\delta n_{y},$ (3) $\displaystyle\partial_{t}[\tilde{\rho}S({\bf r})]\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\frac{D_{n}}{4}(\partial_{x}^{2}-\partial_{y}^{2})\tilde{\rho}+\frac{D_{p}}{2}\partial_{\mu}^{2}(\tilde{\rho}S)$ (4) $\displaystyle+4D_{r}\tilde{\rho}(\tilde{\rho}-1)S-D_{r}\frac{3\tilde{\rho}^{3}S^{3}}{5-\tilde{\rho}},$ where $D_{p}=D_{\\|}+D_{\perp}$ and $D_{n}=D_{\\|}-D_{\perp}$. It is easy to see that the homogeneous state is stable to fluctuations of coupled nematic director and density field. Here we consider the stability of the modes that couple fluctuations of density $\delta\tilde{\rho}({\bf r})=\tilde{\rho}({\bf r})-\tilde{\rho}_{0}$ and magnitude of nematic order $\delta S_{\rho}({\bf r})=\tilde{\rho}({\bf r})S({\bf r})-\tilde{\rho}_{0}S_{0}$ with $\delta n_{y}=0$ around the homogeneous state $(\tilde{\rho}_{0},S_{0})$, where $S_{0}=\sqrt{4(5-\tilde{\rho}_{0})(\tilde{\rho}_{0}-1)/3\tilde{\rho}_{0}^{2}}$. The mode of fluctuations in Fourier components with wave vector ${\bf k}$, defined by $\delta\tilde{\rho}({\bf r})=\int{\rm d}{\bf k}\tilde{\rho}_{\bf k}e^{-i{\bf k}\cdot{\bf r}}$ and $\delta S_{\rho}({\bf r})=\int{\rm d}{\bf k}S_{\rho{\bf k}}e^{-i{\bf k}\cdot{\bf r}}$, is governed by $\displaystyle\partial_{t}\\!\\!\begin{bmatrix}\tilde{\rho}_{{\bf k}}\\\ \\\ S_{\rho\bf k}\end{bmatrix}\\!\\!=\\!-\frac{1}{2}\\!\\!\begin{bmatrix}D_{p}k^{2}&D_{n}\cos\negmedspace 2\theta k^{2}\\\ \\\ \begin{array}[]{ll}D_{n}\cos\negmedspace 2\theta k^{2}/2\\\ -8D_{r}\tilde{\rho}_{0}S_{0}\end{array}&\begin{array}[]{ll}D_{p}k^{2}\\\ +16D_{r}\delta\end{array}\end{bmatrix}\\!\\!\\!\begin{bmatrix}\tilde{\rho}_{{\bf k}}\\\ \\\ S_{\rho\bf k}\end{bmatrix},$ (5) where $\delta=\tilde{\rho}-1$, $\theta$ is the angle between wave vector ${\bf k}$ and nematic director ${\bf n}_{0}$. The eigenvalues of the coefficient matrix in Eq.(5) are given by $D_{r}^{-1}\lambda^{\pm}_{\tilde{\rho},S_{\rho}}=-(8\delta+\kappa^{2})/2\pm\sqrt{D_{0}^{2}\kappa^{4}\cos^{2}\negmedspace 2\theta/8-4\sigma D_{0}\kappa^{2}\cos\negmedspace 2\theta+16\delta^{2}}$, where $\sigma=\sqrt{(4-\delta)\delta/3}$, the rescaled coefficient $D_{0}=D_{n}/D_{p}$, and the wave number $\kappa=\sqrt{D_{p}/D_{r}}k$. The real part of $\lambda^{-}_{\tilde{\rho},S_{\rho}}$ is always negative, representing stable decaying mode. However, the mode $\lambda^{+}_{\tilde{\rho},S_{\rho}}$ becomes positive when $32(D_{0}\sigma\cos\\!2\theta+\delta)\kappa^{2}+(2-D_{0}^{2}\cos^{2}\\!2\theta)\kappa^{4}<0$. The coefficient of $\kappa^{4}$ is always positive since $\|D_{0}\|\leq 1$, signifying that for large enough wave numbers, the fluctuations are always stable. For small wave numbers which describe large-scale fluctuations, the stability is controlled by the coefficient of $\kappa^{2}$. Thus when $(D_{0}\sigma\cos\negmedspace 2\theta+\delta)<0$, the system becomes unstable on large scale. The phase map for $(D_{0},\delta)$ is given in Fig. 1. Between the isotropic and linearly stable nematic phases, there is a region where spatially homogeneous nematic state is unstable. It is denoted as a ‘phase with no stable state’ to emphasize that the only possible form of steady-state solution is unachievable there. For different $(D_{0},\delta)$ within the ‘no stable state’ region, the instability mode structures $(\kappa,\theta)$ are given by the polar plots whose central positions represent $(D_{0},\delta)$. Here, each polar plot is composed of horizontal (black) and vertical (red) branches enclosing unstable fluctuation modes, corresponding to $D_{0}<0$ and $D_{0}>0$, revealing that spatial inhomogeneities are developed parallel and perpendicular to the nematic director, respectively. The maximum values $\kappa_{m}$ of $\kappa$ for the instability regimes are always in the directions $\theta=\pi/2,3\pi/2$ for $D_{0}>0$ and $\theta=0,\pi$ for $D_{0}<0$, respectively. Generally speaking, $\kappa_{m}$ becomes larger when $(D_{0},\delta)$ is far from the phase boundary. The inset of Fig. 1 shows the value of $D_{r}^{-1}\lambda^{+}_{\tilde{\rho},S_{\rho}}$, where for small $\kappa$, $D_{r}^{-1}\lambda^{+}_{\tilde{\rho},S_{\rho}}>0$ corresponds to the long-wavelength instability and the onset of large-scale spatial inhomogeneity. Figure 2: Density and nematic order profiles are plotted. The color scale shows the local relative rescaled density value $\delta({\bf r})=\tilde{\rho}({\bf r})-1$. The length and angle of white segments show the magnitude and direction of nematic order, respectively. The dynamic parameters are $D_{r}=2$, $D_{\perp}=0.4$, and $D_{\\|}=2.4$. The reduced instability dynamic parameter $D_{0}=5/7$ and $\delta=0.01$. The system size is $300\times 300$ with the particle length $l=1$. Periodic boundary condition is implemented. Discrete time step $\Delta_{t}=0.018$ and spatial steps $\Delta_{x}=\Delta_{y}=\Delta_{r}=3$. (a)-(b) The snapshots are taken at times $1.7\times 10^{6}\Delta_{t}$ and $2.3\times 10^{6}\Delta_{t}$. (c)-(k) A close look of the breaking and coalescing processes of a nematic domain, at times $1.6\times 10^{6}$, $1.7\times 10^{6}$, $1.71\times 10^{6}$, $1.72\times 10^{6}$, $1.73\times 10^{6}$, $1.74\times 10^{6}$, $1.75\times 10^{6}$, $1.76\times 10^{6}$ and $1.8\times 10^{6}$ in unit $\Delta_{t}$. What will happen in the phase region where there is no stable steady state? And how the system evolves in time? To answer these questions we directly integrate Eq. (LABEL:dyadd11) numerically in this region using alternative implicit algorithm(see supp ). Starting from an isotropic and spatial- homogeneous initial condition, local ordered nematic domains form at the beginning, accompanied with quick development of density inhomogeneity. Further coarsening of these structures leads to the coexistence of particle- enriched nematic domains and particle-poor isotropic region where $\rho<\rho^{}$ (Fig. 2a). However, such a large-scale spatially inhomogeneous structure is unstable, and it will evolve and become fragmented as shown in Fig. 2b. The fragmented structure will again coalesce and similar process will repeat aperiodically and endlessly (see supp M1.mov). In Fig. 2c-2k, we show how a particle-rich nematic branch breaks into pieces and re-unites into a structure with new morphology. Figure 3: (a)-(c) Three typical steps that an ordered stripe breaks, at times $30000\Delta_{t}$, $60000\Delta_{t}$ and $63000\Delta_{t}$ in sequent. (d) Twisted spindle shaped structure formed after it breaks off from a larger ordered structure. The color scale shows the local relative rescaled density value $\delta({\bf r})=\tilde{\rho}({\bf r})-1$. The length and angle of white segments show the magnitude and direction of nematic order, respectively. The black arrow shows the direction and strength of particle fluxes. (e)-(f) Simulation result shows density inhomogeneity and similar restless evolvement in active nematics for $9.4\times 10^{4}$ and $1.1\times 10^{5}$ simulation sweeps. How does the fragmentation process occur? In Fig. 3a-c, we take a close look at the process that a nematic band breaks up(for a more continuous process, see supp M2.mov). In Fig. 3a, after the spontaneous formation of a nematic band, initially, it is shown that the nematic director in the high-density ordered region is almost parallel to the density stripe boundary. In this case, the nematic director is along the $x$-axis. For $D_{n}>0$, from our previous stability analysis, the term $-\frac{1}{2}D_{n}\partial_{y}^{2}(\rho S)$ in Eq.(2) is directly responsible for the development of such density inhomogeneity. Now, we are interested if such aligned director field is stable to small fluctuations $\delta{\bf n}_{\perp}({\bf r})={\bf n}({\bf r})-{\bf n}_{0}=(0,\delta\hat{n}_{\perp y})$. To linear order, the dynamic equation for $\delta\hat{n}_{\perp y}({\bf r})$ can be obtained from Eq.(LABEL:dyadd11) as $\tilde{\rho}S\partial_{t}\delta n_{\perp y}=\frac{1}{2}D_{p}[\tilde{\rho}S\partial_{x}^{2}\delta n_{\perp y}+\delta n_{\perp y}\partial_{y}^{2}\tilde{\rho}S]$, where we have assumed that there is no spatial variation of $\tilde{\rho}S$ along $x$-axis. The spatial variation of $\tilde{\rho}S$ along the $y$-axis is significant since density inhomogeneity is developed in that direction. Near stripe boundaries, we always have $\partial_{y}^{2}\tilde{\rho}S>0$, which makes the fluctuations $\delta n_{\perp y}$ unstable. Such instability will induce the change of the nematic orientation, and this explains why the nematic directors in Fig. 2 and Fig. 3b are most likely to be oblique to the density profile boundaries. When the nematic directors become oblique to the boundary, as shown in Fig. 3b, there is a leakage of particles from the high-density region. As the particle density in the stripe falls into the ‘no stable state’ region as shown in Fig. 1, the spatial instability takes place again. This leads to a fragmentation event, as shown in Fig. 3c. It is shown that a density crevice forms parallel with the nematic director as spatial instability requires. In Fig. 3d, we show a twisted-spindle shaped high-density region with local nematic order, which is commonly formed after it breaks off from a larger ordered structure in Fig. 2e. We also perform simulations to examine the stability of homogeneous nematic state on the basis of Eq.(LABEL:dyadd11) (see supp ). Fig. 3e shows the formation of a particle-enriched nematic band. We find that such band is also unstable and undergoes similar process (see supp M3.mov ), where the nematic director changes its direction in time and the fragmentation event takes place afterward (Fig. 3f). It is worth to notice that all these highly dynamic structures are surrounded by particle fluxes around the density profile boundaries. The density currents are defined as ${\bf J}({\bf r})=(J_{x}({\bf r}),J_{y}({\bf r}))$, with $J_{\alpha}({\bf r})=-\frac{1}{2}D_{p}\partial_{\alpha}\rho({\bf r})-D_{n}\partial_{\beta}[\rho({\bf r})Q_{\alpha\beta}(\bf r)]$ where the first term is just ordinary diffusive current and the second term is the current generated by coupling nematic directors. As we show in Fig. 3a, inward currents, generated by $(0,\frac{1}{2}D_{n}\partial_{y}(\rho S))$ which is included in the second term of ${\bf J}({\bf r})$, are directly responsible for the development of density inhomogeneity. As the time evolves, in Fig. 3b, along the two sides of stripe boundaries the system generates anti-parallel currents which also originate from $-D_{n}\partial_{\beta}[\rho({\bf r})Q_{\alpha\beta}(\bf r)]$, and the particles seem to move under a self- organized rachet potentialJulicher . When the crevice forms in Fig. 3c, particles flow into the low density region as guided by the nematic directors, accompanied with the growth of nematicly ordered tips near the crevice. In Fig. 3d, as indicated by the density currents, the twisted-spindle shaped nematic region absorbs particles from the medium on both sides and generate outward flux on both tips. In this way it can grow and extend itself quickly into the low-density medium. With the presence of these density fluxes around the ordered structure, it behaves like a creature absorbing, growing, dividing and dissipating into isotropic medium endlessly. In summary, the dynamic equation abstracted from previous simulations and experiments suggests a nematicly ordered phase with no stable steady state. Thus the system must evolve endlessly. We reveal the statistical mechanism that governs the large-scale and seemingly fluctuated collective evolution. We show that, as a result of long-wavelength instability, density and order inhomogeneity develops as guided by nematic director field. The spatial inhomogeneity in turn changes the directions of local nematic directors. The changed nematic directors further guide the fragmentation events, leading to endless evolution of the system. Finally, it would be interesting to extend our analysis to other active fluids. This work was supported by the National Natural Science Foundation of China (No. 10974080). ## References (1) J.K. Parrish and L. Edelstein-Keshet, Science 284, 99 (1999); M. Ballerini et al., Proc. Natl. Acad. Sci. USA 105, 1232 (2008). * (2) T. Feder, Phys. Today 60, 28 (2007). * (3) C. Dombrowski, L. Cisneros, S. Chatkaew, R.E. Goldstein, and J.O. Kessler, Phys. Rev. Lett. 93, 098103 (2004); D. Volfson, S. Cookson, J. Hasty, and L.S. Tsimring, Proc. Natl. Acad. Sci. USA 105, 15346 (2008). * (4) P. Rorth, Trends Cell Biol. 17, 575 (2007). * (5) T. Surrey, F. Nedelec, S. Leibler, and E. Karsenti, Science 292, 1167 (2001); P. Kraikivski, R. Lipowsky, and J. Kierfeld, Phys. Rev. Lett. 96, 250813 (2006); V. Schaller, C. Weber, C. Semmrich, E. Frey, and A.R. Bausch, Nature 467, 73 (2010); X. Shi, Y. Ma, Proc. Natl. Acad. Sci. USA 107, 11709 (2010). * (6) T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen, and O. Shochet, Phys. Rev. Lett. 75, 1226 (1995); J. Toner and Y. Tu, Phys. Rev. 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E 75, 051301 (2007). * (8) See supplementary information for supporting materials and movies. * (9) T. Shimada, M. Doi, and K. Okano, J. Chem. Phys. 88, 7181 (1988); A. Ahmadi, M.C. Marchetti, and T.B. Liverpool, Phys. Rev. E 74, 061913 (2006); A. Baskaran and M.C. Marchetti, Phys. Rev. E 77, 011920 (2008). * (10) E.N. Lorenz, J. Atoms. Sci. 20, 130 (1963). * (11) F. Jülicher, A. Ajdari, and J. Prost, Rev. Mod. Phys. 69, 1269 (1997).	arxiv-papers	2010-11-24T16:02:40	2024-09-04T02:49:15.204088	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Xia-qing Shi and Yu-qiang Ma", "submitter": "Yuqiang Ma", "url": "https://arxiv.org/abs/1011.5408" }
1011.5443	# Corrádi and Hajnal’s theorem for sparse random graphs József Balogh Department of Mathematics, University of Illinois, 1409 W Green Street, Urbana, IL 61801, USA; and Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA. E-mail address: jobal@math.uiuc.edu. This material is based upon work supported by NSF CAREER Grant DMS-0745185, and OTKA Grant K76099. Choongbum Lee Department of Mathematics, UCLA, Los Angeles, CA 90095, USA. E-mail address: choongbum.lee@gmail.com. Research supported in part by Samsung Scholarship. Wojciech Samotij School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel; and Trinity College, Cambridge CB2 1TQ, UK. E-mail address: samotij@post.tau.ac.il. Research supported in part by ERC Advanced Grant DMMCA. ###### Abstract In this paper we extend a classical theorem of Corrádi and Hajnal into the setting of sparse random graphs. We show that if $p(n)\gg(\log n/n)^{1/2}$, then asymptotically almost surely every subgraph of $G(n,p)$ with minimum degree at least $(2/3+o(1))np$ contains a triangle packing that covers all but at most $O(p^{-2})$ vertices. Moreover, the assumption on $p$ is optimal up to the $(\log n)^{1/2}$ factor and the presence of the set of $O(p^{-2})$ uncovered vertices is indispensable. The main ingredient in the proof, which might be of independent interest, is an embedding theorem which says that if one imposes certain natural regularity conditions on all three pairs in a balanced $3$-partite graph, then this graph contains a perfect triangle packing. ## 1 Introduction ### 1.1 Triangle packings in subgraphs of random graphs Let $H$ be a fixed graph on $h$ vertices, let $G$ be a graph on $n$ vertices. An arbitrary collection of vertex-disjoint copies of $H$ in $G$ is called an _$H$ -packing_ in $G$. A _perfect $H$-packing_ (an _$H$ -factor_) is an $H$-packing that covers all vertices of the host graph. In other words, $G$ has an $H$-factor (contains a perfect $H$-packing) if $n$ is divisible by $h$ and $G$ contains $n/h$ vertex-disjoint copies of $H$. It has been long known that for every graph $H$, if the minimum degree of $G$ is sufficiently large, then $G$ contains an $H$-factor. For example, by the Dirac’s Theorem on Hamiltonian cycles [11], if $H$ is a path of length $h-1$, then $\delta(G)\geq n/2$ guarantees that $G$ has an $H$-factor. Corrádi and Hajnal [9] proved that $\delta(G)\geq 2n/3$ is sufficient to guarantee a $K_{3}$-factor and Hajnal and Szemerédi [15] showed that $\delta(G)\geq(1-1/k)n$ suffices to guarantee a $K_{k}$-factor for an arbitrary $k$. Moreover, all these results are easily seen to be best possible. Finding a similar optimal condition on the minimum degree that guarantees an $H$-factor for an arbitrary graph $H$ has turned out to be significantly harder. The first result in this direction was obtained by Alon and Yuster [2], who showed that $\delta(G)\geq(1-1/\chi(H))n$ implies the existence of $n/h-o(n)$ vertex-disjoint copies of $H$ in $G$. Later, the same authors [4] showed that $\delta(G)\geq(1-1/\chi(H))n+o(n)$ guarantees an $H$-factor. Finally, Komlós, Sárközy, and Szemerédi [25] showed that merely $\delta(G)\geq(1-1/\chi(H))n+c(H)$, where $c(H)$ is a (small) constant depending only on $H$, suffices. Moreover, it was observed in [4] that there are graphs $H$ for which the above constant $c(H)$ cannot be omitted. Recently, Kühn and Osthus [30] replaced $\chi(H)$ in the above inequality by another parameter $\chi^{}(H)$, which depends on the relative sizes of the color classes in the optimal colorings of $H$ and satisfies $\chi(H)-1<\chi^{}(H)\leq\chi(H)$. Furthermore, they proved that the ratio $(1-1/\chi^{}(H))$ in the lower bound for $\delta(G)$ is optimal for every $H$. For further information on $H$-factors in graphs with large minimum degree, we refer the reader to [29, 30]. An independent direction of research concerned with $H$-factors has been determining the thresholds for the edge probability $p$ for the property that the Erdős-Rényi random graph $G(n,p)$ contains an $H$-factor. The case $H=K_{2}$ was solved by Erdős and Rényi [12], who proved that $\log n/n$ is the threshold for the existence of a perfect matching in $G(n,p)$. The solution for the case when $H$ is a path is a direct consequence of the result of Pósa [32]. Alon and Yuster [3] and, independently, Ruciński [33] determined the threshold for every $H$ whose fractional arboricity111The fractional arboricity of a graph $H$ is the quantity $\max\left\\{\frac{\|E(H^{\prime})\|}{\|V(H^{\prime})\|-1}\right\\}$, where the maximum is taken over all subgraphs $H^{\prime}$ of $H$ with $\|V(H^{\prime})\|>1$. is larger than its minimum degree. Later, partial results for the case $H=K_{3}$ were obtained by Krivelevich [26] and Kim [19] (a related work of Krivelevich, Sudakov, and Szabó [28] studied this case when the host graph is a sparse pseudo-random regular graph). Finally, Johansson, Kahn, and Vu [18] determined the thresholds for all strictly balanced $H$ and determined them up to a sub-polynomial factor for arbitrary $H$. Much less is known about common extensions of the results of the above two types. To make it precise, we would like to know whether it is true that for sufficiently large $p$, a.a.s. every spanning subgraph of $G(n,p)$ with sufficiently large minimum degree has an $H$-factor. Questions like these can be naturally expressed in the framework of resilience, also called fault tolerance. Following Sudakov and Vu [34], we state the following definition. ###### Definition 1.1. Let $\mathcal{P}$ be a monotone increasing graph property. The local resilience of a graph $G$ with respect to $\mathcal{P}$ is the minimum number $r$ such that by deleting at most $r$ edges at each vertex of $G$, one can obtain a graph without $\mathcal{P}$. Using this terminology, one can restate, e.g., the aforementioned theorem of Corrádi and Hajnal [9] by saying that the local resilience of the complete graph $K_{n}$ with respect to the property of having a triangle-factor is (at least) $n/3$. Rephrasing our previous question, we would like to determine the local resilience of the random graph $G(n,p)$ with respect to the property of containing an $H$-factor for some fixed graph $H$. Sudakov and Vu [34] showed that it is $(1/2+o(1))$ when $H=K_{2}$ and $p\gg\log n/n$ or when $H$ is a path and $p\gg(\log n)^{4}/n$; Lee and Sudakov [31] showed that the assumption $p\gg\log n/n$ suffices also in the latter case. Recently, Huang, Lee, and Sudakov [17] addressed this problem for an arbitrary $H$ in the case when the edge probability $p$ is a constant. ###### Theorem 1.2. Let $H$ be a fixed graph on $h$ vertices, let $p\in(0,1]$, and let $\gamma$ be a positive real. 1. 1. If $H$ has a vertex that is not contained in any triangle, then a.a.s. every spanning subgraph $G\subset G(n,p)$ with $\delta(G)\geq(1-1/\chi(H)+\gamma)np$ has a perfect $H$-packing, provided that $n$ is divisible by $h$. 2. 2. If every vertex of $H$ is contained in a triangle, then a.a.s. every spanning subgraph $G\subset G(n,p)$ with $\delta(G)\geq(1-1/\chi(H)+\gamma)np$ contains an $H$-packing covering all but at most $Dp^{-2}$ vertices of $G$, where $D$ is a constant that depends only on $\chi(H)$. Moreover, it was shown in [17] that in the case when each vertex of $H$ belongs to some triangle, the $Dp^{-2}$ error term cannot be removed as a.a.s. $G(n,p)$ has a spanning subgraph with large minimum degree such that at least $\Omega(p^{-2})$ of its vertices are not contained in a triangle (and hence they are not contained in a copy of $H$). For other results on local resilience of random graphs with respect to the property of containing spanning or nearly spanning subgraphs, see [5, 6, 7, 10, 13, 27]. In this paper, we extend the result of Huang, Lee, and Sudakov to the sparse random graph setting in the case $H=K_{3}$. A rather straightforward argument using the conjecture of Kohayakawa, Łuczak, and Rödl [21, Conjecture 23], which is known to be true for triangles, shows that if $p\gg n^{-1/2}$, then a.a.s. every subgraph of $G(n,p)$ whose minimum degree exceeds $(2/3+o(1))np$ contains a triangle-packing that covers all but at most $\varepsilon n$ vertices, where $\varepsilon$ is an arbitrary positive constant (see Remark 2.8). Our main theorem proves that under the same assumptions, one can make the set of uncovered vertices significantly smaller. More precisely, we prove the following statement. ###### Theorem 1.3. For all positive $\gamma$, there exist constants $C$ and $D$ such that if $p\geq C(\log n/n)^{1/2}$, then a.a.s. every subgraph $G\subset G(n,p)$ with $\delta(G)\geq(2/3+\gamma)np$ contains a triangle packing that covers all but at most $Dp^{-2}$ vertices. Clearly, the ratio $2/3$ in the statement of Theorem 1.3 is best possible as for every positive $\gamma$, a.a.s. $G(n,p)$ has a subgraph $G$ with $\delta(G)\geq(2/3-\gamma)np$ whose largest triangle packing covers no more than $(1-\gamma)n$ vertices (e.g., we may let $G$ be the intersection of $G(n,p)$ with the complete $3$-partite graph with color classes of sizes $(1+\gamma)n/3$, $n/3$, and $(1-\gamma)n/3$). Furthermore, even though it was proved in [18] that $p\gg n^{-2/3}(\log n)^{1/3}$ guarantees that $G(n,p)$ a.a.s. has a triangle-factor, the lower bound on $p$ in Theorem 1.3 cannot be relaxed by more than the $(\log n)^{1/2}$ factor as if $p\ll n^{-1/2}$, then a.a.s. one can remove all triangles from $G(n,p)$ by deleting only $o(np)$ edges incident to every vertex. Finally, the presence of the exceptional set of $Dp^{-2}$ is indispensable, see Proposition 4.6 and [17, Proposition 6.3]. ### 1.2 Embedding theorem for sparse regular triples One of the main ingredients in the proof of Theorem 1.3 is an embedding theorem for large triangle packings in sparse regular triples. Before we state this result (Theorem 1.4 below), we recall a few basic definitions and briefly summarize what is known about embedding large graphs into regular triples. Let $G$ be a graph on a vertex set $V$. Given a pair of disjoint subsets $V_{1},V_{2}\subset V$, let $e(V_{1},V_{2})$ denote the number of edges of $G$ with one endpoint in $V_{1}$ and the other endpoint in $V_{2}$, and let the _density_ $d(V_{1},V_{2})$ of the pair $(V_{1},V_{2})$ be the quantity $e(V_{1},V_{2})/(\|V_{1}\|\|V_{2}\|)$. The pair $(V_{1},V_{2})$ is called _$(\varepsilon,p)$ -regular_ if for all $V_{1}^{\prime}\subset V_{1}$ and $V_{2}^{\prime}\subset V_{2}$ with $\|V_{1}^{\prime}\|\geq\varepsilon\|V_{1}\|$ and $\|V_{2}^{\prime}\|\geq\varepsilon\|V_{2}\|$, we have $\left\|d(V_{1},V_{2})-d(V_{1}^{\prime},V_{2}^{\prime})\right\|\leq\varepsilon p$. An $(\varepsilon,1)$-regular pair is simply called _$\varepsilon$ -regular_. The concept of regularity, first developed by Szemerédi [35], proved to be of extreme importance in modern combinatorics and played a central rôle in proofs of a range of results in extremal graph theory, Ramsey theory, and others. For example, it is well-known that every triple of sets $(V_{1},V_{2},V_{3})$ such that $(V_{i},V_{j})$ is $\varepsilon$-regular and has sufficiently large density for all distinct $i$, $j$ contains a triangle. An $\varepsilon$-regular pair $(V_{1},V_{2})$ is called _$(\delta,\varepsilon)$ -super-regular_ if it satisfies the additional condition that every vertex in $V_{1}$ has at least $\delta\|V_{2}\|$ neighbours in $V_{2}$ and, vice versa, every vertex in $V_{2}$ has at least $\delta\|V_{1}\|$ neighbours in $V_{1}$. Komlós, Sárközy, and Szemerédi [24] proved that super-regular triples are even more powerful than mere regular triples. For instance, every triple $(V_{1},V_{2},V_{3})$ such that $\|V_{1}\|=\|V_{2}\|=\|V_{3}\|$ and $(V_{i},V_{j})$ is $(\delta,\varepsilon)$-super- regular and has sufficiently large density for all distinct $i$, $j$ contains not only a single triangle, but also a family of vertex-disjoint triangles that cover all vertices of the triple. However, if $p\ll 1$, then the power of $(\varepsilon,p)$-regular pairs turns out to be significantly weaker. For example, Łuczak (see [22]) observed that there are $(\varepsilon,p)$-regular triples which do not contain even a single triangle. Still, Kohayakawa, Łuczak, and Rödl [21] proved that most $(\varepsilon,p)$-regular triples contain a triangle provided that $p$ is sufficiently large and conjectured that an analogous result holds for arbitrary graphs (see the survey [14]). It is not much of a surprise that even less is known about embedding large graphs into sparse regular pairs. Böttcher, Kohayakawa, and Taraz [7] proved that if the regular pair is a subgraph of a random graph and each part has size $n$, then (asymptotically almost surely) one can embed into the pair all bipartite graphs with bounded maximum degree whose color classes both have size at most $(1-\eta)n$, where $\eta$ is a fixed positive real. Since in an $(\varepsilon,p)$-regular pair $(V_{1},V_{2})$, each set $V_{i}$ can have as many as $c_{\varepsilon}\|V_{i}\|$ isolated vertices, one cannot hope to embed spanning graphs into the pair without imposing some further restrictions. Let us now consider sparse regular triples. Observe that imposing merely a minimum degree condition as in the dense case is not sufficient since we can remove all triangles that contain a fixed vertex by deleting all edges in its neighbourhood (this will not effect regularity of the triple since the neighbourhoods of this vertex have size $o(n)$). We suggest one possible strengthening of the notion of super-regularity, which we call _strong-super- regularity_ , and show that a sparse strong-super-regular triple in a subgraph of a random graph contains a collection of vertex-disjoint triangles that cover all the vertices of the triple. The definition of a strong-super-regular triple is given in Definition 2.11. ###### Theorem 1.4. For all positive $\delta$ and $\xi$ there exist $\varepsilon(\delta)$ and $C(\delta,\xi)$ such that if $p(n)\geq C(\log n/n)^{1/2}$, then $G(n,p)$ a.a.s. satisfies the following. Every $(\delta,\varepsilon,p)$-strong-super- regular triple $(V_{1},V_{2},V_{3})$ that is a subgraph of $G(n,p)$ with $\|V_{1}\|=\|V_{2}\|=\|V_{3}\|\geq\xi n$ contains a collection of vertex-disjoint triangles that cover all the vertices. It is possible that one can derive the same conclusion from weaker assumptions than strong-super-regularity. However, we will later show that the restriction we imposed is not too strong to make our theorem useless, as Theorem 1.4 will form an essential part in the proof of Theorem 1.3. ### 1.3 Outline of the paper In Section 2, we recall some known definitions and results and introduce a few notions that will be of great importance in all subsequent sections. Section 3 contains an outline of the proof of Theorem 1.3. In Section 4, we establish some properties of the random graph $G(n,p)$ that we will frequently invoke in subsequent sections. In Sections 5, 6, and 7, we prove a series of technical lemmas that culminate in the proof of Theorem 1.3 and 1.4. For a brief outline of this part of the paper, we refer the reader to Section 3. Finally, Section 8 contains a few concluding remarks. ### 1.4 Notation Let $G$ be a graph with vertex set $V$ and edge set $E$. For a vertex $v\in V$, we denote its neighbourhood in $G$ by $N(v)$ and let $\deg(v)$ be its degree. The minimum degree of the graph is denoted by $\delta(G)$. For a set $X\subset V$, we let $e(X)$ be the number of edges of $G$ with both endpoints in the set $X$, and $\deg(v,X)=\|N(v)\cap X\|$. We say that two edges are independent if they do not share a vertex. For two subsets $X,Y\subset V$, we let $e(X,Y)$ be the number of ordered pairs $(x,y)$ such that $x\in X$, $y\in Y$ and $xy$ is an edge of $G$; note that $e(X,X)=2e(X)$. If $X$ and $Y$ are disjoint, we refer to the quantity $e(X,Y)/(\|X\|\|Y\|)$, denoted by $d(X,Y)$, as the density of the pair $(X,Y)$. With a slight abuse of notation, we will sometimes write $(X,Y)$ to denote the set of all edges $xy$ with $x\in X$ and $y\in Y$. Let $X,Y,Z\subset V$ be three pairwise disjoint sets. We say that the triple $(X,Y,Z)$ is balanced if $\|X\|=\|Y\|=\|Z\|$. The minimum density of the triple is the minimum of the numbers $d(X,Y)$, $d(X,Z)$, and $d(Y,Z)$. A triangle across $(X,Y,Z)$ is any triangle with one vertex in each of $X$, $Y$, and $Z$. When the implicit graph we are considering is not clear from the context, we will use subscripts to prevent ambiguity. For example, $\deg_{G}(v)$ is the degree of $v$ in the graph $G$. We write $y=1\pm x$ to abbreviate $y\in[1-x,1+x]$. We omit floor and ceiling signs whenever they are not crucial. Throughout the paper, $\log$ will always denote the natural logarithm. Finally, we often use subscripts such as in $c_{3.6}$ to explicitly indicate that the constant $c_{3.6}$ is defined in Claim/Lemma/Proposition/Theorem 3.6. ## 2 Preliminaries ### 2.1 Sparse regularity lemma Let $G$ be a graph on a vertex set $V$. Recall that a pair $(V_{1},V_{2})$ of disjoint subsets of $V$ is $(\varepsilon,p)$-regular if for all $V_{1}^{\prime}\subset V_{1}$ and $V_{2}^{\prime}\subset V_{2}$ with $\|V_{1}^{\prime}\|\geq\varepsilon\|V_{1}\|$ and $\|V_{2}^{\prime}\|\geq\varepsilon\|V_{2}\|$, $\left\|d(V_{1},V_{2})-d(V_{1}^{\prime},V_{2}^{\prime})\right\|\leq\varepsilon p.$ We call a triple $(V_{1},V_{2},V_{3})$ of disjoint subsets of $V$ _$(\varepsilon,p)$ -regular_ if $(V_{i},V_{j})$ forms an $(\varepsilon,p)$-regular pair for every $\\{i,j\\}\subset\\{1,2,3\\}$. Let $\mathcal{G}(K_{3},(n_{1},n_{2},n_{3}),(d_{12},d_{23},d_{31}),(\varepsilon,p))$ be the collection of all $(\varepsilon,p)$-regular triples $(V_{1},V_{2},V_{3})$ such that $\|V_{i}\|=n_{i}$ for all $i$ and $d(V_{i},V_{j})=d_{ij}p$ for every $\\{i,j\\}\subset\\{1,2,3\\}$. Below we establish two simple hereditary properties of regular pairs. ###### Proposition 2.1. Let positive reals $\varepsilon_{1}$, $\varepsilon_{2}$, and $p$ satisfying $\varepsilon_{1}<\varepsilon_{2}\leq 1/2$ be given. Let $(V_{1},V_{2})$ be an $(\varepsilon_{1},p)$-regular pair and for $i\in\\{1,2\\}$, let $V_{i}^{\prime}\subset V_{i}$ be an arbitrary subset with $\|V_{i}^{\prime}\|\geq\varepsilon_{2}\|V_{i}\|$. Then $(V_{1}^{\prime},V_{2}^{\prime})$ is an $(\varepsilon_{1}/\varepsilon_{2},p)$-regular pair of density $d(V_{1},V_{2})\pm\varepsilon_{1}p$. ###### Proof. By regularity of the pair $(V_{1},V_{2})$, for every pair of subsets $V_{i}^{\prime\prime}\subset V_{i}^{\prime}$ such that $\|V_{i}^{\prime\prime}\|\geq(\varepsilon_{1}/\varepsilon_{2})\|V_{i}^{\prime}\|\geq\varepsilon_{1}\|V_{i}\|$ for $i\in\\{1,2\\}$, we have $\|d(V_{1}^{\prime\prime},V_{2}^{\prime\prime})-d(V_{1}^{\prime},V_{2}^{\prime})\|\leq\|d(V_{1}^{\prime\prime},V_{2}^{\prime\prime})-d(V_{1},V_{2})\|+\|d(V_{1}^{\prime},V_{2}^{\prime})-d(V_{1},V_{2})\|\leq 2\varepsilon_{1}p.$ Since $\max\\{\varepsilon_{1}/\varepsilon_{2},2\varepsilon_{1}\\}=\varepsilon_{1}/\varepsilon_{2}$, the pair $(V_{1}^{\prime},V_{2}^{\prime})$ is $(\varepsilon_{1}/\varepsilon_{2},p)$-regular. The density condition immediately follows from the definition of regularity. ∎ ###### Proposition 2.2. Let $(V_{1},V_{2})$ be an $(\varepsilon,p)$-regular pair in a graph $G$ and let $G^{\prime}$ be a subgraph of $G$ obtained by removing at most $\varepsilon^{3}p\|V_{1}\|\|V_{2}\|$ edges from $(V_{1},V_{2})$. Then $(V_{1},V_{2})$ is $(3\varepsilon,p)$-regular in $G^{\prime}$. ###### Proof. For $i\in\\{1,2\\}$, let $U_{i}$ be a subset of $V_{i}$ of size at least $\varepsilon\|V_{i}\|$. Note that $\|d_{G}(U_{1},U_{2})-d_{G^{\prime}}(U_{1},U_{2})\|\leq\frac{e_{G}(U_{1},U_{2})-e_{G^{\prime}}(U_{1},U_{2})}{\|U_{1}\|\|U_{2}\|}\leq\frac{\varepsilon^{3}p\|V_{1}\|\|V_{2}\|}{\|U_{1}\|\|U_{2}\|}\leq\varepsilon p.$ The conclusion easily follows from the triangle inequality. ∎ An _$(\varepsilon,p)$ -regular partition_ of an $n$-vertex graph $G$ is a partition $(V_{i})_{i=0}^{k}$ of its vertex set such that (i) the exceptional class $V_{0}$ has size at most $\varepsilon n$, (ii) $V_{1},\ldots,V_{k}$ have equal sizes, and (iii) all but at most $\varepsilon k^{2}$ of the pairs $(V_{i},V_{j})$ are $(\varepsilon,p)$-regular. Given a collection of subsets $(W_{i})_{i=0}^{k}$ of the vertex set $V(G)$, the _$(\delta,\varepsilon,p)$ -reduced graph_ $R$ of the collection is the graph on the vertex set $[k]$ such that $i,j\in[k]$ are adjacent if and only if $W_{i}$ and $W_{j}$ form an $(\varepsilon,p)$-regular pair of density at least $\delta p$. Note that when considering reduced graphs, the partition $(W_{i})_{i=0}^{k}$ is not necessarily a regular partition and we ignore the set $W_{0}$. For a graph $R^{\prime}$ on the vertex set $[k]$, we say that $G$ is $(\delta,\varepsilon,p)$-regular over $R^{\prime}$ if for every edge $\\{i,j\\}$ of $R^{\prime}$, the pair $(V_{i},V_{j})$ is $\varepsilon$-regular with density at least $\delta$. Let $\eta$ and $b$ be reals such that $\eta\in(0,1]$, and $b\geq 1$. We say that $G$ is _$(\eta,b,p)$ -upper- uniform_ if $d(V_{1},V_{2})\leq bp$ for all disjoint sets $V_{1}$, $V_{2}$ with $\|V_{1}\|,\|V_{2}\|\geq\eta\|V\|$. With the above definitions at hand, we may now state a version of Szemerédi’s regularity lemma for upper-uniform graphs (see, e.g., [20, 23]). ###### Theorem 2.3. For every positive $\varepsilon$, $b$, and $k_{0}$ with $b,k_{0}\geq 1$, there exist constants $\eta(\varepsilon,b,k_{0})$ and $K(\varepsilon,b,k_{0})$ with $K\geq k_{0}$ such that for every positive $p$, every $(\eta,b,p)$-upper- uniform graph with at least $k_{0}$ vertices admits an $(\varepsilon,p)$-regular partition $(V_{i})_{i=0}^{k}$ such that $k_{0}\leq k\leq K$, and each part forms a regular pair with at least $(1-\varepsilon)k$ other parts. The version of the regularity lemma stated above is slightly different from those given in [20, 23], which say that the total number of irregular pairs is at most $\varepsilon k^{2}$. However, by using some standard techniques, one can derive the ‘minimum degree’ version from the results in [20, 23]. ### 2.2 Typical vertices and super-regularity We start this section by introducing the notions of typical vertices and triples. ###### Definition 2.4. Let $(V_{1},V_{2},V_{3})$ be a triple of sets (not necessarily regular) with densities $d_{ij}p$ between $V_{i}$ and $V_{j}$. 1. (A) Fix a vertex $v\in V_{1}$ and for $i\in\\{2,3\\}$, let $N_{i}=N(v)\cap V_{i}$. We say that $v$ is _$\varepsilon$ -typical_ if for $i\in\\{2,3\\}$, 1. (i) $\|N_{i}\|=(1\pm\varepsilon)d_{1i}p\|V_{i}\|$ and 2. (ii) there exists $N_{i}^{\prime}\subset N_{i}$ satisfying $\|N_{i}^{\prime}\|\geq(1-\varepsilon)\|N_{i}\|$ such that $(N_{2}^{\prime},N_{3}^{\prime})$ is an $(\varepsilon,p)$-regular pair with density $(1\pm\varepsilon)d_{23}p$. 2. (B) The triple $(V_{1},V_{2},V_{3})$ is _$\varepsilon$ -typical_ if it is $(\varepsilon,p)$-regular and for each $i$, all but at most $\varepsilon\|V_{i}\|$ vertices in $V_{i}$ are $\varepsilon$-typical. ###### Remark 2.5. Since the property of being $\varepsilon$-typical depends not only on $\varepsilon$ but also on $p$, we should rather speak of $(\varepsilon,p)$-typical vertices and triples. Nevertheless, since the parameter $p$ will be always clear from the context, we will suppress it from the notation for the sake of brevity. It turns out that an overwhelming majority of all regular triples are also typical. The following lemma, which is a straightforward generalization of [14, Lemma 5.1], makes the above statement precise. We omit its proof as it can be easily read out from the proof of [14, Lemma 5.1]. ###### Lemma 2.6. For all positive $\beta$, $\delta$, $\varepsilon^{\prime}$, and $\xi$, there exist constants $\varepsilon_{0}(\beta,\delta,\varepsilon^{\prime})$ and $C(\delta,\varepsilon^{\prime},\xi)$ such that if $\varepsilon\leq\varepsilon_{0}$, $d_{12},d_{13},d_{23}\geq\delta$, $\xi n\leq n_{1},n_{2},n_{3}\leq n$, and $p\geq Cn^{-1/2}$, then all but at most $\beta^{\delta\xi^{2}n^{2}p}{n_{1}n_{2}\choose d_{12}pn_{1}n_{2}}{n_{1}n_{3}\choose d_{13}pn_{1}n_{3}}{n_{2}n_{3}\choose d_{23}pn_{2}n_{3}}$ graphs in $\mathcal{G}(K_{3},(n_{1},n_{2},n_{3}),(d_{12},d_{13},d_{23}),(\varepsilon,p))$ are $\varepsilon^{\prime}$-typical provided that $n$ is sufficiently large. The following proposition justifies why the notion of $\varepsilon$-typical triples can be useful for our purposes. ###### Proposition 2.7. For every positive $\alpha$, $\delta$, and $p$, there exists an $\varepsilon(\alpha,\delta)$ such that every $\varepsilon$-typical $(\varepsilon,p)$-regular triple $(V_{1},V_{2},V_{3})$ of minimum density at least $\delta p$ contains $(1-\alpha)\min_{i}\|V_{i}\|$ vertex-disjoint triangles. ###### Proof. Note that without loss of generality, we may assume that $\alpha\leq 1/2$. Furthermore, let $\varepsilon=\min\\{\delta/4,\alpha/20\\}$ and let $\varepsilon^{\prime}=2\varepsilon/\alpha$. Let us greedily remove triangles from $(V_{1},V_{2},V_{3})$ until we cannot do it anymore and denote the remaining triple by $(V^{\prime}_{1},V^{\prime}_{2},V^{\prime}_{3})$. If $\|V^{\prime}_{i}\|\leq\alpha\|V_{i}\|$ for some $i$, then there is nothing left to prove, so we may assume that $\|V^{\prime}_{i}\|>\alpha\|V_{i}\|$ for all $i$. Let $W_{i}$ be the set of all those vertices in $V^{\prime}_{i}$ that were $\varepsilon$-typical in the original triple and note that $\|W_{i}\|\geq\|V^{\prime}_{i}\|-\varepsilon\|V_{i}\|\geq(\alpha/2)\|V_{i}\|$. By Proposition 2.1, the triple $(W_{1},W_{2},W_{3})$ is $(\varepsilon^{\prime},p)$-regular and the density of each pair $(W_{i},W_{j})$ is at least $(d_{ij}-\varepsilon)p$. Since $\varepsilon^{\prime}<1/2$, there is a vertex $v\in W_{1}$ with $\deg(v,W_{i})\geq(d_{ij}-\varepsilon-\varepsilon^{\prime})p\|W_{i}\|$ for $i\in\\{2,3\\}$. For $i\in\\{2,3\\}$, let $N_{i}$ and $N^{\prime}_{i}$ be the sets from the definition of an $\varepsilon$-typical vertex for $v$ and let $M_{i}=N_{i}\cap W_{i}$ and $M^{\prime}_{i}=N^{\prime}_{i}\cap W_{i}$. Since $\displaystyle\|M^{\prime}_{i}\|$ $\displaystyle\geq\|M_{i}\|-\|N_{i}\setminus N^{\prime}_{i}\|\geq(d_{1i}-\varepsilon-\varepsilon^{\prime})p\|W_{i}\|-\varepsilon(1+\varepsilon)d_{1i}p\|V_{i}\|$ $\displaystyle\geq\big{[}(1-\varepsilon/\delta-\varepsilon^{\prime}/\delta)(\alpha/2)-\varepsilon(1+\varepsilon)\big{]}d_{1i}p\|V_{i}\|\geq\varepsilon(1+\varepsilon)d_{1i}p\|V_{i}\|\geq\varepsilon\|N^{\prime}_{i}\|$ and $(N^{\prime}_{2},N^{\prime}_{3})$ was $(\varepsilon,p)$-regular with density at least $(1-\varepsilon)\delta p$ and $(1-\varepsilon)\delta p>\varepsilon p$, the pair $(M^{\prime}_{2},M^{\prime}_{3})$ has positive density. It follows that $(W_{1},W_{2},W_{3})$ contains a triangle, but this is impossible. ∎ ###### Remark 2.8. It is quite easy to see that the combination of Theorems 2.3 and 2.16, Lemma 2.6 (see Proposition 4.8), and Proposition 2.7 implies the following statement. For all positive constants $\gamma$ and $\varepsilon$, there exists a $C$ such that if $p(n)\geq Cn^{-1/2}$, then a.a.s. every subgraph $G\subset G(n,p)$ with $\delta(G)\geq(2/3+\gamma)np$ contains a triangle packing that covers all but at most $\varepsilon n$ vertices of $G$. The following concept will serve us as a generalization of super-regularity to the sparse setting. ###### Definition 2.9. A triple $(V_{1},V_{2},V_{3})$ is _$(\delta,\varepsilon,p)$ -super-regular_ if each pair $(V_{i},V_{j})$ is $(\varepsilon,p)$-regular with density at least $\delta p$ and for every $i$, all vertices in $V_{i}$ are $\varepsilon$-typical. We close this section with the following proposition, which tells us how to trim a typical regular triple in order to get a super-regular one. ###### Proposition 2.10. For all positive $\varepsilon^{\prime}$ and $\delta$, there exists an $\varepsilon(\varepsilon^{\prime},\delta)$ such that the following holds. Let $(V_{1},V_{2},V_{3})$ be an $(\varepsilon,p)$-regular triple, where for each $i$ and $j$, the density of $(V_{i},V_{j})$ is $d_{ij}p$, where $d_{ij}\geq\delta$. For each $i$, let $X_{i}$ be an arbitrary subset of $V_{i}$ with $\|X_{i}\|\leq\varepsilon\|V_{i}\|$. Then every vertex $v\in V_{1}\setminus X_{1}$ that is $\varepsilon$-typical and satisfies $\deg(v,X_{j})\leq\varepsilon p\|V_{j}\|$ for every $j\in\\{2,3\\}$ becomes an $\varepsilon^{\prime}$-typical vertex in $(V_{1}\setminus X_{1},V_{2}\setminus X_{2},V_{3}\setminus X_{3})$. ###### Proof. Let $v\in V_{1}$ be any such vertex and for $j\in\\{2,3\\}$, let $N_{j}=N(v)\cap V_{j}$. Since $v$ is $\varepsilon$-typical, $(1-\varepsilon)d_{1j}p\|V_{j}\|\leq\|N_{j}\|\leq(1+\varepsilon)d_{1j}p\|V_{j}\|$. Moreover, there exist subsets $N^{\prime}_{j}\subset N_{j}$ satisfying $\|N^{\prime}_{j}\|\geq(1-\varepsilon)\|N_{j}\|$ such that $(N^{\prime}_{2},N^{\prime}_{3})$ is an $(\varepsilon,p)$-regular pair with density $d_{v}p$, where $d_{v}\in[(1-\varepsilon)d_{23},(1+\varepsilon)d_{23}]$. Moreover, let $M_{j}=N_{j}\setminus X_{j}$ and similarly let $M^{\prime}_{j}=N^{\prime}_{j}\setminus X_{j}$. For each $i$, let $W_{i}=V_{i}\setminus X_{i}$ and recall that $(1-\varepsilon)\|V_{i}\|\leq\|W_{i}\|\leq\|V_{i}\|$. For every $i$ and $j$, let $d^{\prime}_{ij}p$ be the density of the pair $(W_{i},W_{j})$. Since $(V_{i},V_{j})$ is $(\varepsilon,p)$-regular, $d^{\prime}_{ij}\in[d_{ij}-\varepsilon,d_{ij}+\varepsilon]$. It follows that $\|M_{j}\|\leq\|N_{j}\|\leq(1+\varepsilon)d_{1j}p\|V_{j}\|\leq(1+\varepsilon)(1-\varepsilon)^{-1}(d^{\prime}_{1j}+\varepsilon)p\|W_{j}\|.$ And by the given condition $\deg(v,X_{j})\leq\varepsilon p\|V_{j}\|$, we have $\|M_{j}\|\geq\|N_{j}\|-\varepsilon p\|V_{j}\|\geq(1-\varepsilon-\varepsilon/\delta)d_{1j}p\|V_{j}\|\geq(1-\varepsilon-\varepsilon/\delta)(d^{\prime}_{1j}-\varepsilon)p\|W_{j}\|.$ Moreover, since $\|N_{j}\|\geq(1-\varepsilon)\delta p\|V_{j}\|$, we have $\displaystyle\|M^{\prime}_{j}\|$ $\displaystyle\geq\|N^{\prime}_{j}\|-\varepsilon p\|V_{j}\|\geq(1-\varepsilon)\|N_{j}\|-\varepsilon p\|V_{j}\|\geq(1-\varepsilon-\varepsilon/((1-\varepsilon)\delta))\|N_{j}\|$ $\displaystyle\geq(1-\varepsilon-\varepsilon/((1-\varepsilon)\delta))\|M_{j}\|\geq\|M_{j}\|/2.$ By Proposition 2.1, the pair $(M^{\prime}_{2},M^{\prime}_{3})$ is $(2\varepsilon,p)$-regular with density $d^{\prime}_{v}p$ satisfying $(1-\varepsilon)(d^{\prime}_{23}-\varepsilon)-\varepsilon\leq d^{\prime}_{v}\leq(1+\varepsilon)(d^{\prime}_{23}+\varepsilon)+\varepsilon.$ Therefore, if $\varepsilon$ is sufficiently small, then $\deg(v,W_{j})=\|M_{j}\|\in[(1-\varepsilon^{\prime})pd^{\prime}_{1j}\|W_{j}\|,(1+\varepsilon^{\prime})pd^{\prime}_{1j}\|W_{j}\|]$, $\|M^{\prime}_{j}\|\geq(1-\varepsilon^{\prime})\|M_{j}\|$, and $(M^{\prime}_{2},M^{\prime}_{3})$ is $(\varepsilon^{\prime},p)$-regular with density $d^{\prime}_{v}p$, where $d^{\prime}_{v}\in[(1-\varepsilon^{\prime})d^{\prime}_{23},(1+\varepsilon^{\prime})d^{\prime}_{23}]$. It follows that $v$ is $\varepsilon^{\prime}$-typical in $(W_{1},W_{2},W_{3})$. ∎ ### 2.3 Good edges and good vertices As we established in Section 2.2 (see Remark 2.8), imposing certain regularity conditions on the vertices of a regular triple suffices to guarantee the existence of an almost perfect triangle packing. In order to assure that a triangle-factor can be found, we will need to impose some conditions also on the edges of the triple. With hindsight (see the discussion in Section 3.2), we now introduce the notions of _good edges_ and _good vertices_. ###### Definition 2.11. Let $(V_{1},V_{2},V_{3})$ be a triple of sets (not necessarily regular) with densities $d_{ij}p$ between $V_{i}$ and $V_{j}$. 1. (A) We say that an edge between $V_{2}$ and $V_{3}$ is _$\varepsilon$ -good_ if its endpoints have at least $(1-\varepsilon)d_{12}d_{13}p^{2}\|V_{1}\|$ common neighbourhoods in $V_{1}$. 2. (B) We say that an $\varepsilon$-typical vertex $v\in V_{1}$ is $\varepsilon$-good if $(N(v)\cap V_{2},N(v)\cap V_{3})$ contains at most $\varepsilon d_{12}d_{13}d_{23}p^{3}\|V_{2}\|\|V_{3}\|$ edges that are not $\varepsilon$-good. 3. (C) We say that the triple $(V_{1},V_{2},V_{3})$ is _$(\delta,\varepsilon,p)$ -strong-super-regular_ if it is $(\delta,\varepsilon,p)$-super-regular and for every $i$, all vertices in $V_{i}$ are $\varepsilon$-good. Next, we show that super-regular triples are not very far from being strong- super-regular. More precisely, we prove that requiring a triple to be merely typical (recall Definition 2.4) and all pairs in this triple to have non-zero densities forces most of its edges and vertices to be good. ###### Proposition 2.12. Let $\varepsilon$, $\varepsilon^{\prime}$, and $\delta$ be positive constants satisfying $3\varepsilon+\varepsilon/\delta\leq\varepsilon^{\prime}$ and let $(V_{1},V_{2},V_{3})$ be an $\varepsilon$-typical $(\varepsilon,p)$-regular triple, where the density $d_{ij}p$ of each pair $(V_{i},V_{j})$ is at least $\delta p$. Then there are at most $4\varepsilon d_{23}p\|V_{2}\|\|V_{3}\|$ edges between $V_{2}$ and $V_{3}$ which are not $\varepsilon^{\prime}$-good. ###### Proof. For an $\varepsilon$-typical vertex $v\in V_{2}$, let $N_{1}=N(v)\cap V_{1}$ and $N_{3}=N(v)\cap V_{3}$. Recall from Definition 2.4 that $\|N_{i}\|\geq(1-\varepsilon)d_{2i}p\|V_{i}\|$ and that there exist $N_{i}^{\prime}\subset N_{i}$ with $\|N_{i}^{\prime}\|\geq(1-\varepsilon)\|N_{i}\|$ for $i\in\\{1,3\\}$ such that $(N_{1}^{\prime},N_{3}^{\prime})$ is $(\varepsilon,p)$-regular and has density at least $(1-\varepsilon)d_{13}p$. It follows that at least $(1-\varepsilon)\|N_{3}^{\prime}\|$ vertices $w\in N_{3}^{\prime}$ have at least $((1-\varepsilon)d_{13}-\varepsilon)p\|N_{1}^{\prime}\|$ common neighbours with $v$ in $N_{1}^{\prime}$. Since $((1-\varepsilon)d_{13}-\varepsilon)p\|N_{1}^{\prime}\|\geq(1-\varepsilon-\varepsilon/\delta)(1-\varepsilon)d_{13}p\|N_{1}\|\geq(1-3\varepsilon-\varepsilon/\delta)d_{12}d_{13}p^{2}\|V_{1}\|,$ and $3\varepsilon+\varepsilon/\delta\leq\varepsilon^{\prime}$, each such edge $\\{v,w\\}$ is $\varepsilon^{\prime}$-good. Since there are at least $(1-\varepsilon)\|V_{2}\|$ typical vertices in $V_{2}$ and $(1-\varepsilon)\|N_{3}^{\prime}\|\geq(1-\varepsilon)^{3}d_{23}p\|V_{3}\|\geq(1-3\varepsilon)d_{23}p\|V_{3}\|,$ the total number of $\varepsilon^{\prime}$-good edges between $V_{2}$ and $V_{3}$ is at least $(1-4\varepsilon)d_{23}p\|V_{2}\|\|V_{3}\|$. Finally, since the number of edges between $V_{2}$ and $V_{3}$ is exactly $d_{23}p\|V_{2}\|\|V_{3}\|$, the total number of non-$\varepsilon^{\prime}$-good edges is at most $4\varepsilon d_{23}p\|V_{2}\|\|V_{3}\|$. ∎ ###### Proposition 2.13. For every $\varepsilon^{\prime}$ and $\delta$, there exists a positive $\varepsilon(\varepsilon^{\prime},\delta)$ such that the following holds. Let $(V_{1},V_{2},V_{3})$ be an $\varepsilon$-typical $(\varepsilon,p)$-regular triple with minimum density at least $\delta p$. Moreover, assume that the endpoints of no edge in $(V_{2},V_{3})$ have more than $4p^{2}\|V_{1}\|$ common neighbours in $V_{1}$. Then $V_{1}$ contains at most $\varepsilon^{\prime}\|V_{1}\|$ vertices that are not $\varepsilon^{\prime}$-good. ###### Proof. Let $\varepsilon=\min\\{\varepsilon^{\prime}/4,\varepsilon^{\prime}\delta/4,(\varepsilon^{\prime}\delta)^{2}/32\\}$. By Proposition 2.12, at most $4\varepsilon d_{23}p\|V_{2}\|\|V_{3}\|$ edges in $(V_{2},V_{3})$ are not $\varepsilon^{\prime}$-good. Let $\alpha\|V_{1}\|$ be the number of $\varepsilon$-typical vertices in $V_{1}$ that are not $\varepsilon^{\prime}$-good. By definition, the neighbourhood of every such vertex contains at least $\varepsilon^{\prime}d_{12}d_{13}d_{23}p^{3}\|V_{2}\|\|V_{3}\|$ edges that are not $\varepsilon^{\prime}$-good. Therefore, our assumption on the maximum number of common neighbours of the endpoints of edges in $(V_{2},V_{3})$ implies that $\alpha\|V_{1}\|\cdot\varepsilon^{\prime}d_{12}d_{13}d_{23}p^{3}\|V_{2}\|\|V_{3}\|\leq 4p^{2}\|V_{1}\|\cdot 4\varepsilon d_{23}p\|V_{2}\|\|V_{3}\|$ and hence $\alpha\leq 16\varepsilon/(\varepsilon^{\prime}d_{12}d_{13})\leq\varepsilon^{\prime}/2$. Finally, since at most $\varepsilon\|V_{1}\|$ vertices in $V_{1}$ are not $\varepsilon$-typical and $\varepsilon\leq\varepsilon^{\prime}/2$, the number of vertices in $V_{1}$ that are not $\varepsilon^{\prime}$-good is at most $\varepsilon^{\prime}\|V_{1}\|$. ∎ We end this section by showing that the neighbourhood of every typical (good) vertex contains a subgraph with bounded maximum degree and many (good) edges. ###### Proposition 2.14. Let $\varepsilon$, $\delta$, and $p$ be positive constants with $\varepsilon<1/2$. Let $(V_{1},V_{2},V_{3})$ be a triple of sets such that for all $i$ and $j$, the density of $(V_{i},V_{j})$ is $d_{ij}p$, where $d_{ij}\geq\delta$. Then for every $\varepsilon$-typical vertex $v\in V_{1}$, there exist sets $N^{\prime\prime}_{j}\subset N(v)\cap V_{j}$ for $j\in\\{2,3\\}$ such that 1. (i) there are at least $(1-6\varepsilon-2\varepsilon/\delta)d_{12}d_{13}d_{23}p^{3}\|V_{2}\|\|V_{3}\|$ edges in $(N^{\prime\prime}_{2},N^{\prime\prime}_{3})$ and if $v$ is $\varepsilon$-good, then there are at least that many $\varepsilon$-good edges in $(N^{\prime\prime}_{2},N^{\prime\prime}_{3})$, and 2. (ii) for all $j$ and $k$ with $\\{j,k\\}=\\{2,3\\}$, no vertex in $N^{\prime\prime}_{j}$ has more than $(1+2\varepsilon/\delta)(1+\varepsilon^{2})d_{1k}d_{23}p^{2}\|V_{k}\|$ neighbours in $N^{\prime\prime}_{k}$. ###### Proof. Fix an $\varepsilon$-typical vertex $v\in V_{1}$. For each $j\in\\{2,3\\}$, let $N_{j}=N(v)\cap V_{j}$. Since $v$ is $\varepsilon$-typical, there are $N^{\prime}_{j}\subset N_{j}$ with $\|N^{\prime}_{j}\|\geq(1-\varepsilon)\|N_{j}\|\geq(1-\varepsilon)^{2}pd_{1j}\|V_{j}\|$ such that $(N^{\prime}_{2},N^{\prime}_{3})$ is $(\varepsilon,p)$-regular with density $d^{\prime}_{23}p$, where $(1-\varepsilon)d_{23}\leq d^{\prime}_{23}\leq(1+\varepsilon)d_{23}$. Let $N^{\prime\prime}_{j}$ be the set of vertices in $N^{\prime}_{j}$ that have at most $(d^{\prime}_{23}+\varepsilon)p\|N^{\prime}_{k}\|$ neighbours in $N^{\prime}_{k}$ and note that $\|N^{\prime\prime}_{j}\|\geq(1-\varepsilon)\|N^{\prime}_{j}\|$ by $(\varepsilon,p)$-regularity of $(N^{\prime}_{2},N^{\prime}_{3})$. Since $1/d^{\prime}_{23}\leq 1/((1-\varepsilon)d_{23})\leq 2/\delta$, we have $(d^{\prime}_{23}+\varepsilon)p\|N^{\prime}_{k}\|\leq(1+2\varepsilon/\delta)d^{\prime}_{23}p\|N_{k}\|\leq(1+2\varepsilon/\delta)(1+\varepsilon)^{2}d_{1k}d_{23}p^{2}\|V_{k}\|,$ and then (ii) follows. Note that $\|N^{\prime\prime}_{j}\|\geq(1-\varepsilon)\|N^{\prime}_{j}\|\geq(1-\varepsilon)^{2}d_{1j}p\|V_{j}\|$ and $e(N^{\prime\prime}_{2},N^{\prime\prime}_{3})\geq(d^{\prime}_{23}-\varepsilon)p\|N^{\prime\prime}_{2}\|\|N^{\prime\prime}_{3}\|\geq(1-2\varepsilon/\delta)d^{\prime}_{23}p\|N^{\prime\prime}_{2}\|\|N^{\prime\prime}_{3}\|\geq(1-2\varepsilon/\delta)(1-\varepsilon)^{5}d_{12}d_{13}d_{23}p^{3}\|V_{2}\|\|V_{3}\|.$ Moreover, if $v$ is $\varepsilon$-good, then at most $\varepsilon d_{12}d_{13}d_{23}p^{3}\|V_{2}\|\|V_{3}\|$ edges in $(N^{\prime\prime}_{2},N^{\prime\prime}_{3})$ are not $\varepsilon$-good. Now (i) follows. ∎ ### 2.4 Graph theory The following proposition, which we will be using several times in the proof of our main result, is a simple corollary from Hall’s marriage theorem [16] and gives a sufficient condition for a bipartite graph to have a perfect matching. ###### Proposition 2.15. Let $H$ be a bipartite graph on the vertex set $A\cup B$ with $\|A\|=\|B\|$. Suppose that there is an integer $L$ such that 1. (i) $\|N(S)\|\geq\|S\|$ for each $S\subset A$ with $\|A\setminus S\|\geq L$ and 2. (ii) $\|N(T)\|\geq\|T\|$ for each $T\subset B$ with $\|T\|\leq L$. Then $H$ has a perfect matching. Recall that the following theorem was proved by Corrádi and Hajnal [9]. ###### Theorem 2.16. Every graph on $n$ vertices with minimum degree at least $2n/3$ contains a perfect $K_{3}$-packing provided that $n$ is divisible by $3$. ### 2.5 Bounding large deviations Throughout the proof, we will extensively use the following standard estimate on the tail probabilities of binomial random variables, see [1, Appendix A]. We denote by $\operatorname{Bi}(n,p)$ the binomial random variable with parameters $n$ and $p$, i.e., the number of successes in a sequence of $n$ independent Bernoulli trials with success probability $p$. ###### Theorem 2.17 (Chernoff’s inequality). Let $p\in(0,1)$ and let $n$ be a positive integer. Then for every positive $a$ with $a\leq 2np/3$, $P\big{(}\|\operatorname{Bi}(n,p)-np\|>a\big{)}\leq\exp(-a^{2}/(6np)).$ ## 3 Outline of the proof of Theorem 1.3 Let $G$ be a subgraph of $G(n,p)$ with minimum degree at least $(2/3+o(1))np$. Throughout this section, we will tacitly condition on a few events that hold in $G(n,p)$ asymptotically almost surely. The proof of Theorem 1.3 breaks down into the following four simple steps. 1. 1. Apply the sparse regularity lemma (Theorem 2.3) and Theorem 2.16 to partition the vertex set of $G$ into regular triples with positive density and a small exceptional set of vertices. 2. 2. Remove from $G$ a collection of vertex-disjoint triangles so that all but at most $O(p^{-2})$ remaining vertices lie in balanced super-regular triples. 3. 3. Decompose each of those super-regular triples into a triangle packing, a balanced strong-super-regular triple, and a set of $O(p^{-1})$ leftover vertices. 4. 4. Find a triangle-factor in each strong-super-regular triple. Since step 1 is a straightforward application of the regularity lemma (Theorem 2.3) and Theorem 2.16, we will only describe the basic ideas of steps 2, 3, and 4 in this section. The details of these steps will be given in Sections 5, 6, and 7, respectively. ### 3.1 Step 2 In order to construct super-regular triples from the regular triples we obtained in step 1, we first move all non-typical vertices to the exceptional set $V_{0}$. Since we have no control over $V_{0}$ and $\|V_{0}\|$ can be linear in $n$, we need to cover most of it with vertex-disjoint triangles. At the same time, we do not want to use too many vertices from any of the regular triples in order not to destroy their structure, i.e., to keep them close to being super-regular. This will be achieved by an application of Lemma 4.5, which allows us to find such triangles. After we absorb the exceptional vertices into a triangle packing, some triples in the remaining graph might become imbalanced. Since in order for any triple to have a triangle-factor (or at least an almost perfect triangle packing), the sizes of all three of its parts must be equal, we have to balance the sizes of the remaining triples. We will do that by adding to our triangle packing some triangles whose vertices lie in two different triples, see Lemma 5.1. Finally, since at the beginning we removed all non-typical vertices from each triple and later we did not alter it too much, we can make every triple super-regular by deleting at most $O(p^{-1})$ of its vertices (see Proposition 2.10). ### 3.2 Steps 3 and 4 Our general strategy for finding a triangle-factor in a super-regular triple $(V_{1},V_{2},V_{3})$ can be summarized as follows. 1. (i) For each $\\{i,j\\}\subset\\{1,2,3\\}$, randomly select a small set $M_{ij}$ of independent edges in $(V_{i},V_{j})$. 2. (ii) Find an almost perfect triangle packing that does not hit any endpoints of the edges in any $M_{ij}$. 3. (iii) Match the remaining vertices with the edges in the sets $M_{ij}$ in order to extend the triangle packing to a triangle-factor. Assume that the first two steps have been performed. Then, in order to verify Hall’s condition (see Proposition 2.15) to prove that an appropriate matching can be found in (iii), we need to know, in particular, that the endpoints of each edge in $M_{23}$ have many common neighbours in the remaining part of $V_{1}$ and that each vertex in $V_{1}$ is incident to both endpoints of many edges in $M_{23}$ (and that similar conditions hold for other choices of indices). Therefore, it would be convenient if $M_{23}$ consisted only of good edges and $V_{1}$ contained only good vertices (see Section 2.3). Unfortunately, super-regular triples can generally contain vertices that are not good. This is the reason why in step 3, we need to break down each super- regular triple into a triangle packing and a strong-super-regular triple. Therefore, we will perform the above described process twice. First, in step 3, we will absorb all the non-good vertices into a small triangle packing by performing (i) and (iii), see Theorem 6.6. In step 4, once we are left with a balanced strong-super-regular triple (after deleting at most $O(p^{-1})$ further vertices), we can finally perform (i)–(iii), now using only good edges to construct $M_{ij}$s, to find a triangle-factor inside this triple, see Theorem 1.4. ## 4 Properties of Random Graphs In this section we establish several properties of the random graph that will be useful in later sections. ###### Proposition 4.1. For every positive real $\rho$, there exists a constant $C(\rho)$ such that if $p\geq C(\log n/n)^{1/2}$, then $G(n,p)$ a.a.s. satisfies the following properties. 1. (i) Every vertex has degree $(1\pm\rho)np$. 2. (ii) Every pair of distinct vertices has $(1\pm\rho)np^{2}$ common neighbours. 3. (iii) For all $X,Y\subset V$ with $\|X\|,\|Y\|\geq\rho np$, we have $e(X,Y)=(1\pm\rho)\|X\|\|Y\|p$. In particular, $e(X)=e(X,X)/2=(1\pm\rho)\|X\|^{2}p/2$ for all $X$ of size at least $\rho np$. ###### Proposition 4.2. For every $\rho\in(0,1/2)$, $G(n,p)$ satisfies the following. 1. (i) Let $D$ be a positive real. For a fixed set $W\subset V$, with probability $1-e^{(1-\rho^{2}D/12)n}$, all but at most $nDp^{-1}/\|W\|$ vertices in $V\setminus W$ satisfy $\deg(v,W)=(1\pm\rho)\|W\|p.$ 2. (ii) For every positive real $\xi$, there exists a constant $D(\rho,\xi)$ such that a.a.s. the following holds. For all $W\subset V$ with $\|W\|\geq\xi n$, all but at most $Dp^{-1}$ vertices in $V\setminus W$ satisfy $\deg(v,W)=(1\pm\rho)\|W\|p.$ ###### Proof. To prove (i), as a first step, we fix a set $W\subset V$. We may assume that $Dp^{-1}n/\|W\|\leq n$ as otherwise, the claim is vacuously true. Suppose that there are $Dp^{-1}n/\|W\|$ vertices $v\in V\setminus W$ such that $\deg(v,W)\neq(1\pm\rho)\|W\|p$. Then there exists a set $B\subset V\setminus W$ of size $Dp^{-1}n/(2\|W\|)$ such that either $\deg(v,W)>(1+\rho)\|W\|p$ for all $v\in B$ or $\deg(v,W)<(1-\rho)\|W\|p$ for all $v\in B$. This clearly implies that $e(B,W)\neq(1\pm\rho)\|B\|\|W\|p$ for some $B$ as above. Since $e(B,W)$ is a sum of independent binomial random variables and $\mathbb{E}[e(B,W)]=\|B\|\|W\|p=Dn/2$, by Chernoff’s inequality, $P\big{(}e(B,W)\neq(1\pm\rho)\|B\|\|W\|p\big{)}\leq e^{-\rho^{2}Dn/12}.$ By the union bound, the probability that such a set $B$ exists is at most $2^{n}e^{-\rho^{2}Dn/12}$. Now that (i) is proved, we easily get (ii) by applying the union bound. ∎ ###### Proposition 4.3. For all $\xi\in(0,1)$, there exists a $C(\xi)$ such that if $p\geq C(\log n/n)^{1/2}$, then a.a.s. for every $x\in[1,\xi n/2]$, $G(n,p)$ does not contain a set $W$ of $x$ vertices and a set $E$ of $x$ independent edges outside $W$ (i.e., no edge in $E$ has an endpoint in $W$) such that either the endpoints of each edge in $E$ have at least $\xi np^{2}$ common neighbours in $W$ or each vertex in $W$ is adjacent to both endpoints of at least $\xi np^{2}$ edges in $E$. ###### Proof. Fix $x$, $W$, and $E$ as in the statement of this proposition. For a vertex $w\in W$ and an edge $\\{u,v\\}\in E$, let $B(u,v,w)$ denote the event that $w$ is adjacent to both $u$ and $v$. Let $X$ be the random variable denoting the number of events $B(u,v,w)$ that occur in $G(n,p)$. Note that each of the “bad” events described in the statement of this lemma implies that $X\geq\xi np^{2}x\geq 2x^{2}p^{2}$. Moreover, observe that $X\leq x^{2}$, so we can restrict our attention to the case $x\geq\xi np^{2}$. Since all $B(u,v,w)$ are mutually independent, $X$ has binomial distribution with parameters $x^{2}$ and $p^{2}$, and hence by Chernoff’s inequality, $P(X\geq\xi np^{2}x)\leq e^{-c\xi np^{2}x}$ for some absolute positive constant $c$. Since for each $x$, there are at most ${n\choose x}n^{2x}$ pairs $(W,E)$ with $\|W\|=\|E\|=x$, the probability that some “bad” event occurs is at most $\sum_{x=\xi np^{2}}^{\xi n/2}{n\choose x}n^{2x}e^{-c\xi np^{2}x}.$ Finally, note that ${n\choose x}n^{2x}e^{-c\xi np^{2}x}\leq e^{3x\log n-c\xi np^{2}x}\leq n^{-2},$ provided that $np^{2}\geq(4/c\xi)\log n$. ∎ ###### Proposition 4.4. Let $p\gg n^{-1/2}$. For every positive reals $\varepsilon$ and $\rho$, there exists a positive real $D(\varepsilon,\rho)$ such that $G(n,p)$ a.a.s. satisfies the following property. For every set $X$ with $\|X\|\geq Dp^{-2}$, there are at most $\varepsilon n^{2}p$ edges $\\{v,w\\}$ in $G[V\setminus X]$ such that $v$ and $w$ do not have $(1\pm\rho)\|X\|p^{2}$ common neighbours in $X$. ###### Proof. The constant $D(\varepsilon,\rho)$ will be chosen later. Let $X$ be a fixed set of size at least $Dp^{-2}$. Without loss of generality we may assume that $\rho\leq 1/2$. First expose the edges between $X$ and $V\setminus X$ and call a pair of vertices $\\{v,w\\}\in V\setminus X$ bad if $v$ and $w$ do not have $(1\pm\rho)\|X\|p^{2}$ common neighbours in $X$. By Proposition 4.2 (i), with probability $1-e^{(1-\rho^{2}D^{1/2}/50)n}$, there are at most $D^{1/2}p^{-1}n/\|X\|$ vertices that do not satisfy $\deg(v,X)=(1\pm\rho/2)\|X\|p$. Even if each of these vertices forms bad pairs with all $n$ vertices, there are at most $D^{1/2}p^{-1}n^{2}/\|X\|$ such bad pairs. For each vertex that satisfies $\deg(v,X)=(1\pm\rho/2)\|X\|p$, again by Proposition 4.2 (i), with probability $1-e^{(1-\rho^{2}D^{1/2}/50)n}$, there are at most $2D^{1/2}p^{-1}n/(\|X\|p)$ other vertices $w$ which do not have $(1\pm\rho)\|X\|p^{2}$ common neighbours with $w$ in $X$. Therefore, if $D$ is sufficiently large, then with probability at least $1-e^{-2n}$, the total number of bad pairs is at most $\frac{D^{1/2}p^{-1}n^{2}}{\|X\|}+\frac{2D^{1/2}p^{-1}n^{2}}{\|X\|p}\leq 3D^{-1/2}n^{2}\leq\varepsilon n^{2}/2.$ Finally, expose the edges within $V\setminus X$. By Chernoff’s inequality, with probability $1-e^{-\varepsilon n^{2}p/20}$, at most $\varepsilon n^{2}p$ bad pairs will form an edge. Since $n^{2}p\gg n$, if we fix the set $X$, both of the above events happen with probability at least $1-e^{-2n}$. Since there are at most $2^{n}$ choices for $X$, we can take the union bound over all choices of $X$ to derive the conclusion. ∎ Using the above propositions, we now prove the following generalization of [17, Lemma 6.4]. ###### Lemma 4.5. There exist $C$, $D$, and $\varepsilon$ such that if $p\geq C(\log n/n)^{1/2}$, then $G(n,p)$ a.a.s. has the following property. For every spanning subgraph $G^{\prime}\subset G(n,p)$ with $\delta(G^{\prime})\geq(2/3)np$ and every set $T\subset V(G^{\prime})$ with $\|T\|\leq\varepsilon n$, all but at most $Dp^{-2}$ vertices of $V\backslash T$ are contained in a triangle of $G$ which does not intersect $T$. ###### Proof. For the sake of brevity, denote $G(n,p)$ by $G$ and let $V=V(G)$. Let $\varepsilon$ be a small positive constant (we will fix it later), let $C=C_{\ref{prop_randomgraphproperties1}}(\varepsilon)$, and let $D_{0}(\varepsilon)$ be a constant satisfying $D_{0}\geq\max\\{D_{\ref{prop_nontypicalvertices}(ii)}(\varepsilon,\varepsilon),D_{\ref{prop_randomgraphproperties2}}(\varepsilon,\varepsilon),1\\}$. Without loss of generality we may assume that $\|T\|=\varepsilon n$. Let $X_{0}\subset V\setminus T$ be an arbitrary set of size $2D_{0}p^{-2}$. By assuming that the events from Propositions 4.1, 4.2 (ii), and 4.4 hold, we will show that there exists a triangle in $G^{\prime}$ which intersects $X_{0}$ but not $T$. This will prove that there are at most $2D_{0}p^{-2}$ vertices that are not contained in triangles that do not hit $T$. Let $T^{\prime}$ be the collection of all the vertices $v$ that satisfy $\deg(v,X_{0}\cup T)\geq(1+\varepsilon)\|X_{0}\cup T\|p$ and note that $\|T^{\prime}\|\leq D_{0}p^{-1}$ by Proposition 4.2 (ii). Let $T^{\prime\prime}=T\cup T^{\prime}$ and $X=X_{0}\setminus T^{\prime}$. Note that $\|T^{\prime\prime}\|\leq 2\varepsilon n$ and $\|X\|\geq\|X_{0}\|-\|T^{\prime}\|\geq D_{0}p^{-2}$. Let $D$ be the constant defined by $\|X\|=Dp^{-2}$ and note that $D\geq D_{0}$. It suffices to show that there exists a triangle in $G^{\prime}$ which contains a vertex from $X$ but not from $T^{\prime\prime}$. Let $Y=V\setminus(X\cup T^{\prime\prime})$ and fix a vertex $x\in X$. Note that $\displaystyle\deg_{G^{\prime}}(x,Y)$ $\displaystyle=\deg_{G^{\prime}}(x)-\deg_{G^{\prime}}(x,X\cup T^{\prime\prime})\geq\deg_{G^{\prime}}(x)-\deg_{G^{\prime}}(x,X_{0}\cup T)-\|T^{\prime}\|$ $\displaystyle\geq(2/3)np-(1+\varepsilon)\|T\cup X_{0}\|p-D_{0}p^{-1}\geq(2/3-3\varepsilon)np,$ where the last two inequalities follow from the fact that $x\not\in T^{\prime}$ and our assumption on $p$. Finally, let $N_{x}=N_{G}(x)\cap Y$ and fix an arbitrary subset $N_{x}^{\prime}\subset N_{G^{\prime}}(x)\cap Y$ of size $(2/3-3\varepsilon)np$. It suffices to show that the number of triangles $xy_{1}y_{2}$ in $G^{\prime}$ such that $x\in X$ and $y_{1},y_{2}\in N_{x}^{\prime}$ is nonzero. Let this number be $M$. To bound $M$ from below, first bound the number of triangles $xy_{1}y_{2}$ in $G$ such that $x\in X,y_{1},y_{2}\in N_{x}$, and $y_{1}y_{2}$ is an edge of the graph $G^{\prime}$ (we will later subtract the number triangles whose $y_{1}$ or $y_{2}$ is not in $N_{x}^{\prime}$). Let this number be $M_{0}$. Since $\|X\cup T^{\prime\prime}\|,\|Y\|\geq\varepsilon n$, by Proposition 4.1 (iii), we have $e_{G^{\prime}}(Y)\geq e_{G^{\prime}}(V)-e_{G^{\prime}}(Y,X\cup T^{\prime\prime})-e_{G^{\prime}}(X\cup T^{\prime\prime})\geq\frac{2}{3}\cdot\frac{n^{2}p}{2}-4\varepsilon n^{2}p=\left(\frac{2}{3}-8\varepsilon\right)\frac{n^{2}p}{2}.$ By Proposition 4.4, the number of edges $\\{v,w\\}$ in $G^{\prime}[Y]$ that form a triangle in $G$ with fewer than $(1-\varepsilon)D$ vertices in $X$ is at most $\varepsilon n^{2}p$ given that $D_{0}$ is large enough. Thus, $M_{0}\geq\Big{(}e_{G^{\prime}}(Y)-\varepsilon n^{2}p\Big{)}(1-\varepsilon)D\geq\left(\frac{2}{3}-11\varepsilon\right)\frac{Dn^{2}p}{2}.$ To obtain a bound on $M$ from $M_{0}$, we can subtract the number of triangles $xy_{1}y_{2}$ as above such that either $y_{1}$ or $y_{2}$ is not in $N_{x}^{\prime}$. Since $\|N_{x}\|\leq(1+\varepsilon)np$ by Proposition 4.1 (i), we have $\|N_{x}\setminus N_{x}^{\prime}\|=\|N_{x}\|-\|N_{x}^{\prime}\|\leq(1+\varepsilon)np-(2/3-3\varepsilon)np=(1/3+4\varepsilon)np.$ Thus, if $\varepsilon$ is small enough, by Proposition 4.1 (iii) we have, $\displaystyle M$ $\displaystyle\geq M_{0}-\sum_{x\in X}\left(e_{G^{\prime}}(N_{x}\setminus N_{x}^{\prime},N_{x}^{\prime})+e_{G^{\prime}}(N_{x}\setminus N_{x}^{\prime})\right)$ $\displaystyle\geq M_{0}-\sum_{x\in X}\left(1+\varepsilon\right)\left(\left(\frac{1}{3}+4\varepsilon\right)\frac{2}{3}n^{2}p^{3}+\left(\frac{1}{3}+4\varepsilon\right)^{2}\frac{n^{2}p^{3}}{2}\right)$ $\displaystyle\geq\left(\frac{2}{3}-11\varepsilon\right)\frac{Dn^{2}p}{2}-\sum_{x\in X}\left(\frac{5}{9}+10\varepsilon\right)\frac{n^{2}p^{3}}{2}=\left(\frac{1}{9}-21\varepsilon\right)\frac{Dn^{2}p}{2}.$ Therefore there exists a triangle as claimed, provided that $\varepsilon$ is sufficiently small. ∎ The following proposition establishes the fact that it is necessary to have $\Omega(p^{-2})$ vertices not covered by triangles. Its proof closely follows the argument from [17, Proposition 6.3]. ###### Proposition 4.6. Let $\varepsilon>0$. There exists a positive constant $C(\varepsilon)$ such that if $Cn^{-1/2}\leq p\ll 1$, then $G(n,p)$ a.a.s. contains a spanning subgraph of minimum degree at least $(1-\varepsilon)np$ such that $\Omega(p^{-2})$ of its vertices are not contained in a triangle. ###### Proof. Let $C$ be a constant satisfying $C\geq 2$ and $e^{C^{2}/15}\geq 8e/\varepsilon$. If $p\geq(\log n/n)^{1/2}$, then by Proposition 4.1, a.a.s. $\delta(G(n,p))\geq(1-\varepsilon/4)np$ and each pair of vertices of $G(n,p)$ has at most $2np^{2}$ common neighbours. If $Cn^{-1/2}\leq p<(\log n/n)^{1/2}$, then still a.a.s. $\delta(G(n,p))\geq(1-\varepsilon/4)np$, but $G(n,p)$ may contain some edges whose endpoints have more than $2np^{2}$ common neighbours. Let $H$ be the subgraph consisting of all such edges, and let $v$ be an arbitrary vertex. By Chernoff’s inequality, the probability that $v$ and some other vertex have more than $2np^{2}$ common neighbours is at most $e^{-np^{2}/15}$. Therefore, $P(\deg_{H}(v)\geq(\varepsilon/4)np)\leq{n\choose(\varepsilon/4)np}\left(p\cdot e^{-np^{2}/15}\right)^{(\varepsilon/4)np}\leq\left(\frac{4enp}{\varepsilon np}\cdot e^{-C^{2}/15}\right)^{(\varepsilon/4)np}=o(n^{-1}),$ so a.a.s. $\Delta(H)<(\varepsilon/4)np$. Finally, let $G=G(n,p)-H$. Clearly, the endpoints of every edge of $G$ have at most $2np^{2}$ common neighbours. Moreover, by Proposition 4.1 (i), we may assume that $\delta(G)>(1-\varepsilon/2)np$. Let $X$ be an arbitrary fixed set of $(\varepsilon/4)p^{-2}$ vertices of $G$ and let $W=\\{v\notin X\colon\deg(v,X)\geq 2\|X\|p\\}$. By Chernoff’s inequality, the probability that a vertex $v$ belongs to $W$ is $e^{-\Omega(p^{-1})}$ and these events are independent for different vertices. Since $p\gg e^{-\Omega(p^{-1})}$, Chernoff’s inequality implies that a.a.s. $\|W\|\leq(\varepsilon/4)np$. Moreover, since our assumption on $p$ implies that $\|X\|\leq(\varepsilon/8)n$, we can apply Chernoff’s inequality and deduce that a.a.s. $\deg(u,X)\leq(\varepsilon/4)np$ for every vertex $u$. Let $G^{\prime}$ be the subgraph of $G$ obtained by deleting all edges within $X$, all edges between $X$ and $W$, and deleting edges incident to any $y\notin X\cup W$ according to the following rule – for every triangle $xyz$ in $G$ with $x\in X$ and $z\notin X\cup W$, remove the edge $yz$. It is quite easy to see that no vertex of $X$ is contained in a triangle in $G^{\prime}$. Let us now estimate $\delta(G^{\prime})$. Since a vertex $u\in X\cup W$ lost only edges connecting it to $X$ and $W$, we have $\deg_{G^{\prime}}(u)>(1-\varepsilon/2)np-\deg(u,X)-\deg(u,W)\geq(1-\varepsilon/2)np-\deg(u,X)-\|W\|\geq(1-\varepsilon)np.$ Since a vertex $y\notin X\cup W$ is incident to at most $(\varepsilon/2)p^{-1}$ vertices $x\in X$ and it has at most $2np^{2}$ common neighbours with each such $x$, we then have $\deg_{G^{\prime}}(y)>(1-\varepsilon/2)np-(\varepsilon/4)p^{-1}\cdot 2np^{2}\geq(1-\varepsilon)np.$ Thus $G^{\prime}$ has the required properties. ∎ We end this section with two propositions whose proofs are farily standard and are omitted. Proposition 4.7 asserts that in a typical random graph $G(n,p)$, the reduced graph of a regular partition of a subgraph $G\subset G(n,p)$ inherits the minimum degree condition that we impose on $G$. The final proposition, Proposition 4.8 can be proved using Lemma 2.6, and asserts that every regular triple in a random graph is typical. ###### Proposition 4.7. Let $\gamma>0$ and $p\gg n^{-1}$. There exist $\varepsilon_{0}(\gamma)$ and $\delta_{0}(\gamma)$ such that if $\varepsilon\leq\varepsilon_{0}$ and $\delta\leq\delta_{0}$, then the following holds asymptotically almost surely. Given a subgraph $G$ of $G(n,p)$, let $(V_{i})_{i=1}^{k}$ be an $(\varepsilon,p)$-regular partition of $G$ such that $\|V_{i}\|\leq\varepsilon n$ for all $i$ and every part forms an $(\varepsilon,p)$-regular pair with at least $(1-\varepsilon)k$ other parts. Let $R$ be its $(\delta,\varepsilon,p)$-reduced graph. If $G$ has minimum degree at least $(2/3+\gamma)np$, then $R$ has minimum degree at least $(2/3+\gamma/2)k$. ###### Proposition 4.8. Let $p\gg n^{-1/2}$. For all positive $\varepsilon^{\prime}$, $\delta$, and $\xi$, there exists a constant $\varepsilon_{0}(\varepsilon^{\prime},\delta)$ such that a.a.s. in $G(n,p)$, every copy of a graph from $\mathcal{G}(K_{3},(n_{1},n_{2},n_{3}),(d_{12},d_{23},d_{31}),(\varepsilon,p))$ is $\varepsilon^{\prime}$-typical provided that $\varepsilon\leq\varepsilon_{0}$, $d_{12},d_{23},d_{31}\geq\delta$, and $n_{1},n_{2},n_{3}\geq\xi n$. ## 5 Obtaining balanced super-regular triples In Section 3.1, we mentioned that the process of absorbing exceptional vertices into a triangle packing may cause some regular triples in our graph to become slightly unbalanced. The following lemma describes a greedy procedure that finds a small triangle packing which restores the balance in each of these triples. ###### Lemma 5.1. Let $p\gg n^{-1/2}$. For all positive reals $\delta$, $\varepsilon^{\prime}$, and $\gamma$, there exists an $\varepsilon_{0}(\delta,\varepsilon^{\prime},\gamma)$ such that if $\varepsilon<\varepsilon_{0}$, then the following holds asymptotically almost surely. Let $G$ be a subgraph of $G(n,p)$ and let $V_{1},\ldots,V_{3k}$ be disjoint subsets of $V(G)$ satisfying $\|V_{i}\|\in[(1-\varepsilon)m,(1+\varepsilon)m]$ for some $m=\Omega(n)$. Let $R$ be a graph on the vertex set $[3k]$ of minimum degree at least $(2+\gamma)k$ such that $\\{3t-2,3t-1,3t\\}$ forms a triangle for all $t\in[k]$ and assume that $(V_{i})_{i=1}^{3k}$ is $(\delta,\varepsilon,p)$-regular over $R$. Then there exist subsets $B$ and $S$ of $V(G)$ such that the following holds. 1. (i) $\|B\|\leq 4k$, 2. (ii) $G[S]$ contains a perfect triangle packing, 3. (iii) $\|V_{i}\cap(B\cup S)\|\leq\varepsilon^{\prime}m$ for all $i\in[3k]$, and 4. (iv) $V_{i}\setminus(B\cup S)$ have equal sizes for all $i$. ###### Proof. Let $\varepsilon_{1}=\varepsilon_{\ref{prop_findtriangle}}(\frac{1}{2},\delta)$ and $\varepsilon_{0}=\min\\{\frac{\varepsilon^{\prime}}{4(3/\gamma+1)},\varepsilon_{\ref{prop_randomgraphtypical}}(\varepsilon_{1},\frac{\delta}{2}),\frac{\delta}{2},\varepsilon_{1}\\}$. Assume that $\varepsilon\in(0,\varepsilon_{0})$ is given. Let $C_{t}=\\{3t-2,3t-1,3t\\}$ for $t\in[k]$ be triangles of the graph $R$. For each vertex $i\in[3k]$, call an index $t\in[k]$ $i$-rich or rich with respect to $i$ if $i$ is adjacent to all three vertices of $C_{t}$, and assume that there are $g_{i}$ $i$-rich indices. Then by the minimum degree condition on $R$, we have $3g_{i}+2(k-g_{i})\geq(2+\gamma)k,$ which is equivalent to $g_{i}\geq\gamma k$. Thus for each vertex $i\in[3k]$ of $R$, we can assign an $i$-rich index $t\in[k]$ to it so that every index in $[k]$ is chosen by at most $(3k)/(\gamma k)=3/\gamma$ vertices. Consider the following process that adjusts the parts one by one. Throughout the process, we will maintain sets $B\subset V(G)$ and $Z_{i}\subset V_{i}$ for each $i\in[3k]$; they are empty at the beginning. Call a triangle $C_{t}$ balanced if the sets $V_{3t-2}\setminus Z_{3t-2}$, $V_{3t-1}\setminus Z_{3t-1}$, and $V_{3t}\setminus Z_{3t}$ have equal cardinalities. Assume that the triangles $C_{1},\ldots,C_{t-1}$ are already balanced and we are trying to balance the triangle $C_{t}$. Without loss of generality, we may assume that $\|V_{3t-i}\setminus Z_{3t-i}\|-\|V_{3t}\setminus Z_{3t}\|=x_{i}m$ and $0\leq x_{i}\leq 2\varepsilon$ for $i\in\\{1,2\\}$. Thus we have to remove $x_{i}m$ vertices from $V_{3t-i}$ for $i\in\\{1,2\\}$ in order to make $C_{t}$ balanced. By moving at most $2$ arbitrary vertices from each set $V_{3t-1}$ and $V_{3t-2}$ to $B$ and also to $Z_{3t-1}$ and $Z_{3t-2}$, respectively, we may assume that both $x_{1}m$ and $x_{2}m$ are divisible by 3. First consider the set $V_{3t-1}$ and let $s$ be the rich index with respect to $3t-1$ which we have chosen above. As we will later establish, for every $i\in[3k]$, $\|V_{i}\setminus Z_{i}\|\geq m/2$ throughout the process. Therefore by Proposition 2.1, the triple $(V_{3t-1}\setminus Z_{3t-1},V_{3s-j}\setminus Z_{3s-j},V_{3s-k}\setminus Z_{3s-k})$ inherits the regularity of $(V_{3t-1},V_{3s-j},V_{3s-k})$ and is always $(2\varepsilon,p)$-regular of density at least $\delta/2$ for every pair $\\{j,k\\}\subset\\{0,1,2\\}$. By Proposition 4.8, a.a.s. it must also be $\varepsilon_{1}$-typical. Thus by Proposition 2.7 we can find $x_{1}n/3$ triangles across this triple. Do this for each pair $\\{j,k\\}$ and update the sets $Z_{3t-1}$, $Z_{3s-2}$, $Z_{3s-1}$, and $Z_{3s}$ by placing all the vertices of these triangles into corresponding parts. Note that even though the sizes of the sets in $C_{s}$ have decreased, the number by which they decreased is the same for all three of them and thus after performing the same procedure for $V_{3t-2}$, the triangles $C_{1},\ldots,C_{t}$ will be balanced. Note that in the end, $\|B\|\leq 4k$. Moreover, throughout the process, by the restriction that every index is the chosen rich index for at most $3/\gamma$ other indices, we always have, $\|Z_{i}\|\leq 2\varepsilon m\cdot(3/\gamma+1)\leq\min\\{\varepsilon^{\prime}m,m/2\\}$ as claimed. Define $S=\big{(}\cup_{i=1}^{3k}Z_{i}\big{)}\setminus B$ and we have the sets $B$ and $S$ as claimed. ∎ Below is the main theorem of this section. It says that we can partition our graph into balanced super-regular triples, a collection of vertex-disjoint triangles, and a set of at most $O(p^{-2})$ exceptional vertices. We would like to remark that the upper bound imposed on the sizes of the common neighbourhoods in (v) will come in handy in the proof of Theorem 6.6, where we show that the triples $(W_{3t-2},W_{3t-1},W_{3t})$ are close to being strong- super-regular, see Proposition 2.13. ###### Theorem 5.2. For an arbitrary $\gamma$, there exist $\delta(\gamma)$ and $\varepsilon_{0}(\gamma)$ such that for all $\varepsilon\in(0,\varepsilon_{0})$, there exist constants $C(\varepsilon),D(\varepsilon)$, and $\xi(\varepsilon)$ satisfying the following. If $p\geq C(\log n/n)^{1/2}$, then a.a.s. for every spanning subgraph $G^{\prime}\subset G(n,p)$ with $\delta(G^{\prime})\geq(2/3+\gamma)np$, there exist a further subgraph $G^{\prime\prime}\subset G^{\prime}$ and a partition of $V(G)$ into sets $B$, $S$, and $(W_{i})_{i=1}^{3k}$, where $k\leq D$, such that 1. (i) $\|B\|\leq Dp^{-2}$. 2. (ii) $G^{\prime}[S]$ contains a perfect triangle packing. 3. (iii) $(W_{3t-2},W_{3t-1},W_{3t})$ is a $(\delta,\varepsilon,p)$-super-regular triple in $G^{\prime\prime}$ for all $t\in[k]$. 4. (iv) $\|W_{3t-2}\|=\|W_{3t-1}\|=\|W_{3t}\|\geq\xi n$ for all $t\in[k]$. 5. (v) In the graph $G^{\prime\prime}$, for all $t\in[k]$, the endpoints of every edge in $(W_{3t-2},W_{3t-1})$ have at most $4\|W_{3t}\|p^{2}$ common neighbours in $W_{3t}$ and a similar statement holds for other choices of indices. ###### Proof. Given a $\gamma$, let $\delta=\delta_{\ref{prop_reducedmindegree}}(\gamma)$ and $\varepsilon_{0}=\varepsilon_{\ref{prop_reducedmindegree}}(\gamma)$. Moreover, for a given $\varepsilon\in(0,\varepsilon_{0})$, let $\varepsilon_{1}\leq\min\\{\varepsilon/2,\varepsilon_{\ref{prop_typicalstability}}(\varepsilon,\delta),\delta/2\\}$, $\varepsilon_{2}\leq\min\\{\varepsilon_{\ref{lemma_leftovertriangles}}^{2}/36^{2},\varepsilon_{1}^{2}/400,\varepsilon_{\ref{lemma_balancingpartition}}(\delta,\varepsilon_{1}/3,\gamma/2)\\}$, $\varepsilon_{3}\leq(1/27)\min\\{\varepsilon_{\ref{prop_randomgraphtypical}}(\varepsilon_{2},\delta)^{3},\varepsilon_{2}\\}$. Let $K=K_{\ref{thm_regularity}}(\varepsilon_{3},2,1/\varepsilon_{3})$, $\eta=\min\\{\eta_{\ref{thm_regularity}}(\varepsilon_{3},2,1/\varepsilon_{3}),1\\}$, $\xi=\varepsilon_{3}/(4K)$, $C=\max\\{C_{\ref{prop_randomgraphproperties1}}(\eta),C_{\ref{lemma_leftovertriangles}}\\}$, and $D=\max\\{3D_{\ref{lemma_leftovertriangles}},18KD_{\ref{prop_nontypicalvertices}(ii)}(1/3,\xi),24K\\}$. Proposition 4.1 (iii) implies that $G(n,p)$ is a.a.s. $(\eta,2,p)$-upper- uniform. Thus we can apply the regularity lemma, Theorem 2.3, to obtain an $(\varepsilon_{3},p)$-regular partition $V_{0},V_{1},\ldots,V_{3k}$ of the graph $G^{\prime}$, where each part forms a regular pair with at least $(3-3\varepsilon_{3})k$ other parts. Let $m=\|V_{i}\|$, note that $m\geq n/(2K)$, and let $R$ be the reduced graph with parameter $\delta$. Since $G^{\prime}$ has minimum degree at least $(2/3+\gamma)np$, by Proposition 4.7, a.a.s. the reduced graph has minimum degree at least $(2+\gamma/2)k$. Thus by Theorem 2.16, we may assume that $(V_{3t-2},V_{3t-1},V_{3t})$ forms an $(\varepsilon_{3},p)$-regular triple of density at least $\delta$ for all $t\in[k]$. By Proposition 4.4, a.a.s. there are at most $(\varepsilon_{3}/2)pm^{2}$ edges in $(V_{3t-1},V_{3t-2})$ whose endpoints have more than $2\|V_{3t}\|p^{2}$ common neighbours in $V_{3t}$. Similar estimate holds for the edges in $(V_{3t-2},V_{3t})$ and $(V_{3t-1},V_{3t})$. Delete all such edges for all $t\in[k]$ to obtain the subgraph $G^{\prime\prime}$. Then in the graph $G^{\prime\prime}$, each triple $(V_{3t-2},V_{3t-1},V_{3t})$ is $(3\varepsilon_{3}^{1/3},p)$-regular by Proposition 2.2. By Proposition 4.8, we may assume that every $(3\varepsilon_{3}^{1/3},p)$-regular triple of density at least $\delta$ is $\varepsilon_{2}$-typical. Thus for each index $i$, if we let $X_{i}\subset V_{i}$ be the collection of non $\varepsilon_{2}$-typical vertices, then $\|X_{i}\|\leq\varepsilon_{2}\|V_{i}\|$. Furthermore, for each $t\in[k]$, add to $X_{3t}$ the collection of those vertices $v\in V_{3t}$ such that $\deg(v,V_{3t-j})\neq(1\pm\varepsilon_{2})d_{3t,3t-j}p\|V_{3t-j}\|$ for some $j\in\\{1,2\\}$ and define $X_{3t-1}$ and $X_{3t-2}$ accordingly (there are at most $4\varepsilon_{2}n$ such vertices by regularity). By adding arbitrary vertices to $X_{i}$ if necessary, we may assume that $\|X_{i}\|=5\varepsilon_{2}\|V_{i}\|$. Move all the vertices in $X_{i}$ from $V_{i}$ to $V_{0}$ and denote the resulting partition by $(V_{i}^{\prime})_{i=0}^{3k}$. We then have $\|V_{0}^{\prime}\|\leq\|V_{0}\|+\sum_{i}\|X_{i}\|\leq 6\varepsilon_{2}n$. Consider the following process of finding triangles that absorbs the vertices in $V_{0}^{\prime}$. Let $T$ be the empty set; we will update it throughout the process. Apply Lemma 4.5 to find a triangle which hits $V_{0}^{\prime}$ but not $T$ and move all the vertices of this triangle into $T$. If $\|T\cap V_{i}^{\prime}\|\geq\sqrt{\varepsilon_{2}}\|V_{i}^{\prime}\|-3$ for some index $i\in[3k]$, then move all the vertices of $V_{i}^{\prime}$ into $T$. This way, we will have $\|T\|\leq 3\|V_{0}^{\prime}\|\cdot(1/\sqrt{\varepsilon_{2}})+3\|V_{0}^{\prime}\|\leq 36\sqrt{\varepsilon_{2}}n\leq\varepsilon_{\ref{lemma_leftovertriangles}}n$ throughout the process. Terminate the process when we cannot find such triangles anymore. Then, a.a.s. we must have $\|V_{0}^{\prime}\|\leq(D/3)p^{-2}$. Let $B_{0}$ be the collection of all the remaining vertices of $V_{0}^{\prime}$, and $S_{0}$ be the set of vertices in the copies of the triangles that we found. Let $V_{i}^{\prime\prime}=V_{i}^{\prime}\setminus S_{0}=V_{i}\setminus(B_{0}\cup S_{0})$ and note that for all $i$, since $\|V_{i}\cap(B_{0}\cup S_{0})\|\leq\|X_{i}\|+\sqrt{\varepsilon_{2}}\|V_{i}\|\leq(\varepsilon_{1}/6)\|V_{i}\|$, then $\|V_{i}^{\prime\prime}\|\geq(1-\varepsilon_{1}/6)\|V_{i}\|$. By Proposition 2.1, $(V_{3t-2}^{\prime\prime},V_{3t-1}^{\prime\prime},V_{3t}^{\prime\prime})$ forms a $(2\varepsilon_{3},p)$-regular triple of density at least $\delta-\varepsilon_{3}\geq\delta/2$ for all $t\in[k]$. Apply Lemma 5.1 to $(V_{i}^{\prime\prime})_{i=1}^{3k}$ to obtain sets $B_{1}$ and $S_{1}$. Observe that $\|B_{0}\cup B_{1}\|\leq(D/3)p^{-2}+4K\leq(2D/3)p^{-2}$ and $G[S_{0}\cup S_{1}]$ contains a perfect triangle packing. Also, most crucially, if we let $BS=B_{0}\cup S_{0}\cup B_{1}\cup S_{1}$ and $W_{i}=V_{i}\setminus BS=V_{i}^{\prime\prime}\setminus(B_{1}\cup S_{1})$, then all $W_{i}$ have equal sizes and moreover, $\|W_{i}\|\geq(1-\varepsilon_{1}/6-\varepsilon_{1}/3)m\geq(1-\varepsilon_{1}/2)m$. We will remove some vertices from each set $W_{i}$ to make the triples $(W_{3t-2},W_{3t-1},W_{3t})$ super-regular for all $t\in[k]$. Since $\xi n\leq\|X_{i}\|\leq\|BS\cap V_{i}\|\leq(\varepsilon_{1}/2)m$ for all $i$, by Proposition 4.2 (ii), there are at most $(D/(18K))p^{-1}$ vertices which have more than $(2\varepsilon_{1}/3)pm$ neighbours in $BS\cap V_{i}$ for each fixed $i\in[3k]$. Let $Y_{1}$ be the collection of such vertices for the set $BS\cap V_{2}$ and $BS\cap V_{3}$ which lie in $V_{1}$ and similarly define $Y_{2},Y_{3}$. By placing arbitrary vertices into $Y_{1},Y_{2}$, or $Y_{3}$ as necessary, we may assume that $\|Y_{1}\|=\|Y_{2}\|=\|Y_{3}\|\leq(D/(9k))p^{-1}$. Consider the set $W_{i}^{\prime}=W_{i}\setminus Y_{i}$ for $i\in\\{1,2,3\\}$. Then since $\|Y_{i}\|+\|BS\cap V_{i}\|\leq(D/(9K))p^{-1}+(\varepsilon_{1}/2)m\leq\varepsilon_{1}m$, in total we removed at most $\varepsilon_{1}m$ vertices from each part of $(V_{1},V_{2},V_{3})$ to obtain $(W_{1}^{\prime},W_{2}^{\prime},W_{3}^{\prime})$. By the definition of the sets $X_{i}$ at the beginning, all the vertices in $W_{1}^{\prime}$ were $\varepsilon_{1}$-typical in the triple $(V_{1},V_{2},V_{3})$, and by the choice of $Y_{1}$, they have at most $(2\varepsilon_{1}/3)pm+\|Y_{2}\|\leq\varepsilon_{1}pm$ neighbours in the deleted portion in $V_{2}$ (similar for $V_{3}$). Thus by Proposition 2.10, all the vertices in $W_{1}^{\prime}$ are $\varepsilon$-typical in the triple $(W_{1}^{\prime},W_{2}^{\prime},W_{3}^{\prime})$. Also, since all the vertices of $V_{1}$ not in $X_{1}$ had $(1\pm\varepsilon_{1})d_{12}p\|V_{2}\|$ neighbours in $V_{2}$, they will still have $(1\pm 2\varepsilon_{1})d_{12}p\|V_{2}\|$ neighbours in $W_{2}^{\prime}$ and similar for other choices of indices. Moreover, the triple $(W_{1}^{\prime},W_{2}^{\prime},W_{3}^{\prime})$ inherits the regularity of $(V_{1},V_{2},V_{3})$ and is $(2\varepsilon_{3},p)$-regular of density at least $\delta-\varepsilon_{3}>\delta/2$, see Proposition 2.1. Thus by the fact $2\varepsilon_{1}<\varepsilon$ and $2\varepsilon_{3}<\varepsilon$, $(W_{1}^{\prime},W_{2}^{\prime},W_{3}^{\prime})$ is $(\delta/2,\varepsilon,p)$-super-regular. Repeat the above process for all other triples. Let $B$ be the union of $B_{0},B_{1}$, and $Y_{i}$ for all $i\in[3k]$ as above so that $\|B\|\leq(2D/3)p^{-2}+(D/3)p^{-1}=Dp^{-1}$ and $S=S_{0}\cup S_{1}$. We also have the bound $\|W_{i}^{\prime}\|\geq\|V_{i}\|/2\geq\xi n$ for all $i$. Moreover, (v) will hold since in $G^{\prime\prime}$ all the edges between $W_{2}^{\prime}$ and $W_{3}^{\prime}$ have at most $2\|V_{i}\|p^{2}$ common neighbours in $V_{1}$, and therefore at most $4\|W_{1}^{\prime}\|p^{2}$ in $W_{1}^{\prime}$ (similar for other indices). ∎ ## 6 Obtaining balanced strong-super-regular triples In the previous section, we managed to decompose the graph into balanced super-regular triples, a triangle packing, and a small set of exceptional vertices. In this section, we will show how by slightly enlarging the triangle packing and the exceptional set, we can make these triples strong-super- regular. Our main tool, which will also be used in the next section, is the following lemma, which constructs small quasi-random matchings in super-regular triples. For the application in this section, in Theorem 6.6 below, $V_{i}^{\prime}$s will be the sets of non-good vertices in each part of a regular partition of the host graph. We want to find vertex disjoint triangles that cover these sets of non-good vertices. As an intermediate step, we construct random matchings $M_{ij}^{\prime}$ which later can be coupled with the non-good vertices in order to construct vertex-disjoint triangles. See the discussion in Section 3.2 for more detailed description. We would like to remark that even though the stronger assumption (A1) implies the weaker assumption (A2), we state both of them, as (A1) is much simpler and in one of the two applications of Lemma 6.1, we can verify that this stronger condition is satisfied. Also note that the statement of this lemma holds not only for strong-super-regular triples coming from subgraphs of random graphs, but also for general strong-super-regular triples. ###### Lemma 6.1. For all positive $\delta$ and $\eta$ with $\eta<1/140$, there exist $\varepsilon(\delta)$ and $C(\delta,\eta)$ such that the following holds. Let $(V_{1},V_{2},V_{3})$ be a $(\delta,\varepsilon,p)$-super-regular triple with $m=\|V_{1}\|=\|V_{2}\|=\|V_{3}\|$ and $p\geq C(\log m/m)^{1/2}$. For each $i$ and $j$, let $d_{ij}p$ be the density of $(V_{i},V_{j})$, let $q_{ij}=\eta/(d_{ij}pm)$, and let $E_{ij}$ be a subgraph of $(V_{i},V_{j})$ with $\|E_{ij}\|\leq\eta d_{ij}pm^{2}$. Form a set $M^{\prime}_{ij}$ by selecting every edge in $(V_{i},V_{j})\setminus E_{ij}$ independently with probability $q_{ij}$ and let $M_{ij}\subset M^{\prime}_{ij}$ be the set of all selected edges in $(V_{i},V_{j})$ that are not incident to any other edge in $M^{\prime}_{12}\cup M^{\prime}_{13}\cup M^{\prime}_{23}$. Moreover, for each $i$, let $Q_{i}$ be the set of all vertices in $V_{i}$ that are covered by some edge in $M_{12}\cup M_{13}\cup M_{23}$. Assume that for each $i$, $j$, and $k$, there is a set $V^{\prime}_{i}$ such that 1. (A1) the neighbourhood of every $v\in V^{\prime}_{i}$ contains at most $\eta d_{ij}d_{ik}d_{jk}p^{3}m^{2}$ edges of $E_{jk}$ or 2. (A2) for every $v\in V^{\prime}_{i}$, every subgraph $(N^{\prime\prime}_{j},N^{\prime\prime}_{k})$ of $(N(v)\cap V_{j},N(v)\cap V_{k})$ such that $\deg(w,N^{\prime\prime}_{k})\leq 2d_{ik}d_{jk}p^{2}m$ for all $w\in N^{\prime\prime}_{j}$ and $\deg(w,N^{\prime\prime}_{j})\leq 2d_{ij}d_{jk}p^{2}m$ for all $w\in N^{\prime\prime}_{k}$ contains at most $\eta d_{ij}d_{ik}d_{jk}p^{3}m^{2}$ edges of $E_{jk}$. Then $M_{12}\cup M_{13}\cup M_{23}$ is a matching and with probability tending to $1$ as $m$ tends to infinity, for each $i$, $j$, and $k$, 1. (M1) $(\eta/2)m\leq\|M_{ij}\|\leq 2\eta m$, 2. (M2) every $v\in V_{i}$ has at most $3\eta d_{ij}pm$ neighbours in $Q_{j}$, 3. (M3) the neighbourhood of every $v\in V^{\prime}_{i}$ contains at least $(\eta/2)\delta^{2}p^{2}m$ edges of $M_{jk}$, and 4. (M4) the endpoints of each $\eta$-good edge in $(V_{j},V_{k})$ have at least $(1-4\eta)d_{ij}d_{ik}p^{2}m$ common neighbours in $V_{i}\setminus Q_{i}$. ###### Proof. Let $\varepsilon=\min\\{1/100,\delta/50\\}$. Fix $i$, $j$, and $k$ with $\\{i,j,k\\}=\\{1,2,3\\}$. For each vertex $v\in V_{i}$, let $N_{j}=N(v)\cap V_{j}$ and $N_{k}=N(v)\cap V_{k}$. By construction, $M_{12}\cup M_{13}\cup M_{23}$ is a matching. ###### Claim 6.2. With probability $1-o(1)$, $(\eta/2)m\leq\|M_{ij}\|\leq 2\eta m$. ###### Proof. By our assumption on $\|E_{ij}\|$, there are at least $(1-\eta)d_{ij}pm^{2}$ edges in $(V_{i},V_{j})\setminus E_{ij}$, so $\mathbb{E}[\|M^{\prime}_{ij}\|]\geq(1-\eta)d_{ij}pm^{2}q_{ij}=(1-\eta)\eta m$, and Chernoff’s inequality implies that $\|M^{\prime}_{ij}\|\geq(3\eta/4)m$ with probability $1-o(1)$. In order to estimate $\|M_{ij}\|$, note that $\|M^{\prime}_{ij}\|-\|M_{ij}\|$ is at most the number of vertices in $V_{i}\cup V_{j}$ that are incident to an edge of $M^{\prime}_{ij}$ and some other edge in $M^{\prime}_{12}\cup M^{\prime}_{13}\cup M^{\prime}_{23}$. Let $\mathcal{A}_{w}$ denote the event that $w$ is such a “bad” vertex. Since $(V_{1},V_{2},V_{3})$ is $(\delta,\varepsilon,p)$-super-regular, $\deg(v,V_{j})\leq(1+\varepsilon)d_{ij}pm$ and $\deg(v,V_{k})\leq(1+\varepsilon)d_{ik}pm$ for every $v\in V_{i}$. Hence, if $w\in V_{i}$, then $P(\mathcal{A}_{w})\leq\deg(w,V_{j})q_{ij}\cdot(\deg(w,V_{j})q_{ij}+\deg(w,V_{k})q_{ik})\leq(1+\varepsilon)^{2}2\eta^{2}$ and the expected number of such “bad” vertices in $V_{i}$ is at most $(1+\varepsilon)^{2}2\eta^{2}m$. The events $\\{\mathcal{A}_{w}\colon w\in V_{i}\\}$ are mutually independent, so by Chernoff’s inequality, with probability at least $1-o(m^{-1})$, there are at most $3\eta^{2}m$ “bad” vertices in $V_{i}$ and similarly, there are at most $3\eta^{2}m$ “bad” vertices in $V_{j}$. Hence, $\|M_{ij}\|\geq(3/4-6\eta)\eta m\geq(\eta/2)m$ with probability $1-o(1)$. Finally, since the number of edges in $(V_{i},V_{j})\setminus E_{ij}$ is at most $d_{ij}pm^{2}$, we have $\mathbb{E}[\|M^{\prime}_{ij}\|]\leq d_{ij}pm^{2}q_{ij}=\eta m$, and Chernoff’s inequality implies that $\|M_{ij}\|\leq\|M^{\prime}_{ij}\|\leq 2\eta m$ with probability $1-o(1)$. ∎ ###### Claim 6.3. For each fixed vertex $v$, with probability $1-o(m^{-1})$, we have $\deg(v,Q_{j})\leq 3\eta d_{ij}pm$. ###### Proof. Let $Q^{\prime}_{i}$ be the set of vertices in $V_{i}$ that are covered by some edge in $M^{\prime}_{ij}\cup M^{\prime}_{ik}$ and note that $Q^{\prime}_{i}\supset Q_{i}$ (similarly define $Q_{j}^{\prime}$ and $Q_{k}^{\prime}$). For a vertex $w\in V_{j}$, let $\mathcal{B}_{w}$ denote the event that $w\in Q^{\prime}_{j}$. Since $(V_{1},V_{2},V_{3})$ is $(\delta,\varepsilon,p)$-super-regular, $\deg(w,V_{i})\leq(1+\varepsilon)d_{ij}pm$ and $\deg(w,V_{k})\leq(1+\varepsilon)d_{jk}pm$. Hence, $P(\mathcal{B}_{w})\leq\deg(w,V_{j})q_{ij}+\deg(w,V_{k})q_{ik}\leq 2(1+\varepsilon)\eta.$ The events $\\{\mathcal{B}_{w}\colon w\in V_{j}\\}$ are mutually independent and $\|N_{j}\|\geq(\delta/2)pm$, so by Chernoff’s inequality, $\|N_{j}\cap Q^{\prime}_{j}\|\leq(5\eta/2)\|N_{j}\|$ with probability at least $1-e^{-c\eta\delta pm}$ for some absolute positive constant $c$. It follows that $\deg(v,Q_{j})\leq\deg(v,Q^{\prime}_{j})=\|N_{j}\cap Q^{\prime}_{j}\|\leq(5\eta/2)(1+\varepsilon)d_{ij}pm\leq 3\eta d_{ij}pm$ with probability $1-o(m^{-1})$. ∎ ###### Claim 6.4. For each fixed $v\in V^{\prime}_{i}$, with probability $1-o(m^{-1})$, the pair $(N_{j},N_{k})$ contains at least $(\eta/2)d_{ij}d_{ik}p^{2}m$ edges of $M_{jk}$. ###### Proof. Without loss of generality, we may assume that $(i,j,k)=(1,2,3)$. Since $v$ is $\varepsilon$-typical, $(1+\varepsilon^{2})(1+2\varepsilon/\delta)\leq 2$, and $5\varepsilon+2\varepsilon/\delta\leq 1/7$, Proposition 2.14 implies that there are sets $N^{\prime\prime}_{2}\subset N_{2}$ and $N^{\prime\prime}_{3}\subset N_{3}$ such that $(N^{\prime\prime}_{2},N^{\prime\prime}_{3})$ contains at least $(6/7)d_{12}d_{13}d_{23}p^{3}m^{2}$ edges, no vertex in $N^{\prime\prime}_{2}$ has more than $2d_{13}d_{23}p^{2}m$ neighbours in $N^{\prime\prime}_{3}$, and vice versa, no vertex in $N^{\prime\prime}_{3}$ has more than $2d_{12}d_{23}p^{2}m$ neighbours in $N^{\prime\prime}_{2}$. Since at most $\eta d_{12}d_{13}d_{23}p^{3}m^{2}$ edges among $(N^{\prime\prime}_{2},N^{\prime\prime}_{3})$ belong to $E_{23}$ by either (A1) or (A2), it follows that $\mathbb{E}[\|M^{\prime}_{23}\cap(N^{\prime\prime}_{2},N^{\prime\prime}_{3})\|]\geq(5/7)d_{12}d_{13}d_{23}p^{3}m^{2}q_{23}=(5/7)d_{12}d_{13}\eta p^{2}m.$ Since $\|M^{\prime}_{23}\cap(N^{\prime\prime}_{2},N^{\prime\prime}_{3})\|$ is a sum of independent indicator random variables, Chernoff’s inequality implies that for some absolute constant $c$, $P\left(\|M^{\prime}_{23}\cap(N^{\prime\prime}_{2},N^{\prime\prime}_{3})\|\geq(4/7)d_{12}d_{13}\eta p^{2}m\right)\geq 1-e^{-c\eta\delta^{2}p^{2}m}\geq 1-1/m^{2},$ provided that $C$ is sufficiently large. In order to estimate $\|M_{23}\cap(N^{\prime\prime}_{2},N^{\prime\prime}_{3})\|$, note that $\|M^{\prime}_{23}\cap(N^{\prime\prime}_{2},N^{\prime\prime}_{3})\|-\|M_{23}\cap(N^{\prime\prime}_{2},N^{\prime\prime}_{3})\|$ is at most the number of vertices in $N^{\prime\prime}_{2}\cup N^{\prime\prime}_{3}$ that are incident to an edge in $M^{\prime}_{23}\cap(N^{\prime\prime}_{2},N^{\prime\prime}_{3})$ and some other edge in $M^{\prime}_{12}\cup M^{\prime}_{13}\cup M^{\prime}_{23}$. Let $\mathcal{C}_{w}$ denote the event that $w$ is such a “bad” vertex. If $w\in N^{\prime\prime}_{2}$, then $\displaystyle P(\mathcal{C}_{w})$ $\displaystyle\leq\deg(w,N^{\prime\prime}_{3})q_{23}\cdot(\deg(w,V_{3})q_{23}+\deg(w,V_{1})q_{12})$ $\displaystyle\leq 2d_{13}d_{23}p^{2}mq_{23}\cdot((1+\varepsilon)d_{23}pmq_{23}+(1+\varepsilon)d_{12}pmq_{12})=(1+\varepsilon)4\eta^{2}d_{13}\eta^{2}p.$ Since $\|N^{\prime\prime}_{2}\|\leq\|N_{2}\|\leq(1+\varepsilon)d_{12}pm$, the expected number of such “bad” vertices in $N^{\prime\prime}_{2}$ is at most $(1+\varepsilon)^{2}4\eta^{2}d_{12}d_{23}p^{2}m$. The events $\\{\mathcal{C}_{w}\colon w\in N^{\prime\prime}_{2}\\}$ are mutually independent, so by Chernoff’s inequality, for some absolute constant $c$, with probability at least $1-e^{-c\delta^{2}\eta^{2}p^{2}m}$, there are at most $5\eta^{2}d_{12}d_{13}p^{2}m$ “bad” vertices in $N^{\prime\prime}_{2}$ and similarly, there are at most $5\eta^{2}d_{12}d_{13}p^{2}m$ “bad” vertices in $N^{\prime\prime}_{3}$. Hence, with probability $1-o(m^{-1})$, $\|M_{23}\cap(N^{\prime\prime}_{2},N^{\prime\prime}_{3})\|\geq(4/7-10\eta)d_{12}d_{13}\eta p^{2}m\geq(\eta/2)d_{12}d_{13}p^{2}m,$ provided that $C$ is sufficiently large. ∎ ###### Claim 6.5. With probability $1-o(1)$, the endpoints of every $\eta$-good edge in $(V_{j},V_{k})$ have at least $(1-4\eta)d_{ij}d_{ik}p^{2}m$ common neighbours in $V_{i}\setminus Q_{i}$. ###### Proof. For an arbitrary vertex $v\in V_{i}$, let $\mathcal{D}_{v}$ denote the event that $v\in Q^{\prime}_{i}$. Clearly, $P(\mathcal{D}_{v})\leq\deg(v,V_{j})q_{ij}+\deg(v,V_{k})q_{ik}\leq 2(1+\varepsilon)\eta.$ Fix some $\eta$-good edge in $(V_{j},V_{k})$ and let $A\subset V_{i}$ be the set of common neighbours of its endpoints. Then $\mathbb{E}[\|A\cap Q^{\prime}_{i}\|]=\sum_{v\in A}P(\mathcal{D}_{v})\leq\|A\|\cdot 2(1+\varepsilon)\eta$. Moreover by definition, $\|A\|\geq(1-\eta)d_{ij}d_{ik}p^{2}m\geq(1/2)\delta^{2}p^{2}m$. Since the events $\mathcal{D}_{v}$ are mutually independent, Chernoff’s inequality implies that $P(\|A\cap Q^{\prime}_{i}\|\geq 3\eta\|A\|)\leq e^{-c\delta^{2}\eta p^{2}m}$ for some absolute positive constant $c$. Hence, if $C$ is sufficiently large, then with probability at least $1-1/m^{3}$, $\|A\setminus Q_{i}\|\geq\|A\setminus Q^{\prime}_{i}\|\geq(1-3\eta)\|A\|\geq(1-3\eta)(1-\eta)d_{ij}d_{ik}p^{2}m\geq(1-4\eta)d_{ij}d_{ik}p^{2}m.$ Since there are at most $m^{2}$ good edges, the claim is proved. ∎ Finally, note that Claims 6.2–6.5 imply that (M1)–(M4) are satisfied with probability $1-o(1)$ (one needs to apply the union bound over all choices of vertices in order to deduce (M2) and (M3) from 6.3 and 6.4). ∎ Below is the main theorem of this section. It says that we can partition our graph into balanced strong-super-regular triples, a collection of vertex- disjoint triangles, and a set of at most $O(p^{-2})$ exceptional vertices. In the next section, we will prove that each of those strong-super-regular triples contains a triangle-factor. ###### Theorem 6.6. For an arbitrary positive $\gamma$, there exists a positive $\delta$ such that for all $\varepsilon$, there exist constants $C$, $D$, and $\xi$ that satisfy the following. If $p\geq C(\log n/n)^{1/2}$, then a.a.s. every $G\subset G(n,p)$ with $\delta(G)\geq(2/3+\gamma)np$ contains a subgraph $G^{\prime}\subset G$ whose vertex set can be partitioned into sets $B$, $S$, and $(W^{\prime}_{i})_{i=1}^{3k}$, where $k\leq D$, such that 1. (i) $\|B\|\leq Dp^{-2}$, 2. (ii) $G[S]$ contains a perfect triangle packing, 3. (iii) $(W^{\prime}_{3t-2},W^{\prime}_{3t-1},W^{\prime}_{3t})$ is a $(\delta,\varepsilon,p)$-strong-super-regular in $G^{\prime}$ for all $t\in[k]$, and 4. (iv) $\|W^{\prime}_{3t-2}\|=\|W^{\prime}_{3t-1}\|=\|W^{\prime}_{3t}\|\geq\xi n$ for all $t\in[k]$. ###### Proof. Let $\delta=\min\\{\delta_{\ref{thm_cleanstage1}}(\gamma)/2,3/4\\}$. Without loss of generality, we may assume that $\varepsilon\leq 2/\delta$. Furthermore, let $\varepsilon_{3}=\varepsilon_{\ref{prop_typicalstability}}(\varepsilon,2\delta)$, $\varepsilon_{1}=\min\\{\varepsilon\delta^{2}/1180,\varepsilon_{3}\delta^{2}/40\\}$, $\varepsilon_{2}=\min\\{\varepsilon_{\ref{prop_manygoodvertices}}(\varepsilon_{1},2\delta),\varepsilon_{\ref{thm_cleanstage1}}(\gamma),\varepsilon_{\ref{lemma_randommatching}}(\delta),\varepsilon/2\\}$, and $\xi=\xi_{\ref{thm_cleanstage1}}(\varepsilon_{2})/2$. Let $\eta=12\varepsilon_{1}/\delta^{2}$. Moreover, let $D=(6/\xi)\max\\{D_{\ref{thm_cleanstage1}(\varepsilon_{2})},D_{\ref{prop_nontypicalvertices}(ii)}(1/4,\varepsilon_{1}\xi),D_{\ref{prop_randomgraphproperties2}}(\varepsilon_{1}\varepsilon_{2}\delta^{3}\xi^{2}/4,1/4)\\}$ and $C=\max\\{C_{\ref{thm_cleanstage1}}(\varepsilon_{2}),C_{\ref{prop_smallexpansion}}(3\varepsilon_{1}\xi)\\}$. By Theorem 5.2, there exists a further subgraph $G^{\prime}$ of $G$ whose vertex set can be partitioned into sets $B_{0}$, $S_{0}$, and $(W_{i})_{i=1}^{3k}$ such that $\|B_{0}\|\leq(D/2)p^{-2}$, $G[S_{0}]$ contains a perfect triangle packing, and for all $t\in\\{1,\ldots,k\\}$, the triple $(W_{3t-2},W_{3t-1},W_{3t})$ is $(2\delta,\varepsilon_{2},p)$-super-regular in $G^{\prime}$ and satisfies $\|W_{3t-2}\|=\|W_{3t-1}\|=\|W_{3t}\|$. Moreover, the endpoints of no edge in $(W_{3t-2},W_{3t-1})$ have more than $4\|W_{3t}\|p^{2}$ common neighbours in $W_{3t}$ (and a similar statement holds for other choices of indices). We will show that each such triple contains a slightly smaller $(\delta,\varepsilon,p)$-strong-super-regular triple in such a way that all but at most $O(p^{-1})$ leftover vertices can be covered by vertex-disjoint triangles. Obviously, this will imply the assertion of the theorem. Without loss of generality, we will only consider the triple $(W_{1},W_{2},W_{3})$. For the sake of brevity, let $m=\|W_{1}\|=\|W_{2}\|=\|W_{3}\|$ and note that $m\geq 2\xi n$. Without loss of generality, we can condition on the event that $G(n,p)$ satisfies the assertions of • Proposition 4.2 (ii) with $\rho=1/4$ and $\xi=\varepsilon_{1}m/n$, * • Proposition 4.4 with $\varepsilon=\varepsilon_{1}\varepsilon_{2}\delta^{3}m^{2}/(4n^{2})$, and $\rho=1/4$, and * • Proposition 4.3 with $\xi=6\varepsilon_{1}m/(2n)$. For each $i$, let $X_{i}\subset W_{i}$ be the collection of vertices that are not $\varepsilon_{1}$-good. By Proposition 2.13, $\|X_{i}\|\leq\varepsilon_{1}m$ and we may assume that $\|X_{i}\|=\varepsilon_{1}m$. We perform the following cleaning-up procedure. While constantly updating the sets $X_{1}$, $X_{2}$, and $X_{3}$, repeat the following. If there exists an $i$ and a vertex $v\in W_{i}\setminus X_{i}$ such that either 1. (A) $\|N(v)\cap X_{j}\|\geq 4\varepsilon_{1}pm$ for some $j$ or 2. (B) the neighbourhood of $v$ contains more than $\varepsilon_{2}d_{12}d_{13}d_{23}p^{3}m^{2}$ edges whose endpoints have more than $5\varepsilon_{1}p^{2}m$ common neighbours in $X_{i}$, then move $v$ to $X_{i}$. ###### Claim 6.7. The cleaning-up procedure finishes with $\varepsilon_{1}m\leq\|X_{i}\|\leq 3\varepsilon_{1}m$ for all $i$. ###### Proof. Suppose that at some point in time, $\|X_{i}\|>3\varepsilon_{1}m$ for some $i$, and consider the earliest such moment. Without loss of generality, we may assume that $i=1$. Clearly, $3\varepsilon_{1}m<\|X_{1}\|\leq 4\varepsilon_{1}m$ and $\|X_{j}\|\leq 3\varepsilon_{1}m$ if $j\neq 1$. Since at the beginning, every $X_{1}$ contained at most $\varepsilon_{1}m$ vertices, $W_{1}$ contains either $\varepsilon_{1}m$ vertices satisfying (A) or $\varepsilon_{1}m$ vertices satisfying (B). The former is impossible, since $\|X_{j}\|\leq 3\varepsilon_{1}m$ for $j\neq 1$ and we assumed that $G(n,p)$ satisfies the assertion of Proposition 4.2 (ii) with $\xi=\varepsilon_{1}m/n$. Since the endpoints of each edge in $(W_{2},W_{3})$ have at most $4p^{2}m$ common neighbours in $W_{1}$, the latter would imply that $(W_{2},W_{3})$ contains $(\varepsilon_{1}m)\cdot(\varepsilon_{2}d_{12}d_{13}d_{23}p^{3}m^{2})/(4p^{2}m)\geq(\varepsilon_{1}\varepsilon_{2}\delta^{3}/4)pm^{2}$ edges whose endpoints have more than $5\varepsilon_{1}p^{2}m$ common neighbours in $X_{1}$. Since $\|X_{1}\|\leq 4\varepsilon_{1}m$, this is impossible by our assumption that $G(n,p)$ satisfies the assertion of Proposition 4.4 with $\varepsilon=\varepsilon_{1}\varepsilon_{2}\delta^{3}m^{2}/(4n^{2})$, and $\rho=1/4$. ∎ It is not hard to check that $(W_{1}\setminus X_{1},W_{2}\setminus X_{2},W_{3}\setminus X_{3})$ is $(\delta/2,\varepsilon^{\prime},p)$-strong- super-regular. Unfortunately, this conclusion does not help us at the moment as we first need to absorb $X_{1}\cup X_{2}\cup X_{3}$ into vertex-disjoint triangles and in the process of absorbing those vertices, we may use some vertices from the triple $(W_{1}\setminus X_{1},W_{2}\setminus X_{2},W_{3}\setminus X_{3})$. For every $i$, let $Y_{i}\subset X_{i}$ be the set of vertices in $X_{i}$ that have more than $4\varepsilon_{1}pm$ neighbours in $X_{j}$ for some $j$ with $j\neq i$. Since $\|X_{j}\|\leq 3\varepsilon_{1}m$ and we assumed that $G(n,p)$ satisfies the assertion of Proposition 4.2 (ii) with $\xi=\varepsilon_{1}m/n$, then $\|Y_{i}\|\leq Dp^{-1}/(6k)$. By adding arbitrary vertices of $X_{i}$ to $Y_{i}$, we can guarantee that $\|Y_{1}\|=\|Y_{2}\|=\|Y_{3}\|$. For every $i$ and $j$, let $E_{ij}=(X_{i},W_{j})\cup(W_{i},X_{j})$. Since $(W_{i},W_{j})$ is $(\varepsilon_{2},p)$-regular and $\|X_{i}\|\geq\varepsilon_{1}\|W_{i}\|\geq\varepsilon_{2}\|W_{i}\|$, we have $\|E_{ij}\|\leq(d_{ij}+\varepsilon)p(\|X_{i}\|\|W_{j}\|+\|W_{i}\|\|X_{j}\|)\leq(1+\varepsilon/\delta)6\varepsilon_{1}d_{ij}pm^{2}\leq\eta d_{ij}pm^{2}.$ Fix a vertex $v\in W_{i}\setminus Y_{i}$. We check that (A2) in Lemma 6.1 is satisfied. Let $(N^{\prime\prime}_{j},N^{\prime\prime}_{k})$ be as in (A2) in Lemma 6.1. Since $\|N^{\prime\prime}_{j}\cap X_{j}\|\leq\|N(v)\cap X_{j}\|\leq 4\varepsilon_{1}pm$ and similarly, $\|N^{\prime\prime}_{k}\cap X_{k}\|\leq 4\varepsilon_{1}pm$, we have $\|E_{jk}\cap(N^{\prime\prime}_{j},N^{\prime\prime}_{k})\|\leq 4\varepsilon_{1}pm(2d_{ij}d_{jk}p^{2}m+2d_{ik}d_{jk}p^{2}m)\leq(16\varepsilon_{1}/\delta)d_{ij}d_{ik}d_{jk}p^{3}m^{2}\leq\eta d_{ij}d_{ik}d_{jk}p^{3}m^{2}.$ Lemma 6.1 implies that a.a.s. for each $i$ and $j$, there exists an $M_{ij}\subset(W_{i}\setminus X_{i},W_{j}\setminus X_{j})$ such that (M1)–(M4) in Lemma 6.1 are satisfied with $V_{i}=W_{i}$ and $V^{\prime}_{i}=W_{i}\setminus Y_{i}$ for each $i$. Let $Q_{i}$ be defined as in Lemma 6.1. ###### Claim 6.8. The sets $X_{1}\setminus Y_{1}$, $X_{2}\setminus Y_{2}$, and $X_{3}\setminus Y_{3}$ can be covered by vertex-disjoint triangles that use only vertices in $Q_{1}\cup X_{1}$, $Q_{2}\cup X_{2}$, and $Q_{3}\cup X_{3}$. ###### Proof. Since $M_{12}\cup M_{13}\cup M_{23}$ is a matching whose edges are not incident to any vertex in $X_{1}\cup X_{2}\cup X_{3}$, it suffices to show that for each $i$, $j$, and $k$, the vertices of $X_{i}\setminus Y_{i}$ can be paired with some $\|X_{i}\setminus Y_{i}\|$ edges of $M_{jk}$ to form vertex- disjoint triangles. Let $H$ be the bipartite graph on the vertex set $(X_{i}\setminus Y_{i})\cup M_{jk}$, where a vertex $w\in X_{i}\setminus Y_{i}$ is adjacent to an edge $\\{u,v\\}\in M_{jk}$ if and only if $\\{u,v,w\\}$ is a triangle in $(W_{1},W_{2},W_{3})$. Clearly, it suffices to prove that $H$ contains a matching that covers $X_{i}\setminus Y_{i}$. We check that Hall’s condition holds in $H$. Fix an arbitrary non-empty set $S\subset X_{i}\setminus Y_{i}$. If $\|N_{H}(S)\|\leq\|S\|$, then there would be an $x$ with $1\leq x=\|S\|\leq\|X_{i}\|\leq 3\varepsilon_{1}m$ such that $G(n,p)$ contains some $x$ independent edges and $x$ vertices, each of which is adjacent to both ends of at least $(\eta/2)\delta^{2}p^{2}m$ of those edges, see (M3) in Lemma 6.1. This would contradict our assumption that $G(n,p)$ satisfies the assertion of Proposition 4.3 with $\xi=\eta\delta^{2}m/(2n)=6\varepsilon_{1}m/(2n)$. Hence, $\|N_{H}(S)\|>\|S\|$ for all non-empty $S\subset X_{i}\setminus Y_{i}$. ∎ Fix any such triangle packing and for each $i$, let $X^{\prime}_{i}=X_{i}\cup T_{i}$, where $T_{i}\subset Q_{i}$ is the set of vertices in $W_{i}\setminus X_{i}$ that are covered by the triangle packing. Note that $\|T_{i}\|=\|X_{j}\setminus Y_{j}\|+\|X_{k}\setminus Y_{k}\|$. Let $W^{\prime}_{i}=W_{i}\setminus X^{\prime}_{i}$. Since for each $i$, $\|X^{\prime}_{i}\|=\|X_{i}\|+\|T_{i}\|=\|X_{1}\setminus Y_{1}\|+\|X_{2}\setminus Y_{2}\|+\|X_{3}\setminus Y_{3}\|+\|Y_{i}\|$ and $\|Y_{1}\|=\|Y_{2}\|=\|Y_{3}\|$, the sets $W^{\prime}_{1}$, $W^{\prime}_{2}$, and $W^{\prime}_{3}$ have the same number of elements. Denote this number by $m^{\prime}$ and note that $m^{\prime}\geq m-9\varepsilon_{1}m\geq m/2\geq\xi n$. ###### Claim 6.9. The triple $(W^{\prime}_{1},W^{\prime}_{2},W^{\prime}_{3})$ is $(\varepsilon,\delta,p)$-strong-super-regular. ###### Proof. Since $(W_{1},W_{2},W_{3})$ is $(\varepsilon/2,p)$-regular with density at least $2\delta p$ and $m^{\prime}\geq m/2$, Proposition 2.1 implies that $(W^{\prime}_{1},W^{\prime}_{2},W^{\prime}_{3})$ is $(\varepsilon,p)$-regular with density at least $\delta p$. Fix an index $i$, recall that $\|X^{\prime}_{i}\|\leq 9\varepsilon_{1}m\leq\varepsilon_{3}m$, and let $v$ be an arbitrary vertex in $W^{\prime}_{i}$. Without loss of generality, we may assume that $i=1$. Since $v\not\in X_{1}$, (A) implies that $\deg(v,X_{j})\leq 4\varepsilon_{1}pm$ for every $j$. Moreover, (M2) in Lemma 6.1 implies that $\deg(v,Q_{j})\leq 3\eta d_{ij}pm$. Hence, $\deg(v,X^{\prime}_{j})\leq\deg(v,X_{j})+\deg(v,T_{j})\leq\deg(v,X_{j})+\deg(v,Q_{j})\leq\varepsilon_{3}pm,$ and by Proposition 2.10, $v$ becomes $\varepsilon$-typical in $(W^{\prime}_{1},W^{\prime}_{2},W^{\prime}_{3})$. It remains to show that $v$ is also $\varepsilon$-good. Since $v\not\in X_{i}$, it was $\varepsilon_{1}$-good in $(W_{1},W_{2},W_{3})$ and it satisfies (B). Hence, the endpoints of all but at most $(\varepsilon_{1}+\varepsilon_{2})d_{12}d_{13}d_{23}p^{3}m^{2}$ edges in the neighbourhood of $v$ have at least $(1-\varepsilon_{1}-5\varepsilon_{1}/\delta^{2})d_{12}d_{13}p^{2}m$ common neighbours in $W_{1}\setminus X_{1}$. Moreover by (M4), they have at most $4\eta d_{12}d_{13}p^{2}m$ common neighbours in $Q_{1}$. Since $1-\varepsilon_{1}-5\varepsilon_{1}/\delta^{2}-4\eta\geq(1+\varepsilon_{2}/\delta)^{2}(1-\varepsilon)$, each such edge is $\varepsilon$-good in the new triple. It follows that $v$ is $\varepsilon$-good. ∎ Finally, let $B=B_{0}\cup\bigcup_{i=1}^{3k}Y_{i}$ and let $S=S_{0}\bigcup_{i=1}^{3k}(T_{i}\cup(X_{i}\setminus Y_{i}))$. Clearly, the sets $B$, $S$, and $(W^{\prime}_{i})_{i=1}^{3k}$ partition the vertex set of $G$, $\|B\|\leq\|B_{0}\|+\sum_{i=1}^{3k}\|Y_{i}\|\leq Dp^{-2}/2+3k\cdot Dp^{-1}/(6k)\leq Dp^{-2}/2,$ and $G[S]$ contains a perfect triangle packing. Finally, by Claim 6.9, for each $t\in[k]$, the triple $(W^{\prime}_{3t-2},W^{\prime}_{3t-1},W^{\prime}_{3t})$ is $(\varepsilon,\delta,p)$-strong-super-regular and satisfies $\|W^{\prime}_{3t-2}\|=\|W^{\prime}_{3t-1}\|=\|W^{\prime}_{3t}\|\geq\xi n$. ∎ ## 7 Perfect triangle packing in strong-super-regular triples In the previous section, we managed to decompose the graph into balanced strong-super-regular triples, a triangle packing, and a small set of exceptional vertices. In this section, we will show how to find a triangle- factor in each of those triples. We start this section by showing how to construct sets of “buffer” vertices and edges that will allow us to complete an almost-spanning triangle packing into a triangle-factor. ###### Lemma 7.1. For all positive constants $\delta$, $\xi$, and $\eta$ with $\eta\leq 1/140$, there exist constants $C(\delta,\eta,\xi)$ and $\varepsilon(\delta,\eta)$ such that if $p\geq C(\log n/n)^{1/2}$, then $G(n,p)$ a.a.s. satisfies the following. Let $(W_{1},W_{2},W_{3})$ be a $(\delta,\varepsilon,p)$-strong- super-regular triple in a subgraph of $G(n,p)$ such that $\|W_{1}\|=\|W_{2}\|=\|W_{3}\|\geq\xi n$. Then there exist edge sets $M_{12}$, $M_{13}$, $M_{23}$ and vertex sets $X_{1}$, $X_{2}$, $X_{3}$ with the following properties: 1. (P1) $M_{12}\cup M_{13}\cup M_{23}$ is a matching. 2. (P2) For all $j$ and $k$, $M_{jk}\subset(W_{j},W_{k})$ and $(\eta/2)\|W_{j}\|\leq\|M_{jk}\|\leq 2\eta\|W_{j}\|$. 3. (P3) For all $i$, $\|X_{i}\|\leq(\eta/4)\|W_{i}\|$ and $X_{i}\subset W_{i}\setminus Q_{i}$, where $Q_{i}$ is the set of vertices in $W_{i}$ that are covered by some edge in $M_{ij}\cup M_{ik}$. 4. (P4) For all $i$, $j$, and $k$, if $Z_{i}\subset W_{i}$ has size $\|M_{jk}\|$ and contains $X_{i}$, then the subgraph of $(W_{1},W_{2},W_{3})$ induced by $Z_{i}$ and $M_{jk}$ contains a triangle-factor. ###### Proof. For the sake of brevity, let $m=\|W_{1}\|=\|W_{2}\|=\|W_{3}\|$. Without loss of generality, we may assume that $\delta\leq 1$. Let $\alpha=\delta^{2}/24$ and let $\beta$ be a positive constant satisfying $\beta\log(e/\beta)<\alpha\eta/30$. Moreover, let $\varepsilon=\min\\{\delta/2,\eta/4,\varepsilon_{\ref{lemma_randommatching}}(\delta),\beta\cdot(\varepsilon_{0})_{\ref{prop_randomgraphtypical}}(\alpha\delta/16,\delta/2),\alpha\beta\delta/16\\}$ and let $\varepsilon^{\prime}=4\varepsilon/\delta$. Finally, let $C$ be sufficiently large so that $C(\log n/n)^{1/2}\geq C_{\ref{lemma_randommatching}}(\log m/m)^{1/2}$ and without loss of generality we may assume that $G(n,p)$ satisfies the assertion of Proposition 4.3 with $\xi=\eta\delta^{2}m/(2n)$ and $\xi=\eta\delta^{2}m/(12n)$, and Proposition 4.8 with $\varepsilon^{\prime}_{\ref{prop_randomgraphtypical}}=\alpha\delta/16$ and $\delta_{\ref{prop_randomgraphtypical}}=\delta/2$. For all $i$ and $j$, let $q_{ij}=\eta/(d_{ij}mp)$ and select each $\varepsilon^{\prime}$-good edge of $(W_{i},W_{j})$ independently with probability $q_{ij}$. Let $M^{\prime}_{ij}$ be the set of all selected edges in $(W_{i},W_{j})$ and let $M_{ij}\subset M^{\prime}_{ij}$ be the set of all those edges that are not incident to any other selected edge. By Proposition 2.12, $(W_{i},W_{j})$ contains at most $\eta d_{ij}pm^{2}$ edges that are not $\varepsilon^{\prime}$-good. Since each $v\in W_{i}$ is $\varepsilon$-good, its neighbourhood contains at most $\eta d_{12}d_{13}d_{23}p^{3}m^{2}$ edges that are not $\varepsilon^{\prime}$-good. Therefore, Lemma 6.1 applies with $E_{ij}$ being the set of non-$\varepsilon^{\prime}$-good edges in $(W_{i},W_{j})$ and $V_{i}=V^{\prime}_{i}=W_{i}$. ###### Claim 7.2. With probability $1-o(1)$, every set $Y_{i}\subset W_{i}$ of size $\beta m$ satisfies the following. All but at most $\alpha\eta m$ edges of $M_{jk}$ belong to the neighbourhood of some vertex of $Y_{i}$. ###### Proof. Fix a $Y_{i}\subset W_{i}$ of size $\beta m$. By Proposition 2.1, the triple $(Y_{i},W_{j},W_{k})$ is $(\varepsilon/\beta,p)$-regular, and the densities of all three of its parts are at least $\delta/2$. Moreover, since we assumed that $G(n,p)$ satisfies the assertion of Proposition 4.8, $(Y_{i},W_{j},W_{k})$ is $\alpha\delta/16$-typical and by our assumption on $\varepsilon$, it is $(\alpha\delta/16,p)$-regular. By Proposition 2.12, all but at most $(\alpha/2)d_{23}pm^{2}$ edges between $W_{j}$ and $W_{k}$ are $\alpha/2$-good, so in particular all but at most $(\alpha/2)d_{23}pm^{2}$ edges in $(W_{j},W_{k})$ belong to the neighbourhood of some vertex in $Y_{i}$. Hence the expected number of edges chosen among those “bad” edges is at most $(\alpha/2)d_{23}pm^{2}q_{23}=(\alpha\eta/2)m$. Chernoff’s inequality implies that the probability that more than $\alpha\eta m$ of those edges are chosen to $M^{\prime}_{jk}$ is at most $e^{-\alpha\eta m/30}$. Since there are ${m\choose\beta m}$ $\beta m$-subsets of $W_{i}$, ${m\choose\beta m}\leq\left(\frac{em}{\beta m}\right)^{\beta m}=e^{\beta\log(e/\beta)m},$ and $\alpha\eta/30>\beta\log(e/\beta)$, the probability that all sets $Y_{i}$ have the claimed property is $1-o(1)$. ∎ Lemma 6.1 and Claim 7.2 imply that there exist $M_{12}$, $M_{13}$, and $M_{23}$ such that $M_{12}\cup M_{13}\cup M_{23}$ is a matching and for all $i$, $j$, and $k$ (properties 1, 2, and 3 follow from (M1), (M3), and (M4) of Lemma 6.1, respectively, whereas property 4 follows from Claim 7.2): 1. 1. $(\eta/2)m\leq\|M_{jk}\|\leq 2\eta m$, 2. 2. the neighbourhood of every vertex in $V_{i}$ contains at least $(\eta/2)\delta^{2}p^{2}m$ edges of $M_{jk}$, 3. 3. the endpoints of each edge of $M_{jk}$ have at least $(1/2)\delta^{2}p^{2}m$ common neighbours in $W_{i}\setminus Q_{i}$, 4. 4. for every set $Y_{i}\subset W_{i}$ of size $\beta m$, all but at most $\alpha\eta m$ edges of $M_{jk}$ belong to the neighbourhood of some vertex of $Y_{i}$. Fix any such $M_{12}$, $M_{13}$, and $M_{23}$. Next, for each $i$, let $X_{i}$ be a random binomial subset of $W_{i}\setminus Q_{i}$, where each element is included with probability $\eta/5$. A simple application of Chernoff’s inequality combined with Property 3 above shows that if $C$ is sufficiently large, then with probability $1-o(1)$, for all $i$, $j$, and $k$: 1. 5. $\|X_{i}\|\leq(\eta/4)m$, 2. 6. the endpoints of each edge of $M_{jk}$ have at least $(\eta/12)\delta^{2}p^{2}m$ common neighbours in $X_{i}$. Let $X_{1}$, $X_{2}$, and $X_{3}$ be arbitrary sets satisfying 5 and 6 and note that properties (P1)–(P3) are satisfied. It remains to show that (P4) is also satisfied. Fix a $Z_{i}\subset W_{i}$ of size $\|M_{jk}\|$ such that $X_{i}\subset Z_{i}$. Let $H$ be the bipartite graph on the vertex set $Z_{i}\cup M_{jk}$, where a vertex $w\in Z_{i}$ is adjacent to an edge $\\{u,v\\}\in M_{jk}$ if and only if $\\{u,v,w\\}$ is a triangle in $(W_{1},W_{2},W_{3})$. Clearly, it suffices to prove that $H$ contains a perfect matching. We check that $H$ satisfies the assumptions of Proposition 2.15 with $A=Z_{i}$, $B=M_{jk}$, and $L=\alpha\eta m$. Fix an $S\subset Z_{i}$. If $0<\|S\|\leq(\eta/4)\delta^{2}m$, then $\|N_{H}(S)\|>\|S\|$ or otherwise there would be an $x\in[1,(\eta/4)\delta^{2}m]$ such that $G(n,p)$ contains some $x$ independent edges and $x$ vertices, each of which is adjacent to both ends of at least $(\eta/2)\delta^{2}p^{2}m$ of those edges, see 2. This would contradict our assumption that $G(n,p)$ satisfies the assertion of Proposition 4.3 with $\xi=\eta\delta^{2}m/(2n)$. On the other hand, if $\|S\|\geq(\eta/4)\delta^{2}m\geq\beta m$, then by 4, $\|M_{jk}\setminus N_{H}(S)\|\leq\alpha\eta m$. Hence, $\|N_{H}(S)\|\geq\|S\|$ as long as $\|Z_{i}\setminus S\|\geq\alpha\eta m$ Finally, fix a $T\subset M_{jk}$ with $0<\|T\|\leq\alpha\eta m=(\eta/24)\delta^{2}m$. If $\|N_{H}(T)\|<\|T\|$, then there would be an $x\in[1,(\eta/24)\delta^{2}m]$ such that $G(n,p)$ contains $x$ vertices and $x$ independent edges whose endpoints have at least $(\eta/12)\delta^{2}p^{2}m$ common neighbours among those $x$ vertices (recall that $X_{i}\subset Z_{i}$), see 6. This would contradict our assumption that $G(n,p)$ satisfies the assertion of Proposition 4.3 with $\xi=\eta\delta^{2}m/(12n)$. ∎ With Lemma 7.1 at hand, without much effort we can prove Theorem 1.4 which says that a balanced strong-super-regular triple has a triangle-factor. ###### Proof of Theorem 1.4.. Let $\eta=1/140$, $\varepsilon_{1}=\varepsilon_{\ref{prop_findtriangle}}(3\eta/2,\delta/2)$, and $\varepsilon=(\eta/4)\min\\{\varepsilon_{\ref{lemma_main}}(\delta,\eta),\varepsilon_{1}\\}$. Let $C=C_{\ref{lemma_main}}(\delta,\eta,\xi)$. Let $(W_{1},W_{2},W_{3})$ be a $(\delta,\varepsilon,p)$-strong-super-regular triple and $m=\|W_{1}\|=\|W_{2}\|=\|W_{3}\|$. By Lemma 7.1, there exists a matching $M_{12},M_{13},M_{23}$ and sets $Q_{1},Q_{2},Q_{3}$ and $X_{1},X_{2},X_{3}$ satisfying (P1), (P2), (P3), and (P4). Let $W_{i}^{\prime}=W_{i}\setminus(Q_{i}\cup X_{i})$ and note that $\|W_{1}^{\prime}\|=\|W_{1}\|-\|Q_{1}\|-\|X_{1}\|=\|W_{1}\|-\|M_{12}\|-\|M_{13}\|-\|X_{1}\|.$ Let $x=m-\|M_{12}\|-\|M_{13}\|-\|M_{23}\|$ and note that $(1-6\eta)m<x<(1-3\eta/2)m$. By applying Proposition 2.7, we can find $x$ vertex-disjoint triangles inside the triple $(W_{1}^{\prime},W_{2}^{\prime},W_{3}^{\prime})$. Note that the remaining $\|W_{1}^{\prime}\|-x=\|M_{23}\|-\|X_{1}\|$ vertices in $W_{1}^{\prime}$ together with the set $X_{1}$, can be matched with the set $M_{23}$ to construct $\|M_{23}\|$ vertex-disjoint triangles, by property (P4). Similarly, the remaining vertices in $W_{2}^{\prime}\cup X_{2}$ and $W_{3}^{\prime}\cup X_{3}$ can be matched with $M_{13}$ and $M_{12}$, respectively. Therefore, we have found a perfect triangle packing of $(W_{1},W_{2},W_{3})$. ∎ Finally, we briefly summarize Sections 5, 6, and 7 in the proof of our main result, Theorem 1.3. ###### Proof of Theorem 1.3. Let $\delta=\delta_{\ref{thm_decomposition}}(\gamma)$, $\varepsilon=\varepsilon_{\ref{thm_triangleblowup}}(\delta)$, $\xi=\xi_{\ref{thm_decomposition}}(\delta,\varepsilon)$ and $C=\max\\{C_{\ref{thm_decomposition}}(\varepsilon),C_{\ref{thm_triangleblowup}}(\varepsilon,\xi)\\},D=D_{\ref{thm_decomposition}}(\varepsilon)$. By Theorem 6.6, there exist set $B$, $S$, and $(W_{i})_{i=1}^{3k}$ which satisfies (i) - (iv) of Theorem 6.6. Furthermore for each $t\in[k]$, by Theorem 1.4, each $(\delta,\varepsilon,p)$-strong-super-regular triple $(W_{3t-2},W_{3t-1},W_{3t})$ contains a perfect triangle packing. Therefore all the vertices except $B$ can be covered by vertex-disjoint triangles. Since $\|B\|\leq Dp^{-2}$, this completes the proof. ∎ ## 8 Concluding Remarks An immediate question we would like to ask is whether the assumption on $p$ in Theorem 1.3 can be relaxed. Even though our argument breaks down (for a few reasons) if $p\ll(\log n/n)^{1/2}$, we believe that the conclusion of Theorem 1.3 still holds under the (weaker) assumption that $p\gg n^{-1/2}$. If this was true, it would completely resolve the problem of determining the local resilience of $G(n,p)$ with respect to the property of containing an almost spanning triangle packing. We also believe that a similar argument can be used to obtain an extension of the theorem of Hajnal and Szemerédi [15] for larger cliques to the setting of sparse random graphs. Clearly, the edge probability $p$ would have to be sufficiently large so that a corresponding form of Lemma 2.6 holds. However, in our opinion, the importance of such a result does not justify the technical complications one would have to face in order to prove it. The more intriguing and interesting question comes from the attempt to embed general spanning or almost spanning graphs (by general we mean graphs that are not disjoint unions of a fixed graph) into sparse regular pairs. This gives rise to the following question. ###### Question. Can we develop an embedding lemma for general graphs into regular pairs in random graphs for some $p=n^{-o(1)}$? How should the definition of strong- super-regularity be extended? It is quite likely that such an embedding lemma will provide another proof of the theorem of Böttcher, Kohayakawa, and Taraz [7] on embedding almost spanning subgraphs. However, one can hope for a better result where the graph we want to embed is smaller than the host graph by a sublinear number of vertices. To achieve this, one will most likely need to develop a tool similar to that of Theorem 6.6. Another question can be asked regarding embedding of spanning subgraphs. Proposition 4.6 shows that as many as $\Omega(p^{-2})$ vertices have to be left out from the largest triangle packing. More generally, if every vertex of some graph $H$ is contained in a triangle, then we cannot hope to embed $H$ into a sparse host graph of the same order. However, this is no longer the case when $H$ is bipartite. Thus we recall the following question posed by Böttcher, Kohayakawa, and Taraz [8]. ###### Question. Is it possible to have a perfect embedding for bipartite graphs? In fact, it might be true that what actually matters is not that the graph is bipartite, but the fact that there are enough vertices which are not contained in a copy of a triangle. See [17], where such a result is proved for dense graphs. Acknowledgements. We are indebted to the anonymous referee for their extremely careful reading of the paper and many helpful comments and suggestions. The bulk of this work was done when the second and the third authors were visiting the first author in the Department of Mathematics at the University of California, San Diego. ## References * [1] N. Alon and J. Spencer, _The probabilistic method_ , third ed., John Wiley & Sons Inc., Hoboken, NJ, 2008. * [2] N. Alon and R. Yuster, _Almost $H$-factors in dense graphs_, Graphs and Combinatorics 8 (1992), 95–102. * [3] , _Threshold functions for $H$-factors_, Combinatorics, Probability and Computing 2 (1993), 137–144. * [4] , _$H$ -factors in dense graphs_, Journal of Combinatorial Theory. Series B 66 (1996), 269–282. * [5] J. Balogh, B. Csaba, and W. Samotij, _Local resilience of almost spanning trees in random graphs_ , Random Structures & Algorithms 38 (2011), 121–139. * [6] S. Ben-Shimon, M. Krivelevich, and B. 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Krivelevich, _Triangle factors in random graphs_ , Combinatorics, Probability and Computing 6 (1997), 337–347. * [27] M. Krivelevich, C. Lee, and B. Sudakov, _Resilient pancyclicity of random and pseudorandom graphs_ , SIAM Journal on Discrete Mathematics 24 (2010), 1–16. * [28] M. Krivelevich, B. Sudakov, and T. Szabó, _Triangle factors in sparse pseudo-random graphs_ , Combinatorica 24 (2004), 403–426. * [29] D. Kühn and D. Osthus, _Embedding large subgraphs into dense graphs_ , Surveys in combinatorics 2009, London Math. Soc. Lecture Note Ser., vol. 365, Cambridge Univ. Press, Cambridge, 2009, pp. 137–167. * [30] , _The minimum degree threshold for perfect graph packings_ , Combinatorica 29 (2009), 65–107. * [31] C. Lee and B. Sudakov, _Dirac’s theorem for random graphs_ , arXiv:1108.2502v2 [math.CO]. * [32] L. Pósa, _Hamiltonian circuits in random graphs_ , Discrete Mathematics 14 (1976), 359–364. * [33] A. Ruciński, _Matching and covering the vertices of a random graph by copies of a given graph_ , Discrete Mathematics 105 (1992), 185–197. * [34] B. Sudakov and V. H. Vu, _Local resilience of graphs_ , Random Structures & Algorithms 33 (2008), 409–433. * [35] E. Szemerédi, _Regular partitions of graphs_ , Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), Colloq. Internat. CNRS, vol. 260, CNRS, Paris, 1978, pp. 399–401.	arxiv-papers	2010-11-24T17:52:10	2024-09-04T02:49:15.212529	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "J\\'ozsef Balogh, Choongbum Lee, and Wojciech Samotij", "submitter": "Choongbum Lee", "url": "https://arxiv.org/abs/1011.5443" }
1011.5534		arxiv-papers	2010-11-25T00:14:19	2024-09-04T02:49:15.230095	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Dhananjoy Dey, Prasanna Raghaw Mishra1, Indranath Sengupta", "submitter": "Dhananjoy Dey", "url": "https://arxiv.org/abs/1011.5534" }
1011.5553	# On affine rigidity Steven J. Gortler , Craig Gotsman , Ligang Liu and Dylan P. Thurston ###### Abstract. We study the properties of affine rigidity of a hypergraph and prove a variety of fundamental results. First, we show that affine rigidity is a generic property (i.e., depends only on the hypergraph, not the particular embedding). Then we prove that a graph is generically neighborhood affinely rigid in $d$-dimensional space if it is $(d+1)$-vertex-connected. We also show neighborhood affine rigidity of a graph implies universal rigidity of its squared graph. Our results, and affine rigidity more generally, have natural applications in point registration and localization, as well as connections to manifold learning. SJG and CG were partially supported by United States–Israel Binational Science Foundation grant 2006089. LL was partially supported by the National Natural Science Foundation of China (61070071). DPT was supported by the Mathematical Sciences Research Institute and a Sloan Research Fellowship. ## 1\. Introduction Suppose one has a number of overlapping “scans” of a set of points in some space, and that the corresponding points shared between scans have been identified. One naturally may want to register these scans and merge them together into a single configuration [23, inter alia]. Such a merging problem is called a _realization problem_. The study of the uniqueness of the solutions to such realization problems is known as _rigidity_. We model the combinatorics of this problem using a hypergraph $\Theta$, with vertices representing the points, and hyperedges representing the sets of points in each scan. The geometry of the problem is modeled with a configuration $p$, associating each vertex with a point in space. One natural setting is the Euclidean setting, where the scans are known to be related by Euclidean transforms. In this case it is sufficient to study just the case of a graph, where we think of each edge as its own scan with only $2$ points. Unfortunately, many of the Euclidean problems are NP-HARD [18]. In this paper, we study what happens when one relaxes the problem to the affine setting, that is, one assumes that the scans are known to be related by affine transforms. Under this relaxation, much of the analysis reduces to linear algebra, and uniqueness questions reduce to rank calculations. We prove a variety of fundamental results about this type of rigidity and also place it in the context of other rigidity classes such as global rigidity and universal rigidity. We also specifically investigate the case of hypergraphs $\Theta$ that arises by starting with an input graph $\Gamma$, and considering each one-ring in $\Gamma$ as a hyperedge in $\Theta$. We call such a hypergraph the neighborhood hypergraph of $\Gamma$. Such neighborhood hypergraphs naturally arise when studying molecules [13], when applying a divide and conquer approach to sensor network localization [20] and in machine learning [16]. ### 1.1. Summary of Results We start by describing how affine rigidity in $\mathbb{R}^{d}$ is fully characterized by the kernel size of one of its associated “affinity matrices”. (This result was first shown by Zha and Zhang [26].) We show how this implies a number of interesting corollaries including the fact that affine rigidity is a generic property. That is, given a hypergraph $\Theta$ and dimension $d$, either all generic embeddings of $\Theta$ are affinely rigid in $\mathbb{R}^{d}$ or all generic embeddings are affinely flexible in $\mathbb{R}^{d}$. The specific geometric positions of the vertices are irrelevant to this property, as long as they are in sufficiently general position. Thus we call such a hypergraph _generically affinely rigid_ in $\mathbb{R}^{d}$. Next we relate affine rigidity in $\mathbb{R}^{d}$ to the related notion of universal Euclidean rigidity. A framework is universally Euclideanly rigid if it is rigid (in the Euclidean sense) in _any_ dimension. In this context, we prove that affine rigidity in $\mathbb{R}^{d}$ implies universal Euclidean rigidity. We then prove the following sufficiency result: if a graph $\Gamma$ is $d+1$ (vertex) connected, then its neighborhood hypergraph is generically affinely rigid in $\mathbb{R}^{d}$; alternatively, we say that the graph $\Gamma$ itself is generically neighborhood affinely rigid in $\mathbb{R}^{d}$. In particular we will show that almost every _non-symmetric equilibrium stress matrix_ for any generic embedding of $\Gamma$ in $\mathbb{R}^{d}$ will have co-rank $d+1$ (i.e., rank $v-d-1$). Putting these two results together, we show that if a graph is $d+1$ connected, then any generic embedding of its square graph in $\mathbb{R}^{d}$ is universally rigid. This result is interesting, as very few families of graphs have been proven to be generically universally rigid. We give examples showing that many of the implications proved in this paper do not reverse. Finally we discuss some of the motivating applications. The main properties of frameworks of graphs and their implications proven in this paper are summarized below. GGRDP1CGNSESMGNARGNURGNGR Property Graph … GGR is generically globally rigid in $\mathbb{R}^{d}$ DP1C is $d+1$ connected ([10]) GNSESM generically has non-symmetric equilibrium stress matrix of rank $v-d-1$ (Proposition 5.8) GNAR is generically neighborhood affine rigid in $\mathbb{R}^{d}$ (Proposition 5.9) GNUR is generically neighborhood universally rigid in $\mathbb{R}^{d}$ (Corollary 4.2) GNGR is generically neighborhood globally rigid in $\mathbb{R}^{d}$ (by definition) ## 2\. A rigidity zoo In this paper we will consider several different rigidity theories. They all fit in to a unifying framework, which we now explain. Most generally, rigidity questions (of any type) ask if all of the geometric information about a set of points is determined by information from small subsets. In the usual Euclidean rigidity problem, we measure the distances between pairs of points. However, in other cases it is not enough to consider pairs of points for the small subsets; as such, we need to consider hypergraphs rather than just graphs. ###### Definition 2.1. A _hypergraph_ $\Theta$ is a set of $v$ vertices ${\mathcal{V}}(\Theta)$ and $h$ hyperedges ${\mathcal{E}}(\Theta)$, where ${\mathcal{E}}(\Theta)$ is a set of subsets of ${\mathcal{V}}(\Theta)$. We will typically write just ${\mathcal{V}}$ or ${\mathcal{E}}$, dropping the hypergraph $\Theta$ from the notation. There are natural ways to pass from a hypergraph back and forth to a graph. ###### Definition 2.2. Given a hypergraph $\Theta$, define its _body graph_ $B(\Theta)$ as follows. For each vertex in $\Theta$, we have a vertex in $B(\Theta)$. For each hyperedge $h$ in $\Theta$ and each pair of vertices in $h$ we have an edge in $B(\Theta)$. See Figure 1 for an example. Figure 1. Left: A hypergraph with 6 vertices and 4 hyperedges. Each hyperedge is represented by a dotted ellipse enclosing a set of vertices. The hyperedges are: red $\\{1,2,6\\}$, green $\\{2,3,5\\}$, purple $\\{5,6\\}$, blue $\\{4,5\\}$. Right: The body graph of the hypergraph shown in the left. ###### Definition 2.3. Given a graph $\Gamma$, define its _neighborhood hypergraph_ , written as $N(\Gamma)$ as follows. For each vertex in $\Gamma$, we have an associated vertex in $N(\Gamma)$. For each vertex in $\Gamma$ we have a hyperedge in $N(\Gamma)$ consisting of that vertex and its neighbors in $\Gamma$. ###### Definition 2.4. Given a graph $\Gamma$, its _squared graph_ $\Gamma^{2}$ is obtained by adding to $\Gamma$ an edge between two vertices $i$ and $j$ if $i$ and $j$ share some neighbor vertex $k$. ###### Lemma 2.5. For any graph $\Gamma$, $B(N(\Gamma))=\Gamma^{2}$. ###### Proof. Immediate from the definitions. (See Figure 2 for an example.) ∎ Figure 2. Left: A graph with 6 vertices and 7 edges. Middle: Its neighborhood hypergraph with 6 vertices and 6 hyperedges: red $\\{1,2,6\\}$, purple $\\{1,2,3,6\\}$, light blue $\\{2,3,5\\}$, dark blue $\\{4,5\\}$, orange $\\{3,4,5,6\\}$, green $\\{1,2,5,6\\}$. Right: The body graph of the hypergraph in the middle. It is also the squared graph of the graph in the left. ###### Definition 2.6. A _$k$ -hypergraph_ $\Theta$ is a hypergraph where each hyperedge has exactly $k$ vertices. For any $k\in\mathbb{N}$ and hypergraph $\Theta$, let $B_{k}(\Theta)$ be the $k$-hypergraph whose hyperedges are all the subsets $S$ of size $k$ of vertices that are contained together in at least one hyperedge of $\Theta$: ${\mathcal{E}}(B_{k}(\Theta))=\\{\,S\mid\exists h\in{\mathcal{E}}(\Theta),S\subset h,{\lvert S\rvert}=k\,\\}.$ For a vertex set $S$, the _complete $k$-hypergraph on $S$_, written $K_{k}(S)$, is the $k$-hypergraph whose hyperedges are all $\binom{{\lvert S\rvert}}{k}$ subsets of $S$ of size $k$. For instance, a $2$-hypergraph is a graph, and $B_{2}(\Theta)$ is just the body graph $B(\Theta)$. ###### Definition 2.7. A _configuration_ of the vertices ${\mathcal{V}}(\Theta)$ of a hypergraph in $\mathbb{R}^{d}$ is a mapping $p$ from ${\mathcal{V}}(\Theta)$ to $\mathbb{R}^{d}$. Let $C^{d}({\mathcal{V}})$ be the space of configurations in $\mathbb{R}^{d}$. For $p\in C^{d}({\mathcal{V}})$ and $u\in{\mathcal{V}}(\Theta)$, we write $p(u)\in\mathbb{R}^{d}$ for the image of $u$ under $p$. A _framework_ $\rho=(p,\Theta)$ of a hypergraph is the pair of a hypergraph and a configuration of its vertices. $C^{d}(\Theta)$ is the space of frameworks $(p,\Theta)$ with hypergraph $\Theta$ and $p\in C^{d}({\mathcal{V}}(\Theta))$. We may also write $\rho(u)$ for $p(u)$ where $\rho=(p,\Theta)$ is a framework of the configuration $p$. A framework of a hypergraph has also been called a body-and-multipin framework [13]. ###### Definition 2.8. Let $G$ be a group acting on $\mathbb{R}^{d}$, for instance the group $\operatorname{Eucl}(d)$ of all Euclidean isometries of $\mathbb{R}^{d}$. Two frameworks $\rho,\sigma\in C^{d}(\Theta)$ are _$G$ -equivalent_ if for each hyperedge $h\in{\mathcal{E}}(\Theta)$, the positions of the vertices in $\rho$ can be mapped to their positions in $\sigma$ by an element of $G$ depending only on $h$. That is, for each $h\in{\mathcal{E}}$, there is a $g_{h}\in G$ so that for each $u\in h$, we have $g_{h}(\rho(u))=\sigma(u)$. The two are _$G$ -congruent_ if the positions of all the vertices in $\rho$ can be mapped to their positions in $\sigma$ by a single element $g\in G$ (not depending on $h$). A framework $\rho\in C^{d}(\Theta)$ is _globally $G$-rigid_ if for any other framework $\sigma\in C^{d}(\Theta)$ that is $G$-equivalent to $\rho$, we also have that $\rho$ and $\sigma$ are $G$-congruent. Otherwise we say that $\rho$ is globally $G$-flexible in $\mathbb{R}^{d}$. Similarly, a framework $\rho\in C^{d}(\Theta)$ is _locally $G$-rigid_ in $\mathbb{R}^{d}$ if there is a small neighborhood $U$ of $\rho$ in $C^{d}(\Theta)$ so that for any $\sigma\in U$ that is $G$-equivalent to $\rho$, we also have that $\rho$ and $\sigma$ are $G$-congruent. Otherwise we say that $\rho$ is locally $G$-flexible in $\mathbb{R}^{d}$. A related notion of group based rigidity has been explored in the computer aided design literature [19]. In this paper, we are mainly concerned with the cases when $G$ is either $\operatorname{Eucl}(d)$ or $\operatorname{Aff}(d)$, the group of all invertible affine linear maps of $\mathbb{R}^{d}$, in which case we speak about _Euclidean_ or _affine_ rigidity, respectively. But there are other interesting possibilities, like the group of projective transformations. Another interesting case is when $G$ is the group of dilations and translations of $\mathbb{R}^{d}$ (with no rotations); this gives the theory of parallel-line redrawings [25]. In this terminology, Euclidean rigidity is the default: if the group $G$ is not specified, it is the Euclidean group. In much of the rigidity literature, local rigidity is the default, and the qualifier “local” is dropped. However, in this paper this distinction is important and we will write “local” or “global” when the distinction is meaningful. ###### Lemma 2.9. A framework $(p,\Theta)$ is locally (resp. globally) Euclideanly rigid iff the body framework $(p,B(\Theta))$ is locally (resp. globally) Euclideanly rigid. ###### Proof. This easily follows from the fact that, for each hyperedge $h\in{\mathcal{E}}(\Theta)$, the complete graph on ${\lvert h\rvert}$ vertices is globally rigid. ∎ Thus we only need to consider Euclidean rigidity for frameworks of graphs, not hypergraphs. In the next section (Corollary 3.5) we will see that a framework is locally affinely rigid iff it is globally affinely rigid. Thus we can drop the local/global distinction for affine rigidity. ###### Definition 2.10. For $d<d^{\prime}$, we view $C^{d}({\mathcal{V}})$ as contained in $C^{d^{\prime}}({\mathcal{V}})$ by the inclusion of $\mathbb{R}^{d}$ as the first $d$ dimensions of $\mathbb{R}^{d^{\prime}}$. Let $G$ be a family of groups $G_{d}$ acting on $\mathbb{R}^{d}$, so that for $d<d^{\prime}$, $G_{d}$ is the subgroup of $G_{d^{\prime}}$ that fixes $\mathbb{R}^{d}$ as a subset of $\mathbb{R}^{d^{\prime}}$. A framework $\rho\in C^{d}(\Theta)$ is _universally_ locally (resp. globally) $G$-rigid if it is locally (resp. globally) $G$-rigid as a framework in $C^{d^{\prime}}(\Theta)$ for all $d^{\prime}\geq d$. Note that universal rigidity of any sort implies rigidity of the same sort. ###### Lemma 2.11. A framework $p\in C^{d}(\Theta)$ is universally globally Euclideanly rigid iff it is universally locally Euclideanly rigid. ###### Proof. For any two equivalent frameworks $\rho$ in $C^{d}(\Theta)$ and $\rho^{\prime}$ in $C^{d^{\prime}}(\Theta)$, Bezdek and Connelly [3] exhibited an explicit flex between $\rho$ and $\rho^{\prime}$ in $C^{d+d^{\prime}}(\Theta)$. In particular, if $\rho$ is a $d$-dimensional framework with a equivalent but non-congruent framework in $d^{\prime}$ dimensions, then this flex shows that $\rho$ is not locally rigid in $\mathbb{R}^{d+d^{\prime}}$. ∎ Thus we can also drop the local/global distinction in the case of universal Euclidean rigidity. ###### Definition 2.12. A framework $(p,\Gamma)$ of the graph $\Gamma$ is _neighborhood_ rigid (of any of the sorts above) if the corresponding framework $(p,N(\Gamma))$ of the neighborhood hypergraph is rigid (of the same sort). For instance, Lemmas 2.9 and 2.5 tell us that neighborhood Euclidean rigidity of $(\rho,\Gamma)$ is equivalent to the Euclidean rigidity of $(\rho,\Gamma^{2})$. ###### Definition 2.13. A configuration $p$ in $C^{d}({\mathcal{V}})$ is _generic_ if the coordinates do not satisfy any non-zero algebraic equation with rational coefficients. A framework is generic if its configuration is generic. A property is _generic_ in $\mathbb{R}^{d}$ if, for every (hyper)graph, either all generic frameworks in $C^{d}(\Theta)$ have the property or none do. For instance, local and global Euclidean rigidity in $\mathbb{R}^{d}$ are both generic properties of graphs and therefore for hypergraphs as well [2, 8]. For any property $P$ (generic or not) of frameworks, a (hyper)graph $\Theta$ is _generically $P$_ in $\mathbb{R}^{d}$ if every generic framework in $C^{d}(\Theta)$ has $P$. (For a non-generic property like universal Euclidean rigidity, there are (hyper)graphs that are neither generically $P$ or generically not $P$.) The above adjectives are to be read in the reverse of the order we have defined them. Thus, for any framework, we may talk about > (generic/$\emptyset$) (universal/$\emptyset$) (local/global) > (Euclidean/affine) rigidity. where by $\emptyset$ we mean that this term has been dropped. ## 3\. Affine Rigidity in $\mathbb{R}^{d}$ We now move on the main focus of this work, affine rigidity, as defined in the previous section. Though the definitions start from a different point of view, this notion of affine rigidity is identical to the one defined by Zha and Zhang [26] and the concept is also informally mentioned by Brand [4]. Additionally, Theorem 1 below is essentially equivalent to [26, Theorem 5.2]. Our contribution here, described by the corollaries, is showing how affine rigidity fits in to the general scheme of rigidity problems. ###### Lemma 3.1. Any general position framework of a complete $(d+2)$-hypergraph is affinely rigid. ###### Proof. Clear. ∎ ###### Proposition 3.2. A framework $(p,\Theta)$ in general position is affinely locally (resp. globally) rigid iff the associated framework $(p,B_{d+2}(\Theta))$ is affinely locally (resp. globally) rigid. (Compare Lemma 2.9.) ###### Proof. First consider a hyperedge with less than $d+2$ vertices. Any two generic embeddings in $\mathbb{R}^{d}$ of these vertices are related by a $d$-dimensional affine transform. Therefore this hyperedge does not affect affine equivalence of these vertices, and so this hyperedge may be dropped without affecting affine rigidity. Next consider a hyperedge with $k$ vertices, with $k>d+2$. By Lemma 3.1, one can replace this hyperedge with $\binom{k}{d+2}$ hyperedges corresponding to all subsets of $d+2$ vertices. The framework of the new hypergraph will be affinely rigid in $\mathbb{R}^{d}$ iff the original one is. ∎ ###### Definition 3.3. An _affinity matrix_ of a framework $(p,\Theta)$ in $C^{d}(\Theta)$ is any matrix where each row encodes some affine relation between the coordinates of the vertices in a hyperedge of $(p,\Theta)$ as a homogeneous linear equation. That is, there are $v$ columns in the matrix, the only non-zero entries in a row correspond to vertices in some hyperedge, the sum of the entries in a row is $0$, and each of the coordinates of $p$, thought of as a vector of length $v$, is in the kernel of the matrix. An affinity matrix is _strong_ if it encodes all of the affinely independent relations in every hyperedge of $(p,\Theta)$. ###### Lemma 3.4. If the framework $(q,\Theta)$ is affinely equivalent to $(p,\Theta)$ then the coordinates of $q$ are in the kernel of any affinity matrix for $(p,\Theta)$. Additionally, the converse is true if the affinity matrix is strong. ###### Proof. Clear from the definitions. ∎ The kernel of an affinity matrix of a framework $(p,\Theta)\in C^{d}(\Theta)$ always contains the subspace of $\mathbb{R}^{v}$ spanned by the coordinates of $p$ along each axis and the vector $\vec{1}$ of all $1$’s. This corresponds to the fact that any affine image of $p$ is equivalent to $p$. If $p$ is a proper $d$-dimensional configuration (with full $d$-dimensional affine span), these vectors are independent and span a $(d+1)$-dimensional space. In particular, a generic framework of a hypergraph with at least $d+1$ vertices in $\mathbb{R}^{d}$ is proper, so for such frameworks the corank of any of its affinity matrices must be no less than $d+1$. The rank of strong affinity matrices fully characterize affine rigidity. ###### Theorem 1. Let $\Theta$ be a hypergraph with at least $d+1$ vertices. Let $(p,\Theta)$ be any proper, $d$-dimensional framework and let $M$ be any strong affinity matrix for $(p,\Theta)$. Then $(p,\Theta)$ is affinely rigid in $\mathbb{R}^{d}$ iff $\dim(\ker(M))=d+1$. ###### Proof. By Lemma 3.4, for any other configuration $q$ in $C^{d}({\mathcal{V}})$ such that $(q,\Theta)$ is affinely equivalent to $(p,\Theta)$, the coordinates of $q$ must be in the kernel of $M$. When $\dim(\ker(M))=d+1$, the kernel of $M$ contains only one-dimensional projections of $p$ and the all-ones vector. Thus any $(q,\Theta)$ that is affinely equivalent to $(p,\Theta)$ must in fact be affinely congruent to $(p,\Theta)$. Conversely, if the corank is higher, then the kernel must contain an “extra” vector that is not a one-dimensional projection of $p$. Adding any amount of this vector to one of the coordinates of $p$ must, by Lemma 3.4, produce a $q$ such that $(q,\Theta)$ is affinely equivalent to $(p,\Theta)$ but not congruent to it. ∎ It is easy now to prove the following corollaries. ###### Corollary 3.5. If $(p,\Theta)$ is affinely globally flexible in $\mathbb{R}^{d}$ then it is affinely locally flexible. ###### Proof. From the proof of Theorem 1, when $(p,\Theta)$ is affinely globally flexible in $\mathbb{R}^{d}$ there is an extra vector $\delta$ in the kernel of a strong affinity matrix, and we can add any multiple of $\delta$ to one of the coordinates of $p$ to get an affinely equivalent but not congruent framework. ∎ ###### Remark 3.6. In fact, for any two affinely equivalent frameworks $(p,\Theta)$ and $(q,\Theta)$, there is a continuous path of affinely equivalent frameworks in $C^{d}(\Theta)$ connecting them, namely $((1-t)p+tq,\Theta)$. ###### Corollary 3.7. A framework $(p,\Theta)\in C^{d}(\Theta)$ is affinely rigid in $\mathbb{R}^{d}$ iff it is affinely rigid when considered as a (degenerate) framework in $\mathbb{R}^{d^{\prime}}$ for $d^{\prime}\geq d$. ###### Proof. Follows from the proof of Theorem 1. ∎ Thus there is no distinct notion of “universal” affine rigidity. ###### Corollary 3.8. Affine rigidity in $\mathbb{R}^{d}$ is a generic property of a hypergraph. ###### Proof. The condition that $M$ is an affinity matrix for $(p,\Theta)$ is linear in the entries in $M$. The corollary then follows from Proposition A.1. ∎ ###### Remark 3.9. Though we will not pursue the details here, one can use the concept of an affinity matrix to derive an efficient randomized discrete algorithm for testing generic affine rigidity of a hypergraph in $\mathbb{R}^{d}$. To do this, one needs to use integers of sufficiently many bits, and do the arithmetic modulo a suitably large prime. The details parallel those in the global rigidity case [8, Section 5]. ## 4\. Universal Euclidean Rigidity We now turn to universal Euclidean rigidity. To begin, we need the following technical definition: ###### Definition 4.1. We say that the edge directions of a graph framework $(p,\Gamma)\in C^{d}(\Gamma)$ are _on a conic at infinity_ if there exists a symmetric $d$-by-$d$ matrix $Q$ such that for all edges $(u,v)$ of $\Gamma$, we have $[p(u)-p(v)]^{t}Q[p(u)-p(v)]=0.$ The edge directions of $(p,\Gamma)$ are on a conic at infinity iff there is a continuous family of $d$-dimensional non-Euclidean affine transforms that preserve all of the edge lengths [6]. This is a very degenerate situation which is very easy to rule out. For example, if in a hypergraph framework $(p,\Theta)$ some hyperedge in $\Theta$ has vertices whose positions in $p$ affinely span $\mathbb{R}^{d}$, then the edge directions $(p,B(\Theta))$ cannot be on a conic at infinity. ###### Theorem 2. If a framework $(p,\Theta)$ of a hypergraph $\Theta$ with $p\in C^{d}({\mathcal{V}})$ is affinely rigid in $\mathbb{R}^{d}$, and the edge directions of $(p,B(\Theta))$ are not on a conic at infinity, then $(p,\Theta)$ is universally Euclidean rigid. ###### Proof. Let $q\in C^{d^{\prime}}({\mathcal{V}})$ be a configuration with $d^{\prime}>d$ such that $(q,\Theta)$ is Euclidean equivalent in $\mathbb{R}^{d^{\prime}}$ to $(p,\Theta)$. Then $(q,\Theta)$ is affinely equivalent in $\mathbb{R}^{d^{\prime}}$ to $(p,\Theta)$. Since $(p,\Theta)$ is affinely rigid in $\mathbb{R}^{d}$, from Corollary 3.7, we have that $p$ is affinely congruent to $q$ in $\mathbb{R}^{d^{\prime}}$ and the affine span of $q$ must be of the same dimension as that of $p$. Let $R(q)$ be a rotation of $q$ down to $\mathbb{R}^{d}$. Then $R(q)$ is affine congruent in $\mathbb{R}^{d}$ to $p$ and $(R(q),\Theta)$ is Euclidean equivalent in $\mathbb{R}^{d}$ to $(p,\Theta)$. Since the edge directions of $(p,B(\Theta))$ are not on a conic at infinity, any $d$-dimensional transform that is affine but not Euclidean must change some of the inter-vertex lengths within some hyperedge of $\Theta$. Putting this together with the fact that $(R(q),\theta)$ and $(p,\Theta)$ are affinely congruent and Euclidean equivalent, we can conclude that they are Euclidean congruent. Likewise $q$ must be Euclidean congruent to $p$ in $\mathbb{R}^{d^{\prime}}$. Thus we conclude that $(p,\Theta)$ is universally rigid. ∎ ###### Corollary 4.2. Let $\Theta$ be a hypergraph with at least $d+2$ vertices. If a generic framework $(p,\Theta)$ of a hypergraph $\Theta$ with $p\in C^{d}({\mathcal{V}})$ is affinely rigid in $\mathbb{R}^{d}$ then $(p,\Theta)$ is universally rigid. ###### Proof. Any generic framework of a hypergraph $\Theta$ with at least $d+2$ vertices that is affinely rigid in $\mathbb{R}^{d}$ must have at least one hyperedge $h$ with at least $d+2$ vertices, whose vertices in $p$ have a $d$-dimensional affine span. Thus $(p,B(\Theta))$ must include a generic framework of a $(d+1)$-simplex and thus cannot have edge directions at a conic at infinity. Then Theorem 2 applies. ∎ There can be frameworks that are universally rigid but not affinely rigid in $\mathbb{R}^{d}$. (See Figure 3.) Figure 3. The framework in $\mathbb{R}^{2}$ of the hypergraph on the left is not affinely rigid as each hyperedge (shown as a dashed ellipse) has only 3 vertices. But this framework is universally Euclidean rigid, as its body graph (right) is a Cauchy polygon. ## 5\. Neighborhood affine rigidity In this section we prove the following theorem about the generic neighborhood affine rigidity of a graph. ###### Theorem 3. Let $\Gamma$ be a graph with at least $d+1$ vertices. If $\Gamma$ is $(d+1)$-vertex-connected, then $\Gamma$ is generically neighborhood affinely rigid in $\mathbb{R}^{d}$. We will build up the proof of Theorem 3 with a series of lemmas. ###### Definition 5.1. An _equilibrium stress matrix_ of a framework $(p,\Gamma)$ of a graph in $C^{d}(\Gamma)$ is a matrix $\Omega$ indexed by ${\mathcal{V}}\times{\mathcal{V}}$ so that 1. (1) for all $u,w\in{\mathcal{V}}$, we have $\Omega(u,w)=\Omega(w,u)$; 2. (2) for all $u,w\in{\mathcal{V}}$ with $u\neq w$ and $\\{u,w\\}\not\in{\mathcal{E}}$, we have $\Omega(u,w)=0$; 3. (3) for all $u\in{\mathcal{V}}$, we have $\sum_{w\in{\mathcal{V}}}\Omega(u,w)=0$; and 4. (4) for all $u\in{\mathcal{V}}$, we have $\sum_{w\in{\mathcal{V}}}\Omega(u,w)p(w)=0$. A _non-symmetric equilibrium stress matrix_ of a framework $(p,\Gamma)$ is a matrix that satisfies properties (2)–(4) above. Observe first that an equilibrium stress matrix (symmetric or not) $\Omega$ of $(p,\Gamma)$ is an affinity matrix of $(p,N(\Gamma))$. From the properties of affinity matrices, the kernel of $\Omega$ always contains the subspace spanned by the coordinates of $p$ along each axis and the vector $\vec{1}$ of all $1$’s. ###### Definition 5.2. We say that a framework of a graph in $C^{d}(\Gamma)$ has the _convex containment_ property if 1. (1) the one-ring of each vertex has an affine span of dimension $d$, and 2. (2) except for $d+1$ exceptional vertices, each of the other vertices in the framework is contained in the strict $d$-dimensional convex hull of its neighbors. ###### Lemma 5.3. Let $\Gamma$ be a graph with at least $d+1$ vertices. Suppose $\Gamma$ is $(d+1)$-connected. Then there exists a generic framework $(q,\Gamma)$ in $C^{d}(\Gamma)$ with the convex containment property. ###### Proof. Pick any $d+1$ vertices to be exceptional. Constrain the exceptional vertices to fixed generic positions in $\mathbb{R}^{d}$ (at the vertices of a simplex). Associate generic positive weights $\omega_{ij}$ with each undirected edge $ij$. Find the “rubber band” configuration consistent with the constrained vertices and these weights. Namely, find a framework $(r,\Gamma)$ so that each non-exceptional vertex is the weighted linear average of its neighbors: $\sum_{j\in N(i)}\omega_{ij}(r(i)-r(j))=0,$ where $N(i)$ are the neighbors of vertex $i$. This involves solving $d$ systems of linear equations, one for each component of $r$. Note that the resulting configuration $r$ may not be generic. From [15], we know that if $\Gamma$ is $(d+1)$-connected and the constraints on the exceptional vertices and the edge weights $\omega$ are generic, then no set of $d+1$ vertices in $r$ will be contained in a $(d-1)$-dimensional affine plane, giving us the first condition. By construction, any non-exceptional vertex in $(r,\Gamma)$ must be contained in the convex hull of its neighbors. Again, from [15], the convex containment must be strict. Finally, we perturb each vertex in $\mathbb{R}^{d}$ to obtain a generic configuration in $q\in C^{d}({\mathcal{V}})$. By the first convex containment condition, the convex hull of the neighbors of a vertex has non-empty interior, so a sufficiently small perturbation will maintain both conditions. ∎ ###### Definition 5.4. Suppose that $(q,\Gamma)$ has the convex containment property and $\Omega$ is a non-symmetric equilibrium stress matrix for $(q,\Gamma)$. We call a row of $\Omega$ _non-exceptional_ if its corresponding vertex is in the strict $d$-dimensional convex hull of its neighbors. ###### Lemma 5.5. Let $\Gamma$ be a graph with at least $d+1$ vertices. Suppose $\Gamma$ is a $(d+1)$-connected graph, and we have a framework $(q,\Gamma)$ in $C^{d}(\Gamma)$ with the convex containment property. Then there is a non- symmetric equilibrium stress matrix $\Omega$ of $(q,\Gamma)$, such that for every non-exceptional row $i$, we have the following property: If there is an edge connecting vertex $i$ and vertex $j$, then $\Omega_{ij}$ is positive. ###### Proof. All vertices have $d+1$ or more neighbors. For each vertex $i$, we can therefore find “barycentric coordinates”: non-zero edge weights $\omega_{ij}$ on the adjoining edges so that $\sum_{j\in N(i)}\omega_{ij}(q(j)-q(i))=0.$ If $i$ is a non-exceptional vertex, due to the convex containment property we can choose the $\omega_{ij}$ to be positive. We then choose $\Omega_{ij}=\omega_{ij}$ for $i\neq j$ and $\Omega_{ii}=-\sum_{j}\omega_{ij}$. ∎ ###### Remark 5.6. This lemma is false if we require the stress matrix to be symmetric, because this prevents us from choosing $\omega_{ij}$ and $\omega_{ji}$ independently. ###### Lemma 5.7. Let $\Gamma$ be a graph with at least $d+1$ vertices. Suppose $\Gamma$ is $(d+1)$-connected, and we have a framework $(q,\Gamma)$ in $C^{d}(\Gamma)$ with the convex containment property. Then there is a non-symmetric equilibrium stress matrix $\Omega$ of $(q,\Gamma)$ with co-rank $d+1$. ###### Proof. From Lemma 5.5 we find for $(q,\Gamma)$ a non-symmetric equilibrium stress matrix $\Omega$ with the desired positive entries. We now show that $\Omega$ has the stated rank. First remove the $d+1$ rows and columns associated with the exceptional vertices to create a smaller matrix $\Omega^{\prime}$. Due to the sign pattern assumed in $\Omega$, as well as property (3) of any equilibrium stress matrix, $\Omega^{\prime}$ must be weakly diagonally dominant. Let us call a vertex _EN_ if it has an exceptional neighbor and refer to its corresponding row in $\Omega^{\prime}$ as EN. Any EN row must be strictly diagonally dominant (since at least one non-zero off-diagonal entry of $\Omega$ have been removed from this row). Since all entries corresponding to edges are non-zero, the reducible components of $\Omega^{\prime}$ correspond to vertex subsets that remain connected after the exceptional vertices have been removed. Each reducible component of $\Omega^{\prime}$ includes such an EN row, thus $\Omega^{\prime}$ must be full rank. (See, e.g., [24, Theorem 1.21].) Since $\Omega^{\prime}$ has co-rank 0, the co-rank of $\Omega$ must be at most $d+1$. It is no less since any equilibrium stress matrix must have a $(d+1)$-dimensional kernel spanned by the coordinates of $q$ and the all-ones vector. ∎ ###### Proposition 5.8. Let $\Gamma$ be a graph with at least $d+1$ vertices. Suppose $\Gamma$ is $(d+1)$-connected, and $p$ is generic in $C^{d}({\mathcal{V}})$. Then there is a non-symmetric equilibrium stress matrix $\Omega$ of $(p,\Gamma)$ with co- rank $d+1$. ###### Proof. From Lemma 5.3, there must exist a generic framework $(q,\Gamma)$ in $C^{d}(\Gamma)$ that has the convex containment property. From Lemma 5.7, $(q,\Gamma)$ must have a non-symmetric equilibrium stress matrix of co-rank $d+1$. Thus from Proposition A.1, any generic framework $(p,\Gamma)$ must have such a matrix as well. ∎ Figure 4. This framework in $\mathbb{R}^{2}$ is not 3-connected but does have a non-symmetric stress matrix of high rank. See Figure 4 for an example showing that the converse of Proposition 5.8 is not true. Here we show an neighborhood affinely rigid framework in $\mathbb{R}^{2}$ with a non-symmetric equilibrium stress matrix of co-rank $d+1=3$. This framework is not 3-connected. Note that from the proof of Proposition A.1 it is clear that if $\Gamma$ is $(d+1)$-connected, then _almost every_ non-symmetric stress matrix for almost every $(p,\Gamma)$ in $C^{d}(\Gamma)$ will have co-rank $d+1$. Moreover, each row of such a non-symmetric stress matrix of $p$ can be constructed independently from the other rows, and we still expect to find this minimal co-rank. ###### Proposition 5.9. Let $\Gamma$ be a graph with at least $d+1$ vertices. Suppose $(p,\Gamma)$, a framework in $C^{d}(\Gamma)$, has a non-symmetric equilibrium stress matrix $\Omega$ that has co-rank $d+1$. Then $(p,\Gamma)$ is neighborhood affinely rigid in $\mathbb{R}^{d}$. ###### Proof. $\Omega$ is a (not strong) affinity matrix of $(p,N(\Gamma))$ and so the proof follows that of the first direction of Theorem 1. ∎ ###### Proof of Theorem 3. The theorem now follows directly from Propositions 5.8 and 5.9. ∎ Figure 5. This framework in $\mathbb{R}^{2}$ does not have a non-symmetric equilibrium stress matrix of co-rank $d+1=3$, but is (trivially) neighborhood affinely rigid. See Figure 5 for an example showing that the converse of Proposition 5.9 is not true. Here we show a neighborhood affinely rigid framework in $\mathbb{R}^{2}$ that does not have a non-symmetric equilibrium stress matrix of co-rank $d+1=3$. ###### Remark 5.10. Generic global rigidity of a graph $\Gamma$ in $\mathbb{E}^{d}$ can be characterized either using the dimension of the kernel of a single symmetric stress matrix of a generic framework $(p,\Gamma)$ or using the dimension of the _shared symmetric stress kernel_ of a generic $p$: the intersection of the kernels of all stress matrices of $p$ [8]. By contrast, the analogous statement is not true in the affine rigidity case. By “vertically concatenating” a sufficient number of non-symmetric equilibrium stress matrices of $(p,\Gamma)$, we can create a strong affinity matrix for $(p,N(\Gamma))$. The kernel of the vertical concatenation will be the shared non-symmetric stress kernel of $(p,\Gamma)$, and the dimension of this kernel characterizes affine rigidity. Since the converse of Proposition 5.9 is not true, we see that neighborhood affine rigidity cannot in general be characterized by the rank of one single (say, generic) non-symmetric equilibrium stress matrix for $(p,\Gamma)$. Note that there is a different sufficiency condition for affine rigidity given by Zha and Zhang [26]. Their condition is complementary to our condition (neither strictly stronger or weaker), and (like trilateration [7]) is greedy in nature. Their condition on generic frameworks of a hypergraph requires that one can walk between any two hyperedges such that for each sequence $(i,j)$ of two hyperedges along the walk, $h_{i}$ and $h_{j}$ share at least $d+1$ vertices. When translated to a neighborhood hypergraph $N(\Gamma)$, it states that one can walk between any two vertices such that for each sequence $(i,j)$ of two vertices along the walk, the neighborhoods of these vertices share at least $d+1$ vertices in $\Gamma$. Figure 6 shows a graph which clearly fails Zha and Zhang’s condition, but is 3-connected, showing that their condition does not imply Theorem 3. It is not hard to construct examples in the opposite direction, as well. Figure 6. A drawing of a hexagonal lattice on the torus. (The vertices on the top edge should be identified with those on the bottom and similarly with the left and right, as indicated by the arrows.) This graph is $3$-connected but its neighborhood hypergraph does not satisfy the sufficiency condition of Zha and Zhang [26], and its squared graph is not a 2-trilateration graph. From Corollary 4.2, we also have the following corollary of Theorem 3. ###### Corollary 5.11. Let $\Gamma$ be a graph with at least $d+1$ vertices. If $\Gamma$ is $(d+1)$-connected, then any generic framework of $\Gamma^{2}$ in $\mathbb{R}^{d}$ is universally rigid. ###### Remark 5.12. This corollary can also be proven without reference to affine rigidity and Corollary 4.2. In particular, Proposition 5.8 guarantees a non symmetric maximal-rank stress $\Omega$ for $(p,\Gamma)$, and then $\Omega^{t}\Omega$ is a symmetric, positive semi-definite, maximal rank stress for $(p,\Gamma^{2})$. Universal rigidity then follows by a theorem of Connelly [5]. (See also [9].) We wish to highlight this corollary since the only other known (to us) class of graphs that are universally rigid for all generic configurations in $\mathbb{R}^{d}$ are the $d$-trilateration graphs. (A $d$-trilateration graph is one that can be obtained from a complete graph by successively adding vertices, each connected to at least $d+1$ old vertices.) See Figure 6 for an example of a framework whose square is universally rigid by this corollary but is not a trilateration graph. We also mention that a theorem of a related nature, showing a relationship between the connectivity of a graph and global rigidity in the squared graph, has been described by Anderson et al. [1]. ## 6\. Applications ### 6.1. Registration There are many applications where one has multiple views of some underlying configuration, but it is not known how these views all fit together. We assume that these views share some points in common, and this correspondence is known. (Of course in practice, establishing such a correspondence could in itself be a very challenging problem.) For example, in computer vision, one may have multiple uncalibrated laser scans of overlapping parts of some three- dimensional object. In our setting we model all of the points as vertices, and each of the views as a hyperedge. The geometry of the vertices in each hyperedge $h$ is given up to some unknown transform $T_{h}$ from a relevant class. The goal in registration then is to _realize_ the entire hypergraph up to the relevant congruence class. #### Affine case: Suppose we wish to realize a framework $(p,\Theta)$ where we are given as input the geometry of each hyperedge $h$ up to an affine transform $A_{h}$. Theorem 1 tells us that if $(p,\Theta)$ is affine rigid, then we can compute the realization just using linear algebra. In particular, we can use the data for each hyperedge to build its associated rows in a strong affinity matrix. Then we can solve for its kernel, giving us our answer $p$. If our hypergraph $\Theta$ happens to be the neighborhood graph of an underlying graph $\Gamma$, then one could also construct a (smaller) non- symmetric equilibrium stress matrix $\Omega$ for $(p,\Gamma)$. This is not guaranteed to work; even when $(p,\Gamma)$ is neighborhood affinely rigid in $\mathbb{R}^{d}$, the matrix $\Omega$ may have co-rank larger than $d+1$. But Theorem 3 states that if $\Gamma$ is $(d+1)$-connected, this method will indeed work for almost every $p$ in $\mathbb{R}^{d}$ (and, in fact, using almost any non-symmetric equilibrium stress matrix for $(p,\Gamma)$). #### Euclidean case: The Euclidean framework registration problem is perhaps more natural and common. When $(p,\Theta)$ is globally rigid in $\mathbb{R}^{d}$, this problem is well posed, but it is general hard to solve, as the graph case includes the graph realization problem which is strongly NP-HARD [18]. When $(p,\Theta)$ is, in fact, also universally rigid there is an efficient algorithm: we can solve the Euclidean registration problem using semi-definite programming. One simply sets up the program that looks for the Gram matrix of an embedding of the vertices in $\mathbb{R}^{v}$ (a semi-definite constraint on a Gram matrix) subject to the length constraints (linear constraints on the Gram matrix) [14]. Due to universal rigidity, one does not need to explicitly enforce the (non-convex) constraint that the embedding have a $d$-dimensional affine span [22]. When $(p,\Theta)$ is, furthermore, affinely rigid then we can solve the Euclidean registration problem using linear algebra. We can simply reduce this problem to an affine registration problem above, and find $p$ using the kernel vectors of an affinity matrix. This determines $p$ up to some global affine transform. Moreover, for $(p,\Theta)$ that is generically globally rigid in $\mathbb{R}^{d}$, we can solve a second (least squares) linear system to remove the unwanted global affine transform, leaving just the unknown global Euclidean transform (see Appendix B). This approach is morally the same affine relaxation used in the initialization step of the registration method of Krishnan et al. [12] (though in their case, they think of the inter-patch transforms as the unknown variables instead of the point positions). ### 6.2. Global embeddings from edge lengths Similar approaches have been applied to the (NP-HARD) problem of solving for the framework of a graph given its edge lengths. In these approaches one first attempts to find local $d$-dimensional embeddings for each one-ring of the framework up to an unknown local Euclidean transform. This step alone is NP- HARD and can fail. But assuming this step is (approximately) successful one can reduce the rest of the problem to the Euclidean registration problem above. In the As-Affine-As-Possible (AAAP) method [11, 27], this was done using what is essentially a strong affinity matrix. In the Locally-Rigid-Embedding (LRE) method [20] this was done using a non-symmetric equilibrium stress matrix. Both approaches then removed the global affine transform using the least squares linear system described in Appendix B. ### 6.3. Manifold learning Many of the ideas of affine rigidity first appeared in the context of manifold learning. Suppose one has $d$-dimensional smooth manifold $\mathcal{M}$ which is a topological $d$-ball embedded in a larger $D$-dimensional space $\mathbb{R}^{D}$. Also suppose that one has a set ${\mathcal{V}}$ of sample vertices on the manifold. In manifold learning, one first connects nearby samples to form a proximity graph $\Gamma$. One then looks for a framework $(p,\Gamma)$ of this graph in $\mathbb{R}^{d}$ that in some way that preserves some of the geometric relations of the points in $\mathbb{R}^{D}$. This is used to represent a parametrization of $\mathcal{M}$. To compute the coordinates $p$, the Locally Linear Embedding (LLE) method [16] builds a matrix $\Omega$ with structure similar to a non-symmetric equilibrium stress matrix. In particular, row $i$ encodes one affine relation between vertex $i$ and its neighbors in $\mathbb{R}^{D}$. Then (after ignoring the all ones vector) the smallest $d$ eigenvectors of $\Omega^{t}\Omega$ are used to form the coordinates of $p$ in $\mathbb{R}^{d}$. Unfortunately, since the original embedding is in $\mathbb{R}^{D}$, for a graph of high enough valence and assuming no noise, $\Omega$ must have a kernel of size at least $D+1$, which is much larger than $d+1$. Thus it is not clear how the numerically smallest $d+1$ eigenvectors will behave. The paper suggests to add an additional regularization term, possibly to address this issue. A follow up to the LLE paper [17] describes a PCA-LLE variant where a $d$-dimensional local PCA is computed to “flatten” each one-ring before calculating its corresponding row in the matrix $\Omega$. Thus $\Omega$ is designed to represent $d$-dimensional affine relations between the points. The Local-Tangent-Space-Alignment (LTSA) method [28] is an interesting variant of PCA-LLE. In this method, a $v\times v$ matrix $N$ is formed that is the Hessian of a quadratic energy. This energy vanishes only for vectors that are affinely equivalent to each of the flattened one-rings. Thus this matrix plays the role of a strong affinity matrix. It is in this context that Zha and Zhang investigated the rank of this matrix and affine rigidity [26]. In all of these methods, an understanding of affine rigidity is important. In particular it tells us what the rank of the computed matrix would be if the original $d$-dimensional manifold was in fact embedded in $\mathbb{R}^{d}$. For example, if such a framework was not affinely rigid in $\mathbb{R}^{d}$, then the kernel would be too big, and we would not expect a manifold learning technique to succeed. However, in manifold learning the embedding has an affine span greater than $d$ and the analysis becomes more difficult. The kernel of a strong $D$-dimensional affinity matrix is too large, while the kernel of a strong affinity matrix for the locally flattened configurations contains only the all-ones vector but it is hoped that the numerically next smallest $d$ eigenvectors are somehow geometrically meaningful. For an analysis along these lines, see [21]. ## Appendix A Matrix rank For completeness, we recall the necessary material for determining the generic matrix rank. This material is standard. For a more detailed treatment, see, e.g., [8, Section 5]. We will consider the general setting where there is a set of linear constraints that must be satisfied by a vector $m\in\mathbb{R}^{n}$. The entries of $m$ are then arranged in some fixed manner as the entries of a matrix $M$, whose rank we wish to understand. The linear constraints are described by a constraint matrix with $n$ columns: $C(p)$. Each of the entries of $C(p)$ is defined by some polynomial function, with coefficients in $\mathbb{Q}$, of the coordinates of an input configuration $p$. We wish to study the behavior of the rank of $M$ as one changes $p$. We apply this in the proof of Corollary 3.8 where the constraints $C$ specify that the matrix $M$ is an affinity matrix for $(p,\Theta)$ and in the proof of Proposition 5.8 where the constraints $C$ specify that the matrix $M$ is a non-symmetric equilibrium stress matrix for $(p,\Gamma)$. ###### Proposition A.1. Suppose that for some generic $p$, there is a matrix $M$ of rank $s$ consistent with $C(p)m=0$. Then for all generic $p$, there is some matrix $M$ of rank $\geq s$ consistent with $C(p)m=0$. ###### Proof. To prove this proposition we first need the following lemma. ###### Lemma A.2. Let $M(\pi)$ be a matrix whose entries are polynomial functions with rational coefficients in the variables $\pi\in\mathbb{R}^{n}$. Let $r$ be a rank achieved by some $M(\pi_{0})$. Then $\operatorname{rank}(M(\pi))\geq r$ for all points $\pi$ that are generic in $\mathbb{R}^{n}$. ###### Proof. The rank of the $M(\pi)$ is less than $r$ iff the determinants of all of the $r\times r$ submatrices vanish. Let $\pi_{0}\in\mathbb{R}^{n}$ be a choice of parameters so $M(\pi_{0})$ has rank $r$. Then there is an $r\times r$ submatrix $T(\pi_{0})$ of $M(\pi_{0})$ with non-zero determinant. Thus $\det(T(\pi))$ is a non-zero polynomial of $\pi$. For any $\pi$ with $\operatorname{rank}(M(\pi))<r$, this determinant must vanish. Thus, any such $\pi$ cannot be generic. ∎ Next we recall that for a non-singular $n\times n$ matrix $\hat{C}$, (1) $\operatorname{adj}(\hat{C})=\det(\hat{C})\hat{C}^{-1},$ where $\operatorname{adj}\hat{C}$ is the _adjugate matrix_ of $\hat{C}$, the conjugate of the cofactor matrix of $\hat{C}$. In particular, $\operatorname{adj}{\hat{C}}$ is a polynomial in $\hat{C}$. For a given $p$, let $t(p)$ be the rank of $C(p)$. Let $t:=\max_{p}t(p)$. By Lemma A.2 this maximum is obtained for generic $p$. For each $p$ we add a set $H$ of $n-t$ additional rows to $C(p)$ to obtain a matrix $C(p,H)$, and determine $m$ by solving the linear system $C(p,H)m=b$ where $b\in\mathbb{R}^{n}$ is a vector of all zeros except for a single $1$ in one of the positions of a row in $H$ (if there are any rows in $H$). This $m$ is then converted to a matrix $M(p,H)$. $M(p,H)$ is well-defined iff this linear system has a unique solution, i.e., iff $C(p,H)$ has rank $n$. Note that this happens for generic $p$ and $H$. Let $p_{0}$ be generic and have a compatible matrix $M_{0}$ with rank $s$, as in the hypotheses of the proposition. Find a set $H_{0}$ of additional rows so that $C(p_{0},H_{0})$ has rank $n$ and $C(p_{0},H_{0})m_{0}=b$. Let $\hat{C}(p,H)$ be an $n\times n$ submatrix of $C(p,H)$ so that $\hat{C}(p_{0},H_{0})$ is invertible. ($\hat{C}$ necessarily uses $t$ rows from $C(p)$ and all rows of $H$.) Define $\hat{b}$ similarly, let $\tilde{m}(p,H):=\operatorname{adj}(\hat{C})\hat{b}$, and let $\tilde{M}(p,H)$ be the associated matrix. By Lemma A.2, the rank of $\hat{C}(p,H)$ is equal to its maximum value, $n$, at all points $(p,H)$ that are not zeros of a polynomial $P_{1}(p,H):=\det\hat{C}(p,H)$. Moreover, when $P_{1}(p,H)\neq 0$, the linear equation defining $M(p,H)$ has a unique solution and the adjugate matrix $\tilde{M}(p,H)$ is a scalar multiple of $M(p,H)$. In particular we have assumed $(p_{0},H_{0})$ is not a zero of $P_{1}$ and thus $\tilde{M}(p_{0},H_{0})$ has rank $s$. By Lemma A.2 again, the rank of $\tilde{M}(p,H)$ is less than $s$ only at the zeros of a non-zero polynomial $P_{2}(p,H)$. For any generic $p$, there must be some generic point $(p,H)$. Such a generic $(p,H)$ cannot be a zero of $P_{1}$ or $P_{2}$ and thus $\tilde{M}(p,H)$ and $M(p,H)$ must have rank no less than $s$. ∎ ## Appendix B Removing the Affine Transform Suppose one has solved for $q$ – a configuration in $\mathbb{R}^{d}$ – up to an unknown global affine transform $A$ of the true configuration $p$: $p=A(q)$. Given a set of edge lengths for $p$, it is possible to compute $A$ up to an unknown global Euclidean transform. This approach was described by Singer [20]. In particular, let $L$ be a $d\times d$ matrix representing the linear portion of $A$ and let $G:=L^{T}L$. Now consider the following set of linear equations (in the $\frac{d(d+1)}{2}$ unknowns of $G$): For each pair of vertices $i,j$, whose edge lengths are known, we require (2) $(q(i)-q(j))^{T}G(q(i)-q(j))=(p(i)-p(j))^{T}(p(i)-p(j))$ (Since we have more constraints than unknowns, for numerical purposes we would solve this as a least squares linear system in the unknown $G$.) 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ACM Transactions on Sensor Networks (TOSN) 6, 4 (2010), 1–21. * [28] Zhang, Z., and Zha, H. Principal manifolds and nonlinear dimensionality reduction via tangent space alignment. SIAM Journal of Scientific Computing 26 (2004), 313–338.	arxiv-papers	2010-11-25T05:11:34	2024-09-04T02:49:15.234953	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Steven J. Gortler, Craig Gotsman, Ligang Liu, and Dylan P. Thurston", "submitter": "Steven Gortler", "url": "https://arxiv.org/abs/1011.5553" }
1011.5781	# Sulfate attack in sewer pipes: Derivation of a concrete corrosion model via two-scale convergence Tasnim Fatima Adrian Muntean Centre for Analysis, Scientific computing and Applications (CASA), Department of Mathematics and Computer Science, Technical University Eindhoven, Eindhoven, The Netherlands. E-mail: t.fatima@tue.nl Centre for Analysis, Scientific computing and Applications (CASA), Institute for Complex Molecular Systems (ICMS), Department of Mathematics and Computer Science, Technical University Eindhoven, Eindhoven, The Netherlands. E-mail: a.muntean@tue.nl ###### Abstract We explore the homogenization limit and rigorously derive upscaled equations for a microscopic reaction-diffusion system modeling sulfate corrosion in sewer pipes made of concrete. The system, defined in a periodically-perforated domain, is semi-linear, partially dissipative and weakly coupled via a non- linear ordinary differential equation posed on the solid-water interface at the pore level. Firstly, we show the well-posedness of the microscopic model. We then apply homogenization techniques based on two-scale convergence for an uniformly periodic domain and derive upscaled equations together with explicit formulae for the effective diffusion coefficients and reaction constants. We use a boundary unfolding method to pass to the homogenization limit in the non-linear ordinary differential equation. Finally, besides giving its strong formulation, we also prove that the upscaled two-scale model admits a unique solution. ###### keywords: Sulfate corrosion of concrete, periodic homogenization, semi-linear partially dissipative system, two-scale convergence, periodic unfolding method, multiscale system. and ## 1 Introduction This paper treats the periodic homogenization of a semi-linear reaction- diffusion system coupled with a nonlinear differential equation arising in the modeling of the sulfuric acid attack in sewer pipes made of concrete. The concrete corrosion situation we are dealing with here strongly influences the durability of cement-based materials especially in hot environments leading to spalling of concrete and macroscopic fractures of sewer pipes. It is financially important to have a good estimate on the moment in time when such pipe systems need to be replaced, for instance, at the level of a city like Los Angeles. To get good such practical estimates, one needs on one side easy- to-use macroscopic corrosion models to be used for a numerical forecast of corrosion, while on the other side one needs to ensure the reliability of the averaged models by allowing them to incorporate a certain amount of microstructure information. The relevant question is: How much of this oscillatory-type information is needed to get a sufficiently accurate description of the heterogeneous reality? Due to the complexity of possible shapes of the microstructure, averaging concrete materials is far more difficult than averaging metallic composites with rigorously defined well- packed structure. In this paper, we imagine our concrete piece to be made of a periodically-distributed microstructure. Based on this assumption, we provide here a rigorous justification of the formal asymptotic expansion performed by us (in [1]) for this reaction-diffusion scenario. Note that in [1] upscaled models are derived for a more general situation involving a locally-periodic distribution of perforations111The word ”perforation” is seen here as a synonym for ”pore” or ”microstructure”.. Locally periodic geometries refer to a special case of $x$-dependent microstructures, where, inherently, the outer normals to (microscopic) inner interfaces are dependent on both spatial slow variable, say $x$, and fast variable, say $y$. In the framework of this paper, we combine two-scale convergence concepts with the periodic unfolding of interfaces to pass to the homogenization limit (i.e. to $\varepsilon\to 0$, where $\varepsilon$ is a small parameter linked to the relative size of the perforation) for the uniformly periodic case. Here, the outer normals to the inner interfaces are dependent only on the spatial fast variable. For more details on the mathematical modeling of sulfate corrosion of concrete, we refer the reader to [2, 3] (a moving-boundary approach: numerics and formal matched asymptotics), [4] (a two-scale reaction-diffusion system modeling sulfate corrosion), as well as to [5], where a nonlinear Henry-law type transmission condition (modeling $H_{2}S$ transfer across all air-water interfaces present in this sulfatation problem) is analyzed. Mathematical background on periodic homogenization can be found in e.g., [6, 7, 8], while a few relevant (remotely resembling) worked-out examples of this averaging methodology are explained, for instance, in [9, 10, 11, 12, 13, 14]. It is worth noting that, since it deals with the homogenization of a linear Henry-law setting, the paper [11] is related to our approach. The major novelty here compared to [11] is that we now need to pass to the limit in a non-dissipative object, namely a nonlinear ordinary differential equation (ode). The ode is describing sulfatation reaction at the inner water-solid interface – place where corrosion localizes. This aspect makes a rigorous averaging challenging. For instance, compactness-type methods do not work in the case when the nonlinear ode is posed on $\epsilon$-dependent surfaces. We circumvent this issue by ”boundary unfolding” the ode. Thus we fix, as independent of $\epsilon$, the reaction interface similarly as in [15], and only then we pass to the limit. Alternatively, one could use varifolds (cf. e.g. [16]), since this seems to be the natural framework for the rigorous passage to the limit when both the surface measure and the oscillating sequences depend on $\epsilon$. However, we find the boundary unfolding technique easier to adapt to our scenario than the varifolds. Note that here we approach the corrosion problem deterministically. However, we have reasons to expect that the uniform periodicity assumption can be relaxed by assuming instead a Birkhoff-type ergodicity of the microstructure shapes and positions, and hence, the natural averaging context seems to be the one offered by random fields; see ch. 1, sect. 6 in [17], ch. 8 and 9 in [18], or [19]. But, methodologically, how big is the overlap between homogenizing deterministically locally-periodic distributions of microstructures compared to working in the random fields context? We will treat these and related aspects elsewhere. The paper is organized as follows: We start off in section 2 (and continue in section 3) with the analysis of the microscopic model. In section 4, we obtain the $\varepsilon$-independent estimates needed for the passage to the limit $\varepsilon\to 0$. Section 5 contains the main result of the paper: the set of the upscaled two-scal equations. ## 2 The microscopic model In this section, we describe the geometry of our array of periodic microstructures and briefly indicate the most aggressive chemical reaction mechanism typically active in sewer pipes. Finally, we list the set of microscopic equations. ### 2.1 Basic geometry Fig. 1 (i) shows a cross-section of a sewer pipe hosting corrosion. We assume that the geometry of the porous medium in question consists of a system of pores periodically distributed inside the three-dimensional cube $\Omega:=[a,b]^{3}$ with $a,b\in\mathbb{R}$ and $b>a$. The exterior boundary of $\Omega$ consists of two disjoint, sufficiently smooth parts: ${\Gamma^{N}}$ \- the Neumann boundary and ${\Gamma^{D}}$ \- the Dirichlet boundary. The reference pore, say $Y:=[0,1]^{3}$, has three pairwise disjoint connected domains $Y^{s}$, $Y^{w}$ and $Y^{a}$ with smooth boundaries $\Gamma^{sw}$ and $\Gamma^{wa}$, as shown in Fig. 1 (iii). Moreover, $Y:=\bar{Y}^{s}\cup\bar{Y}^{w}\cup\bar{Y}^{a}$. Figure 1: Left: Cross-section of a sewer pipe pointing out one region. Middle: Periodic approximation of the periodic rectangular domain. Right: Reference pore configuration. Let $\varepsilon$ be a sufficiently small scaling factor denoting the ratio between the characteristic length of the pore $Y$ and the characteristic length of the domain $\Omega$. Let $\chi^{w}$ and $\chi^{a}$ be the characteristic functions of the sets $Y^{w}$ and $Y^{a}$, respectively. The shifted set $Y^{w}_{k}$ is defined by $Y^{w}_{k}:=Y+\Sigma_{j=0}^{3}k_{j}e_{j}\;\mbox{for}\;\;k=(k_{1},k_{2},k_{3})\in\mathbb{Z}^{3},$ where $e_{j}$ is the $j^{th}$ unit vector. The union of all shifted subsets of $Y^{w}_{k}$ multiplied by $\varepsilon$ (and confined within $\Omega$) defines the perforated domain $\Omega^{\varepsilon}$, namely $\Omega^{\varepsilon}:=\cup_{k\in\mathbb{Z}^{3}}\\{\varepsilon Y^{w}_{k}\;\|\;\varepsilon Y^{w}_{k}\subset\Omega\\}.$ Similarly, $\Omega_{1}^{\varepsilon}$, $\Gamma^{sw}_{\varepsilon}$, and $\Gamma^{wa}_{\varepsilon}$ denote the union of the shifted subsets (of $\Omega$) $Y^{a}_{k}$, $\Gamma^{sw}_{k}$, and $\Gamma^{wa}_{k}$ scaled by $\varepsilon$. Since usually the concrete in sewer pipes is not completely dry, we decide to take into account a partially saturated porous material222The solid, water and air parts corresponds to $Y^{s}$, $Y^{w}$ and $Y^{a}$, respectively.. We assume that every pore has three distinct non- overlapping parts: a solid part (grain) which is placed in the center of the pore, the water film which surrounds the solid part, and an air layer bounding the water film and filling the space of $Y$ as shown in Fig. 1. The air connects neighboring pores to one another. The geometry defined above satisfies the following assumptions: 1. 1. Neither solid nor water-filled parts touch the boundary of the pore. 2. 2. All internal (air-water and water-solid) interfaces are sufficiently smooth and do not touch each other. These geometrical restrictions imply that the pores are connected by air- filled parts only which is needed not only to give a meaning to functions defined across interfaces, but also to introduce the concept of extension as given, for instance, in [20]. Furthermore, there are no solid-air interfaces. ### 2.2 Description of the chemistry There are many variants of severe attack to concrete in sewer pipes, we focus here on the most aggressive one – the sulfuric acid attack. The situation can be described briefly as follows: (The anaerobic bacteria in the flowing waste water release hydrogen sulfide gas ($H_{2}S$) within the air space of the pipe. These bacteria are especially active in hot environments. From the air space inside the pipe, $H_{2}S(g)$333$H_{2}S(g)$ and $H_{2}S(aq)$ refer to gaseous, and respectively, aqueous $H_{2}S$. enters the pores of the concrete matrix where it diffuses and then dissolves in the pore water. The aerobic bacteria catalyze some of the $H_{2}S$ into sulfuric acid $H_{2}SO_{4}$. $H_{2}S$ molecules can move between air-filled part and water-filled part the water-air interfaces [21]. We model this microscopic interfacial transfer via Henry’s law [22], (see the boundary conditions at $\Gamma^{wa}_{\varepsilon}$ in (3) and (4)). $H_{2}SO_{4}$ being an aggressive acid reacts with the solid matrix444The solid matrix is assumed here to consist of $CaCO_{3}$ only. This assumption can be removed in the favor of a more complex cement chemistry. at the solid-water interface, which is made up of cement, sand, and aggregate, and produces gypsum (i.e. $CaSO_{4}\cdot 2H_{2}O$). Here we restrict our attention to a minimal set of chemical reactions mechanisms as suggested in [2], namely. $\left\\{\begin{aligned} 10H^{+}+SO_{4}^{-2}+\mbox{org. matter}\;&\longrightarrow\;H_{2}S(aq)+4H_{2}O+\mbox{oxid. matter}\\\ H_{2}S(aq)+2O_{2}\;&{\longrightarrow}\;2H^{+}+SO_{4}^{-2}\\\ H_{2}S(aq)\;&\rightleftharpoons\;H_{2}S(g)\\\ 2H_{2}O+H^{+}+SO_{4}^{-2}+CaCO_{3}\;&{\longrightarrow}\;CaSO_{4}\cdot 2H_{2}O+HCO_{3}^{-}\end{aligned}\right.$ (1) We assume that reactions (1) do not interfere with the mechanics of the solid part of the pores. This is a rather strong assumption since it is known that (1) can actually produce local ruptures of the solid matrix [23]. For more details on the involved cement chemistry and connections to acid corrosion, we refer the reader to [24] (for a nice enumeration of the involved physicochemical mechanisms), [23] (standard textbook on cement chemistry), as well as to [25, 26, 27] and references cited therein. For a mathematical approach of a similar theme related to the conservation and restoration of historical monuments, we refer to the work by R. Natalini and co-workers (cf. e.g. [28]). ### 2.3 Setting of the equations The data and unknown are given by $\displaystyle{u^{\varepsilon}_{1}}_{0}$ $\displaystyle:\Omega\longrightarrow\mathbb{R}_{+}\mbox{ - initial concentration of }H_{2}SO_{4}(aq)$ $\displaystyle{u^{\varepsilon}_{2}}_{0}$ $\displaystyle:\Omega\longrightarrow\mathbb{R}_{+}\mbox{ - initial concentration of }H_{2}S(aq)$ $\displaystyle{u^{\varepsilon}_{3}}_{0}$ $\displaystyle:\Omega\longrightarrow\mathbb{R}_{+}\mbox{ - initial concentration of }H_{2}S(g)$ $\displaystyle{u^{\varepsilon}_{4}}_{0}$ $\displaystyle:\Omega\longrightarrow\mathbb{R}_{+}\mbox{ - initial concentration of moisture}$ $\displaystyle{u^{\varepsilon}_{5}}_{0}$ $\displaystyle:\Omega\longrightarrow\mathbb{R}_{+}\mbox{ - initial concentration of gypsum}$ $\displaystyle u_{3}^{D}$ $\displaystyle:\Gamma_{D}\times(0,T)\longrightarrow\mathbb{R}_{+}\mbox{ - exterior concentration (Dirichlet data) of }H_{2}S(g)$ $\displaystyle u^{\varepsilon}_{1}$ $\displaystyle:\Omega^{\varepsilon}\times(0,T)\longrightarrow\mathbb{R}\mbox{ - concentration of }H_{2}SO_{4}(aq)$ $\displaystyle u^{\varepsilon}_{2}$ $\displaystyle:\Omega^{\varepsilon}\times(0,T)\longrightarrow\mathbb{R}\mbox{ - concentration of }H_{2}S(aq)$ $\displaystyle u^{\varepsilon}_{3}$ $\displaystyle:\Omega^{\varepsilon}_{1}\times(0,T)\longrightarrow\mathbb{R}\mbox{ - concentration of }H_{2}S(g)$ $\displaystyle u^{\varepsilon}_{4}$ $\displaystyle:\Omega^{\varepsilon}\times(0,T)\longrightarrow\mathbb{R}\mbox{ - concentration of moisture}$ $\displaystyle u^{\varepsilon}_{5}$ $\displaystyle:\Gamma^{sw}_{\varepsilon}\times(0,T)\longrightarrow\mathbb{R}\mbox{ - concentration of gypsum }$ All concentrations are viewed as mass concentrations. We consider the following system of mass-balance equations defined at the pore level. The mass-balance equation for $H_{2}SO_{4}$ is $\displaystyle\partial_{t}u^{\varepsilon}_{1}+div(-d^{\varepsilon}_{1}\nabla u^{\varepsilon}_{1})$ $\displaystyle=-k^{\varepsilon}_{1}u^{\varepsilon}_{1}+k^{\varepsilon}_{2}u^{\varepsilon}_{2},\quad x\in\Omega^{\varepsilon},\;t\in(0,T)$ (2) $\displaystyle u^{\varepsilon}_{1}(x,0)$ $\displaystyle={u^{\varepsilon}_{1}}_{0}(x),\quad x\in\Omega^{\varepsilon}$ $\displaystyle-n^{\varepsilon}\cdot d^{\varepsilon}_{1}\nabla u^{\varepsilon}_{1}$ $\displaystyle=0,\quad x\in\Gamma^{wa}_{\varepsilon},\;t\in(0,T)$ $\displaystyle-n^{\varepsilon}\cdot d^{\varepsilon}_{1}\nabla u^{\varepsilon}_{1}$ $\displaystyle=\varepsilon\eta(u^{\varepsilon}_{1},u^{\varepsilon}_{5}),\quad x\in\Gamma^{sw}_{\varepsilon}\;t\in(0,T).$ The mass-balance equation for $H_{2}S(aq)$ is given by $\displaystyle\partial_{t}u^{\varepsilon}_{2}+div(-d^{\varepsilon}_{2}\nabla u^{\varepsilon}_{2})$ $\displaystyle=k^{\varepsilon}_{1}u^{\varepsilon}_{1}-k^{\varepsilon}_{2}u^{\varepsilon}_{2},\quad x\in\Omega^{\varepsilon},\;t\in(0,T),$ (3) $\displaystyle u^{\varepsilon}_{2}(x,0)$ $\displaystyle={u^{\varepsilon}_{2}}_{0}(x),\quad x\in\Omega^{\varepsilon}$ $\displaystyle-n^{\varepsilon}\cdot d^{\varepsilon}_{2}\nabla u^{\varepsilon}_{2}$ $\displaystyle=\varepsilon(a^{\varepsilon}(x)u^{\varepsilon}_{3}-b^{\varepsilon}(x)u^{\varepsilon}_{2}),\quad x\in\Gamma^{wa}_{\varepsilon},\;t\in(0,T)$ $\displaystyle-n^{\varepsilon}\cdot d^{\varepsilon}_{2}\nabla u^{\varepsilon}_{2}$ $\displaystyle=0,\quad x\in\Gamma^{sw}_{\varepsilon},\;t\in(0,T).$ The mass-balance equation for $H_{2}S(g)$ reads $\displaystyle\partial_{t}u^{\varepsilon}_{3}+div(-d^{\varepsilon}_{3}\nabla u^{\varepsilon}_{3})$ $\displaystyle=0,\quad x\in\Omega^{\varepsilon}_{1},\;t\in(0,T)$ (4) $\displaystyle u^{\varepsilon}_{3}(x,0)$ $\displaystyle={u^{\varepsilon}_{3}}_{0}(x),\quad x\in\Omega^{\varepsilon}_{1}$ $\displaystyle-n^{\varepsilon}\cdot d^{\varepsilon}_{3}\nabla u^{\varepsilon}_{3}$ $\displaystyle=0,\quad x\in\Gamma^{N},\;t\in(0,T)$ $\displaystyle u^{\varepsilon}_{3}(x,t)$ $\displaystyle=u^{D}_{3}(x,t),\quad x\in\Gamma^{D},\;t\in(0,T)$ $\displaystyle-n^{\varepsilon}\cdot d^{\varepsilon}_{3}\nabla u^{\varepsilon}_{3}$ $\displaystyle=-\varepsilon(a^{\varepsilon}(x)u^{\varepsilon}_{3}-b^{\varepsilon}(x)u^{\varepsilon}_{2}),\quad x\in\Gamma^{wa}_{\varepsilon},\;t\in(0,T).$ The mass-balance equation for moisture follows $\displaystyle\partial_{t}u^{\varepsilon}_{4}+div(-d^{\varepsilon}_{4}\nabla u^{\varepsilon}_{4})$ $\displaystyle=k^{\varepsilon}_{1}u^{\varepsilon}_{1},\quad x\in\Omega^{\varepsilon},\;t\in(0,T)$ (5) $\displaystyle u^{\varepsilon}_{4}(x,0)$ $\displaystyle={u^{\varepsilon}_{4}}_{0}(x),\quad x\in\Omega^{\varepsilon}$ $\displaystyle-n^{\varepsilon}\cdot d^{\varepsilon}_{4}\nabla u^{\varepsilon}_{4}$ $\displaystyle=0,\quad x\in\Gamma^{wa}_{\varepsilon},\;t\in(0,T)$ $\displaystyle-n^{\varepsilon}\cdot d^{\varepsilon}_{4}\nabla u^{\varepsilon}_{4}$ $\displaystyle=0,\quad x\in\Gamma^{sw}_{\varepsilon},\;t\in(0,T).$ The mass-balance equation for the gypsum produced at the water-solid interface is $\displaystyle\partial_{t}u^{\varepsilon}_{5}$ $\displaystyle=\eta(u^{\varepsilon}_{1},u^{\varepsilon}_{5}),\quad x\in\Gamma^{sw}_{\varepsilon},\;t\in(0,T)$ (6) $\displaystyle u^{\varepsilon}_{5}(x,0)$ $\displaystyle={u^{\varepsilon}_{5}}_{0}(x),\quad x\in\Gamma^{sw}_{\varepsilon},\;t\in(0,T).$ ## 3 Weak formulation and basic results We begin this section with a list of notations and function spaces. Then we indicate our working assumptions and give the weak formulation of the microscopic problem; we bring reader’s attention to the well-posedness of the system (2)–(6). ### 3.1 Notations and function spaces We use $(\alpha,\beta)_{(0,T)\times\Omega^{\varepsilon}}:=\int_{0}^{T}\int_{\Omega^{\varepsilon}}\alpha\beta dxdt$, $(\alpha,\beta)_{(0,T)\times\Gamma_{\varepsilon}}:=\int_{0}^{T}\int_{\Gamma_{\varepsilon}}\alpha\beta d\sigma_{x}dt$. $\langle\cdot\rangle$, $\|\cdot\|$ and $\\|\cdot\\|$ denote the dual pairing of $H^{1}(\Omega^{\varepsilon})$ and $H^{-1}(\Omega^{\varepsilon})$, the norm in $L^{2}(\Omega^{\varepsilon})$, and the norm in $H^{1}(\Omega^{\varepsilon}),$ respectively. $\varphi^{+}$ and $\varphi^{-}$ will point out the positive and respectively the negative part of the function $\varphi$. We denote by $C^{\infty}_{\\#}(Y)$, $H^{1}_{\\#}(Y)$, and $H^{1}_{\\#}(Y)/\mathbb{R}$, the space of infinitely differentiable functions in $\mathbb{R}^{n}$ that are periodic of period $Y$, the completion of $C^{\infty}_{\\#}(Y)$ with respect to $H^{1}-$norm, and the respective quotient space, respectively. Furthermore, $H^{1}_{\Gamma^{D}}(\Omega):=\\{u\in H^{1}(\Omega)\|u=0\mbox{ on }\Gamma^{D}\\}$. The Sobolev space $H^{\beta}(\Omega)$ as a completion of $C^{\infty}(\Omega)$ is a Hilbert space equipped with a norm $\\|\varphi\\|_{H^{\beta}(\Omega)}=\\|\varphi\\|_{H^{[\beta]}(\Omega)}+\left(\int_{\Omega}\int_{\Omega}\frac{\|\varphi(x)-\varphi(y)\|^{2}}{\|x-y\|^{n+2(\beta-[\beta])}}dxdy\right)^{\frac{1}{2}}$ and (cf. Theorem 7.57 in [29]) the embedding $H^{\beta}(\Omega)\hookrightarrow L^{2}(\Omega)$ is continuous. Since we deal with an evolution problem, we need typical Bochner spaces like $L^{2}(0,T;H^{1}(\Omega))$, $L^{2}(0,T;H^{-1}(\Omega))$, $L^{2}(0,T;H^{1}_{\Gamma^{D}}(\Omega))$, and $L^{2}((0,T)\times\Omega;H^{1}_{\\#}(Y)/\mathbb{R})$. In the analysis of the microscopic model, we use frequently the following trace inequality for $\varepsilon-$dependent hypersurfaces $\Gamma^{wa}_{\varepsilon}$: For $\varphi_{\varepsilon}\in H^{1}(\Omega^{\varepsilon})$, there exists a constant $C^{}$, which is independent of $\varepsilon$, such that $\varepsilon\|\varphi_{\varepsilon}\|^{2}_{L^{2}(\Gamma_{\varepsilon})}\leq C^{}(\|\varphi_{\varepsilon}\|^{2}_{L^{2}(\Omega^{\varepsilon})}+\varepsilon^{2}\|\nabla\varphi_{\varepsilon}\|^{2}_{L^{2}(\Omega^{\varepsilon})}).$ (7) The proof of (7) is given in Lemma 3 of [30]. For a function $\varphi^{\varepsilon}\in H^{\beta}(\Omega^{\varepsilon})$ with $\beta\in(\frac{1}{2},1)$, the inequality (7) refines into $\varepsilon\|\varphi_{\varepsilon}\|^{2}_{L^{2}(\Gamma_{\varepsilon})}\leq C^{}_{0}(\|\varphi_{\varepsilon}\|^{2}_{L^{2}(\Omega^{\varepsilon})}+\varepsilon^{2\beta}\int_{\Omega}^{\varepsilon}\int_{\Omega}^{\varepsilon}\frac{\|\varphi^{\varepsilon}(x)-\varphi^{\varepsilon}(y)\|^{2}}{\|x-y\|^{n+2\beta}}dxdy),$ (8) where $C^{}_{0}$ is again a constant independent of $\varepsilon$. For proof of (8), see [15]. To simplify the writing of some of the estimates, we employ the next set of notations: $\displaystyle d_{i}$ $\displaystyle:=\min_{[0,T]\times\bar{\Omega}}\mid d_{i}^{\varepsilon}\mid,\;i\in\\{1,2,3,4\\},\quad\tilde{d}_{i}:=\min_{[0,T]\times\bar{\Omega}}\mid\tilde{d}_{i}^{\varepsilon}\mid,$ $\displaystyle D_{m}$ $\displaystyle:=\max_{[0,T]\times\Omega^{\varepsilon}}\|\partial_{t}d^{\varepsilon}_{m}\|,\;m\in\\{1,2,3\\},\quad k_{j}:=\min_{[0,T]\times\bar{\Omega}}\mid k_{j}^{\varepsilon}\mid,\;j\in\\{1,2\\}$ $\displaystyle K_{j}$ $\displaystyle:=\min_{[0,T]\times\bar{\Omega}}\mid\partial_{t}k_{j}^{\varepsilon}\mid,\quad\tilde{k}_{j}:=\min_{[0,T]\times\bar{\Omega}}\mid{\tilde{k}}_{j}^{\varepsilon}\mid,$ $\displaystyle k^{\infty}_{m}$ $\displaystyle:=\sup_{(0,T)\times\Omega}\mid k_{m}^{\varepsilon}\mid,\quad\tilde{k}^{\infty}_{m}:=\sup_{(0,T)\times\Omega}\mid\tilde{k}_{m}^{\varepsilon}\mid,$ $\displaystyle K^{\infty}_{m}$ $\displaystyle:=\sup_{(0,T)\times\Omega}\mid\partial_{t}k_{m}^{\varepsilon}\mid,\quad M_{i}:=\sup_{(0,T)\times\Omega}\mid u_{i}^{\varepsilon}\mid,\;i\in\\{1,2,3,4,5\\},$ $\displaystyle A^{\infty}$ $\displaystyle:=\sup_{(0,T)\times\Gamma^{wa}_{\varepsilon}}\|a_{\varepsilon}\|,\;B^{\infty}:=\sup_{(0,T)\times\Gamma^{wa}_{\varepsilon}}\|b_{\varepsilon}\|,$ $\displaystyle A^{\infty}$ $\displaystyle:=\sup_{(0,T)\times\Gamma^{wa}_{\varepsilon}}\|\partial_{t}a_{\varepsilon}\|,\;B^{\infty}:=\sup_{(0,T)\times\Gamma^{wa}_{\varepsilon}}\|\partial_{t}b_{\varepsilon}\|,$ $\displaystyle\tilde{a}^{\infty}$ $\displaystyle:=\sup_{(0,T)\times\Gamma^{wa}}\|\tilde{a}\|,\;\tilde{b}^{\infty}:=\sup_{(0,T)\times\Gamma^{wa}}\|\tilde{b}\|,$ $\displaystyle Q^{\infty}$ $\displaystyle:=\sup_{s\in(0,T)\times\Gamma^{sw}_{\varepsilon}}\|Q(s)\|,\quad\bar{\eta}:=\|\|\eta\|\|_{\infty},\quad\hat{\eta}:=\|\|\partial_{t}\eta\|\|_{\infty}.$ ### 3.2 Assumptions on the data and parameters We consider the following restriction on the data and parameters: 1. (A1) $d_{i}\in L^{\infty}((0,T)\times Y)^{3\times 3}$, $\partial_{t}d_{i}\in L^{\infty}((0,T)\times Y)^{3\times 3}$, $\partial_{tt}d_{i}\in L^{\infty}((0,T)\times Y)^{3\times 3}$, $(d_{i}(t,x)\xi,\xi)\geq d_{i0}\mid\xi\mid^{2}$ for $d_{i0}>0$, for every $\xi\in\mathbb{R}^{3}$, $(t,x)\in(0,T)\times Y$, $i\in\\{1,2,3,4\\}$. 2. (A2) $\eta$ is measurable w.r.t. $t$ and $x$ and $\eta(\alpha,\beta)=k_{3}^{\varepsilon}R(\alpha)Q(\beta)$, $R$ is sub-linear and locally Lipschitz function and $Q$ is bounded and locally Lipschitz function such that $\displaystyle R(\alpha)=\left\\{\begin{array}[]{ccc}\mbox{positive},\;\;\mbox{if}\;\;\alpha\geq 0,\\\ 0,\;\;\;\mbox{otherwise}\end{array}\right.\quad\quad Q(\beta)=\left\\{\begin{array}[]{ccc}\mbox{positive},\;\;\mbox{if}\;\;\beta<\beta_{max},\\\ 0,\;\;\;\mbox{otherwise}\end{array}\right.$ Additionally to (A2), we sometimes assume (A2)’, that is 3. (A2)’ $\partial_{t}\eta\leq\hat{\eta}$. 4. (A3) $u^{\varepsilon}_{i0}\in L^{2}(\Omega^{\varepsilon})\cap L_{+}^{\infty}(\Omega^{\varepsilon}),\;i\in\\{1,2,4\\}$, $u^{\varepsilon}_{30}\in L^{2}(\Omega^{\varepsilon}_{1})\cap L_{+}^{\infty}(\Omega^{\varepsilon}_{1})$, $u^{\varepsilon}_{50}\in L^{2}(\Gamma^{sw}_{\varepsilon})\cap L_{+}^{\infty}(\Gamma^{sw}_{\varepsilon})$. 5. (A4) $a^{\infty}M_{3}=b^{\infty}M_{2}$, $k^{\infty}_{1}M_{1}=M_{4}$, $k_{1}M_{1}=k^{\infty}_{2}M_{2}$. 6. (A5) $a,b\in{C}^{1}([0,T];C^{0,\alpha}(\Gamma^{wa})),a,b\geq 0$ in $[0,T]\times\Gamma^{wa},\partial_{t}a,\partial_{t}b\in L^{\infty}((0,T)\times\Gamma^{wa})$. 7. (A6) $\partial_{t}u_{3}^{D}$, $\partial_{tt}u_{3}^{D}$ and $\nabla\partial_{t}u_{3}^{D}$ are bounded. 8. (A7) $k_{3}\in C^{1}([0,T];C^{0,\alpha}(\Gamma^{sw}))$ and $k_{j}\in C^{1}([0,T];C^{0,\alpha}(\bar{Y}))$ for any $j\in\\{1,2\\}$ and $\alpha\in]0,1]$. The assumptions (A1)–(A3), (A5), and (A6) are of technical nature. The first equality in (A4) points out an infinitely fast (equilibrium) Henry law, while the last two equalities remotely resemble a detailed balance in two of the involved chemical reactions. ### 3.3 Weak formulation of the microscopic model ###### Definition 1. Assume (A1) and (A3). We call the vector $u^{\varepsilon}=(u^{\varepsilon}_{1},u^{\varepsilon}_{2},u^{\varepsilon}_{3},u^{\varepsilon}_{4},u^{\varepsilon}_{5})$, a weak solution to (2)–(6) if $u^{\varepsilon}_{j}\in L^{2}(0,T;H^{1}(\Omega^{\varepsilon})),\partial_{t}u^{\varepsilon}_{j}\in L^{2}(0,T;H^{-1}(\Omega^{\varepsilon})),j\in\\{1,2,4\\}$, $u^{\varepsilon}_{3}\in u_{3}^{D}+L^{2}(0,T;H_{\Gamma^{D}}^{1}(\Omega_{1}^{\varepsilon})),\partial_{t}u^{\varepsilon}_{3}\in u_{3}^{D}+L^{2}(0,T;H^{-1}(\Omega_{1}^{\varepsilon})),u^{\varepsilon}_{5}\in L^{\infty}((0,T)\times\Gamma^{sw}_{\varepsilon}),\partial_{t}u^{\varepsilon}_{5}\in L^{\infty}((0,T)\times\Gamma^{sw}_{\varepsilon})$ such that the following identities hold $\displaystyle\langle\partial_{t}u^{\varepsilon}_{1},\varphi_{1}\rangle_{(0,T)\times\Omega^{\varepsilon}}$ $\displaystyle+(d_{1}\nabla u^{\varepsilon}_{1}),\nabla\varphi_{1})_{(0,T)\times\Omega^{\varepsilon}}$ (12) $\displaystyle=-(k_{1}u^{\varepsilon}_{1},\varphi_{1})_{(0,T)\times\Omega^{\varepsilon}}+(k_{2}u^{\varepsilon}_{2},\varphi_{1})_{(0,T)\times\Omega^{\varepsilon}}$ $\displaystyle-\varepsilon(\eta(u^{\varepsilon}_{1},u^{\varepsilon}_{4}),\varphi_{1})_{(0,T)\times\Gamma^{sw}_{\varepsilon}},$ $\displaystyle\langle\partial_{t}u^{\varepsilon}_{2},\varphi_{2}\rangle_{(0,T)\times\Omega^{\varepsilon}}$ $\displaystyle+(d^{\varepsilon}_{2}\nabla u^{\varepsilon}_{2}),\nabla\varphi_{2})_{(0,T)\times\Omega^{\varepsilon}}$ (13) $\displaystyle=(k^{\varepsilon}_{1}u^{\varepsilon}_{1},\varphi_{2})_{(0,T)\times\Omega^{\varepsilon}}-(k^{\varepsilon}_{2}u^{\varepsilon}_{2},\varphi_{2})_{(0,T)\times\Omega^{\varepsilon}}$ $\displaystyle+\varepsilon(a_{\varepsilon}u^{\varepsilon}_{3},\varphi_{2})_{(0,T)\times\Gamma^{wa}_{\varepsilon}}-\varepsilon(a_{\varepsilon}u^{\varepsilon}_{2},\varphi_{2})_{(0,T)\times\Gamma^{wa}_{\varepsilon}},$ $\displaystyle\langle\partial_{t}u^{\varepsilon}_{3},\varphi_{3}\rangle_{(0,T)\times\Omega_{1}^{\varepsilon}}$ $\displaystyle=-(d^{\varepsilon}_{3}\nabla u^{\varepsilon}_{3}),\nabla\varphi_{3})_{(0,T)\times\Omega_{1}^{\varepsilon}}$ (14) $\displaystyle-\varepsilon(a_{\varepsilon}u^{\varepsilon}_{3},\varphi_{3})_{(0,T)\times\Gamma^{wa}_{\varepsilon}}+\varepsilon(a_{\varepsilon}u^{\varepsilon}_{2},\varphi_{3})_{(0,T)\times\Gamma^{wa}_{\varepsilon}},$ $\displaystyle\langle\partial_{t}u^{\varepsilon}_{4},\varphi_{4}\rangle_{(0,T)\times\Omega^{\varepsilon}}$ $\displaystyle=-(d^{\varepsilon}_{4}\nabla u^{\varepsilon}_{4}),\nabla\varphi_{4})_{(0,T)\times\Omega^{\varepsilon}}+(k^{\varepsilon}_{1}u^{\varepsilon}_{1},\varphi_{4})_{(0,T)\times\Omega^{\varepsilon}}$ (15) for all $\varphi_{j}\in L^{2}(0,T;H^{1}(\Omega^{\varepsilon})),j\in\\{1,2,4\\}$ and $\varphi_{3}\in L^{2}(0,T;H_{\Gamma^{D}}^{1}(\Omega_{1}^{\varepsilon}))$ together with the ode $\displaystyle\partial_{t}u^{\varepsilon}_{5}$ $\displaystyle=$ $\displaystyle\eta(u^{\varepsilon}_{1},u^{\varepsilon}_{5})\;\;\mbox{a.e. on}\;\;(0,T)\times{\Gamma^{ws}_{\varepsilon}}$ (16) and the initial conditions $\displaystyle u^{\varepsilon}_{i}(0,x)$ $\displaystyle=u^{\varepsilon}_{i0}(x)\;\;x\in\Omega^{\varepsilon}\mbox{ for all }i\in\\{1,2,4\\},$ (17) $\displaystyle u^{\varepsilon}_{3}(0,x)$ $\displaystyle=u^{\varepsilon}_{30}(x)\;\;x\in\Omega_{1}^{\varepsilon},$ $\displaystyle u^{\varepsilon}_{5}(0,x)$ $\displaystyle=u^{\varepsilon}_{50}(x)\;\;x\in\Gamma^{ws}_{\varepsilon}.$ ### 3.4 Basic results ###### Lemma 2. (Positivity and $L^{\infty}$-estimates) Assume (A1)-(A6), and let $t\in[0,T]$ be arbitrarily chosen. Then the following estimates hold: * (i) $u^{\varepsilon}_{i}(t)\geq 0,\;\;i\in\\{1,2,4\\}$ a.e. in ${\Omega^{\varepsilon}}$, $u^{\varepsilon}_{3}(t)\geq 0$ a.e. ${\Omega^{\varepsilon}_{1}}$ and $u^{\varepsilon}_{5}(t)\geq 0$ a.e. on $\Gamma^{ws}_{\varepsilon}$. * (ii) $u^{\varepsilon}_{i}(t)\leq M_{i}$, $i\in\\{1,2\\}$, $u^{\varepsilon}_{4}(t)\leq(t+1)M_{4}$ a.e. in ${\Omega^{\varepsilon}}$ , $u^{\varepsilon}_{3}(t)\leq M_{3}$ a.e. in ${\Omega^{\varepsilon}_{1}}$ and $u^{\varepsilon}_{5}(t)\leq M_{5}$ a.e. on $\Gamma^{ws}_{\varepsilon}$. Proof (i). We test (12)-(15) with $\varphi=(-{u^{\varepsilon}_{1}}^{-},-{u^{\varepsilon}_{2}}^{-},-{u^{\varepsilon}_{3}}^{-},-{u^{\varepsilon}_{4}}^{-})$ element of the space $[L^{2}(0,T;H^{1}(\Omega^{\varepsilon}))]^{2}\times L^{2}(0,T;H^{1}_{\Gamma^{D}}(\Omega_{1}^{\varepsilon})\times L^{2}(0,T;H^{1}(\Omega^{\varepsilon})$. We obtain the following inequality $\displaystyle\frac{1}{2}\partial_{t}\|{u^{\varepsilon}_{1}}^{-}\|^{2}+d_{1}\|\nabla{u^{\varepsilon}_{1}}^{-}\|^{2}$ $\displaystyle\leq- k_{1}\|{u^{\varepsilon}_{1}}^{-}\|^{2}+k^{\infty}_{2}({u^{\varepsilon}_{1}}^{-},{u^{\varepsilon}_{2}}^{-})$ (18) $\displaystyle-\varepsilon(\eta(u^{\varepsilon}_{1},u^{\varepsilon}_{5}),-{u^{\varepsilon}_{1}}^{-})_{\Gamma^{sw}_{\varepsilon}}.$ Note that the first term on the r.h.s of (18) is negative, while the third term is zero because of (A2). We then get $\partial_{t}\|{u^{\varepsilon}_{1}}^{-}\|^{2}+2d_{1}\|\nabla{u^{\varepsilon}_{1}}^{-}\|^{2}\leq k^{\infty}_{2}\left(\|{u^{\varepsilon}_{1}}^{-}\|^{2}+\|{u^{\varepsilon}_{2}}^{-}\|^{2}\right).$ (19) On the other hand, (13) leads to $\displaystyle\frac{1}{2}\partial_{t}\|{u^{\varepsilon}_{2}}^{-}\|^{2}+d_{2}\|\nabla{u^{\varepsilon}_{2}}^{-}\|^{2}$ $\displaystyle\leq\frac{k^{\infty}_{1}}{2}\left(\|{u^{\varepsilon}_{1}}^{-}\|^{2}+\|{u^{\varepsilon}_{2}}^{-}\|^{2}\right)$ $\displaystyle+\varepsilon a^{\infty}({u^{\varepsilon}_{2}}^{-},{u^{\varepsilon}_{3}}^{-})_{\Gamma^{wa}_{\varepsilon}}+\varepsilon b^{\infty}\|{u^{\varepsilon}_{2}}^{-}\|^{2}_{\Gamma^{wa}_{\varepsilon}}.$ By the trace inequality (7) (with $\varepsilon<1$), we get $\displaystyle\partial_{t}\|{u^{\varepsilon}_{2}}^{-}\|^{2}$ $\displaystyle+2(d_{2}-C^{}b^{\infty})\|\nabla{u^{\varepsilon}_{2}}^{-}\|^{2}\leq{k^{\infty}_{1}}\left(\|{u^{\varepsilon}_{1}}^{-}\|^{2}+\|{u^{\varepsilon}_{2}}^{-}\|^{2}\right)$ (20) $\displaystyle+2C^{}b^{\infty}\|{u^{\varepsilon}_{2}}^{-}\|^{2}+2\varepsilon a^{\infty}({u^{\varepsilon}_{2}}^{-},{u^{\varepsilon}_{3}}^{-})_{\Gamma^{wa}_{\varepsilon}}.$ (14) leads to $\partial_{t}\|{u^{\varepsilon}_{3}}^{-}\|^{2}+2(d_{3}-C^{}a^{\infty})\|\nabla{u^{\varepsilon}_{3}}^{-}\|^{2}\leq 2\varepsilon b^{\infty}({u^{\varepsilon}_{2}}^{-},{u^{\varepsilon}_{3}}^{-})_{\Gamma^{wa}_{\varepsilon}}+2C^{}a^{\infty}\|{u^{\varepsilon}_{3}}^{-}\|^{2},$ (21) while from (15), we see that $\partial_{t}\|{u^{\varepsilon}_{4}}^{-}\|^{2}+2d_{4}\|\nabla{u^{\varepsilon}_{5}}^{-}\|^{2}\leq k^{\infty}_{1}\left(\|{u^{\varepsilon}_{1}}^{-}\|^{2}+\|{u^{\varepsilon}_{5}}^{-}\|^{2}\right).$ (22) Adding up inequalities (19)-(22) gives $\displaystyle\partial_{t}\sum_{i=1}^{4}\|{u^{\varepsilon}_{i}}^{-}\|^{2}$ $\displaystyle+2d_{1}\|\nabla{u^{\varepsilon}_{1}}^{-}\|^{2}+2(d_{2}-C^{}b^{\infty})\|\nabla{u^{\varepsilon}_{2}}^{-}\|^{2}$ (23) $\displaystyle+2(d_{3}-C^{}a^{\infty})\|\nabla{u^{\varepsilon}_{3}}^{-}\|^{2}+2d_{4}\|\nabla{u^{\varepsilon}_{4}}^{-}\|^{2}$ $\displaystyle\leq\left(2k^{\infty}_{1}+k^{\infty}_{2}+2C^{}b^{\infty}+2C^{}a^{\infty}\right)\sum_{{i=1}}^{4}\|{u^{\varepsilon}_{i}}^{-}\|^{2}$ $\displaystyle+2\varepsilon(a^{\infty}+b^{\infty})({u^{\varepsilon}_{2}}^{-},{u^{\varepsilon}_{3}}^{-})_{\Gamma^{wa}_{\varepsilon}},$ and hence, $\displaystyle\partial_{t}\sum_{i=1}^{4}\|{u^{\varepsilon}_{i}}^{-}\|^{2}$ $\displaystyle+2d_{1}\|\nabla{u^{\varepsilon}_{1}}^{-}\|^{2}+2(d_{2}-C^{}b^{\infty})\|\nabla{u^{\varepsilon}_{2}}^{-}\|^{2}$ (24) $\displaystyle+2(d_{3}-C^{}a^{\infty})\|\nabla{u^{\varepsilon}_{3}}^{-}\|^{2}+2d_{4}\|\nabla{u^{\varepsilon}_{5}}^{-}\|^{2}$ $\displaystyle\leq\left(2k^{\infty}_{1}+k^{\infty}_{2}+C^{}(a^{\infty}+b^{\infty})\right)\sum_{i=1}^{4}\|{u^{\varepsilon}_{i}}^{-}\|^{2}$ $\displaystyle+\varepsilon\left(a^{\infty}+b^{\infty})(\delta\|{u^{\varepsilon}_{2}}^{-}\|_{\Gamma^{wa}_{\varepsilon}}^{2}+\frac{1}{\delta}\|{u^{\varepsilon}_{3}}^{-}\|^{2}_{\Gamma^{wa}_{\varepsilon}}\right).$ Applying the trace inequality (7) to estimate the last term on the right side of (24), we finally get $\displaystyle\partial_{t}\sum_{i=1}^{4}\|{u^{\varepsilon}_{i}}^{-}\|^{2}$ $\displaystyle+2d_{1}\|\nabla{u^{\varepsilon}_{1}}^{-}\|^{2}+(2d_{2}-2C^{}b^{\infty}-C^{}\delta(a^{\infty}+b^{\infty}))\|\nabla{u^{\varepsilon}_{2}}^{-}\|^{2}$ $\displaystyle+(2d_{3}-2C^{}a^{\infty}-\frac{C^{}2}{\delta}(a^{\infty}+b^{\infty}))\|\nabla{u^{\varepsilon}_{3}}^{-}\|^{2}+2d_{4}\|\nabla{u^{\varepsilon}_{4}}^{-}\|^{2}$ $\displaystyle\leq C_{1}\sum_{i=1}^{4}\|{u^{\varepsilon}_{i}}^{-}\|^{2}.$ Thus, we have $\displaystyle\partial_{t}\sum_{i=1}^{4}\|{u^{\varepsilon}_{i}}^{-}\|^{2}$ $\displaystyle\leq$ $\displaystyle C_{1}\sum_{i=1}^{4}\|{u^{\varepsilon}_{i}}^{-}\|^{2}.$ where $C_{1}:=2k^{\infty}_{1}+k^{\infty}_{2}+C^{}(a^{\infty}+b^{\infty})+C^{}(\delta+\frac{1}{\delta})(a^{\infty}+b^{\infty})$ and $\delta$ is chosen conveniently. Gronwall’s inequality together with $[u^{\varepsilon}_{i}(0)]^{-}=0$ gives now the desired result. Note that (A2) ensures automatically the positivity of $u^{\varepsilon}_{5}$. (ii). We consider the test function $(\varphi_{1},\varphi_{2},\varphi_{3},\varphi_{4})=((u^{\varepsilon}_{1}-M_{1})^{+},(u^{\varepsilon}_{2}-M_{2})^{+},(u^{\varepsilon}_{3}-M_{3})^{+},(u^{\varepsilon}_{4}-(t+1)M_{4})^{+}).$ Obviously, $\varphi\in[L^{2}(0,T;H^{1}(\Omega^{\varepsilon}))]^{2}\times L^{2}(0,T;H^{1}_{\Gamma^{D}}(\Omega_{1}^{\varepsilon})\times L^{2}(0,T;H^{1}(\Omega^{\varepsilon})$ is allowed as test function. We obtain from (12) that $\displaystyle\frac{1}{2}\partial_{t}\|(u^{\varepsilon}_{1}-M_{1})^{+}\|^{2}$ $\displaystyle+d_{1}\|\nabla(u^{\varepsilon}_{1}-M_{1})^{+}\|^{2}\leq- k_{1}\|(u^{\varepsilon}_{1}-M_{1})^{+}\|^{2}$ $\displaystyle-(k_{1}M_{1},(u^{\varepsilon}_{1}-M_{1})^{+})$ $\displaystyle+k^{\infty}_{2}((u^{\varepsilon}_{1}-M_{1})^{+},(u^{\varepsilon}_{2}-M_{2})^{+})$ $\displaystyle+(k^{\infty}_{2}M_{2},(u^{\varepsilon}_{1}-M_{1})^{+})$ $\displaystyle-\varepsilon(\eta(u^{\varepsilon}_{1},u^{\varepsilon}_{5}),(u^{\varepsilon}_{1}-M_{1})^{+})_{\Gamma^{sw}_{\varepsilon}}.$ Relying on (A4), we get the estimate $\displaystyle\begin{array}[]{ccc}\partial_{t}\|(u^{\varepsilon}_{1}-M_{1})^{+}\|^{2}&\leq&k^{\infty}_{2}(\|(u^{\varepsilon}_{1}-M_{1})^{+}\|^{2}+\|(u^{\varepsilon}_{2}-M_{2})^{+}\|^{2}).\end{array}$ (26) (13) in combination with (A4) gives that $\displaystyle\partial_{t}\|(u^{\varepsilon}_{2}-M_{2})^{+}\|^{2}$ $\displaystyle+2(d_{2}-C^{}b^{\infty})\|\nabla(u^{\varepsilon}_{2}-M_{2})^{+}\|^{2}$ (27) $\displaystyle\leq k^{\infty}_{1}(\|(u^{\varepsilon}_{1}-M_{1})^{+}\|^{2}+\|(u^{\varepsilon}_{2}-M_{2})^{+}\|^{2})$ $\displaystyle+2C^{}b^{\infty}\|(u^{\varepsilon}_{2}-M_{2})^{+}\|^{2}$ $\displaystyle+2\varepsilon a^{\infty}((u^{\varepsilon}_{2}-M_{2})^{+},(u^{\varepsilon}_{3}-M_{3})^{+})_{\Gamma^{wa}_{\varepsilon}}.$ By (14), we obtain $\displaystyle\partial_{t}\|(u^{\varepsilon}_{3}-M_{3})^{+}\|^{2}$ $\displaystyle+2(d_{3}-C^{}a^{\infty})\|\nabla(u^{\varepsilon}_{3}-M_{3})^{+}\|^{2}$ (28) $\displaystyle\leq 2C^{}a^{\infty}\|\nabla(u^{\varepsilon}_{3}-M_{3})^{+}\|^{2}$ $\displaystyle+2\varepsilon b^{\infty}((u^{\varepsilon}_{2}-M_{2})^{+},(u^{\varepsilon}_{3}-M_{3})^{+})_{\Gamma^{wa}_{\varepsilon}}.$ Using again (A4), (15) yields $\displaystyle\begin{array}[]{ccc}\partial_{t}\|(u^{\varepsilon}_{4}-(t+1)M_{4})^{+}\|^{2}&\leq&k^{\infty}_{1}(\|(u^{\varepsilon}_{1}-M_{1})^{+}\|^{2}+\|(u^{\varepsilon}_{4}-(t+1)M_{4})^{+}\|^{2}).\end{array}$ (30) Adding up (26)–(30) side by side, we get $\displaystyle\sum_{j=1}^{3}\partial_{t}\|(u^{\varepsilon}_{j}-M_{j})^{+}\|^{2}$ $\displaystyle+\partial_{t}\|(u^{\varepsilon}_{4}-(t+1)M_{4})^{+}\|^{2}+(2d_{2}-2C^{}b^{\infty})\|\nabla(u^{\varepsilon}_{2}-M_{2})^{+}\|^{2}$ $\displaystyle+(2d_{3}-2C^{}a^{\infty})\|\nabla(u^{\varepsilon}_{3}-M_{3})^{+}\|^{2}$ $\displaystyle\leq(2k^{\infty}_{2}+k^{\infty}_{1}+2C^{}a^{\infty}+2C^{}b^{\infty})(\sum_{j=1}^{3}\|(u^{\varepsilon}_{j}-M_{j})^{+}\|^{2}$ $\displaystyle+\|(u^{\varepsilon}_{4}-(t+1)M_{4})^{+}\|^{2})+\varepsilon(a^{\infty}+b^{\infty})(\delta\|(u^{\varepsilon}_{2}-M_{2})^{+}\|^{2}_{\Gamma^{wa}_{\varepsilon}}$ $\displaystyle+\frac{1}{\delta}\|(u^{\varepsilon}_{3}-M_{3})^{+}\|^{2}_{\Gamma^{wa}_{\varepsilon}}).$ We use the trace inequality (7) (with $\varepsilon<1$) to deal with the boundary terms in (3.4). Then Gronwall’s inequality yields for all $t\in(0,T)$ the following estimate $\displaystyle u^{\varepsilon}_{j}(t)$ $\displaystyle\leq M_{j},\;\;j\in\\{1,2,5\\}\;a.\;e.\;in\;\Omega^{\varepsilon}$ $\displaystyle u^{\varepsilon}_{3}(t)$ $\displaystyle\leq M_{3},\;a.\;e.\;in\;\Omega_{1}^{\varepsilon}$ $\displaystyle u^{\varepsilon}_{4}$ $\displaystyle\leq(t+1)M_{4}\;\mbox{a.e. in}\;\Omega^{\varepsilon}.$ Furthermore, by (A2) $u^{\varepsilon}_{5}$ is bounded. ###### Proposition 3. (Uniqueness) Assume (A1)–(A4). Then there exists at most one weak solution in the sense of Definition 1. Proof. We assume that ${u}^{j,\varepsilon}=(u_{1}^{j,\varepsilon},u_{2}^{j,\varepsilon},u_{3}^{j,\varepsilon},u_{4}^{j,\varepsilon},u_{5}^{j,\varepsilon}),j\in\\{1,2\\}$ are two distinct weak solutions in the sense of Definition 1. We set $u_{i}^{\varepsilon}:=u_{i}^{1,\varepsilon}-u_{i}^{2,\varepsilon}$ for all $i\in\\{1,2,3,4\\}$. Firstly, we deal with (15). We obtain $\displaystyle\partial_{t}{u^{1,\varepsilon}_{5}}-\partial_{t}{u^{2,\varepsilon}_{5}}=\eta(u^{1,\varepsilon}_{1},u^{1,\varepsilon}_{5})-\eta(u^{2,\varepsilon}_{1},u^{2,\varepsilon}_{5}).$ (31) Integrating (31) along (0,T) and using (A2), we get $\displaystyle\|u^{1,\varepsilon}_{5}-u^{2,\varepsilon}_{5}\|\leq k_{3}^{\infty}c_{R}c_{Q}M_{1}\int_{0}^{t}\|{u^{1,\varepsilon}_{5}}-{u^{2,\varepsilon}_{5}}\|d\tau+k_{3}^{\infty}c_{R}Q^{\infty}\int_{0}^{t}\|{u^{1,\varepsilon}_{1}}-{u^{2,\varepsilon}_{1}}\|d\tau.$ Gronwall’s inequality implies $\displaystyle\|{u^{1,\varepsilon}_{5}}(t)-{u^{2,\varepsilon}_{5}(t)}\|\leq C_{2}\int_{0}^{t}\|{u^{1,\varepsilon}_{1}}-{u^{2,\varepsilon}_{1}}\|d\tau\;\;\mbox{for a.e. }t\in(0,T),$ (32) where $C_{2}:=k_{3}^{\infty}c_{R}Q^{\infty}(1+C_{3}te^{C_{3}t})$ and $C_{3}:=k_{3}^{\infty}c_{R}c_{Q}M_{1}$. We calculate $\frac{1}{2}\partial_{t}\|{u^{\varepsilon}_{1}}\|^{2}+d_{1}\|\nabla{u^{\varepsilon}_{1}}\|^{2}\leq- k_{1}\|{u^{\varepsilon}_{1}}\|^{2}+k^{\infty}_{2}({u^{\varepsilon}_{1}},{u^{\varepsilon}_{2}})+\varepsilon(\eta_{1}-\eta_{2},{u^{\varepsilon}_{1}})_{\Gamma^{sw}_{\varepsilon}},$ (33) where we denote $\eta_{1}-\eta_{2}:=\eta(u^{1,\varepsilon}_{1},u^{1,\varepsilon}_{5})-\eta(u^{2,\varepsilon}_{1},u^{2,\varepsilon}_{5})$. We can write $\displaystyle\frac{1}{2}\partial_{t}\|{u^{\varepsilon}_{1}}\|^{2}+d_{1}\|\nabla{u^{\varepsilon}_{1}}\|^{2}$ $\displaystyle\leq- k_{1}\|{u^{\varepsilon}_{1}}\|^{2}+\frac{k^{\infty}_{2}}{2}(\|{u^{\varepsilon}_{1}}\|^{2}+\|{u^{\varepsilon}_{2}}\|^{2})$ (34) $\displaystyle+\varepsilon C_{3}({u^{1,\varepsilon}_{5}}-{u^{2,\varepsilon}_{5}},{u^{\varepsilon}_{1}})_{\Gamma^{sw}_{\varepsilon}}$ $\displaystyle+\varepsilon k_{3}^{\infty}c_{R}Q^{\infty}({u^{1,\varepsilon}_{1}}-{u^{2,\varepsilon}_{1}},{u^{\varepsilon}_{1}})_{\Gamma^{sw}_{\varepsilon}}.$ Now, inserting (32) in (34) yields $\displaystyle\frac{1}{2}\partial_{t}\|{u^{\varepsilon}_{1}}\|^{2}+d_{1}\|\nabla{u^{\varepsilon}_{1}}\|^{2}$ $\displaystyle\leq- k_{1}\|{u^{\varepsilon}_{1}}\|^{2}+\frac{k^{\infty}_{2}}{2}(\|{u^{\varepsilon}_{1}}\|^{2}+\|{u^{\varepsilon}_{2}}\|^{2})$ (35) $\displaystyle+C_{4}\varepsilon\|{u^{\varepsilon}_{1}}\|^{2}_{\Gamma^{sw}_{\varepsilon}}+\frac{\varepsilon C_{3}^{2}}{2\delta}\int_{0}^{t}\|{u^{\varepsilon}_{1}}\|^{2}_{\Gamma^{sw}_{\varepsilon}}d\tau,$ where $C_{4}:=k_{3}^{\infty}c_{R}Q^{\infty}+\frac{C_{3}}{2\delta}$. Using (7), we estimate the last two terms in (35) to obtain the inequality $\displaystyle\frac{1}{2}\partial_{t}\|{u^{\varepsilon}_{1}}\|^{2}+d_{1}\|\nabla{u^{\varepsilon}_{1}}\|^{2}$ $\displaystyle\leq- k_{1}\|{u^{\varepsilon}_{1}}\|^{2}+\frac{k^{\infty}_{2}}{2}(\|{u^{\varepsilon}_{1}}\|^{2}+\|{u^{\varepsilon}_{2}}\|^{2})+C^{}C_{4}(\|{u^{\varepsilon}_{1}}\|^{2}$ (36) $\displaystyle+\varepsilon^{2}\|\nabla{u^{\varepsilon}_{1}}\|)+C^{}\frac{C_{3}^{2}}{2\delta}\int_{0}^{t}(\|{u^{\varepsilon}_{1}}\|^{2}+\varepsilon^{2}\|\nabla{u^{\varepsilon}_{1}}\|^{2})d\tau.$ Note that the constant $C^{}$, arising from in (36), stems from (7). Rearranging now the terms, we have $\displaystyle\partial_{t}\|{u^{\varepsilon}_{1}}\|^{2}$ $\displaystyle+(2d_{1}-2C^{}C_{4}\varepsilon^{2})\|\nabla{u^{\varepsilon}_{1}}\|^{2}+2k_{1}\|{u^{\varepsilon}_{1}}\|^{2}\leq({k^{\infty}_{2}}+C^{}C_{4})(\|{u^{\varepsilon}_{1}}\|^{2}$ (37) $\displaystyle+\|{u^{\varepsilon}_{2}}\|^{2})+C^{}\frac{C_{3}^{2}}{2\delta}\int_{0}^{t}(\|{u^{\varepsilon}_{1}}\|^{2}+\varepsilon^{2}\|\nabla{u^{\varepsilon}_{1}}\|^{2})d\tau.$ Following the same line of arguments as before, we obtain from (13) that $\displaystyle\partial_{t}\|{u^{\varepsilon}_{2}}\|^{2}+2d_{2}\|\nabla{u^{\varepsilon}_{2}}\|^{2}$ $\displaystyle\leq-2k_{2}\|{u^{\varepsilon}_{2}}\|^{2}+{k^{\infty}_{1}}(\|{u^{\varepsilon}_{1}}\|^{2}+{u^{\varepsilon}_{2}}\|^{2})$ (38) $\displaystyle+2\varepsilon a^{\infty}({u^{\varepsilon}_{3}},{u^{\varepsilon}_{2}})_{\Gamma^{wa}_{\varepsilon}}+2\varepsilon b^{\infty}\|{u^{\varepsilon}_{2}}\|^{2}_{\Gamma^{wa}_{\varepsilon}},$ while from (14), we deduce $\displaystyle\partial_{t}\|{u^{\varepsilon}_{3}}\|^{2}+2d_{3}\|\nabla{u^{\varepsilon}_{3}}\|^{2}$ $\displaystyle\leq 2\varepsilon b^{\infty}({u^{\varepsilon}_{2}},{u^{\varepsilon}_{3}})_{\Gamma^{wa}_{\varepsilon}}+2\varepsilon a^{\infty}\|{u^{\varepsilon}_{3}}\|^{2}_{\Gamma^{wa}_{\varepsilon}}.$ (39) Proceeding similarly, (15) yields $\displaystyle\partial_{t}\|{u^{\varepsilon}_{4}}\|^{2}+2d_{4}\|\nabla{u^{\varepsilon}_{4}}\|^{2}$ $\displaystyle\leq{k^{\infty}_{2}}(\|{u^{\varepsilon}_{1}}\|^{2}+\|{u^{\varepsilon}_{4}}\|^{2}).$ (40) Putting together (37)–(40), we get $\displaystyle\partial_{t}\Sigma^{4}_{i=1}\|{u^{\varepsilon}_{i}}\|^{2}$ $\displaystyle+(2d_{1}-C^{}C_{4}\varepsilon^{2})\|\nabla{u^{\varepsilon}_{1}}\|^{2}+2d_{2}\|\nabla{u^{\varepsilon}_{2}}\|^{2}+2d_{3}\|\nabla{u^{\varepsilon}_{3}}\|^{2}$ (41) $\displaystyle+2d_{4}\|\nabla{u^{\varepsilon}_{4}}\|^{2}+2k_{1}\|{u^{\varepsilon}_{1}}\|^{2}$ $\displaystyle\leq\left(2{k^{\infty}_{1}}+{k^{\infty}_{2}}+C^{}C_{2}\right)\Sigma^{4}_{i=1}\|{u^{\varepsilon}_{i}}\|^{2}$ $\displaystyle+C^{}\frac{C_{1}^{2}}{2\delta}\int_{0}^{t}(\|{u^{\varepsilon}_{1}}\|^{2}+\varepsilon^{2}\|\nabla{u^{\varepsilon}_{1}}\|^{2})d\tau$ $\displaystyle+2\varepsilon b\|{u^{\varepsilon}_{2}}\|^{2}_{\Gamma^{wa}_{\varepsilon}}+2\varepsilon a\|{u^{\varepsilon}_{3}}\|^{2}_{\Gamma^{wa}_{\varepsilon}}$ $\displaystyle+\varepsilon(a^{\infty}+b^{\infty})(\delta\|{u^{\varepsilon}_{2}}\|_{\Gamma^{wa}_{\varepsilon}}^{2}+\frac{1}{\delta}\|{u^{\varepsilon}_{3}}\|_{\Gamma^{wa}_{\varepsilon}}^{2}).$ Applying the trace inequality (7) to the boundary terms in (41), we get $\displaystyle\partial_{t}\Sigma^{4}_{i=1}\|{u^{\varepsilon}_{i}}\|^{2}$ $\displaystyle+(2d_{1}-2C^{}C_{4}\varepsilon^{2})\|\nabla{u^{\varepsilon}_{1}}\|^{2}$ $\displaystyle+(2d_{2}-2C^{}b^{\infty}\varepsilon^{2}-C^{}\delta\varepsilon^{2}(a^{\infty}+b^{\infty}))\|\nabla{u^{\varepsilon}_{2}}\|^{2}$ $\displaystyle+(2d_{3}-2C^{}a^{\infty}\varepsilon^{2}-\frac{C^{}\varepsilon^{2}}{\delta}(a^{\infty}+b^{\infty}))\|\nabla{u^{\varepsilon}_{3}}\|^{2}$ $\displaystyle+2d_{4}\|\nabla{u^{\varepsilon}_{4}}\|^{2}+2k_{1}\|{u^{\varepsilon}_{1}}\|^{2}\leq C_{5}\Sigma^{4}_{i=1}\|{u^{\varepsilon}_{i}}\|^{2}$ $\displaystyle+C^{}\frac{C_{1}^{2}}{2\delta}\int_{0}^{t}(\|{u^{\varepsilon}_{1}}\|^{2}+\varepsilon^{2}\|\nabla{u^{\varepsilon}_{1}}\|^{2})d\tau,$ (42) where $C_{5}:=2{k^{\infty}_{1}}+{k^{\infty}_{2}}+C^{}C_{2}+2C^{}(a^{\infty}+b^{\infty})+C^{}(a^{\infty}+b^{\infty})(\delta+\frac{1}{\delta})$. Let us choose $\varepsilon$ and $\delta$ such that $\displaystyle\varepsilon$ $\displaystyle\in\left]0,\frac{2d_{1}}{C_{1}C^{}}\right[$ $\displaystyle\delta$ $\displaystyle\in\left[\frac{C^{}\varepsilon^{2}(a^{\infty}+b^{\infty})}{2d_{3}-C^{}a^{\infty}\varepsilon^{2}},\frac{2d_{2}-C^{}b^{\infty}\varepsilon^{2}}{C^{}\varepsilon^{2}(a^{\infty}+b^{\infty})}\right].$ With this choice of $(\varepsilon,\delta)$, (42) takes the form $\partial_{t}\Sigma^{4}_{i=1}\|{u^{\varepsilon}_{i}}\|^{2}+\bar{C}\|\nabla{u^{\varepsilon}_{1}}\|^{2}+\bar{C}\|{u^{\varepsilon}_{1}}\|^{2}\leq{C_{6}}{}(\Sigma^{4}_{i=1}\|{u^{\varepsilon}_{i}}\|^{2}+\int_{0}^{t}(\|{u^{\varepsilon}_{1}}\|^{2}+\varepsilon^{2}\|\nabla{u^{\varepsilon}_{1}}\|^{2})d\tau),$ where $C_{6}:=2{k^{\infty}_{1}}+{k^{\infty}_{2}}+C^{}C_{2}+C^{}(a^{\infty}+b^{\infty})+C^{}\frac{C_{1}^{2}}{2\delta}$ and $\bar{C}:=\min\\{2d_{1}-2C^{}C_{2}\varepsilon^{2},2k_{1}\\}$. Taking in (42) the supremum along $t\in(0,T)$ and applying Gronwall’s inequality, we obtain the following estimate $\Sigma^{4}_{i=1}\|{u^{\varepsilon}_{i}}\|^{2}+\bar{C}\int_{0}^{T}\|\nabla{u^{\varepsilon}_{1}}\|^{2}dt+\bar{C}\int_{0}^{T}\|{u^{\varepsilon}_{1}}\|^{2}dt\leq 0.$ (43) Thus, the proof of Proposition 3 is completed. ###### Theorem 4. (Global Existence) Assume $(A1)-(A3)$. Then there exists at least a global-in- time weak solution in the sense of Definition 1. Proof. The proof is based on the Galerkin argument. Since the proof is rather standard, and here we wish to focus on the passage to the limit $\varepsilon\to 0$, we omit it. ## 4 A priori estimates for microscopic solutions This section includes the $\varepsilon-$ independent estimates. ###### Lemma 5. Assume (A1)-(A6). Then the weak solution of the microscopic model (12)-(17) satisfies the following a priori bounds: $\displaystyle\parallel u^{\varepsilon}_{j}\parallel_{L^{2}(0,T;H^{1}(\Omega^{\varepsilon}))}\leq C,\;j\in\\{1,2,3,4\\}$ (44) $\displaystyle\parallel\nabla\partial_{t}u^{\varepsilon}_{2}\parallel_{L^{2}(0,T;L^{2}(\Omega^{\varepsilon}))}\leq C,$ (45) $\displaystyle\parallel\partial_{t}u^{\varepsilon}_{j}\parallel_{L^{2}(0,T;L^{2}(\Omega^{\varepsilon}))}\leq C,$ (46) $\displaystyle\parallel u^{\varepsilon}_{3}\parallel_{L^{2}(0,T;H^{1}(\Omega^{\varepsilon}_{1}))}\leq C,$ (47) $\displaystyle\parallel\nabla\partial_{t}u^{\varepsilon}_{3}\parallel_{L^{2}(0,T;L^{2}(\Omega^{\varepsilon}_{1}))}\leq C,$ (48) $\displaystyle\parallel\partial_{t}u^{\varepsilon}_{3}\parallel_{L^{2}(0,T;L^{2}(\Omega^{\varepsilon}_{1}))}\leq C,$ (49) $\displaystyle\parallel u^{\varepsilon}_{5}\parallel_{L^{\infty}((0,T)\times\Gamma^{sw}_{\varepsilon})}\leq C,$ (50) $\displaystyle\parallel\partial_{t}u^{\varepsilon}_{5}\parallel_{L^{2}((0,T)\times\Gamma^{sw}_{\varepsilon})}\leq C.$ (51) In (44)–(51), the generic constant $C$ is independent of $\varepsilon$. Proof. We test (12) with $\varphi_{1}=u^{\varepsilon}_{1}$ to get $\displaystyle\frac{1}{2}\partial_{t}\|u^{\varepsilon}_{1}\|^{2}+d_{1}\|\nabla u^{\varepsilon}_{1}\|^{2}$ $\displaystyle\leq- k_{1}\|u^{\varepsilon}_{1}\|^{2}+k_{2}^{\infty}(u^{\varepsilon}_{1},u^{\varepsilon}_{2})-\varepsilon(\eta,u^{\varepsilon}_{1})_{\Gamma^{sw}_{\varepsilon}},$ (52) $\displaystyle\leq\frac{k_{2}^{\infty}}{2}(\|u^{\varepsilon}_{1}\|^{2}+\|u^{\varepsilon}_{2}\|^{2})+\varepsilon k^{\infty}_{3}Q^{\infty}c_{R}(u^{\varepsilon}_{1},u^{\varepsilon}_{1})_{\Gamma^{sw}_{\varepsilon}}.$ After applying the trace inequality to the last term on r.h.s of (52), we get $\displaystyle\frac{1}{2}\partial_{t}\|u^{\varepsilon}_{1}\|^{2}+d_{1}\|\nabla u^{\varepsilon}_{1}\|^{2}$ $\displaystyle\leq\frac{k_{2}^{\infty}}{2}(\|u^{\varepsilon}_{1}\|^{2}+\|u^{\varepsilon}_{2}\|^{2})+C^{}k^{\infty}_{3}Q^{\infty}c_{R}(\|u^{\varepsilon}_{1}\|^{2}+\varepsilon^{2}\|\nabla u^{\varepsilon}_{1}\|^{2})_{\Gamma^{sw}_{\varepsilon}}.$ $\displaystyle\frac{1}{2}\partial_{t}\|u^{\varepsilon}_{1}\|^{2}$ $\displaystyle+(d_{1}-\varepsilon^{2}C^{}k^{\infty}_{3}Q^{\infty}c_{R})\|\nabla u^{\varepsilon}_{1}\|^{2}\leq C_{7}(\|u^{\varepsilon}_{1}\|^{2}+\|u^{\varepsilon}_{2}\|^{2}),$ (53) where $C_{7}:=\frac{k_{2}^{\infty}}{2}+C^{}k^{\infty}_{3}Q^{\infty}c_{R}.$ Taking $\varphi_{2}=u^{\varepsilon}_{2}$ in (13), we get $\displaystyle\frac{1}{2}\partial_{t}\|u^{\varepsilon}_{2}\|^{2}+d_{2}\|\nabla u^{\varepsilon}_{2}\|^{2}$ $\displaystyle\leq\frac{k_{1}^{\infty}}{2}(\|u^{\varepsilon}_{1}\|^{2}+\|u^{\varepsilon}_{2}\|^{2})-k_{2}\|u^{\varepsilon}_{2}\|^{2}$ $\displaystyle+\varepsilon a^{\infty}(u^{\varepsilon}_{3},u^{\varepsilon}_{2})_{\Gamma^{wa}_{\varepsilon}}+\varepsilon b^{\infty}\|u^{\varepsilon}_{2}\|^{2}_{\Gamma^{wa}_{\varepsilon}}.$ Application of the trace inequality (7) only to the last term leads to $\displaystyle\frac{1}{2}\partial_{t}\|u^{\varepsilon}_{2}\|^{2}+(d_{2}-C^{}b^{\infty}\varepsilon^{2})\|\nabla u^{\varepsilon}_{2}\|^{2}$ $\displaystyle\leq\frac{k_{1}^{\infty}}{2}(\|u^{\varepsilon}_{1}\|^{2}+\|u^{\varepsilon}_{2}\|^{2})+2$ (54) $\displaystyle+\varepsilon a^{\infty}(u^{\varepsilon}_{3},u^{\varepsilon}_{2})_{\Gamma^{wa}_{\varepsilon}}.$ We choose $\varphi_{3}=u^{\varepsilon}_{3}$ as a test function in (14) to calculate $\displaystyle\frac{1}{2}\partial_{t}\|u^{\varepsilon}_{3}\|^{2}+(d_{3}-C^{}a^{\infty}\varepsilon^{2})\|\nabla u^{\varepsilon}_{3}\|^{2}$ $\displaystyle\leq\varepsilon b^{\infty}(u^{\varepsilon}_{3},u^{\varepsilon}_{2})_{\Gamma^{wa}_{\varepsilon}}+C^{}a^{\infty}\|u^{\varepsilon}_{3}\|^{2}.$ (55) Setting $\varphi_{4}=u^{\varepsilon}_{4}$ in (15), we are led to $\displaystyle\frac{1}{2}\partial_{t}\|u^{\varepsilon}_{4}\|^{2}+d_{4}\|\nabla u^{\varepsilon}_{4}\|^{2}$ $\displaystyle\leq$ $\displaystyle\frac{k_{1}^{\infty}}{2}(\|u^{\varepsilon}_{1}\|^{2}+\|u^{\varepsilon}_{4}\|^{2}).$ (56) Putting together (53)-(56), we obtain $\displaystyle\frac{1}{2}\Sigma_{i=1}^{4}\partial_{t}\|u^{\varepsilon}_{i}\|^{2}$ $\displaystyle+(d_{1}-\varepsilon^{2}C^{}k^{\infty}_{3}Q^{\infty}c_{R})\|\nabla u^{\varepsilon}_{1}\|^{2}+d_{4}\|\nabla u^{\varepsilon}_{4}\|^{2}$ (57) $\displaystyle+(d_{2}-C^{}b^{\infty}\varepsilon^{2})\|\nabla u^{\varepsilon}_{2}\|^{2}+(d_{3}-C^{}a^{\infty}\varepsilon^{2})\|\nabla u^{\varepsilon}_{3}\|^{2}$ $\displaystyle\leq({k_{1}^{\infty}}{}+\frac{k_{2}^{\infty}}{2}+C^{}b^{\infty}+C^{}a^{\infty})\Sigma_{i=1}^{4}\|u^{\varepsilon}_{i}\|^{2}$ $\displaystyle+\varepsilon(a^{\infty}+b^{\infty})(u^{\varepsilon}_{3},u^{\varepsilon}_{2})_{\Gamma^{wa}_{\varepsilon}}.$ Combing Young’s inequality and the trace inequality to the boundary term, (57) turns out to be $\displaystyle\frac{1}{2}\Sigma_{i=1}^{4}\partial_{t}\|u^{\varepsilon}_{i}\|^{2}$ $\displaystyle+(d_{1}-\varepsilon^{2}C^{}k^{\infty}_{3}Q^{\infty}c_{R})\|\nabla u^{\varepsilon}_{1}\|^{2}$ $\displaystyle+(d_{2}-C^{}b\varepsilon^{2}-\frac{C^{}\varepsilon^{2}\delta}{2}(a^{\infty}+b^{\infty}))\|\nabla u^{\varepsilon}_{2}\|^{2}$ $\displaystyle+(d_{3}-C^{}a\varepsilon^{2}-\frac{C^{}\varepsilon^{2}}{2\delta}(a^{\infty}+b^{\infty}))\|\nabla u^{\varepsilon}_{3}\|^{2}+d_{4}\|\nabla u^{\varepsilon}_{4}\|^{2}$ $\displaystyle\leq({k_{1}^{\infty}}{}+\frac{k_{2}^{\infty}}{2}+C^{}(a^{\infty}+b^{\infty})(\delta+\frac{1}{\delta}))\Sigma_{i=1}^{4}\|u^{\varepsilon}_{i}\|^{2}.$ Choosing $\varepsilon$ small enough and $\delta$ conveniently such that the coefficients of the terms involving $\|\nabla u^{\varepsilon}_{2}\|^{2}$ and $\|\nabla u^{\varepsilon}_{3}\|^{2}$ stay positive, we are led to $\displaystyle\Sigma_{i=1}^{4}\partial_{t}\|u^{\varepsilon}_{i}\|^{2}+d^{\prime}_{1}\|\nabla u^{\varepsilon}_{1}\|^{2}+d^{\prime}_{2}\|\nabla u^{\varepsilon}_{2}\|^{2}+d^{\prime}_{3}\|\nabla u^{\varepsilon}_{3}\|^{2}$ $\displaystyle+$ $\displaystyle 2d_{4}\|\nabla u^{\varepsilon}_{4}\|^{2}\leq C_{7}\Sigma_{i=1}^{4}\|u^{\varepsilon}_{i}\|^{2},$ where $\displaystyle d^{\prime}_{1}$ $\displaystyle:=2(d_{1}-\varepsilon^{2}C^{}k^{\infty}_{3}Q^{\infty}c_{R}),$ $\displaystyle d^{\prime}_{2}$ $\displaystyle:=2(d_{2}-C^{}b^{\infty}\varepsilon^{2}-\frac{C^{}\varepsilon^{2}\delta}{2}(a^{\infty}+b^{\infty})),$ $\displaystyle d^{\prime}_{3}$ $\displaystyle:=2(d_{3}-C^{}a^{\infty}\varepsilon^{2}-\frac{C^{}b\varepsilon^{2}}{2\delta}(a^{\infty}+b^{\infty})),$ while the constant $C$ is given by $C_{8}:=2{k_{1}^{\infty}}{}+\frac{k_{2}^{\infty}}{2}+C^{}a^{\infty}+C^{}b^{\infty}+C^{}(a^{\infty}+b^{\infty})(\delta+\frac{1}{\delta}).$ Summarizing, we have $\displaystyle\Sigma_{i=1}^{4}\partial_{t}\|u^{\varepsilon}_{i}\|^{2}+d_{0}\Sigma_{j=1}^{3}\|\nabla u^{\varepsilon}_{j}\|^{2}+d_{0}\|\nabla u^{\varepsilon}_{3}\|^{2}\leq C\Sigma_{i=1}^{4}\|u^{\varepsilon}_{i}\|^{2},$ (58) where $d_{0}:=min\\{d^{\prime}_{1},d^{\prime}_{2},d^{\prime}_{3},d^{\prime}_{4}\\}$. By Gronwall’s inequality, we have $\displaystyle\Sigma_{i=1}^{4}\|u^{\varepsilon}_{i}\|^{2}\leq C\Sigma_{i=1}^{4}\|u_{i}(0)\|^{2},$ and hence, $\displaystyle\parallel u^{\varepsilon}_{j}\parallel_{L^{2}(0,T;L^{2}(\Omega^{\varepsilon}))}\leq C\mbox{ for all }i\in\\{1,2,4\\}\;\mbox{and}\;\\|u^{\varepsilon}_{3}\\|_{{L^{2}(0,T;L^{2}(\Omega^{\varepsilon}_{1}))}}\leq C,$ (59) where $C$ depends on initial data and model parameters but is independent of $\varepsilon$. Integrating (58) along $(0,T)$, we get $\displaystyle\parallel u^{\varepsilon}_{j}\parallel_{L^{2}(0,T;H^{1}(\Omega^{\varepsilon}))}$ $\displaystyle\leq C,\;j\in\\{1,2,4\\},$ (60) $\displaystyle\parallel u^{\varepsilon}_{3}\parallel_{L^{2}(0,T;H^{1}(\Omega^{\varepsilon}_{1}))}$ $\displaystyle\leq C.$ With the help of (A2) together with the boundedness of $u^{\varepsilon}_{1}$, we conclude from (16) that $\displaystyle\parallel u^{\varepsilon}_{5}\parallel_{L^{\infty}((0,T)\times\Gamma^{sw}_{\varepsilon})}\leq C.$ Multiplying (16) by $\partial_{t}u^{\varepsilon}_{5}$ and using (A2), we get $\displaystyle\\|\partial_{t}u^{\varepsilon}_{5}\\|_{L^{2}((0,T)\times\Gamma^{sw}_{\varepsilon})}\leq C.$ Now, we focus on obtaining $\varepsilon-$independent estimates on the time derivative of the concentrations. Firstly, we choose $\varphi_{1}=\partial_{t}u^{\varepsilon}_{1}$ and get $\displaystyle\int_{0}^{t}\int_{\Omega^{\varepsilon}}\partial_{t}u^{\varepsilon}_{1}\partial_{t}u^{\varepsilon}_{1}dxd\tau$ $\displaystyle+\int_{0}^{t}\int_{\Omega^{\varepsilon}}d_{1}^{\varepsilon}\nabla u^{\varepsilon}_{1}\nabla\partial_{t}u^{\varepsilon}_{1}dxd\tau$ (61) $\displaystyle=-\int_{0}^{t}\int_{\Omega^{\varepsilon}}k^{\varepsilon}_{1}u^{\varepsilon}_{1}\partial_{t}u^{\varepsilon}_{1}dxd\tau+\int_{0}^{t}\int_{\Omega^{\varepsilon}}k^{\varepsilon}_{2}u^{\varepsilon}_{2}\partial_{t}u^{\varepsilon}_{1}dxd\tau$ $\displaystyle-\varepsilon\int_{0}^{t}\int_{\Gamma^{sw}_{\varepsilon}}\eta\partial_{t}u^{\varepsilon}_{1}d\sigma_{x}d\tau.$ Consequently, it holds $\displaystyle\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{1}\|^{2}dxd\tau$ $\displaystyle+\int_{0}^{t}\int_{\Omega^{\varepsilon}}\left(\frac{1}{2}\partial_{t}(d_{1}^{\varepsilon}\|\nabla u^{\varepsilon}_{1}\|^{2})-(\partial_{t}d_{1}^{\varepsilon})\|\nabla u^{\varepsilon}_{1}\|^{2}\right)dxd\tau$ $\displaystyle\leq-\frac{k_{1}}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\partial_{t}\|u^{\varepsilon}_{1}\|^{2}dxd\tau$ $\displaystyle+\frac{k_{2}^{\infty}}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\left(\frac{1}{\delta}\|u^{\varepsilon}_{2}\|^{2}+\delta\|\partial_{t}u^{\varepsilon}_{1}\|^{2}\right)dxd\tau$ $\displaystyle-{\varepsilon}{}\int_{0}^{t}\int_{\Gamma^{sw}_{\varepsilon}}(\partial_{t}(\eta u^{\varepsilon}_{1})-(\partial_{t}\eta)u^{\varepsilon}_{1})d\sigma_{x}d\tau,$ $\displaystyle(1-\frac{k_{2}^{\infty}\delta}{2})\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{1}\|^{2}dxd\tau$ $\displaystyle\leq D_{1}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\nabla u^{\varepsilon}_{1}\|^{2}dxd\tau$ (62) $\displaystyle+\frac{d_{1}^{\infty}}{2}\int_{\Omega^{\varepsilon}}\|\nabla{u_{1}}_{0}\|^{2}dx+\frac{k_{2}^{\infty}}{2\delta}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|u^{\varepsilon}_{2}\|^{2}dxd\tau$ $\displaystyle+\frac{\varepsilon}{2}\int_{\Gamma^{sw}_{\varepsilon}}\left(\|\eta\|^{2}+\|u^{\varepsilon}_{1}\|^{2}+\|\eta(0)\|^{2}+\|u^{\varepsilon}_{1}(0)\|^{2}\right)d\sigma_{x}$ $\displaystyle+\frac{\varepsilon}{2}\int_{0}^{t}\int_{\Gamma^{sw}_{\varepsilon}}\left(\|\partial_{t}\eta\|^{2}+\|u^{\varepsilon}_{1}\|^{2}\right)d\sigma_{x}d\tau,$ where $\eta(0):=\eta(u^{\varepsilon}_{1}(0),u^{\varepsilon}_{5}(0))$. Applying (7) and recalling (60), we have $\displaystyle\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{1}\|^{2}dxd\tau$ $\displaystyle\leq$ $\displaystyle C_{9},$ (63) where $\displaystyle C_{9}$ $\displaystyle:=D_{1}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\nabla u^{\varepsilon}_{1}\|^{2}dxd\tau+\frac{k_{1}}{2}\int_{\Omega^{\varepsilon}}\|u^{\varepsilon}_{1}(0)\|^{2}dx+\frac{d_{1}^{\infty}}{2}\int_{\Omega^{\varepsilon}}\|\nabla{u_{1}}_{0}\|^{2}dx$ $\displaystyle+\frac{k_{2}^{\infty}}{2\delta}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|u^{\varepsilon}_{2}\|^{2}dxd\tau+\frac{\varepsilon}{2}\int_{\Gamma^{sw}_{\varepsilon}}(\|\overline{\eta}\|^{2}+\|{\eta}(0)\|^{2}+\|\hat{\eta}\|^{2})$ $\displaystyle+\frac{C^{}}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}(\|u^{\varepsilon}_{1}\|^{2}+\varepsilon^{2}\|\nabla u^{\varepsilon}_{1}\|^{2})dxd\tau+\frac{C^{}}{2}\int_{\Omega^{\varepsilon}}(\|u^{\varepsilon}_{1}\|^{2}+\varepsilon^{2}\|\nabla u^{\varepsilon}_{1}\|^{2})dxd\tau,$ and $\delta\in\left]0,\frac{2}{k_{2}^{\infty}}\right[$. Testing (13) with $\varphi_{2}=\partial_{t}u^{\varepsilon}_{2}$ gives $\displaystyle\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{2}\|^{2}dxd\tau$ $\displaystyle+$ $\displaystyle\int_{0}^{t}\int_{\Omega^{\varepsilon}}(\frac{1}{2}\partial_{t}(d_{2}^{\varepsilon}\|\nabla u^{\varepsilon}_{2}\|^{2})-(\partial_{t}d_{2}^{\varepsilon})\|\nabla u^{\varepsilon}_{2}\|^{2})dxd\tau$ $\displaystyle\leq$ $\displaystyle-\frac{k_{2}}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\partial_{t}\|u^{\varepsilon}_{2}\|^{2}dxd\tau+\frac{k_{1}^{\infty}}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}(\frac{1}{\delta}\|u^{\varepsilon}_{1}\|^{2}$ $\displaystyle+$ $\displaystyle\delta\|\partial_{t}u^{\varepsilon}_{2}\|^{2})dxd\tau+\frac{\varepsilon a^{\infty}}{2}\int_{0}^{t}\int_{\Gamma^{wa}_{\varepsilon}}\left(\|u^{\varepsilon}_{3}\|^{2}+\|\partial_{t}u^{\varepsilon}_{2}\|^{2}\right)d\sigma_{x}d\tau$ $\displaystyle+$ $\displaystyle\frac{\varepsilon b^{\infty}}{2}\int_{0}^{t}\int_{\Gamma^{wa}_{\varepsilon}}\partial_{t}\|u^{\varepsilon}_{2}\|^{2}d\sigma_{x}d\tau,$ and hence, $\displaystyle\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{2}\|^{2}dx\tau$ $\displaystyle+$ $\displaystyle\frac{d_{2}}{2}\int_{\Omega^{\varepsilon}}\|\nabla{u_{2}}^{\varepsilon}\|^{2}dxd\tau$ $\displaystyle\leq$ $\displaystyle\frac{d_{2}^{\infty}}{2}\int_{\Omega^{\varepsilon}}\|\nabla u^{\varepsilon}_{2}(0)\|^{2}dx+D_{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\nabla u^{\varepsilon}_{2}\|^{2}dxd\tau$ $\displaystyle+$ $\displaystyle\frac{k_{1}^{\infty}}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}(\frac{1}{\delta}\|u^{\varepsilon}_{1}\|^{2}+\delta\|\partial_{t}u^{\varepsilon}_{2}\|^{2})dxd\tau$ $\displaystyle+$ $\displaystyle\frac{C^{}a^{\infty}}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\left(\|u^{\varepsilon}_{3}\|^{2}+\varepsilon^{2}\|\nabla u^{\varepsilon}_{3}\|^{2}+\varepsilon^{2}\|\nabla\partial_{t}u^{\varepsilon}_{2}\|^{2}\right)dxd\tau$ $\displaystyle+$ $\displaystyle\frac{\varepsilon b^{\infty}}{2}\int_{\Gamma^{wa}_{\varepsilon}}(\|u^{\varepsilon}_{2}\|^{2}-\|u^{\varepsilon}_{2}(0)\|^{2})d\sigma_{x}.$ By (7) and (60), we get $\displaystyle\left(1-\frac{C^{}a^{\infty}}{2}-\frac{k_{1}^{\infty}\delta}{2}\right)\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{2}\|^{2}dxd\tau$ $\displaystyle\leq$ $\displaystyle C_{10}\left(1+\varepsilon^{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\nabla\partial_{t}u^{\varepsilon}_{2}\|^{2}dxd\tau\right).$ Consequently, choosing $\delta\in]0,\frac{2-C^{}a^{\infty}}{k_{1}^{\infty}}[$, we are led to $\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{2}\|^{2}dxd\tau\leq C_{10}(1+\varepsilon^{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\nabla\partial_{t}u^{\varepsilon}_{2}\|^{2}dxd\tau),$ (64) where $\displaystyle C_{10}$ $\displaystyle:=$ $\displaystyle D_{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\nabla u^{\varepsilon}_{1}\|^{2}dxd\tau+\frac{d_{2}^{\infty}}{2}\int_{\Omega^{\varepsilon}}\|\nabla{u^{\varepsilon}}_{2}(0)\|^{2}dx+\frac{k_{1}^{\infty}}{2\delta}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|u^{\varepsilon}_{2}\|^{2}dxd\tau$ $\displaystyle+$ $\displaystyle\frac{C^{}b^{\infty}}{2}\int_{\Omega^{\varepsilon}}\left(\|u^{\varepsilon}_{2}\|^{2}+\varepsilon^{2}\|\nabla u^{\varepsilon}_{2}\|^{2}+\|u^{\varepsilon}_{2}(0)\|^{2}+\varepsilon^{2}\|\nabla u^{\varepsilon}_{2}(0)\|^{2}\right)dx$ $\displaystyle+$ $\displaystyle\frac{C^{}a^{\infty}}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}(\|u^{\varepsilon}_{3}\|^{2}+\varepsilon^{2}\|\nabla u^{\varepsilon}_{3}\|^{2})dxd\tau.$ The initial data $u^{\varepsilon}_{30}$ holding in $\Omega^{\varepsilon}_{1}$ and the Dirichlet data $u_{3}^{D}$ acting on the exterior boundary of $\Omega_{1}^{\varepsilon}$ are considered here as restrictions of the respective functions defined on whole of $\overline{\Omega}$. Testing now (14) with $\varphi_{3}=\partial_{t}(u^{\varepsilon}_{3}-u_{3}^{D})$ leads to $\displaystyle\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{3}\|^{2}dxd\tau$ $\displaystyle+\frac{d_{3}}{2}\int_{\Omega^{\varepsilon}}\|\nabla{u^{\varepsilon}}_{3}\|^{2}$ $\displaystyle\leq\frac{d_{3}}{2}\int_{\Omega^{\varepsilon}}\|\nabla{u^{\varepsilon}_{3}}(0)\|^{2}+\frac{1}{2}(\|\partial_{t}u^{\varepsilon}_{3}\|^{2}+\|\partial_{t}u^{D}_{3}\|^{2})$ $\displaystyle+D_{3}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\nabla{u^{\varepsilon}_{3}}\|^{2}+\frac{d_{3}^{\infty}}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}(\|\nabla{u^{\varepsilon}_{3}}\|^{2}+\|\nabla\partial_{t}u^{D}_{3}\|^{2})$ $\displaystyle+\frac{{\varepsilon a^{\infty}}}{\delta}\int_{0}^{t}\int_{\Gamma^{wa}_{\varepsilon}}\|u^{\varepsilon}_{3}\|^{2}+\frac{\varepsilon\delta}{2}(a^{\infty}+a^{\infty})\int_{0}^{t}\int_{\Gamma^{wa}_{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{3}\|^{2}$ $\displaystyle+\frac{\varepsilon}{2}(a^{\infty}+b^{\infty})\int_{0}^{t}\int_{\Gamma^{wa}_{\varepsilon}}\|\partial_{t}u^{D}_{3}\|^{2}+\frac{\varepsilon b^{\infty}}{\delta}\int_{0}^{t}\int_{\Gamma^{wa}_{\varepsilon}}\|u^{\varepsilon}_{2}\|^{2}.$ Using (7) and (A6), we obtain $\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{3}\|^{2}dxd\tau\leq C_{11}(1+\varepsilon^{2}\delta\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\nabla\partial_{t}u^{\varepsilon}_{3}\|^{2}dxd\tau),$ (65) where $\delta\in]0,\frac{2}{C^{}(a^{\infty}+b^{\infty})}[$ and $\displaystyle C_{11}$ $\displaystyle:=D_{3}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\nabla u^{\varepsilon}_{3}\|^{2}dxd\tau+\frac{d_{3}}{2}\int_{\Omega^{\varepsilon}}\|\nabla{u^{\varepsilon}_{3}}(0)\|^{2}dx+\frac{1}{2\delta}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\nabla u^{D}_{3}\|^{2}$ $\displaystyle+\frac{d^{\infty}_{3}}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}(\|\nabla u^{\varepsilon}_{3}\|^{2}+\|\nabla\partial_{t}u^{D}_{3}\|^{2})+\frac{C^{}a^{\infty}}{\delta}\int_{0}^{t}\int_{\Omega^{\varepsilon}}(\|u^{\varepsilon}_{3}\|^{2}+\varepsilon^{2}\|\nabla u^{\varepsilon}_{3}\|^{2})$ $\displaystyle+\frac{C^{}b^{\infty}}{\delta}\int_{0}^{t}\int_{\Omega^{\varepsilon}}(\|u^{\varepsilon}_{2}\|^{2}+\varepsilon^{2}\|\nabla u^{\varepsilon}_{2}\|^{2})dxd\tau$ $\displaystyle+\frac{C^{}(a^{\infty}+b^{\infty})}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}(\|\partial_{t}u^{D}_{3}\|^{2}+\varepsilon^{2}\|\nabla\partial_{t}u^{D}_{3}\|^{2})dxd\tau.$ From (15), we get $\displaystyle\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{4}\|^{2}dxd\tau$ $\displaystyle\leq$ $\displaystyle C_{12}.$ (66) In order to estimate (64) and (65), we proceed first with differentiating (13) with respect to time and then testing the result with $\partial_{t}u^{\varepsilon}_{2}$. Consequently, we derive $\displaystyle\frac{1}{2}\int_{\Omega^{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{2}\|^{2}dx$ $\displaystyle+$ $\displaystyle{d_{2}}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\nabla\partial_{t}{u_{2}}^{\varepsilon}\|^{2}dxd\tau$ (67) $\displaystyle+$ $\displaystyle\int_{0}^{t}\int_{\Omega^{\varepsilon}}\left(\frac{1}{2}(\partial_{t}d_{2}\|\nabla{u_{2}}^{\varepsilon}\|^{2}-(\partial_{t}\partial_{t}d_{2})\|\nabla{u_{2}}^{\varepsilon}\|^{2}\right)$ $\displaystyle\leq$ $\displaystyle\frac{k_{1}^{\infty}}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}(\|\partial_{t}u^{\varepsilon}_{1}\|^{2}+\|\partial_{t}u^{\varepsilon}_{2}\|^{2})+\frac{K_{1}^{\infty}}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}(\|u^{\varepsilon}_{1}\|^{2}+\|\partial_{t}u^{\varepsilon}_{2}\|^{2})$ $\displaystyle-$ $\displaystyle k_{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{2}\|^{2}-\frac{K_{2}}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\partial_{t}\|u^{\varepsilon}_{2}\|^{2}$ $\displaystyle+$ $\displaystyle\frac{{\varepsilon a^{\infty}}}{2}\int_{0}^{t}\int_{\Gamma^{wa}_{\varepsilon}}(\frac{1}{\delta}\|\partial_{t}u^{\varepsilon}_{2}\|^{2}+\delta\|\partial_{t}u^{\varepsilon}_{3}\|^{2})d\sigma_{x}d\tau$ $\displaystyle+$ $\displaystyle\frac{\varepsilon b^{\infty}}{2}\int_{0}^{t}\int_{\Gamma^{wa}_{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{2}\|^{2}d\sigma_{x}d\tau+\frac{\varepsilon B}{2}\int_{0}^{t}\int_{\Gamma^{wa}_{\varepsilon}}\partial_{t}\|u^{\varepsilon}_{2}\|^{2}d\sigma_{x}d\tau.$ Using (7), it yields $\displaystyle\frac{1}{2}\int_{\Omega^{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{2}\|^{2}dx$ $\displaystyle+\left({d_{2}}-\frac{C^{}A^{\infty}\varepsilon^{2}}{2}-\frac{C^{}B^{\infty}\varepsilon^{2}}{2}-\frac{C^{}a^{\infty}\varepsilon^{2}}{2\delta}\right)\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\nabla\partial_{t}{u_{2}}^{\varepsilon}\|^{2}dxd\tau$ (68) $\displaystyle\leq C_{13}+\frac{C^{}a^{\infty}\delta}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}(\|\partial_{t}u^{\varepsilon}_{3}\|^{2}+\varepsilon^{2}\|\nabla\partial_{t}u^{\varepsilon}_{3}\|^{2})$ $\displaystyle+\left(\frac{k_{1}^{\infty}}{2}+\frac{K_{1}^{\infty}}{2}-k_{2}+\frac{C^{}A^{\infty}}{2}+\frac{C^{}B^{\infty}\varepsilon^{2}}{2}+\frac{C^{}a^{\infty}}{2\delta}\right)\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{2}\|^{2},$ where $C_{13}$ depends on the bounded terms of r.h.s of (67). Differentiating now (14) with respect to time and then testing the result with $\partial_{t}(u^{\varepsilon}_{3}-u^{D}_{3})$, we get $\displaystyle\frac{1}{2}\int_{\Omega^{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{3}\|^{2}dx$ $\displaystyle+$ $\displaystyle{d_{3}}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\nabla\partial_{t}{u_{3}}^{\varepsilon}\|^{2}dxd\tau$ $\displaystyle+$ $\displaystyle\int_{0}^{t}\int_{\Omega^{\varepsilon}}\left(\frac{1}{2}(\partial_{t}d_{3}\|\nabla{u_{3}}^{\varepsilon}\|^{2}-(\partial_{t}\partial_{t}d_{3})\|\nabla{u_{3}}^{\varepsilon}\|^{2}\right)$ $\displaystyle\leq$ $\displaystyle\frac{D_{3}}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\nabla{u_{3}}^{\varepsilon}\|^{2}dxd\tau+\frac{d^{\infty}_{3}}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\nabla\partial_{t}{u_{3}}^{\varepsilon}\|^{2}dxd\tau$ $\displaystyle+$ $\displaystyle\frac{d^{\infty}_{3}+D_{3}}{2}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\nabla\partial_{t}{u_{3}}^{D}\|^{2}dxd\tau+\frac{\varepsilon A^{\infty}}{2}\int_{0}^{t}\int_{\Gamma^{wa}_{\varepsilon}}\partial_{t}\|{u_{3}}^{\varepsilon}\|^{2}dxd\tau$ $\displaystyle+$ $\displaystyle{\varepsilon a^{\infty}}{}\int_{0}^{t}\int_{\Gamma^{wa}_{\varepsilon}}\|\partial_{t}{u_{3}}^{\varepsilon}\|^{2}dxd\tau+\frac{\varepsilon A^{\infty}}{2}\int_{0}^{t}\int_{\Gamma^{wa}_{\varepsilon}}(\|{u_{3}}^{\varepsilon}\|^{2}+\|\partial_{t}{u_{3}}^{D}\|^{2})dxd\tau$ $\displaystyle+$ $\displaystyle\frac{\varepsilon a^{\infty}}{2}\int_{0}^{t}\int_{\Gamma^{wa}_{\varepsilon}}(\|\partial_{t}{u_{3}}^{\varepsilon}\|^{2}+\|\partial_{t}{u_{3}}^{D}\|^{2})dxd\tau$ $\displaystyle+$ $\displaystyle\frac{\varepsilon B^{\infty}}{2}\int_{0}^{t}\int_{\Gamma^{wa}_{\varepsilon}}(\|{u_{2}}^{\varepsilon}\|^{2}+\|\partial_{t}{u_{3}}^{\varepsilon}\|^{2}+\|{u_{2}}^{\varepsilon}\|^{2}+\|\partial_{t}{u_{3}}^{D}\|^{2})dxd\tau$ $\displaystyle+$ $\displaystyle\frac{\varepsilon b^{\infty}}{2}\int_{0}^{t}\int_{\Gamma^{wa}_{\varepsilon}}(\frac{1}{\delta}\|\partial_{t}{u_{2}}^{\varepsilon}\|^{2}+\delta\|\partial_{t}{u_{3}}^{\varepsilon}\|^{2}+\|\partial_{t}{u_{2}}^{\varepsilon}\|^{2}+\|\partial_{t}{u_{3}}^{D}\|^{2})dxd\tau.$ Using (7) to deal with the boundary terms, we obtain $\displaystyle\frac{1}{2}\int_{\Omega^{\varepsilon}}\|\partial_{t}u^{\varepsilon}_{3}\|^{2}dx$ $\displaystyle+$ $\displaystyle\left({d_{3}}-\frac{d_{3}^{\infty}}{2}-\frac{C^{}\varepsilon^{2}}{2}(3a^{\infty}+B^{\infty}+b^{\infty}+a^{\infty}\delta)\right)\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\nabla\partial_{t}{u_{3}}^{\varepsilon}\|^{2}dxd\tau$ $\displaystyle\leq$ $\displaystyle C_{14}+C_{15}\int_{0}^{t}\int_{\Omega^{\varepsilon}}\|\partial_{t}{u_{3}}^{\varepsilon}\|^{2}dxd\tau$ (69) $\displaystyle+$ $\displaystyle C^{}b^{\infty}\int_{0}^{t}\int_{\Omega^{\varepsilon}}(\|\partial_{t}{u_{3}}^{\varepsilon}\|^{2}+\varepsilon^{2}\|\nabla\partial_{t}{u_{3}}^{\varepsilon}\|^{2})dxd\tau$ (70) Adding (68) and (4) and using (64) and (65) to get the desired result. ### 4.1 Extension step Since we deal here with an oscillating system posed in a perforated domain, the natural next step is to extend all concentrations to the whole $\Omega$. We do this by following a two-steps procedure: In Step 1, we rely on the standard extension results indicated in section 4.2 to extend all active concentrations $u^{\varepsilon}_{\ell}$ ($\ell\in\\{1,\dots,4\\}$) to $\Omega$. In step 2, we unfold the ode for $u^{\varepsilon}_{5}$ such that the unfolded concentration is defined on the fixed boundary $\Gamma$; see section 5.1. ### 4.2 Extension lemmas Since all the concentrations are defined in $\Omega^{\varepsilon}$ and $\Omega^{\varepsilon}_{1}$, to get macroscopic equations we need to extend them into $\Omega$. ###### Remark 6. Take $\varphi^{\varepsilon}\in L^{2}(0,T;H^{1}(\Omega^{\varepsilon}))$. Note that since our microscopic geometry is sufficiently regular, we can speak in terms of extensions. Recall the linearity of the extension operator $\mathcal{P}^{\varepsilon}:L^{2}(0,T;H^{1}(\Omega^{\varepsilon}))\rightarrow L^{2}(0,T;H^{1}(\Omega))$ defined by $\mathcal{P}^{\varepsilon}\varphi^{\varepsilon}=\tilde{\varphi}^{\varepsilon}$. To keep notation simple, we denote the extension $\tilde{\varphi}^{\varepsilon}$ again by $\varphi^{\varepsilon}$. ###### Lemma 7. (Extension) Consider the geometry described in Section 2.1. There exists an extension $\tilde{u}^{\varepsilon}$ of $u^{\varepsilon}$ such that 1. 1. $\parallel\tilde{u}^{\varepsilon}\parallel_{L^{2}(Y)}\leq\hat{C}\parallel u^{\varepsilon}\parallel_{L^{2}(Y^{w})},$ for $u^{\varepsilon}\in{L^{2}(Y^{w})}$ 2. 2. $\parallel\nabla\tilde{u}^{\varepsilon}\parallel_{L^{2}(Y)}\leq\hat{C}\parallel\nabla u^{\varepsilon}\parallel_{L^{2}(Y^{w})},$ for $\nabla u^{\varepsilon}\in{L^{2}(Y^{w})}$ 3. 3. $\parallel\tilde{u}^{\varepsilon}\parallel_{H^{1}(\Omega)}\leq\hat{C}\parallel u^{\varepsilon}\parallel_{H^{1}(\Omega^{\varepsilon})}$, for $u^{\varepsilon}\in H^{1}(\Omega^{\varepsilon})$ Proof. For the proof of this Lemma, see Section 2 in [20] or compare Lemma 5, p.214 in [30]. ###### Definition 8. (Two-scale convergence cf. [31, 32]) Let $\\{u^{\varepsilon}\\}$ be a sequence of functions in $L^{2}((0,T)\times\Omega)$ ($\Omega$ being an open set of $\mathbb{R}^{N}$) where $\varepsilon$ being a sequence of strictly positive numbers that tends to zero. $\\{u^{\varepsilon}\\}$ is said to two-scale converge to a unique function $u_{0}(t,x,y)\in L^{2}((0,T)\times\Omega\times Y)$ if and only if for any $\psi\in C^{\infty}_{0}((0,T)\times\Omega,C^{\infty}_{\\#}(Y))$, we have $\displaystyle lim_{\varepsilon\rightarrow 0}\int_{0}^{T}\int_{\Omega}u^{\varepsilon}\psi(t,x,\frac{x}{\varepsilon})dxdt=\int_{\Omega}\int_{Y}u_{0}(t,x,y)\psi(t,x,y)dydxdt.$ (71) We denote (71) by $u^{\varepsilon}\stackrel{{\scriptstyle 2}}{{\rightharpoonup}}u_{0}$. ###### Theorem 9. 1. (i) From each bounded sequence $\\{u^{\varepsilon}\\}$ in $L^{2}((0,T)\times\Omega)$, one can extract a subsequence which two-scale converges to $u_{0}(t,x,y)\in L^{2}((0,T)\times\Omega\times Y)$. 2. (ii) Let $\\{u^{\varepsilon}\\}$ be a bounded sequence in $H^{1}((0,T)\times\Omega)$, which converges weakly to a limit function $u_{0}(t,x,y)\in H^{1}((0,T)\times\Omega\times Y)$. Then there exists $\tilde{u}\in L^{2}(\Omega;H^{1}_{\\#}(Y)/\mathbb{R})$ such that up to a subsequence $\\{u^{\varepsilon}\\}$ two-scale converges to $u_{0}(t,x,y)$ and $\nabla u^{\varepsilon}\stackrel{{\scriptstyle 2}}{{\rightharpoonup}}\nabla_{x}u_{0}+\nabla_{y}\tilde{u}.$ 3. (iii) Let $\\{u^{\varepsilon}\\}$ and $\\{\varepsilon\nabla u^{\varepsilon}\\}$ be bounded sequences in $L^{2}((0,T)\times\Omega)$, then there exists${u_{0}}\in L^{2}((0,T)\times\Omega;H^{1}_{\\#}(Y))$ such that up to a subsequence $u^{\varepsilon}$ and $\varepsilon\nabla u^{\varepsilon}$ two-scale converge to $u_{0}(t,x,y)$ and $\nabla_{y}u_{0}(t,x,y)$ respectively. ###### Definition 10. (Two-scale convergence for $\varepsilon-$periodic hypersurfaces [33]) A sequence of functions $\\{u^{\varepsilon}\\}$ in $L^{2}((0,T)\times\Gamma_{\varepsilon})$ is said to two-scale converge to a limit $u_{0}\in L^{2}((0,T)\times\Omega\times\Gamma)$ if and only if for any $\psi\in C^{\infty}_{0}((0,T)\times\Omega,C^{\infty}_{\\#}(\Gamma))$ we have $\displaystyle lim_{\varepsilon\rightarrow 0}\varepsilon\int_{0}^{T}\int_{\Gamma_{\varepsilon}}u^{\varepsilon}\psi(t,x,\frac{x}{\varepsilon})d\sigma_{x}dt=\int_{\Omega}\int_{\Gamma}u_{0}(t,x,y)\psi(t,x,y)d\sigma_{y}dxdt.$ ###### Theorem 11. 1. (i) From each bounded sequence $\\{u^{\varepsilon}\\}\in L^{2}((0,T)\times\Gamma_{\varepsilon})$, one can extract a subsequence $u^{\varepsilon}$ which two-scale converges to a function $u_{0}\in L^{2}((0,T)\times\Omega\times\Gamma)$. 2. (ii) If a sequence of functions $\\{u^{\varepsilon}\\}$ is bounded in $L^{\infty}((0,T)\times\Gamma_{\varepsilon})$, then $u^{\varepsilon}$ two- scale converges to a function $u_{0}\in L^{\infty}((0,T)\times\Omega\times\Gamma)$. Proof. For proof of (i), see [33] and the one for (ii), see [15]. ###### Lemma 12. Assume the hypotheses of Lemma 5 and Lemma 7 to hold. The a priori estimates lead to the following convergence results: 1. (a) $u^{\varepsilon}_{i}\rightharpoonup u_{i}$ in $L^{2}(0,T;H^{1}(\Omega)$ for all $i\in\\{1,2,3,4\\}$, 2. (b) $u^{\varepsilon}_{i}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}u_{i}$ in $L^{\infty}((0,T)\times\Omega)$, 3. (c) $\partial_{t}u^{\varepsilon}_{i}\rightharpoonup\partial_{t}u_{i}$ in $L^{2}((0,T)\times\Omega)$, 4. (d) $u^{\varepsilon}_{i}\rightarrow u_{i}$ in $L^{2}(0,T;H^{\beta}(\Omega))$ for $\frac{1}{2}<\beta<1$, also $\parallel u^{\varepsilon}_{i}-u_{i}\parallel_{L^{2}((0,T)\times\Gamma_{\varepsilon})}\rightarrow 0$ as $\varepsilon\rightarrow 0$, 5. (e) $u^{\varepsilon}_{i}\stackrel{{\scriptstyle 2}}{{\rightharpoonup}}u_{i},\nabla u^{\varepsilon}_{i}\stackrel{{\scriptstyle 2}}{{\rightharpoonup}}\nabla_{x}u_{i}+\nabla_{y}u_{i1}$, $u_{i1}\in L^{2}((0,T)\times\Omega;H^{1}_{\\#}(Y)/\mathbb{R})$, 6. (f) $u^{\varepsilon}_{5}\stackrel{{\scriptstyle 2}}{{\rightharpoonup}}u_{5}$, and $u_{5}\in L^{\infty}((0,T)\times\Omega\times\Gamma^{sw})$, 7. (g) $\partial_{t}u^{\varepsilon}_{5}\stackrel{{\scriptstyle 2}}{{\rightharpoonup}}\partial_{t}u_{5}$, and $u_{5}\in L^{2}((0,T)\times\Omega\times\Gamma^{sw})$. Proof. (a) and (b) are obtained as a direct consequence of the fact that $u^{\varepsilon}_{i}$ is bounded in $L^{2}(0,T;H^{1}(\Omega))\cap L^{\infty}((0,T)\times\Omega)$; up to a subsequence (still denoted by $u^{\varepsilon}_{i}$) $u^{\varepsilon}_{i}$ converges weakly to $u_{i}$ in $L^{2}(0,T;H^{1}(\Omega))\cap L^{\infty}((0,T)\times\Omega)$. A similar argument gives (c). To get (d), we use the compact embedding $H^{\beta^{\prime}}(\Omega)\hookrightarrow H^{\beta}(\Omega)$, for $\beta\in(\frac{1}{2},1)$ and $0<\beta<\beta^{\prime}\leq 1$ (since $\Omega$ has Lipschitz boundary). We have $W:=\\{u_{i}\in L^{2}(0,T;H^{1}(\Omega))\;\mbox{and}\;\partial_{t}u_{i}\in L^{2}((0,T)\times\Omega)\mbox{ for all}\;i\in\\{1,2,3,4\\}\\}$ For a fixed $\varepsilon$, $W$ is compactly embedded in $L^{2}(0,T;H^{\beta}(\Omega))$ by the Lions-Aubin Lemma; cf. e.g. [34]. Using the trace inequality (8) $\displaystyle\parallel u^{\varepsilon}_{i}-u_{i}\parallel_{L^{2}((0,T)\times\Gamma_{\varepsilon})}$ $\displaystyle\leq$ $\displaystyle{C^{}_{0}}\parallel u^{\varepsilon}_{i}-u_{i}\parallel_{L^{2}(0,T;H^{\beta}(\Omega^{\varepsilon}))},$ $\displaystyle\leq$ $\displaystyle C\parallel u^{\varepsilon}_{i}-u_{i}\parallel_{L^{2}(0,T;H^{\beta}(\Omega))},$ where $\parallel u^{\varepsilon}_{i}-u_{i}\parallel_{L^{2}(0,T;H^{\beta}(\Omega))}\rightarrow 0$ as $\varepsilon\rightarrow 0.$ To investigate (e), (f) and (g), we use the notion of two-scale convergence as indicated in Definition 8 and 10. Since $u^{\varepsilon}_{i}$ are bounded in $L^{2}(0,T;H^{1}(\Omega)$, up to a subsequence $u^{\varepsilon}_{i}\stackrel{{\scriptstyle 2}}{{\rightharpoonup}}u_{i}$ in $L^{2}((0,T)\times\Omega\times Y)$, and $\nabla u^{\varepsilon}_{i}\stackrel{{\scriptstyle 2}}{{\rightharpoonup}}\nabla_{x}u_{i}+\nabla_{y}\tilde{u}_{i}$, $\tilde{u}_{i}\in L^{2}((0,T)\times\Omega;H^{1}_{\\#}(Y)/\mathbb{R})$. By Theorem 11, $u^{\varepsilon}_{5}$ in $L^{\infty}((0,T)\times\Omega\times\Gamma)$ converges two-scale to $u_{5}$ in the same space and $\partial_{t}u^{\varepsilon}_{5}$ converges two-scale to $\partial_{t}u_{5}$ in $L^{2}((0,T)\times\Omega\times\Gamma)$. Due to the presence of the non-linear reaction rate on the interface $\Gamma^{sw}_{\varepsilon}$, the convergences listed in Lemma 12 are still not sufficient to pass to the limit $\varepsilon\rightarrow 0$ in the microscopic model. To be more precise, we can pass to $\varepsilon\rightarrow 0$ in the pde’s, but not in the ode. ### 4.3 Cell problems In order to be able to formulate the upscaled equations, we define two classes of cell problems very much in the spirit of [9]. One class of problems will refer to the water-filled parts of the pore, while the second class will refer to the air-filled part of the pores. ###### Definition 13. (Cell problems) The cell problems in water-filled part are given by $\displaystyle-\nabla_{y}.(D_{\ell}(t,y)\nabla_{y}\chi_{i})$ $\displaystyle=\sum_{k=1}^{3}\partial_{y_{k}}{D_{\ell}}_{ki}(t,y),\;\mbox{in}\;\;Y^{w},$ $\displaystyle-D_{\ell}(t,y)\frac{\partial\chi_{i}}{\partial n}$ $\displaystyle=\sum_{k=1}^{3}{D_{\ell}}_{ki}(t,y)n_{k}\;\;\mbox{on}\;\Gamma^{sw},$ $\displaystyle-D_{\ell}(t,y)\frac{\partial\chi_{i}}{\partial n}$ $\displaystyle=\sum_{k=1}^{3}{D_{\ell}}_{ki}(t,y)n_{k}\;\;\mbox{on}\;\Gamma^{wa},$ for all $i,\ell\in\\{1,2,4\\}$ and $\chi_{i}$ are Y-periodic in y. The cell problems in air-filled part are given by $\displaystyle-\nabla_{y}.(D_{3}(t,y)\nabla_{y}\varsigma_{i})$ $\displaystyle=\sum_{k=1}^{3}\partial_{y_{k}}{D_{3}}_{ki}(t,y),\;\mbox{in}\;\;Y^{a},$ $\displaystyle-D_{3}(t,y)\frac{\partial\varsigma_{i}}{\partial n}$ $\displaystyle=\sum_{k=1}^{3}{D_{3}}_{ki}(t,y)n_{k}\;\;\mbox{on}\;\Gamma^{wa},$ $\displaystyle-D_{3}(t,y)\frac{\partial\varsigma_{i}}{\partial n}$ $\displaystyle=\sum_{k=1}^{3}{D_{3}}_{ki}(t,y)n_{k}\;\;\mbox{on}\;\partial Y^{a}-\Gamma^{wa},$ for all $i\in\\{1,2,3\\}$ and $\varsigma_{i}$ are $Y$-periodic in $y$. ## 5 Two-scale limit equations ###### Theorem 14. The sequences of the solutions of the weak formulation (12)-(16) converges to the weak solution $u_{i},i\in\\{1,2,3,4,5\\}$ as $\varepsilon\rightarrow 0$ such that $u_{i}\in H^{1}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{1}(\Omega))\cap L^{\infty}((0,T)\times\Omega)$ and $u_{5}\in H^{1}(0,T;L^{2}(\Omega\times\Gamma))\cap L^{\infty}((0,T)\times\Omega\times\Gamma))$. The weak formulation of the two- scale limit equations is given by $\displaystyle\int_{0}^{T}\int_{\Omega}\partial_{t}u_{i}(t,x)\phi_{i}(t,x)dxdt$ $\displaystyle+$ $\displaystyle\int_{0}^{T}\int_{\Omega}\tilde{d}_{i}(t)\nabla u_{i}(t,x)\nabla\phi_{i}dxdt$ $\displaystyle=$ $\displaystyle\int_{0}^{T}\int_{\Omega}F_{i}(u)\phi_{i}dxdt\mbox{ for all }i\in\\{1,2,3,4\\},$ where $\displaystyle F_{1}(u)$ $\displaystyle:=$ $\displaystyle-\tilde{k}_{1}(t)u_{1}(t,x)+\tilde{k}_{2}(t)u_{2}(t,x)$ $\displaystyle\;-\frac{1}{\|Y\|}\int_{\Gamma}k_{3}(t,y)R(u_{1}(t,x))Q(u_{5}(t,x,y))d\sigma_{y},$ $\displaystyle F_{2}(u)$ $\displaystyle:=$ $\displaystyle\tilde{k}_{1}(t)u_{1}(t,x)-\tilde{k}_{2}(t)u_{2}(t,x)+\tilde{a}(t)u_{3}(t,x)-\tilde{b}(t)u_{2}(t,x),$ $\displaystyle F_{3}(u)$ $\displaystyle:=$ $\displaystyle-\tilde{a}(t)u_{3}(t,x)+\tilde{b}(t)u_{2}(t,x),$ $\displaystyle F_{4}(u)$ $\displaystyle:=$ $\displaystyle\tilde{k}_{1}(t)u_{1}(t,x),$ with the initial values $u_{i}(0,x)={u_{i}}_{0}(x)$ for $x\in\Omega$, and $\displaystyle\int_{0}^{T}\int_{\Omega\times\Gamma}$ $\displaystyle\partial_{t}u_{5}(t,x,y)\phi_{5}(t,x,y)dtdxd\sigma_{y}$ (73) $\displaystyle=\int_{0}^{T}\int_{\Omega\times\Gamma}k_{3}(t,y)R(u_{1}(t,x))Q(u_{5}(t,x,y))\phi_{5}(t,x,y)dtdxd\sigma_{y},$ with $u_{5}(0,x,y)={u_{5}}_{0}(x,y)$ for $\;x\in\Omega,\;y\in\Gamma^{sw}$. Also $\phi:=(\phi_{1},\phi_{2},\phi_{3},\phi_{4})\in[C^{\infty}((0,T)\times\Omega)]^{4}$, $\psi:=(\psi_{1},\psi_{2},\psi_{3},\psi_{4})\in[C^{\infty}((0,T)\times\Omega);C^{\infty}_{\\#}(Y)]^{4}$, $\displaystyle\tilde{k}_{j}(t)$ $\displaystyle:=$ $\displaystyle\frac{1}{\|Y\|}\int_{Y}k_{j}(t,y)dy,\;j\in\\{1,2\\},$ (74) $\displaystyle\tilde{a}(t)$ $\displaystyle:=$ $\displaystyle\frac{1}{\|Y\|}\int_{\Gamma^{wa}}a(t,y)d\sigma_{y},$ (75) $\displaystyle\tilde{b}(t)$ $\displaystyle:=$ $\displaystyle\frac{1}{\|Y\|}\int_{\Gamma^{wa}}b(t,y)d\sigma_{y},$ (76) $\displaystyle{\tilde{d}_{\ell ij}}$ $\displaystyle:=$ $\displaystyle\sum_{k=1}^{3}\int_{Y}(d_{\ell ij}(t,y)+d_{\ell ik}(t,y)(\delta_{n\ell}\partial_{y_{k}}\chi_{j}+\delta_{3\ell}\partial_{y_{k}}\varsigma_{j})dy,$ $\displaystyle\ell\in\\{1,2,3,4\\},\,n\in\\{1,2,4\\}$ with $\chi_{j},\varsigma_{j}$ being solutions of the cell problems defined in Definition 13, while $\delta$ denotes here the Kronecker’s symbol. Proof. We apply two-scale convergence techniques together with Lemma 12 to get macroscopic equations. We take test functions incorporating the following oscillating behavior $\varphi_{i}(t,x)=\phi_{i}(t,x)+\varepsilon\psi_{i}(t,x,\frac{x}{\varepsilon}),\phi_{i}\in C^{\infty}((0,T)\times\Omega),\phi_{i}\in C^{\infty}((0,T)\times\Omega,;C^{\infty}_{\\#}(Y)),i\in\\{1,2,3,4\\}$. Applying two-scale convergence yields $\displaystyle\|Y\|\int_{0}^{T}\int_{\Omega}$ $\displaystyle\partial_{t}u_{i}\phi_{i}(t,x)dxdt+\int_{0}^{T}\int_{\Omega}\int_{Y}d_{i}(t,y)(\nabla_{x}u_{i}(t,x)$ (78) $\displaystyle+$ $\displaystyle\nabla_{y}\tilde{u}_{i}(t,x,y))(\nabla_{x}\phi_{i}(t,x)+\nabla_{y}\psi_{i}(t,x,\frac{x}{\varepsilon}))dydxdt$ $\displaystyle=$ $\displaystyle\int_{0}^{T}\int_{\Omega}f_{i}(u)\phi_{i}(t,x)dxdt.$ $\displaystyle\int_{0}^{T}\int_{\Omega}f_{1}(u)$ $\displaystyle\phi_{1}(t,x)dxdt=-\lim_{\varepsilon\rightarrow 0}\int_{0}^{T}\int_{\Omega^{\varepsilon}}k_{1}^{\varepsilon}u^{\varepsilon}_{1}(\phi_{1}(t,x)+\varepsilon\psi_{1}(t,x,\frac{x}{\varepsilon}))dxdt$ $\displaystyle+$ $\displaystyle\lim_{\varepsilon\rightarrow 0}\int_{0}^{T}\int_{\Omega^{\varepsilon}}k_{2}^{\varepsilon}u^{\varepsilon}_{2}(\phi_{1}(t,x)+\varepsilon\psi_{1}(t,x,\frac{x}{\varepsilon}))dxdt$ $\displaystyle-$ $\displaystyle\lim_{\varepsilon\rightarrow 0}\varepsilon\int_{0}^{T}\int_{\Gamma^{sw}_{\varepsilon}}\eta(R(u^{\varepsilon}_{1}),Q(u^{\varepsilon}_{5}))(\phi_{1}(t,x)+\varepsilon\psi_{1}(t,x,\frac{x}{\varepsilon}))d\sigma_{x}dt.$ Using Lemma 12, we have $\displaystyle\int_{0}^{T}\int_{\Omega}f_{1}(u)\phi_{1}(t,x)dxdt$ $\displaystyle=$ $\displaystyle-\int_{0}^{T}\int_{\Omega}\int_{Y}k_{1}(t,y)u_{1}(t,x)\phi_{1}(t,x)dydxdt$ $\displaystyle+$ $\displaystyle\int_{0}^{T}\int_{\Omega}\int_{Y}k_{2}(t,y)u_{2}(t,x)\phi_{1}(t,x)dydxdt$ $\displaystyle-$ $\displaystyle\lim_{\varepsilon\rightarrow 0}\varepsilon\int_{0}^{T}\int_{\Gamma^{sw}_{\varepsilon}}\partial_{t}u^{\varepsilon}_{5}(\phi_{1}(t,x)+\varepsilon\psi_{1}(t,x,\frac{x}{\varepsilon}))d\sigma_{x}dt.$ $\displaystyle\int_{0}^{T}\int_{\Omega}f_{1}(u)\phi_{1}(t,x)dxdt$ $\displaystyle=$ $\displaystyle-\|Y\|\int_{0}^{T}\int_{\Omega}\tilde{k}_{1}(t)u_{1}(t,x)\phi_{1}(t,x)dxdt$ $\displaystyle+$ $\displaystyle\|Y\|\int_{0}^{T}\int_{\Omega}\tilde{k}_{2}(t)u_{2}(t,x)\phi_{1}(t,x)dxdt$ $\displaystyle-$ $\displaystyle\int_{0}^{T}\int_{\Omega}\int_{\Gamma}\partial_{t}u_{5}\phi_{1}(t,x)d\sigma_{y}dxdt.$ $\displaystyle\int_{0}^{T}\int_{\Omega}f_{2}(u)\phi_{2}(t,x)dxdt$ $\displaystyle=$ $\displaystyle\lim_{\varepsilon\rightarrow 0}\int_{0}^{T}\int_{\Omega^{\varepsilon}}k_{1}^{\varepsilon}u^{\varepsilon}_{1}(\phi_{2}(t,x)+\varepsilon\psi_{2}(t,x,\frac{x}{\varepsilon}))dxdt$ $\displaystyle-$ $\displaystyle\lim_{\varepsilon\rightarrow 0}\int_{0}^{T}\int_{\Omega^{\varepsilon}}k_{2}^{\varepsilon}u^{\varepsilon}_{2}(\phi_{2}(t,x)+\varepsilon\psi_{2}(t,x,\frac{x}{\varepsilon}))dxdt$ $\displaystyle+$ $\displaystyle\lim_{\varepsilon\rightarrow 0}\varepsilon\int_{0}^{T}\int_{\Gamma^{wa}_{\varepsilon}}a_{\varepsilon}u^{\varepsilon}_{3}(\phi_{2}(t,x)+\varepsilon\psi_{2}(t,x,\frac{x}{\varepsilon}))d\sigma_{x}dt$ $\displaystyle-$ $\displaystyle\lim_{\varepsilon\rightarrow 0}\varepsilon\int_{0}^{T}\int_{\Gamma^{wa}_{\varepsilon}}b_{\varepsilon}u^{\varepsilon}_{2}(\phi_{2}(t,x)+\varepsilon\psi_{2}(t,x,\frac{x}{\varepsilon}))d\sigma_{x}dt.$ $\displaystyle\int_{0}^{T}\int_{\Omega}f_{2}(u)\phi_{2}(t,x)dxdt$ $\displaystyle=$ $\displaystyle\|Y\|\int_{0}^{T}\int_{\Omega}\tilde{k}_{1}(t)u_{1}(t,x)\phi_{2}(t,x)dxdt$ $\displaystyle-$ $\displaystyle\|Y\|\int_{0}^{T}\int_{\Omega}\tilde{k}_{2}(t)u_{2}(t,x)\phi_{2}(t,x)dxdt$ $\displaystyle+$ $\displaystyle\|Y\|\int_{0}^{T}\int_{\Omega}\tilde{a}(t)u_{3}(t,x)\phi_{2}(t,x)dxdt$ $\displaystyle-$ $\displaystyle\|Y\|\int_{0}^{T}\int_{\Omega}\tilde{b}(t)u_{2}(t,x)\phi_{2}(t,x)dxdt.$ We also have $\displaystyle\int_{0}^{T}\int_{\Omega}f_{3}(u)\phi_{3}(t,x)dxdt$ $\displaystyle=$ $\displaystyle-\|Y\|\int_{0}^{T}\int_{\Omega}\tilde{a}(t)u_{3}(t,x)\phi_{3}(t,x)dxdt$ $\displaystyle+$ $\displaystyle\|Y\|\int_{0}^{T}\int_{\Omega}\tilde{b}(t)u_{2}(t,x)\phi_{3}(t,x)dxdt$ and $\displaystyle\int_{0}^{T}\int_{\Omega}f_{4}(u)\phi_{4}(t,x)dxdt$ $\displaystyle=$ $\displaystyle\|Y\|\int_{0}^{T}\int_{\Omega}\tilde{k}_{1}(t)u_{1}(t,x)\phi_{4}(t,x)dxdt.$ We set $\phi_{i}=0,i\in\\{1,2,3,4\\}$ in (78) to calculate the expression of the known function $\tilde{u}_{1}$ and obtain $\int_{0}^{T}\int_{\Omega}\int_{Y}d_{i}(t,y)(\nabla_{x}u_{i}(t,x)+\nabla_{y}\tilde{u}_{i}(t,x,y))\nabla_{y}\psi_{i}(t,x,\frac{x}{\varepsilon})dydxdt=0,\mbox{ forall }\psi_{i}.$ Since $\tilde{u}_{1}$ depends linearly on $\nabla_{x}u_{1}$, it can be defined as $\tilde{u}_{i}:=\sum_{j=1}^{3}\partial_{x_{j}}u_{i}(\delta_{in}\chi_{j}(t,y)+\delta_{3i}\varsigma_{j}(t,y))\;\mbox{for}\;n\in\\{1,2,4\\}$ where the function $\chi_{j},\varsigma_{j}$ are the unique solutions of the cell problems defined in Definition 13. Setting $\psi_{i}=0$ in (78), we get $\displaystyle\int_{0}^{T}\int_{\Omega}\int_{Y}$ $\displaystyle\sum_{j,k=1}^{3}{d_{i}}_{jk}(t,y)(\partial_{x_{k}}u_{i}(t,x)$ $\displaystyle+\sum_{m=1}^{3}(\delta_{in}\partial_{y_{k}}\chi_{m}+\delta_{3i}\partial_{y_{k}}\varsigma_{m})\partial_{x_{m}}u_{i}(t,x))\partial_{x_{j}}\phi_{k}(t,x)dydxdt$ $\displaystyle=\|Y\|\int_{0}^{T}\int_{\Omega}\sum_{j,k=1}^{3}{\tilde{d}_{ijk}}\partial_{x_{k}}u_{i}(t,x)\partial_{x_{j}}\phi_{i}(t,x)dxdt.$ Hence, the coefficients (entering the effective diffusion tensor) are given by ${\tilde{d}_{ijk}}:=\frac{1}{\|Y\|}\sum_{k=1}^{3}\int_{Y}(d_{\ell ij}(t,y)+d_{\ell ik}(t,y)(\delta_{in}\partial_{y_{k}}\chi_{j}+\delta_{3i}\partial_{y_{k}}\varsigma_{j})dy.$ (79) $i\in\\{1,2,3,4\\}$, $n\in\\{1,2,4\\}$ and $j,k\in\\{1,2,3\\}$. ### 5.1 Passing to the limit $\varepsilon\to 0$ in (16) It is not yet possible to pass to the limit $\varepsilon\rightarrow 0$ with the convergence results stated in Lemma 12. To overcome this difficulty, we use the notion of periodic unfolding. It si worth mentioning that there is an intimate link between the two-scale convergence and weak convergence of the unfolded sequences; see [35, 15]. The key idea is: Instead of getting strong convergence for $u^{\varepsilon}_{5}$, obtain strong convergence for the periodic unfolding of $u^{\varepsilon}_{5}$. ###### Definition 15. For $\varepsilon>0$, the boundary unfolding of a measurable function $\varphi$ posed on oscillating surface $\Gamma_{\varepsilon}$ is defined by $T^{b}_{\varepsilon}\varphi(x,y)=\varphi(\varepsilon y+\varepsilon k),\;\;y\in\Gamma,x\in\Omega$ where $k:=[\frac{x}{\varepsilon}]$ denotes the unique integer combination $\Sigma_{j=1}^{3}k_{j}e_{j}$ of the periods such that $x-[\frac{x}{\varepsilon}]$ belongs to $Y$. Note that the oscillation due to the perforations are shifted into the second variable $y$ which belongs to fixed surface $\Gamma$. ###### Lemma 16. If $u_{\varepsilon}$ converges two-scale to $u$ and $T_{b}^{\varepsilon}u_{\varepsilon}$ converges weakly to $u^{}$ in $L^{2}((0,T)\times\Omega;L^{2}_{\\#}(\Gamma))$, then $u=u^{}$ a.e. in $(0,T)\times\Omega\times\Gamma$. Proof. The proof details for this statement can be found in Lemma 4.6 of [15]. ###### Lemma 17. If $\varphi\in L^{2}((0,T)\times\Gamma^{\varepsilon})$, then the following identity holds $\frac{1}{\|Y\|}\\|T^{\varepsilon}_{b}\varphi\\|_{L^{2}((0,T)\times\Omega\times\Gamma)}=\varepsilon\\|\varphi\\|_{L^{2}((0,T)\times\Gamma^{\varepsilon})}.$ Proof. Consider $\displaystyle\frac{1}{\|Y\|}\|T^{\varepsilon}_{b}\varphi\|^{2}_{L^{2}(\Omega\times\Gamma)}$ $\displaystyle=\frac{1}{\|Y\|}\int_{\Omega\times\Gamma}\|T^{\varepsilon}_{b}\varphi\|^{2}dxd\sigma_{y}=\frac{1}{\|Y\|}\int_{\Omega\times\Gamma}T^{\varepsilon}_{b}\varphi^{2}dxd\sigma_{y},$ $\displaystyle=\frac{1}{\|Y\|}\Sigma^{3}_{k=1}\int_{\varepsilon(Y+k)}\int_{\Gamma}T^{\varepsilon}_{b}\varphi^{2}dxd\sigma_{y}=\frac{1}{\|Y\|}\Sigma^{3}_{k=1}\int_{\varepsilon(Y+k)}dx\int_{\Gamma}\varphi^{2}d\sigma_{y},$ $\displaystyle=\Sigma^{3}_{k=1}\varepsilon^{3}\int_{\Gamma}\varphi^{2}d\sigma_{y}.$ Changing variable $z=\varepsilon(y+k)$, where $k=[\frac{x}{\varepsilon}]$, we get $\displaystyle\frac{1}{\|Y\|}\|T^{\varepsilon}_{b}\varphi\|^{2}_{L^{2}(\Omega\times\Gamma)}$ $\displaystyle=$ $\displaystyle\Sigma^{3}_{k=1}\varepsilon^{3}\int_{\Gamma}\varphi^{2}d\sigma_{y}=\Sigma^{3}_{k=1}\varepsilon\int_{\varepsilon(\Gamma+k)}\varphi^{2}d\sigma_{z}=\varepsilon\int_{\Gamma^{\varepsilon}}\varphi^{2}d\sigma_{z}.$ This completes the proof of (17). ###### Lemma 18. If $\varphi\in L^{2}(\Omega)$, then $T^{\varepsilon}_{b}\varphi\rightarrow\varphi$ as $\varepsilon\rightarrow 0$ strongly in $L^{2}(\Omega\times\Gamma)$. Proof. See in [36, 37] for proof details. Using the boundary unfolding operator $T^{\epsilon}_{b}$, we unfold the ode (16). Changing the variable, $x=\varepsilon y+\varepsilon k$ (for $x\in\Gamma^{sw}_{\varepsilon}$) to the fixed domain $(0,T)\times\Omega\times\Gamma$, we have $\displaystyle\partial_{t}T^{\epsilon}_{b}u^{\epsilon}_{5}(t,x,y)=\eta(T^{\epsilon}_{b}u^{\epsilon}_{1}(t,x,y),T^{\epsilon}_{b}u^{\epsilon}_{5}(t,x,y)).$ (80) In the remainder of this section, we prove that $T^{\varepsilon}_{b}u^{\varepsilon}_{5}$ converges strongly to $u_{5}$ in $L^{2}(\Omega\times\Gamma)$. From the two-scale convergence of $u^{\epsilon}_{5}$, we obtain weak convergence of $T^{\epsilon}u^{\epsilon}_{5}$ to $u_{5}$ in $L^{\infty}((0,T)\times\Omega;L^{2}_{per}(\Gamma))$. We start with showing that $\\{T^{\varepsilon}_{b}u^{\varepsilon}_{5}\\}$ is a Cauchy sequence in $L^{2}(\Omega\times\Gamma)$. To this end, we choose $m,n\in\mathbb{N}$ with $n>m$ arbitrary. Writing down (80) for the two different choices of $\varepsilon$ (i.e. $\varepsilon_{i}=\varepsilon_{n}$ and $\varepsilon_{i}=\varepsilon_{m}$), we obtain after subtracting the corresponding equations that $\displaystyle\partial_{t}$ $\displaystyle\int_{\Omega\times{\Gamma}}\|T^{\epsilon_{n}}_{b}u^{\epsilon_{n}}_{5}-T^{\epsilon_{m}}_{b}u^{\epsilon_{m}}_{5}\|^{2}d\sigma_{y}dx$ (81) $\displaystyle=$ $\displaystyle\int_{\Omega\times{\Gamma}}[k^{\varepsilon}_{3}R(T^{\epsilon_{n}}_{b}u^{\epsilon_{n}}_{1})Q(T^{\epsilon_{n}}_{b}u^{\epsilon_{n}}_{5})-k^{\varepsilon}_{3}R(T^{\epsilon_{m}}_{b}u^{\epsilon_{m}}_{1})Q(T^{\epsilon_{m}}_{b}u^{\epsilon_{m}}_{5}))$ $\displaystyle\times$ $\displaystyle(T^{\epsilon_{n}}_{b}u^{\epsilon_{n}}_{5}-T^{\epsilon_{m}}_{b}u^{\epsilon_{m}}_{5})d\sigma_{y}dx,$ $\displaystyle\leq$ $\displaystyle k^{\infty}_{3}c_{R}(\frac{Q^{\infty}}{2}+c_{Q}sup_{\Omega\times{\Gamma}}\|T^{\epsilon_{n}}_{b}u^{\epsilon_{n}}_{1}\|)\int_{\Omega\times{\Gamma}}\|T^{\epsilon_{n}}_{b}u^{\epsilon_{n}}_{5}-T^{\epsilon_{m}}_{b}u^{\epsilon_{m}}_{5}\|^{2}d\sigma_{y}dx$ $\displaystyle+$ $\displaystyle\frac{k^{\infty}_{3}c_{R}Q^{\infty}}{2}\int_{\Omega\times{\Gamma}}\|T^{\epsilon_{n}}_{b}u^{\epsilon_{n}}_{1}-T^{\epsilon_{m}}_{b}u^{\epsilon_{m}}_{1}\|^{2}d\sigma_{y}dx.$ To get (81), we have used the uniform boundedness of $T^{\epsilon_{n}}_{b}u^{\epsilon_{n}}_{1}$. We consider now $\displaystyle\int_{\Omega\times{\Gamma}}$ $\displaystyle\|T^{\epsilon n}_{b}u^{\epsilon n}_{1}-T^{\epsilon n}_{b}u^{\epsilon m}_{1}\|^{2}d\sigma_{y}dx$ (82) $\displaystyle\leq$ $\displaystyle\int_{\Omega\times{\Gamma}}(\|T^{\epsilon n}_{b}u^{\epsilon n}_{1}-T^{\epsilon m}_{b}u_{1}\|^{2}+\|T^{\epsilon n}_{b}u_{1}-u_{1}\|^{2})d\sigma_{y}dx$ $\displaystyle+$ $\displaystyle\int_{\Omega\times{\Gamma}}(\|T^{\epsilon m}_{b}u_{1}-u_{1}\|^{2}+\|T^{\epsilon m}_{b}u^{\varepsilon m}_{1}-T^{\epsilon m}_{b}u_{1}\|^{2})d\sigma_{y}dx.$ Since $u_{1}$ is constant w.r.t. $y$, we have that $T^{\epsilon_{m}}_{b}u_{1}\rightarrow u_{1}$ strongly in $L^{2}((0,T)\times\Omega\times\Gamma)$ as $\varepsilon\rightarrow 0$. From Lemma 17, we conclude that $\int_{\Omega\times{\Gamma}}\|T^{\epsilon}_{b}u^{\epsilon}_{1}-T^{\epsilon}_{b}u_{1}\|^{2}d\sigma_{y}dx\leq\varepsilon\int_{{\Gamma}^{\varepsilon}}\|u^{\epsilon}_{1}-u_{1}\|^{2}d\sigma_{y}dx\leq\varepsilon C.$ (82) turns out to be $\displaystyle\int_{\Omega\times{\Gamma}}$ $\displaystyle\|T^{\epsilon_{n}}_{b}u^{\epsilon_{n}}_{1}-T^{\epsilon_{n}}_{b}u^{\epsilon_{m}}_{1}\|^{2}d\sigma_{y}dxdt\leq C(\varepsilon_{n}+\varepsilon_{m}),$ while (81) becomes $\partial_{t}\int_{\Omega\times{\Gamma}}\|T^{\epsilon n}_{b}u^{\epsilon n}_{5}-T^{\epsilon m}_{b}u^{\epsilon m}_{5}\|^{2}d\sigma_{y}dx\leq C_{15}\int_{\Omega\times{\Gamma}}\|T^{\epsilon_{n}}_{b}u^{\epsilon_{n}}_{5}-T^{\epsilon_{m}}_{b}u^{\epsilon_{m}}_{5}\|^{2}d\sigma_{y}dx+\frac{C_{16}}{n},$ where $C_{15}:=k^{\infty}_{3}c_{R}(\frac{Q^{\infty}}{2}+c_{Q}sup_{\Omega\times{\Gamma}}\|T^{\epsilon_{n}}_{b}u^{\epsilon_{n}}_{1}\|)$ and $C_{16}:=\frac{k^{\infty}_{3}c_{R}Q^{\infty}}{2}C$. The Gronwall’s inequality gives $\displaystyle\parallel T^{\epsilon n}_{b}u^{\epsilon n}_{5}-T^{\epsilon m}_{b}u^{\epsilon m}_{5}\parallel_{L^{2}(\Omega\times\Gamma)}\leq\frac{C_{16}}{n}.$ (83) By (83), $\\{T^{\epsilon}_{b}u^{\epsilon}_{5}\\}$ is a Cauchy sequence. Now, we take the two-scale limit in the ode (80) to get $\displaystyle\lim_{\varepsilon\rightarrow 0}\varepsilon\int^{T}_{0}\int_{\Gamma^{sw}_{\varepsilon}}\partial_{t}T^{\epsilon}_{b}u^{\varepsilon}_{5}\phi_{1}(t,x,\frac{x}{\varepsilon})d\sigma_{x}dt=\lim_{\varepsilon\rightarrow 0}\varepsilon\int_{0}^{T}\int_{\Gamma^{sw}_{\varepsilon}}\eta(T^{\epsilon}_{b}u^{\varepsilon}_{1},T^{\epsilon}_{b}u^{\varepsilon}_{5})\phi_{1}(t,x,\frac{x}{\varepsilon})d\sigma_{x}dt.$ Consequently, we have $\displaystyle\int^{T}_{0}\int_{\Omega\times\Gamma^{sw}}$ $\displaystyle\partial_{t}u_{5}\phi_{5}(t,x,y)dxd\sigma_{y}dt$ $\displaystyle=$ $\displaystyle\lim_{\varepsilon\rightarrow 0}\varepsilon\int^{T}_{0}\int_{\Gamma^{sw}_{\varepsilon}}T^{\epsilon}_{b}k^{\varepsilon}_{3}R(T^{\epsilon}_{b}u^{\varepsilon}_{1})Q(u^{\varepsilon}_{5})\phi_{5}(t,x,\frac{x}{\varepsilon})d\sigma_{x}dt,$ $\displaystyle=$ $\displaystyle\lim_{\varepsilon\rightarrow 0}\varepsilon\int^{T}_{0}\int_{\Gamma^{sw}_{\varepsilon}}T^{\epsilon}_{b}k^{\varepsilon}_{3}R(T^{\epsilon}_{b}u^{\varepsilon}_{1})Q(u_{5})\phi_{5}(t,x,\frac{x}{\varepsilon})d\sigma_{x}dt$ $\displaystyle+$ $\displaystyle\lim_{\varepsilon\rightarrow 0}\varepsilon\int^{T}_{0}\int_{\Gamma^{sw}_{\varepsilon}}T^{\epsilon}_{b}k^{\varepsilon}_{3}R(T^{\epsilon}_{b}u^{\varepsilon}_{1})(Q(T^{\epsilon}_{b}u^{\varepsilon}_{5})-Q(u_{5}))\phi_{5}(t,x,\frac{x}{\varepsilon})d\sigma_{x}dt.$ By (A2) and the strong convergence of $u^{\varepsilon}_{1}$, the first term on the right hand side of (5.1) converges two-scale to $\int^{T}_{0}\int_{\Omega}\int_{\Gamma^{sw}}k_{3}(t,y)R(u_{1})Q(u_{5})\phi_{5}(t,x,y)d\sigma_{y}dxdt,$ while the second integral of (5.1) $\displaystyle\varepsilon$ $\displaystyle\int^{T}_{0}\int_{\Gamma^{sw}_{\varepsilon}}T^{\epsilon}_{b}k^{\varepsilon}_{3}R(T^{\epsilon}_{b}u^{\varepsilon}_{1})(Q(T^{\epsilon}_{b}u^{\varepsilon}_{5})-Q(u_{5}))\phi_{5}(t,x,\frac{x}{\varepsilon})d\sigma_{x}dt$ $\displaystyle\leq$ $\displaystyle\varepsilon\left(\int^{T}_{0}\int_{\Gamma^{sw}_{\varepsilon}}\|T^{\epsilon}_{b}k^{\varepsilon}_{3}R(T^{\epsilon}_{b}u^{\varepsilon}_{1})\phi_{5}(t,x,\frac{x}{\varepsilon})\|^{2}d\sigma_{x}dt\right)^{\frac{1}{2}}\cdot$ $\displaystyle\cdot$ $\displaystyle\left(\int^{T}_{0}\int_{\Gamma^{sw}_{\varepsilon}}\|Q(T^{\epsilon}_{b}u^{\varepsilon}_{5})-Q(u_{5})\|^{2}d\sigma_{x}dt\right)^{\frac{1}{2}},$ $\displaystyle\rightarrow$ $\displaystyle 0\;\mbox{as}\;\varepsilon\rightarrow 0.$ At this point, we have used again (A2) in combination with the strong convergence of $T^{\epsilon}_{b}u^{\varepsilon}_{5}$. So, as result of passing to the limit $\varepsilon\rightarrow 0$ in (16) we get (73). It is worth noting that the weak solution to the two-scale model inherits a.e. the positivity and boundedness properties from the corresponding properties of the weak solution of the microscopic model. Now, it only remains to ensure the uniqueness of weak solutions to the upscaled model. ###### Lemma 19. (Uniqueness of solutions of (14)-(73) Assume (A1)-(A6). There exists at most one weak solution to the two-scale limit problem (14) and (73). Proof. Suppose there are two weak solutions to the two-scale limit problem $(u^{j}_{1},u^{j}_{2},u^{j}_{3},u^{j}_{4},u^{j}_{5})$ with $j\in\\{1,2\\}$. We denote $u_{\ell}=u^{1}_{\ell}-u^{2}_{\ell},\;\ell\in\\{1,2,3,4\\}$ and choose as test function $\phi_{\ell}=u_{\ell}$. After straightforward calculations, we have from (73) $\displaystyle\|u^{1}_{5}-u^{2}_{5}\|\leq C\int_{0}^{t}\|u^{1}_{1}-u^{2}_{1}\|d\tau.$ (85) Take $\phi_{1}=u_{1}$ in (14) to obtain $\displaystyle\frac{1}{2}\int^{t}_{0}\int_{\Omega}\partial_{t}\|u_{1}\|^{2}dxdt$ $\displaystyle+$ $\displaystyle\tilde{d}_{1}\int^{t}_{0}\int_{\Omega}\|\nabla u_{1}\|^{2}dxdt$ (86) $\displaystyle\leq$ $\displaystyle-\tilde{k}_{1}\int^{t}_{0}\int_{\Omega}\partial_{t}\|u_{1}\|^{2}dxdt+\frac{\hat{k}_{2}^{\infty}}{2}\int^{t}_{0}\int_{\Omega}(\|u_{1}\|^{2}+\|u_{2}\|^{2})dxdt$ $\displaystyle+$ $\displaystyle\frac{k_{3}^{\infty}}{\|Y\|}c_{R}c_{Q}M_{1}\int^{t}_{0}\int_{\Omega\times\Gamma^{sw}}(u^{1}_{5}-u^{2}_{5})u_{1}dxd\sigma_{y}dt$ $\displaystyle+$ $\displaystyle\frac{k_{3}^{\infty}}{\|Y\|}c_{R}Q^{\infty}\int^{t}_{0}\int_{\Omega\times\Gamma^{sw}}\|u_{1}\|^{2}dxd\sigma_{y}dt.$ Using (85) together with the trace inequality for fixed domains, see section 5.5 Theorem 1 in [38] and also the fact that $u_{1}$ is independent of y in (86), we get $\displaystyle\int^{t}_{0}\int_{\Omega}\partial_{t}\|u_{1}\|^{2}dxdt$ $\displaystyle+$ $\displaystyle(2\tilde{d}_{1}-{k_{3}^{\infty}c_{R}}{}C^{}(\delta M_{1}+Q^{\infty}))\int^{t}_{0}\int_{\Omega}\|\nabla u_{1}\|^{2}dxd\tau$ $\displaystyle+$ $\displaystyle 2\tilde{k}_{1}\int^{t}_{0}\int_{\Omega}\partial_{t}\|u_{1}\|^{2}dxd\tau$ $\displaystyle\leq$ $\displaystyle({\tilde{k}_{2}^{\infty}}+{k_{3}^{\infty}c_{R}}{}C^{}(\delta M_{1}+Q^{\infty}))\int^{t}_{0}\int_{\Omega}(\|u_{1}\|^{2}+\|u_{2}\|^{2})dxd\tau$ $\displaystyle+$ $\displaystyle{k_{3}^{\infty}}{\delta}c_{R}c_{Q}M_{1}C^{}\int^{t}_{0}\int_{\Omega}\int^{\tau}_{0}(\|u_{1}\|^{2}+\|\nabla u_{1}\|^{2})dsdxd\tau.$ For suitable choice of $\delta\in]0,\frac{2d_{1}-k_{3}^{\infty}c_{R}C^{}Q^{\infty}}{k_{3}^{\infty}c_{R}C^{}M_{1}}[$, we have $\displaystyle\int^{T}_{0}\int_{\Omega}\partial_{t}\|u_{1}\|^{2}dxdt$ $\displaystyle+$ $\displaystyle\tilde{d}_{1}\int^{T}_{0}\int_{\Omega}\|\nabla u_{1}\|^{2}dxd\tau+2\tilde{k}_{1}\int^{T}_{0}\int_{\Omega}\partial_{t}\|u_{1}\|^{2}dxd\tau$ (87) $\displaystyle\leq$ $\displaystyle({\tilde{k}_{2}^{\infty}}+k_{3}^{\infty}c_{R}C^{}(\delta M_{1}+Q^{\infty}))\int^{T}_{0}\int_{\Omega}(\|u_{1}\|^{2}+\|u_{2}\|^{2})dxd\tau$ $\displaystyle+$ $\displaystyle\frac{k_{3}^{\infty}}{\delta}c_{R}c_{Q}M_{1}C^{}\int^{T}_{0}\int_{\Omega}\int^{\tau}_{0}(\|u_{1}\|^{2}+\|\nabla u_{1}\|^{2})dsdxd\tau.$ Take $\phi_{2}=u_{2}$ in (14), we get $\displaystyle\frac{1}{2}\int^{t}_{0}\int_{\Omega}\partial_{t}\|u_{2}\|^{2}dxd\tau$ $\displaystyle+$ $\displaystyle\tilde{d}_{2}\int^{t}_{0}\int_{\Omega}\|\nabla u_{2}\|^{2}dxd\tau$ $\displaystyle\leq$ $\displaystyle-\tilde{k}_{2}\int^{t}_{0}\int_{\Omega}\partial_{t}\|u_{2}\|^{2}dxd\tau+\frac{\tilde{k}_{1}^{\infty}}{2}\int^{t}_{0}\int_{\Omega}(\|u_{1}\|^{2}+\|u_{2}\|^{2})dxd\tau$ $\displaystyle+$ $\displaystyle{\tilde{a}^{\infty}}\int^{t}_{0}\int_{\Omega}u_{2}u_{3}dxdt-{\tilde{b}}\int^{T}_{0}\int_{\Omega}\|u_{2}\|^{2}dxd\tau.$ $\displaystyle\int^{t}_{0}\int_{\Omega}\partial_{t}\|u_{2}\|^{2}dxd\tau$ $\displaystyle+$ $\displaystyle\tilde{d}_{2}\int^{t}_{0}\int_{\Omega}\|\nabla u_{2}\|^{2}dxd\tau$ (88) $\displaystyle\leq$ $\displaystyle({\tilde{k}_{1}^{\infty}}+\tilde{a}^{\infty})\int^{t}_{0}\int_{\Omega}(\|u_{1}\|^{2}+\|u_{2}\|^{2}+\|u_{3}\|^{2})dxd\tau.$ Similarly, we obtain from (14) $\displaystyle\int^{t}_{0}\int_{\Omega}\partial_{t}\|u_{3}\|^{2}dxd\tau$ $\displaystyle+$ $\displaystyle\tilde{d}_{3}\int^{t}_{0}\int_{\Omega}\|\nabla u_{3}\|^{2}dxd\tau\leq\tilde{b}^{\infty}\int^{t}_{0}\int_{\Omega}(\|u_{2}\|^{2}+\|u_{3}\|^{2})dxd\tau,$ (89) $\displaystyle\int^{t}_{0}\int_{\Omega}\partial_{t}\|u_{4}\|^{2}dxd\tau$ $\displaystyle+$ $\displaystyle\tilde{d}_{4}\int^{t}_{0}\int_{\Omega}\|\nabla u_{4}\|^{2}dxd\tau\leq\tilde{k}^{\infty}_{1}\int^{T}_{0}\int_{\Omega}(\|u_{1}\|^{2}+\|u_{3}\|^{2})dxd\tau.$ (90) Adding side by side (87)-(90) and applying Gronwall’s inequality to the corresponding result, we receive $\displaystyle\Sigma_{i=1}^{4}\int_{\Omega}\|u_{i}\|^{2}dx+\hat{d}\Sigma_{i=1}^{4}\int^{t}_{0}\int_{\Omega}\|\nabla u_{i}\|^{2}dxd\tau+\hat{d}\int^{t}_{0}\int_{\Omega}\|u_{1}\|^{2}dxd\tau\leq 0.$ (91) In (91), we have $\hat{d}:=min\\{\tilde{d}_{1},\tilde{d}_{2},\tilde{d}_{3},\tilde{d}_{4},\tilde{k}_{1}\\}>0$. Taking in (92) supremum over $(0,T)$, we obtain $\displaystyle\Sigma_{i=1}^{4}\int_{\Omega}\|u_{i}\|^{2}dx+\hat{d}\Sigma_{i=1}^{4}\int^{T}_{0}\int_{\Omega}\|\nabla u_{i}\|^{2}dxd\tau\leq 0,$ (92) which concludes the proof of the Lemma. ###### Lemma 20. (Strong formulation of the two-scale limit equations) Assume the hypothesis of Lemma 12 to hold. Then the strong formulation of the two-scale limit equations (for all $t\in(0,T)$) reads $\displaystyle\partial_{t}u_{1}(t,x)$ $\displaystyle+\nabla\cdot(-\tilde{d}_{1}\nabla u_{1}(t,x))$ (93) $\displaystyle=-\tilde{k}_{1}(t)u_{1}(t,x)+\tilde{k}_{2}(t)u_{2}(t,x)$ $\displaystyle-\frac{1}{\|Y\|}\int_{\Gamma^{sw}}k_{3}(t,y)R(u_{1}(t,x))Q(u_{5}(t,x,y))d\sigma_{y},\;x\in\Omega$ $\displaystyle u_{1}(0,x)$ $\displaystyle=u_{10}(x),\;x\in\bar{\Omega},$ $\displaystyle n\cdot(-\tilde{d}_{1}\nabla u_{1}(t,x))$ $\displaystyle=0,\;x\in\partial\Omega$ $\displaystyle\partial_{t}u_{2}(t,x)$ $\displaystyle+\nabla\cdot(-\tilde{d}_{2}\nabla u_{2}(t,x))=\tilde{k}_{1}(t)u_{1}(t,x)-\tilde{k}_{2}(t)u_{2}(t,x)$ (94) $\displaystyle+\tilde{a}(t)u_{3}(t,x)-\tilde{b}(t)u_{2}(t,x),\;x\in\Omega,$ $\displaystyle u_{2}(0,x)$ $\displaystyle=u_{20}(x),\;x\in\bar{\Omega},$ $\displaystyle n\cdot(-\tilde{d}_{2}\nabla u_{2}(t,x))$ $\displaystyle=0,\;x\in\partial\Omega,$ $\displaystyle\partial_{t}u_{3}(t,x)$ $\displaystyle+\nabla\cdot(-\tilde{d}_{3}\nabla u_{3}(t,x))=-\tilde{a}(t)u_{3}(t,x)+\tilde{b}(t)u_{2}(t,x),\;x\in\Omega,$ (95) $\displaystyle u_{3}(0,x)$ $\displaystyle=u_{30}(x),\;x\in\bar{\Omega},$ $\displaystyle u_{3}(t,x)$ $\displaystyle=u^{D}_{3}(x),\;x\in\Gamma^{D},$ $\displaystyle n\cdot(-\tilde{d}_{3}\nabla u_{3}(t,x))$ $\displaystyle=0,\;x\in\Gamma^{N}$ $\displaystyle\partial_{t}u_{4}(t,x)$ $\displaystyle+\nabla\cdot(-\tilde{d}_{4}\nabla u_{4}(t,x))=\tilde{k}_{1}(t)u_{1}(t,x),\;x\in\Omega,$ (96) $\displaystyle u_{4}(0,x)$ $\displaystyle=u_{40}(x),\;x\in\bar{\Omega},$ $\displaystyle n\cdot(-\tilde{d}_{4}\nabla u_{4}(t,x))$ $\displaystyle=0,\;x\in\partial\Omega$ $\displaystyle\partial_{t}u_{5}(t,x,y)$ $\displaystyle=k_{3}(t,y)R(u_{1}(t,x))Q(u_{5}(t,x,y)),\;x\in\Omega,y\in\Gamma^{sw},$ (97) $\displaystyle u_{5}(0,x,y)$ $\displaystyle=u_{50}(x,y)\;x\in\bar{\Omega},y\in\Gamma^{sw},$ where $\tilde{d}_{i},i\in\\{1,2,3,4\\}$ and $\tilde{k}_{j},j\in\\{1,2\\}$ are defined in Theorem 14. ### Acknowledgements We would like to thank M. Ptashnyk (RWTH Aachen) and M. A. Peletier (TU Eindhoven) for fruitful discussions on this subject. ## References [1] T. Fatima, N. Arab, E. P. Zemskov, A. Muntean, Homogenization of a reaction-diffusion system modeling sulfate corrosion in locally-periodic perforated domains, J. Eng. Math. (2010) to appear. * [2] M. Böhm, F. Jahani, J. Devinny, G. Rosen, A moving-boundary system modeling corrosion of sewer pipes, Appl. Math. Comput. 92 (1998) 247–269. * [3] C. V. Nikolopoulos, A mushy region in concrete corrosion, Appl. Math. Model. 34 (2010) 4012–4030. * [4] V. Chalupeck${\rm\acute{y}}$, T. Fatima, A. Muntean, Numerical study of a fast micro-macro mass transfer limit: The case of sulfate attack in sewer pipes, J. of Math-for-Industry 2B (2010) 171–181. * [5] A. Muntean, M. Neuss-Radu, A multiscale Galerkin approach for a class of nonlinear coupled reaction-diffusion systems in complex media, J. Math. Anal. Appl. 371 (2) (2010) 705–718. * [6] A. Bensoussan, J. L. Lions, G. 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C. Evans, Partial Differential Equations, Vol. 19, AMS, Providence, Rhode Island USA, 1998.	arxiv-papers	2010-11-26T13:37:44	2024-09-04T02:49:15.250550	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Tasnim Fatima and Adrian Muntean", "submitter": "Tasnim Fatima", "url": "https://arxiv.org/abs/1011.5781" }
1011.6235	# Estimating black hole masses of blazars Xue-Bing Wu1 e-mail:wuxb@pku.edu.cn F. K. Liu1 M. Z. Kong2 R. Wang3 J. L. Han4 1Department of Astronomy Peking University Beijing 100871 China 2 Department of Physics Hebei Normal University Shijiazhang 050016 China 3 Steward Observatory University of Arizona Tucson AZ 85721 USA 4 National Astronomical Observatories Chinese Academy of Sciences Beijing 100012 China ###### Abstract Estimating black hole masses of blazars is still a big challenge. Because of the contamination of jets, using the previously suggested size – continuum luminosity relation can overestimate the broad line region (BLR) size and black hole mass for radio-loud AGNs, including blazars. We propose a new relation between the BLR size and $H_{\beta}$ emission line luminosity and present evidences for using it to get more accurate black hole masses of radio-loud AGNs. For extremely radio-loud AGNs such as blazars with weak/absent emission lines, we suggest to use the fundamental plane relation of their elliptical host galaxies to estimate the central velocity dispersions and black hole masses, if their velocity dispersions are not known but the host galaxies can be mapped. The black hole masses of some well-known blazars, such as OJ 287, AO 0235+164 and 3C 66B, are obtained using these two methods and the M - $\sigma$ relation. The implications of their black hole masses on other related studies are also discussed. ###### keywords: Black holes – galaxies:active – blazars – jets ## 1 Introduction In the last two decades, dynamical measurements have clearly revealed that supermassive black holes exist in the center of nearby galaxies. However, dynamical methods can not be applied to most of AGNs because they are too bright. Currently the most reliable method for AGN black hole mass estimation is the reverberation mapping. Using this technique, the BLR size can be measured using the time lag between the variations of continuum and emission line fluxes. The black hole mass can be derived from the BLR size and the characteristic velocity using the virial law. So far, reverberation mapping studies have yielded black hole masses of about 40 Seyfert 1 galaxies and nearby quasars ( Kaspi et al. 2000, 2005; Peterson et al. 2004). Using the observed data of these reverberation mapping AGNs, an empirical relation between the BLR size ($R$) and continuum luminosity at 5100$\AA$ ($L_{5100\AA}$) has been derived by Kaspi et al. (2000, 2005), which has been frequently adopted to estimate the BLR size and the black hole masses for large samples of AGNs, including radio-loud quasars. However, the optical continuum luminosity of radio-loud AGNs may not be a good indicator of ionizing luminosity. Powerful jets of blazar-type AGNs may significantly contribute to the optical continuum luminosity. Therefore, using the $R-L_{5100\AA}$ relation may significantly overestimate the actual BLR size and the black hole mass of radio-loud AGNs. In addition, another tight correlation between black hole mass and bulge velocity dispersion ($\sigma$) has been found for nearby galaxies (Tremaine et al. 2002) and for a few Seyfert galaxies as well (Ferrarese et al. 2001). Such a relation suggests a possibility to estimate the black hole masses of AGNs using the measured bulge velocity dispersions. Especially for BL Lacertae objects, the reverberation mapping technique cannot be applied because they show no or weak emission lines in their optical spectra. Using the MBH-$\sigma$ relation may be the only way to derive their black hole masses, though measuring $\sigma$ is possible only for nearby sources. We have to seek for other methods for most blazars because the velocity dispersions of their host galaxies are too difficult to be measured. In this paper we report our progress on estimating the black hole masses of radio-loud AGNs using a new BLR size – $H_{\beta}$ emission line luminosity relation and the fundamental plane relation of the elliptical host galaxies. The results on some well-known blazars are also presented. ## 2 The BLR size – Hβ luminosity relation Using the available data of BLR sizes and Hβ fluxes for 34 AGNs in the reverberation mapping studies (Kaspi et al. 2000), we investigated the relation between the BLR size and the Hβ emission line luminosity (Wu et al. 2004). An empirical relation between the BLR size and Hβ luminosity was derived as: $\rm{Log}~{}R~{}(\rm{lt-days})=(1.381\pm 0.080)+(0.684\pm 0.106)Log~{}(L_{H_{\beta}}/10^{42}~{}ergs~{}s^{-1}).$ (1) The Spearman’s rank correlation coefficient of this relation is 0.91. In Fig. 1 we show the dependence of the BLR size on $L_{H_{\beta}}$ and $L_{5100\AA}$. Obviously these two relations are similar, which means that the $R-L_{H_{\beta}}$ relation can be an alternative of the $R-L_{5100\AA}$ relation in estimating the BLR size for radio-quiet AGNs. We applied both the $R-L_{H_{\beta}}$ and $R-L_{5100\AA}$ relations to estimate the black hole masses of 87 radio-loud quasars and compare them in Fig. 2. Evidently the masses obtained with the $R-L_{H_{\beta}}$ relation are systematically lower that those obtained with the $R-L_{5100\AA}$ relation for some extremely radio-loud quasars. The difference between two black hole mass estimates is smaller when the radio-loudness is small but becomes larger as the radio- loudness increases. For some individual quasars with higher radio-loudness, the black hole mass estimated with the $R-L_{5100\AA}$ relation can be 3$\sim$10 times larger than that estimated with the $R-L_{H_{\beta}}$ relation. Recently Kong et al. (2006) also extended such a study to the broad UV emission lines MgII and CIV, and obtained the BLR size – UV emission line luminosity relations. Our results demonstrated that using the BLR size – emission line luminosity relations can avoid the overestimations of the black hole masses for blazar-like radio-loud AGNs. Figure 1: The $R-L_{H_{\beta}}$ relation and the $R-L_{5100\AA}$ relation. The open and filled symbols denote Seyferts and quasars respectively. The figure is taken from Wu et al. (2004). Figure 2: Comparison of the black hole masses of radio-loud quasars estimated with two R–L relations, and the dependence of the black hole mass difference on radio loudness. The figure is taken from Wu et al. (2004). ## 3 Black hole masses estimated from the fundamental plane relation of AGN elliptical host galaxies For radio-loud AGNs such as BL Lacs with weak/absent emission lines, the emission line based methods for black hole mass estimations can not apply. Directly measuring their stellar velocity dispersion is also difficult. However, the host galaxies of BL Lacs are virtually ellipticals. It is well known for ellipticals that three observables, namely the effective radius ($R_{e}$), the average surface brightness ($<\mu_{e}>_{R}$ in R-band) and the central velocity dispersion ($\sigma$), follow a tight fundamental plane relation. For about 300 normal ellipticals and radio galaxies, Bettoni et al. (2001) found that the fundamental plane can be robustly described as $\log R_{e}=(1.27\pm 0.04)\log\sigma+(0.326\pm 0.007)<\mu_{e}>_{R}-8.56\pm 0.06,$ (2) This relation provides us another way to estimate the central velocity dispersions and then the black hole masses of AGNs (Wu, Liu & Zhang 2002). Figure 3: Histograms of the derived black hole mass distribution of HBLs and LBLs. The figure is taken from Wu et al. (2002). Figure 4: Histograms of the derived black hole mass distribution of radio galaxies, radio-loud and radio- quiet quasars. The figure is taken from Wu et al. (2002). Using the imaging data of BL Lacs obtained from the HST snapshot survey (Urry et al. 2000), we adopted the fundamental plane relation to estimate the central velocity dispersions and black hole masses of 51 high-frequency peaked BL Lacs (HBLs) and 12 low-frequency peaked BL Lacs (LBLs). Our results show no significant difference in the black hole masses between HBLs and LBLs (see Fig. 3). We also applied the same method to 10 radio galaxies (RGs), 10 radio- loud quasars (RLQs) and 13 radio-quiet quasars (RQQs) which have been imaged by HST (McLure et al. 1999), we found that there are no significant differences in the black hole masses among these different types of AGNs with elliptical host galaxies (see Fig. 4). ## 4 Black hole masses of three well-known blazars ### 4.1 AO 0235+164 Liu, Zhao, & Wu (2006) applied the $R-L_{H_{\beta}}$ relation suggested by Wu et al. (2004) to a well studied BL Lac object AO 0235+164 and estimated its black hole mass as $5.8\times 10^{8}M_{\odot}$, which is consistent with the mass ($3.6\times 10^{8}M_{\odot}$) estimated from the $M_{BH}-\sigma$ relation by taking the narrow emission line width as a surrogate of $\sigma$, but is much smaller than the mass ($1.5\times 10^{9}M_{\odot}$) obtained from the $R-L_{5100\AA}$ relation. This again demonstrates that using the previous $R-L_{5100\AA}$ relation can overestimate the black hole masses of blazar-like AGNs. Our study indicates that the black hole mass of AO 0235+164 is most likely around $5\times 10^{8}M_{\odot}$. ### 4.2 OJ 287 We applied the fundamental plane relation of ellipticals to a well-known BL Lac object OJ 287, which is a possible supermassive black hole binary system. Liu & Wu (2002) estimated its primary black hole mass to be about $4\times 10^{8}M_{\odot}$, which is consistent with the upper limit ($10^{9}M_{\odot}$) obtained by Valtaoja et al. (2000) based on a binary black hole model for OJ 287. This value is also within the range of typical black hole masses for BL lac objects (Wu, Liu & Zhang 2002). We noticed that our derived black hole mass is much smaller than $1.8\times 10^{10}M_{\odot}$, which is required in a new precessing binary black hole model of OJ 287 (Valtonen 2007). ### 4.3 3C 66B 3C 66B is a well-known nearby FR I radio galaxy. The central velocity dispersion of 3C 66B has been obtained by Balcells et al. (1995) as 348$\pm$29 km/s based on their spectroscopic observations. Using the M - $\sigma$ relation suggested by Tremaine et al. (2002), we estimate that $\log(M/M_{\odot})=9.097\pm 0.175$, namely the mass of the central black hole of 3C 66B is about $1.25\times 10^{9}M_{\odot}$. This is within the range of typical black hole masses for radio galaxies (Wu, Liu & Zhang 2002), but one- order of magnitude smaller than the maximum mass estimated by Sudou et al. (2003), $5\times 10^{10}M_{\odot}$, which was obtained by modeling the observed orbital motion with a binary black hole model. A very recent study by Iguchi et al. (2010) supported that the larger black hole mass is $(1.2^{+0.5}_{-0.2})\times 10^{9}$ $M_{\odot}$, which is consistent with the value obtained by the M - $\sigma$ relation. Janet et al. (2004) adopted the black hole mass of 3C 66B obtained by Sudou et al. (2003) to calculate the gravitational wave emission from 3C 66B and argued that the binary black hole model can be excluded because observations from the Pulsar Timing Array (PTA) did not detect the gravitational wave from 3C 66B. However, if the smaller black hole mass is adpted, the gravitational wave emitted from 3C 66B will be weaker than the current PTA detection limit and the binary black hole model can not be excluded. ## 5 Summary We proposed to use the BLR size – emission line luminosity relation and the fundamental plane relation of the elliptical host galaxies to estimate the black hole masses of radio-loud AGNs. We demonstrated that with the first relation we can get more accurate black hole mass estimates for radio-loud AGNs than using the usual $R-L_{5100\AA}$ relation, and for some radio-loud AGNs such as BL Lacs the second method is probably the only available one for their black hole mass estimations in the case when directly measuring the stellar velocity dispersions is difficult. We have adopted these methods, as well as the M - $\sigma$ relation, to estimate the black hole masses of three well-known blazars, AO 0235+164, OJ 297 and 3C 66B, and the results are helpful to other related study about their physics nature. Finally, we would like to mention that these two methods can be also applied to estimate the black hole masses of high redshift AGNs, including blazars, with high quality spectroscopic and imaging observations. This work is supported by the NSFC Grants (No. 10473001, 10573001, 10525313 and 11033001) and the 973 program (No. 2007CB815405). ## References * [1] Balcells, M. et al. 1995, A&A, 302, 665 * [2] Bettoni, D. et al. 2001, A&A, 380, 471 * [3] Ferrarese, L. et al. 2001, ApJ, 555, L79 * [4] Iguchi, S., Takeshi, O., & Sudou, H. 2010, ApJ, 724, L166 * [5] Jenet, F.A., et al. 2004, ApJ, 606, 799 * [6] Kaspi, S. et al. 2000, ApJ, 533, 631 * [7] Kaspi, S. et al. 2005, ApJ, 629, 61 * [8] Kong, M. Z., Wu, X.-B., Wang, R., Han, J. L. 2006, Chin. J. Astron. Astrophys. 6, 396 * [9] Liu, F. K., & Wu, X.-B., 2002, A&A, 388, L48 * [10] Liu, F. K., Zhao, G., & Wu, X.-B., 2006, ApJ, 650, 749 * [11] McLure, R. J., et al. 1999, MNRAS, 308, 377 * [12] Peterson, B. M. et al. 2004, ApJ, 613, 682 * [13] Sudou, H. et al., et al. 2003, Science, 300, 1263 * [14] Tremaine, S. et al. 2002, ApJ, 574, 740 * [15] Urry, C. M. et al. 2000, ApJ, 532, 816 * [16] Valtaoja, E. et al. . 2000, ApJ, 531, 744 * [17] Valtonen, M.J., 2007, ApJ, 659, 1074 * [18] Wu, X.-B., Liu, F. K., Zhang, T. Z. 2002, A&A, 389, 742 * [19] Wu, X.-B., Wang, R., Kong, M. Z., Liu, F. K., Han, J. L., 2004, A&A, 424, 793	arxiv-papers	2010-11-29T14:18:43	2024-09-04T02:49:15.272926	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Xue-Bing Wu, F. K. Liu, M. Z. Kong, R. Wang, J. L. Han", "submitter": "Xue-Bing Wu", "url": "https://arxiv.org/abs/1011.6235" }
1011.6262	# Viscous lock-exchange in rectangular channels J.MARTIN N.RAKOTOMALALA L.TALON D.SALIN Univ. Pierre et Marie Curie- Paris6, Univ. Paris-Sud, CNRS. Lab FAST, Bat. 502, Rue du Belvedere, Campus Univ., Orsay, F-91405, France. ###### Abstract In a viscous lock-exchange gravity current, which describes the reciprocal exchange of two fluids of different densities in a horizontal channel, the front between two Newtonian fluids spreads as the square root of time. The resulting diffusion coefficient reflects the competition between the buoyancy driving effect and the viscous damping, and depends on the geometry of the channel. This lock-exchange diffusion coefficient has already been computed for a porous medium, a $2D$ Stokes flow between two parallel horizontal boundaries separated by a vertical height, $H$, and, recently, for a cylindrical tube. In the present paper, we calculate it, analytically, for a rectangular channel (horizontal thickness $b$, vertical height $H$) of any aspect ratio ($H/b$) and compare our results with experiments in horizontal rectangular channels for a wide range of aspect ratios ($1/10-10$). We also discuss the $2D$ Stokes-Darcy model for flows in Hele-Shaw cells and show that it leads to a rather good approximation, when an appropriate Brinkman correction is used. ## 1 Introduction The lock-exchange configuration refers to the release, under gravity, of the interface between two fluids of different densities, confined in the section of a horizontal channel. This physical process has prompted renewed interest, as a part of the carbon dioxide sequestration issues ([Neufeld & Huppert(2009)]). The top of Fig. 1 shows the initial lock-exchange situation of a so-called full-depth release. The two fluids, initially separated by a vertical barrier (the lock gate), fill the whole section of the tank. When the gate is withdrawn (bottom of Fig. 1), buoyancy drives the denser fluid along the bottom wall, while the lighter one flows in the opposite direction at the top of the channel. The so-called lock-exchange results in the elongation of the interface between the two fluids along the horizontal direction. Different regimes have been reported for the velocity and shape of the elongating interface. The slumping phase refers to the initial regime where inertia dominates over viscous forces, which typically applies for the case of salted and fresh water in a tank. In this regime, [Benjamin(1968)], and more recently [Shin et al.(2004)Shin, Dalziel & Linden] showed that in the presence of a small density contrast (i.e. in the Boussinesq approximation $\Delta\rho<<\rho$), the two opposite currents traveled at the same constant velocity. When the Boussinesq approximation does not apply, [Lowe et al.(2005)Lowe, Rottman & Linden], [Birman et al.(2005)Birman, Martin & Meiburg], [Cantero et al.(2007)Cantero, Lee, Balachandar & Garcia] and [Bonometti et al.(2008)Bonometti, Balachandar & Magnaudet] showed that the two opposite fronts did travel at constant, but with different velocities. However this interface elongation, proportional to the time, is slowed down at later stages, in the viscous phase, where dissipation prevails over inertia. In the latter regime, the interface elongates as $t^{\alpha}$, where the exponent $\alpha$, smaller than unity, may take different values depending on the geometry and the confinement of the flow ([Didden & Maxworthy(1982), Huppert(1982), Gratton & Minotti(1990), Cantero et al.(2007)Cantero, Lee, Balachandar & Garcia, Takagi & Huppert(2007), Hallez & Magnaudet(2009)]). In porous media, [Bear(1988)] and [Huppert & Woods(1995)] predicted an interface spreading proportionally to the square root of time that [Séon et al.(2007)Séon, Znaien, Salin, Hulin, Hinch & Perrin] observed in a horizontal cylindrical tube. Such a spreading can be quantified with a diffusion coefficient, which reflects the balance between the buoyancy driving and the viscous damping. This coefficient, which depends on the nature and the geometry of the flow, has been computed for a porous medium by [Huppert & Woods(1995)], for a $2D$ Stokes flow between two parallel horizontal boundaries separated by a vertical height, $H$, by [Hinch(2007)] and [Taghavi et al.(2009)Taghavi, Seon, Martinez & Frigaard], and for a cylindrical tube by [Séon et al.(2007)Séon, Znaien, Salin, Hulin, Hinch & Perrin]. However, to our knowledge, such a diffusion coefficient has not been derived for a rectangular channel (horizontal thickness $b$, vertical height $H$, Fig. 1), for which one expects to recover the porous medium regime for $b\ll H$, and, possibly, the Stokes flow regime for $b\gg H$. In order to gather the limiting cases in the same paper, we first recall the results for porous media and $2D$ Stokes flows, together with the tube case, for the sake of comparison. Then we compute, for a rectangular channel of aspect ratio, $H/b$, the dependence of the interface $h(x,t)$ and the corresponding viscous lock-exchange diffusion coefficient. We also test the so-called Stokes-Darcy $2D$ model to this lock- exchange configuration. Finally, we test and validate our theoretical results with experiments in horizontal rectangular channels for a wide range of aspect ratios ($1/10-10$). Figure 1: Sketch of the rectangular cell of height $H$ in the gravity direction and width $b$. Top: Initial configuration where the heavy fluid of density, $\rho$, is separated from the light fluid of density, $\rho-\Delta\rho$, by a vertical gate (plane $y-z$). Bottom: After removing the gate, buoyancy differences cause the denser fluid to flow in one direction along the bottom of vessel, while the lighter one flows in the opposite direction at the top of the vessel. The goal is to determine the space and time dependencies of the pseudo-interface, $h(x,t)$. ## 2 Lock-exchange in different geometries Let us first recall the basic hypotheses on the viscous gravity currents, common to the different geometries, used for instance by [Huppert & Woods(1995)] or [Hinch(2007)]). As sketched in Fig. 1, the interface between the two fluids is assumed independent on the $y$ direction. Its distance from the bottom boundary of the vessel is denoted $h(x,t)$. This interface can be a pseudo-interface between two miscible fluids for which molecular diffusion can be neglected or between two immiscible fluids, provided that the interfacial tension can be neglected. The flow is assumed to be quasi-parallel to the horizontal $x$ axis. This is a key hypothesis. Neglecting accordingly the vertical component of the fluid velocity implies that the vertical pressure gradient follows the hydrostatic variations: $\partial p/\partial z=-\rho g$. This hypothesis is violated at short times, immediately after the opening of the gate, but should become valid at later stages, as soon as the interface has slumped over a distance larger than $H$, thus ensuring a small enough local slope $\partial h/\partial x$. Then, the pressures, $P_{+}$ and $P_{-}$, in the lower layer, $0<z<h(x,t)$, and in the upper one, $h(x,t)<z<H$, respectively, write $P_{+}=p(x,t)-\rho_{+}gz\;\;;\;P_{-}=P_{+}+\Delta\rho g(z-h(x,t))$ (1) where $p(x,t)$ denotes the pressure at the lower wall, $z=0$. The difference between the horizontal pressure gradients in the two fluids is therefore linked to the interface slope by: $\frac{\partial P_{+}}{\partial x}-\frac{\partial P_{-}}{\partial x}=\Delta\rho g\frac{\partial h}{\partial x}$ (2) The time evolution of the interface, $h(x,t)$, is governed by the mass conservation of each fluid (see Fig. 1 for notations). For instance, for the heavier bottom layer we have: $\frac{\partial h}{\partial t}+\frac{\partial q}{\partial x}=0$ (3) where $q(x,t)=q_{+}(x,t)$ is the horizontal flux ($m^{2}/s$) of the denser fluid at the location $x$: $q(x,t)=q_{+}(x,t)=\int_{0}^{h}\frac{1}{b}\int u_{x}(x,y,z,t)dydz$ (4) with $u_{x}(x,y,z,t)$ the $x-$velocity component and $b$ the spanwise length (the $y$ integration is along this spanwise length). Moreover, in our configuration of uniform section along the horizontal axis $x$, $u_{x}(x,y,z,t)$ must also satisfy the no net flux condition: $q_{+}(x,t)+q_{-}(x,t)=\int_{0}^{H}\frac{1}{b}\int u_{x}(x,y,z,t)dydz=0$ (5) We will see in the following that, in the viscous regime of interest, the horizontal velocity component $u_{x}$, solution of either a Darcy or a Stokes equation, is proportional to the pressure gradient in each fluid layer. Such solutions, combined with eq. (2), eq. (4) and eq. (5) allow then to eliminate the pressure gradients and to derive an expression of the flux $q$, of the form: $q=-Df(\frac{h}{H})\frac{\partial h}{\partial x}$ (6) where $D$ writes $D=\tau\frac{\Delta\rho g}{\eta}$ (7) and where the constant $\tau$ (scaling with a volume) and the function $f$ depend on the geometry and the flow equation and $\eta$ is the dynamic viscosity. Using the expression (6) for the flux, eq. (3) admits a self- similar solution $h(\zeta)=H\,\psi(\zeta)$ with the similarity variable $\zeta=x/\sqrt{Dt}$, which obeys: $-\zeta\frac{d\psi}{d\zeta}=2\,\frac{d}{d\zeta}(f(\psi)\frac{d\psi}{d\zeta})$ (8) This equation may alternatively be rewritten, in terms of $\zeta(\psi)$: $\zeta(\frac{d\zeta}{d\psi})^{2}-2\,f\frac{d^{2}\zeta}{d\psi^{2}}+2\frac{df}{d\psi}(\frac{d\zeta}{d\psi})=0$ (9) The solution of the above equations can be found analytically or numerically, depending on the complexity of the normalized flux function $f(\psi)$. In the following, we will first recall the case of porous media, treated by [Huppert & Woods(1995)] and the $2D$ Stokes flow, addressed by [Hinch(2007)] (unpublished) and [Taghavi et al.(2009)Taghavi, Seon, Martinez & Frigaard]. We note that the latter paper included the effects of the rheological properties of the fluids. However, in order to focus on the geometrical aspects, we will assume in the following that both fluids are Newtonian and have the same viscosity. ### 2.1 Lock-exchange in porous media For a homogeneous layer of porous medium of permeability $\kappa$ (see for instance [Huppert & Woods(1995)]), the flow in each fluid is given by Darcy’s law which relates the velocity in each phase to the local pressure gradient : $u_{x\pm}=-\frac{\kappa}{\eta}\frac{\partial P_{\pm}}{\partial x}$ (10) At a given location $x$, the velocity is then uniform in each layer, and the no net flux condition (eq. (5)) simply writes: $hu_{x+}+(H-h)u_{x-}=0$. The latter equation, combined with eq. (10) and eq. (2), leads to eq. (6), and thus (combined with eq. (3)) to eq. (8) with a diffusion coefficient and a flux function: $\displaystyle D_{PM}=\kappa\,H\frac{\Delta\rho g}{\eta}$ (11) $\displaystyle f_{PM}(\psi)=\psi(1-\psi)$ (12) The solution of eq. (8), in the similarity variable $\zeta=x/\sqrt{D_{PM}t}$, is then a linear profile ([Huppert & Woods(1995)]): $\psi=h(x,t)/H=(1+\zeta)/2$ (13) The so-obtained front profile in homogeneous porous media is displayed in Fig. 2 (straight line) together with the ones for rectangular cells (referred to in subsection 2.3). The leading ($\psi=0$, $\zeta=-1$) and trailing ($\psi=1$, $\zeta=1$) edges spread as $\sqrt{D_{PM}t}$. Therefore, the lock-exchange diffusion coefficient for porous media is $D_{PM}$. It should be noticed that [Bear(1988)] reported a numerical integration of eq. (8) indicating that the gravity current spreads as the square root of time. Figure 2: Lock-exchange interface $\psi(\zeta)$ between the two fluids for rectangular cells of different aspect ratios $\Gamma=H/b$. The straight line of slope 0.5 corresponds to the Darcy porous media limit (eq. (13)). The other curves correspond to the case of a rectangular cross-section channel (subsection 2.3). From the bottom left to right, $\Gamma=10,4,2,1,0.5$ and $0.2$. Note that the corresponding result for a Hele-Shaw cell, that is two parallel plates of height, $H$, separated by a tiny gap $b$ ($b\ll H$), is obtained using the permeability $\kappa=b^{2}/12$: $D_{HS}=\frac{b^{2}H\Delta\rho g}{12\eta}$ (14) ### 2.2 Lock-exchange for a $2D$ Stokes flow between two horizontal boundaries For a $2D$ Stokes flow between two horizontal parallel boundaries, separated by a height $H$ in the plane ($z-x$) (assuming invariance along the $y-$direction), the flow in each fluid is given by the Stokes equation: $\eta\;\nabla^{2}u_{x\pm}(x,z)=\frac{\partial P_{\pm}}{\partial x}$ (15) At a given location $x$, the velocity profile consists of two parabola profiles matching the no slip boundary conditions at the bottom and the top boundaries ($u_{x+}(x,0)=u_{x-}(x,H)=0$) and the continuity of the velocity ($u_{x-}(x,h)=u_{x+}(x,h)$) and of the shear stress ($\eta\;\partial u_{x-}(x,h)/\partial z=\eta\;\partial u_{x+}(x,h)/\partial z$) at the interface, $z=h$. Using the no net flux condition (eq. (5)) and eq. (2), we obtain eq. (6), which enables to rewrite eq. (3), in the form of eq. (8) or eq. (9), with: $\displaystyle D=\frac{H^{3}\Delta\rho g}{3\eta}$ (16) $\displaystyle f(\psi)=\psi^{3}(1-\psi)^{3}$ (17) Note that the polynomial development of the solution of eq. (9) around $\psi=0$ gives: $\zeta=-\zeta_{0}+2\psi^{3}/(3\,\zeta_{0})$. Thus, the location, $\zeta(\psi=0)=-\zeta_{0}$, of the leading edge of the interface is indeed constant in the similarity variable. Moreover, the development shows that the slope of the interface is vertical at the bottom wall ($\psi=0$). This is also the case at the upper wall ($\psi=1$), as the problem is symmetric with respect to the centre of the cell. We note that in the presence of such a vertical slope, our (horizontal) quasi-parallel flow assumption falls locally, but it is still valid upstream and downstream, where the slope of the interface remains small. The solution of eq. (9) can be found numerically using a shooting method similar to the one used by [Hinch(2007)] and [Taghavi et al.(2009)Taghavi, Seon, Martinez & Frigaard]. It was computed starting the integration of eq. (9) from $(\psi=0.5,\zeta=0)$ and matching the asymptotic development in the vicinity of $\psi=0$. From the so-obtained solution, one can deduce the spreading diffusion coefficient between the leading edge ($h=0$, $-\zeta_{0}=-0.1607$) and the trailing edge ($h=H$, $\zeta_{0}$) of the front, from $[0.5\,(x(h=0)-x(h=H))]^{2}=D\,t$, which gives $D_{2D}=D\,\zeta_{0}^{2}$, so that: $D_{2D}=0.0086\,H^{3}\,\frac{\Delta\rho g}{\eta}$ (18) This result is in agreement with the one found by [Hinch(2007)]. [Taghavi et al.(2009)Taghavi, Seon, Martinez & Frigaard] provide five $\psi(\zeta)$ plots in their Fig. 9, corresponding to different viscosity ratios and including our case. From that figure we may obtain a value of their similarity variable, $\eta_{0}\sim 0.09$, which is consistent with our finding $\zeta_{0}=0.1607$, when taking into account their definition of the similarity variable, $\zeta=\eta\,\sqrt{3}$. For completeness, $D_{2D}$ may be compared to the result for a cylindrical tube of diameter $d$ ([Séon et al.(2007)Séon, Znaien, Salin, Hulin, Hinch & Perrin]): $D_{T}=0.0054\,d^{3}\,\frac{\Delta\rho g}{\eta}$ (19) The above expression is indeed very close to the $2D$ result, with $d$ playing the role of $H$. ### 2.3 Lock-exchange in a rectangular cross-section channel This article aims to extend the computation of the lock-exchange diffusion coefficient to rectangular cells of arbitrary cross-sections $H\times b$ (see Fig. 1). In the following, the cross-section aspect ratio will be denoted $\Gamma=H/b$. As in the previous section, a quasi-parallel flow approximation is assumed (_i.e._ small interface slope) which leads to eq. (2) for the pressure gradient. We will also assume the invariance of the interface location along the gap direction $y$. This requires that the deformation of the interface, induced by the flow profile along the direction $y$, relaxes much more quickly than the gradient along $x$. The flow in each fluid obeys a $3D$ Stokes equation: $\eta\;\nabla^{2}u_{x\pm}(x,y,z)=\frac{\partial P_{\pm}}{\partial x}$ (20) In order to solve this equation, we follow the series decomposition in Fourier modes of the velocity field used by [Gondret et al.(1997)Gondret, Rakotomalala, Rabaud, Salin & Watzky]. This paper addressed the issue of the parallel flow of two fluids of different viscosities in a rectangular cell. This issue is very closed to ours, as it requires to solve the Poisson equation (eq. (20)), but with different viscosities and the same pressure gradient for both fluids in [Gondret et al.(1997)Gondret, Rakotomalala, Rabaud, Salin & Watzky]. The method used was to split the velocity into two terms, $u_{x\pm}(x,y,z)=u_{x\pm}^{}(x,y)+u_{x\pm}^{}(x,y,z)$. Here, the first term, $u_{x\pm}^{}(x,y)=\frac{b^{2}}{8\eta}\frac{\partial P_{\pm}}{\partial x}[1-(\frac{2y}{b})^{2}]$ is the Poiseuille-like unperturbed velocity far away from the interface. The second term satisfies the Laplace equation, $\nabla^{2}u_{x\pm}^{*}(x,y,z)=0$ and vanishes far away from the interface. Its expression in terms of a sum of Fourier modes leads to a velocity profile of the form: $u_{x\pm}(x,y,z)=\frac{b^{2}}{8\eta}\frac{\partial P_{\pm}}{\partial x}{\left\\{1-(\frac{2y}{b})^{2}\right.}\\\ \left.{+\sum_{n=1}^{\infty}\frac{32(-1)^{n}\,(a_{\pm n}\,e^{(2n-1)\frac{\pi(z-H/2)}{b}}+b_{\pm n}\,e^{-(2n-1)\frac{\pi(z-H/2)}{b}})}{{\left(2\,n-1\right)}^{3}\,{\pi}^{3}}\cos[(2n-1)\frac{\pi y}{b}]}\right\\}$ (21) in which the no slip boundary conditions at the two vertical walls ($u_{x\pm}(x,y=\pm b/2,z)=0$) have been taken into account. Each Fourier mode, ($(2n-1)\pi/b)$, involves two constants for each fluid, $a_{\pm n}$ and $b_{\pm n}$. These four constants are determined by using the no slip boundary conditions at the bottom and top of the cell ($u_{x+}(x,y,z=0)=u_{x-}(x,y,z=H)=0$) and the continuity of the velocity ($u_{x+}(x,y,z=h)=u_{x-}(x,y,z=h)$) and of the shear stress ($\eta\,\partial u_{x+}(x,y,z=h)/\partial z=\eta\,\partial u_{x-}(x,y,z=h)/\partial z$) at the interface. Figure 3: Log-log plot of the normalized lock-exchange diffusion coefficient, $D_{R}/D_{HS}$, versus the aspect ratio $\Gamma=H/b$, for rectangular cells. $D_{R}$ is normalized by the diffusion coefficient $D_{HS}=b^{2}\,H\Delta\rho g/(12\,\eta)$ (eq. (14), obtained for the Hele-Shaw cell limit, also corresponding to a $2D$ porous medium. Accordingly, the latter regime corresponds to the dashed straight line ($D_{R}/D_{HS}=1$) in this representation. The solid straight line corresponds to the $2D$ Stokes limit (eq. (18)), leading to a slope $2$ in this representation. Combining the so-obtained expressions for the velocity with eq. (2) and the no net flux condition (eq. (5)), one obtains the horizontal flux of the heavy fluid (eq. (6)): $q=-D_{HS}f_{\Gamma}(h/H)\frac{\partial\,h}{\partial\,x}$ (22) with $\displaystyle D_{HS}=\frac{b^{2}H\Delta\rho g}{12\eta}$ (23) $\displaystyle f_{\Gamma}(\psi)=\frac{\psi+\alpha_{\Gamma}(\psi)}{1-\gamma_{\Gamma}}(1-\psi-\alpha_{\Gamma}(\psi)-\gamma_{\Gamma})-\delta_{\Gamma}(\psi)$ (24) where $\displaystyle\alpha_{\Gamma}(\psi)$ $\displaystyle=$ $\displaystyle\frac{1}{\Gamma}\sum_{n=1}^{\infty}\frac{96\,\left(1+e^{\Gamma\left(1-\psi\right)\,\left(2\,n-1\right)\,\pi}\right)\,\left(1-e^{\Gamma\;\psi\,\left(2\,n-1\right)\,\pi}\right)}{\left(1+e^{\Gamma\,\left(2\,n-1\right)\,\pi}\right)\,{\left(2\,n-1\right)}^{5}\,{\pi}^{5}}$ (25) $\displaystyle\delta_{\Gamma}(\psi)$ $\displaystyle=$ $\displaystyle\frac{1}{\Gamma}\sum_{n=1}^{\infty}\frac{48\,{\left(1-e^{\Gamma\left(1-\psi\right)\,\left(2\,n-1\right)\,\pi}\right)}^{2}\,{\left(1-e^{\Gamma\;\psi\,\left(2\,n-1\right)\,\pi}\right)}^{2}}{\left(-1+e^{2\,\Gamma\,\left(2\,n-1\right)\,\pi}\right)\,{\left(2\,n-1\right)}^{5}\,{\pi}^{5}}$ (26) $\displaystyle\gamma_{\Gamma}$ $\displaystyle=$ $\displaystyle\frac{1}{\Gamma}\sum_{n=1}^{\infty}\frac{192\,\tanh\,(\frac{\Gamma\,\left(2\,n-1\right)\,\pi}{2})}{{\left(2\,n-1\right)}^{5}\,{\pi}^{5}}$ (27) Eq. (22) admits a self-similar solution, $h(\zeta)=H\,\psi(\zeta)$, with the similarity variable $\zeta=x/\sqrt{D_{HS}\,t}$, which obeys eq. (8) or eq. (9). As previously, it is easier to compute the solution $\zeta(\psi)$ of eq. (9) subject to the corresponding asymptotics, $\zeta=-\zeta_{0}+8\,\Gamma^{2}\,\psi^{3}/(3\,\zeta_{0})$ in the vicinity of the boundary, $\psi=0$. We solve this equation using the shooting method previously described and using Mathematica Software. The solutions $h(\zeta)$ are plotted in Fig. 2 for different values of the cell aspect ratio $\Gamma$. We notice that, in contrast with Darcy predictions (straight line in Fig. 2), but similarly to the case of the $2D$ Stokes flow, the profiles, $h(\zeta)$, exhibit vertical slopes at the edges of the cell. We note also that such vertical slopes were observed in the experiments by [Séon et al.(2007)Séon, Znaien, Salin, Hulin, Hinch & Perrin] and [Huppert & Woods(1995)]. When comparing their experiments in a Hele-Shaw cell with Darcy predictions, the latter authors reported that ”Some discrepancies develop near the leading edge of the current as a result of the increasing importance of the bottom friction at the nose” (Fig. 2 of [Huppert & Woods(1995)]). This mismatch will be addressed in the next section. According to the so-obtained profiles, stationary in the similarity variable $\zeta$, the leading and trailing edges of the front spread as the square root of time, and a lock-exchange diffusion coefficient, dependent on the cell aspect ratio, can be defined: $D_{R}=\frac{b^{2}H\Delta\rho g}{12\eta}\,F(\frac{H}{b})=D_{HS}\,F(\frac{H}{b})$ (28) Fig. 3 displays a log-log plot of the normalized rectangular cell lock- exchange diffusion coefficient, $D_{R}/D_{HS}=F(H/b)$, versus the aspect ratio $\Gamma=H/b$. At small aspect ratios, $\Gamma<1$, the diffusion coefficient falls on top of the full line of slope $2$, which corresponds to the $2D$ Stokes flow between boundaries distant of $H$ ($b\rightarrow\infty$, eq. (18)). At large aspect ratios, $\Gamma\rightarrow 100$, the diffusion coefficient approaches the dashed line, $D_{R}/D_{HS}=1$, obtained for the $2D$ homogeneous porous medium case (eq. (14)). We note that the latter case, which corresponds to a Hele-Shaw cell of infinite aspect ratio, overestimates the lock-exchange diffusion coefficient, by a relative amount of about $30\%$, for aspect ratios as large as $\Gamma=10-20$. ## 3 $2D$ Stokes-Darcy model for lock-exchange in a rectangular cross-section channel The above-mentioned failures of the $2D$ Darcy model at finite aspect ratios may come from the velocity slip condition at the bottom and top edges of the cell ($z=0$ and $z=H$, respectively). This non physical condition is indeed required by the use of Darcy equation for the flow, which neglects the momentum diffusion in the presence of velocity gradients, in the plane of the cell ($z-x$ plane). The momentum diffusion may however be taken into account in $2D$, through the so-called Stokes-Darcy equation (see [Bizon et al.(1997)Bizon, Werne, Predtechensky, Julien, McCormick, Swift & Swinney, Ruyer-Quil(2001), Martin et al.(2002a)Martin, Rakotomalala & Salin, Zeng et al.(2003)Zeng, Yortsos & Salin]), which is similar to the Darcy-Brinkman equation used in porous media (see [Brinkman(1947)]). This $2D$ model enables to handle discontinuities such as cell edges, gap heterogeneities and fluid interfaces ([Ruyer-Quil(2001), Martin et al.(2002a)Martin, Rakotomalala & Salin, Zeng et al.(2003)Zeng, Yortsos & Salin, Talon et al.(2003)Talon, Martin, Rakotomalala, Salin & Yortsos]) and was successfully applied in the study of Rayleigh-Taylor instability ([Martin et al.(2002a)Martin, Rakotomalala & Salin, Fernandez et al.(2002)Fernandez, Kurowski, Petitjeans & Meiburg, Graf et al.(2002)Graf, Meiburg & Härtel]), of dispersion in heterogeneous fractures ([Talon et al.(2003)Talon, Martin, Rakotomalala, Salin & Yortsos]) and of chemical reaction fronts ([Martin et al.(2002b)Martin, Rakotomalala, Salin & Böckmann]). Although our present case of interest can be handled with $3D$ Stokes calculations, it is of interest to test the applicability of the $2D$ Stokes-Darcy model to the case of deep and narrow cells. Indeed, such a $2D$ model, once validated, could be a useful tool to address the issue of more complicated cases, such as gravity currents in the presence of viscosity contrasts, or in fractures with aperture heterogeneities. In this model, the flow in the rectangular cell (Fig. 1) is assumed to be parallel to the plates ($\vec{u}(x,y,z)=(u_{x}(x,y,z),0,u_{z}(x,y,z)$) with a Poiseuille parabolic profile across the gap (the key assumption). Using the Stokes equation with this $y$ dependency, the gap-averaged fluid velocity $\vec{U}(x,z)=\frac{1}{b}\int_{-b/2}^{b/2}\vec{u}(x,y,z)dy$, follows a Stokes- Darcy (SD) equation which reads here for the horizontal component of the velocity: $-\frac{12\eta}{b^{2}}U_{x\pm}(x,z)+\beta\;\eta\nabla^{2}U_{x\pm}(x,z)=\frac{\partial P_{\pm}}{\partial x}$ (29) The first term on the left hand side of eq. (29) and the pressure gradient correspond to the Darcy’s law (eq. (10)) with a permeability $\kappa=\frac{\displaystyle{b^{2}}}{\displaystyle{12}}$ for the Hele-Shaw cell as mentioned above (eq. (14)). The second term on the left hand side of eq. (29) is the Brinkman correction to the Darcy equation (see [Brinkman(1947)]), which involves an effective viscosity, $\beta\eta$. This effective viscosity may be taken equal to the one of the fluid ($\beta=1$) for the sake of simplicity (or to enable the matching with a $2D$ Stokes regime at $\Gamma\rightarrow 0$). However, [Zeng et al.(2003)Zeng, Yortsos & Salin] showed that in the Hele-Shaw cell regime (at large $\Gamma$), the effective viscosity was slightly higher, with $\beta=12/\pi^{2}\simeq 1.215$. Figure 4: Log-log plot of the normalized lock-exchange diffusion coefficient, versus the aspect ratio $\Gamma=H/b$, for rectangular cells. The data points, obtained using the $2D$ Stokes-Darcy model (eq. (4)) with Brinkman viscosity factors $\beta=1$ (open squares) and $\beta=12/\pi^{2}$ (open triangles) are compared to the full $3D$ results already shown in Fig. 3 and depicted here as a solid line. The inset gives the relative difference between the $2D$ Stokes- Darcy model ($\beta=1$, open squares and $\beta=12/\pi^{2}$, open triangles) and the $3D$ calculations. Note that these data were obtained by the difference between values of accuracy of the order of a few $10^{-3}$, which results in the small dispersion observed in the figure. At a given location $x$, integrating eq. (29) leads to the two velocity profiles matching the no slip boundary conditions at the bottom and the top boundaries ($u_{x+}(x,0)=u_{x-}(x,H)=0$) and the continuity of the velocity ($u_{x-}(x,h)=u_{x+}(x,h$)) and of the shear stress ($\beta\,\eta\,\partial u_{x-}(x,h)/\partial z=\beta\,\eta\,\partial u_{x+}(x,h)/\partial z$) at the interface. Using the no net flux condition, $\int_{0}^{h}u_{x+}dz+\int_{h}^{H}u_{x-}dz=0$, and eq. (2), we obtain the horizontal flux (eq. (4)): $q(x)=-D_{HS}\,f_{SD}(h/H)\,\frac{\partial\,h}{\partial\,x}$ (30) where $D_{HS}$ was already given in eq. (14) and the reduced flux function is equal to: $f_{SD}(\psi)=\frac{1}{4d(d-\tanh d)}\left\\{2+4\,d^{2}\,(1-\psi)\,\psi-d\,\frac{3\,\cosh(2\,d)+\cosh(2\,d\,(1-2\,\psi))}{\sinh(2\,d)}\right.\\\ \left.+4\,d\,\frac{(1-\psi)\,\cosh(2\,d\,(1-\psi))+\psi\,\cosh(2\,d\,\psi)}{\sinh(2\,d)}\,-2\,\frac{\cosh(d\,(1-2\,\psi))}{\cosh(d)}\right\\}$ (31) where $d=\sqrt{\frac{H^{2}}{4\kappa\beta}}=\sqrt{\frac{3}{\beta}}\,\Gamma$ (32) and $\kappa=b^{2}/12$. A comparison of the full $3D$ calculations for a rectangular channel of aspect ratio $\Gamma=H/b$ with this $2D$ approximation can be performed on the flux functions, $f_{\Gamma}(\psi)$ (eq. (24)) and $f_{SD}(\psi)$ (eq. (31)). These two flux functions are close to each other, within a few per cents. In order to address the comparison in the range of interest for the Hele-Shaw assumption, i.e. $\Gamma\gg 1$, let us analyze the limit $\Gamma\rightarrow\infty$ ($d\rightarrow\infty$), which gives $f_{SD,\Gamma\rightarrow\infty}\simeq\psi(1-\psi)-(\frac{3}{4}-\psi(1-\psi))\sqrt{\frac{\beta}{3}}\;\frac{b}{H}+O\left((\frac{b}{H})^{2}\right)$ (33) for the Stokes-Darcy flux and $f_{\Gamma,\Gamma\rightarrow\infty}\simeq\psi(1-\psi)-(\frac{3}{4}-\psi(1-\psi))\frac{186\,Zeta(5)}{\pi^{5}}\;\frac{b}{H}+O\left((\frac{b}{H})^{2}\right)$ (34) for the full $3D$ rectangular cell flux (with $Zeta(5)=\sum_{1}^{\infty}n^{-5}=1.03693$, the value of the Riemann-Zeta function). We note that the leading term of both series corresponds to the expected porous media Darcy limit (eq. (12)) with a permeability $\kappa=b^{2}/12$. However, the next order term ($O(b/H)$) is not the same, unless one chooses for the factor $\beta$, $\beta=3\;(\frac{186\,Zeta(5)}{\pi^{5}})^{2}\simeq 1.192$ (35) which is very close to the value $12/\pi^{2}\simeq 1.215$ found by [Zeng et al.(2003)Zeng, Yortsos & Salin] and to the value $6/5$ proposed by [Ruyer- Quil(2001)]. The lock-exchange diffusion coefficient has been computed, with the same procedure as above, by integrating eq. (9), using $f_{SD}(\psi)$, from $\psi=0.5$, and matching the asymptotics, $\zeta=-\zeta_{0}+8\,d^{2}\,\psi^{3}/(9\,\zeta_{0})$ in the vicinity of the boundary, $\psi=0$. The so-obtained lock-exchange diffusion coefficients, calculated for two different values of $\beta$ ($\beta=1$ and $\beta=12/\pi^{2}$) are compared to the $3D$ calculations in Fig. 4. The data for both values of $\beta$ are indeed very close to the $3D$ data over the whole range of aspect ratios, $\Gamma=1-100$. The inset of Fig. 4 gives the percentage of error for the two values of $\beta$. We note that these data were obtained by the difference between values of accuracy of the order of a few $10^{-3}$, which results in the small dispersion observed in the inset of Fig. 4. As expected, the results at large $\Gamma$ (in the Hele-Shaw regime) are closer to the $3D$ full problem for $\beta=12/\pi^{2}$ than for $\beta=1$. We point out that, whereas the Brinkman term does bring a significant correction, the exact value of the Brinkman viscosity factor $\beta$ is however not crucial: For instance, for $\Gamma=10$, we obtained a diffusion coefficient $3.5\%$ smaller than the $3D$ value for $\beta=1$ and $0.1\%$ larger for $\beta=12/\pi^{2}$, to be compared to the $30\%$ of error if the cell was assumed to be of infinite aspect ratio (Hele-shaw limit) as in [Huppert & Woods(1995)]. In conclusion of this comparison, we have shown that the $2D$ Stokes-Darcy model for lock-exchange in a rectangular cell captures quite accurately the effect of the finiteness of the cross-section aspect ratio. By using the correct $\beta$ value, the error in the model is smaller than $5\%$ for aspect ratios larger than $\Gamma>1$. ## 4 Experiments In this section, we will present experimental measurements of the diffusion coefficient in Hele-Shaw cells of different aspect ratios and we will compare them with our computed values. Figure 5: Side view (plane $z-x$) of the lock-exchange interface for rectangular cells of aspect ratios $\Gamma=H/b=10$, $6$, $2.5$, $1.5$, $1$, $2/3$, $0.4$, $1/3$, from top to bottom, respectively. The horizontal axis is scaled with $\sqrt{D_{HS}\,t}$, which allows to observe the decrease of $\zeta_{0}$ as $\Gamma$ decreases. The vertical dashed line corresponds to $\zeta=0$. We used borosilicate rectangular cells of height $H$ and thickness $b$ and typical length $30\,cm$ (Fig. 1). The rectangular cross-sections of the cells were (in $mm^{2}$): $2\times 6$, $2\times 12$, $2\times 20$, $3\times 3$, $3\times 9$, $3\times 30$, $4\times 6$, $4\times 10$, $6\times 6$. Each cell was used with one side or the other held vertically, leading to two aspect ratios per cell. With such values, we covered a wide range of aspect ratios, from $\Gamma=H/b=1/10$ to $10$. We used, as Newtonian miscible fluids, aqueous solutions of natrosol and calcium chlorite. The fluids had equal viscosities, which were fixed by the polymer concentration and measured with an accuracy of $1\%$. The fluid densities were adjusted by addition of salt and measured with an accuracy of $0.01\%$. Figure 6: Plot of the square of the spreading distance of the leading edge of the front versus time for five cells: ($\bullet$) $H=3\,mm$, $b=30\,mm$;($\square$) $H=6\,mm$, $b=2\,mm$; ($\ast$) $H=6\,mm$, $b=4\,mm$; ($\circ$) $H=b=6\,mm$; ($+$) $H=30\,mm$, $b=3\,mm$. The solid line is a linear fit to the data, the slope of which gives the diffusion coefficient plotted in Fig. 7 (top). The overall accuracy in $D_{HS}$ was typically $5\%$, when taking into account the above accuracies in viscosities and densities and the inherent temperature variations during the experiments. The viscosities and the densities of the fluids were chosen to satisfy two experimental requirements. The experiments must be fast enough in order to prevent any significant molecular mixing of the fluids and one should be able to put the two fluids in contact without mixing. The latter condition requires a rather large density contrast and large viscosities. With our cell sizes, a good compromise was obtained with a density contrast of about $1\%$ and typical viscosities in the range, $10-50\,mPa.s$, leading to a lock-exchange diffusion coefficient ranging from $10^{-3}cm^{2}/s$ to $1cm^{2}/s$. The typical Reynolds number, built with the gap of the cell of these experiments is smaller than $0.1$. For each experiment, the cell was first held with its axis $Ox$ vertical. The fluids were successively slowly injected, with the lighter fluid on top of the heavier. Then the cell was closed and put in the desired position, with its axis $Oz$ vertical in a few seconds. The development of the lock-exchange pseudo-interface was then recorded thanks to a video camera. Figure 7: Top: Log-log plot of the normalized measured lock-exchange diffusion coefficient, $D_{exp}/D_{HS}$ (open squares), versus the aspect ratio, $\Gamma=H/b$, of the cell. The solid line corresponds to the $3D$ results (same as in Fig. 4). Bottom : Superimposition of the measured pseudo-interface between the two fluids (grey fuzzy line) and of the theoretical profile (full dark line) corresponding to Fig. 2. The aspect ratio is $\Gamma=4$. Typical pictures (side view in the plane $z-x$) are given in Fig. 5 for cells of different aspect ratios. The horizontal axis is scaled with $\sqrt{D_{HS}\,t}$, so that one can see the decrease of $\zeta_{0}$ as $\Gamma$ decreases. With this representation using the self-similar variable $\zeta=x/\sqrt{D_{HS}\,t}$, the profiles are stationary. One may notice that the trailing edge is fuzzy. This can be attributed to the stick condition at the upper wall: The dark dense fluid does stay at the walls for a long time, in particular in the corners of the cross-section. The same phenomenon takes place at the bottom of the cell, but the presence of transparent light fluid has little effect on the turbidity of the heavy dark fluid, and is therefore not noticeable on the pictures. It is worth noting that the shape of the leading edge evolves from an edge at large aspect ratios $\Gamma$ to a more and more step-like shape as $\Gamma$ decreases. It should be noticed that for small aspect ratios, although it is rather difficult to take pictures, a top view of the cell reveals a mild spanwise dependency of the interface, but we do not observe the spanwise lobe-and-cleft instability reported by [Simpson(1972)]. For each experiment, the locations of the leading and trailing edges of the front were measured in time. Fig. 6 gives the variations of the square of the spreading distance versus time for five cells. It is worth noting that the dependency is almost linear: Therefore a linear fit provided the lock-exchange diffusion coefficient, with a typical accuracy of $20\%$. Fig. 7 (top) displays the so-obtained normalized lock-exchange diffusion coefficient as a function of $\Gamma$. One can see that the agreement with the $3D$ calculations over the two decades of our measurements is rather good. We note that for the large aspect ratio limit of the experiments (up to $\Gamma=10$), the Hele-Shaw cell limit is not reached, and would underestimate, by $30\%$, the lock-exchange diffusion coefficient. This result thus confirms that for such aspect ratios, one should either compute the full $3D$ Stokes equation or use the Stokes-Darcy model to obtain the correct behaviour. We also note that our calculation still holds for aspect ratios as small as $\Gamma=0.1$. This result is quite unexpected since for such aspect ratios, some spanwise dependency of the profile was observed, and the hypothesis of the interface surface, $h(x,y,z)$, invariant in the $y$ direction is certainly broken. The bottom of Fig. 7 displays the superimposition of the theoretical and the experimental interfaces between the fluids, for an aspect ratio $\Gamma=4$. The agreement between the two is rather good, thus validating our model. Such an agreement is rather surprising as our small slope assumption is violated at the edges of the gravity current. This agreement, already emphasized by [Huppert(1982)] and [Séon et al.(2007)Séon, Znaien, Salin, Hulin, Hinch & Perrin], is likely to be common to viscosity dominated gravity current without surface tension. ## 5 Conclusion The viscous lock-exchange diffusion coefficient reflects the competition between the buoyancy driving effect and the viscous damping, and depends on the geometry of the channel. We give the backbone to calculate this coefficient in different configurations: We recall its computation for a porous medium already found by [Huppert & Woods(1995)], and compute it for a $2D$ Stokes flow between two parallel horizontal boundaries separated by a vertical height, $H$. This result is in agreement with [Hinch(2007)] (unpublished) and in reasonable agreement with recent computations by [Taghavi et al.(2009)Taghavi, Seon, Martinez & Frigaard]. Using a quasi-parallel flow assumption, we have calculated the pseudo-interface profile between the two fluids and the diffusion coefficient of viscous lock-exchange gravity currents for a rectangular channel (horizontal thickness $b$, vertical height $H$) of any aspect ratio ($H/b$). This analysis provides a cross-over between the $2D$ Stokes flow between two parallel horizontal boundaries separated by a vertical height, $H$, and the Hele-Shaw cell limit (applying for $H/b>100$). Moreover, the shape of our profiles allows to account for the discrepancy observed at the nose of the gravity current in the experiments by [Huppert & Woods(1995)]. The agreement, obtained despite the failure of the lubrication assumption at the edges of the current, should deserve however further theoretical investigation. Our calculations of the diffusion coefficient and of the shape of the profile have also been convincingly compared to new experiments carried out in cells of various aspect ratios ($1/10-10$). We have also calculated the lock-exchange diffusion coefficient for the same rectangular cells, using the $2D$ Stokes-Darcy model. This model is shown to apply to aspect ratios $H/b>1$, provided that the appropriate Brinkman correction is used. Such a $2D$ model may be useful to describe gravity currents with a finite volume of release, with fluids of different viscosities, or in heterogeneous vertical fractures. ## 6 Acknowledgement This work was partly supported by CNES (No 793/CNES/00/8368), ESA (No AO-99-083), by Réseaux de Thématiques de Recherches Avancées ”Triangle de la physique”, by the Initial Training Network (ITN) ”Multiflow” and by French Research National Agency (ANR) through the ”Captage et Stockage du CO${}_{\mbox{2}}$” program (projet CO-LINER No ANR-08-PCO2-XXX). 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1011.6275	# Cancelation of dispersion and temporal modulation with non-entangled frequency-correlated photons Víctor Torres-Company Electrical and Computer Engineering Department, Purdue University, West Lafayette, 47906-IN, USA Alejandra Valencia ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain Martin Hendrych ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain Juan P. Torres juan.perez@icfo.es ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain Department of Signal Theory and Communications, Universitat Politecnica Catalunya, Campus Nord D3, 08034 Barcelona, Spain ###### Abstract The observation of the so-called dispersion cancelation and temporal phase modulation of paired photons is generally attributed to the presence of frequency entanglement between two frequency anticorrelated photons. In this paper, it is shown that by introducing the appropriate amount of chromatic dispersion or phase modulation between non-entangled photons, it is also possible to observe these effects. Indeed, it is found that the relevant characteristic for the observation of dispersion cancelation or the cancelation of temporal phase modulation is the presence of certain frequency correlations between the photons. ###### pacs: 42.25.Kb, 42.50.Dv, 42.50.Ar ## I Introduction The theory of quantum coherence glauber1963 ; glauber2006 describes the temporal and frequency characteristics of a stream of photons through a hierarchy of correlation functions. In particular, the second-order correlation function determines the probability to detect a photon at a certain location and instant time in coincidence with a companion photon at another location and time. This magnitude is often used to test the quantum or classical nature of a light source mandel_book . For instance, the normalized correlations of classical fields (i.e., fields with a positive $P$-representation) obey certain inequalities, whose violation constitutes an unequivocal signature of the non-classical properties of the light kimble . A particular type of correlation between photons is entanglement. Paired photons that show frequency entanglement can be generated by means of the process of spontaneous parametric downconversion (SPDC). Interestingly, in an SPDC process, classical-like features of the fluctuations of the beams coexist with the strong nonclassical photon-pair correlations loudon . Two important effects have been attributed to the existence of frequency entanglement and its demonstration have made use of the correlations existing between signal and idler photons generated in SPDC. The first effect is dispersion cancelation franson1992 ; gisin1998 ; kim2009 , which is observed in the temporal domain. Briefly, if the signal and idler photons generated in SPDC pumped by a continuous-wave (CW) beam are sent through two separate dispersive optical elements, such as single-mode fibers, with corresponding group-delay-dispersion (GDD) coefficients $\Phi_{1}$ and $\Phi_{2}$, the temporal width of the second-order correlation function increases as $(\Phi_{1}+\Phi_{2})^{2}$, in a similar way to the broadening of a pulse that propagates in an optical fiber due to chromatic dispersion valencia2002 . So, if the GDD parameters of both dispersive media are identical but of opposite sign, the broadening of the second-order correlation function can be suppressed. This is in contrast to the case where two identical broadband coherent light pulses propagate through two dispersive media such as single- mode optical fibers. In this case, the cross-correlation width broadens as $(\Phi_{1}^{2}+\Phi_{2}^{2})$ and therefore the cancelation of the dispersion effects is never possible franson1992 . The second effect is remote temporal modulation of entangled photons harris2008 ; harris2009 , which is observed in the frequency domain. For a CW- pumped SPDC process, the detection of a signal photon at frequency $\omega_{1}$ would only coincide with the detection of an idler photon at frequency $\omega_{2}$, given that $\omega_{1}+\omega_{2}=2\omega_{0}$, where $2\omega_{0}$ is the frequency of the CW pump beam. When synchronously driven temporal modulators are placed in the signal and idler paths, respectively, new frequency correlations appear. In a similar manner to dispersion cancelation, if the two identical modulators are driven in opposite phase, their global effect is to negate each other and the spectral correlations appear as those when there are no phase modulators present. It becomes of fundamental relevance to determine whether these effects are due to the classical or the quantum-like behavior of the SPDC source. Several authors have shown that similar dispersion cancelation effects can be obtained with non-entangled light victor2009 ; shapiro2010 . In particular, the work in victor2009 considers classical thermal light equally split in two dispersive arms and demonstrates that the broadening of the second-order correlation function increases as $(\Phi_{1}-\Phi_{2})^{2}$, so that it remains unaffected if $\Phi_{1}=\Phi_{2}$. Later, the work in shapiro2010 shows that by introducing a phase conjugator element in one arm of the intensity interferometer (producing a Gaussian-state light source), the broadening of the correlation function increases as $(\Phi_{1}+\Phi_{2})^{2}$, similar to the signal-idler photon pairs from SPDC, and thus the same dispersion compensation rules apply. In this work, we provide new insights into the effects of dispersion cancelation and temporal modulation applying the Heisenberg picture. This formalism allows us to include the case of multiphoton pair generation in a straightforward way, and thus to calculate all relevant coherence functions (signal-signal and signal-idler correlations). Two important findings are revealed: First, there is a background term in the second-order correlation function for all types of correlations, but the signal-to-background ratio for signal-signal correlations is lower than for the signal-idler correlations franson2009 ; franson2010 . Second and more importantly, dispersion cancelation and temporal modulation can also be observed with the correlation existing among photons from an individual beam of SPDC (i.e., either signal- signal or idler-idler), which do not show entanglement. This last point illustrates that it is the existence of certain frequency correlations between photons, rather than the entanglement, that is the key enabling factor that allows the observation of dispersion cancelation and remote temporal modulation. ## II Quantum description of the light generated in Spontaneous Parametric Downconversion The effects studied in this work, i.e., remote dispersion cancelation and cancelation of temporal modulation are observed in the corresponding second- order correlation functions. To calculate these magnitudes one needs to establish the dependence of the creation and annihilation operators with the physical parameters set by the SPDC process. In this section, we review the quantum theory of the SPDC process pumped by a CW pump and establish this dependence. Let us consider degenerate SPDC produced by a CW plane-wave pump with a central frequency $2\omega_{0}$ in a crystal of length $L$ and nonlinear coefficient $\chi^{(2)}$. Within the Heisenberg formalism, the propagation equations describing the evolution of the signal and idler creation $\tilde{a}_{s,i}^{{\dagger}}(\omega_{0}+\Omega)$ and annihilation $\tilde{a}_{s,i}(\omega_{0}+\Omega)$ operators can be written as loudon ; barak $\displaystyle\frac{\partial\tilde{a}_{s}\left(z,\Omega\right)}{\partial z}$ $\displaystyle=$ $\displaystyle\sigma\tilde{a}_{i}^{{\dagger}}\left(z,-\Omega\right)\exp\left[i\Delta(\Omega)z\right],$ (1) $\displaystyle\frac{\partial\tilde{a}_{i}\left(z,\Omega\right)}{\partial z}$ $\displaystyle=$ $\displaystyle\sigma\tilde{a}_{s}^{{\dagger}}\left(z,-\Omega\right)\exp\left[i\Delta(\Omega)z\right],$ (2) where $\Omega$ is the frequency deviation from the central frequency $\omega_{0}$, $\Delta\left(\Omega\right)=k_{p}^{0}-k_{s}\left(\Omega\right)-k_{i}\left(-\Omega\right)$ is the phase matching function, $\sigma$ is a constant parameter proportional to the number of pump photons traversing the nonlinear crystal, $k_{j}(\Omega)=(\omega_{0}+\Omega)n_{j}/c$ denotes the signal and idler wavenumbers, with $n_{j}$ being the refractive index at the corresponding central frequencies, $k_{p}^{0}=2\omega_{0}n_{p}/c$ the wavenumber of the pump beam and $c$ the speed of light in a vacuum. Solving Eqs. (1) and (2) and writing $\tilde{a}_{s}(z,\Omega)$ and $\tilde{a}_{i}(z,\Omega)$ in terms of the vacuum field operators at the input face of the nonlinear crystal, $b_{s}(\Omega)$ and $b_{i}(\Omega)$, we obtain navez ; brambilla $\displaystyle\tilde{a}_{s}(z,\Omega)$ $\displaystyle=$ $\displaystyle U(z,\Omega)b_{s}(\Omega)+V(z,\Omega)b_{i}^{\dagger}(-\Omega),$ (3) $\displaystyle\tilde{a}_{i}(z,\Omega)$ $\displaystyle=$ $\displaystyle U(z,\Omega)b_{i}(\Omega)+V(z,\Omega)b_{s}^{\dagger}(-\Omega)$ (4) with $\displaystyle U(z,\Omega)$ $\displaystyle=$ $\displaystyle\exp\left[\frac{i\Delta(\Omega)z}{2}\right]\left\\{\cosh\left[\Gamma(\Omega)z\right]\right.$ (5) $\displaystyle\left.-\frac{i\Delta(\Omega)}{2\Gamma(\Omega)}\,\sinh\left[\Gamma(\Omega)z\right]\right\\},$ $\displaystyle V(z,\Omega)$ $\displaystyle=$ $\displaystyle-\frac{i\sigma}{\Gamma(\Omega)}\exp\left[i\frac{\Delta(\Omega)z}{2}\right]$ (6) $\displaystyle\times\sinh\left[\Gamma(\Omega)z\right],$ where $\Gamma(\Omega)=(\|\sigma\|^{2}-\Delta^{2}(\Omega)/4)^{1/2}$ and $\sigma$ is the nonlinear coefficient of the medium. From these results, we can calculate the required correlations in the temporal and frequency domains. We will refer to the second-order coherence function between signal and idler photons as to _interbeam correlations_. Analogously, we will refer to the second-order coherence function between signal-signal, or idler-idler, streams of photons as to _intrabeam correlations_. We will make use of the commutation rules $[b_{j}(\Omega_{1}),\,b_{k}(\Omega_{2})]=\delta(\Omega_{1}-\Omega_{2})\delta_{jk}$, where $\delta_{jk}$ is the Kronecker’s delta and $(j,k)=(s,i)$. In particular, for the effect of dispersion cancelation, two quantities are of interest. For the interbeam second-order correlations, we have $G^{(2)}_{\textrm{inter}}(\tau)=\langle a_{s}^{\dagger}(t)a_{i}^{\dagger}(t+\tau)a_{i}(t+\tau)a_{s}(t)\rangle.$ (7) Here, the operators $a_{s,i}(t)$ form a Fourier transform pair with the creation operators $\tilde{a}_{s,i}(\Omega)$. The above magnitude is related to the probability of detecting a signal photon at time $t$ in coincidence with an idler photon at $t+\tau$. For the intrabeam correlations, the corresponding magnitude is $G^{(2)}_{\textrm{intra}}(\tau)=\langle a_{s}^{\dagger}(t)a_{s}^{\dagger}(t+\tau)a_{s}(t+\tau)a_{s}(t)\rangle,$ (8) where signal photons have been arbitrarily chosen. This function provides the probability of detecting a signal photon in coincidence with another signal photon at $t+\tau$. Recent experiments suggest blauensteiner that intrabeam correlations in SPDC can be measured when high-power CW beams are used to pump the nonlinear crystal. Then the intrabeam higher-order correlations show classical-like features similar to thermal light yurke . In such a case, the interbeam correlations must be also adequately accounted for to include the multiphoton pair effects. For the effects of cancelation of temporal modulation, the relevant second- order correlation functions are calculated in the spectral domain. In particular, for the interbeam photon correlations, one has $\tilde{G}^{(2)}_{\textrm{inter}}(\Omega_{1},\Omega_{2})=\langle\tilde{a}_{s}^{\dagger}(\Omega_{1})\tilde{a}_{i}^{\dagger}(\Omega_{2})\tilde{a}_{i}(\Omega_{2})\tilde{a}_{s}(\Omega_{1})\rangle,$ (9) which determines the probability of detecting a signal photon at frequency $\Omega_{1}$ in coincidence with an idler photon at frequency $\Omega_{2}$. Analogously, we consider the intrabeam second-order correlation function $\tilde{G}^{(2)}_{\textrm{intra}}(\Omega_{1},\Omega_{2})=\langle\tilde{a}_{s}^{\dagger}(\Omega_{1})\tilde{a}_{s}^{\dagger}(\Omega_{2})\tilde{a}_{s}(\Omega_{2})\tilde{a}_{s}(\Omega_{1})\rangle,$ (10) which establishes the probability of detecting in coincidence two signal photons, one at frequency $\Omega_{1}$ and the other at $\Omega_{2}$. ## III Interbeam configuration: Cancelation of dispersion and temporal modulation in the multiphoton regime In this section, we calculate the change in the interbeam correlations, $G^{(2)}_{\textrm{inter}}(\tau)$ and $G^{(2)}_{\textrm{inter}}(\Omega_{1},\Omega_{2})$, when the photons propagate through the optical systems corresponding to dispersion and temporal modulation cancelations, respectively. The goal of this study is to extend the previous theoretical results franson1992 ; harris2008 to the case in which multiphoton pairs are generated in the SPDC process. In the interbeam configuration, signal and idler photons are separated (see Fig. 1), following afterwards paths $1$ and $2$. This arrangement can be achieved by generating SPDC photons that propagate in different directions (noncollinear type I) or that show orthogonal polarizations (collinear type II) and are divided by a polarizing beam splitter. Figure 1: (Color online) General configurations for observing interbeam correlations in (a) the temporal domain and (b) the frequency domain. PBS: Polarizing beam splitter; SPCM: Single photon counting module; cc: coincidence counting; Mono: Monochromator. In (a) there are two single-mode optical fibers in paths $1$ and $2$ with transfer functions $H_{1}(\Omega)$ and $H_{2}(\Omega)$, respectively. In (b), there are two temporal phase modulators with transfer functions $m_{1}(t)$ and $m_{2}(t)$. Red and blue lines describe photons with orthogonal polarizations. ### III.1 Dispersion cancelation in the interbeam configuration Let us start by considering remote dispersion cancelation. As depicted in Fig. 1(a), before reaching the detectors, the signal photons traverse path $1$, with dispersive transfer function $H_{1}(\omega)$, and the idler photons traverse the dispersive medium $2$, described by $H_{2}(\omega)$. In this case, the annihilation operators for the signal and idler photons at $D_{1}$ and $D_{2}$ are given by $a^{\prime}_{s}(t)=\int\textrm{d}\Omega H_{1}(\omega_{0}+\Omega)\tilde{a}_{s}(\Omega)\exp(-i\Omega t)$ and $a^{\prime}_{i}(t)=\int\textrm{d}\Omega H_{2}(\omega_{0}+\Omega)\tilde{a}_{i}(\Omega)\exp(-i\Omega t)$. Substituting these expressions into Eq. (7) and taking into account Eqs. (3) and the commutation rules, it is easy to show that $\displaystyle G_{\textrm{inter}}^{(2)}(\tau)=N^{2}+\frac{1}{(2\pi)^{2}}$ $\displaystyle\left\|\int\textrm{d}\Omega\exp(i\Omega\tau)R(\Omega)H_{1}(\omega_{0}+\Omega)H_{2}(\omega_{0}-\Omega)\right\|^{2},$ (11) where $R(\Omega)=U(\Omega)V(-\Omega)$. From Eq. (III.1), we observe that $G_{\textrm{inter}}^{(2)}(\tau)$ consists of two terms: The first one is just a constant background provided by the flux of photon pairs, $N=N_{s}(t)=N_{i}(t)=\langle a_{s}^{\dagger}(t)a_{s}(t)\rangle$, and the second contains the temporal structure of the second-order correlation function that is indeed affected by the spectral transfer functions of the elements placed in the photon arms. In the absence of any dispersive medium in the propagation paths of both streams of photons, one would get $G^{(2)}_{\textrm{inter}}(\tau)=N^{2}+\left\|1/(2\pi)\,\int\textrm{d}\Omega\exp(i\Omega\tau)R(\Omega)\right\|^{2}.$ (12) Thus, the dispersive media only affect the temporal structure of the second- order correlation function, not the background term, which is always present. For the particular case in which the media can be assumed to be first-order dispersive, such as single-mode fibers, we have $H_{1}(\Omega)=\exp(i\Phi_{1}\Omega^{2}/2)$ and $H_{2}(\omega)=\exp(i\Phi_{2}\Omega^{2}/2)$, with the GDD parameters $\Phi_{k}=\beta_{2k}L_{k}$, $k=(1,2)$. $\beta_{2k}$ is the group velocity dispersion (GVD) coefficient of fiber $k$ and $L_{k}$ its length. Therefore, the dispersion effect can be suppressed if $\Phi_{1}=-\Phi_{2}$, i.e., if the GDD parameters in the two paths have opposite signs franson1992 . The type of frequency correlation between signal and idler photons, i.e. frequency anticorrelation $\Omega_{s}+\Omega_{i}=0$, also implies that odd-order dispersion terms cannot be canceled, but on the contrary, their effects are added. The effect of increasing the pair generation rate is to degrade the signal-to- background ratio. With the aid of Eqs. (5) and (6), it can be shown that this ratio scales as $1/(NL)$ and tends to infinity as N decreases harris2005 . ### III.2 Temporal modulation cancelation in the interbeam configuration The configuration to study remote temporal modulation cancelation is depicted in Fig. 1(b). The signal photons travel through the modulator $m_{1}(t)$ and the idler photons through modulator $m_{2}(t)$. In order to measure frequency correlations, signal and idler photons pass through ideal scanning monochromators before reaching the photodetectors harris2008 ; harris2009 . For simplicity, we consider the photons to be phase modulated with sinusoidal phase modulators with temporal complex transfer function $m_{1,2}(t)=\exp[i\Delta\theta_{1,2}\sin(\Omega_{m}t)]$, where $\Omega_{m}$ is the modulation frequency (in the microwave regime) and $\Delta\theta_{1,2}$ are the modulation indexes of the corresponding modulators. In order to calculate the interbeam second-order correlation function, $\tilde{G}^{(2)}_{\textrm{inter}}(\Omega_{1},\Omega_{2})$, we need to calculate the evolution of the operators $\tilde{a}_{s,i}(\Omega)$ after phase modulation. This is done by making the convolution of the operators with the frequency response of the corresponding modulation function $m_{1,2}(t)$ harris2008 . In the frequency domain, the action of a sinusoidal phase modulator can be written as $M(\Omega)=\sum_{n}J_{n}(\Delta\theta)\delta(\Omega-n\Omega_{m})$, where $J_{n}$ are the Bessel functions of the first kind. Thus one obtains that $\tilde{G}_{\textrm{inter}}^{(2)}(\Omega_{1},\Omega_{2})=N_{s}(\Omega_{1})N_{i}(\Omega_{2})+\left\|T(\Omega_{1},\Omega_{2})\right\|^{2},$ (13) with $\displaystyle T\left(\Omega_{1},\Omega_{2}\right)=\frac{1}{2\pi}\int\textrm{d}\Omega R(\Omega)$ $\displaystyle\times M_{1}(\Omega_{1}+\Omega)M_{2}(\Omega_{2}-\Omega).$ (14) In an analogous way to the dispersive case, there is a constant background term proportional to the flux of photons at the considered frequencies and another intricate non-factorizable term that gets affected by the spectral transfer functions of the modulators. State-of-the-art electro-optic phase modulators can provide a maximum optical bandwidth of tens of GHz, yet much smaller than the bandwidth of most common SPDC sources with typical values around $10$-$20$ THz. Under these conditions, we can safely write $\displaystyle T(\Omega_{1},\Omega_{2})=R(\Omega_{-}/2)$ $\displaystyle\times\sum_{n=-\infty}^{\infty}J_{n}(\Delta\theta_{1}+\Delta\theta_{2})\delta(\Omega_{+}-n\Omega_{m}),$ (15) where $\Omega_{-}=\Omega_{1}-\Omega_{2}$ and $\Omega_{+}=\Omega_{1}+\Omega_{2}$. Before proceeding further, let us write the corresponding expression in the absence of temporal modulation: $\displaystyle\tilde{G}_{\textrm{inter}}^{(2)}(\Omega_{1},\Omega_{2})=N_{s}(\Omega_{1})N_{i}(\Omega_{2})+$ $\displaystyle\left\|R(\Omega_{1})\right\|^{2}\delta(\Omega_{1}+\Omega_{2}).$ (16) This equation indicates that, besides a background term, there is a strong correlation between frequencies satisfying $\Omega_{1}=-\Omega_{2}$. From Eq. (III.2), we conclude that it is possible to recover this result if the modulation depths of the two modulators fulfill the condition $\Delta\theta_{1}=-\Delta\theta_{2}$. In this case, the only non-zero Bessel term is $n=0$, so that $T(\Omega_{1},\Omega_{2})=R(\Omega_{1})\delta(\Omega_{1}+\Omega_{2})$, i.e., the effect of the modulators is canceled harris2008 . Again, the effect of having multiphoton pairs is only to degrade the signal-to-background ratio, which decreases with the flux-rate of the generated pairs. We remark that both dispersion and temporal modulation cancelations effects are due to the presence of frequency anti-correlation between signal and idler photons, which can be expressed by the basic relationship $\langle\tilde{a}_{s}(\Omega_{1})\tilde{a}_{i}(\Omega_{2})\rangle=R(\Omega_{1})\delta(\Omega_{1}+\Omega_{2})$. ## IV Intrabeam configuration: Remote cancelation of dispersion and temporal modulation with non-entangled photon pairs Let us now consider the intrabeam configuration depicted in Fig. 2. Without loss of generality, one of the beams produced by the source is discarded and the attention is centered on one of the beams only, for example, the signal. Figure 2: (Color online) General configurations for observing intrabeam correlations in (a) the temporal domain and (b) the frequency domain. PBS: Polarizing beam splitter; BS: Beam Splitter; SPCM: Single photon counting module; cc: coincidence counting; Mono: Monochromator. In (a) there are two single-mode fibers in paths $1$ and $2$ with transfer functions $H_{1}(\Omega)$ and $H_{2}(\Omega)$, respectively. In (b), there are two temporal phase modulators with transfer functions $m_{1}(t)$ and $m_{2}(t)$. Red and blue lines describe photons with orthogonal polarizations. Photons with a given polarization are discarded. ### IV.1 Remote dispersion compensation in the intrabeam configuration Let us consider the situation depicted in Fig. 2(a). After traversing the dispersive media located in paths $1$ and $2$, the probability to detect a photon in path $1$ at time $t$ in coincidence with a photon in path $2$ at time $t+\tau$ is given by the intrabeam second-order correlation function, $G^{(2)}_{\textrm{intra}}(\tau)=\langle a^{\dagger}_{s1}(t)a^{\dagger}_{s2}(t+\tau)a_{s2}(t+\tau)a_{s1}(t)\rangle.$ (17) By analogy to the case of the previous section, the corresponding operators are calculated as $a_{s(1,2)}(t)=\int\textrm{d}\Omega\tilde{a}_{s}(\Omega)H_{1(2)}(\Omega)\exp(-i\Omega t)$, where $H_{1(2)}(\Omega)$ denotes the complex transfer function placed in the path of photon 1(2). Substituting these expressions and taking into account Eqs. (5) and (6), it is easy to show that $\displaystyle G_{\textrm{intra}}^{(2)}(\tau)=N^{2}+\frac{1}{(2\pi)^{2}}\left\|\int\textrm{d}\Omega\,S(\Omega)\right.$ $\displaystyle\left.\times H_{1}^{}(\omega_{0}+\Omega)H_{2}(\omega_{0}+\Omega)\exp\left(i\Omega\tau\right)\right\|^{2},$ (18) where $S(\Omega)=\|V(\Omega)\|^{2}$. We observe that the joint probability detection contains the same background term as in the interbeam case. Even more, the functional dependence on $\tau$ depends on the multiplication of the transfer functions of the dispersive media in such a way, that their phase difference may alter completely the output. However, now the previous role of the spectrum $R(\Omega)=U(\Omega)V(-\Omega)$ is replaced by the signal’s photon spectrum $S(\Omega)$. For the sake of comparison, we derive the expression of $G^{(2)}_{\textrm{intra}}(\tau)$ in the absence of dispersive media: $\displaystyle G_{\textrm{intra}}^{(2)}(\tau)=N^{2}+\left\|\Gamma(\tau)\right\|^{2},$ (19) where $\Gamma(\tau)=1/(2\pi)\,\int\textrm{d}\Omega\exp(i\Omega\tau)S(\Omega)$. Notice that a similar suppression of the effects of dispersion can be obtained if both dispersive media are identical, i.e., $H_{1}\equiv H_{2}$. In contrast to the interbeam case, the suppression of the dispersive effects is not limited to even-order dispersion terms, but it affects all the terms victor2009 . On the other hand, the signal-to-background ratio is maximally bounded to $2$, thus challenging the observation of the effect in an experiment. The factor of $2$ is a manifestation of the thermal-like character of the intrabeam correlations. Note that a reduced signal-to-background ratio could also be attained in the interbeam correlations provided that sufficient multiple pairs were generated in the SPDC process. ### IV.2 Remote temporal modulation compensation in the intrabeam configuration Finally, we revisit the temporal modulation cancelation scheme with the photons from only one of the downconverted beams (signal, for example), as depicted in Fig. 2(b). After modulation, the probability to detect a photon in path $1$ with frequency $\omega_{0}+\Omega_{1}$ in coincidence with a photon in path $2$ with frequency $\omega_{0}+\Omega_{2}$ is provided by the magnitude $\tilde{G}^{(2)}_{\textrm{intra}}(\Omega_{1},\Omega_{2})=\langle\tilde{a}_{s1}^{\dagger}(\Omega_{1})\tilde{a}_{s2}^{\dagger}(\Omega_{2})\tilde{a}_{s2}(\Omega_{2})\tilde{a}_{s1}(\Omega_{1})\rangle.$ (20) To calculate the evolution of the operators, one just has to proceed as in section III(B), i.e., to calculate the convolution of the signal operator with the modulator transfer function in the spectral domain. Then $\displaystyle\tilde{G}_{\textrm{intra}}^{(2)}(\Omega_{1},\Omega_{2})$ $\displaystyle=$ $\displaystyle N_{s}(\Omega_{1})N_{s}(\Omega_{2})$ (21) $\displaystyle+\left\|T^{\prime}(\Omega_{1},\Omega_{2})\right\|^{2},$ where $\displaystyle T^{\prime}(\Omega_{1},\Omega_{2})=\frac{1}{2\pi}\int\textrm{d}\Omega^{\prime}S(\Omega^{\prime})$ $\displaystyle\times M_{1}^{}(\Omega_{1}-\Omega^{\prime})M_{2}(\Omega_{2}-\Omega^{\prime}).$ (22) As before, we obtain a background term proportional to the number of photons in modes $\omega_{0}+\Omega_{1}$ and $\omega_{0}+\Omega_{2}$. The second term is a non-separable function in frequency which is affected by the transfer functions of the modulators. Analogously to the derivation of Eq. (III.2), we can consider low-bandwidth modulators, so that this function reduces to $\displaystyle T^{\prime}(\Omega_{1},\Omega_{2})=S(\Omega_{+}/2)$ $\displaystyle\times\sum_{n=-\infty}^{\infty}J_{n}(\Delta\theta_{1}-\Delta\theta_{2})\delta(\Omega_{-}-n\Omega_{m}).$ (23) To close the loop, we calculate the second-order correlation function that would be achieved in the absence of modulators, $\displaystyle\tilde{G}_{\textrm{intra}}^{(2)}(\Omega_{1},\Omega_{2})=N_{s}(\Omega_{1})N_{s}(\Omega_{2})+$ $\displaystyle\left\|S(\Omega_{1})\right\|^{2}\delta(\Omega_{1}-\Omega_{2}).$ (24) Similarly to the interbeam case, there is a strong correlation between photons in paths $1$ and $2$ with the same frequency. Giving a further step, from Eq. (IV.2), if $\Delta\theta_{1}=\Delta\theta_{2}$, i.e., photons in paths $1$ and $2$ are equally phase modulated, the effects of the temporal phase modulation are suppressed. Now, the only non-zero term is $n=0$, and therefore only the detection of photons with frequencies $\Omega_{1}=\Omega_{2}$ is enhanced. The origin of these suppression (cancelation) effects when the photons that follow paths $1$ and $2$ are equally phase-modulated, or traverse equal dispersive media, is the existence of frequency correlation between the signal photons, i.e., $\langle\tilde{a}_{s}^{\dagger}(\Omega_{1})\tilde{a}_{s}(\Omega_{2})\rangle=S(\Omega_{2})\delta(\Omega_{2}-\Omega_{1})$. Eqs. (IV.1) and (IV.2), which allow for the remote dispersion and temporal modulation cancelation, make an explicit use of the existence of these characteristic frequency correlations. This strong frequency correlation could be revealed measuring, before any temporal phase modulation takes place, the number of photons in path $1$ with frequency $\omega_{0}+\Omega_{1}$ in coincidence with the signal photons with frequency $\omega_{0}+\Omega_{2}$ that traverse path $2$, which is given by Eq. (IV.2). ## V Summary and conclusions We have considered the effects of cancelation of dispersion and temporal modulation in the regime of multiphoton pair generation in the SPDC process. The two schemes are studied with two kinds of photon correlations, those arising from the downconverted signal and idler photons (interbeam correlations), which show entanglement, and those from the individual photons in a single beam (intrabeam correlations), which show strong correlations but not entanglement. In the intrabeam regime, it is possible to achieve the suppression of the effects of either dispersion or temporal modulation at all orders. An important point to remark is that the observation of the remote cancelation of the dispersion and modulation effects in the interbeam configuration happens even in the high-flux regime ($\sigma L\gg 1$). In this case, an inequality of normalized second-order correlation functions howell2006 ; mancini2002 which is fulfilled by classical-like fields, is no longer violated. This highlights the role of the frequency anticorrelation and correlation effects as the reason for the observation of any dispersion or modulation cancelation effects. The observation in a particular setting of dispersion and modulation cancelation effects depends on the type of frequency correlations present. In the interbeam case, where the photons show frequency anticorrelation, the observation of dispersion and modulation cancelation effects requires that both photons suffer only even-order dispersion of the opposite sign, or identical phase modulation but with opposite phases. On the other hand, if the photons are frequency correlated (intrabeam correlations with $\Omega_{1}-\Omega_{2}=0$), the observation of dispersion and modulation cancelation effects requires that both photons suffer equal dispersion, for all dispersion terms, or identical phase modulation, with the same phases. The effects described here bear important similarities with the description of two-photon imaging experiments with two different types of two-photon sources: paired photons entangled in the spatial degree of freedom and bunched pairs of photons coming from a thermal light source scarcelli2004 . In pittman1995 , a two-photon optical imaging experiment was performed based on the spatial correlations of the signal and idler photon pairs produced in SPDC. The possibility to use thermal (or pseudothermal) radiation for two-photon imaging experiments has also been demonstrated valencia2005 . In both cases, the important point that enables the observation of similar effects with dissimilar sources is the presence of spatial correlations between the paired photons. Again, the spatial correlations between signal and idler photons show different characteristics than the spatial correlations of signal pairs. But the observation of two-photon imaging, as well as the capacity to cancel diffraction effects when measuring two-photon coincidences, are due to some of the common characteristics that both types of sources share: the presence of certain correlations between paired photons. ## Acknowledgements This work was supported by the Government of Spain (Consolider Ingenio CSD2006-00019, FIS2010-14831), and was supported in part by FONCICYT project 94142\. 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Tombesi, Phys. Rev. Lett. 88, 120401 (2002). * (24) G. Scarcelli, A. Valencia, and Y. Shih, Phys. Rev. A 70 051802 (2004). * (25) T. B. Pittman, Y. Shih, D. V. Strekalov, and A. V. Sergienko, Phys. Rev. A, 52, 3429(R) (1995). * (26) A. Valencia, G. Scarcelli, M. D Angelo, and Y. Shih, Phys. Rev. Lett. 94 063601 (2005).	arxiv-papers	2010-11-29T16:06:31	2024-09-04T02:49:15.287004	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Victor Torres-Company, Alejandra Valencia, Martin Hendrych and Juan P.\n Torres", "submitter": "Juan Torres P.", "url": "https://arxiv.org/abs/1011.6275" }
1011.6283	Like a bird on the wire, Like a drunk in a midnight choir I have tried in my way to be free111Leonard Cohen: Bird on the wire.. To Stephen Smale, at his $80$ \- th birthday # Turning Washington’s heuristics in favor of Vandiver’s conjecture Preda Mihăilescu Mathematisches Institut der Universität Göttingen preda@uni-math.gwdg.de (Date: Version 2.0 ) ###### Abstract. A famous conjecture bearing the name of Vandiver states that $h_{p}^{+}=1$ in the $p$ \- cyclotomic extension of $\mathbb{Q}$. Heuristics arguments of Washington, which have been briefly exposed in [La], p. 261 and [Wa], p. 158 suggest that the Vandiver conjecture should be false, if certain conditions of statistical independence are fulfilled. In this note we assume that Greenberg’s conjecture is true for the $p{\rm-th}$ cyclotomic extensions and prove an elementary consequence of the assumption that Vandiver’s conjecture fails for a certain value of $p$: the result indicates that there are deep correlations between this fact and the defect $\lambda^{-}i(p)$, where $i(p)$ is like usual the irregularity index of $p$, i.e. the number of Bernoulli numbers $B_{2k}\equiv 0\bmod p,1<k<(p-1)/2$. As a consequence, if one combines the various assumptions in Washington’s heuristics, these turn, on base of the present result, into an argument in favor of the Vandiver’s conjecture. ## 1\. Introduction Let $p$ be an odd prime and $\mathbb{K}=\mathbb{Q}[\zeta]$ be the $p{\rm-th}$ cyclotomic field and $G=\mbox{ Gal }(\mathbb{K}/\mathbb{Q})$. If $X$ is a finite abelian group, we denote by $X_{p}$ its $p$ \- Sylow group; let $A=\mathcal{C}(\mathbb{K})_{p}$, the $p$ \- Sylow subgroup of the class group $\mathcal{C}(\mathbb{K})$ and $h^{+},h^{-}$ the sizes of $A^{+}$ respectively $A^{-}$. In a letter to Kronecker from 1857, Kummer refers to $p\nmid h^{+}$ as a noch zu beweisender Satz, a theorem yet to prove (see also [Wa], p. 158). The fact was stated later as a conjecture by Vandiver. In [La], p. 261 Washington gives an heuristic argument which suggests that there might be an asymptotic amount of $O(\log\log(N))$ of primes $p\leq N$ for which $\lambda(A_{\infty}^{-})=i(p)+1$, where $i(p)$ is the irregularity index of $p$, i.e. the number of Bernoulli numbers $B_{2k},1<k<(p-1)/2$ that vanish modulo $p$. In [Wa], p.158, Washington starts with a naive argument, on base of which the cyclotomic unit $\eta_{2k}:=e_{2k}(1-\zeta)^{\sigma-1}$ (see below for the definition of the idempotents $e_{j}\in\mathbb{F}_{p}[G]$) may be a $p{\rm-th}$ power with probability $1/p$: this yields a probability of more than one half, for the failure of Vandiver’s conjecture, so the argument is obviously too crude. Washington considers then that the probabilities that a Bernoulli number vanishes modulo $p$ and the one that the corresponding cyclotomic unit $\eta_{2k}$ is a $p$-power are independent: this heuristic leads to a frequence of $O(\log\log(n))$ primes $p<n$ for which the conjecture fails. As a consequence, various specialists in the field expect that the conjecture should not always hold. Our result in this note, shows that if Vandiver’s conjecture fails, then one has the additional condition $\lambda^{-}>i(p)$. If one considers this condition also as statistically independent (!) from the two conditions in Washington’s heuristiscs, then the same argument suggest that there may be $O(1)$ primes $p<n$ for which Vandiver’s conjecture fails, thus possibly none. Therefore the elementary result in this note may be understood as one that turns Washington’s heuristics into an argument in favor of Vandiver’s conjecture. In this paper we prove an elementary fact, which implies that the failure of Vandiver’s conjecture has an impact on the value of $\lambda$, and thus two events which were supposed to be uncorrelated in the heuristic approach: namely $h_{p}^{+}\neq 0$ and $\lambda^{-}>i(p)$ are not independent. No direct consequence can be drawn as to the truth of the Kummer - Vandiver conjecture; however we have an explicite theorem which indicates an unknown dependence, and also a method of investigation which may be extended for the purpose of investigating more possible consequences of the assumption that the Kummer- Vandiver conjecture is false. The result of this paper is the following: ###### Theorem 1. Let $p$ be an odd prime with irregularity index $i(p)=1$. If $h_{p}^{+}>1$, then $\lambda^{-}\geq 2$. Since $\lambda^{-}>1$ is an implication of $h_{p}^{+}>1$, the two events cannot be considered as independent events, each one with probability $1/p$. But this implication can also suggest that the probability that $\eta_{2k}$ is a $p{\rm-th}$ power has rather the probability $1/p^{2}$ than $1/p$, since it implies the vanishing of a higher order Bernoulli number. Either way, we consider that our elementary result should suggest that it is worthwhile to consider that Vandiver’s conjecture might be true, and pursue the investigation for the reasons why this may be the case. For this purpose, the central idea of our proof can be extended, with additional detail, to the general case, and this shall be done in a subsequent paper. Note that we restrict our analysis, for simplicity, to the case of irregularity index $1$. However, this is the critical case in Washington’s heuristics, and if the assumption of “statistical independence” is close to reality222Washington mentions explicitly that this is the crucial and critical in the various heuristics of this kind., then the probability of failure of the conjecture for higher values of $i(p)$ can only be smaller, so the argument stays valid. ## 2\. Proof of the Theorem We let $\mathbb{K}=\mathbb{Q}[\zeta]$ be the $p{\rm-th}$ cyclotomic extension and $\mathbb{K}_{n}=\mathbb{K}[\zeta^{1/p^{n}}],n\geq 1$, the $p^{n}{\rm-th}$ extension. The galois groups are $\displaystyle G=\mbox{ Gal }(\mathbb{K}/\mathbb{Q})$ $\displaystyle=$ $\displaystyle\\{\sigma_{a}\ :\ a=1,2,\ldots p-1,\ \zeta\mapsto\zeta^{a}\\}\cong(\mathbb{Z}/p\cdot\mathbb{Z})^{},$ $\displaystyle G_{n}=\mbox{ Gal }(\mathbb{K}_{n}/\mathbb{Q})$ $\displaystyle=$ $\displaystyle G\times\langle\tau\rangle,\quad\tau(\zeta_{p^{n}})=\zeta_{p^{n}}^{1+p},$ so $\tau$ generates $\mbox{ Gal }(\mathbb{K}_{n}/\mathbb{K})$, in particular. If $g\in\mathbb{F}_{p}$ is a generator of $(\mathbb{Z}/p\cdot\mathbb{Z})^{}$, then $\sigma=\sigma_{g}$ generates $G$ multiplicatively. We write $\jmath\in G$ for complex multiplication. For $\sigma\in G$ and $R\in\\{\mathbb{F}_{p},\mathbb{Z}_{p},\mathbb{Z}/(p^{m}\cdot\mathbb{Z})\\}$ we let $\varpi(\sigma)\in R$ be the value of the Theichmüller character on $\sigma$; for $R=\mathbb{F}_{p}$ we may also write $\hat{\sigma}$ for this values. The orthogonal idempotents $e_{k}\in R[G]$ are $e_{k}=\frac{1}{p-1}\sum_{a=1}^{p-1}\varpi^{k}(\sigma_{a})\cdot\sigma_{a}^{-1}.$ If $X$ is a finite abelian $p$ \- group on which $G$ acts, then $e_{k}(\mathbb{Z}_{p})$ acts via its approximants to the $p^{m}{\rm-th}$ order; we shall not introduce additional notations for these approximants. A fortiori, complex conjugation acts on $X$ splitting it in the canonical plus and minus parts: $X=X^{+}\oplus X^{-}$, with $X^{+}=X^{1+\jmath},X^{-}=X^{1-\jmath}$. The units of $\mathbb{K}$ and $\mathbb{K}_{n}$ are denoted by $E,E_{n}$ and the cyclotomic units by $C,C_{n}$. The Iwasawa invariants $\lambda,\lambda^{-}$ are related to the cyclotomic $\mathbb{Z}_{p}$-extension $\mathbb{K}_{\infty}=\cup_{n}\mathbb{K}_{n}$ and $A_{n}=(\mathcal{C}(\mathbb{K}_{n}))_{p}$ are the $p$-parts of the ideal class groups of $\mathbb{K}_{n}$. They form a projective sequence with respect to the relative norms $N_{m,n}=\mbox{\bf N}_{\mathbb{K}_{m}/\mathbb{K}_{n}},m>n\geq 1$ and $\mbox{\bf A}=\varprojlim_{n}A_{n}$. We also write $A$ for $A_{1}$. We shall write for simplicity $A(\mathbb{L})=(\mathcal{C}(\mathbb{L}))_{p}$ for the $p$-part of the class group of an arbitrary number field $\mathbb{L}$, so $A=A(\mathbb{K})$, etc. We fix now an odd prime $p$ such that * 1\. Greenberg’s conjecture holds for $p$, so $A^{+}$ is finite and $\lambda^{+}=0$. * 2\. Vandiver’s conjecture fails for $p$. * 3\. There is a unique irregular index $2k$ such that $A_{p-2k}=\varepsilon_{p-2k}A\neq\\{1\\}$. Additionally $A_{2k}\neq\\{1\\}$, as a consequence of 2. Under these premises, we show that $\mathbb{Z}_{p}\hbox{-rk}(\varepsilon_{p-2k}\mbox{\bf A}>1$, which is the statement of the theorem. We prove the statement by contraposition, so we assume that $\mathbb{Z}_{p}\hbox{-rk}(\varepsilon_{p-2k}\mbox{\bf A})=1$. Since there is a unique irregular index, the minimal polynomial of A is linear. Let $\mathbb{H}_{n}/\mathbb{K}_{n}$ be the maximal $p$-abelian unramified extensions. They split in plus and minus parts according to $A_{n}=A_{n}^{+}\oplus A_{n}^{-}$ and our assumption implies that $\mathbb{H}_{n}^{+}/\mathbb{K}_{n}$ are cyclic extensions of degree $d_{n}:=[\mathbb{H}_{n}^{+}:\mathbb{K}_{n}]=\|A_{n}^{+}\|.$ We may also consider $\mathbb{H}_{n}^{+}$ as the compositum of $\mathbb{K}_{n}$ with the full $p$-part of the Hilbert class field of $\mathbb{K}_{n}^{+}\subset\mathbb{K}_{n}$, the maximal reals subextension of $\mathbb{K}_{n}$: thus $\mathbb{H}_{n}^{+}$ is a canonical subfield, corresponding by the Artin map to $A_{n}^{+}$. It follows that $\mathbb{H}_{n}^{+}/\mathbb{K}_{n}^{+}$ is an abelian extension, and thus $\mathbb{H}_{n}^{+}$ is a CM field (see also [Wa], Lemma 9.2 for a detailed proof). There is a canonic construction of radicals from $A_{n}^{-}$, such that $\mathbb{H}_{n}\cdot\mathbb{K}_{m}\subset\mathbb{K}_{m}[(A_{m}^{-})^{1/p^{m}}]$ for sufficiently large $m$. As a consequence of Greenberg’s conjecture holding for $\mathbb{K}$, there is an $n_{0}\geq 1$ such that $\|A_{n}^{+}\|=\|A_{n_{0}}^{+}\|$ for all $n\geq n_{0}$ and for such $n$, let $a_{n}\in A_{n}^{-}$ generated this cyclic group. Let $\mathfrak{Q}\in a_{n}$ and $\alpha_{0}\in\mathbb{K}_{n}^{\times}$ with $(\alpha_{0})=\mathfrak{Q}^{{\rm ord}\ (a_{n})}$; there is an $\alpha=\eta\cdot\alpha_{0}^{1-\jmath},\eta\in\mu_{p^{n}}$ which is well defined up to roots of unity, such that $\mathbb{H}_{n}^{+}\subset\mathbb{K}_{n}[\alpha^{1/p^{n}}]$. The radical $B_{n}$ of $\mathbb{H}_{n}^{+}$ is then the multiplicative group generated by $\alpha$ and $(\mathbb{K}_{n}^{\times})^{d_{n}}$. Since $\mathbb{H}_{n}^{+}/K_{n}$ is cyclic, a folklore result, which we prove for completeness in Lemma 3 of the Appendix below, implies that (1) $\displaystyle A(\mathbb{H}_{n}^{+})=(\mathcal{C}(\mathbb{H}_{n}^{+}))_{p}=\\{1\\}.$ A classical result, proved by Iwasawa [Iw] in a general cohomological language, states that for an arbitrary galois extension $\mathbb{L}/\mathbb{F}$ of finite number fields, there is a canonical isomorphism (2) $\displaystyle H^{1}(\mbox{ Gal }(\mathbb{L}/\mathbb{F}),E(\mathbb{L}))\cong\mathcal{A}(\mathbb{L}),$ where $\mathcal{A}(\mathbb{F})$ are Hilbert’s ambig ideals, i.e. the ideals of $\mathbb{L}$ which are invariant under $\mbox{ Gal }(\mathbb{L}/\mathbb{F})$, factored by the principal ideals of $\mathbb{F}$. These can be either totally ramified ideals or ideals from $\mathbb{F}$ that capitulate completely (become principal) in $\mathbb{L}$. We shall in the sequel often consider the homology groups $H^{0},H^{1}$ for the unit groups. We can then write, for simplicity $H^{i}(\mathbb{L}/\mathbb{F}):=H^{i}(\mbox{ Gal }(\mathbb{L}/\mathbb{F}),E(\mathbb{L}))\quad\hbox{for}\quad i=0,1.$ The isomorphism above restricts also to one of $p$-parts of the respective groups; furthermore, complex conjugation also induces canonical isomorphisms of the plus and minus parts of $H^{i}$. The extensions $\mathbb{H}_{n}^{+}/\mathbb{K}_{n}$ being cyclic of degree $d_{n}$, the Herbrand quotient is $d_{n}$ and thus $H^{1}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})=d_{n}\cdot H^{0}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n}).$ We claim that $\left(H^{0}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})\right)^{+}=\\{1\\}$. Indeed, the ambig ideals in an unramified extension are capitulated ideals. In our case, since $d_{n}=\|A_{n}^{+}\|$ by definition, we have exactly $\|\mathcal{A}(\mathbb{H}_{n}^{+})\|=d_{n}^{2}$. This follows from the fact that the plus part capitulates completely, while the minus part is cyclic too and generates the radical of the extension. Consequently $\mathfrak{Q}^{{\rm ord}\ (a_{n})/d_{n}}$ becomes principal in $\mathbb{H}_{n}^{+}$, which confirms that $\|\mathcal{A}(\mathbb{H}_{n}^{+})\|=d_{n}^{2}\quad\hbox{ and }\|\mathcal{A}(\mathbb{H}_{n}^{+})\|^{-}=\|\mathcal{A}(\mathbb{H}_{n}^{+})\|^{+}=d_{n}.$ Therefore, $\|H^{0}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})\|=d_{n}$. The roots of unity $\zeta_{p^{n}}\not\in\mbox{\bf N}_{\mathbb{H}_{n}^{+}/\mathbb{K}_{n}}(E(\mathbb{H}_{n}^{+}))$: indeed, if $\zeta_{p^{m}}=N(\delta)$ for $\delta\in E(\mathbb{H}_{n}^{+})$ and $m\leq n$, then $\varepsilon=\delta/\overline{\delta}$ is well defined in the CM field $\mathbb{H}_{n}^{+}$ and it is a root of unity, by Dedekind’s unit Theorem – so $\varepsilon\in\mathbb{K}_{n}$. Moreover, we have $\mbox{\bf N}_{\mathbb{H}_{n}^{+}/\mathbb{K}_{n}}(\varepsilon)=\varepsilon^{d_{n}}=\zeta_{p^{m}}^{2}$. Since $p$ is odd, it follows that $\mu_{p^{n}}/\mu_{p^{n}/d_{n}}\subset H^{0}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})$; by comparing orders of the groups, we conclude that $H^{0}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})=\mu_{p^{n}}/\mu_{(p^{n}/d_{n})}=\left(H^{0}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})\right)^{-}.$ We have proved: ###### Lemma 1. Notations being like above, $\left(H^{0}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})\right)^{+}=\\{1\\}.$ In particular (3) $\displaystyle\mbox{\bf N}_{\mathbb{H}_{n}^{+}/\mathbb{K}_{n}}(E^{+}(\mathbb{H}_{n}^{+}))=E^{+}(\mathbb{K}_{n}),$ where for a CM field $\mathbb{F}$ we write $E^{+}(\mathbb{F})=\\{e\cdot\overline{e}:e\in E(\mathbb{F})\\}$. In our case, $E^{+}$ are the real units and the units of $\mathbb{K}_{n}^{+}$, resp. $\mathbb{H}(\mathbb{K}_{n}^{+})\subset\mathbb{H}_{n}^{+}$; the prime $p$ is odd and we are interested in $p$-parts, so the implicit exponent $2$ in the above definition has no further consequences: the norm is surjective on the real units in our class field. Then $\mathbb{H}_{n+1}^{+}=\mathbb{K}_{n+1}\cdot\mathbb{H}_{n}^{+}$ and we have a commutative diagram of fields. By computing $H^{0}(\mathbb{H}_{n+1}^{+}/\mathbb{K}_{n})$ in two ways, over the intermediate field $\mathbb{K}_{n+1}$ and respectively over $\mathbb{H}_{n}^{+}$, we obtain from (3) that (4) $\displaystyle\left(H^{0}(\mathbb{K}_{n+1}/\mathbb{K}_{n})\right)^{+}=\left(H^{0}(\mathbb{H}^{+}_{n+1}/\mathbb{H}^{+}_{n})\right)^{+}.$ The core observation of this proof is ###### Proposition 1. Let $\lambda_{n}=1-\zeta_{p^{n}}$. Then the ramified prime $\wp_{n}=(\lambda_{n})\subset\mathbb{K}_{n}$ above $p$ splits totally in $\mathbb{H}^{+}_{n}$ in $p$-principal ideals. Moreover, if $\nu\in\mbox{ Gal }(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})$ is a generator of this cyclic group, then there is a prime $\pi_{n}\in\mathbb{H}_{n}^{+}$ with $\mbox{\bf N}_{\mathbb{H}_{n}^{+}/\mathbb{K}_{n}}(\pi_{n})=\lambda_{n}$. ###### Proof. Since $\wp_{n}=(\lambda_{n})$ is principal, the Principal Ideal Theorem implies that it splits completely in the unramified extension $\mathbb{H}_{n}^{+}/\mathbb{K}_{n}$ and since $A(\mathbb{H}_{n}^{+})=\\{1\\}$ by (3), the primes above $\wp_{n}$ are $p$-principal. Let $E^{\prime}(\mathbb{F})$ denote the $p$-units of the number field $\mathbb{F}$, i.e. the units of the smallest ring containing $E(\mathbb{K})$ and in which all the primes above $p$ are invertible. In particular $E_{n}^{\prime}=E_{n}[1/\lambda_{n}]$; it is customary to denote by $A^{\prime}_{n}$ the $p$-part of the ideal class group of the $p$-integers. Since $\mathbb{H}_{n}^{+}/\mathbb{K}_{n}$ splits the prime above $p$ and $A_{n}^{+}=(A^{\prime}_{n})^{+}$ and $H^{0}(\mbox{ Gal }(\mathbb{H}_{n}^{+}/\mathbb{K}_{n}),E^{\prime}(\mathbb{H}_{n}^{+}))=H^{0}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})$. In particular, the norm $\mbox{\bf N}_{\mathbb{H}_{n}^{+}/\mathbb{K}_{n}}:E^{\prime}(\mathbb{H}_{n}^{+})\rightarrow E^{\prime}(\mathbb{K}_{n})$ is surjective, so there is a prime $\pi_{n}\in\mathbb{H}_{n}^{+}$ mapping on $\lambda_{n}$. The proof of this proposition is made particularly simple by the use of (3). However, a more involved proof shows that the facts hold in more generality and the primes above $\lambda$ are principal in any subfield of the Hilbert class field $\mathbb{H}_{n}$. ∎ As a consequence of the proposition, we see that $\mathcal{A}(\mathbb{H}_{n}^{+})=\\{1\\}$. Indeed, by Lemma 3, the class group $A(\mathbb{H}_{n}^{+})=\\{1\\}$, and the only primes that ramify in $\mathbb{H}_{n+1}^{+}/\mathbb{H}_{n}^{+}$ lay above $p$, so they are principal by Lemma 1. There are consequently no real ambig ideals in $\mathbb{H}_{n}^{+}$. On the other hand, for $n$ such that $\|A^{+}_{n}\|=\|A_{n+1}^{+}\|$, the capitulation kernel $P_{n}:=\mbox{ Ker }(\iota_{n,n+1}:A_{n}^{+}\rightarrow A_{n+1}^{+})$ is an $\mathbb{F}_{p}$-space of dimension $d=p-{\rm rank}\ (\mbox{\bf A}^{+})=p-{\rm rank}\ (A_{n})=1$. We obtain a contradiction with (4), which shows that if $i(p)=1$, then either $\mathbb{Z}_{p}\hbox{-rk}(A^{-})>1$ or $h_{p}^{+}=1$. This completes the proof of the Theorem. ## 3\. Three detailed variants of the proof Let $m$ be the first index for which $\|A_{m}^{(}\mathbb{K})\|=\|A_{m+1}^{+}(\mathbb{K})\|=\|A^{+}(\mathbb{K})\|$. Consider now $\mathbb{H}_{n}^{+}=\mathbb{H}_{m}\cdot\mathbb{K}_{n}^{+}$ for $n\geq m$. Since $[\mathbb{H}_{n}^{+}:\mathbb{K}_{n}^{+}]=q=\|A(\mathbb{K})^{+}\|$ for all $n>m$, we see that $\mbox{ Ker }(A^{+}(\mathbb{H}_{m})\rightarrow A^{+}(\mathbb{H}_{\infty})=1$. Iwasawa’s Theorem 12 in [Iw1] implies that $H^{1}(\mbox{ Gal }(\mathbb{H}_{n}/\mathbb{H}_{m}),E^{\prime}(\mathbb{H}_{n}))=1$, and the fact that the primes above $p$ are principal leads by a computation of Herbrand quotients to $E(\mathbb{H}_{m})=N_{n,m}(E(\mathbb{H}_{n}))$ for all $n>m$. We have $\displaystyle N_{\mathbb{H}_{n},\mathbb{K}^{+}_{m}}(E(\mathbb{H}_{n}))$ $\displaystyle=$ $\displaystyle N_{\mathbb{H}_{m}/\mathbb{K}^{+}_{m}}(E(\mathbb{H}_{m}))=E(\mathbb{K}_{m}^{+})$ $\displaystyle=$ $\displaystyle N_{n,m}(N_{\mathbb{H}_{n}/\mathbb{K}_{n}^{+}}(E(\mathbb{H}_{n}))=N_{n,m}(E(\mathbb{K}_{n}^{+})).$ It follows thus that $N_{n,m}(E(\mathbb{K}_{n}^{+}))=E(\mathbb{K}_{m}^{+})$ for all $n>m$. However, for sufficiently large $n$ we have $N_{n,m}(E(\mathbb{K}_{n}^{+}))=C(\mathbb{K}_{m}^{+})$ and this would require that $E(\mathbb{K}_{m}^{+})=C(\mathbb{K}_{m}^{+})$, which is the contradiction above. We give below333I thank Hiroki Takahashi for some critical questions and an interesting dialogue, which suggested that the details provided in this section may be useful for readers interested to understand better the specific conditions of the $p{\rm-th}$ cyclotomic extensions, compared for instance with some real quadratic extensions in which Greenberg’s conjecture is known to hold, yet $A^{+}=p$ and $A^{-}$ is cyclic. some additional detail about the units in $\mathbb{H}_{n}$ which lead to two variants of this proof and reveal interesting details. ### 3.1. Metacyclotomic units Let $E(\mathbb{H}_{n})$ be the units of $\mathbb{H}_{n}^{+}$ and $E(\mathbb{K}_{n}),C(\mathbb{K}_{n})$ the units, resp. cyclotomic units of $\mathbb{K}_{n}$. We shall construct some norm coherent sequences of units in $E(\mathbb{H}_{n})$ and relate them to the cyclotomic. For this, fix some large integer $N$. We assume for simplicity that the primes above $p$ in $\mathbb{H}_{n}$ are principal and derive some specific units on base of this assumption. Since we are only interested in $p$-parts, raising to a power coprime to $p$ can be neglected. We shall denote the norms $\mbox{\bf N}_{\mathbb{H}_{n}/\mathbb{H}_{m}}=\mbox{\bf N}_{\mathbb{K}_{n}/\mathbb{K}_{m}}$ by $N_{n,m}$ and let $\mathcal{N}=\mbox{\bf N}_{\mathbb{H}_{n}/\mathbb{K}_{n}}$ for all $n$ sufficiently large. We choose $\pi_{2N}\in\mathbb{H}_{2N}$ a prime with $\mathcal{N}(\pi_{2N})=1-\zeta_{p^{2N}}$ and let $\pi_{n}=N_{2N,n}(\pi_{2N})$ for all $n<2N$. The inertia group $I(\pi_{2N})\cong\mbox{ Gal }(\mathbb{K}_{2N}/\mathbb{Q})$. We let $\sigma^{\prime},\tau^{\prime}\in I(\pi_{2N})$ be lifts of $\sigma,\tau\in\mbox{ Gal }(\mathbb{K}_{2N}/\mathbb{Q})$, the generators of the subgroups with $p-1$ and $p^{2N-1}$ elements, respectively. We identify $T=\tau^{\prime}-1$ a lift of $T$ to $\mbox{ Gal }(\mathbb{H}_{2N}/\mathbb{H}_{m})$, where $m$ is an integer such that $\|A^{+}(\mathbb{K}_{m})\|=\|A^{+}(\mathbb{K}_{m+1})\|=q$, say. We fix an other large integer $M>>N$ and define the lifted idempotents $\varepsilon^{\prime}_{j}=\frac{1}{p-1}\sum_{\varsigma\in<\sigma^{\prime}>}\omega^{j}(\varsigma)\varsigma^{-1}\in\mathbb{Z}[\mbox{ Gal }(\mathbb{H}_{n}/\mathbb{Q})],$ where $\omega:<\sigma^{\prime}>\rightarrow\mathbb{Z}$ approximates the Teichmüller character to the power $p^{M}$. We also let $\tilde{\varepsilon}_{j}=\frac{1}{p-1}\sum_{\varsigma\in<\sigma^{\prime}>}\varpi^{j}(\varsigma)\varsigma^{-1}\in\mathbb{Z}_{p}[\mbox{ Gal }(\mathbb{H}_{n}/\mathbb{Q})],$ with $\varpi$ the $p$-adic Teichmüller character. Then we define $E^{(2j)}(\mathbb{K}_{n}^{+})=\varepsilon_{2j}E(\mathbb{K}_{n}^{+})$, so $\tilde{\varepsilon}_{2}j(E(\mathbb{K}_{n}^{+})/E^{p^{M}}(\mathbb{K}_{n}^{+}))=E^{(2j)}(\mathbb{K}_{n}^{+})/(E^{(2j)}(\mathbb{K}_{n}^{+}))^{p^{M}}$. Moreover, $m$ verifies for all $n>m$ (5) $\displaystyle E(\mathbb{K}_{n})=C(\mathbb{K}_{n})\cdot E(\mathbb{K}_{m}),\quad\forall n>m.$ Indeed, this condition is equivalent to $[E(\mathbb{K}_{n}):C(\mathbb{K}_{n})]=\|A_{n}^{+}\|=[E(\mathbb{K}_{m}):C(\mathbb{K}_{m})]=\|A_{m}^{+}\|$ for all $n>m$, a condition which is implied by our assumption (e.g. Fukuda). It follows also that $[E(\mathbb{K}_{l}):C(\mathbb{K}_{l})]=\|A^{+}_{l}\|=p\|A_{l-1}^{+}\|$ for $1<l\leq m$, and if $\varepsilon_{l}\in e_{2k}E(\mathbb{K}_{l})$ generates this component up to $p^{M}{\rm-th}$ powers, then $E(\mathbb{K}_{n})=\langle\varepsilon_{m},\eta_{n}\rangle_{\mathbb{Z}[T]}.$ We show ###### Lemma 2. Notations being like above, there is a module $\overline{C}(\mathbb{H}_{n})\subset E(\mathbb{H}_{n})$, of finite index and norm coherent for all $n\leq 2N$. Moreover, if $m$ is the smallest index such that $\|A_{m}^{+}\|=\|A^{+}_{m+1}\|$, then (6) $\displaystyle E(\mathbb{H}_{n})=E(\mathbb{H}_{m})\cdot\overline{C}(\mathbb{H}_{n})\quad\forall n\geq m.$ ###### Proof. Let $\eta_{n}=(\pi_{n}\cdot\overline{\pi_{n}})^{\sigma^{\prime}-1}$ with $\pi_{n}$ defined above, let $\delta_{n}=(\pi_{n}\cdot\overline{\pi_{n}})^{T}$ and $C(\mathbb{H}_{n})=\eta_{n}^{\mathbb{Z}[\mbox{\small Gal }(\mathbb{H}_{n}/\mathbb{Q})]}$ be the $\mathbb{Z}$-module generated by $\eta_{n}$, for all444We define the metacyclotomic units only for $n<N$, in order to make sure that they are norms from a large extension above $\mathbb{H}_{n}$. By avoiding the limit process $\cap_{N}N_{N,n}(E^{\prime}(\mathbb{H}_{n}))$ we obtain in addition explicit uniformizors $\pi_{n}$, which are norm coherent. $n<N$. By definition, $\mathcal{N}(C(\mathbb{H}_{n}))=C(\mathbb{K}_{n})$ and the modules $C(\mathbb{H}_{n})$ are norm coherent. In particular, if $e\in C(\mathbb{H}_{l})$ then there is a unit $e^{\prime}\in C(\mathbb{H}_{2N})$ with $e=N_{2N,l}(e^{\prime})$. Let further $(a_{n})_{n\in\mathbb{N}}\in A^{+}_{n}\setminus(A_{n}^{+})^{p}$ be a norm coherent sequence of classes; thus $a_{n}^{q}=1$ for all $n\geq m$ and $q=\exp(A^{+})$. Let $\mathfrak{Q}\in a_{2N}$ be a totally split prime and $\gamma\in\mathbb{H}_{2N}$ be such that $\mathfrak{Q}\mathcal{O}(\mathbb{H}_{2N})=(\gamma)$; if $\mathfrak{Q}_{n}=N_{2N,n}(\mathfrak{Q})$ and $\gamma_{n}=N_{2N,n}(\gamma)$, then $\mathfrak{Q}_{n}\mathcal{O}(\mathbb{H}_{n})=(\gamma_{n})$ for all $n<2N$. We let $c_{n}=\gamma_{n}^{s}\in E(\mathbb{H}_{n})$, a norm coherent sequence of units for $n\leq 2N$. Let $\mathcal{C}(\mathbb{H}_{n})=c_{n}^{\mathbb{Z}[s]}\cdot\delta_{n}^{\mathbb{Z}[T,s]}$. Then $\mathcal{C}(\mathbb{H}_{n})$ also form a norm coherent sequence and we define $\overline{C}(\mathbb{H}_{n})=C(\mathbb{H}_{n})\cdot\mathcal{C}(\mathbb{H}_{n})$, which is a further norm coherent sequence of units in $E(\mathbb{H}_{n})$. We note that $\mbox{ Ker }(\mathcal{N}:E(\mathbb{H}_{n})\rightarrow E(\mathbb{K}_{n}^{+}))=E^{s}(\mathbb{H}_{n})\cdot\mathcal{C}(\mathbb{H}_{n})$. Letting $\mathcal{E}(\mathbb{H}_{n})=E(\mathbb{H}_{n})/(E^{s}(\mathbb{H}_{n})\cdot\mathcal{C}(\mathbb{H}_{n}))$, it follows from (3) that $\mathcal{E}(\mathbb{H}_{n})\cong E(\mathbb{K}_{n}),$ and the norm $\mathcal{N}$ is an isomorphism between these two modules. As a consequence, if $c_{0}\in\mathcal{C}(\mathbb{H}_{n})$ generates this module and $e_{2j}\in E(\mathbb{H}_{n}),j=0,1,\ldots,\frac{p-3}{2}$ have non trivial image in $\mathcal{E}$ and map to a set of units $d_{j};j=0,1,\ldots,(p-3)/2$ which generate $E(K_{n}^{+})$ as a $\mathbb{Z}[T]$-module and such that $d_{j}$ generates $\varepsilon_{2j}E(\mathbb{K}_{n}^{+})/E(\mathbb{K}_{n}^{+})^{p^{M}}$ for fixed, arbitrarily large $M>0$, then $\langle c_{0};e_{0},\ldots,e_{(p-3)/2}\rangle_{\mathbb{Z}[T,s]}=E(\mathbb{H}_{n}).$ We let $e_{n}\in E(\mathbb{H}_{n})$ be such that $\mathcal{N}(e_{n})$ generates $\varepsilon_{2j}E(\mathbb{K}_{n})/E^{p^{M}}(\mathbb{K}_{n})$ for some fixed, large $M$. We then consider the systems $\displaystyle\mathcal{H}_{n}$ $\displaystyle=$ $\displaystyle\left\\{c_{n}^{s^{j}}\ :\ j=0,1,\ldots,p-2\right\\}\bigcup\left\\{\delta_{n}^{s^{j}T^{l}}\ :\ j=0,1,\ldots,p-1;l=0,\ldots,p^{n-1}-1\right\\}$ $\displaystyle\bigcup\left\\{\varepsilon^{\prime}_{2j}\eta_{n}^{s^{i}\cdot T^{l}}\ :\ i=0,1,2,\ldots,p-1;j=1,2,\ldots,\frac{p-3}{2};l=0,\ldots,p^{n-1}-1\right\\},$ $\displaystyle\overline{\mathcal{H}}_{n}$ $\displaystyle=$ $\displaystyle\left\\{c_{n}^{s^{j}}\ :\ j=0,1,\ldots,p-2\right\\}\bigcup\left\\{\delta_{n}^{s^{j}T^{l}}\ :\ j=0,1,\ldots,p-1;l=0,\ldots,p^{n-1}-1\right\\}$ $\displaystyle\bigcup\left\\{(e_{n}^{(2j)})^{s^{i}\cdot T^{l}}\ :\ i=0,1,2,\ldots,p-1;j=1,2,\ldots,\frac{p-3}{2};l=0,\ldots,p^{n-1}-1\right\\}.$ One verifies that $\|\mathcal{H}_{n}\|=\mathbb{Z}\hbox{-rk}(E(\mathbb{H}_{n}))$; an application of the Nakayama lemma to $\overline{C}(\mathbb{H}_{n})/\overline{C}(\mathbb{H}_{n})^{p^{M}}$ implies that the system is also a generating system for $\overline{C}(\mathbb{H}_{n})$. Moreover $\overline{\mathcal{H}}_{n}$ generates $E(\mathbb{H}_{n})/E^{p^{M}}(\mathbb{H}_{n})$ for $n\leq m$. These systems are reminiscent of Hilbert’s relative units in his Theorem 91. It follows that $\overline{C}(\mathbb{H}_{n})$ has finite index in $E(\mathbb{H}_{n})$ for all $n<2N$. Assume now that (6) is false, and there is some $m^{\prime}>m$ and $e_{m^{\prime}}\in E(\mathbb{H}_{m^{\prime}})\setminus E(\mathbb{H}_{m})\cdot\overline{C}(\mathbb{H}_{m^{\prime}})$; we can assume without restriction of generality that $e_{m^{\prime}}\in E^{(2j)}(\mathbb{H}_{m^{\prime}})$ (see below for the definition of the components $E^{(2j)}(\mathbb{H}_{m^{\prime}})$), since at least one component verifies the condition. Then, by (5) $\mathcal{N}(e_{m^{\prime}})\in E(\mathbb{K}_{m})\cdot\overline{C}(\mathbb{K}_{m^{\prime}})=\mathcal{N}(E(\mathbb{H}_{m})\cdot\overline{C}(\mathbb{H}_{m^{\prime}})).$ There is thus a unit $e\in E(\mathbb{H}_{m})\cdot\overline{C}(\mathbb{H}_{m^{\prime}})$ such that $e_{m^{\prime}}=e\cdot w,w\in\mbox{ Ker }(\mathcal{N}:E(\mathbb{H}_{m^{\prime}})\rightarrow E(\mathbb{K}_{m^{\prime}}))$. Since $\mathcal{C}_{n}$ are norm coherent and $(1-\varepsilon^{\prime}_{0})\mbox{ Ker }(\mathcal{N}:E(\mathbb{H}_{m^{\prime}})\rightarrow E(\mathbb{K}_{m^{\prime}}))=(1-\varepsilon^{\prime}_{0})E(\mathbb{H}_{m^{\prime}})^{s}$, we can assume that $w\in(1-\varepsilon^{\prime}_{0})E(\mathbb{H}_{m^{\prime}})$, so $w=(e_{m^{\prime}}^{\prime})^{s}$ with $e_{m^{\prime}}^{\prime}\in(1-\varepsilon^{\prime}_{0})E(\mathbb{H}_{m^{\prime}})$. By repeating the same argument, we find that $e_{m^{\prime}}^{\prime}=e^{\prime\prime}\cdot w^{\prime}$, where $w^{\prime}\in\mbox{ Ker }(\mathcal{N}:E(\mathbb{H}_{m^{\prime}})\rightarrow E(\mathbb{K}_{m^{\prime}}))$. It follows by induction that $e_{m^{\prime}}\in E(\mathbb{H}_{m})\cdot\overline{C}(\mathbb{H}_{m^{\prime}})\cdot E(\mathbb{H}_{m^{\prime}})^{s^{u}}$ for any $u>0$. Since $s^{p}=pv,v\in(\mathbb{Z}_{p}[s])^{\times}$, it follows that $e_{m^{\prime}}\in E(\mathbb{H}_{m})\cdot\overline{C}(\mathbb{H}_{m^{\prime}})\cdot E(\mathbb{H}_{m^{\prime}})^{p^{M^{\prime}}}$ for arbitrary $M^{\prime}>0$. But the index $[E(\mathbb{H}_{m^{\prime}}):\overline{C}(\mathbb{H}_{m^{\prime}})]$ is finite, so the claim must be true. Therefore $m$ is defined as the smallest index after which $\|A_{n}^{+}\|$ stabilizes ∎ We obtain the following explicite variant of the first proof given above. Since $E(\mathbb{H}_{m})\neq\overline{C}(\mathbb{H}_{m})$ while the quotient of these modules is finite, there is a unit $e\in E(\mathbb{H}_{m})\setminus\overline{C}(\mathbb{H}_{m})$, such that $e^{p}\in\overline{C}(\mathbb{H}_{m})$. Consequently, there is a metacyclotomic unit $c_{m+1}\in\overline{C}(\mathbb{H}_{m+1})$ such that $N_{m+1,m}(c_{m+1})=e^{p}$. Let $d=c_{m+1}/e\not\in\overline{C}(\mathbb{H}_{m+1})$, by choice of $e$. By definition, $N_{m+1,m}(d)=1$ and thus Hilbert 90 implies that $d=\gamma^{\omega_{m}}$ for some $\gamma\in\mathbb{H}_{m+1}$. The ideal $(\gamma)$ is ambig, and since $A^{+}(\mathbb{H}_{m+1})=1$, it follows that $\gamma$ must be a product of principal ramified primes. Thus $\gamma=\pi_{m+1}^{x}\cdot\delta$, for some $x\in\mathbb{Z}[\mbox{ Gal }(\mathbb{H}_{m+1}/\mathbb{Q})]$ and $\delta\in E(\mathbb{H}_{m+1}$. But then $\pi_{m+1}^{x\omega_{m+1}}\in\overline{C}(\mathbb{H}_{m+1})$ by definition of $\overline{C}$. Moreover, the structure identity (6) implies that $\delta^{\omega_{m}}\in\overline{C}(\mathbb{H}_{m+1})^{\omega_{m}}\subset\overline{C}(\mathbb{H}_{m+1})$. Altogether, we find that $d\in\overline{C}(\mathbb{H}_{m+1})$ and thus $e\in\overline{C}(\mathbb{H}_{m+1})$ too. This contradicts the choice $e\not\in\overline{C}(\mathbb{H}_{m})$, given the (6): units which are metacyclotomic in $\mathbb{H}_{m+1}$ must be so already in $\mathbb{H}_{m}$. Note that this contradiction makes explicite the argument of Iwasawa used in the first proof. ### 3.2. Local units and components We consider the structure of the local units $U(\mathbb{K}_{n}),U(\mathbb{H}_{n})$. For the first, it is known that $U(\mathbb{K}_{n}^{+})=\bigoplus_{k}\varepsilon_{2k}x_{n}^{\Lambda}$ are $\Lambda$-cyclic modules, generated by some $x_{n}\in U(\mathbb{K}^{+}_{n})$ (e.g. [Wa], §8). Since $p$ is totally split, we also have $U(\mathbb{H}_{n})\cong(U(\mathbb{K}_{n}^{+}))^{p}$, as cartesian product. Let $\xi_{n}=(x_{n},1,1,\ldots,1)$ under the Chinese Remainder Theorem, with the first component being the projection to $\mathbb{H}_{n,\wp_{n}}$ and $\wp_{n}$ a ramified prime above the initially chosen $\wp=(\pi)$. Then $\tilde{\varepsilon}_{2j}\xi_{n}$ defines the $2j$-component of $U(\mathbb{H}_{n})$ as a $\Lambda[s]$-module. Since $\nu$ acts transitively on the projections in $U(\mathbb{H}_{n})$, it follows that these modules are also canonically given by (7) $\displaystyle U_{2j}(\mathbb{H}_{n})=(U(\mathbb{K}_{n}^{+}))^{p}$ where the right hand side is a cartesian of copies corresponding to the completions at the various primes above $p$. The action of $\mbox{ Gal }(\mathbb{K}_{n}^{+}/\mathbb{Q})$ by conjugation on $\nu$ implies $\displaystyle\nu^{\sigma}=\nu^{\varpi(\sigma)^{2k}},\quad\nu^{\tau}=\nu^{p+1},$ with $\varpi$ being the Teichmüller character. From this, one verifies without using the construction above, that $(U(\mathbb{K}_{n}^{+}))^{p}$ is a canonic $\mathbb{Z}_{p}[\mbox{ Gal }(\mathbb{H}/\mathbb{Q})]$-module which well defined by (7). We say that $U_{2j}(\mathbb{H}_{n})$ is the $2j{\rm-th}$ component of $U(\mathbb{H}_{n})$ and have $U(\mathbb{H}_{n})=\bigoplus_{j=0}^{(p-3)/2}U_{2j}(\mathbb{H}_{n}).$ Let $\mbox{\bf N}_{l}=\cap_{n}N_{n,l}(U(\mathbb{H}_{n}))$ for $l\geq 1$. Then class field theory implies that $U(\mathbb{H}_{l})/\mbox{\bf N}_{l}\cong\prod U^{(1)}(\mathbb{Z}_{p})$ is the product of the $p$ copies of $U^{(1)}(\mathbb{Z}_{p})$ in $U(\mathbb{H}_{l})$. Since $\mathbb{Z}_{p}\subset U_{0}(\mathbb{K}_{n}^{+})$ it follows from the definition (7) that $U(\mathbb{H}_{l})/\mbox{\bf N}_{l}\hookrightarrow U_{0}(\mathbb{H}_{n})$, and therefore (8) $\displaystyle U_{2j}(\mathbb{H}_{m})\subset\mbox{\bf N}_{m}\quad\hbox{for all}\quad j\neq 0.$ We identify the units $E(\mathbb{H}_{n})$ with their diagonal embedding $E(\mathbb{H}_{n})\hookrightarrow U(\mathbb{H}_{n})$. Since $E(\mathbb{H}_{n})/E(\mathbb{H}_{n})^{p^{M}}$ is a $\mathbb{Z}_{p}$-module, we can use the components of the local units for defining the components of global units by $\displaystyle E^{(2j)}(\mathbb{H}_{n})$ $\displaystyle=$ $\displaystyle E(\mathbb{H}_{n})\cap\left(U_{2j}(\mathbb{H}_{n})\cdot E(\mathbb{H}_{n})^{p^{M}}\right),$ $\displaystyle C^{(2j)}(\mathbb{H}_{n})$ $\displaystyle=$ $\displaystyle E^{(2j)}(\mathbb{H}_{n})\cap\overline{C}(\mathbb{H}_{n})$ By definition of $U_{2j}$, we have $E^{(2l)}(\mathbb{H}_{n})\cap E^{(2j)}(\mathbb{H}_{n})=E(\mathbb{H}_{n})^{p^{M}}$ for $l\neq j$. Let now $e\in E^{(2k)}(\mathbb{H}_{m})\setminus N_{m+1,m}(E^{(2k)}(\mathbb{H}_{m+1}))$. Note that by (6), the existence of such a unit is granted. But then, by applying $\mathcal{N}$ we deduce that $\varepsilon_{2k}E(\mathbb{K}_{m})=\varepsilon_{2k}C(\mathbb{K}_{m})$. However, our initial assumption implies that $\varepsilon_{2k}A^{+}_{m}\neq\\{1\\}$ and the main conjecture thus leads to $[\varepsilon_{2k}E(\mathbb{K}_{m}):\varepsilon_{2k}C(\mathbb{K}_{m})]\neq 1$. Therefore, we conclude that there is unit $e$ with the claimed properties. The Hasse norm principle implies by the above that $e\in N_{m+1,m}(\mathbb{H}_{m+1}^{\times})$. Let thus $x\in\mathbb{H}_{m+1}^{\times}$ with $N_{m+1,m}(x)=e$. Then $(x)=\mathfrak{X}^{\omega_{m}}$, since $H^{1}(\langle\nu\rangle,I(\mathbb{H}_{n}))$ vanishes for the ideals in the cyclic extension $\mathbb{H}_{n}/\mathbb{K}_{n}$. Moreover, $\mathfrak{X}$ is not $p$-principal: otherwise, if $(t,p)=1$ and $\mathfrak{X}^{t}=(\xi)$ is principal, then $x^{t}=\xi^{\omega_{m}}\cdot e_{1}$ for some unit $e_{1}\in E(\mathbb{H}_{m+1})$. If the $ut\textasciiacute+vp=1$ we have $e=N_{m+1,m}(e^{v}\cdot\xi^{\omega_{m}}e_{1})=N_{m+1,m}(e^{v}\cdot e_{1})\in N_{m+1,m}E(\mathbb{H}_{m+1}),$ and we had assumed $e\not\in N_{m+1,m}E(\mathbb{H}_{m+1})$. Consequently $A(\mathbb{H}_{m+1})\neq\\{1\\}$, which is a contradiction to the Lemma 3 that completes the proof. Note that universal norms behave differently in the positive components and in the zero component, the latter being by (8) the one which localizes the norm residue defects along the cyclotomic $\mathbb{Z}_{p}$-extension of any base field. This fact may indicate the major distinction between zero components and positive indexed components. ## 4\. Appendix For the sake of completeness, we give a proof of the following ###### Lemma 3. Let $\mathbb{K}$ be a number field and $A$ be the $p$ \- part of its class group, while $\mathbb{H}$ is the $p$ \- part of its Hilbert class group. If $A$ is cyclic, then $A(\mathbb{H}):=\mathcal{C}(\mathbb{H})_{p}=\\{1\\}$. ###### Proof. Since $A$ is cyclic, $\mbox{ Gal }(\mathbb{H}/\mathbb{K})\cong A$ is a cyclic group and the ideals of a generating class $a\in A$ are inert and become principal in $\mathbb{H}$. Let $\sigma=\varphi(a)\in\mbox{ Gal }(\mathbb{H}/\mathbb{K})$ be a generator and $s=\sigma-1$. Suppose that $b\in A(\mathbb{H})\setminus A(\mathbb{H})^{(s,p)}$ is a non trivial class and let $\mathfrak{Q}\in b$ be an ideal above a rational prime $\mathfrak{q}\subset\mathbb{K}$, which splits completely in $\mathbb{H}/\mathbb{K}$: such a prime must exist, by Tchebotarew’s Theorem. If $b^{\prime}=[\mathfrak{q}]$, then the order of $b^{\prime}$ must be a power of $p$, since this holds for $\mathfrak{Q}$; thus $b^{\prime}\in A(\mathbb{K})=\langle a\rangle$. But we have seen that the ideals from $a$ capitulate in $\mathbb{H}$, so $b=1$, in contradiction with our choice. Therefore $b^{\prime}=1$ and thus $\mbox{\bf N}_{\mathbb{H}/\mathbb{K}}(b)=1$. Furtwängler’s Hilbert 90 Theorem for ideal class groups in cyclic extensions [Fu] says that $\mbox{ Ker }(\mbox{\bf N}:A(\mathbb{H})\rightarrow A(\mathbb{K}))\subset A(\mathbb{H})^{s}$, and this implies that $b\in A(\mathbb{H})^{s}$, which contradicts the choice of $b$ and completes the proof. ∎ ## References * [Fu] P. Furtwängler: Über die Reziprozitätsgesetze zwischen $l$-ten Potenzresten in algebraischen Zahlkörpern, wenn $l$ eine ungerade Primzahl bedeutet, (German) Math. Ann. 58, 1-50 (1904). * [Hi] David Hilbert: The Theory of Algebraic Numbers (Zahlbericht), Translated by Iain Adamson, Edited by Iain Adamson, Franz Lemmermayer and Norbert Schappacher; Springer (1998). * [La] S. Lang: Cyclotomic fields I and II, First Edition, Springer (1978,80) * [Iw] K. Iwasawa: A note on the group of units of an algebraic number fields, Journal de Math. Pures et Appl. 35/121 (1956), pp. 189-192. * [Iw1] K. Iwasawa: On $\mathbb{Z}_{\ell}$ \- extensions of number fields, Ann. Math. Second Series, (1973), 98, 247 – 326. * [Wa] Lawrence Washington: Introduction to cyclotomic fields, Springer, Graduate Texts in Mathematics 83, 2-nd edition (1996).	arxiv-papers	2010-11-29T16:33:22	2024-09-04T02:49:15.295212	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Preda Mih\\u{a}ilescu", "submitter": "Preda Mihailescu", "url": "https://arxiv.org/abs/1011.6283" }
1011.6471	From Uniform Continuity to Absolute Continuity Kai Yang, Chenhong Zhu The notion of uniform continuity emerged slowly in the lectures of Dirichlet(1854) and of Weierstrass(1861) [2]. Then in 1905, Vitali established the absolute continuity for a class of functions in the paper “Sulle funzioni integrali” [3]. Sooner or later, several equivalent and sufficient conditions for a function to be absolutely continuous were derived (see [4]), which depend on several results of measure theory and integration. For example, Banach-Zarecki Criterion[5] states that, “$f$ is absolutely continuous on $[a,b]$ if and only if $f$ is continuous and of bounded variation on $[a,b]$ and maps null sets to null sets. ”, and in particular, every Lipschitz continuous function is absolutely continuous. Absolute continuity implies uniform continuity, but generally not vice versa. However, under certain conditions (piecewise convexity), uniform continuity will also imply absolute continuity. In this short note, we will present a sufficient condition for a uniformly continuous function to be absolutely continuous. ###### Definition 1 (piecewise convex function) A function $f$ defined on an interval $I_{a,b}$ of the real line is piecewise convex, if there exits a finite partition $P=\\{a_{i}\\}_{i=0}^{N}$ such that on each subinterval $[a_{i},a_{i+1}]$ $(i=0,\dots,$ $N-1)$, $f$ is concave or convex. Remark: Here $I_{a,b}$ is an interval of $\mathbb{R}$ with $a,b$ as the endpoints and it can be open, closed or half-open; moreover, $a,b$ can also take the value $\pm\infty$ when $a$ or $b$ is not contained in $I_{a,b}$ (in this case, we use $(a_{0},a_{1}]$ or $[a_{N-1},a_{N})$ instead of the closed subinterval). ###### Theorem 1 For a uniformly continuous function $f$ defined on $I_{a,b}$ of $\mathbb{R}$, if it is piecewise convex, then it is also absolutely continuous on $I_{a,b}$. Remark: We can verify that if $I_{a,b}=[a,b]$, then the conditions in Theorem 1 satisfy the Banach-Zarecki Criterion. However, our attempt is based on some elementary properties of uniform continuity and convexity. In particular, one simple example is that $f(x)=\sqrt{x}$ is absolutely continuous (not Lipschitz continuous) on $[0,c]$. Moreover, the converse statement of this theorem is false. One proper counterexample is $f(x)=x^{2}\sin(1/x)$ on $[0,1]$ (define $f(0)=0$). In addition, the cantor function[7] is uniformly continuous but not absolutely continuous. Figure 1: graph of $f(x)=\sqrt{x}$ when $\sigma=0.5$ This picture gives us the idea of Lemma 1 that the slope of an increasing concave function of two points with the same distance of the $x$ coordinate will decrease. In addition, the monotonicity of the change of slopes will still be valid for other monotone convex or concave functions. ###### Lemma 1 If $f$ is monotone and concave or convex on an interval $I$ of $\mathbb{R}$, then $G_{\sigma}(x)=\|f(x+\sigma)-f(x)\|$ is monotone with respect to $x$, where $\sigma$ is any positive real number. Proof. We only demonstrate the case that $f(x)$ is monotone increasing and concave, other situations can be proved similarly. Obviously, in this case, $G_{\sigma}(x)=f(x+\sigma)-f(x)$ for $\sigma>0$. We will show that $G_{\sigma}(x)$ is decreasing on $I$ with respect to $x$. Suppose $x,y+\sigma\in I$, and $x<y$. Since $f(x)$ is concave, for any $\theta\in(0,1)$, we get $f(\theta a+(1-\theta)b)\geq\theta f(a)+(1-\theta)f(b),\,\,\,\,\forall a,b\in I.$ (1) 1) Set $a=x$, $b=y+\sigma$, $\theta=\frac{y-x}{y-x+\sigma}$: By (1) , we have $f(x+\sigma)\geq\frac{y-x}{y-x+\sigma}f(x)+\frac{\sigma}{y-x+\sigma}f(y+\sigma),$ which is equivalent to $\frac{f(y+\sigma)-f(x)}{y-x+\sigma}\leq\frac{f(x+\sigma)-f(x)}{\sigma}.$ (2) 2) Set $a=x$, $b=y+\sigma$, $\theta=\frac{\sigma}{y-x+\sigma}$: Similarly, we can get $\frac{f(y+\sigma)-f(y)}{\sigma}\leq\frac{f(y+\sigma)-f(x)}{y-x+\sigma}.$ (3) By (2) and (3) , we obtain $\frac{f(y+\sigma)-f(y)}{\sigma}\leq\frac{f(x+\sigma)-f(x)}{\sigma}.$ Thus, $G_{\sigma}(y)\leq G_{\sigma}(x)$, which implies that $G_{\sigma}(x)$ is a monotone decreasing function. Remark: Lemma 1 is one special case of a classical property of convex or concave functions on an interval of $\mathbb{R}$, see [6]. ###### Lemma 2 For a monotone concave or convex function $f(x)$ defined on an interval $I$ of $\mathbb{R}$, if $f$ is uniformly continuous, then $f$ is absolutely continuous. Proof. Since $f$ is uniformly continuous on $I$, for any $\epsilon>0$, there exists $\delta>0$ such that for any $x,y\in I$ and $\|x-y\|<\delta$, we have $\|f(x)-f(y)\|<\epsilon.$ Now, we consider every finite collection $\\{(x_{i},y_{i})\\}_{i=1}^{n}$ of nonoverlapping subintervals of $I$ with $x_{i}<x_{i+1}$, and $\sum_{i=1}^{n}(y_{i}-x_{i})<\delta.$ Define $\sigma_{i}=y_{i}-x_{i}>0$, then $\|f(y_{i})-f(x_{i})\|=G_{\sigma_{i}}(x_{i}).$ By Lemma 1, we know that $G_{\sigma}(x)$ is monotone. 1) $G_{\sigma}(x)$ is monotone deceasing. Since $(x_{2},y_{2})$ and $(x_{1},y_{1})$ are nonoverlapping, we get $G_{\sigma_{2}}(x_{2})\leq G_{\sigma_{2}}(y_{1})=G_{\sigma_{2}}(x_{1}+\sigma_{1}).$ Inductively, for $i\geq 2$, we will obtain $G_{\sigma_{i}}(x_{i})\leq G_{\sigma_{i}}(x_{1}+\sum_{j=1}^{i-1}\sigma_{j}).$ Therefore, we have $\displaystyle\sum_{i=1}^{n}\|f(y_{i})-f(x_{i})\|$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{n}G_{\sigma_{i}}(x_{i})\leq G_{\sigma_{1}}(x_{1})+\sum_{i=2}^{n}G_{\sigma_{i}}(x_{1}+\sum_{j=1}^{i-1}\sigma_{j}).$ Define $z_{1}=x_{1}$, and $z_{i}=x_{1}+\sum_{j=1}^{i-1}\sigma_{j}$ $(2\leq i\leq n+1)$. Since $f$ is monotone, the above inequality is equivalent to $\sum_{i=1}^{n}\|f(y_{i})-f(x_{i})\|\leq\sum_{i=1}^{n}\|f(z_{i+1})-f(z_{i})\|=\|f(z_{n+1})-f(z_{1})\|.$ In addition, $\|z_{n+1}-z_{1}\|=\sum_{i=1}^{n}\sigma_{i}=\sum_{i=1}^{n}(y_{i}-x_{i})<\delta$, since $f$ is uniformly continuous, we obtain $\sum_{i=1}^{n}\|f(y_{i})-f(x_{i})\|\leq\|f(z_{n+1})-f(z_{1})\|<\epsilon.$ Hence, $f$ is also absolutely continuous on $I$. 2) $G_{\sigma}(x)$ is monotone increasing. The strategy is quite similar to the previous one; however, we just fix $(x_{n},y_{n})$ first and define $z_{n+1}=y_{n}$, $z_{i}=y_{n}-\sum_{j=i}^{n}\sigma_{j}$ for $1\leq i\leq n$. Similarly, we have $\sum_{i=1}^{n}\|f(y_{i})-f(x_{i})\|\leq\sum_{i=1}^{n}\|f(z_{i+1})-f(z_{i})\|=\|f(z_{n+1})-f(z_{1})\|<\epsilon.$ Therefore, $f$ is absolutely continuous on $I$. Remark: The idea of this lemma is to glue disjoint subintervals $(x_{i},y_{i})$ together as one subinterval then apply the property of uniform continuity. By Lemma 2, it is clear that $f(x)=\sqrt{x}$ is absolutely continuous on $[0,c]$. In addition, we can consider more general functions that oscillate finite times, and on each monotone subinterval, they also admit convexity. If these functions are uniformly continuous, then they are also absolutely continuous by utilizing the same strategy in the proof of Lemma 2 on each subinterval. Proof of Theorem 1. If $f$ is not monotone on $[a_{i},a_{i+1}]$, then we can split $[a_{i},a_{i+1}]$ into two subintervals such that on each of them, $f$ is monotone, since $f$ assumes convexity. If we relabel them, then on each $[a_{i},a_{i+1}]$, $f$ is monotone and concave or convex. The following proof is based on this situation. Since $f$ is uniformly continuous on $I$, for any $\epsilon/N>0$, there exists $\delta>0$ such that for any $x,y\in I$ and $\|x-y\|<\delta$, we have $\|f(x)-f(y)\|<\epsilon/N.$ Choose $\delta_{1}<\min\\{a_{1}-a_{0},\dots,a_{N}-a_{N-1},\delta\\}$, then for every finite collection $\\{(x_{i},y_{i})\\}_{i=1}^{n}$ of nonoverlapping subintervals of $I$ with $x_{i}<x_{i+1}$ and $\sum_{i=1}^{n}(y_{i}-x_{i})<\delta_{1},$ each subinterval $(x_{i},y_{i})$ can contain at most one $a_{j}$. For such interval $(x_{i},y_{i})$ containing $a_{j}$, by triangle inequality, we have $\|f(y_{i})-f(x_{i})\|\leq\|f(y_{i})-f(a_{j})\|+\|f(a_{j})-f(x_{i})\|.$ (4) Now, we still treat $(x_{i},a_{j})$ and $(a_{j},y_{i})$ as two “nonoverlapping subintervals”. Then, we relabel all the subintervals as $\\{(x_{j},y_{j})\\}_{j=1}^{m}$ ($n\leq m\leq 2n$). Through the above strategy, each new $(x_{j},y_{j})$ will lie exactly in one $[a_{i},a_{i+1}]$ and $\sum_{j=1}^{m}(y_{j}-x_{j})=\sum_{i=1}^{n}(y_{i}-x_{i})<\delta_{1}.$ For all the new subintervals that lie in $[a_{i},a_{i+1}]$ $(i=0,\dots,N-1)$, since $f$ is also monotone and concave or convex and $\sum_{(x_{j},y_{j})\subset[a_{i},a_{i+1}]}(y_{j}-x_{j})<\delta_{1}<\delta,$ by the method in Lemma 2, we obtain that $\sum_{(x_{j},y_{j})\subset[a_{i},a_{i+1}]}\|f(y_{j})-f(x_{j})\|<\epsilon/N.$ Therefore, $\sum_{j=1}^{m}\|f(y_{j})-f(x_{j})\|=\sum_{i=0}^{N-1}{\sum_{(x_{j},y_{j})\subset[a_{i},a_{i+1}]}\|f(y_{j})-f(x_{j})\|}<\epsilon.$ (5) By (4) and (5) , we get $\sum_{i=1}^{n}\|f(y_{i})-f(x_{i})\|\leq\sum_{j=1}^{m}\|f(y_{j})-f(x_{j})\|<\epsilon.$ Hence, $f$ is also absolutely continuous on $I_{a,b}$. Question: We consider the case on the real line; however, one can think about the situation for the multidimensional Euclidean space. ## References * [1] * [2] E. Hairer, G. Wanner, Analysis by Its History, Springer-Verlag, New York, 1996. * [3] M. B. Porter, Concerning Absolutely Continuous Functions, Bulletin of the American Mathematical Sociaty, Vol.22, No.3, p109-111, 1915. * [4] E. V. Dale, On Absolutely Continuous Functions, American Mathematical Monthly, Vol.72, No.8, p831-841, 1965\. * [5] J. Yeh, Lectures on Real Analysis, World Scientific, Singapore, 2000. * [6] H. L. Royden, Real Analysis, 3rd ed., Macmillan Publishing Company, New York, 1988. * [7] D. R. Chalice, A Characterization of the Cantor Function, Amererican Mathematical Monthly, Vol.98, No.3, p255-258, 1991. Department of Mathematics and Statistics, Memorial University of Newfoundland, NL, Canada. kyang@mun.ca Applied Mathematical and Computational Sciences, University of Iowa, IA, USA. chenhzhu@math.uiowa.edu	arxiv-papers	2010-11-30T07:32:55	2024-09-04T02:49:15.306432	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Kai Yang, Chenhong Zhu", "submitter": "Kai Yang", "url": "https://arxiv.org/abs/1011.6471" }
1011.6516	020006 2010 I. Ippolito A. Coniglio, Universitá di Napoli “Federico II”, Napoli, Italy. 020006 As a fragile construction, a granular pile is very sensitive to minute external perturbations. In particular, it is now well established that a granular assembly is sensitive to variations of temperature. Such variations can produce localized rearrangements as well as global static avalanches inside a pile. In this review, we sum up the various observations that have been made concerning the effect of temperature on a granular assembly. In particular, we dwell on the way controlled variations of temperature have been employed to generate the compaction of a granular pile. After laying emphasis on the key features of this compaction process, we compare it to the classic vibration-induced compaction. Finally, we also review other granular systems in a large sense, from microscopic (jammed multilamellar vesicles) to macroscopic scales (stone heave phenomenon linked to freezing and thawing of soils) for which periodic variations of temperature could play a key role in the dynamics at stake. # Invited review: Effect of temperature on a granular pile Thibaut Divoux [inst1] E-mail: Thibaut.Divoux@ens-lyon.fr (10 August 2010; 28 October 2010) ††volume: 2 99 inst1 Université de Lyon, Laboratoire de Physique, École Normale Supérieure de Lyon, CNRS UMR 5672, 46 Allée d’Italie, 69364 Lyon cedex 07, France. ## 1 Introduction: a granular pile as a fragile construction A granular pile can be described as a bunch of hard and frictional grains for which the thermal ambient agitation is negligible [1]. Indeed, the potential energy of a grain of density $\rho$ assessed over a displacement equivalent to its diameter $d$ satisfies $\rho gd^{4}/k_{B}T\simeq 10^{11}\gg 1$. Thus a granular pile is an athermal system, and one needs to inject energy inside the pile to trigger any reorganization of the packing. The accessible packings can then be seen as jammed states, i.e. minima of the potential energy, in some energy landscape (Fig. 1) and the energy one injects makes it possible to overcome the energy barrier that separates two configurations from one another. Various methods have been used, over the passed 20 years, to provide energy to a granular pile, among which mechanical vibrations [2, 3, 4] and shear [5, 6] are the most widespread. Figure 1: Sketch of the energy landscape associated to the dynamics of a glass, below the glass transition. In the case of a granular pile, the thermal agitation is negligible and one has to inject energy to overcome the energy barrier separating the jammed states. Reproduced with permission from [85]. Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Nevertheless, this description of a granular pile in terms of an athermal system hinders one of its major characteristics: namely its fragility [1, 7]. The sensitivity of a granular pile to minute external perturbations has first been pointed out in the case of frictionless hard spheres [8, 9, 10, 11]. A heap of such spheres presenting a slight polydispersity has been shown to be isostatic and thus very sensitive to external perturbations. In the case of frictional spheres, jamming and isostaticity no longer go hand in hand and the way the packing has been build plays a crucial role on its stability [12]. Nevertheless, the pile pictured as a contact network, can be decomposed in two subnetworks: a network gathering strong contacts (weak contacts resp.) involving grains which carry a force larger (lower resp.) than the average force in the packing [13]. The key contribution of both this “strong contact” network and the surface roughness of the grains to the fragility of the pile is very well illustrated by the Scalar Arching Model (SAM) [14]. This model, inspired from the q-model developed by Liu, Coppersmith et al. [15, 16] only takes into account the weight of the grains and the solid friction between the grains following Coulomb’s law. The control parameter of the simulation is the friction coefficient between two grains, denoted $R_{c}$. In this model, looking at a static pile, Claudin and Bouchaud demonstrated that a relative variation of $R_{c}$ as small as $10^{-7}$ triggers large scale reorganizations inside the pile, named static avalanches (Fig. 2), emphasizing the fragility of a pile to minute perturbations, despite its athermal nature [14]. Of course, in a laboratory experiment, controlling and varying the friction coefficient between the grains during an experiment is impossible. Nonetheless, as suggested in [14], variations of temperature might perturb the pile at the scale of the surface roughness of the grains, having an effect equivalent to a change of the friction coefficient. Indeed, one can assess that a granular heap of size $L$ (typically a few centimeters) submitted to variations of temperature of amplitude $\Delta T$ experiences a dilation $\delta L=\kappa_{g}L\Delta T$, where $\kappa_{g}$ stands for the thermal expansion coefficient of the grains. Dilations corresponding to the surface roughness scale lead to $\Delta T\simeq 0.1^{\circ}$C for standard glass beads ($\kappa_{g}=10^{-6}$ K-1). Such an amplitude is easily accessible and for instance daily variations of temperature in the lab are already of roughly a few degrees. This is the topic of this brief review, where we focus on the effect of temperature on a granular assembly. The content of this review goes as follows: In section II we sum up the first experimental observations of uncontrolled temperature variations that have pushed for further experiments under controlled variations (section III). In section IV, we dwell on the use of cycles of temperature to induce the compaction of a granular pile in a delicate way, in particular we discuss the role of both the amplitude and the frequency of the imposed cycles. Section V deals with enlarged “granular” systems and extend the scope of the results presented in previous parts. Finally, section VI proposes some outlooks to this method of thermal cycling as a general method to probe granular systems. Figure 2: Force chains in a two-dimensional granular pile (200$\times$200) following the Scalar Arching Model. The white dots label the grains involved in a force chain in the initial force network, whereas the black dots label the grains involved in the force chains network after a relative change of $10^{-7}$ of the friction coefficient $R_{c}$. One observes that the reorganization takes place in the whole pile, despite the perturbation taking place at the scale of the grain surface roughness. Reprinted figure with permission from [14]. Copyright (1997) by the American Physical Society. ## 2 From undesired variations of temperature … The effect of temperature variations over a granular pile has first been reported as a hindrance to perform reproducible measurements. Those experiments were not dedicated to probe the consequences of temperature variations on a granular assembly, and the role of temperature is simply assessed. Nonetheless, several issues were raised in this seminal work, and are highlighted here. ### 2.1 Sound in Sand The influence of temperature is first mentioned in the early 90’s in a work dealing with sound propagation in sand [17, 18, 19]. An acoustic wave is generated inside a granular pile and recorded a few centimeters away. C. Liu and S. Nagel observed that “a temperature change of only 0.04 K inside the pile, produced by the change of the ambient temperature, or by a local heater, could cause a factor of 3 reversible change in the measured vibration transmission” [17]. One can indeed assess the grain dilation $\delta d$ associated to such variations of temperature $\Delta T$ to be $\delta d=\kappa_{g}\Delta Td\simeq 2$ nm. In agreements with the reversibility of the effect, such an amplitude is on the one hand negligible compared to the typical surface roughness of the beads used in their experiments (probably 100 nm for glass beads [32]) and, on the other hand less than the typical deformation of the beads inside the heap (roughly 10 nm assuming a Hertz-like contact). However, at the time of these experiments, it was still unclear whether the temperature or the gradient of temperature were leading to such observations. Experiments performed by Clément and co-workers later confirmed the key role of the gradient, and brought to the fore the role of the container dilation on the reorganization process [20]. Figure 3: Apparent mass and temperature variations. The system (see text) consists in a vertical cylinder (radius 2.0 cm) filled with glass beads (typical diameter 3 mm). A drift in the temperature of 0.4∘C leads to several reorganizations inside the pile and thus to several variations of the apparent mass of the system. Reprinted from [20]. ### 2.2 Apparent mass fluctuations In 1997, E. Clément and co-workers reproduced Jansen experiments [20, 21], which consists in measuring the apparent mass of a granular pile confined in a tube. A piston at the tube bottom, implemented on an electronic scale, makes it possible to measure the apparent mass of the pile. Although the experiment is in good agreement with Jansen’s prediction [22], they noticed that their data presented fluctuations as high as 20 % in the saturated limit, where the measured mass becomes independent of the amount of beads poured inside the tube. Part of this fluctuations were attributed to temperature fluctuations and the authors emphasized that “the origin is mixed since it can be due to the dilation of the boundaries and the resulting action on the piston or it can be due to the dilation of the grains, themselves inducing spontaneous rearrangements of the force network” [20]. In particular, the contribution of the boundary dilation will be addressed in section IV. Figure 3 illustrates that even a drift in the temperature of 0.4∘C is sufficient to trigger rearrangements which lead to measurable mass fluctuations. However, no systematic measurements were performed on this Jansen configuration and it would still be of great interest to assess the effect of temperature variations on apparent mass of a pile [23]. Figure 4: Sketch of the experimental setup used to measure the effective thermal conductivity $\lambda_{g}$. The granular material consists in glass beads (typical diameter $300~{}\mu$m). The copper tube dimensions are the followings: inner diameter 1 cm; outer diameter 5 cm; length 15 cm. Reprinted from [26]. ## 3 … to controlled variations of temperature Figure 5: Thermal conductivity $\lambda_{g}$ vs. number of cycles $n$. The upper curve corresponds to a pile which has been gently tapped prior to the experiment. The lower curve corresponds to a loose pile. Note that the conductivity is larger for a larger density. Moreover, the conductivity of the loose material increases significantly when one imposes the thermal cycles (typical grain diameter 300 $\mu$m, $\Delta T=40^{\circ}$C). Reprinted from [26]. . Since then, temperature variations have been used to generate minute perturbations inside a granular medium. Local heaters placed at different positions inside a static pile have been shown to produce very different disturbances on sound propagation underlining the key role of force chains in the propagation process [18]. Moreover, small thermal perturbations have been used to generate large non-Gaussian conductance fluctuations in a 2D packing of metallic beads. Perturbations are induced by a 75 W light bulb standing a few centimeters away from the packing and experiments were performed with light turned on for packing prepared with light off, and vice versa. Gathered in bursts, such conductance fluctuations were interpreted as the signature of individual bead creep rather than collective vault reorganizations [24, 25]. In particular, the probability density function $P(\Delta t)$ where $\Delta t$ denotes the waiting time between two successive bursts of conductivity, follows a power law: $P(\Delta t)\sim\Delta t^{-(1+\alpha_{t})}$ The exponent for stainless steal beads is found to be $\alpha_{t}\simeq 0.6$ independent of both the strength of the perturbation and of the external stress. The value of $\alpha_{t}$ is only governed by the surface roughness of the beads which has been confirmed by AFM measurements of the Hurst exponent of the bead surface [24]. In agreement with the SAM [14] discussed in section I, the origin of the fluctuations of conductivity originate in local micro- contact rearrangements. More recently [26], temperature variations have been applied by means of a nickel wire (diameter $r_{w}$ =100 $\mu$m) connected to a power supply, and which crosses a granular medium partially filling a copper tube (Fig. 4). The tube is maintained at a constant temperature $T_{e}$ (precision $0.1^{\circ}$C). The imposed current $I$ and the resulting voltage $U$ are simultaneously measured in order to estimate the heating power $P=UI$ and the wire temperature $T_{w}$ which is deduced from the resistance $R_{w}=U/I$ (Fig. 4). This setup makes it possible to measure the effective thermal conductivity $\lambda_{g}$ of the granular material as $T_{w}-T_{e}\propto P/\lambda_{g}$. One obtains $\lambda_{g}=0.162$ W/m/K which is compatible with the value expected for a pile of glass beads ($\lambda_{glass}=1.4$ W/m/K) surrounded by air ($\lambda_{air}=0.025$ W/m/K) [27]. More interestingly, the authors also observe that for the same material $\lambda_{g}$ depends on the preparation : One measures $\lambda_{g}\simeq 0.162$ W/m/K if the system is tapped prior to the measurement and $\lambda_{g}\simeq 0.156$ W/m/K if not. It is then particularly interesting to consider the behavior of the sample when subjected to several temperature cycles. When the measurements are repeated several times, one observes that the thermal conductivity of the loose sample along with its packing fraction significantly increase with the number $n$ of imposed cycles. By contrast, the conductivity of the tapped sample only slightly fluctuates around a constant value (Fig. 5). Such results clearly emphasize that one can induce compaction by thermal cycling. This issue is addressed in the following section. Figure 6: Change in packing fraction after multiple thermal cycles (cycle temperatures: red, $\Delta T=107\pm 2^{\circ}$C; blue, $\Delta T=41\pm 1^{\circ}$C) for glass spheres in a plastic cylinder. Lines represent fits of the data by the sum of two exponentials introducing a short (resp. long) timescale associated to individual (resp. collective) rearrangements of the grains [34, 35]. Reprinted with permission from [28]. Copyright (2006) by the Nature Publishing Group. ## 4 Compaction induced by thermal cycling The compaction of a granular pile has been studied in two different configurations: (i) both the grains and the container are submitted to cycles of temperature [28, 29, 30], and (ii) the grains alone experience the cycles [31, 32]. In both cases, the packing fraction increases under thermal cycling. We first review the role of the amplitude and frequency of the cycles for both cases. Second, we gather some insights on the dynamics at the grain scale. ### 4.1 The role of the cycling amplitude and frequency Chen and co-workers [28, 30] examined the change in packing fraction for glass, polystyrene or high density polyethylene spheres (typical diameter 1 mm) contained in polymethylpentene plastic or borosilicate glass cylinders (diameter ranging from 14 to 102 mm) in response to thermal cycling. One thermal cycle is conducted by placing both the grains and the container into an oven until the thermal equilibrium is reached, before letting them relax at ambient temperature. The typical cycling period is about 10 hrs and has not been varied. They observed that the packing fraction increases, even after one cycle [28]. This effect is independent of the sample depth and width (14-102 mm), more efficient for higher cycling amplitudes (Fig. 6) and, contrary to their first guess, also independent of the relative coefficients of thermal expansion of the grains and the container [30], as confirmed by numerical simulations [33]. This indicates that the compaction mechanism is local as suggested by the observations discussed in section III and thus linked to the solid friction between the beads, and between the beads and the container. Figure 7: Height variation $h_{n}$ vs. number of cycles $n$. One observes first an exponential behavior at short time followed by a logarithmic creep at long time. The black curve corresponds to the test function $h_{n}^{t}\equiv\,h_{0}+h_{e}\,\exp{(-n/n_{c})}+h_{l}\,\ln(n)$. Inset - Oscillations of the column height associated with the temperature cycles : $A_{n}$ and $\delta_{n}$ are respectively defined to be the amplitude of the increase and the drift of $h_{n}$ at the cycle $n$ ($H=140$ cm, $2\pi/\omega=600$ s and $\Delta T=10.8^{\circ}$C.). Reprinted figure with permission from [29]. Copyright (2008) by the American Physical Society. Two other setups have been used to impose cycles of temperature: the first one consists of a vertical glass tube filled with spherical glass beads [26, 29]. The temperature cycles are imposed by means of a heating cable (40 W/m) directly taped on the outer surface of the tube wall. Here again both the container and the grains are thus submitted to the cycles. The free surface of the material is imaged from the side with a video camera which makes it possible to measure accurately the column height $H$ from the images and to perform a time resolved study of its dynamic. In such a setup one can control both the amplitude $\Delta T$ of the cycles, and the penetration length $l_{p}\equiv\sqrt{2\lambda/(\rho\,C\omega)}$ through the frequency $\omega/2\pi$ of the cycles ($\lambda$ and $C$ respectively denote the thermal conductivity and heat capacity of a typical glass-grains pile). Prior to each experiment, the granular column is prepared in a low-density state thanks to a dry-nitrogen upward flow. The top of the column is then higher than the field imaged by the camera and one sets the amplitude of the cycles, $\Delta T$, to the largest accessible value, $\Delta T=27.1^{\circ}$ C. The column flows under thermal cycling, and the preparation of the sample ends when the top of the column enters the observation field. At this point, the granular column is “quenched”: the amplitude of the cycles is set to the chosen value $\Delta T$ lying between $0$ and $27.1^{\circ}$C, which defines the origin of time $t=0$. The granular column is subsequently submitted to at least $1000$ cycles. Figure 8: Amplitude $A_{n}$ (defined in Fig. 7) versus the number of cycles n. The data are successfully accounted for by $A_{n}=\Delta T\,[a_{0}+b_{0}\ln(n)]$ with $a_{0}=1.0$ $\mu$m$\cdot$K-1 and $a_{0}=0.13$ $\mu$m$\cdot$K-1 (initial column height: 140 cm, cycling period: $2\pi/\omega=600$ s and, from top to bottom, $\Delta T$=27.1, 16.2 and 10.8∘C). Such an increase in the dilation amplitude with the cycles of temperature is a signature of the aging the granular pile experiences under thermal cycling [26]. Reprinted figure with permission from [29]. Copyright (2008) by the American Physical Society. Role of cycling amplitude\- Here, the cycling period (10 minutes) is chosen to ensure that the associated thermal penetration length $l_{p}\simeq 6$ mm is about the tube radius so that all the grains experience the dilation process. Figure 7 reports the variation of the column height defined as $h_{n}\equiv H(2\pi n/\omega)-H(0)$, where $n$ denotes the number of imposed cycles. For a large $\Delta T$ (typically more than 3∘C), the column systematically compacts (Fig. 7) at each cycle [Fig. 7, (inset)]. Following Barker and Mehta [34, 35] like Chen and co-workers [28, 30], one can try to fit the height decay by the sum of two exponentials: the result is not satisfying and the compaction dynamics is better accounted for by the following test function: $h_{n}^{t}\equiv\,h_{0}+h_{e}\,\exp{(-n/n_{c})}+h_{l}\,\ln(n)$ (Fig. 7) [32]. First, this response to the thermal quenching is strikingly similar to the one the system exhibits to vanishing step strain perturbations [36]. Second, the long time logarithmic behavior of the column height leads to an inverse logarithmic evolution for the packing fraction $\phi_{n}$ which is strongly reminiscent of the way the packing fraction of the granular pile submitted to tapping evolves in the absence of any convection [2]. Another remarkable feature of this time resolved study is that one can also observe that $A_{n}$, defined as the amplitude of the increase of $h_{n}$ at the $n$-th cycle (Fig. 7), is proportional to the cycling amplitude $\Delta T$ and increases logarithmically with $n$ (Fig. 8). The more the column compacts, the higher is the dilation amplitude of the total height of the column at each cycle. This is a signature of the aging of the pile under thermal cycling, like the increase of the thermal conductivity (section III). In the limit of small cycling amplitude $\Delta T$ (here below $\Delta T_{c}=3^{\circ}$C), one observes that the column is not flowing regularly anymore, but evolves by successive collapses (typical amplitude of a tenth of grain diameter) separated by rest periods (randomly distributed) [Fig. 10 (top)]. Such a different compaction process from the one observed for high cycling amplitude has been linked to the surface roughness of the glass beads. Indeed, for $\Delta T<\Delta T_{c}$ the dilation of a grain is smaller than the surface roughness, and thus, the beads behave as smooth particles whose dilation induces only localized rearrangements. Thus one has to wait several cycles of temperature before observing a large scale collapse of the column level, as a cumulative effect of several local reorganizations. On the contrary, for $\Delta T>\Delta T_{c}$ the dilation of a grain is larger than the surface roughness, and thus the beads behave as rough particles. One cycle of temperature is more likely to generate a large scale reorganization [29]. Role of cycling frequency\- The influence of the frequency is discussed in [32], but still remains to be fully explored. On the one hand, for frequencies shorter than the one discussed above, the penetration length is larger than the tube radius. The compaction process is more efficient, and the dilation amplitude $A_{n}$ is constant, independent of the age of the system. On the other hand, for larger frequencies, the penetration length is shorter than the tube radius. The column is observed to flow continuously while small amplitude settlings may occur (Fig. 9). The interpretation proposed in [32] is the following: the penetration length is shorter than the tube radius, the grains in the center of the column are not submitted to the cycles of temperature and thus behave as a solid body that experiences some stick-slip motion due to the periodic dilations of the grains close from the walls of the container. Indeed, the column dynamics over a few cycles [Fig. 9, (inset)] displays a very similar behavior to a tilted monolayer of grains on an inclined plane (see Fig. 3 in [37]). In this case, the flow results from a competition between a gravitational flow and the formation of arches [37]. However, a full study on the role of the frequency, and in particular on the influence of harmonics in the shape of the cycles (squares, tooth-like signals, etc.) on the compaction dynamics still remains to be done. Figure 9: Evolution of the column height $h_{n}$ versus the number $n$ of cycles of temperature. Inset: zoom on the evolution of $h_{n}$ over 24 cycles of temperature: the column settles by jumps separated by periods of continuous flowing ($H=140$ cm, $T=2\pi/\omega=150$ s et $\Delta T=9.5^{\circ}$C). Reprinted from [32]. The second setup that has been used to impose temperature cycles is very similar to the one previously discussed in figure 4. A thin wire crosses a granular column in its center, while this time the container is a thin glass tube [31]. For large enough frequencies, the penetration length is shorter than the tube radius which makes it possible to cycle the grains close to the wire and let the container at rest. It is here relevant to compare the way the dilation of the container modifies the compaction process in the limit of low amplitude cycles. In the case where both the grains and the container are submitted to variations of temperature, the compaction process is linear in the number $n$ of applied cycles [Fig. 10 (top)], whereas, in the case where the grains alone are submitted to cycles of temperature and the container is fixed, the compaction process goes logarithmically with $n$ [Fig. 10 (bottom)]. Both experiments were conducted under the same cycling frequency, and for comparable cycling amplitudes (resp. $\Delta T$ = 2.8 and 1∘C). Those results clearly indicate that the dilation of the container plays a key role in the global compaction dynamics of the granular column. However, up to now, the full contribution of the container dilation has not been characterized. The answer should be easily obtained with the set-up presented in figure 4. Figure 10: Height variation $h_{n}$ of the column versus the number of cycles of temperature. (Top) Curve obtained in the case where temperature cycles were applied to both the container and the grains. On average, $h_{n}$ decreases linearly with the number of cycles and the column settles by jumps separated by rest periods ($H=140$ cm, $2\pi/\omega=600$ s, and $\Delta T=$2.8∘C). Reprinted figure with permission from [29]. Copyright (2008) by the American Physical Society. (Bottom) Curve obtained in the case where cycles of temperature are applied to the grains alone by means of a hot wire crossing the pile (see Fig. 4). The column also settles by jumps, but the height variations goes as the logarithm of the number of cycles. Reprinted from [31]. ### 4.2 Dynamics at the grain scale under thermal cycling One of the key features of the motion induced by thermal cycling in a granular pile is that a grain stays in contact with its neighbors and that the global motion is really slow (hours and days typically). It makes it possible to use experimental techniques developed to study creep motion, like dynamic light scattering (DLS) [38], successive snapshots [29] or 3D scanning [39] to follow the individual grain trajectories. Recently, L. Djaoui and J. Crassous used a DLS setup to follow the evolution of a granular pile under thermal cycling [40]. This method was successful at extracting both linear and non-affine displacement of the grains [41], as recently done under mechanical shear [42], thus confirming that controlled temperature variations are a delicate way to generate small displacements. Slotterback and co-workers have been following the dynamics induced by thermal cycling at the grain scale [39]. In their experiment, the column of grains is immersed in an index-matching oil containing a laser dye which makes it possible to produce 3-D images of the system at the end of each cycle by means of a laser sheet scanning method. Particle displacements in this jammed fluid correlate strongly with rearrangements of the Voronoi cells defining the local environment about the particles. This might be a proof for stringlike cooperative motion as already pointed out in quasi-2D air driven granular flows [43], and more generally in supercooled liquids [44, 45]. However, it is rather difficult to compare these results to those obtain for dry grains [28, 29]. First, because in the case of the immersed experiment, a weight is placed on top of the granular column to apply a controlled vertical force, whereas in the dry case the top column is free of any weight. Second, the presence of the interstitial fluid lubricates the contacts between the grains, and certainly leads to a more homogeneous propagation of temperature front through the pile [46]. Indeed, the thermal conductivity of oil is roughly 10 times higher than the thermal conductivity of air which reduces the role played by force chains in the heat conduction. Those three effects speed up the compaction process compared to the dry case: the steady state packing fraction is obtained after only 10 cycles in the immersed system (Fig. 11). Figure 11: Packing fraction vs the number of cycles of temperature for various temperature differentials. The system consisting of glass beads immersed in an index matching oil inside a cylinder, is subjected to thermal cycling via a water bath. A weight is placed on top of the bead packing to apply a controlled vertical force. Both this weight and the interstitial fluid might explain the rapid compaction compared to what is observed in Fig. 7. Reprinted figure with permission from [39]. Copyright (2008) by the American Physical Society. ### 4.3 A comparison with the tapping experiments Let us here recall that tapping experiments consist in imposing vertical vibrations to a container full of grains. In practical terms, a low frequency signal (usually $10<f=\omega/2\pi<100$ Hz) [47] is used to generate a “tap” of amplitude $a$, two successive taps being separated by a duration $\Delta t$ long enough to be considered as independent (roughly $\Delta t\simeq 1$ s). The key control parameter is the reduced acceleration $\Gamma$ defined as $\Gamma\equiv a\omega^{2}/g$ [48] and one can distinguish between three different regimes. For low values of the reduced acceleration ($0\leq\Gamma\leq\Gamma^{}\simeq 1.2$), where $\Gamma^{}$ denotes the critical acceleration at which the grains lose contact with the bottom of the container, the compaction process is extremely slow [49]. This regime is very similar to that observed for thermal cycling: the geometry of the pile is essentially frozen and the force network can still evolve by slowly depleting the most fragile contacts [50, 51]. Besides, under such small vibrations, the packing is aging: the behavior of the pile is first dominated by grain motion and then by the contact force variations at larger timescales [51]. However, in this range of acceleration, no phenomenological law was proposed to describe the evolution of the packing fraction and/or the column height that could be compared to the results obtained under thermal cycling. For intermediate values of the acceleration ($\Gamma^{}\simeq 1.2\leq\Gamma\leq\Gamma_{c}\simeq 2$), the compaction process is more efficient and takes place over timescales accessible in the lab [2, 49]. At each tap, the packing loses contact with the bottom of the container, and crushes back which generates a shockwave responsible for the compaction of the pile. A convection roll might take place inside the container - roughly, if the ratio of the container size to the typical bead diameter is large enough, a signature of which can be found on the free surface of the pile that presents a slope [52]. Such convection rolls are not observed in a pile submitted to cycles of temperature as confirmed by the results observed under tapping for higher range of acceleration. Indeed, for larger values of the acceleration ($\Gamma>\Gamma_{c}\simeq 2$), the compaction takes place homogeneously in the whole shaken sample [2, 53], but the dynamics strongly depends on the presence or the absence of convection rolls inside the pile [4, 53, 54]. Without any convection roll, the packing fraction $\phi_{n}$ evolves in a similar way to what was found for thermal cycling in [29]: $\phi_{\infty}-\phi_{n}\propto\frac{1}{\ln(n)}$ (1) where $\phi_{\infty}$ is the steady state packing fraction obtained at long time. This phenomenological expression, first proposed in [2], has been justified theoretically using various approaches: analogy to a parking process [55, 56], “tetris-like” models [57, 58], excluded volume approach [59] and void diffusion models [60, 61]. A common feature of these models is the geometrical frustration: the denser the packing, the harder it becomes to insert grains from the top. Such a frustration process seems to be also at stake in the compaction induced by thermal cycling, which still remains to be properly put into equations. Still, under tapping, if convection rolls develop inside the pile, the compaction dynamics follows a Kohlraush-Williams-Watts (KWW) law: $\phi_{\infty}-\phi_{n}\propto\exp\left[-(n/n_{f})^{\beta}\right]$ (2) where $n_{f}$ and $\beta$ are two parameters which depend on the acceleration $\Gamma$ [49, 53, 62]. Such a law pops up naturally if one considers the global dynamics at stake as a superposition of several processes, each with a proper and well defined characteristic time. Indeed, one can see here the compaction process as the sum of the reorganization of several groups of beads, each with different sizes and different relaxation times. One can also emphasize that the global dynamics is not controlled by the biggest (and thus slowest) groups of beads, but by the fastest individual beads. These are the grains that can quickly relax individually and jump over large distances (roughly their radius), and so during a single tap, that control the compaction dynamics [63]. This result indicates once again that there are certainly no convection rolls inside a granular pile submitted to cycles of temperature as neither the KWW law nor rare jumps of individual beads are observed experimentally in this case [64]. Figure 12: (Top) Scheme of the capillary containing the onion gel, the location of the field of view and the orientation of the parallel and perpendicular axis. (Bottom) Portion of a typical image of the sample taken by light microscopy between cross polarizers; The arrow points to a large onion. Reprinted figure with permission from [66]. Copyright (2009) by the American Physical Society. ## 5 Enlarged “granular” systems In this section, we tackle two other systems different from a simple dry granular assembly and for which temperature cycles might play a crucial role in the observed dynamics. ### 5.1 Thermally driven aging in a polydisperse vesicle assembly The closest case from what we have been presenting in this article consists in a dense packing of polydisperse multilamellar vesicles, or “onions”, submitted to (unavoidable) temperature fluctuations [65, 66]. Such a system, a water- based mixture of surfactants and block-copolymer, is fluid below 8∘C but forms a phase of jammed vesicles at ambient temperature (here 23.4∘C) which is known to experience aging [67, 68]. The system is loaded in a glass capillary which is flamed sealed and placed under a microscope. Images are taken every 15 sec during 24 h (Fig. 12). The temperature is controlled within 0.09∘C. Mazoyer and co-workers observed that the unavoidable temperature fluctuations induce local mechanical shears in the whole sample due to local thermal expansion and contraction, which is a scenario very similar to the one proposed in [29] for dry grains. Here, each image [Fig. 12 (Bottom)] is divided into small regions of interest (ROI) and the authors look at the translation motion $\Delta R(t_{w},\tau)$ of each ROI for pairs of images taken at two different times ($t_{w}$ and $t_{w}+\tau$). Both the global parallel displacement $\langle\Delta R_{//}\rangle(t_{w})$ [Fig. 13 (b)] and the relative parallel displacement $[\langle\Delta r_{//}^{2}\rangle]^{1/2}(t_{w})$ [Fig. 13 (a)] present strong correlations with the temperature fluctuations $\Delta T$ [Fig. 13 (c)] [65]. Furthermore, looking at individual trajectories of ROIs, the authors identify two kinds of dynamical events: reversible and irreversible rearrangements. The first class is due to the contraction-elongation of the sample because of temperature fluctuations, and correspond to a shear deformation along the long axis of the capillary. The second class of events that are irreversible occurs as the result of repeated shear cycles. The motion resulting from these ultraslow rearrangements is ballistic (the initial growth of $\Delta r_{//}$ is proportional to $\tau$), with a velocity that decreases exponentially as the sample ages. Such results could be easily checked in dry and immersed granular systems [29, 39] to see how far the analogy between a jammed vesicle assembly and a granular pile goes. Also, it raises the generality of such dynamics governed by temperature fluctuations. In the case of dilute colloidal systems [69], this scenario does not hold, as no correlation could be observed between temperature fluctuations and rearrangements. Indeed, one expects temperature fluctuations to play a key role in divided and athermal systems being rather concentrated or solid-like. It might therefore be relevant to find out other systems presenting temperature-induced strain fluctuations, in order to find how general the properties of these ultraslow rearrangements are. Figure 13: Age dependence of (a) the square root relative parallel displacement, (b) the global parallel displacement and (c) the temperature variation $\Delta T$ over a lag time $\tau$. Reprinted figure with permission from [65]. Copyright (2006) by the American Physical Society. ### 5.2 Stone heave phenomena Cyclic freezing and thawing of soils can cause stones and particles embedded in those soils to move and relocate mainly depending on the initial void ratio (defined as the fraction of sample filled by empty space). The stones move vertically upward (Fig. 14) in dense soils (low void ratio) and vertically downward in loose soils (high void ratio). A full interpretation of this phenomenon is still lacking, but in the case of stone heave, the mechanism invoked lays on the propagation of the frost front in the soil: the freezing from the top leads to the growth of the ice beneath the stone, due to its higher thermal conductivity than that of the surrounding soil and in turn, causes the stone heave. A review of this topic can be found in [71]. A remarkable effect is that the void ratio also evolves toward a critical value (in the range of 0.29-0.34) under the successive freezing and thawing cycles [71]. Thus, a dense (loose resp.) soil will become looser (denser resp.) under cycles of freezing and thawing. This is exactly what experiences a granular assembly submitted to a mechanical shear: its packing fraction will tend to a critical value [72]. Here again, as for onions and dry granular assemblies, the temperature-induced shear controls the dynamics. Such an analogy suggests that one could shed some new lights on the stone heave phenomenon by looking at the way a large intruder (simulating the stone) in a granular pile (simulating the soil) behaves under thermal cycling (which produces a shear [29]). Such a work has been tackled by Chen and co-workers in [30]. They observed that, under thermal cycles, the intruder (aluminum or brass) either does not move, or sinks inside the pile (polystyrene beads in a borosilicate glass container) if the intruder density overcomes a certain threshold. The pressure exerted by the intruder on the pile seems to be the relevant parameter, despite the ratio of the container size to the grains diameter which might play a crucial role on the force network inside the pile, has not been varied. Also, no experiments have been performed varying the initial packing fraction or the initial position of the intruder in the granular packing to mimic the observations performed on stone heave [71, 73]. Still, such a simple setup is certainly relevant to extract the key ingredients of the stone heave phenomena, and deserves further use. A comparison of the results with those obtained for vibrated grains (Brazil nut effect [3, 74]) would also be interesting. Figure 14: Raised stone in a paved parking lot in Luleå, Sweden. Reprinted from [70], Copyright (2000), with permission from Elsevier. ## 6 Summary, open questions and outlooks ### 6.1 Summary Temperature variations, even of low amplitude, induce the reorganization of a granular assembly and its slow compaction, and lead to the aging of the system, a signature of which can be found in the slowing down of the dynamics and in the evolution of the key properties of the system (increase of the effective thermal expansion and thermal conduction coefficients, etc). In this sense, temperature variations induce aging in a pile the same way moisture [75, 76], constant applied stress [77] and chemical reaction [78] do. However, the mechanism at stake lays on a pinning-depinning transition at the grain scale generated by the shear-induced successive contractions and dilations cycles. Temperature variations are thus a local and very delicate way to perturb and induce displacement in a granular assembly. ### 6.2 Open questions and outlooks First, the role of several parameters still remains to be assessed: the mean temperature $T$ around which the variations of amplitude $\Delta T$ are imposed might play a role on the reorganization process inside the pile. Because of the presence of the container and the fact that the granular pile is a deformable medium, the mean temperature impacts on the initial stress distribution and on the space available for the grains. Its role fully remains to be investigated. Also, the parameters relevant in the case of vibrated grains are to be studied here. Namely, the shape of the grains [79, 80, 81, 82], their surface roughness [83, 84], their polydispersity [3], etc. But also, the size ratio of the grains and the container [53]. Such a study might help to compare more accurately tapping and thermal cycling experiments. It could then be relevant to mix the two types of solicitation on a granular assembly to test the statistical framework proposed by Edwards [4, 85]. Indeed, recent experimental results lead to think that the steady state packing fraction reached by a shaken granular column (and function of $\Gamma$) is a genuine thermodynamic state, within this theoretical framework [86, 87]. It might thus be relevant to study the fluctuations of packing fraction [88] around its equilibrium value, induced by cycles of temperature. Also, moisture is another relevant parameter that would deserve an exhaustive experimental study. Indeed, the existence of a liquid bridge between two adjacent grains is expected to increase the effective thermal conductivity [46] and thus to reinforce the role of the force chains inside the pile. Second, the aging phenomenon observed under thermal cycling needs to be better characterized. In particular, how can we compare the aging effects observed in vibrated granular piles and described in [89] to the one described here ? Another way to put it would be to ask if we can loosen a sample by means of cycles of temperature. Third, more insights on the individual grain dynamics is needed in the dry case. Do the results of Slotterback and co-workers discussed in section IV for immersed grains still hold in the dry case ? The techniques already used for vibrated dry granular assembly like X-rays tomography [90] or $\gamma$-rays absorption [49] would certainly provide some answers to these questions. Also, numerical simulations either simply based on successive dilation/contraction of the grains to mimic the effect of temperature, or taking into account the heat conduction through the granular media [33, 91, 94] would spare time and unravel the key parameters in the dynamics at the grain scale. ###### Acknowledgements. I am indebted to my PhD advisor, J.-C. 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Lett. 91, 104302 (2003). * [93] J Dijksmann, M van Hecke, The role of tap duration for the steady-state density of vibrated granular media, Europhys. Lett. 88, 44001 (2009). * [94] W L Vargas, J J McCarthy, Stress effects on the conductivity of particulate beds, Chem. Eng. Sci. 57, 3119 (2002).	arxiv-papers	2010-11-30T11:41:03	2024-09-04T02:49:15.313367	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Thibaut Divoux", "submitter": "Luis Ariel Pugnaloni", "url": "https://arxiv.org/abs/1011.6516" }
1011.6552	# Patterns in the sine map bifurcation diagram C.C. Lalescu clalescu@ulb.ac.be Statistical and Plasma Physics, Université Libre de Bruxelles, Campus Plaine, CP 231, B-1050 Brussels, Belgium Faculty of Physics, University of Craiova, A.I. Cuza 13, 200585 Craiova, Romania Chaos, Sine map, Bifurcation diagram ## I Introduction A onedimensional map $f$ is defined here as any $f:[a,b]\rightarrow[a,b]$. The logistic map $\phi_{r}$ $\phi_{r}:[0,1]\rightarrow[0,1],\phi_{r}(x)=rx(1-x)$ (1) is well known for it’s chaotic properties. As $r$ is varied through different intervals, the logistic map goes from having a fixed point to having stable cycles of different orders $n$ (there is at least an $x$ so that $\phi_{r}^{n}(x)=x$, where the exponent refers to composition). This type of behaviour is generic for a large number of chaotic systems, and it is best viewed in a bifurcation diagram, see figure 1. The initial fixed point degenerates into a cycle with period 2, and then gradually the period of the stable cycle keeps doubling, up to a point where the map becomes fully chaotic. A reference paper discussing the universality of the period doubling cascade is Feigenbaum_1978 . One obvious feature of this bifurcation diagram is that patterns can be observed in it — discontinuities in the density of points. Figure 1: Bifurcation diagram for the logistic map ## II Sine map Consider the map $\xi_{r}:\mathbb{R}\rightarrow\mathbb{R},\xi_{r}(x)=r\sin(x)$ (2) This map generates a bifurcation diagram that is symmetric with respect to both axis of the $(r,x)$ plane. Figure 2: Bifurcation diagram for the sine map, for $r\in[-2\pi,2\pi]$. A twodimensional map can be defined: $\psi:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2},\psi(x,y)=(x,x\sin(y))$ (3) The bifurcation diagram (viewed as a subset of $\mathbb{R}^{2}$) is simply the largest subset of $\mathbb{R}^{2}$ that is invariant to $\psi$: $\mathcal{B}=\lim_{n\rightarrow\infty}\psi^{n}(\mathbb{R}^{2})$ (4) ## III Patterns Figure 3: The curve pairs $C^{(n)}$ ($n=1,2,3,4$) superposed over the bifurcation diagram. It can be seen directly in figure 2 that the following two functions appear as patterns: $\pm r\sin(r),\textrm{ with the graphs given by }\psi(\pm r,r)$ (5) More generally, the bifurcation diagrams suggest that the array of curve pairs $C^{(n)}=\left\\{\psi^{n}(\pm x,x)\|x\in\mathbb{R}\right\\}$ (6) is in fact the succession of all patterns. The convention $\psi^{0}(x,y)\equiv(x,y)$ is adopted here, so $C^{(0)}=\\{(x,x),(-x,x)\|x\in\mathbb{R}\\}$. Figure 3 further suggests that $C^{(n)}\cap C^{(m)}\subset\mathcal{B}$, for all $n\neq m\in\mathbb{N}$. ## IV Exact results Let $(a,b)\in C^{(n)}\cap C^{(m)}$. It is convenient to introduce the notation $\psi^{n}_{\pm}(x,y)\equiv\psi^{n}(\pm x,y)$. This means that $(a,b)\in\\{\psi^{n}_{\pm}(a,a),\psi^{m}_{\pm}(a,a)\\}$ (7) Consider the case $(a,b)=\psi^{n}_{+}(a,a)=\psi^{m}_{+}(a,a)$. Assume $m>n$, and $m-n=p$. This means that $\psi^{n}_{+}(a,a)=\psi^{p}_{+}(\psi^{n}_{+}(a,a))$ (8) so, by definition, $\psi^{n}_{+}(a,a)\equiv(a,b)$ is part of a cycle with period $p$, which means it is in $\mathcal{B}$. The alternative is that $\psi^{n}_{+}(a,a)=\psi^{m}_{-}(a,a)$. This means that $\displaystyle\psi^{n}_{+}(a,a)$ $\displaystyle=\psi^{p}_{+}(\psi^{n}_{-}(a,a))$ (9) $\displaystyle\psi^{p}_{-}(\psi^{n}_{+}(a,a))$ $\displaystyle=\psi^{p}_{-}(\psi^{p}_{+}(\psi^{n}_{-}(a,a)))$ (10) $\displaystyle\psi^{p}_{+}(\psi^{n}_{-}(a,a))$ $\displaystyle=\psi^{p}_{+}(\psi^{p}_{+}(\psi^{n}_{+}(a,a)))$ (11) $\displaystyle\psi^{n}_{+}(a,a)$ $\displaystyle=\psi^{2p}_{+}(\psi^{n}_{+}(a,a))$ (12) so, again by definition, $(a,b)$ is part of a cycle with period $2p$, so $(a,b)\in\mathcal{B}$. Concerning $\lim_{n\rightarrow\infty}C^{(n)}$, it is sufficient to remark that $C^{(n)}=\psi^{n}(C^{(0)}$, so the limit $C^{(\infty)}$ is a subset of $\mathcal{B}$. ## V Generalization Consider a general mapping $\theta:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2},\theta(x,y)=(x,xf(y))$ (13) For $f:\mathbb{R}\rightarrow[-1,1]$ (with $f(\mathbb{R})=[-1,1]$), the invariant set of $\theta$ is well defined as $\mathcal{D}\equiv\lim_{n\rightarrow\infty}\theta^{n}\left(\mathbb{R}^{2}\right)$ (14) and the reasoning from the previous section can be applied. Call $\theta_{\pm}(x,y)\equiv\theta(\pm x,y)$ and $D^{(n)}\equiv\left\\{\theta^{n}_{\pm}(x,x)\big{\|}x\in\mathbb{R}\right\\}$ (15) Then for any $n\neq m\in\mathbb{N}$, $D^{(n)}\cap D^{(m)}\subset\mathcal{D}$ and the limit $D^{(\infty)}$ is a subset of $\mathcal{D}$. ## VI Notes The author would like to thank Gy. Steinbrecher and D. Constantinescu from the University of Craiova for their valuable comments. The author is unaware if the ideas presented are new; the webpage Evgeny_Demidov does discuss a family of curves that appears to have the properties mentioned in this work. The figures in this article were generated with a simple Python script111http://chichi.lalescu.ro/files/bc.py. ## References * (1) Mitchell J. Feigenbaum. Quantitative universality for a class of nonlinear transformations. Journal of Statistical Physics, 19:25–52, 1978. 10.1007/BF01020332. * (2) Evgeny Demidov. http://www.ibiblio.org/e-notes/Chaos/bifurcat.htm	arxiv-papers	2010-11-30T14:13:56	2024-09-04T02:49:15.323959	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Cristian Constantin Lalescu", "submitter": "Cristian Constantin Lalescu", "url": "https://arxiv.org/abs/1011.6552" }
1011.6590	# Extending ArXiv.org to Achieve Open Peer Review and Publishing Axel Boldt Department of Mathematics Metropolitan State University Saint Paul, Minnesota, U.S.A. Axel.Boldt@metrostate.edu (Date: November 18, 2010) ###### Abstract. Today’s peer review process for scientific articles is unnecessarily opaque and offers few incentives to referees. Likewise, the publishing process is unnecessarily inefficient and its results are only rarely made freely available to the public. In this article we outline a comparatively simple extension of arXiv.org, an online preprint archive widely used in the mathematical and physical sciences, that addresses both of these problems. Under the proposal, editors invite referees to write public and signed reviews to be attached to the posted preprints, and then elevate selected articles to “published” status. ## 1\. The status quo In the system of peer review that is currently used in the sciences, an editor invites one or more referees to review an article submitted to a scientific journal. Based on the referees’ recommendations, the editor will accept the article, demand modifications, or reject it. Referee reports are generally made available to the article’s author in anonymized form only and are not otherwise published. (Some journals also anonymize the article to be refereed, even though ascertaining the true author of a submission is usually a simple matter of using an internet search engine.) The system as described is completely opaque to outside observers. Neither the quality and timeliness of reviews, nor the standards of a journal’s editors, nor the extent of modifications made after initial review, nor the number of times an article has been rejected by other journals are publicly available. Other than professional integrity, referees have little legitimate incentive to produce timely, fair and high-quality reviews. Since the reviews are not published, referees are not accountable for their work and cannot use it to bolster a case for professional advancement or to improve their general standing in the academic community. Probably the biggest (and most problematic) incentive for referees is the accumulation of editor goodwill, to be expended during future article submissions. It is also conceivable that some referees reject articles whose authors they dislike or whose approach or results interfere with their own research agenda. Finally, editors may circumvent the peer review process altogether in order to promote their own or their associates’ work. Several case reports of dysfunction and breakdown of the peer review process in the mathematical and physical sciences have recently appeared in the literature. [Baez 06, Schiermeier 08, Trabino 09]. Authors, editors and referees are not paid for their work in this publication process. Nevertheless, publishers often charge exorbitant amounts for the resulting product, journals which have typically ended up being hidden away in university libraries, inaccessible to the public who funded the research in the first place. Independent workers as well as researchers in poor countries thus have often been cut out of the research loop entirely. The need for a system of open electronic publishing of scientific articles has long been recognized (see e.g. [Odlyzko 95]). Several electronic journals have now been created. Some of these charge readers for access, others are free to read but charge authors for publication, and still others are free for all parties involved. Perhaps the biggest success of the Open Access movement was a 2007 U.S. law requiring all NIH-supported research to be submitted to an openly accessible archive one year after publication. [Weiss 07] Internet-based alternatives to the prevalent peer review and publishing process have been discussed in [Harnard 00] and [Nielsen 08]. A trial in open peer review at the journal _Nature_ in 2006 generated widespread debate of the concept [Nature 06-1]; the final report concluded that, while the general concept was received enthusiastically, participation in and satisfaction with their particular model of open commentary were disappointing. [Nature 06-2] ## 2\. ArXiv.org The website arXiv.org (formerly xxx.lanl.gov) is an electronic archive of freely accessible research preprints. [Ginsparg 97] It was started by physicist Paul Ginsparg in August 1991 and has since become an indispensable tool for researchers in physics, mathematics and, increasingly, computer science and quantitative biology. Authors submit their articles to the archive prior to peer review and official publication by a scientific journal; the preprints are posted on the website in perpetuity after superficial moderator review. To participate, authors need an affiliation with a recognized academic institution or an endorsement by an established author. Interested parties can sign up for regular e-mail announcements containing the abstracts of new preprints in their chosen fields. Once a manuscript has been peer reviewed and accepted for publication, authors should ideally post an updated version to the archive. Not all authors remember to do this, and some journals explicitly prohibit the practice, claiming a copyright on the final result of peer review.111See for instance Elsevier’s policy on electronic preprints at http://www.elsevier.com/wps/find/authorshome.authors/preprints (accessed 26 December 2008) Consequently, arXiv.org in its present incarnation and similar preprint archives in other fields do not serve as authoritative Open Access repositories of peer reviewed research. ## 3\. A proposed solution To address the problems outlined in section 1, I propose the following extension to the arXiv.org preprint archive. A new class of users is created, the “editors”. Each editor works for an electronic journal. Authors, after having uploaded a preprint to the archive, may elect to submit their article for review and official publication in one of these electronic journals. An editor of that journal then decides whether the article is appropriate for the journal in terms of scope and quality. If it is not, this decision is publicly attached to the article and the process ends; if it is, the editor invites one or more referees to write public reviews, to be attached to the article. The article author may subsequently post a public rebuttal to the reviews. Based on the referee reports and rebuttals, the editor decides whether to accept, demand changes to, or reject the article. The original article, reviews, rebuttal and publication decision are published in perpetuity. If accepted, the author posts a final version of the article to arXiv.org; as a peer reviewed and officially published article, it is visibly set apart from mere preprints and added to the electronic journal’s collection of published articles. Rejected articles may be submitted to another electronic journal. Reviews should be signed with the referee’s full name and affiliation. This maximizes transparency and allows referees to receive academic credit for their work. However, some reviewers might be reluctant to participate in such a system, for instance because they hesitate to openly reject the work of friends or influential researchers, or because they do not want to call attention to their ignorance of some of the issues discussed in the reviewed article. Thus it is probably necessary to offer referees the option to publish their reviews under a pseudonym. Over time, such a pseudonym might naturally develop a reputation as a solid reviewer, completely divorced from the writer’s real-world identity. Using a straightforward cryptographic scheme, a referee could prove to selected others that he or she owns a certain pseudonym; in this way even pseudonymous referees could receive academic credit for their work at the time of tenure or promotion decisions. Some electronic journals may wish to develop a process for attaching notes to published articles, for instance to point out prior work, mistakes or scientific misconduct discovered after publication. It will also be desirable to attach a moderated discussion forum to each article, as a natural gathering place of interested researchers. The quality of these forums would serve as a criterion to differentiate electronic journals from each other. The pseudonyms used for refereeing could also be used to sign forum contributions. One may hope that the proposed system will engender several desirable consequences. The act of refereeing will rise in prestige in accordance with its importance for the scientific process. The quality of referee reports will improve. Outside evaluations and comparisons of the standards and practices of different electronic journals will become possible. The process becomes completely transparent and its results are made freely available. ## References * [Baez 06] John Baez. _The Bogdanoff Affair._ 21 June 2006. http://math.ucr.edu/home/baez/bogdanov.html (accessed 26 December 2008) * [Ginsparg 97] Paul Ginsparg. First Steps towards Electronic Research Communication. In: _Gateways to Knowledge: The Role of Academic Libraries in Teaching, Learning, and Research_ by Lawrence Dowler. MIT Press, 1997 * [Harnard 00] Stevan Harnad. The Invisible Hand of Peer Review. _Exploit Interactive_ , issue 5, April 2000. http://eprints.ecs.soton.ac.uk/2862/1/nature2.html * [Nature 06-1] _Nature_ ’s peer review debate, http://www.nature.com/nature/peerreview/debate/index.html (accessed 14 August 2010) * [Nature 06-2] _Overview: Nature’s peer review trial_. December 2006. http://www.nature.com/nature/peerreview/debate/nature05535.html (accessed 14 August 2010) * [Nielsen 08] Michael Nielsen. _The Future of Science._ 17 July 2008. http://michaelnielsen.org/blog/the-future-of-science-2/ (accessed 14 August 2010) * [Odlyzko 95] Andrew M. Odlyzko. Tragic Loss or Good Riddance? The Impending Demise of Traditional Scholarly Journals. _Notices of the American Mathematical Society_ , vol 42, no 1, pp. 49 - 53. January 1995. http://www.ams.org/notices/199501/forum.pdf * [Schiermeier 08] Quirin Schiermeier. Self-publishing editor set to retire. _Nature_ 456 (2008), 432 * [Trabino 09] Rick Trabino. _How to Publish a Scientific Comment in 1 2 3 Easy Steps._ 18 August 2009. http://www.scribd.com/doc/18773744/How-to-Publish-a-Scientific-Comment-in-1-2-3-Easy-Steps(accessed 13 November 2010) * [Weiss 07] Rick Weiss. Measure Would Require Free Access To Results of NIH-Funded Research. _The Washington Post,_ 21 December 2007. http://www.washingtonpost.com/wp-dyn/content/article/2007/12/20/AR2007122002115.html	arxiv-papers	2010-11-23T14:23:07	2024-09-04T02:49:15.329128	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Axel Boldt", "submitter": "Axel Boldt", "url": "https://arxiv.org/abs/1011.6590" }
1011.6627	# Combining independent, arbitrarily weighted $P$-values: a new solution to an old problem using a novel expansion with controllable accuracy Gelio Alves National Institutes of Health, Bethesda, USA and Yi-Kuo Yu† National Institutes of Health, Bethesda, USA ###### Abstract Good’s formula and Fisher’s method are frequently used for combining independent $P$-values. Interestingly, the equivalent of Good’s formula already emerged in 1910 and mathematical expressions relevant to even more general situations have been repeatedly derived, albeit in different context. We provide here a novel derivation and show how the analytic formula obtained reduces to the two aforementioned ones as special cases. The main novelty of this paper, however, is the explicit treatment of nearly degenerate weights, which are known to cause numerical instabilities. We derive a controlled expansion, in powers of differences in inverse weights, that provides both accurate statistics and stable numerics. ###### keywords: combine $P$-values; nearly degenerate weights; exponential variable; gamma variable; gamma distribution; Erlang distribution 00footnotetext: ${\dagger}$Address for correspondence: National Center for Biotechnology Information, National Library of Medicine, National Institutes of Health, 8600 Rockville Pike, Bethesda, MD 20894, USA. E-mail: yyu@ncbi.nlm.nih.gov ## 1 Introduction The question of how to obtain an overall significance level for the results of independent runs of studies has been investigated since the 1930s (Tippett, 1931; Fisher, 1932; Pearson, 1933, 1938). In fact, forming a single statistical significance out of multiple independent tests has been an important subject of study in numerous area of scientific disciplines, including social psychology (Stouffer et al., 1949; Mosteller and Bush, 1954), medical research (Olkin, 1995), genetics (Loesgen et al., 2001), proteomics (Alves et al., 2008), genomics (Hess and Iyer, 2007), bioinformatics (Bailey and Gribskov, 1998; Yu et al., 2006) and others. Frequently used methods for combining independent $P$-values fall into numeric and analytic categories. This classification is not totally precise since method such as Fisher’s started out with the necessity of inverting the $\chi^{2}$ cumulative distribution function and thus seemed like a numerical approach (Pearson, 1938). The method mentioned in (Bahrucha-Reid, 1960), although not in the context of combining $P$-values, brought out an analytical expression for combined $P$-value using Fisher’s method, thus effectively brought Fisher’s method into analytic category. In the context of combining $P$-values, by mapping to a known result by Feller (Feller, 1966), Bailey and Gribskov (1998) also provided an analytic formula for Fisher’s combined $P$-value. Numerical approaches typically involve inverting cumulative distribution functions. For example, Stouffer’s z-methods (Stouffer et al., 1949), whether unweighted (Whitlock, 2005) or weighted (Liptak, 1958; Koziol and Tuckwell, 1994), require inverting the error function. Lancaster’s generalization (Lancaster, 1961; Koziol, 1996) of Fisher’s formalism also requires inverting gamma distribution function to incorporate unequal weighting for $P$-values combined. In this paper, we focus on analytic methods only. Two existing analytic approaches, Fisher’s (Fisher, 1932; Bahrucha-Reid, 1960; Bailey and Gribskov, 1998; Alves et al., 2008) and Good’s (Good, 1955; Likes, 1967) , are frequently employed. Fisher’s method combines $L$ independent tail-area probabilities democratically to form a single significance assignment while Good’s formula weights each tail-area probability differently to form a different single significance assignment. Since Good’s expression involves, in the denominator, pairwise differences between weights, he cautiously remarked that the expression may become ill-conditioned and thus the calculations should be done by holding more decimal places when weights of similar magnitudes exist. This statement has been paraphrased by many authors (Bhoj, 1992; Olkin and Saner, 2001; Hou, 2005). In addition to the cases considered by Fisher and Good, it is foreseeable that one may wish to categorize obtained independent $P$-values into different groups so that one would like to weight $P$-values within the same group democratically and weight different group differently. We will call this scenario the general case (GC). The GC naturally occurs since one may wish to categorize data obtained from the same type of experimental instruments into the same group, and data collected from different instrument types may justify the use of different weights. When there is only one instrument type, the GC reduces to Fisher’s consideration. When there exist no replicates within each instrument types, the GC coincides with the consideration of Good. In principle, the weighted version of the Stouffer’s method can also be used for this purpose. Since the main scope here is to pursue analytic results, we won’t delve into methods in numerical category. It is important, however, to point out that the mathematical problem of combining $P$-values is also related to other areas of research. For example, the equivalent of Good’s formula had already emerged in 1910 in the context of sequential radioactive decay (Bateman, 1910), while analytical expression for Fisher’s combined $P$-value emerged in 1960 as a special case of the former when all the decay constants are identical (Bahrucha-Reid, 1960). After Good’s formula (Good, 1955), the same distribution function was rederived by McGill and Gibbon (1965) and later on by Likes (1967). As for the GC, Fisher’s method included, the mathematical equivalents appear in different areas of studies mainly under the consideration of sum of exponential/gamma variables. The distribution function of linear combination of exponential/gamma variables are useful in various fields. When limited to exponential variables, it results in the Erlang distribution that is often encountered in queuing theory (Morse, 1958). It is also connected to the renewal theory (Cox, 1962), time series problem (MacNeill, 1974), and can be applied to model reliability (Jasiulewicz and Kordecki, 2003). The intimate connections between these seemingly different problems are not obvious at first glance. Consequently, it is not surprising that the distribution function has been rediscovered/rederived many times and that some information about it has not been widely circulated. Our literature searches show that the first explicit result (without further derivatives involved) for the distribution function was obtained by Mathai (1983). Subsequently, motivated by different contexts, Harrison (1990), Amari and Mirsa (1997), and Jasiulewicz and Kordecki (2003) all rederived the same distribution function. Employing a complex variable integral formulation, we are able to provide a different derivation of the distribution function and become the first to make connection to the GC of combining $P$-values. Since both Fisher’s and Good’s considerations arise as special limiting cases of the GC, we also illustrate that our cumulative probability distribution for GC indeed reduces to the appropriate limiting formulas upon taking appropriate parameters. The main novelty of this paper, however, is the explicit treatment of cases where nearly degenerate weights exist. These cases are known for numerical instabilities, which motivated many authors to pursue uncontrolled approximations (Solomon and Stephens, 1977; Gabler, 1987; Bhoj, 1992; Hou, 2005). We have derived a controlled expansion, in power of differences in inverse weights, that provides both accurate statistics and stable numerics. In the following section, we will first summarize Fisher’s and Good’s methods for combining $P$-values, followed by the mathematical definition of the GC. A section devoted to derivation of the probability distribution function and cumulative probability for the GC then follows. We then delve into the case of nearly degenerate weights and provides a formula with controllable accuracy for combining $P$-values. A few examples of using the main results are then provided. This paper concludes with future directions. A C++ program, which computes the combined $P$-values with equal numerical stability regardless of whether weights are (nearly) degenerate or not, is available upon request from the authors. ## 2 Summary of Fisher’s and Good’s methods for combining $P$-values Imagine that one wishes to combine $L$ independent $P$-values $p_{1},p_{2},\ldots,p_{L}$, each of which is drawn from an uniform, independent distribution over $(0,1]$. For later convenience, let us define $\displaystyle{\tau_{\scriptscriptstyle F}}$ $\displaystyle\equiv$ $\displaystyle p_{1}\cdot p_{2}\cdots p_{L}\;,$ (1) $\displaystyle{\tau_{\scriptscriptstyle G}}$ $\displaystyle\equiv$ $\displaystyle p_{1}^{w_{1}}\cdot p_{2}^{w_{2}}\cdots p_{L}^{w_{L}}\;.$ (2) To form a unified significance, Fisher and Good considered respectively the stochastic quantities $Q_{F}$ and $Q_{G}$, defined by $\displaystyle Q_{F}$ $\displaystyle\equiv$ $\displaystyle x_{1}\cdot x_{2}\cdots x_{L}\;,$ (3) $\displaystyle Q_{G}$ $\displaystyle\equiv$ $\displaystyle x_{1}^{w_{1}}\cdot x_{2}^{w_{2}}\cdots x_{L}^{w_{L}}\;,$ (4) where each $x_{i}$ represents a random variable drawn from an uniform, independent distribution over $(0,1]$. The following probabilities $\displaystyle{\rm Prob}(Q_{F}\leq{\tau_{\scriptscriptstyle F}})$ $\displaystyle=$ $\displaystyle{\tau_{\scriptscriptstyle F}}\sum_{l=0}^{L-1}{[\;\ln(1/{\tau_{\scriptscriptstyle F}})\,]^{\,l}\over l!}$ (5) $\displaystyle{\rm Prob}(Q_{G}\leq{\tau_{\scriptscriptstyle G}})$ $\displaystyle=$ $\displaystyle\sum_{l=1}^{L}\,\Lambda_{l}\;{\tau_{\scriptscriptstyle G}}^{1/w_{l}}$ (6) provide the unified statistical significances, corresponding respectively to Fisher’s and Good’s considerations, from combining $L$ independent $P$-values. In eq. (6), the prefactor $\Lambda_{l}$ is given by $\Lambda_{l}=\frac{w_{l}^{L-1}}{\prod_{k\neq l}(w_{l}-w_{k})}.$ (7) Apparently, $\Lambda_{l}$ is ill-defined when the weight $w_{l}$ coincides with or is numerically close to any other weights $w_{k}$. Although Fisher did not derive (5), from this point on, we shall refer to (5) as Fisher’s formula and (6) as the Good’s formula. ## 3 General case including Fisher’s and Good’s formulas Let us divide the $L$ $P$-values into $m$ groups with $1\leq m\leq L$. Within each group $k$, we weight the $n_{k}$ $P$-values within equally; while $P$-values in different groups are weighted differently. Therefore, when $m=L$ and $n_{k}=1$ $\forall~{}k$, we have the Good’s case and when $m=1$ and $n_{1}=L$, we reach Fisher case. We will therefore define the following quantities of interest $\displaystyle\tau$ $\displaystyle\equiv$ $\displaystyle\prod_{k=1}^{m}\left[\prod_{j=1}^{n_{k}}p_{k;j}\right]^{w_{k}}$ (8) $\displaystyle Q$ $\displaystyle\equiv$ $\displaystyle\prod_{k=1}^{m}\left[\prod_{j=1}^{n_{k}}x_{k;j}\right]^{w_{k}}$ (9) where each $x_{k;j}$ represents again a random variable drawn from an uniform, independent distribution over $(0,1]$. The quantity of interest ${\rm Prob}(Q\leq\tau)$, if obtained, should cover both results of Fisher’s and Good’s as the limiting cases. In the next section, we will derive an exact expression for ${\rm Prob}(Q\leq\tau)$ and describe how to recover the results of Fisher’s and Good’s. ## 4 Derivation of ${\rm Prob}(Q\leq\tau)$ Let $F(\tau)\equiv{\rm Prob}(Q\leq\tau)$, we then may write $F(\tau)=\int_{0}^{1}\cdots\int_{0}^{1}\theta\left(\tau-\prod_{k=1}^{m}\left[\prod_{i=1}^{n_{k}}x_{k;i}\right]^{w_{k}}\right)\prod_{k=1}^{m}\prod_{j=1}^{n_{k}}dx_{k;j}\;,$ (10) where the heaviside step function $\theta(x)$ takes value $1$ when $x>0$ and value $0$ when $x<0$. Upon taking a derivative with respect to $\tau$, we obtain $f(\tau)\equiv\frac{dF(\tau)}{d\tau}=\int_{0}^{1}\cdots\int_{0}^{1}\delta\left(\tau-\prod_{k=1}^{m}\left[\prod_{j=1}^{n_{k}}x_{k;j}\right]^{w_{k}}\right)\prod_{k=1}^{m}\prod_{j=1}^{n_{k}}dx_{k;j}\;,$ (11) where $\delta(x)$ represents Dirac’s delta function that takes zero value everywhere except at $x=0$ and that $\forall~{}a>0$, $\int_{-a}^{a}\delta(x)dx=1$. To proceed, let us make the following change of variables $\displaystyle\tau$ $\displaystyle=$ $\displaystyle e^{-t}$ $\displaystyle x_{k;j}$ $\displaystyle=$ $\displaystyle e^{-u_{k;j}}$ and remember that if $y_{0}$ is the only root of $f$ ($f(y_{0})=0$) $\delta(f(y))=\frac{\delta(y-y_{0})}{\|f^{\prime}(y_{0})\|}\;,$ we may then rewrite (11) as $f(\tau)=\int_{0}^{\infty}\cdots\int_{0}^{\infty}e^{t}e^{-\sum_{k,j}u_{k;j}}\delta\left(t-\sum_{k=1}^{m}w_{k}\left[\sum_{j=1}^{n_{k}}u_{k;j}\right]\right)\prod_{k=1}^{m}\prod_{j=1}^{n_{k}}du_{k;j}\;.$ (12) Note that the right hand side of (12), except for the additional factor $e^{t}$, is the probability density function of a weighted, linear sum of exponential variables. By introducing the integral representation of the $\delta$ function $\delta(t-c)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dq\;e^{-iq(t-c)}\;,$ we may re-express (12) as $\displaystyle f(\tau)$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{\infty}\frac{dq}{2\pi}\,e^{-it(q+i)}\prod_{k=1}^{m}\left[\int_{0}^{\infty}e^{-u}e^{iqw_{k}u}du\right]^{n_{k}}$ (13) $\displaystyle=$ $\displaystyle\int_{-\infty}^{\infty}\frac{dq}{2\pi}\,e^{-it(q+i)}\prod_{k=1}^{m}\left[\frac{1}{1-iqw_{k}}\right]^{n_{k}}$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{\infty}\frac{dq}{2\pi}\,e^{-it(q+i)}\prod_{l=1}^{m}\left(\frac{i}{w_{l}}\right)^{n_{l}}\prod_{k=1}^{m}\left[\frac{1}{q+ir_{k}}\right]^{n_{k}}$ $\displaystyle=$ $\displaystyle\left[\prod_{l=1}^{m}\,r_{l}^{n_{l}}\right]\left(i\right)^{\sum_{k=1}^{m}n_{k}}\int_{-\infty}^{\infty}\frac{dq}{2\pi}\,e^{-it(q+i)}\prod_{k=1}^{m}\left[\frac{1}{q+ir_{k}}\right]^{n_{k}}$ $\displaystyle\equiv$ $\displaystyle\left[\prod_{l=1}^{m}\,r_{l}^{n_{l}}\right]\tilde{f}(\tau;n_{1},n_{2},\ldots,n_{m})\;,$ (14) where $r_{k}\equiv 1/w_{k}$ is introduced for the ease of analytical manipulation and $\tilde{f}$ is introduced for later convenience. Since all $w_{k}>0$, implying that all $r_{k}>0$ and thus the poles of the integrand in (13) lie completely at the lower half of the $q$-plane. The integral of $q$ may be extended to enclose the lower half $q$-plane to result in $\displaystyle f(\tau)$ $\displaystyle=$ $\displaystyle e^{t}\,\left[\prod_{l=1}^{m}(i\,r_{l})^{n_{l}}\right]\;\left(\frac{-2\pi i}{2\pi}\right)\sum_{k=1}^{m}\,\frac{\left(\partial/\partial q\right)^{n_{k}-1}}{(n_{k}-1)!}\left[\,e^{-itq}\prod_{j=1,j\neq k}^{m}\left(\frac{1}{q+ir_{j}}\right)^{n_{j}}\right]_{q=-ir_{k}}$ (15) $\displaystyle=$ $\displaystyle e^{t}\,\left[\prod_{l=1}^{m}(i\,r_{l})^{n_{l}}\right]\sum_{k=1}^{m}\left\\{(-i)\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}g_{1},g_{2},\ldots,g_{m}=0\\\ \sum g_{i}=n_{k}-1\end{subarray}}\\!\\!\\!\\!\frac{(-1)^{n_{k}-1}(it)^{g_{k}}}{g_{k}!\;e^{r_{k}\,t}}\prod_{j=1,j\neq k}^{m}\frac{(n_{j}-1+g_{j})!}{(n_{j}-1)!g_{j}!}\left(\frac{-i}{r_{j}-r_{k}}\right)^{n_{j}+g_{j}}\right\\}$ $\displaystyle=$ $\displaystyle e^{t}\,\left[\prod_{l=1}^{m}r_{l}^{n_{l}}\right]\sum_{k=1}^{m}\left\\{\sum_{\begin{subarray}{c}g_{1},g_{2},\ldots,g_{m}=0\\\ \sum g_{i}=n_{k}-1\end{subarray}}\;\frac{(t)^{g_{k}}}{g_{k}!}e^{-r_{k}\,t}\prod_{j=1,j\neq k}^{m}\frac{(n_{j}-1+g_{j})!}{(n_{j}-1)!g_{j}!}\frac{(-1)^{g_{j}}}{(r_{j}-r_{k})^{n_{j}+g_{j}}}\right\\}\;.$ Comparing eq. (15) with eq. (12), we see that the right hand side of (15) is composed of the product of the factor $e^{t}$ and the probability density function of a weighted sum of exponential variables. In fact, the explicit expression of latter, in addition to the new derivation presented here, was derived much earlier (Mathai, 1983) under different context and was rediscovered/rederived multiple times (Harrison, 1990; Amari and Mirsa, 1997; Jasiulewicz and Kordecki, 2003) by different means. Its connection to combining $P$-values, however, was never made explicit till now. From (10), we know that $F(\tau=0)=0$, implying that $\displaystyle F(\tau)$ $\displaystyle=$ $\displaystyle\int_{0}^{\tau}f(\tau^{\prime})d\tau^{\prime}=\int_{t}^{\infty}f(e^{-t^{\prime}})\;e^{-t^{\prime}}dt^{\prime}$ (16) $\displaystyle=$ $\displaystyle\left[\prod_{l=1}^{m}r_{l}^{n_{l}}\right]\sum_{k=1}^{m}\sum_{\begin{subarray}{c}g_{1},g_{2},\ldots,g_{m}=0\\\ \sum g_{i}=n_{k}-1\end{subarray}}\left(\prod_{j=1,j\neq k}^{m}\frac{(n_{j}-1+g_{j})!}{(n_{j}-1)!g_{j}!}\frac{(-1)^{g_{j}}}{(r_{j}-r_{k})^{n_{j}+g_{j}}}\right)\int_{t}^{\infty}\frac{(t^{\prime})^{g_{k}}}{g_{k}!}e^{-r_{k}\,t^{\prime}}dt^{\prime}$ $\displaystyle=$ $\displaystyle\left[\prod_{l=1}^{m}r_{l}^{n_{l}}\right]\sum_{k=1}^{m}\sum_{\begin{subarray}{c}g_{1},g_{2},\ldots,g_{m}=0\\\ \sum g_{i}=n_{k}-1\end{subarray}}\left(\prod_{j=1,j\neq k}^{m}\frac{(n_{j}-1+g_{j})!}{(n_{j}-1)!g_{j}!}\frac{(-1)^{g_{j}}}{(r_{j}-r_{k})^{n_{j}+g_{j}}}\right)\left(\sum_{l=0}^{g_{k}}\frac{t^{g_{k}-l}}{r_{k}^{l+1}(g_{k}-l)!}e^{-r_{k}\,t}\right)$ $\displaystyle=$ $\displaystyle\sum_{k=1}^{m}\sum_{\begin{subarray}{c}g_{1},g_{2},\ldots,g_{m}=0\\\ \sum g_{i}=n_{k}-1\end{subarray}}\left(\prod_{j=1,j\neq k}^{m}\frac{(n_{j}-1+g_{j})!}{(n_{j}-1)!g_{j}!}\frac{(-r_{k})^{g_{j}}r_{j}^{n_{j}}}{(r_{j}-r_{k})^{n_{j}+g_{j}}}\right)\left(\sum_{l=0}^{g_{k}}\frac{(r_{k}\,t)^{g_{k}-l}}{(g_{k}-l)!}e^{-r_{k}\,t}\right)$ $\displaystyle=$ $\displaystyle\sum_{k=1}^{m}\sum_{g_{k}=0}^{n_{k}-1}\sum_{\begin{subarray}{c}g_{i\neq k}=0\\\ \sum_{i}g_{i}=n_{k}-1\end{subarray}}^{n_{k}-1-g_{k}}\left(\prod_{j=1,j\neq k}^{m}\frac{(n_{j}-1+g_{j})!}{(n_{j}-1)!g_{j}!}\frac{(-r_{k})^{g_{j}}r_{j}^{n_{j}}}{(r_{j}-r_{k})^{n_{j}+g_{j}}}\right)H(r_{k}\,t,\,g_{k})\;,$ where the function $H$ is defined as $H(x,n)\,\equiv\;e^{-x}\,\sum_{k=0}^{n}\frac{x^{k}}{k!}\;.$ (17) Eq. (16) represents the most general formula that interpolates the scenarios considered by Fisher and Good. Let us take the limiting case from (16). For Fisher’s formula, one weights every $P$-value equally, and thus correspond to $m=1$ and $n_{1}=L$. The constraint in the sum of (16) forces $g_{1}=n_{1}-1=L-1$. Consequently, we have (by calling $r_{1}$ by $r$ for simplicity) ${\rm Prob}(Q_{F}\leq{\tau_{\scriptscriptstyle F}})=H(rt,\,L-1)=e^{-rt}\,\sum_{l=0}^{L-1}\frac{(rt)^{l}}{l!}$ (18) Notice that regardless whatever the weight $w$ one assigns to all the $P$-values, the final answer is independent of the weight. This is because $t=-\ln\tau=-w\ln{\tau_{\scriptscriptstyle F}}=(-\ln{\tau_{\scriptscriptstyle F}})/r$ and therefore $rt=\ln(1/{\tau_{\scriptscriptstyle F}})$. This results in ${\rm Prob}(Q_{F}\leq{\tau_{\scriptscriptstyle F}})={\tau_{\scriptscriptstyle F}}\sum_{l=0}^{L-1}\frac{\left[\;\ln(1/{\tau_{\scriptscriptstyle F}})\,\right]^{\,l}}{l!}\;,$ (19) exactly what one anticipates from (5). To obtain the results of Good, one simply makes $m=L$ and $n_{k}=1$ $\forall~{}k$, implying all $g_{i}=0$. In this case, (16) becomes (with $r_{l}=1/w_{l}$, $e^{-t}={\tau_{\scriptscriptstyle G}}$ and $H(x,0)=1$ $\forall~{}x$) ${\rm Prob}(Q_{G}\leq{\tau_{\scriptscriptstyle G}})=\sum_{l=1}^{L}\left(\prod_{k\neq l}\frac{r_{k}}{r_{k}-r_{l}}\right){\tau_{\scriptscriptstyle G}}^{1/w_{l}}\;=\;\sum_{l=1}^{L}\Lambda_{l}\;{\tau_{\scriptscriptstyle G}}^{1/w_{l}}\;,$ (20) reproducing exactly (6) One may also re-express eq. (16) in a slightly different form $\displaystyle F(\tau)$ $\displaystyle=$ $\displaystyle\left[\prod_{l=1}^{m}r_{l}^{n_{l}}\right]\sum_{k=1}^{m}\sum_{g_{k}=0}^{n_{k}-1}\frac{1}{r_{k}^{g_{k}+1}}H(r_{k}\,t,\,g_{k})\times$ (21) $\displaystyle\hskip 30.0pt\times\sum_{\begin{subarray}{c}g_{i\neq k}=0\\\ \sum_{i}g_{i}=n_{k}-1\end{subarray}}^{n_{k}-1-g_{k}}\left(\prod_{j=1,j\neq k}^{m}\frac{(n_{j}-1+g_{j})!}{(n_{j}-1)!g_{j}!}\frac{(-1)^{g_{j}}}{(r_{j}-r_{k})^{n_{j}+g_{j}}}\right)$ $\displaystyle\equiv$ $\displaystyle\left[\prod_{l=1}^{m}r_{l}^{n_{l}}\right]\tilde{F}(\tau;n_{1},n_{2},\ldots,n_{m})\;.$ Note that in the expression (21), we have isolated an overall multiplying factor and keeps explicit the $n_{1\leq k\leq m}$ dependence for later convenience. ## 5 Cases of nearly degenerate weights In our derivation of (19), it is explicitly shown in section 4 that the final $P$-value obtained is independent of the weight $w$ that was used to assign to all the individual $P$-values, $p_{1},p_{2},\ldots,p_{L}$. It is thus natural to ask, if one starts by weighting each $P$-value differently, upon making the weights closer to one another, will one recover Fisher’s formula (5) from Good’s formula (6)? By continuity, the answer is expected to be affirmative. More generally, one would like to have a formal protocol to compute the combined $P$-value when the weights may be categorized into several subsets, within each subset the weights are almost degenerate. In this section, we first illustrate the transition from Good’s formula to Fisher’s formula by combining two almost degenerate $P$-values. We will then provide a general protocol to explicitly, when there exist nearly degenerate weights, deal with the possible numerical instability that was first cautioned by Good (1955) and subsequently by many authors (Bhoj, 1992; Olkin and Saner, 2001; Hou, 2005). Let us consider combining $p_{1}$ and $p_{2}$ with weights $w_{1}$ and $w_{2}$ using Good’s formula. One has ${\rm Prob}(Q_{G}\leq{\tau_{\scriptscriptstyle G}})=\frac{1}{w_{1}-w_{2}}\left[w_{1}\,p_{1}p_{2}^{\frac{w_{2}}{w_{1}}}-w_{2}\,p_{1}^{\frac{w_{1}}{w_{2}}}p_{2}\right]\;.$ (22) Without loss of generality, one assumes $w_{1}>w_{2}$ and hence writes $w_{1}/w_{2}=1+\epsilon$ with $\epsilon>0$. We are interested in the case when the weights get close to each other, or when $\epsilon\to 0$. We now rewrite eq. (22) as ${\rm Prob}(Q_{G}\leq{\tau_{\scriptscriptstyle G}})=\frac{w_{2}}{w_{1}-w_{2}}\left[\frac{w_{1}}{w_{2}}\,p_{1}p_{2}^{\frac{w_{2}}{w_{1}}}-\,p_{1}^{\frac{w_{1}}{w_{2}}}p_{2}\right]=\frac{1}{\epsilon}\left[(1+\epsilon)\,p_{1}p_{2}^{\frac{1}{1+\epsilon}}-\,p_{1}^{1+\epsilon}p_{2}\right]\;.$ (23) In the limit of small $\epsilon$, we may rewrite (23)as $\displaystyle{\rm Prob}(Q_{G}\leq{\tau_{\scriptscriptstyle G}})$ $\displaystyle=$ $\displaystyle\frac{p_{1}p_{2}}{\epsilon}\left[(1+\epsilon)\,p_{2}^{-\frac{\epsilon}{1+\epsilon}}-\,p_{1}^{\epsilon}\right]=\frac{p_{1}p_{2}}{\epsilon}\left[(1+\epsilon)\,e^{-\frac{\epsilon}{1+\epsilon}\ln p_{2}}-\,e^{\epsilon\ln p_{1}}\right]$ (24) $\displaystyle=$ $\displaystyle\frac{p_{1}p_{2}}{\epsilon}\left[\epsilon-\epsilon(\ln p_{2}+\ln p_{1})+{\cal O}(\epsilon^{2})\right]$ $\displaystyle=$ $\displaystyle p_{1}p_{2}\left[1-\ln(p_{1}p_{2})+{\cal O}(\epsilon)\right]$ $\displaystyle\xrightarrow[\epsilon\to 0]{}$ $\displaystyle p_{1}p_{2}\left[1-\ln(p_{1}p_{2})\right]\;=\;{\rm Prob}(Q_{F}\leq{\tau_{\scriptscriptstyle F}})\;.$ Note that when the small weight difference $w_{1}-w_{2}$ is near the machine precision of a digital computer, using formula (6) directly will inevitably introduce numerical instability caused by rounding errors. To construct a general protocol to deal with nearly degenerate weights, one first observes from eqs. (13-21) that it is the inverse of weights $r_{k}\equiv 1/w_{k}$ that permeates the derivation of the unified $P$-value. The closeness between weights is thus naturally defined by closeness in the inverse weights. As shown in eqs. (2) and (6), the combined $P$-value by Good’s formula is independent of the absolute size of the weights but only on the relative weights. Making the observation that $r_{k}\,t$ in eq. (16) only depend on the ratios $r_{j\neq k}/r_{k}$, one also sees explicitly that the most general combined $P$-values (see (16)) only depend on the relative weights as well. We are thus free to choose any scale we wish. For simplicity, we normalize the inverse weight associated with each method by demanding the sum of inverse weights equal the total number of methods $\sum_{j=1}^{M}r_{j}=\sum_{j=1}^{M}1=M\;,$ (25) where $1/r_{j}$ represents the weight associated method $j$ and $M$ represents the total number of $P$-values (or methods) to be combined. For the GC described in section 4, $M=\sum_{k=1}^{m}n_{k}$. This normalization choice makes the average inverse weight of participating methods be $1$. The next step is to determine, for a given list of inverse weights and the radius of clustering, the number of clusters needed. This task may be achieved in a hierarchical manner. After normalizing the inverse weights $r_{k}$ using eq. (25), one may sort the inverse weights in either ascending or descending order. For a given radius $\eta>0$, one starts to seek the pair of inverse weights that are closest but not identical, and check if it is smaller than the radius $\eta$. If yes, one will merge that pair of inverse weights by using their average, weighted by number of occurrences, as the new center and continue the process till every inverse weights in the list is separated by a distance farther than $\eta$. We use an example of $M=8$ to illustrate the idea. Let the normalized inverse weights $\\{r_{j}\\}_{j=1}^{8}$ be $0.50,~{}0.70(2),~{}0.71,~{}0.74,~{}1.03~{},1.80~{},1.82$ where the number $2$ inside the pair of parentheses after $0.70$ simply indicates that there are two identical inverse weights $0.70$ to start with. Assume that one chooses the radius of cluster $\eta$ to be $0.005$, since every adjacent inverse weights are separated by more than $0.005$, no further clustering procedures is needed and one ends up having seven effective clusters: one cluster with two identical inverse weights $0.70$, and the rest of six clusters are all singletons. This corresponds to $m=7$, $n_{1}=1$, $n_{2}=2$, $n_{3}=n_{4}=\cdots=n_{7}=1$. Suppose one chooses the clustering radius $\eta$ to be $0.05$. In the first step, we identify that $0.70$ and $0.71$ are the closet pair of inverse weights. The weighted average between them is $\frac{2\cdot 0.70+0.71}{3}=\frac{2.11}{3}=0.70\bar{3}\;.$ The list of inverse weights then appears as $0.50,~{}0.70\bar{3}(3),~{}0.74,~{}1.03~{},1.80~{},1.82\;.$ The closest pair of inverse weights is now between $1.80$ and $1.82$, and upon merging them we have the list now appears as $0.50,~{}0.70\bar{3}(3),~{}0.74,~{}1.03~{},1.81(2)\;.$ Next pair of closest inverse weights is then $0.70\bar{3}$ and $0.74$. The weighted average leads to $(2.11+0.74)/4=0.7125$. After this step, the list of inverse weights appears as $0.50,~{}0.7125(4),~{}1.03~{},1.81(2)\;,$ indicating that we have $m=4$ ( four clusters), with number of members being $n_{1}=1$, $n_{2}=4$, $n_{3}=1$ and $n_{4}=2$. The centers of the four clusters are specified by the average inverse weights: $0.50,~{}0.7125,~{}1.03~{},1.81$. The distance between any two of the average inverse weights is now larger than $0.05$. This is a good place for us to introduce some notation. We shall denote by $r_{k}+\eta_{k;j}$ the $j$th inverse weights of cluster $k$, whose averaged inverse weight is $r_{k}$. With this definition, for the example above, we have $\eta_{1;1}=0$, $\eta_{2;1}=\eta_{2;2}=-0.0125$, $\eta_{2;3}=-0.0025$, $\eta_{2;4}=0.0275$, $\eta_{3;1}=0$, $\eta_{4;1}=-0.01$, and $\eta_{4;2}=0.01$. Using the hierarchical protocol mentioned above, the number of clusters $m$ and the numbers of members $n_{k}$ associated with cluster $k$ are all obtained along with $\\{\eta_{k;j}\\}$. Following the derivation in section 4, we obtain a probability density function very similar to (13) $f(\tau)=\left[\prod_{l=1}^{m}\prod_{j=1}^{n_{l}}(r_{l}+\eta_{l;j})\right]\left(i\right)^{\sum_{k=1}^{m}n_{k}}\int_{-\infty}^{\infty}\frac{dq}{2\pi}\,e^{-it(q+i)}\prod_{k=1}^{m}\left[\prod_{j=1}^{n_{k}}\frac{1}{q+i(r_{k}+\eta_{k;j})}\right]\;.$ (26) From section 4, we see that the ill-conditioned situations emerge when some weights are nearly degenerate and the source of difference in inverse weights comes from obtaining $\tilde{F}(\tau;n_{1},n_{2},\ldots,n_{m})$ in (21) from $\tilde{f}(\tau;n_{1},n_{2},\ldots,n_{m})$ in (14). Therefore, one may leave the prefactor $\left[\prod_{l=1}^{m}\prod_{j=1}^{n_{l}}(r_{l}+\eta_{l;j})\right]$ untouched and focus on the rest of the right hand side of eq. (26). To proceed, we write $\displaystyle\frac{1}{q+i(r_{k}+\eta_{k;j})}$ $\displaystyle=$ $\displaystyle\frac{1}{q+ir_{k}}\left(1+\frac{i\,\eta_{k;j}}{q+ir_{k}}\right)^{-1}=\frac{1}{q+ir_{k}}\,e^{-\ln\left(1+\frac{i\,\eta_{k;j}}{q+ir_{k}}\right)}$ $\displaystyle=$ $\displaystyle\frac{1}{q+ir_{k}}\,\exp\left[\sum_{g=1}^{\infty}\frac{1}{g}\left(\frac{-i\,\eta_{k;j}}{q+ir_{k}}\right)^{g}\right]\;.$ Consequently, we may write $\prod_{j=1}^{n_{k}}\frac{1}{q+i(r_{k}+\eta_{k;j})}=\frac{1}{(q+ir_{k})^{n_{k}}}\exp\left[\sum_{g=1}^{\infty}\frac{Y_{k;g}\,(i)^{g}}{\left(q+ir_{k}\right)^{g}}\right]$ (27) where $Y_{k;g}\equiv\sum_{j=1}^{n_{k}}\frac{(-\eta_{k;j})^{g}}{g}\;.$ (28) The product in eq. (26) may now be formally written as $\prod_{k=1}^{m}\left[\prod_{j=1}^{n_{k}}\frac{1}{q+i(r_{k}+\eta_{k;j})}\right]=\left[\prod_{k=1}^{m}\frac{1}{(q+ir_{k})^{n_{k}}}\right]\exp\left[\sum_{g=1}^{\infty}(i)^{g}\sum_{k=1}^{m}\frac{Y_{k;g}}{\left(q+ir_{k}\right)^{g}}\right]\;.$ (29) We now note a simplification by choosing $r_{k}$ to be the average inverse weight of the $k$th cluster. In this case, we have $\sum_{j=1}^{n_{k}}\eta_{k;j}=0$ $\forall~{}k$. That is, $Y_{k;1}=0$ always. This allows us to write eq. (29) as $\prod_{k=1}^{m}\left[\prod_{j=1}^{n_{k}}\frac{1}{q+i(r_{k}+\eta_{k;j})}\right]=\left[\prod_{k=1}^{m}\frac{1}{(q+ir_{k})^{n_{k}}}\right]\exp\left[\sum_{g=2}^{\infty}(i)^{g}\sum_{k=1}^{m}\frac{Y_{k;g}}{\left(q+ir_{k}\right)^{g}}\right]\;.$ (30) The key idea here is to Taylor expand the exponential and collect terms of equal number of $1/(q+ir)$. Evidently, the first correction term starts with $1/(q+ir)^{2}$. Furthermore, before the $1/(q+ir)^{4}$ order, there is no mixing between different clusters. Below, we rewrite eq. (26) to include the first few orders of correction terms $\displaystyle\frac{f(\tau)}{\prod_{l=1}^{m}\prod_{j=1}^{n_{l}}(r_{l}+\eta_{l;j})}=\left(i\right)^{\sum_{k=1}^{m}n_{k}}\int_{-\infty}^{\infty}\frac{dq}{2\pi}\,e^{-it(q+i)}\frac{\exp\left[\sum_{g=2}^{\infty}(i)^{g}\sum_{k=1}^{m}\frac{Y_{k;g}}{\left(q+ir_{k}\right)^{g}}\right]}{\prod_{k=1}^{m}(q+ir_{k})^{n_{k}}}$ $\displaystyle=\tilde{f}(\tau;\\{n_{l}\\}_{l=1}^{m})+\sum_{k=1}^{m}Y_{k;2}\;\tilde{f}(\tau;\\{n_{l\neq k},n_{k}+2\\})$ $\displaystyle+\sum_{k=1}^{m}Y_{k;3}\;\tilde{f}(\tau;\\{n_{l\neq k},n_{k}+3\\})+\sum_{k=1}^{m}\left(Y_{k;4}+\frac{(Y_{k;2})^{2}}{2!}\right)\;\tilde{f}(\tau;\\{n_{l\neq k},n_{k}+4\\})$ $\displaystyle+\frac{1}{2!}\sum_{\begin{subarray}{c}k,k^{\prime}=1\\\ k\neq k^{\prime}\end{subarray}}^{m}Y_{k;2}Y_{k^{\prime};2}\;\tilde{f}(\tau;\\{n_{l\neq k,k^{\prime}},n_{k}+2,n_{k^{\prime}}+2\\})+{\cal O}(\eta^{5})\;.$ (31) This immediately leads to $\displaystyle\frac{F(\tau)}{\prod_{l=1}^{m}\prod_{j=1}^{n_{l}}(r_{l}+\eta_{l;j})}=\tilde{F}(\tau;\\{n_{l}\\}_{l=1}^{m})+\sum_{k=1}^{m}Y_{k;2}\;\tilde{F}(\tau;\\{n_{l\neq k},n_{k}+2\\})$ $\displaystyle+\sum_{k=1}^{m}Y_{k;3}\;\tilde{F}(\tau;\\{n_{l\neq k},n_{k}+3\\})+\sum_{k=1}^{m}\left(Y_{k;4}+\frac{(Y_{k;2})^{2}}{2!}\right)\;\tilde{F}(\tau;\\{n_{l\neq k},n_{k}+4\\})$ $\displaystyle+\frac{1}{2!}\sum_{\begin{subarray}{c}k,k^{\prime}=1\\\ k\neq k^{\prime}\end{subarray}}^{m}Y_{k;2}Y_{k^{\prime};2}\;\tilde{F}(\tau;\\{n_{l\neq k,k^{\prime}},n_{k}+2,n_{k^{\prime}}+2\\})+{\cal O}(\eta^{5})\;.$ (32) Note that when the cluster radius $\eta$ is chosen to be zero, the only clusters are from sets of identical weights, and all $\eta_{k;j}$ must be zero. In this case, only the first term on the right hand side of (32) exists and the result derived in section 4 is recovered exactly. Since all $\tilde{F}$ are finite positive quantities, the errors resulting from truncating the expression in eq. (32) at certain order of $\eta$ can be easily bounded. Therefore, any desired precision may be obtained via including the corresponding number of higher order terms. As the main result of the current paper, our expansion provides a systematic, numerically stable method to achieve desired accuracy in computing combined $P$-values. ## 6 Examples Example (a): This example, assuming $m=4$, demonstrates how to compute the $\tilde{F}(\tau;\\{n_{l}\\})$ function present in eq. (21). Let $r_{k}$ be the inverse weights associated with cluster $k$. When combining multiple $P$-values with a prescribed clustering radius on the inverse weights, see (25) and the previously described clustering procedure, the $r_{k}$s and the deviations $\eta_{k;j}$ are obtained once and for all. The $\eta_{k;j}$, through eq. (28), constitute the key expansion parameters $Y_{k;g}$ that yield, upon multiplying by $\tilde{F}(\tau;{n_{l}})$ with different $\\{n_{l}\\}$, the higher order terms in our key result (32). Note that in eq. (32), in the zeroth order term, the argument $n_{l}$ of $\tilde{F}$ represents the number of members associated with cluster $l$. However, for higher order correction terms, the $n_{l}$s entering $\tilde{F}$ no longer carry the same meaning. Therefore, in the example shown here, one does not assume that $n_{j}$ is the number of methods associated with cluster $k$. We now illustrate how to open up the sum in eq. (21). The constraint $\sum_{i}g_{i}=n_{k}-1$ implies that one only has $m-1$ independent $g_{i}$s. Once $m-1$ $g_{i}$s are specified, the remaining one is also determined. To simplify the exposition, let us introduce the following notation $\alpha(g_{j};j,k)\equiv\frac{(n_{j}-1+g_{j})!}{(n_{j}-1)!g_{j}!}\frac{(-1)^{g_{j}}}{(r_{j}-r_{k})^{n_{j}+g_{j}}}\;.$ One then expand the sum in (21) as $\displaystyle\tilde{F}(\tau)=\sum_{g_{1}=0}^{n_{1}-1}\frac{H(r_{1}\,t,\,g_{1})}{r_{1}^{g_{1}+1}}\sum_{g_{2}=0,}^{n_{1}-1-g_{1}}\alpha(g_{2};2,1)\;\sum_{g_{3}=0}^{n_{1}-1-g_{1}-g_{2}}\alpha(g_{3};3,1)\;\;\alpha(g_{4};4,1)$ $\displaystyle+\sum_{g2=0}^{n_{2}-1}\frac{H(r_{2}\,t,\,g_{2})}{r_{2}^{g_{2}+1}}\sum_{g_{1}=0}^{n_{2}-1-g_{2}}\alpha(g_{1};1,2)\;\sum_{g_{3}=0}^{n_{2}-1-g_{2}-g_{1}}\alpha(g_{3};3,2)\;\;\alpha(g_{4};4,2)$ $\displaystyle+\sum_{g3=0}^{n_{3}-1}\frac{H(r_{3}\,t,\,g_{3})}{r_{3}^{g_{3}+1}}\sum_{g_{1}=0}^{n_{3}-1-g_{3}}\alpha(g_{1};1,3)\;\sum_{g_{2}=0}^{n_{3}-1-g_{3}-g_{1}}\alpha(g_{2};2,3)\;\;\alpha(g_{4};4,3)$ $\displaystyle+\sum_{g4=0}^{n_{4}-1}\frac{H(r_{4}\,t,\,g_{4})}{r_{4}^{g_{4}+1}}\sum_{g_{1}=0}^{n_{4}-1-g_{4}}\alpha(g_{1};1,4)\;\sum_{g_{2}=0}^{n_{4}-1-g_{4}-g_{1}}\alpha(g_{2};2,4)\;\;\alpha(g_{3};3,4)\;.$ (33) Example (b): This example illustrates the possibility of numerical instability associated with eqs. (6) and (21) when they are used to combine P-values with nearly equal weights. We also show how such instabilities are resolved by using eq. (32). Consider the case of combining five $P$-values, $\\{0.008000257,0.008579261,0.0008911761,0.006967988,0.004973110\\}$, weighted respectively by $\\{0.54531152,0.54532057,0.54531221,0.54531399,0.54531776\\}$. Using eq. (2), one obtains $\tau_{G}=4.30656196\times 10^{-7}$. The combined $P$-value is then obtained as the probability of attaining a random variable $Q_{G}$, defined in eq. (4), such that is less than or equal to $\tau_{G}$. Combining $P$-values using eq. (6) gives $\displaystyle{\rm Prob}(Q_{G}\leq\tau_{G})$ $\displaystyle=1923475672.53812003+134195847.49348195$ $\displaystyle-3271698577.16100168+1726093852.57087326-512066795.44147670$ $\displaystyle=-0.00000322\;.$ When one uses equation (21), $\tau$ takes the value of $\tau_{G}$ and the random variable $Q$ is simply $Q_{G}$, and the combined $P$-value becomes $\displaystyle{\rm Prob}(Q\leq\tau)$ $\displaystyle=170090507.09336647+21761086.68190728$ $\displaystyle-972903041.25101399+941269625.31004059-512066795.44252247$ $\displaystyle=-0.00000006\;.$ Apparently, probability can’t be negative and the negative values shown above illustrate how eqs. (6) and (21) may suffer from numerical instability when the weights are nearly degenerate. This numerical instability is removed by applying equation (32) which combines weighted $P$-values using a controlled expansion and yields, for this example, $\displaystyle{\rm Prob}(Q\leq\tau)$ $\displaystyle=5.379093\times 10^{-8}+1.407305\times 10^{-16}$ $\displaystyle-1.066323\times 10^{-21}+1.634917\times 10^{-25}+{\cal O}(10^{-29})$ $\displaystyle=5.37909\times 10^{-8}$ Example (c): One natural question to ask is that how well does eq. (32) work when one chooses a larger clustering radius and group weights that are clearly distinguishable into a few clusters? To consider this case, let us use the five $P$-values from example (b) above but with weights chosen differently. Let us assume that the inverse weights ($r_{k}\equiv 1/w_{k}$) associated with these five $P$-values are $\\{0.6,0.65,1.2,1.25,1.3\\}$. For this case, $\tau=\tau_{G}=1.935663\times 10^{-13}$. Combining $P$-value using formulas (6) yields $\displaystyle{\rm Prob}(Q\leq\tau)$ $\displaystyle=$ $\displaystyle\;\;2.187324\times 10^{-6}-5.946040\times 10^{-7}+2.131226\times 10^{-13}$ $\displaystyle-8.011644\times 10^{-14}+7.639290\times 10^{-15}$ $\displaystyle=$ $\displaystyle 1.59272\times 10^{-6}\;,$ while combining $P$-values using (21) yields identical results $\displaystyle{\rm Prob}(Q\leq\tau)$ $\displaystyle=$ $\displaystyle\;\;1.725699\times 10^{-6}-3.049251\times 10^{-7}+1.311524\times 10^{-13}$ $\displaystyle-6.162803\times 10^{-14}+7.639290\times 10^{-15}$ $\displaystyle=$ $\displaystyle 1.59272\times 10^{-6}\;.$ When one uses $\eta=0.1$ as the clustering radius, one obtains two clusters: one with average inverse weight $0.625$ and the other with average inverse weight $1.25$. If one then uses eq. (32) to combine $P$-values, one attains the following results $\displaystyle{\rm Prob}(Q\leq\tau)$ $\displaystyle=$ $\displaystyle\;\;1.472453\times 10^{-6}+1.171521\times 10^{-7}+0$ (34) $\displaystyle+2.584710\times 10^{-9}+4.889899\times 10^{-10}+{\cal O}(10^{-12})$ $\displaystyle=$ $\displaystyle 1.59268\times 10^{-6}\;,$ which contains no sign alternation and agrees well with the results from both (6) and (21). This illustrates the robustness of eq. (32) in combining $P$-values. Note that the third term on the right hand side of (34) is zero. This is because the multiplying factor $Y_{k;3}$ is zero for both clusters. In general, $Y_{k;3}$ measures the skewness of inverse weights associated with cluster $k$ and for our case here both cluster of inverse weights are perfectly symmetrical with respect to their centers, leading to zero skewness. If the inverse weights of cluster $k$ distribute perfectly symmetrically with respect to its center, it is evident from eq. (28) that $Y_{k;g}=0$ for odd $g$. Evidently if one chooses a large clustering radius $\eta$ and then uses eq. (32) to combine $P$-values, many higher order terms in the expansion will be required to achieve high accuracy in the final combined $P$-value. ## 7 Future directions Although the expression (16) provides access to exact statistics for a broader domain of problems and our expansion formula (32) provides accurate and stable statistics even when nearly degenerate weights are present, there remain a few unanswered questions that should be addressed by the community in the near future. For example, even though we can accommodate any reasonable $P$-value weighting, thanks to (32), the more difficult question is how does one choose the right set of weights when combining statistical significance (Zelen and Joel, 1959; Koziol and Perlman, 1978; Hedges, 1985; Pepe and Fleming, 1989; Forrest, 2001). The weights chosen reflects how much does one wish to trust various obtained $P$-values. Ideally, a fully systematic method should also provide a metric for choosing appropriate weights. How to obtain the best set of weights remains an open problem and definitely deserves further investigations. Another limitation of (16) and (32), and consequently of Fisher’s and Good’s formulas, is that one must assume the $P$-values to be combined as independent. In real applications, it is foreseeable that $P$-values reported by various methods may exhibit non-negligible correlations. 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The Annals of Mathematical Statistics 30(4), pp. 885–895.	arxiv-papers	2010-11-30T17:56:19	2024-09-04T02:49:15.335151	{ "license": "Public Domain", "authors": "Gelio Alves and Yi-Kuo Yu", "submitter": "Yi-Kuo Yu", "url": "https://arxiv.org/abs/1011.6627" }
1012.0007	# Quantum state tomography of an itinerant squeezed microwave field F. Mallet JILA, National Institute of Standards and Technology and the University of Colorado, Boulder, CO 80309, USA M. A. Castellanos-Beltran JILA, National Institute of Standards and Technology and the University of Colorado, Boulder, CO 80309, USA Department of Physics, University of Colorado, Boulder, CO 80309, USA H. S. Ku JILA, National Institute of Standards and Technology and the University of Colorado, Boulder, CO 80309, USA Department of Physics, University of Colorado, Boulder, CO 80309, USA S. Glancy National Institute of Standards and Technology, Boulder, Colorado 80305, USA E. Knill National Institute of Standards and Technology, Boulder, Colorado 80305, USA K. D. Irwin National Institute of Standards and Technology, Boulder, Colorado 80305, USA G. C. Hilton National Institute of Standards and Technology, Boulder, Colorado 80305, USA L. R. Vale National Institute of Standards and Technology, Boulder, Colorado 80305, USA K. W. Lehnert konrad.lehnert@jila.colorado.edu JILA, National Institute of Standards and Technology and the University of Colorado, Boulder, CO 80309, USA Department of Physics, University of Colorado, Boulder, CO 80309, USA ###### Abstract We perform state tomography of an itinerant squeezed state of the microwave field prepared by a Josephson parametric amplifier (JPA). We use a second JPA as a pre-amplifier to improve the quantum efficiency of the field quadrature measurement (QM) from 2 % to $36\pm 4~{}\%$. Without correcting for the detection inefficiency we observe a minimum quadrature variance which is $68^{+9}_{-7}~{}\%$ of the variance of the vacuum. We reconstruct the state’s density matrix by a maximum likelihood method and infer that the squeezed state has a minimum variance less than 40 % of the vacuum, with uncertainty mostly caused by calibration systematics. Josephson parametric amplifier, squeezed state, quantum state tomography ###### pacs: 42.50.Dv, 42.50.Lc, 03.67.Bg Fundamental quantum optics experiments at microwave frequencies have been recently performed with superconducting qubits or Rydberg atoms inside high- quality microwave cavities. Examples include the reconstruction of the Wigner functions of Fock states from one Houck et al. (2007) to a few photons and coherent superpositions of few photons Deléglise et al. (2008); Hofheinz et al. (2008, 2009). States such as these, which are manifestly nonclassical light states, are crucial for quantum information processing, because they can be used to generate entanglement. However, in the cited experiments, these states are confined in cavities. Therefore distributing entanglement to separate parties, as required in quantum communication protocols, remains challenging for microwave implementations. In contrast to the discrete Fock state approach, continuous variables quantum information (CVQI) strategy uses another type of nonclassical states, the squeezed states, which are readily created in itinerant modes. These states exhibit reduced noise, below the vacuum fluctuations, in one of their quadrature components and amplified noise in the other one. They are also easily generated at optical frequencies in the itinerant output modes of parametric amplifiers made of optically nonlinear crystals. At optical frequencies, CVQI has progressed rapidly from the initial creation of squeezed states Slusher et al. (1985) and tomographic reconstruction Smithey et al. (1993); Schiller et al. (1996); Breitenbach et al. (1997) of those states to teleportation Furusawa et al. (1998); Yonezawa et al. (2007) and quantum error correction Aoki et al. (2009); Lassen et al. (2010). At microwave frequencies, the field is less advanced. The generation of microwave squeezed states using the nonlinear electrical response of superconducting Josephson junctions has been reported Yurke et al. (1988), with inferred squeezing down to $10~{}\%$ of vacuum variance Castellanos- Beltran et al. (2008). Such states can be powerful tools for quantum information processing and communication because microwaves and superconducting qubits can mimic useful light–atom interactions, as demonstrated in Wallraff et al. (2004). Furthermore, these devices are made of compact and integrable electrical circuits, with much promise for building complex quantum information processors. The lack of an efficient quadrature measurement (QM) for itinerant modes has slowed the advancement of CVQI. However, as demonstrated recently in Teufel et al. (2009), it is possible with a JPA to realize an efficient single QM. In this Letter, we report the tomography of an itinerant squeezed microwave field. We demonstrate that our JPA based measurement scheme has a quantum efficiency $20$ times greater than a QM employing state-of-art semiconductor amplifiers. We infer the quantum state prepared by maximum likelihood tomography, correcting for inefficiency in our QM. We discuss the achieved degree of squeezing, from the perspective of generating entanglement on chip. Homodyne tomography is a standard experimental tool to infer the quantum state of a single mode of light. It was proposed in Vogel and Risken (1989) and pioneered on a squeezed optical field in Smithey et al. (1993). Its principle is depicted in the Fig. 1. A homodyne detection apparatus measures the value of the quadrature $X_{\theta}$, where $\theta$ is set by adjusting the phase of the local oscillator. The probability density function $\textrm{pr}(X_{\theta})$ for measuring a particular value of $X_{\theta}$ is the marginal density function of the Wigner function, i.e. $\textrm{pr}(X_{\theta})=\int dX_{\theta+\pi/2}W(X_{\theta},X_{\theta+\pi/2})$, as shown in Fig. 1 (b). Thus by performing measurements of $X_{\theta}$ on many identical copies of the state and varying $\theta$, the “hidden” quantum object can be seen from different angles and its state inferred. Losses and other Gaussian noise sources in the homodyne detector can be modeled with the insertion of a fictitious beam splitter of transmissivity $\eta$, as shown in Fig. 1 (a). In such a case, the measured $\textrm{pr}(X_{\theta})$ are no longer the projections of the desired Wigner function $W$, but of a smoother distribution which is the convolution of $W$ with a Gaussian Wigner function Leonhardt and Paul (1993). However methods like maximum likelihood quantum state tomography can be used to deconvolve the effect of inefficiency Lvovsky and Raymer (2009). Figure 1: Principle of the experiment. (a-left): The squeezer (SQ, in red) prepares a squeezed state whose quadrature distributions are measured for different phases $\theta$ with an efficiency $\eta$. (a-right): Simulated measurement results for 20,000 realizations of creating the squeezed state and measuring it at a single $\theta$. The top graph shows the measured quadrature value versus realization number. The bottom plot is a histogram (blue circles) and Gaussian probability distribution $\textrm{pr}(X_{\theta})$ (red curve) of this random process. (b): Graphical Interpretation: the probability distribution $\textrm{pr}(X_{\theta})$ is simply the projection of the Wigner function. At optical frequencies, $\eta\geq 90~{}\%$ is routinely obtained using a pair of balanced photodiodes Lvovsky and Raymer (2009). Such detectors are not available for microwaves and until recently the best setup was a chain of phase-insensitive amplifiers followed by a mixer, or two such chains in parallel Menzel et al. (2010); Mariantoni et al. (2010); Bozyigit et al. (2010). In such a case, noise $A_{n}$ greater than $1/2$ (the vacuum variance) must be added to the QM Caves (1982). This noise can be modeled as an effective efficiency by the relation $\eta=1/(1+2A_{n})$ Leonhardt and Paul (1994), so the QM efficiency using phase-insensitive amplifiers is limited to $50~{}\%$. State of the art microwave amplifiers, high-electron-mobility transistors (HEMTs), have $A_{n}\approx 10-20$. In practice, the unavoidable losses present in a microwave experiment typically result in $\eta\approx 2$%. However, as demonstrated in Teufel et al. (2009), inserting a JPA used as a single quadrature preamplifier before the HEMT increases the experimentally achieved $\eta$ by a factor of approximately 20. To perform a high-quality reconstruction of the Wigner function of a squeezed microwave state we operate two JPAs in series, as shown in Fig. 2 (b). The first JPA, referred to as the squeezer (SQ), prepares the squeezed state. The second JPA, referred to as the pre-amplifier (AMP), amplifies the quadrature of the squeezed state determined by the phase difference $\theta$ between the AMP and the SQ pump tones. We vary $\theta$ by applying to the two cavities pump tones slightly detuned from one another. The SQ stage is pumped at $7.45$ GHz, while the AMP stage is pumped at $100$ kHz higher frequency; therefore, sweeping $\theta$ through $2\pi$ every $10$ µs. Our implementation of an SQ or an AMP at microwaves, as shown in Fig. 2 (a), requires three elements: (i) a JPA used in reflection, (ii) a directional coupler and (iii) a circulator. As described in Castellanos-Beltran et al. (2008), the JPAs are nonlinear resonant cavities built from coplanar waveguides whose central conductor has been replaced by a series of many Josephson junctions. The Josephson junctions’ nonlinearity causes the cavity’s phase velocity to be intensity dependent. Therefore when the cavity is pumped it becomes a phase sensitive amplifier for input modes whose frequencies lie within the bandwidth of the JPA centered on the pump frequency. Such microwave modes incident on the JPA are reflected and exit the cavity with one quadrature amplified and the other squeezed, depending on their phase relative to the pump’s phase. A directional coupler is used to add the pump tone to the incident signal and remove the pump tone from the reflected signal. Finally the incident and reflected modes are separated into different cables using a circulator. Figure 2: (a): To implement a SQ or AMP at microwaves, three microwaves components are required: (i) a JPA, (ii) a directional coupler and (iii) a circulator. Taking port (1) of the directional coupler as reference, (2) is the weakly coupled port, (3) the isolated port and (4) the direct port. Port (2) is used to pump the JPA. Port (3) is used to apply a cancelation tone (adjusted with a room temperature attenuator and phase shifter) that nulls the pump and displaces the output of the JPA back to the origin of the phase space. (b): Schematic of the experiment. In this figure, all the microwave components and cables are considered lossless; their imperfections are absorbed into the experimentally determined total transmissivities $\xi$, $\alpha$ and $\beta$. Following Fig. 2 (b), in the limit of large HEMT power gain $G_{H}$, our quantum efficiency can be cast as $\eta=\frac{\alpha}{2+2A_{A}-\alpha+[2A_{H}-(1-\beta)]/G_{A}\beta},$ (1) where $A_{A}$ ($A_{H}$) is the AMP (HEMT) added noise, $\alpha$ ($\beta$) is the fraction of power transmitted by the microwave circuitry between the SQ and the AMP (the AMP and the HEMT), and $G_{A}$ is the power gain of the AMP stage. A detailed description of how we calibrate each of these parameters is in the supplementary information. Briefly, we inject different amounts of thermal noise into the amplifier chain while operating each JPA either as an amplifier (ON) or as a noiseless element with unit gain (OFF). We then infer the added noise and loss of the elements by observing the variation in the noise at the output of the measurement chain. The thermal noise is varied by connecting the input of the SQ through a switch to either a “hot load” (50 $\Omega$ microwave termination at $4.1$ K) or a “cold load” (at $20$ mK). Although the tomography is only performed with the “cold load”, both are required for calibration. We obtain $A_{A}=0.25\pm 0.06$, $A_{H}=17.3\pm 0.1$, $\alpha=68\pm 2~{}\%$ and $\beta=74\pm 5~{}\%$. However, as the switch is operated at the 4.1 K stage and is slightly lossy, the state presented at the input of the SQ with the “cold load” is not pure quantum vacuum, but a low occupancy thermal state with average photon number $\overline{n}\simeq 0.15\pm 0.15$. One quadrature of the resulting squeezed state is then amplified at the AMP stage with sufficient gain $G_{A}=180$ such that the noise in the amplified quadrature exceeds $A_{H}$ for any $\theta$. From Eq. (1), we obtained an overall quantum efficiency of $36\pm 4~{}\%$, which can be compared to $\eta\approx 2~{}\%$ without the AMP stage. In this experiment our uncertainty in $\eta$ and $\overline{n}$ create a systematic source of error. We thus perform our data analysis under three assumptions (1) high efficiency ($\eta=0.40$) and high mean photon number ($\overline{n}=0.30$), (2) best estimate for both efficiency ($\eta=0.36$) and mean photon number ($\overline{n}=0.15$), and (3) low efficiency ($\eta=0.33$) and low mean photon number ($\overline{n}=0$). These three cases give us “pessimistic”, “best-guess”, and “optimistic” analyses, in terms of the purity of the squeezed state estimated by the tomography. Using a lower estimate for $\eta$ and $\overline{n}$ as inputs to the tomography algorithm causes it to return a more pure, more squeezed, and therefore a more “optimistic” estimate of the squeezed state. Associated with each of these three cases, we also have statistical uncertainty, so the given error bounds cover an interval that includes both uncertainties around the “best-guess” estimate. They are reported in the form $X_{-L}^{+U}$, where $X$ is the statistical mean using the “best-guess” calibration and $L$ and $U$ are respectively the lower and upper bounds of the one standard deviation uncertainty in the “pessimistic” and “optimistic” cases. We must also calibrate the QM to convert the measured voltage noise into units of noise quanta. In optical homodyne tomography, this is usually done by inserting the vacuum and observing the quadrature noise. Analogously, we insert the weak thermal state with mean photons $\overline{n}$ (by simply turning the SQ stage OFF) and measure voltages proportional to quadrature values at many $\theta$, as shown (in blue) in Fig. 3. As expected this voltage noise is $\theta$ independent, with a variance $\Delta V_{\mathrm{SQ,OFF}}^{2}=3.2\times 10^{-5}~{}\textrm{mV}^{2}$. Under the convention that vacuum has variance $1/2$ in unitless quadrature space (or in units of “quanta”), we calibrate this voltage variance to $\Delta X_{\mathrm{SQ,OFF}}^{2}=(1-\eta)/2+\eta(1/2+\overline{n})=0.55_{-0.05}^{+0.07}$ quanta. Therefore the desired conversion factor $\Delta X_{\mathrm{SQ,OFF}}^{2}/\Delta V_{\mathrm{SQ,OFF}}^{2}=1.71_{-0.17}^{+0.20}\times 10^{4}~{}\mathrm{quanta/mV}^{-2}$ is used to rescale the variances in Fig. 3 (c). Figure 3: (a): Density plot of number of occurrences in a $1~{}$µV bin size of the amplified quadrature voltage $V_{\theta}$ versus $\theta/2\pi$, with the SQ pump OFF (top) and ON (bottom). (b): In particular, histograms of $V_{\theta}$ at the maximum of squeezing: data ($\circ$) and Gaussian fit (continuous lines) for the SQ pump OFF (blue) and ON (red). (c): Noise variance $\Delta X^{2}_{\theta}$ in quanta units on a log scale versus $\theta/2\pi$ for the SQ pump ON (red) and OFF (blue). The (black) line indicates our estimate of the vacuum noise level under the “best-guess” calibration. In Fig. 3 (a), we show QM data of the squeezed state. With SQ ON (red) we observe the characteristic phase dependent noise for a squeezed state. At the phase for which the variance is minimum, we show the histogram of quadrature measurements in Fig. 3 (b). The SQ OFF histogram is clearly wider than the SQ ON histogram, demonstrating our ability to observe squeezing directly at the output of our measurement chain. In Fig. 3 (c) we plot the variance of the QM with SQ ON and OFF as a function of $\theta$, expressed in units of quanta, clearly showing squeezing below the vacuum level. Without correcting for $\eta$, we observe a minimum quadrature variance which is $\Delta X^{2}_{\mathrm{SQ,MIN}}=68_{-7}^{+9}~{}\%$ of the vacuum variance. To infer the quantum state created by the squeezer, correcting for loss during the QM, we used maximum likelihood quantum state tomography Hradil et al. (2004). For each of the three calibration cases, we performed 35 reconstructions using independent subsets each containing 10,000 QMs of the total measured data. We estimated statistical uncertainty from the spread of properties (such as fidelity or minimum variance) of the set of 35 reconstructions. The statistical uncertainty was significantly lower than the systematic uncertainty. In Fig. 4 we show the Wigner function of the “best- guess” reconstructed state $\rho$. The pure squeezed vacuum state $\|\psi\rangle$ that has the highest fidelity with $\rho$ has minimum quadrature variance $6.0_{-1.1}^{+1.4}~{}\%$ of the vacuum variance, and that maximum fidelity is $F=\langle\psi\|\rho\|\psi\rangle=0.81_{-0.17}^{+0.16}$. As explained in the supplementary information, the minimum variance of $\rho$ is biased by an amount comparable to our systematic uncertainty, so we infer the minimum variance $\Delta x^{2}_{\mathrm{SQ,MIN}}$ directly from the observed minimum variance as $\Delta x^{2}_{\mathrm{SQ,MIN}}=(1/\eta)(\Delta X^{2}_{\mathrm{SQ,MIN}}-(1-\eta)/2)$. We find $\Delta x^{2}_{\mathrm{SQ,MIN}}=12^{+30}_{-12}~{}\%$ of the vacuum variance. For comparison, the most highly squeezed optical state ever made has a variance of only $7~{}\%$ of the vacuum variance Mehmet et al. (2010). Figure 4: Mean of 35 reconstructions of the Wigner function of the state exiting the SQ, inferred by maximum likelihood under the “best-guess” assumption, in quanta units. The faint pattern of ripples extending from the origin is caused by truncation at 30 photons of the density matrix used to represent the state. The white circle at the origin shows the full-width at half-maximum of the vacuum state. Producing squeezed states of itinerant modes allows the generation of distributable entanglement by sending two copies of a squeezed vacuum state through the two input ports of a balanced beam splitter. The coherent information Schumacher and Nielsen (1996) is one useful way to characterize the entanglement between the two output modes. The asymptotic number of maximally entangled qubit pairs (e-bits) that can be distilled per copy of the noisy entangled state, by using local operations and one-way classical communication, is at least as large as the coherent information Devetak and Winter (2005). Given two copies of $\rho$, one could make two entangled modes with $2.5^{+1.0}_{-0.4}$ e-bits of coherent information. In conclusion, we have reconstructed the Wigner function of an itinerant squeezed microwave field generated at the output of a Josephson Parametric Amplifier. Using a second JPA as a preamplifier has increased the quantum efficiency of the microwave homodyne detection from approximately $2~{}\%$ to $36~{}\%$. The level of squeezing is primarily limited by noise added to the squeezed state by the JPA. Improving the performance of the JPAs (as both squeezers and phase-sensitive amplifiers) will require more detailed investigation of the source of this noise. We used maximum likelihood quantum state tomography to deconvolve the QM inefficiency in order to precisely characterize the state generated. This is an important step toward generating easily distributable microwave entanglement on chip. Notes: A different method was recently used to obtain a similar state reconstruction Eichler et al. (2010). ###### Acknowledgements. The authors acknowledge support from the DARPA/MTO QuEST program. ## References * Houck et al. (2007) A. A. Houck et al., Nature 449, 328 (2007). * Deléglise et al. (2008) S. 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The crucial aspect that makes this calibration possible is that the JPA cavities have widely-tunable resonance frequencies, adjusted by imposing a magnetic flux Castellanos-Beltran et al. (2008, 2009). Far from resonance the JPA cavities behave as open circuits. They are simply mirrors that reflect the microwave field without otherwise transforming it; therefore, either the SQ or AMP or both stages can effectively be bypassed. We begin with both JPA stages bypassed, so that they have $G_{\rm{S}}=G_{\rm{A}}=1$. If the switch were lossless, when it is connected to the cold load, the noise power exiting the HEMT amplifier would be $S=G_{H}(A_{H}+S_{f})$, where $S_{f}=(1/2)+n_{f}=(1/2)+[\exp(\hbar\omega/k_{B}T_{f})-1]^{-1}$ and $T_{f}$ is the refrigerator’s temperature. Notice that the result doesn’t depend on the transmissivities $\alpha$, $\beta$, or $\xi$ because these are at the same temperature as the cold load, consequently each loss component emits as much power as it absorbs. However, with the switch connected to the hot load, the expression for the total power at the output becomes $S=G_{H}(A_{H}+(\xi\alpha\beta)S_{h}+(1-\xi\alpha\beta)S_{f})$, with $S_{h}=(1/2)+n_{h}=(1/2)+[\exp(\hbar\omega/k_{B}T_{h})-1]^{-1}$ and $T_{h}=4.1$ K. In both cases, we expect and observe that $S$ depends linearly on $S_{f}$ with an offset. By fitting these linear dependencies we can extract $G_{H}$, $A_{H}$, and the product $\xi\alpha\beta$. We cannot assume that the switch is lossless. Because its loss sits at 4.1 K, it will always emit noise power $S_{h}(1-\lambda)+S_{in}\lambda$, where $S_{in}$ is the incident noise and $\lambda$ is the switch transmissivity. So, even when $n_{f}\ll 1/2$, the state presented at the SQ stage will have average thermal occupancy $\bar{n}=(1-\lambda)\xi n_{h}$. We write the noise power at the output as function of $S_{f}$, for switches in both positions as $S_{1c}=G_{H}A_{H}+S_{h}G_{H}(1-\lambda)\xi\alpha\beta+S_{f}[G_{H}\lambda\xi\alpha\beta+G_{H}(1-\xi\alpha\beta)]=b_{1c}+m_{1c}S_{f}$ (2) $S_{1h}=G_{H}A_{H}+S_{h}G_{H}(\xi\alpha\beta)+S_{f}[G_{H}(1-\xi\alpha\beta)]=b_{1h}+m_{1h}S_{f},$ (3) where the subscript $1c$ ($1h$) corresponds to the switch connected to the cold (hot) load. Fitting our noise data to the right hand side of Eq. 2 and 3, we can obtain the four parameters $b_{1h},b_{1c},m_{1h}$ and $m_{1c}$. However as these parameters are not independent, $S_{h}=(b_{1h}-b_{1c})/(m_{1c}-m_{1h})$, we cannot extract the switch loss independently. We can nevertheless bound this unknown loss by taking a worst case estimate as the manufacturers minimum specified transmission (at room temperature) $\lambda=0.83$ and assuming it is less lossy at 4.1 K. We moreover confirmed that at room temperature the frequency dependent loss of the switch is within the manufacturer’s specification. Then by using $1<\lambda<0.83$, we can bound the desired parameters using Eq. 2 and 3, with the expressions $(\xi\alpha\beta)^{-1}=1+m_{1h}S_{h}\lambda/(b_{1h}-b_{1c})$, and $G_{H}=m_{1h}/(1-\xi\alpha\beta)$, and $A_{H}=(b{1c}/G_{H})-(1-\lambda)S_{h}(\xi\alpha\beta)$. We then perform the same analysis, finding the linear dependence of the output noise on $S_{f}$ and on the switch position, with the AMP ON and SQ OFF. From these fits and knowledge of $A_{H}$ and $G_{H}$ we find $\xi\alpha$, $A_{A}$, and$G_{A}\beta$. Finally, we operate the experiment with AMP OFF and SQ ON. A third time we fit the linear dependence of $S$ on $S_{f}$ with the switch in both positions, determining $\xi$, $\alpha$ and $\beta$ separately (Fig. 5b). We evaluate the expressions for $\alpha$, $\beta$, $A_{A}$, $A_{H}$, $G_{H}$ and $\bar{n}$ at the bounds on $\lambda$, finding the range of values in the main text. We also find $\xi=-9.9\pm 1$ dB, of which 6 dB arises from an attenuator that has been placed at the input of the SQ stage. Figure 5: The noise density $S$ in arbitrary units at the output of the measurement versus refrigerator temperature $T_{f}$. a.) Data acquired with the AMP and SQ OFF and the switch connected to the cold load (circles) and hot load (squares). The lines are linear fits to $S$ versus $S_{f}$ for the case of the switch connected to the cold load (solid) and hot load (dashed). b.) Data acquired with the AMP ON and SQ OFF (blue) and AMP OFF and SQ ON (red), with the switch connected to the cold load (circles) with a linear fit (solid) and hot load (squares) with a linear fit (dashed). The arbitrary y-scale is consistent between the six plots. The linear fits do not appear as lines because we plot $S$ versus $T_{f}$ rather than $S_{f}$. To acquire these calibration data sets, we regulate the refrigerator’s temperature at 10 values between base temperature ($T<50$ mK) and 800 mK, which requires about 7 hours to complete. For each temperature point we measure the noise at the output under all six conditions, 2 switch positions, and 3 amplifier configurations (AMP OFF SQ OFF, AMP ON SQ OFF, and AMP OFF SQ ON). We inject a tone detuned from the AMP pump by 20 kHz. By dividing the noise power at the output of the chain by the power in this tone, we become insensitive to any variation in $G_{H}$ over the time needed to acquire the data. At the end of the calibration, we immediately operate the experiment with SQ ON and AMP ON, to acquire the data in the paper. In addition, we use the tone to ensure that we do not saturate the amplifier chain. Data is acquired by digitizing the output (IF port) of the mixer at rate of $10^{7}$ samples per second. We filter the IF port with a 5 MHz anti-aliasing low-pass filter. The digitized data is digitally filtered with a 3rd-order Butterworth high-pass filter with a 500 kHz corner (3 dB) frequency. The noise density $S$ is the average noise density in the frequency range between 500 kHz and 5 MHz. ## III Maximum likelihood analysis of the squeezed state Table 1: Inferred properties of the squeezed state, upon our three analysis assumptions. \| Pessimistic \| Best guess \| Optimistic ---\|---\|---\|--- Fidelity \| $0.66\pm 0.02$ \| $0.807\pm 0.016$ \| $0.960\pm 0.005$ Min. var. of comparison pure state $\left\|\psi\right\rangle$111Ratio of the variance of the squeezed quadrature of the pure squeezed vacuum state with highest fidelity to the variance of the vacuum.: \| $0.065\pm 0.009$ \| $0.060\pm 0.003$ \| $0.0493\pm 0.0006$ $\rho$’s purity \| $0.62\pm 0.02$ \| $0.74\pm 0.02$ \| $0.96\pm 0.01$ $\rho$’s sq. var.222Ratio of variance of most likely state $\rho$’s squeezed or anti-squeezed quadrature to the variance of the vacuum. \| $0.918\pm 0.002$ \| $0.484\pm 0.013$ \| $0.304\pm 0.008$ $\rho$’s anti-sq. var.222Ratio of variance of most likely state $\rho$’s squeezed or anti-squeezed quadrature to the variance of the vacuum. \| $25.54\pm 0.07$ \| $20.17\pm 0.06$ \| $19.18\pm 0.05$ Coherent info.333Coherent information (in e-bits) that could be produced with two copies of the squeezed state and a beam splitter. \| $2.19\pm 0.08$ \| $2.46\pm 0.09$ \| $3.42\pm 0.05$ Linear sq. var.444Direct linear inference of the squeezed state’s minimum variance, relative to vacuum variance. \| $0.40\pm 0.02$ \| $0.12\pm 0.02$ \| $-0.18\pm 0.02$ Table 1 shows the statistical errors in our estimates of inferred parameters characterizing the squeezed state, for the three analysis cases, based upon our systematic calibration uncertainties. The first line presents the fidelity $F=\langle\psi\|\rho\|\psi\rangle$, where $\rho$ is the maximum likelihood reconstructed density matrix of the field exiting the SQ and $\|\psi\rangle$ is the pure vacuum squeezed state that maximizes the fidelity. The second line gives the ratio of the minimum variance of $\|\psi\rangle$ to the variance of vacuum. The third line gives the purity $\mathrm{Tr}({\rho^{2}})$ of $\rho.$ The fourth and fifth lines give the ratios of the squeezed and anti-squeezed variances of the reconstructed state to the variance of vacuum. The sixth line presents the coherent information that could be obtained by combining on a beam splitter two copies $\rho$. The last line gives our estimate the the experimental states’ minimum variance based on direct linear inference. We have stated three variances that characterize the state created in this experiment: the linear estimate of the experimental state’s minimum variance ($12~{}\%$), the most likely state $\rho$’s minimum variance ($48~{}\%$), and the minimum variance of the pure squeezed vacuum state $\left\|\psi\right\rangle$ that maximizes the fidelity with $\rho$ ($6.0~{}\%$). Here we give more discussion of these variances. The quadrature measurements we observe are the linear combination of the quantum state created by the squeezer and vacuum fluctuations: $X_{\theta}=\sqrt{\eta}x_{\theta}+\sqrt{(1-\eta)}y_{\theta},$ where $x_{\theta}$ is the quadrature of the squeezed state, and $y_{\theta}$ is the quadrature of the vacuum state. Solving for $x_{\theta}$ gives $x_{\theta}=\frac{1}{\sqrt{\eta}}\left(X_{\theta}-\sqrt{\left(1-\eta\right)}y_{\theta}\right).$ Therefore the inferred variance of the squeezed state’s quadrature $\Delta x_{\theta}^{2}$ is $\Delta x_{\theta}^{2}=\frac{1}{\eta}\left[\Delta X_{\theta}^{2}-\left(1-\eta\right)\Delta y_{\theta}^{2})\right].$ The vacuum variance $\Delta y_{\theta}^{2}=1/2$, and we can easily calculate an unbiased estimate of $\Delta X_{\theta}^{2}$ for every phase $\theta$. This gives us an unbiased estimate of $\Delta x_{\theta}^{2}$ that does not depend on the details (for example, Gaussianity) of the quantum state. We calculate $\Delta x_{\theta}^{2}$ using 20,000 quadrature measurements at each of 100 evenly spaced $\theta$ and calculate the minimum value $\Delta x_{\mathrm{SQ,MIN}}^{2}$, in Table 1. The statistical uncertainties show one standard deviation in the estimate of $\Delta x_{\mathrm{SQ,MIN}}^{2}$. For the “optimistic case” we calculate a negative variance, which is clearly unphysical. This is a sign of inconsistency in the “optimistic” calibration parameters. Because the “optimistic” estimate for the squeezed state is computed using the lower bounds on $\eta$ and $\overline{n}$, this negative variance is evidence that the detector’s true $\eta$ and / or effective gain ($\Delta X_{\mathrm{SQ,OFF}}^{2}/\Delta V_{\mathrm{SQ,OFF}}^{2}$) must be larger than the lower bounds set by calibration. The minimum variance of $\rho$ is significantly higher than this linear estimate. This is caused by bias in the maximum likelihood method. Quantum state estimation by maximum likelihood is biased toward more mixed states, and the amount of bias increases with increasing purity of the state from which the measurements are drawn Glancy et al. (2009). Based on numerical experiments, the bias in our estimates of the fidelity should be well below the uncertainty level set by systematic effects. However, the bias in our estimates of the minimum variance of the inferred state could be larger. To attempt to quantify this effect, we simulated measuring and performing maximum likelihood tomography on a Gaussian state. This Gaussian state is chosen to have minimum and maximum variances equal to those calculated by the linear method described above for the “best-guess” case. By computer we simulate 10,000 quadrature measurements (the same number we used for ML analysis of the true experiment) from this Gaussian state and perform maximum likelihood tomography on those measurements. The inferred state has minimum variance $40~{}\%$. Therefore it is possible that the experimental state has smaller minimum variance than the most likely state inferred from only 10,000 measurements. Because we have some independent evidence for non-Gaussian effects in the experiment, we cannot quantify this size of this bias using this Gaussian simulation. Other numerical simulations have confirmed that this bias decreases as the number of measurements analyzed increases and that this bias is not caused by truncation of the Hilbert space at 30 photons. The apparent discrepancy between the $6~{}\%$ for the variance of $\|\psi\rangle$ and the $48~{}\%$ for the variance of $\rho$ also deserves some comments. It is important to note that one would not expect the minimum variance of a mixed state to equal the minimum variance of its highest fidelity pure state. The fidelity between a mixed Gaussian state (centered at the origin of phase space) whose minimum and maximum variances are $v_{x}$ and $v_{p}$ and a pure squeezed vacuum state with minimum variance $v_{s}$ is given by $F_{\mathrm{Gauss}}=\frac{2}{\sqrt{\frac{(1+4v_{s}v_{p})(v_{s}+v_{x})}{v_{s}}}}.$ The highest fidelity pure state has minimum variance $v_{s}=\frac{1}{2}\sqrt{\frac{v_{x}}{v_{p}}}$, and the fidelity between these two states is $F_{\mathrm{Gauss,max}}=\frac{2}{1+2\sqrt{v_{x}v_{p}}}$ Consider the state $\sigma$ to be a Gaussian state with minimum variance of $48~{}\%$ and maximum variance $2017~{}\%$. ($\sigma$ has variances equal to those of our state $\rho$, but unlike $\rho$, $\sigma$ is guaranteed to be Gaussian.) Then let $\|\psi\rangle$ be the pure squeezed vacuum state that has maximum fidelity with $\sigma$. $F_{\mathrm{Gauss,max}}=\langle\psi\|\sigma\|\psi\rangle=0.49$, and the minimum variance of $\|\psi\rangle$ is $7.7~{}\%$. The difference between the minimum variances of $\rho$ and $\left\|\psi\right\rangle$ is to be expected. However, the maximum fidelity of $\rho$ is significantly larger than we would expect if it was perfectly Gaussian. This non-Gaussianity could be caused by bias in the maximum likelihood inference and / or genuine non-Gaussian effects in the experiment. Tomographic reconstruction of a quantum state requires that the experimental device always creates the same (potentially mixed) quantum state, that the measurements are well described by inefficient quadrature measurements, and that the calibration of those measurements is consistent. In this experiment we have observed some evidence that at least one of these assumptions is violated. The likelihood of the maximum likelihood state is significantly lower than one should expect from simulated measurements on that state. That is, if the tomographic assumptions above were true, we expect to find a significantly higher value for the maximum likelihood. We believe this effect could be caused by an interaction between the state preparation and measurement stages of the experiment, such as a phase dependent efficiency of the measurement JPA, and/or non linear processes in the measurement.	arxiv-papers	2010-11-30T21:01:51	2024-09-04T02:49:15.349960	{ "license": "Public Domain", "authors": "F. Mallet, M. A. Castellanos-Beltran, H. S. Ku, S. Glancy, E. Knill,\n K. D. Irwin, G. C. Hilton, L. R. Vale, K. W. Lehnert", "submitter": "Konrad Lehnert", "url": "https://arxiv.org/abs/1012.0007" }
1012.0143	††thanks: Author to whom correspondence should be addressed; fwwang@aphy.iphy.ac.cn # First-principles study of pressure-induced phase transition and electronic property of PbCrO3 Bao-Tian Wang Institute of Theoretical Physics and Department of Physics, Shanxi University, Taiyuan 030006, People’s Republic of China State Key Laboratory of Magnetism, Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China Wen Yin State Key Laboratory of Magnetism, Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China Wei-Dong Li Institute of Theoretical Physics and Department of Physics, Shanxi University, Taiyuan 030006, People’s Republic of China Fangwei Wang State Key Laboratory of Magnetism, Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China ###### Abstract We have performed a systematic first-principles investigation to calculate the structural, electronic, and magnetic properties of PbCrO3, CrPbO3 as well as their equiproportional combination. The local density approximation (LDA)$+U$ and the generalized gradient approximation$+U$ theoretical formalisms have been used to account for the strong on-site Coulomb repulsion among the localized Cr $3d$ electrons. By choosing the Hubbard _U_ parameter around 4 eV within LDA$+U$ approach, ferromagnetic, and/or antiferromagnetic ground states can be achieved and our calculated volumes, bulk moduli, and equation of states for PCO-CPO in $R3$ phase and PCO in $Pm\bar{3}m$ or $R3c$ phases are in good agreement with recent experimental Phase I and Phase II [W. Xiao _et al._ , PNAS 107, 14026 (2010)], respectively. Under pressure, phase transitions of $R3$ PCO-CPO to $Pm\bar{3}m$ PCO at 1.5 GPa and $R3$ PCO-CPO to $R3c$ PCO at -6.7 GPa have been predicted. The abnormally large volume and compressibility of Phase I is due to the content of CrPbO3 in the experimental sample and the transition of PbO6/2 octahedron to CrO6/2 upon compression. Our electronic structure study showed that there will occur an insulator-metal transition upon the phase transitions. Clear hybridization of Cr 3 _d_ and O 2 _p_ orbitals in wide energy range has been observed. ###### pacs: 61.50.Ah, 71.15.Mb, 75.25.+z ## I INTRODUCTION Strongly correlated electron systems of transition-metal oxides with ABO3 cubic perovskite or pseudo cubic perovskite structures exhibit particular interesting physical properties. Zhong ; Imada ; Tokura ; Dagotto Their ferroelectric, ferromagnetic, ferroelastic, multiferroic, and/or magnetoresistive features originate from the mutual interplay of various degrees of freedom, including lattice, spin, charge and orbital, in their partially filled B site 3 _d_ electrons. The multiple chemical characters of the A ion with lone pair electrons, especially for Bi3+ and Pb2+, also play an important role. Neaton ; Picozzi Correctly describing of their electronic and magnetic structures are critical. In late 1960s, a few groups Roth1 ; Roth2 ; Chamberland1 ; Chamberland2 ; Goodenough ; Weiher successfully synthesized some perovskites containing Cr4+ (CaCrO3, SrCrO3, and PbCrO3) under high temperature and high pressure. For PbCrO3 (PCO), a lattice constant of about 4.00 Å for the cubic structure was determined by X-ray diffraction on single crystal and powder samples and powder neutron diffraction Roth1 ; Roth2 ; Chamberland1 . It was reported that the PCO is an antiferromagnetic (AFM) G-type semiconductor with a magnetic moment of $\mathtt{\sim}$1.9 $\mu_{B}$ on each Cr ion Roth1 ; Roth2 and with 0.27 eV activation energy. Chamberland1 The Neel temperature ($T_{\text{N}}$) of about 240 K was obtained by examining the magnetic and transport properties. Roth1 ; Roth2 For CaCrO3 and SrCrO3, preliminary structural, magnetic, and conductive properties were also investigated. Chamberland2 ; Goodenough ; Weiher From then on, in a long period of more than thirty years little works had focused on these systems due to difficulty of synthesis. However, a renewed interest on them has been inspired on their transport and magnetic properties, insulator-metal transition, and pressure behaviors. Zhou ; Ortega ; Komarek ; ArevaloJSSC ; ArevaloJPCM ; ArevaloInorg ; Xiao Anomalous properties of Seebeck coefficient, thermal conductivity, magnetic susceptibility, and room-temperature compressibility have been observed for SrCrO3 by Zhou _et al._ They concluded that SrCrO3 is nonmagnetic (NM) insulator and CaCrO3 is also an insulator at low temperature. But more recent studies claimed that both these perovskites are AFM metals. Ortega ; Komarek An orbital ordering transition from $t_{2g}^{2}$ to $d_{xy}^{1}(d_{xz}d_{yz})^{1}$ and electronic phase coexistence of $C$-AFM tetragonal and NM cubic phases have been discovered in SrCrO3. Ortega Komarek _et al._ reported that CaCrO3 is an intermediately correlated metal with similar $C$-type AFM ground state. Thus, whether these systems are metallic, strongly correlated, and spin ordered is still controversial. Streltsov ; Lee Recent experimental works of PCO concentrated on its structure, electron energy loss spectroscopy, magnetic structure, and high pressure phase transition. ArevaloJSSC ; ArevaloJPCM ; ArevaloInorg ; Xiao Electron diffraction and high-resolution electron microscopy study revealed that the microstructure of “PbCrO3” is a rather complex perovskite with a compositionally modulated $a_{p}\times 3a_{p}\times(14\mathtt{\sim}18)a_{p}$ superlattice structure, where $a_{p}=4.002$ Å is the lattice constant of the cubic PCO perovskite. ArevaloJSSC The magnetic structure of PCO is also complex. Alario-Franco _et al._ reported AFM ordering of the chromium moments at $T_{\text{N}}$$\mathtt{\sim}$245 K with a spin-reorientation at temperature range of 185 K to 62 K and their magnetic hysteresis loops for “PbCrO3” suggested weak ferromagnetism at low temperature. ArevaloInorg Their resistivity measurements indicated two activation energies ranges with 0.11 and 0.26 eV in different temperatures. As for pressure study, recent work performed by Xiao _et al._ Xiao observed a large volume collapse in the isostructural transition of cubic PCO perovskite at $\mathtt{\sim}$1.6 GPa from Phase I to Phase II. They concluded that the transition seems not related with any change of electronic state, but probably has tight relation with the abnormally large volume and compressibility of the Phase I. The real Phase I might be a kind of mixture of PbCrO3-CrPbO3 (PCO-CPO) combination due to the fact that the cubic lattice constant is enlarged if the CrO6/2 octahedron could be replaced by PbO6/2 Xiao . In present study, we focus our sight on PCO, PCO-CPO, and CrPbO3 (CPO) in the cubic perovskite structure (space group $Pm\bar{3}m$) and some possible distorted perovskite structures, such as $R3c$, $R3$, and $P4/mmm$ phases. Electronic and magnetic properties as well as pressure behaviors have been systematically investigated by the first-principles electronic structure calculations based on density functional theory (DFT) and DFT+_U_ schemes due to Dudarev _et al_. Dudarev The validity of the ground-state calculation is carefully tested. Our calculated lattice parameter and bulk modulus _B_ for cubic PCO are well consistent with previous local density approximation (LDA) and generalized gradient approximation (GGA) results Xiao . The total energy, lattice constant, bulk modulus _B_ , and spin moment of Cr ion for NM, ferromagnetic (FM), and AFM phases calculated in wide range of effective Hubbard _U_ parameter are presented and our calculated results within LDA+_U_ with _U_ =3-4 eV for PCO in $Pm\bar{3}m$ or $R3c$ phases and PCO-CPO in $R3$ phase accord well with experimental Xiao Phase II and Phase I, respectively. Our calculated spin moment by LDA+_U_ is in good agreement with recent experimental value of saturation moment $M_{\rm{sat}}$=1.70 $\mu_{B}$, which is deduced from the effective moment $M_{\rm{eff}}$=2.51 $\mu_{B}$ ArevaloInorg according to the relation $M_{\rm{eff}}$=2$[S(S+1)]^{1/2}$=$[M_{\rm{sat}}(M_{\rm{sat}}+2)]^{1/2}$. The P-V relations of PCO and PCO-CPO are calculated to compare with experiment. The insulting property of “PbCrO3” at ambient condition is successfully predicted. ## II computational methods First-principles DFT calculations on the basis of the frozen-core projected augmented wave (PAW) method of Blöchl PAW are performed within the Vienna ab initio simulation package (VASP) Kresse3 , where the exchange and correlation effects are described by the DFT within LDA and GGA LDA ; GGA . For the plane- wave set, a cutoff energy of 500 eV is used. The _k_ -point meshes in the full wedge of the Brillouin zone (BZ) are sampled by 12$\times$12$\times$12 and 6$\times$6$\times$6 grids according to the Monkhorst-Pack Monk scheme for PCO, CPO, and PCO-CPO in their cubic and rhombohedral unit cells, respectively. The cubic perovskite structure are built for nonmagnetic (NM) and ferromagnetic (FM) calculations and the rhombohedral unit cells (Fig. 1) for G-type antiferromagnetic (AFM) configuration, which is ($\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$) order in terms of the original perovskite cell. The rhombohedral unit cell for PCO in its cubic structure is constructed using the $R3c$ space group ($\alpha$=60∘) with Pb atoms in $2a$(0, 0, 0) site, Cr in $2a$($\frac{1}{4}$, $\frac{1}{4}$, $\frac{1}{4}$) site, and O in $6b$($\frac{1}{2}$, 0, $\frac{1}{2}$) site. Moving away slightly atoms from these high symmetry positions will lower the symmetry and consequently build the cell in real $R3c$ or $R3$ phase. In this study, the Pb 5$d$106$s$26$p$2, Cr 3$d$54$s$1, and O 2$s$22$p$4 orbitals are explicitly included as valence electrons. The strong on-site Coulomb repulsion among the localized Cr 3 _d_ electrons is described by using the formalism formulated by Dudarev _et al._ Dudarev . In this scheme, the total LDA (GGA) energy functional is of the form $\displaystyle E_{\mathrm{{LDA(GGA)}+U}}$ $\displaystyle=E_{\mathrm{{LDA(GGA)}}}$ $\displaystyle+\frac{U-J}{2}\sum_{\sigma}[\mathrm{{Tr}\rho^{\sigma}-{Tr}(\rho^{\sigma}\rho^{\sigma})],}$ (1) where $\rho^{\sigma}$ is the density matrix of _d_ states with spin $\sigma$, while _U_ and _J_ are the spherically averaged screened Coulomb energy and the exchange energy, respectively. In this work, the Coulomb _U_ is treated as one variable, while the parameter _J_ is set to 0.5 eV. Since only the difference between _U_ and _J_ is meaningful in Dudarev’s approach, therefore, we label them as one single parameter _U_ for simplicity. To obtain the energy data under different pressures, we perform the structure-relaxation calculations at a series of fixed volumes. The corresponding pressure values are deduced from the energy-volume data by $P$=$-\partial$$E$/$\partial$$V$. ## III results ### III.1 Structural and magnetic properties of cubic PCO Both spin-unpolarized and spin-polarized calculations are performed for cubic PCO. For NM, FM, and AFM configurations, the total energies (-39.685 eV, -39.706 eV, and -39.818 eV, respectively) calculated within the DFT (_U_ =0) have no visible differences. After turning on the Hubbard _U_ , the NM phase is not energetically favorable both in the LDA+_U_ and GGA+_U_ formalisms compared with FM and AFM phases. In the following, we only present results of FM and AFM configurations. The dependences of total-energy differences (per formula unit at respective optimum geometries) between FM and AFM ($E_{\text{FM}}\mathtt{-}E_{\text{AFM}}$), lattice parameter, bulk modulus, and spin moments of Cr ions on _U_ for cubic PCO are shown in Fig. 2. The theoretical equilibrium volume, bulk modulus _B_ , and pressure derivative of the bulk modulus _B ′_ are obtained by fitting the third-order Birch-Murnaghan equation of state (EOS) Birch . For comparison, recent experimental values Xiao of $a_{0}$ and _B_ for Phase I and Phase II as well as the experimental results of spin magnetic moment of Cr ions are also shown. In LDA or GGA (_U_ =0 eV), the total energy of AFM phase is lower than that of FM phase. However, as shown in Fig. 2(a), the total energy of the FM phase decreases to become lower than that of the AFM phase when increasing _U_. At a typical value of _U_ =4 eV, the total-energy differences between FM and AFM ($E_{\text{FM}}\mathtt{-}E_{\text{AFM}}$) reach their minimums of $-$74 meV and $-$91 meV within the LDA+_U_ and GGA+_U_ formalisms, respectively. Note that we have also considered other types of AFM configurations and our results show that the _G_ -AFM in ($\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$) order is the most stable state. These results are consistent with recent experimental observations ArevaloInorg , where they reported the AFM ordering of the chromium moments at $T_{\text{N}}$$\mathtt{\sim}$245 K with a spin- reorientation at temperature range of 185 K to 62 K and their magnetic hysteresis loops for “PbCrO3” suggested weak ferromagnetism at low temperature. As clearly shown in Fig. 2(b), our calculated lattice parameters of cubic PCO within two DFT+_U_ approaches are all by large smaller than the experimental value $a_{0}$=4.013 Å Roth1 ; Roth2 ; Chamberland1 ; Xiao of Phase I. Corresponding results of bulk modulus are dramatically bigger than that obtained by experiment (_B_ =59 GPa) for Phase I [Fig. 2(c)]. But the experimental lattice parameter $a_{0}$=3.862 Å and bulk modulus _B_ =187 GPa of Phase II well lies in the range of our calculated results of cubic PCO, which supports their conclusion of cubic PCO perovskite Xiao that the real Phase I might be a kind of mixture of random PCO-CPO combination. In next subsection, we will discuss carefully the structural properties of PCO-CPO. As shown in Fig. 2, the tendencies of $a_{0}$, _B_ , and spin moments of Cr ions for FM phase with _U_ are similar to that of the AFM phase. Results of NM phase (not presented) further indicates its unstable nature at low temperature. As shown in Fig. 2(b), LDA underestimates the lattice parameter with respect to the experimental value while GGA coincides well with experiment. Our LDA and GGA results for NM phase accords well with previous first-principle calculations Xiao . Result of LDA is due to its typical overbinding character. The LDA+_U_ method will lead to relative larger equilibrium volume compared to the LDA and therefore improves the agreement with experiment, especially for FM and AFM phases. At a typical value _U_ =4 eV, the LDA+_U_ gives $a_{0}$=3.822 Å for AFM cubic PCO which is very close to the experimental value. On the other hand, the GGA+_U_ enlarges the underbinding effect with increasing Hubbard _U_. As a comparison, at _U_ =4 eV, the GGA+_U_ gives $a_{0}$=3.921 Å. As for the dependence of bulk modulus _B_ on _U_ shown in Fig. 2(c), it is clear that the LDA results (230-153 GPa) are always higher than the GGA results (187-113 GPa), which is due to the above mentioned underbinding effect of the GGA approach. At a typical value _U_ =4 eV, the LDA+_U_ and GGA+_U_ give _B_ =198 (_B ′_=4.6) and 148 GPa (_B ′_=5.1) for AFM cubic PCO, respectively. Obviously, result calculated within LDA+_U_ with _U_ =4 eV is consistent with the experimental value of _B_ =187 GPa Xiao . For the dependence of spin moments of Cr ions on _U_ shown in Fig. 2(d), we see clear increasing amplitude of magnetic moments with _U_ for both FM and AFM phases. Our calculated value of 1.47 $\mu_{B}$ using LDA for AFM phase is in good agreement with previous LMTO calculation value of 1.414 $\mu_{B}$ Jaya . At a typical value _U_ =4 eV, the LDA+_U_ and GGA+_U_ give magnetic moments of 2.46 and 2.62 $\mu_{B}$ for AFM PCO, respectively, both of which exceed recent experimental value of 1.70 $\mu_{B}$ ArevaloInorg . Similar trend has also been exhibited in study of BiFeO3 Neaton . Therefore, inclusion of on-site Coulomb energy for adequately describing the structural and magnetic properties is crucial. In study of CrO2, x-ray absorption and resonant photoemission spectroscopy support the importance of Coulomb correlations Dedkov . Huang _et al._ concluded that the on-site Coulomb interaction energy of CrO2 is 3-4 eV through comparing their experimental measurements and LDA+_U_ calculations. They found that the shift of Cr 3 _d_ spin-up DOS slightly away from the Fermi level increases the Cr spin moment. In our present study, similar shift of Cr 3 _d_ DOS is also observed (see below). Overall, comparing with the experimental data, the accuracy of our atomic-structure prediction for AFM cubic PCO is quite satisfactory by tuning the effective Hubbard parameter _U_ in a range of 3-4 eV within the LDA+_U_ approach, which supplies the safeguard for our following study of electronic structure and pressure behaviors of PCO. In following study, we present our results within LDA and LDA+_U_ with constant _U_ =4 eV. ### III.2 Phase transition analysis After previous analysis, one question arises: What’s the ground state of experimentally observed “PbCrO3” at ambient condition. In this section, we will test carefully the ground state of “PbCrO3” by calculating the total energies of PCO, PCO-CPO, and CPO in cubic perovskite structure and some possible distorted perovskite structures. Their relative energies in $G$-AFM configuration calculated within LDA+$U$ are shown in Fig. 3. Clearly, although the equilibrium lattice parameter (3.998 Å) for $Pm\bar{3}m$ phase of PCO-CPO accords well with experimental data of Phase I Xiao , the $Pm\bar{3}m$ phase of PCO-CPO and CPO are not energetically favorable in the whole range of volume. After testing various possible crystal structures and atomic arrangements, we find that $R3c$ phase of PCO and $R3$ phase of PCO-CPO possess same energy level with that of cubic PCO in some volumes. In the whole range of volume, $R3c$ PCO is more energetically favorable than cubic PCO. Considering this material only can be synthesized under high temperature and high pressure, the high-pressure phase of PCO can be understood as crystallizing in $Pm\bar{3}m$ or $R3c$ structure. Increasing the cell volume, as shown in Fig. 3, the $R3$ PCO-CPO becomes more energetically favorable than the two high-pressure phases. This clearly indicates that the structure of “PbCrO3” at ambient pressure is $R3$ PCO-CPO. Therefore, upon compression there will occur structural phase transition among the three phases. As shown in the inset of Fig. 3, phase transitions of $R3$ PCO-CPO to $Pm\bar{3}m$ PCO at 1.5 GPa and $R3$ PCO-CPO to $R3c$ PCO at -6.7 GPa can be predicted by the slopes of the common tangent rule. The former value coincides well with recent experimentally observed value of 1.6 GPa, while the latter value is smaller than that. As a result, the crystal structures of PCO under high pressure need more works to clarify. Our present study give two most possible structures: $Pm\bar{3}m$ or $R3c$. Concerning the energetics of the transitions from $R3$ PCO-CPO to $Pm\bar{3}m$ PCO at 1.5 GPa and from $R3$ PCO-CPO to $R3c$ PCO at -6.7 GPa, Fig. 3 clearly shows that per formula unit of PCO-CPO needs 97 meV and 458 meV of energy, respectively. In Table I, we present the calculated lattice parameters for PCO in $R3c$ phase and PCO-CPO in $R3$ structure. It can be found that the equilibrium volume of $R3c$ PCO is consistent with that of experimental Phase II and $R3$ PCO-CPO comparable to Phase I Xiao . Although Xiao _et al._ reported that both the Phase I and Phase II crystallize in cubic perovskite structure, our calculations are different from their observations. Our present results need more experimental works to test. For $R3c$ PCO and $R3$ PCO-CPO, our calculated values of $E_{\text{FM}}\mathtt{-}E_{\text{AFM}}$ (per formula unit) are $-$80 meV and $-$62 meV within the LDA+_U_ formalism, respectively, the corresponding bulk moduli _B_ =164 (_B ′_=7.9) and 153 GPa (_B ′_=4.3) for AFM phase, respectively, _B_ =157 (_B ′_=5.6) and 150 GPa (_B ′_=4.4) for FM phase, respectively, and spin moments of Cr ions are 2.79 and 2.92 $\mu_{B}$ for AFM configuration, respectively. Averaged to every Cr4+ ion, the $E_{\text{FM}}\mathtt{-}E_{\text{AFM}}$ of $R3$ PCO-CPO is about $-$31 meV, which is almost consistent with recent experimental report ArevaloInorg of the AFM ordering at $T_{\text{N}}$$\mathtt{\sim}$245 K. For the bulk moduli, values of $R3c$ PCO are slightly smaller than the experimental value of Phase II _B_ =187 GPa Xiao , while values of $R3$ PCO-CPO are prominently larger than the experimental value of Phase I. The abnormal high compressibility of Phase I is due to the fact that the CrO6/2 to PbO6/2 transition has been compressed to occur under low pressure (see below). For spin moments, results of FM phase are almost equal to the AFM phase for PCO-CPO in $R3$ phase and PCO in both $Pm\bar{3}m$ and $R3c$ phases. Table 1: Calculated lattice constant $a$, rhombohedral angle $\alpha$, volume $V$, and Wyckoff parameters for PCO in $R3c$ phase and PCO-CPO in $R3$ structure. For $R3c$ phase, the Wyckoff positions 2$a$ ($x,x,x$) and 6$b$ ($x,y,z$) refer to the cations and anions, respectively. In case of the $R3$ structure, the corresponding Wyckoff labels are 1$a$ ($x,x,x$) and 3$b$ ($x,y,z$). \| \| PCO \| PCO-CPO ---\|---\|---\|--- space group \| \| $R3c$ \| $R3$ $a$ [Å] \| \| 5.381 \| 5.800 $\alpha$ [∘] \| \| 60.86 \| 55.12 $V$ [Å3] \| \| 112.33 \| 122.31 Pb \| $x$ \| 0.977 \| 0.987/0.729 Cr \| $x$ \| 0.218 \| 0.209/0.525 O \| $x$ \| 0.542 \| 0.533/0.436 \| $y$ \| 0.958 \| 0.924/0.139 \| $z$ \| 0.392 \| 0.377/0.815 ### III.3 Pressure behaviors The equation of states of AFM PCO-CPO in $Pm\bar{3}m$ and $R3$ phases, AFM PCO in $Pm\bar{3}m$ and $R3c$ phases, and the experimental measured pressure- volume data from Ref. [Xiao ] are presented in Fig. 4. For $Pm\bar{3}m$ and $R3c$ phases of PCO ($R3$ phase of PCO-CPO), the relative smaller volumes calculated in our scheme compared with experimental Phase II (Phase I) originates from the typical overbinding character of LDA. From Fig. 4, one can find that our calculated volume collapses of $Pm\bar{3}m$ PCO-CPO to $Pm\bar{3}m$ PCO and $R3$ PCO-CPO to $R3c$ PCO at experimental phase transition pressure 1.6 GPa is about 12.4% and 8.0%, respectively. The former value is larger than the measured value (9.8%) in recent experiments Xiao , while the latter value is smaller than that. Underestimation of the volume collapse value, from $R3$ PCO-CPO to $R3c$ PCO, can be attributed to the experimental fact that the CrO6/2 to PbO6/2 transition has been compressed to occur under low pressure of around 0.1-1.6 GPa in the experimental compound “Phase I”. This kind of partial transition leads to abnormal high compressibility of Phase I compared with CaCrO3, SrCrO3, and high-pressure Phase II of PbCrO3 Zhou ; Xiao . ### III.4 Electronic structure Figure 5 shows the total density of states (DOS) as well as the projected DOS for the Cr 3 _d_ , Cr 4 _s_ , and O 2 _p_ orbitals for AFM PCO in $Pm\bar{3}m$ and $R3c$ phases and AFM PCO-CPO in $R3$ phase at selective values of _U_ within LDA+_U_ formalism. Corresponding band-structures calculated with _U_ =4 eV are presented in Fig. 6, where both spin-up and spin-down results are plotted. Since spin-down results for PCO in $Pm\bar{3}m$ and $R3c$ phases are same with their spin-up results, as indicated in Figs. 6(a)-6(b), we only plot in Figs. 5(a)-5(d) the spin-up results. For AFM PCO-CPO in $R3$ phase, slight differences between spin-up and spin-down can be seen in Figs. 5(e) and 6(c). Overall, results of PCO in both $Pm\bar{3}m$ and $R3c$ phases indicate that the AFM PCO is metallic without accounting for or after switching on the on- site Coulomb repulsion [see Figs. 5(a)-5(d) and 6(a)-6(b)]. This fact conflicts with the experimental observations that the “PbCrO3” is AFM semiconductor with 0.27 eV or 0.11 eV activation energy in different temperature ranges Chamberland1 ; ArevaloInorg . In our calculations even increasing the amplitude of _U_ up to 8 eV, the metallic state has not changed for these two phases of PCO. The metallic ground states have also been observed for NM and FM phases. Besides, inclusion of the spin-orbit coupling (SOC) and noncollinearity also can not open a gap at the Fermi level. Thus, we only can conclude that the high-pressure phase of PbCrO3 is a conductor. Additionally, results of CPO and PCO-CPO in $Pm\bar{3}m$ phase also show that they are conductors. Although a gap is opened with the Hubbard _U_ =6 eV for $Pm\bar{3}m$ PCO-CPO (not shown), the calculated insulating band gap (2.12 eV) is prominently larger than the experimental values Chamberland1 ; ArevaloInorg . In present study, we prefer to believe that the LDA+_U_ with _U_ =4 eV can give a correct depictions of the ground state electronic structures for PCO and PCO-CPO. For AFM PCO-CPO in $R3$ phase, our LDA+_U_ with _U_ =4 eV calculation open an insulating band gap of about 0.48 eV [see Fig. 5(e) and 6(c)]. Figure 6(c) clearly indicates that the valence band maximum (VBM) appear at Z point and conduction band minimum (CBM) at $\Gamma$ point in BZ. We find that both VBM and CBM have predominant O 2 _p_ state character mixed with significant Cr 3 _d_ contribution. Although the calculated value of band gap (0.48 eV) is almost two times of the experimental value 0.27 eV Chamberland1 ; ArevaloInorg , since the activation energy of the Phase I perovskite increases with lowing temperature ArevaloInorg , we believe that our calculation (valid only at 0 K) can give a proper depictions of the electronic structures for low-pressure phase of PbCrO3. Our calculations clearly illustrate that the “PbCrO3” will occur an insulator-metal transition together with the phase transition of $R3$ PCO-CPO to $Pm\bar{3}m$/$R3c$ PCO upon compression. We note that the conductivity of SrCrO3 and CaCrO3 exists controversial Zhou ; Ortega ; Komarek ; Streltsov ; Lee . Theoretical calculation for CaCrO3 Streltsov predicted metallic ground state with LDA and insulating state with LDA+_U_. For SrCrO3 Lee , the metallic ground sate was observed either with LDA or LDA+_U_. Besides, using LDA+_U_ method with _U_ =4 eV, Lee _et al._ Lee successfully predicted an orbital-ordering transition from $t_{2g}^{2}$ to $d_{xy}^{1}(d_{xz}d_{yz})^{1}$ for SrCrO3. No evidence of orbital ordering within the $t_{2g}$ shell for CaCrO3 was observed Komarek . In our study of PCO, CPO, and PCO-CPO compounds, this kind of orbital ordering in Cr $t_{2g}$ states has also not been found. As shown in Fig. 5, inclusion of the on-site Coulomb repulsion will lower the occupation energy of Cr 3 _d_ electrons from the Fermi level and elevate Cr 3 _d_ , Cr 4 _s_ , and O 2 _p_ orbitals occupation levels near $-$7.0 eV to high level. As a result, localization pictures of electrons occupation appear after introducing the Coulomb repulsion. From Figs. 5(c)-5(e), one can find that the Cr 4 _s_ contribution is limited. When Cr ions combining with O ions to form covalent/ionic bonds, part of Cr 3 _d_ and Cr 4 _s_ electrons will transfer to O 2 _p_ orbital. This kind of electron transfer behavior can be read from the partial DOS pictures. Considering the effect of _d_ -hole creation due to 3 _d_ -4 _s_ hybridization, we have also examined the effect of Coulomb repulsion on Cr 4 _s_ -shell. However, no difference of Cr 4 _s_ orbital has been found for introducing the Coulomb repulsion into the _d_ -shell or the _s_ -shell. In Figs. 5 and 6, only _d_ -shell is considered to participate in the Coulomb exchange. In the whole energy domain, electronic structures of AFM PCO in $Pm\bar{3}m$ and $R3c$ phases have no evident differences. The main occupation at the Fermi level is from Cr 3 _d_ and O 2 _p_ orbitals. Our results calculated at _U_ =0 eV are consistent with previous calculations Jaya . A clear hybridization of Cr 3 _d_ and O 2 _p_ orbitals in the energy range from $-$7.3 to 0.3 eV can be observed at _U_ =0 eV. After switching on the _U_ to 4 eV, this hybridization energy range is moved to from $-$6.2 to 0.2 eV. A well resolved peak of Cr 3 _d_ state at around $-$0.3 eV at _U_ =0 eV is flatted when the Habburd _U_ parameter being increased to about 4 eV. In addition, a band gap in the conduction band is apparent under _U_ =0 to 6 eV. This band gap increases from 0.6 eV at _U_ =0 eV to about 2.0 eV at _U_ =4 eV. The main occupation in the conduction band is from Cr 3 _d_ orbital with some contribution from O 2 _p_ states. For AFM PCO-CPO in $R3$ phase, hybridization of Cr 3 _d_ and O 2 _p_ orbitals in the energy range from $-$6.3 to $-$1.0 eV is clear. One narrow peak locates just below the Fermi level. ## IV CONCLUSIONS In conclusion, the ground state properties as well as the high pressure behaviors of PCO, CPO, and PCO-CPO compounds were studied by means of the first-principles DFT+_U_ method. By choosing the Hubbard _U_ parameter around 4 eV within the LDA+_U_ approach, FM and/or AFM ground states were achieved and our calculated volumes, bulk moduli, spin moments, and equation of states are in good agreement with recent experiments. While the PCO-CPO in $R3$ phase is consistent with the experimental low-pressure Phase I, both $Pm\bar{3}m$ and $R3c$ phases of PCO coincide well with high-pressure Phase II. Specially, the semiconductor nature of $R3$ PCO-CPO is in good agreement with experiments. These observations explicitly indicate the existence of strongly correlated electronic behaviors in these compounds. Our electronic spectrums illustrate a clear hybridization of Cr 3 _d_ and O 2 _p_ orbitals in wide energy range. In contrast to SrCrO3, the orbital-ordering transition from $t_{2g}^{2}$ to $d_{xy}^{1}(d_{xz}d_{yz})^{1}$ has not been found in these materials. ###### Acknowledgements. We are grateful to O. Eriksson for useful discussions. 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One can find that along [111] direction in PCO the Pb and Cr alternate while in PCO-CPO the Pb and Cr alternate in pairs. Compared with PCO, in PCO-CPO half percent of the A site and B site atoms are exchanged while in CPO the A site and B site atoms are totally exchanged. Fig. 2: (Color online) Dependences of (a) total-energy differences (per formula unit) between FM and AFM ($E_{\text{FM}}\mathtt{-}E_{\text{AFM}}$), (b) lattice parameter, (c) bulk modulus, and (d) spin moments of Cr ions on _U_ for AFM PCO in $Pm\bar{3}m$ structure. Fig. 3: (Color online) Comparison of relative energies of two unit cells of AFM PCO in $Pm\bar{3}m$ and $R3c$ phases, one formula unit of AFM PCO-CPO in $Pm\bar{3}m$ and $R3$ phases, and two unit cells of AFM CPO in $Pm\bar{3}m$ phase vs the volume. All results are calculated within LDA+_U_ at _U_ =4 eV. Phase transitions of $R3$ PCO-CPO to $Pm\bar{3}m$ PCO at 1.5 GPa and $R3$ PCO- CPO to $R3c$ PCO at -6.7 GPa can be predicted by the slopes of the common tangent rule, as shown in the inset. Fig. 4: (Color online) The P-V relations of the AFM PCO-CPO in $Pm\bar{3}m$ and $R3$ phases as well as AFM PCO in $Pm\bar{3}m$ and $R3c$ phases computed in the LDA+_U_ formalism. Experimental results from Ref. Xiao are also presented. The volume collapses at experimental phase transition pressure 1.6 GPa are labeled. Fig. 5: (Color online) The total DOS for AFM PCO in $Pm\bar{3}m$ and $R3c$ phases as well as AFM PCO-CPO in $R3$ phase computed in the LDA+_U_ formalism with selective values of _U_. The projected DOSs for the Cr 3 _d_ , Cr 4 _s_ , and O 2 _p_ orbitals are also shown. In panel (e), both spin-up and spin-down results are presented. The Fermi energy level is set at zero. Fig. 6: (Color online) Band-structures of AFM PCO in $Pm\bar{3}m$ and $R3c$ phases as well as AFM PCO-CPO in $R3$ phase computed in the LDA+_U_ formalism with _U_ =4 eV. While the solid lines show the spin-up results, the dashed lines stand for spin-down. The Fermi energy level is set at zero as shown by the short-dashed lines.	arxiv-papers	2010-12-01T10:08:57	2024-09-04T02:49:15.359987	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Bao-Tian Wang, Wen Yin, Wei-Dong Li and Fangwei Wang", "submitter": "Bao Wang", "url": "https://arxiv.org/abs/1012.0143" }
1012.0153	eurm10 msam10 # Experimental Investigation of Longitudinal Space-Time Correlations of the Velocity Field in Turbulent Rayleigh-Bénard Convection Quan ZHOU Email address for correspondence: qzhou@shu.edu.cn Chun-Mei LI Zhi- Ming LU Yu-Lu LIU Shanghai Key Laboratory of Mechanics in Energy and Environment Engineering, Shanghai Institute of Applied Mathematics and Mechanics, E-Institutes of Shanghai Universities, Shanghai University, Shanghai 200072, China (?? and in revised form ??) ###### Abstract We report an experimental investigation of the longitudinal space-time cross- correlation function of the velocity field, $C(r,\tau)$, in a cylindrical turbulent Rayleigh-Bénard convection cell using the particle image velocimetry (PIV) technique. We show that while the Taylor’s frozen-flow hypothesis does not hold in turbulent thermal convection, the recent elliptic model advanced for turbulent shear flows He & Zhang (2006) is valid for the present velocity field for all over the cell, i.e., the isocorrelation contours of the measured $C(r,\tau)$ have a shape of elliptical curves and hence $C(r,\tau)$ can be related to $C(r_{E},0)$ via $r_{E}^{2}=(r-U\tau)^{2}+V^{2}\tau^{2}$ with $U$ and $V$ being two characteristic velocities. We further show that the fitted $U$ is proportional to the mean velocity of the flow, but the values of $V$ are larger than the theoretical predictions. Specifically, we focus on two representative regions in the cell: the region near the cell sidewall and the cell’s central region. It is found that $U$ and $V$ are approximately the same near the sidewall, while $U\simeq 0$ at cell center. ###### keywords: Rayleigh-Bénard Convection, velocity space-time correlations, Taylor’s frozen- turbulence hypothesis ## 1 Introduction Turbulent flows contain eddies with various scales. Turbulent kinetic energy is transferred from eddies with the largest scale of turbulence, $L$, at which energy is injected into the turbulence system, to eddies with the smallest scale of turbulence, $\eta$, at which energy is dissipated by fluid viscosity. Such cascade processes are usually characterized by the velocity structure functions, $S_{p}(r)=\langle\|\delta_{r}v\|^{p}\rangle$, defined as moments of velocity increments over a space separation $r$, where $\langle\cdots\rangle$ denotes a time average. Since the pioneering work of Kolmogorov (1941), various theories and models have been put forwards to predict the scaling behaviors of the velocity structure functions in the so-called inertial range $\eta<<r<<L$ (see, for reviews, Frisch, U., 1995; Sreenivasan & Antonia, 1997). From experimental aspects, velocity measurements are usually carried out at a single fixed location, based on which time series of fluctuating velocities are obtained and the velocity structure functions, $S_{p}(\tau)=\langle\|\delta_{\tau}v\|^{p}\rangle$, are calculated as moments of velocity increments over a time separation $\tau$. To relate the properties of the experimentally measured $S_{p}(\tau)$ in time domain to theoretical predictions of $S_{p}(r)$ in space domain, one needs to invoke the Taylor’s frozen-flow hypothesis Taylor (1938). The validity of such hypothesis demands low turbulent intensity and weak shear rates Lumley (1965). However, these conditions are not always met by actual flows of interest Pinton & Labbe (1994). Another quantity that can also be used to characterize the cascade processes is the velocity space-time correlation function, defined as $C(r,\tau)=\frac{\langle v(\textbf{\emph{x}}+\textbf{\emph{r}},t+\tau)v(\textbf{\emph{x}},t)\rangle}{v_{rms}(\textbf{\emph{x}})v_{rms}(\textbf{\emph{x}}+\textbf{\emph{r}})},$ (1) where $v(\textbf{\emph{x}})$ is one of the components of the velocity vector at position x and $v_{rms}(\textbf{\emph{x}})$ is the root-mean-square (r.m.s.) velocity at x. For simplicity, we consider here only the situation that r is in the direction of $v(\textbf{\emph{x}})$, i.e. the longitudinal velocity correlations. [Studies of other kinds of velocity correlations, such as those between the wall-normal velocity component and the streamwise component, could be found in Jachens et al. (2006).] Again, the velocity temporal auto-correlation function $C(0,\tau)$, based on the pointwise measurements, is the quantity most often studied in experiments and Taylor’s frozen-flow hypothesis is usually used to translate $C(0,\tau)$ in time domain to $C(r,0)$ in space domain. When Taylor’s hypothesis holds, we have $C(r,\tau)=C(r_{T},0)$ with $r_{T}=r-U_{0}\tau,$ (2) where $U_{0}$ is the mean velocity of the flow. Such a relation implies that $C(r,\tau)$ would keep constant at increasing $r$ and $\tau$ once the value of $r-U_{0}\tau$ remains constant, which violates the basic properties of the correlations that $C(r,\tau)$ decays to zero at sufficient large separations. Therefore, Taylor hypothesis holds only for the limited ranges of $r$ and $\tau$. Recently, based on a second order approximation to $C(r,\tau)$, He & Zhang (2006) advanced an elliptic model for turbulent shear flows and proposed that $C(r,\tau)$ could be related to $C(r_{E},0)$ via $r_{E}^{2}=(r-U\tau)^{2}+V^{2}\tau^{2},$ (3) where $U$ is a characteristic convection velocity proportional to the mean velocity $U_{0}$ and $V$ is a characteristic velocity associated with the r.m.s. velocity and the shear-induced velocity Zhao & He (2009). Specifically, when $V$ vanishes, (3) is degenerated to the Taylor’s hypothesis (2), while Kraichnan’s sweeping-velocity hypothesis Kraichnan (1964) is obtained if $U$ vanishes. The He’s elliptic model provides a useful tool for a large set of flow systems where the Taylor’s hypothesis does not hold and hence the validity of the model in various practical flows needs to be tested. In this paper, we want to validate the elliptic model in turbulent convection, an important class of turbulent flows that play an central role in many natural and engineering processes. The flow at hand is turbulent Rayleigh- Bénard (RB) convection, i.e. the convective motion of an enclosed fluid layer heated from below and cooled from above, which has received tremendous attention during the past few decades (Ahlers, Grossmann $\&$ Lohse 2009; Lohse $\&$ Xia 2010). Although it has long been recognized that the conditions for Taylor’s hypothesis are often not met in the system (see, e.g., Shang & Xia, 2001), single-point or time-domain measurements are employed by most experimental investigators for the studies of turbulent cascade processes of both the velocity and temperature fields (see the recent review paper, Lohse $\&$ Xia 2010, and references therein). To translate correctly the quantities measured in time domain to those in space domain, it is thus essential to validate the elliptic model in turbulent RB convection. Recently, He, He $\&$ Tong (2010) verified indirectly the elliptic model via the local temperature data as in the bulk region of turbulent RB convection temperature behaves as a passive scalar (Calzavarini, Toschi $\&$ Tripiccione 2002; Xi _et al._ 2009), which is driven by the velocity field via a linear equation. The authors showed that the elliptic relation (3) is valid for the temperature space-time correlations measured in the cell’s sidewall region, but at the cell center they did not observe the predicted relation (3) for the temperature field. However, a passive additive may display characteristics so different from those of the advecting velocity field Warhaft (2000). In addition, the temperature field is nearly homogeneous in the cell’s central region. The lack of temperature contrast would make the local temperature not follow the behaviors of the velocity field, e.g. the local velocity fluctuations show strong oscillations at the cell center Qiu et al. (2004); Zhou et al. (2009), while the oscillation is absent for temperature measured at the same location Qiu & Tong (2002), and hence it is not surprising that (3) does not hold for temperature in the cell’s central region. Therefore, it is highly desirable to verify the elliptic model _directly_ via the velocity field, which is the object of the present experimental investigation. The remainder of this paper is organized as follows. We give detailed descriptions of the experimental apparatus and conditions and the measuring technique in $\S$2\. Experimental results are presented and analyzed in $\S$3, which is divided into three parts. In $\S\S$3.1 and 3.2, we study the properties of longitudinal space-time correlation functions of the vertical velocity near the cell sidewall and at the cell center, respectively. Section 3.3 presents results of longitudinal space-time correlations of the vertical velocity along the cell’s diameter at the middle height of the cell and of longitudinal space-time correlations of the horizontal velocity along the cell’s central vertical axis. The properties of the characteristic velocities $U$ and $V$ are also investigated in detail in $\S$3.3. We summarize our findings and conclude in $\S$4. ## 2 Experimental setup and parameters The convection cell is similar to that used in previous experiments Sun et al. (2005a), but has a different size Zhou & Xia (2010). It is a vertical cylinder of height $H=50$ cm and inner diameter $D=50$ cm and hence of unit aspect ratio. Deionized and degassed water was used as the convecting fluid. The cell’s sidewall is a plexiglas tube of 5 mm in wall thickness and a square- shaped jacket made of flat plexiglas plates and filled with water is fitted round the sidewall, which greatly reduced the distortion effect to the PIV images caused by the curvature of the cylindrical sidewall. The top and bottom plates are made of pure copper with nickel-plated fluid-contact surfaces. The thickness of the top plate is 3 cm and that of the bottom one is 1.5 cm. Four spiral channels of 1.2 cm in width and 1.5 cm in depth are machined into the top plate and the separation between adjacent channels is 1.1 cm. The channels start from the center and end near the edge of the plate. A silicon rubber sheet and a Plexiglas plate are fixed on the top to form the cover and also to prevent interflow between the channels. Each channel is connected to a separate refrigerated circulator (Polyscience 9712) that has a temperature stability of 0.01 ∘C. The channels and the circulators are connected such that the incoming cooler fluid and the outgoing warmer fluid in adjacent channels always flow in opposite directions. Four quarter-circular Kapton film heaters, connected in parallel to a dc power supply (Xantrex XDC 300-20) with 99.99$\%$ long-term stability, are sandwiched to the back side of the bottom plate to provide constant and uniform heating. Therefore, the experiments were conducted under constant heating of the bottom plate while maintaining a constant temperature at the top plate. Eight thermistors are embedded beneath the fluid-contact surface of each conducting plate, equally spaced azimuthally at about one-third radius from the edge. The measured relative temperature differences among eight thermistors in the same plate are found to be smaller than 3$\%$ of that across the fluid layer. During the experiment the entire cell was wrapped by several layers of Styrofoam and the cell was tilted by a small angle of about 0.5∘ (Ahlers, Brown $\&$ Nikolaenko 2006) so that the measurements were carried out within the vertical plane of the large-scale circulation. The mean temperature of water was kept at 29∘, corresponding to a Prandtl number $Pr=\nu/\kappa=5.5$. The experiment covered the range $5.9\times 10^{9}\lesssim Ra\lesssim 1.1\times 10^{11}$ of the Rayleigh number $Ra=\alpha g\Delta TH^{3}/\nu\kappa$, with $g$ being the gravitational acceleration, $\Delta T$ the temperature difference across the fluid layer, and $\alpha$, $\nu$ and $\kappa$ being, respectively, the thermal expansion coefficient, the kinematic viscosity, and the thermal diffusivity of water. As all measurements give the same qualitative results, only that for $Ra=9.5\times 10^{10}$ will be presented in this paper. At this $Ra$, the period of the large-scale circulation is around 62 sec. Figure 1: Sketch of the convection cell, the Cartesian coordinates used in the experiment, and the time-averaged vector map of the whole velocity field measured in the vertical plane of the large-scale circulation at $Ra=9.5\times 10^{10}$ and $Pr=5.5$. For clarity, a coarse-grained vector map of size $32\times 32$ is shown. The magnitude of the velocity $\sqrt{u_{0}^{2}+w_{0}^{2}}$ is coded in both color and the length of the arrows in units of cm/s. The time average is taken over a period of 145 min corresponding to 20 000 velocity frames. The two black dots in the figure mark the positions (x=-21.73cm, z=0cm) (see $\S$ 3.1) and (x=0cm, z=0cm) (see $\S$ 3.2), respectively. The details of the particle image velocimetry (PIV) measurements in turbulent RB convection have been described and discussed by Xia, Sun $\&$ Zhou (2003) and Sun, Xia $\&$ Tong (2005), here we give only its main features. The laser lightsheet thickness is $\sim 2$ mm and the seed particles are 50-$\mu$m-diameter polyamid spheres (density 1.03 g/cm3). As the seed particles are neutrally buoyant, they are assumed to follow the motion of the fluid. The measuring region has an area of $49\times 49$ cm2 with a spatial resolution of 7.76 mm, corresponding to $63\times 63$ velocity vectors. We chose the vertical rotation plane of the large-scale circulation as the laser- illuminated plane, defined as the $(x,z)$ plane. The Cartesian coordinate is defined such that the origin $(0,0)$ coincides with the cell center, the $x$ axis points to the right, and the $z$ axis points upwards. The horizontal velocity component $u(x,z)$ and the vertical one $w(x,z)$ were obtained. The experiment lasted 145 minutes in which a total of 20000 two-dimensional vector maps were acquired with a sampling rate of $\sim 2.3$ Hz. Figure 1 shows the measured mean flow field, together with the Cartesian coordinates used in the experiment. In the figure, the magnitude of the mean velocity $\sqrt{u_{0}^{2}+w_{0}^{2}}$ are coded both by color and by the length of the arrows, where $u_{0}=\langle u(t)\rangle$ and $w_{0}=\langle w(t)\rangle$ are the time-averaged horizontal and vertical components of the velocity vector, respectively. One sees clearly that the mean flow is a clockwise rotatory motion with a relatively quiet central region and high velocity regions concentrated along the perimeter of the cell. To see whether the Taylor’s frozen-flow hypothesis is valid or not in turbulent RB system, we plot in figure 2 the spatial distributions of the ratios between the time-averaged and the r.m.s. velocities, $\|u_{0}/u_{rms}\|$ [figure 2(_a_)] and $\|w_{0}/w_{rms}\|$ [figure 2(_b_)], where $u_{rms}=\sqrt{\langle[u(t)-u_{0}]^{2}\rangle}$ and $w_{rms}=\sqrt{\langle[w(t)-w_{0}]^{2}\rangle}$ are the r.m.s. horizontal and vertical velocities. If Taylor hypothesis holds, the values of $\|u_{0}/u_{rms}\|$ or $\|w_{0}/w_{rms}\|$ should be much larger than 1. In the figures, it is seen that both $\|u_{0}/u_{rms}\|$ and $\|w_{0}/w_{rms}\|$ are nearly zero in the cell’s central region and $\|u_{0}/u_{rms}\|$ and $\|w_{0}/w_{rms}\|$ increase, respectively, in the vertical and horizontal directions. The maximization of $\|u_{0}/u_{rms}\|$ occurs near the top and bottom plates with a maximum value $\|u_{0}/u_{rms}\|_{max}\simeq 2.1$, and $\|w_{0}/w_{rms}\|$ reaches its maximum value near the cell sidewall with $\|w_{0}/w_{rms}\|_{max}\simeq 2.4$. For such large r.m.s. velocities, Taylor’s frozen-flow hypothesis is then not expected to be valid. Figure 2: Color-coded contour maps of $\|u_{0}/u_{rms}\|$ (_a_) and $\|w_{0}/w_{rms}\|$ (_b_) obtained at $Ra=9.5\times 10^{10}$ and $Pr=5.5$. Here, $u_{0}$ and $u_{rms}$ are the time-averaged and the r.m.s. horizontal velocities, respectively, and $w_{0}$ and $w_{rms}$ are the time-averaged and the r.m.s. vertical velocities, respectively. The PIV technique provides us a convenient tool to directly and simultaneously measure the local fluctuating velocities at multi-points in a particular plane of interest. With the measured $u(x,z,t)$ and $w(x,z,t)$, one can obtain the longitudinal space-time cross-correlation functions for both the horizontal and vertical velocities, respectively, defined as $C_{u}(r,\tau;z)=\frac{\langle[u(x+r,z,t+\tau)-u_{0}(x+r,z)][u(x,z,t)-u_{0}(x,z)]\rangle}{u_{rms}(x+r,z)u_{rms}(x,z)}$ (4) and $C_{w}(r,\tau;x)=\frac{\langle[w(x,z+r,t+\tau)-w_{0}(x,z+r)][w(x,z,t)-w_{0}(x,z)]\rangle}{w_{rms}(x,z+r)w_{rms}(x,z)}.$ (5) ## 3 Results and discussions ### 3.1 Near the cell sidewall Figure 3: (_a_) Space-time correlations $C_{w}(r,\tau;x)$ as a function of $r$ and $\tau$ measured near the cell sidewall ($x=-21.73$ cm and $z=0$ cm). Here, the amplitude of $C_{w}(r,\tau;x)$ is coded by color. (_b_) The corresponding isocorrelation contours of $C_{w}(r,\tau;x)$ with the color-coded correlation amplitude varying from 0.4 to 0.85 at increments of 0.05 (outer to inner contours). We first study the properties of longitudinal space-time correlations for the vertical velocity, $C_{w}(r,\tau;x)$, near the cell sidewall ($x=-21.73$ cm) at the middle height of the cell ($z=0$ cm), where the mean and the r.m.s velocities are of the same order [see figure 1(_b_)]. Figure 3(_a_) shows the flood contours of the measured $C_{w}(r,\tau;x)$ as a function of separations $r$ and $\tau$ and the corresponding isocorrelation contours are plotted in figure 3(_b_). By definition, the maximization of $C_{w}(r,\tau;x)$ occurs at the origin with a maximum value of $C_{w}(0,0;x)=1$. As $C_{w}(r,\tau;x)$ decays fast near the origin, our present resolution could not resolve properly the isocorrelation contours when $C_{w}(r,\tau;x)\gtrsim 0.85$. If the Taylor’s hypothesis relation (2) is valid for the present flow field, the isocorrelation contours of $C_{w}(r,\tau;x)$ should be straight lines. However, for the $r$\- and $\tau$-range studied, one sees in figure 3(b) that the isocorrelation contours of the measured $C_{w}(r,\tau;x)$ are the elongated and closed curves, rather than straight lines, and seem to have a shape of elliptical curves that can be described well by (3). Furthermore, all isocorrelation contours appear to be self-similar, i.e., they share the same preferred orientation and the same aspect ratio. One also sees that $C_{w}(r,\tau;x)$ decays relatively slowly in the preference direction, but drops much faster in the direction that is perpendicular to the preference direction. Figure 4: (_a_) The measured peak position $r_{p}$ as a function of $\tau$. The solid line shows the fitted linear function, $r_{p}=U\tau$, with $U=1.52$ cm/s. (_b_) The measured peak position $\tau_{p}$ as a function of $r$. The solid line shows the fitted linear function, $\tau_{p}=[U/(U^{2}+V^{2})]r$, with $U/(U^{2}+V^{2})=0.33$ s/cm. All data were obtained near the cell sidewall ($x=-21.73$ cm and $z=0$ cm). To determine the characteristic velocities $U$ and $V$ in (3), note that from the conditions $\partial r_{E}/\partial r\|_{\tau}=0$ and $\partial r_{E}/\partial\tau\|_{r}=0$ we have, respectively, $r_{p}=U\tau\mbox{\ \ and\ \ }\tau_{p}=[U/(U^{2}+V^{2})]r,$ (6) where $r_{p}$ maximizes $C_{w}(r,\tau;x)$ for a given $\tau$ and $\tau_{p}$ is the peak position at which $C_{w}(r,\tau;x)$ reaches its maximum value for a fixed separation $r$ He et al. (2010). Our results for the measured $r_{p}$ as a function of time separation $\tau$ are shown in figure 4(_a_). One sees clearly that $r_{p}$ increases with increasing $\tau$ because a longer time lag is needed to move velocity fluctuations across a larger separation. It is further seen that the increasing manner may indeed be described by a simple linear function, $r_{p}=U\tau$, with $U=1.52$ cm/s. How the measured $\tau_{p}$ varies with separation $r$ is shown in figure 4(_b_). Again, one sees that $\tau_{p}$ increases linearly with increasing $r$. A linear fit to the data yields $\tau_{p}=[U/(U^{2}+V^{2})]r$ with $U/(U^{2}+V^{2})=0.33$ s/cm. Note that the similar relation has also been observed for the scalar fields in the same system Zhou & Xia (2008); He et al. (2010). Taken together, we have $U=1.52$ cm/s and $V=1.51$ cm/s at $x=-21.73$ cm and $z=0$ cm. Comparisons between the fitted velocities $U$ and $V$ and the mean and the r.m.s. velocities of the flow will be presented in $\S$ 3.3. Figure 5: The space-time correlations $C_{w}(r,\tau;x)$ obtained near the cell sidewall ($x=-21.73$ cm and $z=0$ cm) for various values of $r=$0, 0.78, 2.33, 3.88, 5.43, 6.98, and 9.31 cm as functions of (_a_) time separation $\tau$, (_b_) the Taylor’s separation $r_{T}=r-U_{0}\tau$ with the mean flow velocity $U_{0}=1.95$ cm/s, and (_c_) the separation $r_{E}=\sqrt{(r-U\tau)^{2}+V^{2}\tau^{2}}$ with $U=1.52$ cm/s and $V=1.51$ cm/s. With the obtained $U$ and $V$, we can now test the relation between $C_{w}(r,\tau;x)$ and $C_{w}(r_{E},0;x)$. Figure 5(_a_) shows the evolution of $C_{w}(r,\tau;x)$ as a function of time separation $\tau$ for several different values of $r$. One sees that the measured $C_{w}(r,\tau;x)$ all have a single peak at the position $\tau_{p}$. The peak position $\tau_{p}$ increases with increasing $r$ [see also figure 4(_b_)], meanwhile the correlation amplitude $C_{w}(r,\tau_{p};x)$ decreases. This is because velocity fluctuations at two points decorrelate gradually when the separation between these two points increases. For comparison, we first plot in figure 5(_b_) the measured $C_{w}(r,\tau;x)$ as a function of the Taylor’s separation $r_{T}$ with $r_{T}$ calculated from (2) with the mean velocity $U_{0}=1.95$ cm/s, and then shows in figure 5(_b_) the measured $C_{w}(r,\tau;x)$ as a function of the separation $r_{E}$ with $r_{E}$ calculated from (3). In the figure, only positive parts of $C_{w}(r,\tau;x)$ are plotted because $C_{w}(r_{E},0;x)$ is a symmetric function with respect to $r_{E}=0$. It is seen that when using the Taylor’s hypothesis [see figure 5(_b_)] the correlations could not collapse on top of each other, whereas, when using the elliptic model [see figure 5(_c_)] the correlations all collapse well on top of each other, indicating that the space-time correlations $C_{w}(r,\tau;x)$ can be only determined by the space correlations $C_{w}(r_{E},0;x)$ and the solution $r_{E}$ of (3). ### 3.2 At the cell center Figure 6: (_a_) Space-time correlations $C_{w}(r,\tau;x)$ as a function of $r$ and $\tau$ measured at the cell center ($x=0$ cm and $z=0$ cm). Here, the amplitude of $C_{w}(r,\tau;x)$ is coded by color. (_b_) The corresponding isocorrelation contours of $C_{z}(x,r,\tau)$ with the color-coded correlation amplitude varying from 0.4 to 0.85 at increments of 0.05 (outer to inner contours). Let’s now turn to the velocity field at the cell center ($x=0$ cm and $z=0$ cm), where the mean horizontal and vertical velocities are both nearly zero. As the velocity field in the cell’s central region is approximately locally homogeneous and isotropic (Sun, Zhou $\&$ Xia 2006; Zhou, Sun $\&$ Xia 2008) and we also find that space-time correlations of the horizontal and vertical velocities share the same qualitative properties at the cell center, only the results of $C_{w}(r,\tau;x)$ will be presented in this subsection. Figure 6(_a_) shows the flood contours of space-time correlation $C_{w}(r,\tau;x)$ as a function of space separation $r$ and time separation $\tau$ and figure 6(_b_) shows the corresponding isocorrelation contours. Three features are worthy of note. (i) The measured $C_{w}(r,\tau;x)$ is a single-peak function with the peak locating at the origin and decays with increasing separations $r$ or $\tau$. (ii) All isocorrelation contours of $C_{w}(r,\tau;x)$ are closed curves and have an elliptic shape. (iii) The elliptic isocorrelation contours can be well described by a standard elliptic equation $\frac{\tau^{2}}{a^{2}}+\frac{r^{2}}{b^{2}}=1,$ (7) i.e., the isocontours are aligned with the coordinate axis. Here, $a$ and $b$ are two parameters, determining the lengths of the major and minor axes of the standard ellipse. Figure 7: (_a_) The measured peak position $r_{p}$ as a function of $\tau$. The solid line marks $r_{p}=0$. (_b_) The measured peak position $\tau_{p}$ as a function of $r$. The solid line marks $\tau_{p}=0$. All data were obtained at the cell center ($x=0$ cm and $z=0$ cm). Figure 8: (_a_) The blue solid curves are the isocorrelation contours of $C_{z}(x,r,\tau)$ measured at the cell center ($x=0$ cm and $z=0$ cm) with the correlation amplitude varying from 0.4 to 0.85 at increments of 0.05 (outer to inner contours). Here, we use the same date sets as figure 6. The red dashed curves are the elliptic fittings of (7) to the isocorrelation contours. (_b_) The fitted parameters of the standard elliptic equation (7): $b$ vs $a$. The solid line shows the best fit of $b=Va$, with $V=1.39$ cm/s. Figure 7(_a_) shows the measured $r_{p}$ as a function of time separation $\tau$ and figure 7(_b_) shows the obtained $\tau_{p}$ as a function of space separation $r$. It is seen that $r_{p}$ and $\tau_{p}$ both vary around zero, implying $U\simeq 0$ and $[U/(U^{2}+V^{2})]\simeq 0$ [see (6)]. To yield $V$, we note that when $U=0$, equation (3) can be rewritten as $r_{E}^{2}=r^{2}+V^{2}\tau^{2}.$ (8) Comparing with the standard elliptic equation (7), we have $b=Va$. Therefore, the value of $V$ can be estimated from the values of $a$ and $b$. In figure 8(_a_), we plot again the isocorrelation contours of $C_{w}(r,\tau;x)$ as the blue solid curves. In the figure, the best fittings of (7) to the contours are plotted as the red dashed curves. It is seen that the fitted curves collapse well on top of the contours, further confirming that the isocorrelation contours have a shape of standard elliptic curves. Figure 8(_b_) shows the lengths of the $r$-axes of the fitted elliptic curves, $b$, as a function of the lengths of their $\tau$-axes, $a$. The solid line in the figure shows the best fit of $b=Va$ to the data, which gives $V=1.39$ cm/s. Taken together, we have $U\simeq 0$ cm/s and $V=1.39$ cm/s at the cell center. Figure 9: The space-time correlations $C_{w}(r,\tau;x)$ obtained at the cell center ($x=0$ cm and $z=0$ cm) for various values of $r=$0, 0.78, 2.33, 3.88, 5.43, 6.98, and 9.31 cm as functions of (_a_) time separation $\tau$ and (_b_) the separation $r_{E}=\sqrt{r^{2}+V^{2}\tau^{2}}$ with $V=1.39$ cm/s. Figure 9(_a_) shows the space-time correlations $C_{w}(r,\tau;x)$ as a function of time separation $\tau$ for several different values of $r$. It is seen that $C_{w}(r,\tau;x)$ obtained at the cell center is also a single-peak function. However, unlike the case near the cell sidewall [see figure 5(_a_)], the measured peak positions $\tau_{p}$ here do not vary with $r$, but all locate at positions around $\tau=0$ [see also figure 7(_b_)]. This is because of the zero mean velocity at the cell center. Figure 9(_b_) shows the measured $C_{w}(r,\tau;x)$ as a function of $r_{E}$. One sees that when the solution $r_{E}$ of (3) is used, reasonable collapses among these correlations are achieved, further confirming the validity of the elliptic model for the present flow. ### 3.3 Along the cell’s horizontal and vertical central lines Figure 10: The isocorrelation contours of $C_{w}(r,\tau;x)$ as functions of time separation $\tau$ and space separation $r$ measured at the middle height of the cell ($z=0$ cm) at $x=-21.73$ (_a_), $-13.97$ (_b_), $-6.98$ (_c_), and $0$ (_d_) cm. The amplitude of the contours is coded by color and varies from 0.4 to 0.85 at increments of 0.05 (outer to inner contours). Note that (_a_) is the same as figure 3(_b_) and (_d_) is the same as figure 6(_b_). We replot them here for comparison. Figures 10(_a_)-(_d_) show the evolution of the isocorrletion contours of the longitudinal space-time correlations $C_{w}(r,\tau;x)$ measured at the middle height of the cell ($z=0$ cm) at four different values of $x$ from near the cell sidewall to at the cell center. Figures 10(_a_) and (_d_) are the same as figure 3(_b_) and figure 6(_b_), respectively. We replot these figures here for comparison. One sees that the contours are all elliptic closed curves with their preferred orientations. Furthermore, the slopes of the preferred orientations become smaller as the reference position movies from the wall towards the center. This is because the preferred orientations of the contours are directly related to the mean velocity of the flow He & Zhang (2006); Zhao & He (2009) and the mean vertical velocity $w_{0}$, after reaching its maximum value near the cell sidewall, decreases with the increasing distance from the wall Qiu & Tong (2001) (see also figure 1). Figure 11: Comparison of the magnitudes of the characteristic velocities $U$ (circles) and $V$ (triangles) for $C_{w}(r,\tau;x)$ along the $x$-axis. Direct comparison is made in figure 11 between the magnitudes of the characteristic velocities $U$ (circles) and $V$ (triangles) for $C_{w}(r,\tau;x)$ along the $x$-axis. It is seen that $\|U\|\simeq 0$ at the cell center and increases along the cell’s diameter at the middle height of the cell from the cell center to the sidewall and the maximization of $\|U\|$ occurs near the cell sidewall, while the variation of $V$ is much weaker. Note that the validity of Taylor’s frozen-flow hypothesis requires $\|U\|\gg V$. However, the figure shows clearly that $\|U\|\gg V$ is not the case, i.e., $\|U\|$ and $V$ are approximately the same near the sidewall, while at the cell center the value of $V$ is even much larger than that of $\|U\|$. This further conforms that Taylor hypothesis does not hold in the present system. Figure 12: (_a_) Comparison of the characteristic velocity $U$ (solid circles) for $C_{w}(r,\tau;x)$ and the mean velocity of the flow $w_{0}$ (open circles) along the $x$-axis. (_b_) $U$ vs $w_{0}$. The solid line shows the linear fit to the data, $U=0.82w_{0}$. (_c_) Comparison of the characteristic velocity $V$ for $C_{w}(r,\tau;x)$ (solid circles) and the theoretical prediction $V_{t}$ (open circles) of the elliptic model along the $x$-axis. (_d_) $V/V_{t}$ vs $x$. The solid line marks the mean value 1.87 of the ratios. To further test the elliptic model, we compare in figure 12 the measured values of the characteristic velocities $U$ and $V$ and their theoretical predictions $U_{t}$ and $V_{t}$. Based on Navier-Stokes equation, Zhao & He (2009) showed that $U_{t}$ is a characteristic convection velocity proportional to the mean velocity of the flow and $V_{t}$ is the sum of the random sweeping velocity and the shear-induced velocity, i.e. $V_{t}=\sqrt{S^{2}\lambda^{2}+w^{2}_{rms}}$, where the subscript ”t” indicates theoretical predications and $S$ and $\lambda$ are, respectively, the shear rate and the Taylor microscale of the flow. For the vertical velocity profile along the $x$-axis Qiu & Tong (2001); Sun et al. (2005b), the shear rate was evaluated as $S\simeq 2(W_{0})_{max}/D$ He et al. (2010), where $(W_{0})_{max}$ is the maximal value of the vertical velocity along the $x$-axis. The Taylor microscale $\lambda(x)$ was estimated using the equation $C_{w}(r_{E},0;x)\simeq 1-(r/\lambda(x))^{2}$ for $\|r\|<1$ cm, where the space autocorrelation function $C_{w}(r_{E},0;x)$ was obtained from the time autocorrelation function $C_{w}(0,\tau;x)$ using the relation (3) with $r=0$. We note that $S\lambda(x)\ll w_{rms}(x)$ for all measuring positions. Comparison is made in figure 12(_a_) between $U$ (solid circles) and the mean vertical velocity $w_{0}$ (open circles). One sees that both $U$ and $w_{0}$ decrease with increasing $x$, but the magnitude of $U$ seems to be systematically smaller than that of $w_{0}$. Figure 12(_b_) shows the measured $U$ as a function of $w_{0}$. It is seen that $U$ increases with increasing $w_{0}$ and the increasing manner may be described by a simple linear function $U=0.82w_{0}$. The solid line in figure 12(_b_) shows the fitting linear function. Figure 12(_c_) shows the comparison of $V$ and $V_{t}$. One sees that the experimentally measured $V$ is larger than its theoretical predictions $V_{t}$ for all measuring positions. The ratio of $V$ to $V_{t}$ along the $x$-axis is plotted in figure 12(_d_). In the figure, the dashed line marks the mean value 1.87 of the ratios. One sees that all data points vary around the dashed line. This seems to suggest a constant ratio between $V$ and $V_{t}$, i.e., the experimentally measured $V$ is proportional to its theoretical predications $V_{t}$. Figure 13: The isocorrelation contours of $C_{u}(r,\tau;z)$ as functions of time separation $\tau$ and space separation $r$ measured along the cell’s central vertical axis ($x=0$ cm) at $z=-21.73$ (_a_), $-13.97$ (_b_), $-6.98$ (_c_), and $0$ (_d_) cm. The amplitude of the contours is coded by color and varies from 0.4 to 0.85 at increments of 0.05 (outer to inner contours). Figure 14: Comparison of the magnitudes of the characteristic velocities $U$ (circles) and $V$ (triangles) for $C_{u}(r,\tau;z)$ along the $z$-axis. Figure 15: (_a_) Comparison of the characteristic velocity $U$ (solid circles) for $C_{u}(r,\tau;z)$ and the mean velocity of the flow $u_{0}$ (open circles) along the $z$-axis. (_b_) $U$ vs $u_{0}$. The solid line shows the linear fit to the data, $U=0.97u_{0}$. (_c_) Comparison of the characteristic velocity $V$ for $C_{u}(r,\tau;z)$ (solid circles) and the theoretical prediction $V_{t}$ (open circles) of the elliptic model along the $z$-axis. (_d_) $V/V_{t}$ vs $z$. The solid line marks the mean value 4.21 of the ratios. Finally, we study the longitudinal space-time correlations $C_{u}(r,\tau;z)$ for the horizontal velocity along the cell’s central vertical axis. Figures 13(_a_)-(_d_) show the evolution of the isocorrelation contours of $C_{u}(r,\tau;z)$ obtained at $x=0$ and four different values of $z$. Again, one sees that all contours are elliptic closed curves with their preferred orientations. The preferred orientations guide along the second and fourth quadrants, which is due to the negative mean velocities at the measuring positions (see figure 1), and the slopes of the preferred orientations decrease with the increasing distance from the plate, which is due to the decrease of the magnitude of the mean horizontal velocity (see figure 1). Figure 14 shows the comparison of $\|U\|$ and $V$. It is also seen that $\|U\|$ and $V$ are nearly the same near the top and bottom plates, while $V$ is much larger than $\|U\|$ at the cell center. The fact that $\|U\|\gg V$ does not hold for the horizontal velocity again indicates the invalidity of Taylor’s frozen- flow hypothesis in the present flow. Figure 15(_a_) shows the comparison of the measured $U$ and the mean horizontal velocity $u_{0}$ along the cell’s central vertical axis. It is seen that $U$ and $u_{0}$ are approximately the same for all values of $z$. Figure 15(_b_) shows $U$ as a function of $u_{0}$. The best linear fit to the data yields $U=0.97u_{0}$, again indicating that $U$ is proportional to $w_{0}$. Direct comparison is made in figure 15(_c_) between $V$ and $V_{t}$ ($=\sqrt{S^{2}\lambda^{2}+u^{2}_{rms}}$). Here, the shear rate was estimated as $S\simeq 2(U_{0})_{max}/H$ with $(U_{0})_{max}$ being the maximal horizontal velocity along the $z$-axis and the Taylor microscale $\lambda(z)$ was evaluated from the space autocorrelation function $C_{u}(r_{E},0;z)$. Again, we find that $S\lambda(z)\ll u_{rms}(z)$ for all measuring positions. In figure 15(_c_), one sees that similar to the case of $C_{w}(r,\tau;x)$, the values of $V$ are also larger than those of $V_{t}$. Nevertheless, the ratio $V/V_{t}$ shown in figure 15(_d_) varies around its mean value 4.21, suggesting that $V$ is proportional to $V_{t}$. Taken together, our results reveal that $C_{u}(r,\tau;z)$ shares the same qualitative properties as $C_{w}(r,\tau;x)$. ## 4 Conclusion To conclude, we have presented an systematic experimental study of the velocity field in a cylindrical turbulent Rayleigh-Bénard (RB) convection cell with unit aspect ratio using water as working fluid. The two-dimensional velocity field in the vertical circulation plane of the large-scale circulation was measured via the particle image velocimetry (PIV) technique and the longitudinal space-time cross-correlation functions for both the horizontal and vertical velocities, $C_{u}(r,\tau;z)$ and $C_{w}(r,\tau;x)$, were investigated in great detail. Our results show that the isocorrelation contours of space-time correlations are elliptic closed curves and the space- time correlations $C_{u}(r,\tau;z)$ and $C_{w}(r,\tau;x)$ can be related to the space correlations $C_{u}(r_{E},0;z)$ and $C_{w}(r_{E},0;x)$, respectively, via the elliptic relation (3), i.e. $r_{E}^{2}=(r-U\tau)^{2}+V^{2}\tau^{2}$. The characteristic velocities $U$ and $V$ were then calculated and studied. We find that the magnitude of $U$ reaches its maximum value near the sidewall and plates and decreases when away from the walls and plates, while the position-dependence of $V$ is much weaker. Specifically, at the cell center we have $U\simeq 0$ and hence the relation (3) becomes $r_{E}^{2}=r^{2}+V^{2}\tau^{2}$ with its major and minor axes coinciding with the $\tau$\- and $r$-axis. Note that such relation has the same form as Kraichnan’s sweeping-velocity hypothesis Kraichnan (1964). Direct comparison of the values of $U$ and $V$ and their theoretical predictions further show that $U$ is proportional to the mean velocity of the flow, while $V$ is systematically larger than its prediction. Our results validate the elliptic model in turbulent RB convection, where Taylor’s frozen-flow hypothesis does not hold due to the relatively large values of the r.m.s. velocity. As pointed out by He & Zhang (2006), the elliptic model is developed based on a second-order approximation, while Taylor’s hypothesis implies a first-order approximation. Thus, the elliptic model is a generation of Taylor’s frozen-flow hypothesis and Taylor’s hypothesis is only a special case of the elliptic model [i.e., when $V=0$ the elliptic model’s relation (3) is degenerated to the Taylor’s relation (2)]. Like Taylor’s hypothesis, the elliptic model could also be used to translate time series to space series. This is because the correlation function is a basic quantity and most statistical properties interested in the field of turbulence, such as structure function and power spectrum, can be obtained theoretically from the correlation functions. In fact, previous work by He et al. (2010) has used the model to translate the temperature power spectrum from time domain to space domain. An important implication of the elliptic model is that when $r=0$, the relation (3) becomes $r_{E}=(U^{2}+V^{2})^{1/2}\tau.$ (9) In this case, $r$ is still proportional to $\tau$ but just that the proportionality constant is $(U^{2}+V^{2})^{1/2}$, rather than the mean velocity $U_{0}$ as stated in the Taylor’s hypothesis [see (2) with $r=0$]. This implies that if one is ONLY interested in the scaling exponents of $C(r,0)$ in space domain, such scaling properties can still be correctly obtained by studying the scaling of $C(0,\tau)$ in time domain in the elliptic model even though Taylor’s hypothesis is not valid. We note that Taylor’s hypothesis has been widely used to study the scaling behaviors of structure functions and power spectrum of the velocity and temperature fields in turbulent RB convection Lohse & Xia (2010). However, it has long been known in the field that the condition for Taylor’s hypothesis is not often net Lohse & Xia (2010), and hence the results based on Taylor’s hypothesis is questionable and not convincible. Here, the relation (9) implies that one does not really need the validity of Taylor’s hypothesis to reconstruct the space series from the measured time series. However, Taylor’s hypothesis could not yield the correct scaling ranges in space domain, which could only be obtained upon the transform of (9) in the elliptic model. It should be noted that in the original elliptic model He & Zhang (2006) the correlations $R(r,\tau)=\langle v(\textbf{\emph{x}}+\textbf{\emph{r}},t+\tau)v(\textbf{\emph{x}},t)\rangle$ were considered, while we studied the normalized quantities $C(r,\tau)=R(r,\tau)/v_{rms}(\textbf{\emph{x}})v_{rms}(\textbf{\emph{x}}+\textbf{\emph{r}})$ here. The elliptic relation (3) holds for both $R(r,\tau)$ and $C(r,\tau)$. However, if the r.m.s. velocity $v_{rms}(\textbf{\emph{x}})$ depends on position, the characteristic velocities $U$ and $V$ for $R(r,\tau)$ may be different from those for $C(r,\tau)$. We note that previous works usually focused on $C(r,\tau)$ (see, e.g., He et al., 2010). In the present work, we have also checked the properties of $R(r,\tau)$ and the similar results were obtained. ###### Acknowledgements. The experiments were carried out in the Chinese University of Hong Kong. We gratefully acknowledge Prof. Ke-Qing Xia for making the PIV data available to us. We thank also Prof. Guo-Wei He and Dr. Xiao-Zhou He for helpful discussions. This work was supported by Natural Science Foundation of China (Nos. 11002085, 11072139), “Pu Jiang” project of Shanghai (No. 10PJ1404000), “Chen Guang” Project of Shanghai (No. 09CG41), E-Institutes of Shanghai Municipal Education Commission, and Shanghai Program for Innovative Research Team in Universities. ## References * Ahlers et al. (2006) Ahlers, G., Brown, E. & Nikolaenko, A. 2006 The search for slow transient, and the effect of imperfect vertical alignment, in turbulent Rayleigh-Bénard convection. J. Fluid Mech. 557, 347–367. * Ahlers et al. (2009) Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection. Rev. Mod. Phys. 81, 503–537. * Calzavarini et al. 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1012.0233	# J/$\psi$ production at high $p_{T}$ at STAR Zebo Tang (for the STAR Collaboration) Department of Modern Physics, University of Science and Technology of China, 96 Jinzhai Road, Hefei, Anhui, China 230026 Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA ###### Abstract We report results on $J/\psi$-hadron azimuthal angular correlations in 200 GeV $p$+$p$ collision in the STAR experiment at RHIC. The extracted $B$-hadron feed-down contribution to inclusive $J/\psi$ yield is found to be 10-25% in $4<p_{T}<12~{}\textrm{GeV}/c$ and has no significant center-of-mass energy dependence from RHIC to LHC. The $p_{T}$ spectrum of charged hadron associated with high-$p_{T}$ $J/\psi$ triggers on the away side is found to be consistent with that from di-hadron correlations. $J/\psi$ signal from partially produced Au+Au 39 GeV data will also be presented to demonstrate STAR’s $J/\psi$ capability at RHIC low energy run. ###### keywords: $J/\psi$ , high $p_{T}$ , color screening , correlation ## 1 Introduction The dissociation of $J/\psi$ due to color-screening of their constituent quarks in a Quark-Gluon Plasma (QGP) is a classic signature for deconfinement in relativistic heavy-ion collisions [1]. Results from the PHENIX experiment at RHIC show that the suppression of $J/\psi$ as a function of centrality (the number of participants) is similar to that observed by NA50 and NA60 at the CERN-SPS, even though the temperature and energy density reached in these collisions is significantly lower than at RHIC [2]. This indicates that additional mechanisms, such as recombination of charm quarks in the later stage of the collision and/or suppression of feed-down contribution from charmonium excited states or $B$-hadrons, may play an important role; they will need to be studied systematically before conclusion from the observed suppression pattern can be drawn. Recently, the STAR experiment has extended $J/\psi$ suppression measurement to high $p_{T}$ in Cu+Cu collisions and found that the $J/\psi$ nuclear modification factor $R_{AA}$ is consistent with no $J/\psi$ suppression at $p_{T}>5~{}\textrm{GeV}/c$, in contrast to the prediction from a theoretical model of quarkonium dissociation in a strongly coupled liquid using an AdS/CFT approach [3, 4]. The project is not yet complete and we need to increase the statistics, investigate the meachanism of $J/\psi$ formation, and perform the same measurement with a larger system (Au+Au). On the other hand, measurements from CDF shows that the contribution of $B$-hadrons relative to the inclusive $J/\psi$ yield in $p+\bar{p}$ collisions at 1.96 TeV significantly increases with increasing $p_{T}$. The same measurement at RHIC energy will be also essentially needed to disentangle the physics origin of the high-$p_{T}$ $J/\psi$ suppression measurements [5]. $B$ was rarely studied at RHIC in the past ten years. The $B\rightarrow J/\psi$ measurements in heavy-ion collisions at STAR are still difficult without a precise vertex detector. But it can be done in $p$+$p$ collisions through $J/\psi$-hadron correlations, originally proposed and studied by UA1 [6]. Furthermore, $J/\psi$-hadron correlations can be also used to study the hadronic activity produced in association with a high-$p_{T}$ $J/\psi$ to investigate its production mechanism which is still poorly understood more than 30 years after the discovery of $J/\psi$. In this paper we present the measurement of the correlation between high-$p_{T}$ $J/\psi$’s and charged hadrons at mid-rapidity with the STAR experiment in $p+p$ collisions at $\sqrt{s}$ $=200~{}\textrm{GeV}$ in RHIC year 2009 high luminosity run. We also report the status of measurement of $J/\psi$ in Au+Au collisions at $\sqrt{s_{{}_{\mathrm{NN}}}}$ $=39~{}\textrm{GeV}$ (an energy between CERN-SPS and RHIC top energies) at STAR with newly fully-installed Time-Of-Flight (TOF) detector [7, 8, 9]. ## 2 high-$p_{T}$ $J/\psi$ production in $p$+$p$ collisions at 200 GeV Figure 1: Invariant mass distribution for unlike-sign (solid circles) and like-sign (grey band) electron pairs at mid-rapidity ($\|y\|<1$) in $p$+$p$ collisions at $\sqrt{s_{{}_{\mathrm{NN}}}}$ =200 GeV. Figure 2: $J/\psi$-hadron azimuthal angular correlations in the $J/\psi$ $p_{T}$ range of $8<p_{T}<12~{}\textrm{GeV}/c$ at mid-rapidity ($\|y\|<1$) in $p$+$p$ collisions at $\sqrt{s_{{}_{\mathrm{NN}}}}$ =200 GeV. Figure 3: Fraction of $B\rightarrow J/\psi$ over the inclusive $J/\psi$ yield from two sets of run at STAR. The same ratios measured by UA1, D0, CDF and CMS collaborations are also shown for comparison. Figure 4: Associated charged hadron $p_{T}$ distributions on the away side with respect to high-$p_{T}$ $J/\psi$ triggers and charged hadron triggers at mid-rapidity in $p$+$p$ collisions at $\sqrt{s_{{}_{\mathrm{NN}}}}$ =200 GeV. In this analysis, the $J/\psi$ is reconstructed through its decay into electron-position pairs, $J/\psi\rightarrow e^{+}e^{-}$ (Branching ratio (B) = 5.9%). The data sample used was triggered at level-0 by the STAR Barrel Electromagnetic Calorimeter (BEMC) by requiring the transverse energy deposited in any tower ($\Delta\eta\times\Delta\phi=0.05\times 0.05$) above a given high-energy threshold to enrich high-$p_{T}$ electrons. This effectively enriches high-$p_{T}$ $J/\psi$ with limited data acquisition rate. The integrated luminosity is 1.8 $pb^{-1}$, 3.2 $pb^{-1}$ and 23.1 $pb^{-1}$ with transverse energy threshold $2.6~{}\textrm{GeV}<E_{T}<4.3~{}\textrm{GeV}$, $E_{T}>4.3~{}\textrm{GeV}$ and $E_{T}>6.0~{}\textrm{GeV}$ respectively. The reconstruction method is similar as what we used in year 2005 and year 2006 data. We tightened the $dE/dx$ cut slightly to enhance the signal-to- background (S/B) ratio for the correlation study [3, 10]. In year 2009, STAR installed 72% TOF trays at mid-rapidity ($\|\eta\|<0.9$). This detector combined with the Time Projection Chamber (TPC) can clearly identify electrons from low to high $p_{T}$ by rejecting hadrons at low and intermediate $p_{T}$ range. To further improve the S/B ratio of $J/\psi$, we also require the electron which does not trigger the BEMC to have 1/$\beta$ measured by TOF within 0.97-1.03 when its $p_{T}$ is less than 1 GeV/$c$ [11]. Figure 2 shows the invariant mass distribution for unlike-sign (solid circles) and like-sign (shaded band) electron pairs. We reconstructed 376 $J/\psi$ with $3.0<M<3.2~{}\textrm{GeV}/c^{2}$ at $p_{T}>4~{}\textrm{GeV}/c$. The S/B ratio in this range is 22. Such high S/B ratio is very suitable for the $J/\psi$-hadron correlation study. We do the correlation in 3 $J/\psi$ $p_{T}$ slices: $4-6~{}\textrm{GeV}/c$, $6-8~{}\textrm{GeV}/c$ and $8-12~{}\textrm{GeV}/c$. Figure 2 shows the azimuthal angle correlations between high-$p_{T}$ $J/\psi$ of $8-12~{}\textrm{GeV}/c$ and charged hadrons. The correlated yield on the near-side is not as significant at that in the di- hadron correlation measurements [12]. The lines show the results of a PYTHIA calculation. The dot-dashed line exhibits a strong near-side correlation compared to the away-side dominantly from the decay $B\rightarrow J/\psi+X$. The solid line shows a $\chi^{2}$ fit with the two simulated components to extract the relative contribution of $B$-hadron feed-down to the inclusive $J/\psi$ yield. This ratio is 10%-25% in the measured $p_{T}$ range, shown in Fig. 4 in red solid circles, increases with increasing $p_{T}$. The results are consistent with STAR’s previous measurement (solid star symbol), but with better precision [3]. The same ratios measured by UA1 in $p$+$p$ collisions at 630 GeV, by D0 (CDF) in $p+\bar{p}$ collisions at 1.8 (1.96) TeV and by CMS in $p+p$ collisions at 7 TeV in various rapidity ranges are also shown for comparison [5, 6, 13, 14]. They are consistent with each other even though the center-of-mass energies differ by an order of magnitude. The ATLAS and LHCb collaborations also observed a similar behavior [15, 16]. The physics origin of this consistency is still unclear. With such an amount of $B$-hadron feed- down fraction, combined with this $J/\psi$-hadron correlation study, further study of $J/\psi$ cross-section will allow us to constrain the $B$ cross- section substantially in the future. Figure 4 shows the associated charged hadron $p_{T}$ distribution on the away side with respect to high-$p_{T}$ $J/\psi$ triggers and high-$p_{T}$ charged hadron triggers. The $p_{T}$ spectra of charged hadron associated with high-$p_{T}$ $J/\psi$ are consistent from different runs, but year 2009 results have a better precision. To compare the results with those from di- hadron correlation, we require $J/\psi$ triggers in year 2009 run within the same $p_{T}$ window as charged hadron triggers: $4-6~{}\textrm{GeV}/c$. The $p_{T}$ spectra of the associated charged hadrons with respect to both kinds of triggers are consistent with each other, which indicates that the hadrons on the away side of $J/\psi$ triggers are dominantly from light quark or gluon fragmentation, instead of heavy quark fragmentation. ## 3 $J/\psi$ production in Au+Au collisions at 39 GeV Figure 5: Invariant mass distribution of electron pairs in BEMC triggered (left) and minimum-bias (right) triggered Au+Au events at $\sqrt{s_{{}_{\mathrm{NN}}}}$ =39 GeV. The solid and dashed histograms represent background reproduced using like-sign and mixed-event technique respectively. The consistency of $J/\psi$ $R_{AA}$ at midrapidity at RHIC and SPS top energies is still a puzzle. Two kinds of models with very different physics origins (recombination models and sequential dissociation models) can qualitatively explain this feature. The measurements of $R_{AA}$ in heavy-ion collisions at a center-of-mass energy between RHIC and SPS top energies are crucial to test these models. The RHIC Beam Energy Scan (BES) program enables such measurements (the reference data for $R_{AA}$ determination already exist). STAR has recorded hundreds of million Au+Au events at $\sqrt{s_{{}_{\mathrm{NN}}}}$ = 39, 62 and 200 GeV respectively during year 2010 run. Figure 5 shows $J/\psi$ signal from partially produced 39 GeV Au+Au data to demonstrate STAR’s $J/\psi$ capability at RHIC low energy run. The left panel of Fig. 5 shows the invariant mass distributions for electron pairs in BEMC triggered events. The electron identification and $J/\psi$ reconstruction is similar as what we used in year 2009 $p$+$p$ data. The S/B ratio is lower than that in $p$+$p$ collisions as expected, but still very high. To improve the statistics, we also reproduce the combinatorial background using mixed-event technique. It is consistent with that from like- sign technique in the mass range shown in the figure. We observed $82\pm 13$ (6 $\sigma$) $J/\psi$ from this dataset, mainly at $p_{T}>2$ GeV/$c$. To study $J/\psi$ production at low $p_{T}$, we also analyzed minimum-bias (MB) triggered data. In this analysis, we excluded BEMC from electron identification due to its inefficiency at low $p_{T}$. The signal is shown in the right panel of Fig. 5. $91\pm 22$ (4 $\sigma$) $J/\psi$ were observed from this 9% of full dataset, 52 in $p_{T}$ range 0-2 GeV/$c$ and 39 in $p_{T}$ range 2-4 GeV/$c$. We expect $\sim 1000$ (13 $\sigma$) $J/\psi$ signal from the full MB dataset. Our projection shows STAR even has the capability to measure $J/\psi$ at 27 and 18 GeV with 1-2 weeks beam time in RHIC year 2011 run. ## 4 Summary In summary, we reported results on $J/\psi$-hadron correlation in $p$+$p$ collisions at $\sqrt{s}$ =200 GeV and $J/\psi$ signal in Au+Au collisions at $\sqrt{s_{{}_{\mathrm{NN}}}}$ =39 GeV from the STAR experiment at RHIC. The fraction of $B$-hadron feed-down contribution to inclusive $J/\psi$ yield in $p$+$p$ collisions was extracted from the $J/\psi$-hadron correlation and found to be 10-25% in $4<p_{T}<12~{}\textrm{GeV}/c$, with no significant dependence on center-of-mass energy. The $p_{T}$ spectra of charged hadron associated with both high-$p_{T}$ $J/\psi$ triggers and high-$p_{T}$ charged hadron triggers on the away side were found to be consistent, which indicates the hadron production on the away side is not dominantly from heavy quark fragmentation. STAR observed 6 $\sigma$ $J/\psi$ signal (mainly at $p_{T}>2~{}\textrm{GeV}/c$) in BEMC triggered 39 GeV Au+Au events, and 4 $\sigma$ signal in 9% produced MB 39 GeV Au+Au events. ## Acknowledgement The author is supported in part by the National Natural Science Foundation of China under Grant No. 11005103 and the China Fundamental Research Funds for the Central Universities. ## References * [1] T. Matsui, H. Satz, Phys. Lett. B178 (1986) 416. doi:10.1016/0370-2693(86)91404-8. * [2] F. Karsch, D. Kharzeev, H. Satz, Phys. Lett. B637 (2006) 75–80. doi:10.1016/j.physletb.2006.03.078. * [3] B. I. Abelev, et al., Phys. Rev. C80 (2009) 041902. arXiv:0904.0439, doi:10.1103/PhysRevC.80.041902. * [4] H. Liu, K. Rajagopal, U.A.Wiedemann, Phys. Rev. Lett. 98 (2007) 182301. * [5] D. E. Acosta, et al., Phys. Rev. D71 (2005) 032001. doi:10.1103/PhysRevD.71.032001. * [6] C. Albajar, et al., Phys. Lett. B256 (1991) 112–120. doi:10.1016/0370-2693(91)90227-H. * [7] B. Bonner, et al., Nucl. Instrum. Meth. A508 (2003) 181–184. doi:10.1016/S0168-9002(03)01347-0. * [8] M. Shao, et al., Nucl. Instrum. Meth. A492 (2002) 344–350. doi:10.1016/S0168-9002(02)01355-4. * [9] J. Wu, et al., Nucl. Instrum. Meth. A538 (2005) 243–248. doi:10.1016/j.nima.2004.08.105. * [10] Z. Tang, Ph.D. thesis, University of Science and Technology and China (2009). * [11] J. Adams, et al., Phys. Rev. Lett. 94 (2005) 062301. doi:10.1103/PhysRevLett.94.062301. * [12] J. Adams, et al., Phys. Rev. Lett. 95 (2005) 152301. doi:10.1103/PhysRevLett.95.152301. * [13] S. Abachi, et al., Phys. Lett. B370 (1996) 239–248. doi:10.1016/0370-2693(96)00067-6. * [14] CMS Collaboration (2010). arXiv:1011.4193. * [15] H. K. Woehri, in: Contribution to 4th Internation Conference on Hard and Electromagnetic Probs of High-Energy Nuclear Collisions (Hard Probes 2010), 2010\. * [16] C. Maiani, in: Contribution to 4th Internation Conference on Hard and Electromagnetic Probs of High-Energy Nuclear Collisions (Hard Probes 2010), 2010\.	arxiv-papers	2010-12-01T16:05:10	2024-09-04T02:49:15.377695	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Zebo Tang (for the STAR Collaboration)", "submitter": "Zebo Tang", "url": "https://arxiv.org/abs/1012.0233" }
1012.0360	# The 2-3 mixing and mass split: atmospheric neutrinos and magnetized spectrometers Abhijit Samantaa,b and A. Yu. Smirnovb aRamakrishna Mission Vivekananda University, Belur Math, Howrah 711 202, India bThe Abdus Salam International Centre for Theoretical Physics Strada Costiera 11, I-34014 Trieste, Italy email: abhijit.samanta@gmail.comemail: smirnov@ictp.it ###### Abstract We study dependence of the atmospheric $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ fluxes on the deviations of the 2-3 mixing from maximal, $\|45^{\circ}-\theta_{23}\|$, on the $\theta_{23}$-octant and on the neutrino mass splitting $\Delta m_{32}^{2}$. Analytic expressions for the $\theta_{23}-$deviation effect and the octant asymmetry are derived. We present conservative estimations of sensitivities of the iron (magnetized) calorimeter detectors (ICAL) to these parameters. ICAL can establish the $\theta_{23}$-deviation at higher than 3$\sigma$ confidence level if $\|45^{\circ}-\theta_{23}\|>6^{\circ}$ with the exposure of 1 Mton$\cdot$yr. Sensitivity to the octant is low for zero or very small 1-3 mixing, but it can be substantially enhanced for $\theta_{13}>3^{\circ}$. ICAL can measure the difference of $\Delta m_{32}^{2}$ in $\nu$ and $\bar{\nu}$ channels (the CPT test) with accuracy $0.8\times 10^{-4}$ eV2 (3$\sigma$) with 1 Mton$\cdot$yr exposure, and the present MINOS result can be excluded at $>5\sigma$ confidence level. We discuss possible ways to further improve sensitivity of the magnetized spectrometers. ###### pacs: 14.60.Pq,14.60Lm ## I Introduction Determination of the 2-3 mass splitting and leptonic mixing, and in particular, the deviation of $\theta_{23}$ from the maximal mixing angle, $\delta_{23}\equiv 45^{\circ}-\theta_{23},$ (1) is of fundamental importance 111$\delta_{23}$ is related to another deviation parameter, $D_{23}\equiv 1/2-\sin^{2}\theta_{23}$ used in literature as $D_{23}=\sin 2\delta_{23}$.. Here we use the standard parameterization of the PMNS mixing matrix: $U_{PMNS}=U_{23}(\theta_{23})\Gamma_{\delta}U_{13}(\theta_{13})U_{12}(\theta_{12}),$ (2) where $U_{ij}$ is the matrix of rotation in the $ij-$plane, and $\Gamma_{\delta}\equiv{\rm diag}(0,0,e^{i\delta})$. Being maximal or close to maximal, the 2-3 mixing testifies for existence of certain underlying symmetry Mohapatra:2006gs . Comparison of the values of $\delta_{23}$ and $\theta_{13}$ as well as the mixing angles in the quarks and lepton sectors can shed some light on the origins of fermion mass and mixing in general. The existing results on $\theta_{23}$ and $\Delta m^{2}_{23}$ are summarized in the Table 1. Note that the global fits of oscillation data GonzalezGarcia:2010er (see also Fogli:2008jx ) show some deviation of the 2-3 mixing from maximal: $\delta_{23}=2-3^{\circ}$ (1$\sigma$). Although the data agree well with maximal mixing, large deviation, $\delta_{23}=\pm 9^{\circ}$, is still possible. Concerning the 2-3 mass splitting, the global fit values are in agreement with the results of SuperKamiokande (SK) Hosaka:2006zd as well as the MINOS measurement in the $\nu$ channel minos . Recently MINOS has reported the values of $\Delta m_{31}^{2}$ and $\theta_{23}$ in the $\bar{\nu}$ channel minos which differ from those in the $\nu$ channel (see tables 1 and 2). If confirmed, this result will testify for an effective (due to existence of some new interactions Kopp:2010qt ) or fundamental CPT violation. The analysis of the atmospheric neutrino data does not confirm MINOS result although the sensitivity of SK to CPT violation is not high since SK sum up effects of neutrinos and antineutrinos Kopp:2010qt . Iron calorimeters (ICAL) ino can perform very sensitive search for the CPT violation and check MINOS result. New accelerator experiments T2K Kato:2008zz and NO$\nu$A Ayres:2004js will improve precision of measurements of $\Delta m_{32}^{2}$ by factor 2, but their accuracy of measurements of $\theta_{23}$ will be only slightly better than that of the present global fit (see table 1). There are two aspects of the $\theta_{23}$-measurements: * • determination of the absolute value of the deviation $\|\delta_{23}\|$, and * • identification of the $\theta_{23}$-octant, i.e. the sign of $\delta_{23}$, or in other words, resolution of the octant degeneracy. The problem of determination of $\delta_{23}$ and the octant with atmospheric neutrinos has been addressed in a number of publications before Kim:1998bv ; Peres:2003wd ; Peres:2009xe ; GonzalezGarcia:2004cu ; Choubey:2005zy ; Indumathi:2006gr ; Bandyopadhyay:2007kx . It was realized Peres:2003wd ; Peres:2009xe ; GonzalezGarcia:2004cu that at low energies oscillation effects on the electron neutrino flux are proportional to this deviation, and therefore searches for an excess (or suppression) of events in the sub-GeV range would testify for $\delta_{23}$. For water Cherenkov detectors both aspects of the 2-3 mixing determinations have been explored in Peres:2009xe ; GonzalezGarcia:2004cu . The study was mainly concentrated on effects in the electron neutrino flux. Magnetized calorimeters are mainly aimed at a detection of the muon neutrinos, but they can distinguish neutrinos and antineutrinos, and this composes their main advantage. These detectors provide better energy and angular resolution of the charged leptons, and consequently, neutrinos. A possibility to disentangle neutrinos and antineutrinos reinforces the following features: i) The energy and angular resolutions (reconstruction) are different for neutrinos and antineutrinos. ii) Sensitivities of the neutrino and antineutrino channels to the oscillation parameters are substantially different. Sensitivity of a magnetized calorimeter to the 2-3 mixing and mass splitting has been explored in Choubey:2005zy and Indumathi:2006gr . It has been shown that at nonzero value of the 1-3 mixing the octant discrimination is more feasible with the magnetized detector than with the water Cherenkov detector since the former can directly measure the matter effect Choubey:2005zy . In these studies, however, various simplifications have been made which do not allow for realistic estimations of potential of the experiments. In the analysis Samanta:2008af a possibility of the octant discrimination has been evaluated for two benchmark values of $\theta_{23}$ and relatively high $\theta_{13}=7.5^{\circ}$. Here we present a comprehensive study of sensitivities of the ICAL to the parameters of 2-3 sector. We assume that by the time of operation of this detector certain information about $\theta_{13}$ will be obtained. The paper is organized as follows. In sec. II we study dependence of the $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ fluxes on parameters of the 2-3 sector: $\|\delta_{23}\|,$ octant and $\Delta m_{23}^{2}$. In sec. III we describe details of our statistical analysis. We evaluate physics potential of the magnetized calorimeter to measure these parameters in sec. IV. In sec. V we consider further improvements of sensitivities of the magnetized calorimeters. Conclusions are given in sec. VI. ## II Dependence of the atmospheric neutrino fluxes on 2-3 mixing The original atmospheric neutrino flux contains both the muon, $F_{\mu}^{0}$, and electron, $F_{e}^{0}$, neutrino components, so that the muon neutrino flux at a detector equals $\displaystyle F_{\mu}$ $\displaystyle=$ $\displaystyle F_{\mu}^{0}P_{\nu_{\mu}\rightarrow\nu_{\mu}}+F_{e}^{0}P_{\nu_{e}\rightarrow\nu_{\mu}}$ (3) $\displaystyle=$ $\displaystyle F_{\mu}^{0}\left[P_{\nu_{\mu}\rightarrow\nu_{\mu}}+\frac{1}{r}P_{\nu_{e}\rightarrow\nu_{\mu}}\right].$ Here the flavor ratio $r(E,\theta_{Z})\equiv\frac{F_{\mu}^{0}(E,\theta_{Z})}{F_{e}^{0}(E,\theta_{Z})}\,$ (4) is function of the neutrino energy $E$ and zenith angle $\theta_{Z}$. For the standard parameterization of the mixing matrix dependence of the oscillation probabilities on the 2-3 mixing $\theta_{23}$ and CP-phase $\delta$ is explicit for an arbitrary density profile. This follows from the order of rotation in eq. (2) and the fact that the matrix of matter potentials has the form $V=diag\\{V_{e},0,0\\}$ in the flavor basis, i.e. it is invariant under the 2-3 rotations. Indeed, the neutrino evolution can be considered in the propagation basis, $\tilde{\nu}\equiv(\nu_{e},\tilde{\nu}_{2},\tilde{\nu}_{3})$, defined via the following relation with the flavor basis: $\nu_{f}\equiv U_{23}\Gamma_{\delta}\tilde{\nu}$. Consequently, $\tilde{\nu}=U_{13}U_{12}\nu_{mass}$, where $\nu_{mass}\equiv(\nu_{1},\nu_{2},\nu_{3})$ is the basis of mass eigenstates. In the propagation basis the Hamiltonian, and therefore the amplitudes of transitions depend on $\theta_{13},\theta_{12},V_{e}$, and mass squared differences: $A_{\alpha\beta}=A_{\nu_{\alpha}\rightarrow\nu_{\beta}}(\theta_{13},\theta_{12},V_{e}),~{}~{}~{}~{}\alpha,\beta=e,\tilde{2},\tilde{3},$ (5) and they do not depend on $\theta_{23}$ and $\delta$. In the flavor basis, dependence of the amplitudes $A_{f}$ on these parameters appears via projections of the matrix of amplitudes (5) from the propagation basis to the flavor basis: $\hat{A}_{f}=U_{23}\Gamma_{\delta}\hat{A}\Gamma_{-\delta}U_{23}^{T},$ (6) where $\hat{A}^{f}$ is the matrix of amplitudes in the flavor basis. In terms of the amplitudes $A_{\alpha\beta}$ using eq. (6) one can rewrite the expression for the muon neutrino flux, eq. (3), in the following form Akhmedov:2008qt : $\displaystyle\frac{F_{\mu}}{F_{\mu}^{0}}$ $\displaystyle\approx$ $\displaystyle 1-\sin^{2}2\theta_{23}\sin^{2}\frac{\phi}{2}$ (7) $\displaystyle-$ $\displaystyle\frac{1}{2}\sin^{2}2\theta_{23}\cos\phi\left[1-{\rm Re}(A_{\tilde{2}\tilde{2}}^{}A_{\tilde{3}\tilde{3}})\right]$ $\displaystyle-$ $\displaystyle\left(s_{23}^{4}-\frac{s_{23}^{2}}{r}\right)\tilde{P}_{A}-\left(c_{23}^{4}-\frac{c_{23}^{2}}{r}\right)\tilde{P}_{S}$ $\displaystyle-$ $\displaystyle\sin 2\theta_{23}P_{\delta},$ where $\tilde{P}_{A}\equiv\|A_{e\tilde{3}}\|^{2}~{}~{}{\rm and}~{}~{}\tilde{P}_{S}\equiv\|A_{e\tilde{2}}\|^{2}$ (8) are the probabilities of transitions $\nu_{e}\rightarrow\tilde{\nu}_{2}$ and $\nu_{e}\rightarrow\tilde{\nu}_{3}$ correspondingly, and $P_{\delta}$ is the function which depends on the CP-violation phase $\delta$. In eq. (7) $\phi$ is the oscillation phase due to the 2-3 mass splitting: $\phi=\Delta m_{32}^{2}x/2E$, $c_{23}\equiv\cos^{2}\theta_{23}$, etc.. Notice that $\tilde{P}_{S}$, $\tilde{P}_{A}$, $P_{\delta}$ and $\phi$ do not depend on $\theta_{23}$. In eq. (7) the first two terms are due to vacuum oscillations driven by the 2-3 mixing and mass splitting; the second line describes interference of the 2-3 and 1-2 modes of oscillations. The product of amplitudes in this term can be approximated as $\|A_{\tilde{2}\tilde{2}}A_{\tilde{3}\tilde{3}}\|\approx\sqrt{(1-\tilde{P}_{A})(1-\tilde{P}_{S})}.$ (9) The terms in the third line describe effects of oscillations due to the 1-2 and 1-3 mixing. The last term in (7) describes the CP-violation. The leading (second) term as well as the interference and CP-violating terms are symmetric with respect to change of sign of the deviation: $\delta_{23}\rightarrow-\delta_{23}$. The octant symmetry (degeneracy) is broken by the terms in the third line of eq. (7); these terms vanish for the maximal 2-3 mixing and $r=2$. For antineutrinos one obtains the same formula as in eq. (7) with substitution $\tilde{P}_{S}\rightarrow\bar{\tilde{P}}_{S},$ $\tilde{P}_{A}\rightarrow\bar{\tilde{P}}_{A}$ and $r\rightarrow\bar{r}$. The octant effect can be characterized by the octant asymmetry defined as $\frac{\Delta^{oct}F_{\mu}}{F^{0}_{\mu}}\equiv\frac{1}{F^{0}_{\mu}}\left[F_{\mu}(45^{\circ}+\delta_{23})-F_{\mu}(45^{\circ}-\delta_{23})\right].$ (10) For such symmetric deviations from the maximal 2-3 mixing one has $\displaystyle\Delta(\sin^{2}2\theta_{23})$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle~{}~{}~{}\Delta(\cos^{2}\theta_{23})$ $\displaystyle=$ $\displaystyle\Delta(\cos^{4}\theta_{23})=-\sin 2\delta_{23}.$ (11) Then according to eq. (7) the octant asymmetry equals $\frac{\Delta^{oct}F_{\mu}}{F_{\mu}^{0}}=\sin 2\delta_{23}\,\left(1-\frac{1}{r}\right)\,(\tilde{P}_{A}-\tilde{P}_{S}).$ (12) Notice that $\tilde{P}_{S}$ and $\tilde{P}_{A}$ enter the asymmetry with opposite signs, and therefore partly cancel each other. To get an idea about dependences of the probabilities on the neutrino parameters one can use expressions for the amplitudes in medium with constant density: $\displaystyle A_{e\tilde{2}}$ $\displaystyle=$ $\displaystyle c_{13}^{m}\sin 2\theta_{12}^{m}\sin\phi_{21}^{m}~{},$ $\displaystyle A_{e\tilde{3}}$ $\displaystyle=$ $\displaystyle\sin 2\theta_{13}^{m}\left(\sin\phi_{32}^{m}e^{-i\phi_{31}^{m}}+\cos^{2}\theta_{12}^{m}\sin\phi_{21}^{m}\right).$ (13) The probability $\tilde{P}_{S}\propto\sin^{2}2\theta^{m}_{12}$ decreases with energy, whereas $\tilde{P}_{A}\propto\sin^{2}2\theta^{m}_{13}$ increases being resonantly enhanced in the neutrino channel (for the normal mass hierarchy) at high energies. Here sensitivity of ICAL to the sign of muon charge will play important role. The two probabilities become comparable at 1 GeV for $\sin^{2}\theta_{13}\sim 0.01$. In the limit of zero 1-3 mixing $\tilde{P}_{A}=0$, $\tilde{P}_{S}\rightarrow{P}_{S}$, and one finds from eq. (7) $\displaystyle\frac{F_{\mu}}{F_{\mu}^{0}}\approx 1-\sin^{2}2\theta_{23}\sin^{2}\frac{\phi}{2}-\frac{1}{2}\sin^{2}2\theta_{23}\cos\phi\times$ $\displaystyle\times\left(1-\sqrt{1-P_{S}}\right)-\left(c_{23}^{4}-\frac{c_{23}^{2}}{r}\right)P_{S},$ (14) where ${P}_{S}=\tilde{P}_{S}(\theta_{13}=0)$, which is the $2\nu$ probability of oscillations driven by $\Delta m^{2}_{21}$ and $\theta_{12}$. For the octant asymmetry we obtain $\frac{\Delta^{oct}F_{\mu}}{F_{\mu}^{0}}=-\sin 2\delta_{23}\,\left(1-\frac{1}{r}\right)\,P_{S}.$ (15) At low energies: $r\approx 2,$ and therefore the asymmetry equals $\sim 1/2\,\sin 2\delta_{23}\,P_{S}$. In fig. 1 we show the oscillograms for the octant asymmetry, that is, the lines of equal asymmetry $\Delta^{oct}F_{\mu}/F_{\mu}^{0}$ in the $E-\cos\theta_{Z}$ plane in neutrino and antineutrino channels. According to eq. (15) these oscillograms coincide with the oscillograms for $P_{S}$ up to coefficient which weakly depends on $E$ and $\theta_{Z}$ at $E<1$ GeV. We use $\delta_{23}=5^{\circ}$, $\theta_{13}=0,$ and other parameters are set at their best-fit values. The asymmetry increases with decrease of the neutrino energy. Maximal asymmetry is achieved in the 1-2 resonance ($E\sim 0.1$ GeV): $\Delta^{oct}F_{\mu}/F_{\mu}^{0}\approx 0.087$. For realistic threshold $E_{th}=0.3$ GeV and $\delta_{23}=5^{\circ}$ the averaged asymmetry is about (2 - 3)% and for $E_{th}=0.8$ GeV it is below 1%. Notice that the octant asymmetry of $\nu_{e}-$flux is about 4 times larger than the $\nu_{\mu}-$flux asymmetry: $\frac{\Delta^{oct}F_{e}}{F_{e}^{0}}=-\sin 2\delta_{23}\,r\,P_{S}.$ (16) Here, however, the original $\nu_{e}-$flux is 2 times smaller. Furthermore, detection of muons provide better energy and direction resolutions. Since $P_{S}$ and $\bar{P}_{S}$ are of the same order, separation of the neutrino and antineutrino signals has no sense here. Let us consider variations of the $\nu_{\mu}$-flux due to deviation of the 2-3 mixing from maximal. According to (7) the relative change of the flux, which we will call the $\theta_{23}-$deviation effect, equals $\displaystyle\frac{\Delta^{dev}F_{\mu}}{F_{\mu}^{0}}$ $\displaystyle\equiv$ $\displaystyle\frac{F_{\mu}(45^{\circ})-F_{\mu}(45^{\circ}-\delta_{23})}{F_{\mu}^{0}}\approx$ (17) $\displaystyle-$ $\displaystyle\Delta(\sin^{2}2\theta_{23})\sin^{2}\frac{\phi}{2}\cong-\frac{1}{2}\sin^{2}2\delta_{23},$ where in the last equality we have averaged the oscillations due to large mass splitting. Ratio of the octant asymmetry and the 2-3 deviation effect equals $\frac{\Delta^{oct}F_{\mu}}{\Delta^{dev}F_{\mu}}=-\frac{1}{\sin 2\delta_{23}}\left(1-\frac{1}{r}\right)(\langle\tilde{P_{A}}\rangle-\langle\tilde{P_{S}}\rangle),$ (18) where $\langle\tilde{P_{S}}\rangle$ and $\langle\tilde{P_{A}}\rangle$ are the probabilities averaged over the experimental $E-\cos\theta_{Z}$ ranges. Although $\Delta^{dev}F_{\mu}^{0}$ is proportional to square of the deviation parameter, for not very small $\delta_{23}$ ($>5^{\circ}$) and $\theta_{13}=0$, the integral effect of the deviation from maximal mixing is stronger than the effect of octant. The reason is that the deviation effect does not change with energy, whereas the octant asymmetry being proportional to $P_{S}$ quickly decreases with $E$ ($\langle\tilde{P_{S}}\rangle\ll\langle\tilde{P_{A}}\rangle).$ For zero 1-3 mixing and $\delta_{23}=5^{\circ}$ we have $\Delta^{dev}F_{\mu}/F_{\mu}^{0}=0.015$, and $\Delta^{oct}F_{\mu}/F_{\mu}^{0}=0.087\langle P_{S}\rangle$. Figure 1: The oscillograms for the octant asymmetry: The contours of equal asymmetry $\Delta^{oct}F_{\mu}/F_{\mu}^{0}$ for neutrino (left) and antineutrino (right) with $\delta_{23}=5^{\circ}$ at $\theta_{13}=0.$ The other parameters are set at their best-fit values. For non-zero 1-3 mixing at high energies the ratio of the flux differences is $\frac{\Delta^{oct}F_{\mu}}{\Delta^{dev}F_{\mu}}\cong-\left(1-\frac{1}{r}\right)\frac{\langle\tilde{P}_{A}\rangle}{\sin 2\delta_{23}},$ (19) and since $\tilde{P}_{A}$ does not decrease with energy the ratio is not small in contrast to the previous case. In our studies of sensitivities to the parameters of 2-3 sector we obtain the oscillation probabilities solving numerically full three flavor evolution equation and using the Preliminary Reference Earth Model (PREM) Dziewonski:1981xy for the density profile of the Earth. We will use the consideration presented in this section for interpretation of the numerical results. ## III The $\chi^{2}$ analysis for ICAL To evaluate physics potential of the atmospheric neutrino studies with a magnetized ICAL detector we generated the atmospheric neutrino events and considered the muon energy and direction (directly measurable quantities) using event generator NUANCE-v3 Casper:2002sd . The GEANT geant simulation of ICAL detector shows that the energy and angular resolutions of muons are very high and the corresponding uncertainties are negligible compared to differences between the angles as well as energies of the neutrino and muon in the scattering (production) process. $\chi^{2}$ is calculated according to the Poisson distribution. The term due to contribution of prior information on the oscillation parameters measured by other experiments is not added to $\chi^{2}$ here for conservative estimation; the effect of priors will be considered in sec. V. The data have been binned in cells of equal size in the ($\log_{10}E$ \- $L^{0.4}$) plane, where $L=2R\cos\theta_{Z}$ is the length of neutrino trajectory. Choice of this binning is motivated by pattern of the oscillation probability $P(\nu_{\mu}\rightarrow\nu_{\mu})$ in the $L-E$ plane Samanta:2008ag . The distance between two consecutive oscillation peaks in this plane increases (decreases) as one goes to lower $L$ ($E$) values for a given $E$ ($L$). The binning of $L$ has been optimized to get better sensitivity to the oscillation parameters. To maintain $\chi^{2}/d.o.f.\approx 1$ in Monte Carlo simulation study, number of events should be $>4$ per cell Samanta:2008af . If the number is smaller than 4 (which happens in the high energy bins), we combine results from the nearest cells. For each set of oscillation parameters we integrate the atmospheric neutrino flux at the detector over the energy and zenith angle folding it with the cross-section, the exposure time, the target mass, the efficiency of detection and the two dimensional energy-angle correlated resolution functions to obtain data for the $\chi^{2}$ analysis. We use the charge current cross section of NUANCE-v3 Casper:2002sd and the neutrino flux of the 3-dimensional scheme Honda:2006qj . The systematic uncertainties of the atmospheric neutrino flux are crucial for determination of the oscillation parameters. We have divided them into two categories: (i) the overall flux normalization uncertainties which are independent of the neutrino energy and zenith angle, and (ii) the spectral tilt uncertainties which depend on $E$ and $\theta_{Z}$. The flux with uncertainties included can be written as $\displaystyle\Phi(E,\theta_{Z})=\Phi_{0}(E)\left[1+\delta_{E}\log_{10}\frac{E}{E_{0}}\right]$ $\displaystyle\times\left[1+\delta_{Z}(\|\cos\theta_{Z}\|-0.5)\right]\times\left[1+\delta_{f_{N}}\right].$ (20) For $E<1$ GeV we take the energy-dependent uncertainty: $\delta_{E}=15\%$ and $E_{0}=1$ GeV, and for $E>10$ GeV correspondingly, $\delta_{E}=5\%$ and $E_{0}=10$ GeV. The uncertainty is $\delta_{E}\sim 7\%$ in the range $E=1-10$ GeV. The overall flux uncertainty as function of the zenith angle is parameterized by $\delta_{Z}$. According to Honda:2006qj we use $\delta_{Z}=4\%$ which leads to 2% vertical/horizontal flux uncertainty. We take for the overall flux normalization uncertainty $\delta_{f_{N}}=10\%$ and for the neutrino cross-section uncertainty: $\delta_{\sigma}=10\%$. In our $\chi^{2}$ analysis the numbers of events have been computed for the theoretical (fit) values and experimental (true) values of parameters in the same way by migrating the number of events from the neutrino to muon energy and zenith angle bins. The resolution functions have been taken from the previous work Samanta:2009qw . In studies of sensitivity to the 2-3 mixing we marginalize $\chi^{2}$ over $\Delta m_{32}^{2},~{}\theta_{13}$, $\delta_{CP}$ for $\nu$’s and $\bar{\nu}$’s separately. Then we find the total $\chi^{2}$ as $\chi^{2}=\chi^{2}_{\nu}+\chi^{2}_{\bar{\nu}}$. We have chosen the following benchmark values of the neutrino parameters: $\Delta m_{32}^{2}=2.5\times 10^{-3}$ eV2, $\delta_{CP}=0$, $\Delta m_{21}^{2}=7.9\times 10^{-5}$ eV2 and $\theta_{12}=34.24^{\circ}$. In marginalization we use, first, flat distributions of values of the parameters in the following ranges: $\Delta m_{32}^{2}=(2.3-2.7)\times 10^{-3}$eV2, $\theta_{23}=36^{\circ}-54^{\circ}$ and $\theta_{13}=0^{\circ}-10.5^{\circ}$. The range of $\theta_{13}$ is changed for some particular analyses. (The non-flat distributions of parameters will be considered in sec. V.) The parameters $\Delta m_{21}^{2}$ and $\theta_{12}$ produce subleading effects on the atmospheric neutrino fluxes for $E>1$ GeV. Moreover, effect of these parameters in marginalization is very small due to their narrow allowed regions. Therefore we have taken fixed values of $\Delta m_{21}^{2}$ and $\theta_{12}$ in our analysis. Figure 2: Dependence of $\Delta\chi^{2}$ on fit value of $\theta_{23}$ for fixed input (true) values $\theta_{23}=37^{\circ}$ (left) and $40^{\circ}$ (right). We used $\theta_{13}=0$, ${\mathcal{E}=}$ 1, 2, and 4 Mton$\cdot$yr, and the energy threshold 0.141 GeV. The $\chi^{2}$ is marginalized with respect to all the oscillation parameters except $\theta_{23}$. The range of marginalization for $\theta_{13}$ is $0-10.5^{\circ}$. Figure 3: Dependence of the ICAL sensitivity to the deviation from maximal mixing on the input value of $\theta_{23}$ for different exposure times: ${\mathcal{E}=}$ 1, 2, and 4 Mton$\cdot$yr. We use the threshold 0.141 GeV. Here $\Delta^{dev}\chi^{2}=\chi^{2}(45^{\circ})-\chi^{2}(\theta_{23}^{true})$. $\chi^{2}$ is marginalized with respect to all the oscillation parameters except $\theta_{23}$. The range of marginalization for $\theta_{13}$ is $0-10.5^{\circ}$. Figure 4: Dependence of the ICAL sensitivity to the $\theta_{23}$-octant on the input (true) value of $\theta_{23}$ for different values of ${\mathcal{E}}$, and threshold 0.141 GeV. Here $\Delta^{oct}\chi^{2}=\chi^{2}(90^{\circ}-\theta_{23})-\chi^{2}(\theta_{23}).$ $\chi^{2}$ is marginalized with respect to all the oscillation parameters except $\theta_{23}$. The range of marginalization for $\theta_{13}$ is $0-10.5^{\circ}$. Figure 5: Dependence of the ICAL sensitivity to the deviation from maximal 2-3 mixing on the input value of $\theta_{23}$ for different marginalization ranges of $\theta_{13}$. We use ${\mathcal{E}}=$ 1 Mton$\cdot$yr and threshold 0.806 GeV. Here $\Delta^{dev}\chi^{2}\equiv\chi^{2}(45^{\circ})-\chi^{2}(\theta_{23}^{true}).$ $\chi^{2}$ is marginalized with respect to all oscillation parameters except $\theta_{23}$. Figure 6: Dependence of the ICAL sensitivity to the $\theta_{23}$-octant on the input value of $\theta_{23}$ with ${\mathcal{E}}=$ 1 Mton$\cdot$yr and threshold of 0.806 GeV. Here $\Delta^{oct}\chi^{2}\equiv\chi^{2}(90^{\circ}-\theta_{23})-\chi^{2}(\theta_{23}).$ $\chi^{2}$ is marginalized with respect to all oscillation parameters except $\theta_{23}$. ## IV Sensitivities of ICAL In our computations we explored the neutrino energy range (0.141 -15) GeV, we used different energy thresholds and different exposures, ${\mathcal{E}}$, of 0.25, 1, 2, and 4 Mton$\cdot$yr. ### IV.1 Determination of 2-3 mixing for $\theta_{13}=0$ The sensitivity of ICAL experiment to $\theta_{23}$ is shown in fig. 2. We plot $\Delta\chi^{2}\equiv\chi^{2}(\theta_{23})-\chi^{2}(\theta^{true}_{23})$ (21) as function of the fit value for fixed input values $\theta_{23}^{true}=37^{\circ}$, (left) and $40^{\circ}$ (right) with ${\mathcal{E}}=$ 1, 2, and 4 Mton$\cdot$yr. We have marginalized $\chi^{2}$ with respect to all the oscillation parameters except $\theta_{23}$. The figure shows high sensitivity to the deviation $\delta_{23}$: it would be possible to discriminate between a given $\theta_{23}$ and maximal mixing at 99% C.L., if $\|\delta_{23}\|>5^{\circ}$. For instance, after 1 Mton$\cdot$yr the angle $\theta_{23}=37^{\circ}$ can be distinguished from $45^{\circ}$ at $8\sigma$ level. The figure shows also low sensitivity of the experiment to the octant. Indeed, $\Delta\chi^{2}$ is higher in the right minima which correspond to the wrong octant. After 1 Mton$\cdot$yr exposure the difference of $\Delta\chi^{2}$ in the true and wrong octants is smaller than 1. The difference becomes more than 2 (90% CL) only after 4 Mton$\cdot$yrs. Identification of the octant becomes even more difficult for smaller $\delta_{23}$. If $\delta_{23}=5^{\circ},$ we find $\Delta\chi^{2}=1.2$ for ${\mathcal{E}}=4$ Mton$\cdot$yr. This result can be readily seen from our analytical consideration in sec. II. The probability $P_{S}$ averaged over the energy interval $(0.14-15)$ GeV equals $\langle P_{S}\rangle\sim 0.02$, so that for $\delta_{23}=8^{\circ}$: $\Delta^{oct}F_{\mu}/F^{0}_{\mu}\sim 4\cdot 10^{-3}$, whereas $\Delta^{dev}F_{\mu}/F^{0}_{\mu}\sim 0.04$ \- an order of magnitude larger. As we mentioned before, this big difference of sensitivities to the deviation and octant (degeneracy) is because the octant asymmetry is collected only at very low energies where $P_{S}$ is unsuppressed, whereas whole energy range contributes to the sensitivity to the deviation $\delta_{23}$. The sensitivity to $\delta_{23}$ drops down substantially with decrease of $\delta_{23}$. Reducing $\delta_{23}$ from $8^{\circ}$ to $5^{\circ}$ (compare the left and right panel of fig. 2) leads to decrease of the flux difference of eq. (17) by factor 2.5, and correspondingly, significance of discrimination from maximal mixing becomes $2\sigma$ (for ${\mathcal{E}}=1$ Mton$\cdot$yr). Sensitivity to the octant at the $1\sigma$ level appears only if ${\mathcal{E}}>4$ Mton$\cdot$yr. In fig. 3 we show the marginalized $\Delta\chi^{2}$, calculated at the maximal mixing (fit value) for different input (true) values of $\theta_{23}$: $\Delta^{dev}\chi^{2}\equiv\chi^{2}(45^{\circ})-\chi^{2}(\theta_{23}^{true}).$ (22) The picture is complementary to that in fig. 2, and there is an approximate symmetry with respect to $\theta_{23}=45^{\circ}$. The reason of sharp increase of $\Delta\chi^{2}$ at 42 and $48^{\circ}$ is related to weak dependence of the oscillation probability on $\delta_{23}=0$ around $\delta_{23}=0$ and to overall flux uncertainty. In fig. 4 we illustrate dependence of the sensitivity to the octant on $\theta_{23}$. For different input (true) values of $\theta_{23}$ we plotted $\Delta^{oct}\chi^{2}\equiv\chi^{2}(90^{\circ}-\theta_{23})-\chi^{2}(\theta_{23}).$ (23) According to fig. 4, it will be possible to discriminate the octant at 90% CL if $\theta_{23}\mathrel{\hbox to0.0pt{\raise 2.15277pt\hbox{$<$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}38^{\circ}$ or $\mathrel{\hbox to0.0pt{\raise 2.15277pt\hbox{$>$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}52^{\circ}$ after ${\mathcal{E}}=4$ Mton$\cdot$yr. Notice that the curves are nearly symmetric with respect to $\theta_{23}=45^{\circ}$. Due to fast decrease of $P_{S}$ with increase of energy (see fig. 1) the sensitivity to the octant disappears for high values of the threshold. ### IV.2 Determination of $\theta_{23}$ in the presence of non-zero 1-3 mixing. Analysis of the oscillation data testifies for the non-zero 1-3 mixing, although significance of this result is not high and zero value of $\theta_{13}$ is not yet excluded. The global fit at 1$\sigma$ C.L. gives $\theta_{13}={7.3^{\circ}}\,^{+2.1^{\circ}}_{-3.2^{\circ}}$, with 3$\sigma$ upper bound of $\theta_{13}<13^{\circ}$, and $\delta_{\text{CP}}\in[0,\,360]$ GonzalezGarcia:2010er . Analysis of the solar and KAMLAND data by SNO collaboration leads to $\theta_{13}={8.13^{\circ}}\,^{+3.53^{\circ}}_{-7.03^{\circ}}$ at 95% CL Aharmim:2009gd . New and forthcoming experiments Double Chooz, Daya Bay, RENO, T2K, NO$\nu$A can confirm this result with higher confidence level or put new stringent upper bound Mezzetto:2010zi which would correspond approximately to a situation with zero 1-3 mixing considered in the previous section. By the time when ICAL will collect significant statistics the angle $\theta_{13}$ will be known with relatively good accuracy. To clarify an impact of this information on the determination of parameters of the 2-3 sector we have performed analysis for non-zero $\theta_{13}$. For illustration purpose we use $\theta_{13}=5^{\circ}$ as the true value and different fit intervals with flat distribution (which could reflect errors in measurements of $\theta_{13}$). In fig. 5 we show dependence of $\Delta^{dev}\chi^{2}$ defined in eq. (22) on the true (input) value of $\theta_{23}$ for the fit value $\theta_{23}=45^{\circ}$ and ${\mathcal{E}}=$ 1 Mton$\cdot$yr. Different curves correspond to different marginalization intervals of $\theta_{13}$. Comparing these results with the results of fig. 3 we find that inclusion of the 1-3 mixing does not change significantly the sensitivity to $\delta_{23}$. The reason is that this sensitivity follows from the main mode of the $\nu_{\mu}-$oscillations; large probability for this mode extends to higher energies and 1-3 mixing produces just additional distortion of the oscillatory pattern. However, inclusion of the 1-3 mixing makes the curve less symmetric with respect to $45^{\circ}$ which reflects an increase of sensitivity to the octant. Also sensitivity to the deviation weakly depends on the marginalization interval for $\theta_{13}$. Fig. 6 illustrates the sensitivity of ICAL to the octant in the presence of non-zero 1-3 mixing. We show dependence of $\Delta^{oct}\chi^{2}$ defined in eq. (23) on the true value of $\theta_{23}^{true}$ for the fit value $\theta_{23}^{fit}=90^{\circ}-\theta_{23}$. Different curves correspond to different marginalization intervals for $\theta_{13}$. There are two important features of the fig. 6. First, the sensitivity to the octant is substantially better for non-zero value of $\theta_{13}$ than for vanishing 1-3 mixing (fig. 6) as was also shown in previous publications GonzalezGarcia:2004cu ; Hiraide:2006vh ; Kajita:2006bt ; Meloni:2008it . This is related to the fact that the octant asymmetry of the flux is determined now by $\frac{\Delta^{oct}F}{F_{\mu}^{0}}\approx\sin 2\delta_{23}\left(1-\frac{1}{r}\right)\langle P_{A}\rangle$ (24) and $\langle P_{A}\rangle\sim 0.1$ in the interval $E=(0.14-15)$ GeV; it is much larger than $\langle P_{S}\rangle$ being enhanced in the energy range E = (3 - 10) GeV (in the resonance channel). Furthermore, at high energies $r\sim(3-4)$ and value of the coefficient in eq. (24) becomes larger. As a result, for $\theta_{23}=51^{\circ}$ the octant can be identified at $2\sigma$ level ($\Delta^{oct}\chi^{2}=4$) with ${\mathcal{E}}=$ 1 Mton$\cdot$yr, as compared to $\Delta^{oct}\chi^{2}=0.3$ for $\theta_{13}=0$. The second feature is significant asymmetry of the curves with respect to $\theta_{23}=45^{\circ}$. The asymmetry is practically absent for fixed input value of $\theta_{13}$ but it increases with broadening of the marginalization interval, and more importantly, with increase of the lower border of this interval. For $\theta_{23}>45^{\circ}$ the curves are practically not changed with change of the interval, whereas for $\theta_{23}<45^{\circ}$ the sensitivity substantially decreases. For instance, taking $\theta_{23}=40^{\circ}$ we obtain $\Delta\chi^{2}=1$ for the interval $\theta_{13}=(3^{\circ}-12.5^{\circ})$ instead of $\Delta^{oct}\chi^{2}=3$ for fixed value $\theta_{13}=5^{\circ}$. This asymmetry can be readily understood from the analytic consideration of sec. II. Neglecting the effect of 1-2 mixing the $\nu_{\mu}-$flux can be presented as $\frac{F_{\mu}}{F_{\mu}^{0}}\approx K(\sin 2\theta_{23})-f(\theta_{23})\left(1-\frac{1}{r}\right)P_{A}(\theta_{13}),$ (25) where $K(\sin 2\theta_{23})$ is an even function of the deviation (symmetric with respect to change of the octant), and $f(\theta_{23})\equiv\left(s_{23}^{4}-\frac{s^{2}_{23}}{r}\right)$ (26) quickly increases with $\theta_{23}$, so that for $r=3-4$ one has $f(\theta_{23}=40^{\circ})\ll f(\theta_{23}>50^{\circ})$. Therefore for $\theta_{23}<45^{\circ}$ the flux $F_{\mu}$ has much weaker dependence on $\theta_{13}$ than for $\theta_{23}>45^{\circ}$. In the process of marginalization over $\theta_{13}$ we compare the true value of the flux, $F_{\mu}^{true}$, with the fit value, $F_{\mu}^{fit}$, and $\Delta^{oct}\chi^{2}$ is proportional to their difference: $\displaystyle\frac{1}{F_{\mu}^{0}}\left\|F_{\mu}^{true}-F_{\mu}^{fit}\right\|=\left(1-\frac{1}{r}\right)\times$ $\displaystyle\left\|f(\theta_{23})\langle P_{A}(\theta_{13}^{true})\rangle-f(90^{\circ}-\theta_{23})\langle P_{A}(\theta_{13})\rangle\right\|.$ (27) If the fit value of 1-3 mixing is fixed: $\theta_{13}^{fit}=\theta_{13}^{true}$, the curve is approximately symmetric with respect to change $\theta_{23}\leftrightarrow(90^{\circ}-\theta_{23})$. Indeed, in this case we have from eq. (27) $\displaystyle\frac{1}{F_{\mu}^{0}}\left\|F_{\mu}^{true}-F_{\mu}^{fit}\right\|=\left(1-\frac{1}{r}\right)$ $\displaystyle\times\langle P_{A}(\theta_{13}^{true})\rangle\left\|f(\theta_{23})-f(90^{\circ}-\theta_{23})\right\|.$ (28) The situation is different if $\theta_{13}$ varies in certain interval $\theta_{13}=[\theta_{13}^{min}-\theta_{13}^{max}]$, and we perform marginalization over $\theta_{13}$ in this interval. Marginalization minimizes the difference of fluxes (eq. (27)) over $\theta_{13}$ for a given value of $\theta_{23}$. If $\theta_{23}<45^{\circ}$, then $f(\theta_{23})\ll f(90^{\circ}-\theta_{23})$. In this case the difference of fluxes is minimal if $\theta_{13}\sim\theta_{13}^{min}$. Indeed, since $\langle P_{A}\rangle$ decreases with $\theta_{13}$, a small value of $\langle P_{A}\rangle$ partially compensates large value of $f(90^{\circ}-\theta_{23})$ in the second term on the right hand side of eq. (27). Furthermore, the smaller $\theta_{13}^{min}$ the stronger compensation, and therefore the smaller $\Delta\chi^{2}$ can be obtained. If $\theta_{23}>45^{\circ}$, then $f(\theta_{23})\gg f(90^{\circ}-\theta_{23})$. Now to compensate the first term in eq. (27) one should take $\langle P_{A}(\theta_{13}^{fit})\rangle\gg\langle P_{A}(\theta_{13})\rangle$. This is, however, not possible for the considered values of $\theta_{13}$. Thus, in the case of unprecise determination of $\theta_{13}$ sensitivity to the octant is higher for $\theta_{23}>45^{\circ}$. Figure 7: Dependence of $\Delta\chi^{2}$ on the fit value of $\Delta m_{32}^{2}$ for the true value $\Delta m_{32}^{2}=2.35\times 10^{-3}$ eV2 in the neutrino and antineutrino channels. We use ${\mathcal{E}}=$ 0.25 and 1 Mton$\cdot$yr and the threshold 0.8 GeV. The marginalization is done over all the oscillation parameters except $\Delta m_{32}^{2}$. The $\chi^{2}$ values 1, 4, 9 correspond to 1$\sigma$ (68.3%), 2$\sigma$ (95.4%), and 3$\sigma$ (99.73%), respectively. Figure 8: The iso-$\chi^{2}$ contours in the $\Delta m^{2}_{32}-\theta_{23}$ plane for neutrinos (left) and antineutrinos (right) obtained with ${\mathcal{E}}=$ 0.25 Mton$\cdot$yr. The inner contour corresponds to $\chi^{2}=1$ and others with increment 1. We use the input value $\Delta m^{2}_{32}=-2.35\times 10^{-3}$eV2, $\theta_{23}=45^{\circ}$, $\theta_{13}=5^{\circ}$ and the threshold 0.8 GeV. Here $\Delta\chi^{2}$ values 2.3, 4.6, 9.2 correspond to 1$\sigma$ (68.3%), 2$\sigma$ (90%), and 3$\sigma$ (99%), respectively. Intensity of color reflects value of $\Delta\chi^{2}$ (see column on the right hand sides of the panels for identification). Figure 9: The same as in fig. 8 but for ${\mathcal{E}}=$ 1 Mton$\cdot$yr. The inner contour corresponds to $\chi^{2}=2$ and for other contours the increment $\Delta\chi^{2}=2$. ### IV.3 Determination of the 2-3 mass split. CPT test Important advantage of a magnetized calorimeter is that it allows one to measure the neutrino mass differences and mixing angles in the neutrino and antineutrino channel separately. A difference of results can be related to some effective or fundamental violation of the CPT symmetry. In fig. 7 we show dependence of $\Delta\chi^{2}$ on the fit value of $\Delta m^{2}_{32}$ for the true value $\Delta m_{32}^{2}=2.35\cdot 10^{-3}$ eV2 in the neutrino and antineutrino channels. We take $\theta_{13}=5^{\circ}$ and $\theta_{23}=45^{\circ}$. According to this figure with ${\mathcal{E}}=0.25$ Mton$\cdot$yr the value $\Delta m_{32}^{2}=3.3\cdot 10^{-3}$ eV2 can be discriminated from the true value at about 2$\sigma$ level. The accuracy of measurement of $\Delta m_{32}^{2}$ is better in the $\nu-$channel. For ${\mathcal{E}}=0.25$ Mton$\cdot$yr the error in $\nu$-channel is about two times smaller than that in $\bar{\nu}$ channel. The difference of accuracies decreases with increase of exposure and e.g. for 1 Mton$\cdot$yr it becomes about 25%. The curves $\Delta\chi^{2}$ are asymmetric with respect to ${\Delta m_{32}^{2}}^{true}$, which is related to the dependence of oscillation probability on $\Delta m_{32}^{2}$. The $1\sigma$ error for $\Delta m^{2}_{23}$ could be $0.15\cdot 10^{-3}$ eV2 and $0.04\cdot 10^{-3}$ eV2 after 0.25 Mton$\cdot$yr and 1 Mton$\cdot$yr exposures correspondingly. With ${\mathcal{E}}=1$ Mton$\cdot$yr the error $0.15\cdot 10^{-3}$ eV2 can be achieved at $3\sigma$ level. With ${\mathcal{E}}=1$ Mton$\cdot$yr one can obtain an accuracy $0.08\cdot 10^{-3}$ eV2 ($90\%$ C.L.) which is better than the present MINOS accuracy. In figs. 8 and 9 we show $\Delta\chi^{2}$ as function of $\Delta m_{32}^{2}$ and $\theta_{23}$ in the neutrino and antineutrino channels for ${\mathcal{E}}=$ 0.25 and 1 Mton$\cdot$yr. As true values we take $\theta_{23}=45^{\circ}$ and $\Delta m_{32}^{2}=2.35\cdot 10^{-3}$ eV2. According to fig. 9 the present MINOS result for $\bar{\nu}$ ($\|\Delta m_{31}^{2}\|=3.36\cdot 10^{-3}~{}{\rm eV}^{2},$ $\theta_{23}=34^{\circ}$) can be excluded by ICAL at about $6\sigma$ level with ${\mathcal{E}}=$ 1 Mton$\cdot$yr. ## V Further improvements of sensitivities There are several directions in which sensitivity of ICAL can be further improved. ### V.1 Adding information about hadrons Figure 10: Effect of inclusion in the analysis the information on hadrons. The iso-$\chi^{2}$ contours (from inner side $\chi^{2}=2$ with increment 2) in the $\Delta m^{2}_{32}-\theta_{23}$ plane without hadrons (left) and with hadrons (right). We take 1 Mton$\cdot$yr, the threshold 0.8 GeV for both muons and hadrons. and sum up the signals from neutrinos and antineutrinos. Measurements of the hadron energy in ICAL in addition to the muon energy is expected to improve reconstruction of the neutrino energy for $E\mathrel{\hbox to0.0pt{\raise 2.15277pt\hbox{$>$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}2$ GeV. However, the total hadron energy in an event is carried out by multiple low energy hadrons. The average energy per hadron per event is $\mathrel{\hbox to0.0pt{\raise 2.15277pt\hbox{$<$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}1$GeV and the average number of hadrons per event is $\mathrel{\hbox to0.0pt{\raise 2.15277pt\hbox{$>$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}2$ At $E\mathrel{\hbox to0.0pt{\raise 2.15277pt\hbox{$<$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}1$GeV the energy resolution is very poor, roughly $80\%$, and the number of hits (number of active detector layers in which signal is detected) increases only logarithmically with $E$. The resolution of hadron energy (for all pions and kaons) at ICAL has been obtained from GEANT4 simulation and parametrized as $\sigma_{had}/E_{had}=a/\sqrt{E_{had}}+b,$ (29) where, $a\approx 0.60$ and $b\approx 0.1$ for the thickness of iron layer 5.6 cm and after averaging over all directions. For each hadron in an event we find the reconstructed hadron energy ($E_{had}^{rec}$) by a random number method using the value of $\sigma_{had}$ from eq. 29. Then we find the final resolution as a function of $[E_{\nu}-(E_{\mu}+E_{had}^{recS})]/E_{\nu}$; where, $E_{had}^{recS}$ is the sum of the energies of all reconstructed hadrons in an event. We use the atmospheric neutrino events without oscillations for an exposure of $50\times 1000$ kTon$\cdot$yr generated by Nuance to reconstruct the resolution functions for each energy and zenith angle bins. In fig. 10 we show the iso-$\chi^{2}$ contours with and without inclusion of information about hadrons. We find that improvement of the sensitivity to $\theta_{23}$, and therefore $\delta_{23}$, is marginal. Also there is a very small change in the sensitivity to $\Delta m_{32}^{2}$: $\Delta(\Delta m_{32}^{2})=0.02\cdot 10^{-3}$ eV2. ### V.2 Cross-sections and fluxes Figs. 11 and 12 illustrate improvements of the sensitivities with reduction of different systematic uncertainties. Decrease of uncertainties of the overall flux normalization, the ratio of horizontal/vertical flux, the neutrino cross- section, and the tilt (below 1 GeV) from 10%, 10%, 2%, and 15% to 2%, 2%, 2%, and 3%, respectively, do not lead to significant improvement of the sensitivities. The reason is that the up-going neutrinos oscillate and down- going neutrinos remain practically unchanged. In $\chi^{2}$ analysis these down-going neutrinos allow to reduce the effect of systematic uncertainties in fluxes. Significant improvement occurs only when all systematic uncertainties are zero. Figure 11: The same as in fig. 5, but with reduced systematic uncertainties. Figure 12: The same as in fig. 6, but with reduced systematic uncertainties. ### V.3 Adding priors to $\chi^{2}$ In $\chi^{2}$ analysis in sec. 4 for simplicity we used flat distributions of uncertainties of the oscillation parameters in marginalization. The prior contribution of the parameters to $\chi^{2}$. should improve the sensitivity. Proper procedure would require the use of the best-fit values and variations of parameters (especially $\theta_{13}$) which will be possible after results of forthcoming experiments will be known. Figure 13: The same as in fig. 5, but with the contributions from priors for the oscillation parameters added to $\chi^{2}$ (see text). The marginalization range for $\theta_{13}$ is $3^{\circ}-12.5^{\circ}(0^{\circ}-12.5^{\circ})$ in absence (presence) of prior contribution. Figure 14: The same as in fig. 6, but with the contribution from priors for the oscillation parameters added to the $\chi^{2}$. The marginalization range for $\theta_{13}$ is $3^{\circ}-12.5^{\circ}(0^{\circ}-12.5^{\circ})$ in absence (presence) of prior contribution. In figs. 13 and 14 we show improvements of the sensitivities to the $\theta_{23}$ deviation and the octant due to inclusion of the prior contribution. We have assumed the Gaussian distribution of the uncertainties around the best fit with width $\sigma(\sin^{2}2\theta_{13})=0.01$, $\sigma(\sin^{2}2\theta_{23})=0.015$ and $\sigma(\Delta m_{32}^{2})/\Delta m_{32}^{2}=0.015$ following Huber:2009cw ; Huber:2004ug . The $2\sigma$ errors in measurements of $\Delta m_{32}^{2}$ are $\pm 0.09$ with flat uncertainties of oscillation parameters, $\pm 0.07$ with prior information from present global-fit, and $\pm 0.018$ with prior information from possible T2K result. The asymmetry in the sensitivity to the octant (which was due to the uncertainty of $\theta_{13}$ in absence of prior contribution) now disappears, and the result does not depend on marginalization range. Of course, larger values of $\theta_{13}$ can substantially enhance the sensitivity to the octant since $P_{A}\sim\sin^{2}\theta_{13}$ at low energies. Table 1: Results of determination of $\theta_{23}$ $\theta_{23}$ \| CL \| Source ---\|---\|--- ${42.9^{\circ}}^{+4.1^{\circ}}_{-2.8^{\circ}}$ \| 1$\sigma$ \| global-fitGonzalezGarcia:2010er $35.7^{\circ}-54^{\circ}$ \| 3$\sigma$ \| global-fitGonzalezGarcia:2010er ${45^{\circ}}^{+10^{\circ}}_{-7.8^{\circ}}$ \| 99% \| SK Hosaka:2006zd $45^{\circ}\pm 9^{\circ}$ \| 90% \| MINOS ($\nu$) Hosaka:2006zd ${34^{\circ}}^{+6^{\circ}}_{-4^{\circ}}$ or ${56^{\circ}}^{+4^{\circ}}_{-6^{\circ}}$ \| 90% \| MINOS ($\bar{\nu}$) minos $39^{\circ}-51^{\circ}$ \| 2$\sigma$ \| T2K Huber:2004dv $36^{\circ}-54^{\circ}$ \| 2$\sigma$ \| NO$\nu$A Huber:2004dv $40^{\circ}-50^{\circ}$ \| 2$\sigma$ \| INO (1 Mton$\cdot$yr) Table 2: Results of determination of $\Delta m^{2}_{31}$ $\Delta m_{32}^{2}(10^{-3}$eV2) \| CL \| Source ---\|---\|--- $-2.36\pm 0.07(\pm 0.36)$ \| 1 (3)$\sigma$ \| global-fit GonzalezGarcia:2010er $+2.47\pm 0.12(\pm 0.37)$ \| 1 (3)$\sigma$ \| global-fit GonzalezGarcia:2010er $2.5^{+0.52}_{-0.60}$ \| 99% \| SK 3$\nu$ Hosaka:2006zd $2.35^{+0.11}_{-0.08}$ \| 90% \| MINOS $\nu$ minos $3.36^{+0.45}_{-0.40}$ \| 90% \| MINOS $\bar{\nu}$ minos $2.5\pm 0.04$ \| 2$\sigma$ \| T2K Huber:2004dv $2.5_{-0.04}^{+0.07}$ \| 2$\sigma$ \| NO$\nu$A Huber:2004dv $2.5\pm 0.07$ \| 2$\sigma$ \| INO (1 Mton$\cdot$yr) ## VI Conclusion We have studied analytically the dependence of the $\theta_{23}-$deviation effect and octant asymmetry of $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ fluxes on the neutrino parameters $\theta_{23}$ and $\theta_{13}$. We explored numerically a sensitivities of a magnetized calorimeter to the $\theta_{23}$-deviation, to the octant and to $\Delta m_{32}^{2}$. We show that for $\theta_{13}=0$ the sensitivity of ICAL to the octant is low even for maximally allowed values of the deviation of the 2-3 mixing from maximal. This is related to the fact that the octant asymmetry is proportional to the “solar” probability $P_{S}$ which is large: $O(1)$ at $E\sim 0.1$ GeV but quickly, as $\propto E^{-2}$, decreases with energy. The situation can be improved by lowering the threshold, increasing exposure and reducing systematic errors (especially in spectral index). We find that $sign(\delta_{23})$ (octant) can be established at $90\%$ C.L. if $\|\delta_{23}\|=7^{\circ}$, ${\mathcal{E}}=$ 4 Mton$\cdot$yr and $E_{th}=0.141$ GeV. ICAL has good sensitivity to the $\theta_{23}$-deviation from maximal 2-3 mixing: the effect is proportional to the probability of the main channel of oscillations, $\nu_{\mu}-\nu_{\tau}$, which is unsuppressed in whole considered neutrino energy range. As a result, dependence of the sensitivity on the energy threshold is weak and it does not change substantially when the effect of 1-3 mixing is included. We find that with the 1 Mton$\cdot$yr exposure the $3\sigma$ accuracy of determination of the deviation will be $\|\delta_{23}\|\approx 6^{\circ}$, which is better than the present global fit result and slightly better than expected sensitivity of T2K ($\approx 9^{\circ}$). The oscillations driven by non-zero 1-3 mixing substantially improve the sensitivity to the octant. One can determine the octant for $\delta_{23}=5^{\circ}$ and $\theta_{13}=5^{\circ}$ at 90% C.L. with 1 Mton$\cdot$yr exposure. We find that this sensitivity depends crucially on the uncertainty range of $\theta_{13}$. For a given nonzero $\theta_{13}$, the sensitivity to octant discrimination is symmetric in $\theta_{23}$ with respect to $\theta_{23}=45^{\circ}.$ However, the asymmetry arises (smaller sensitivity for $\theta_{23}<45^{\circ}$) if value of $\theta_{13}$ can vary in large range. The symmetry is restored if prior for the 1-3 mixing is added. The accuracy of measurements of $\Delta m_{23}^{2}$ by ICAL, $\Delta(\Delta m_{23}^{2})=0.15\cdot 10^{-3}$ eV2 ($3\sigma$, 1 Mton$\cdot$yr exposure), is two times better than the accuracy of the present global fit and it is worthier than the expected sensitivity of T2K. ICAL can measure the difference of $\Delta m_{32}^{2}$ in $\nu$ and $\bar{\nu}$ channels (the CPT test) with accuracy $0.8\times 10^{-4}$ eV2 at 3$\sigma$ confidence level with 1 Mton$\cdot$yr exposure and the present MINOS result can be excluded at $>5\sigma$ confidence level. 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Winter, Nucl. Phys. Proc. Suppl. 145, 190 (2005) [arXiv:hep-ph/0412133].	arxiv-papers	2010-12-02T01:35:36	2024-09-04T02:49:15.386000	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Abhijit Samanta, A. Yu. Smirnov", "submitter": "Abhijit Samanta", "url": "https://arxiv.org/abs/1012.0360" }
1012.0383	# Thermodynamics of apparent horizon and modified Friedman equations Ahmad Sheykhi111sheykhi@mail.uk.ac.ir Department of Physics, Shahid Bahonar University, P.O. Box 76175, Kerman, Iran Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha, Iran ###### Abstract Abstract Starting from the first law of thermodynamics, $dE=T_{h}dS_{h}+WdV$, at apparent horizon of a FRW universe, and assuming that the associated entropy with apparent horizon has a quantum corrected relation, $S=\frac{A}{4G}-\alpha\ln\frac{A}{4G}+\beta\frac{4G}{A}$, we derive modified Friedmann equations describing the dynamics of the universe with any spatial curvature. We also examine the time evolution of the total entropy including the quantum corrected entropy associated with the apparent horizon together with the matter field entropy inside the apparent horizon. Our study shows that, with the local equilibrium assumption, the generalized second law of thermodynamics is fulfilled in a region enclosed by the apparent horizon. ## I Introduction The pioneer study on the deep connection between gravity and thermodynamics was done by Jacobson Jac who showed that the gravitational Einstein equation can be derived from the relation between the horizon area and entropy, together with the Clausius relation $\delta Q=T\delta S$. Further studies on the connection between gravity and thermodynamics has been investigated in various gravity theories Elin ; Cai1 ; Pad . In the cosmological context, attempts to disclose the connection between Einstein gravity and thermodynamics were carried out in Cai2 ; Cai3 ; CaiKim ; Fro ; verlinde . It was shown that the differential form of the Friedmann equation in the Friedmann-Robertson-Walker (FRW) universe can be written in the form of the first law of thermodynamics on the apparent horizon. The profound connection provides a thermodynamical interpretation of gravity which makes it interesting to explore the cosmological properties through thermodynamics. Investigations on the deep connection between gravity and thermodynamics have recently been extended to braneworld scenarios Cai4 ; Shey1 ; Shey2 . It is interesting to note that Friedmann equations, in Einstein’s gravity, can be derived by applying the Clausius relation to the apparent horizon of FRW universe, in which entropy is assumed to be proportional to its horizon area, $S={A}/{4G}$ CaiKim . However, this definition for entropy can be modified from the inclusion of quantum effects, motivated from the loop quantum gravity (LQG). The quantum corrections provided to the entropy-area relationship leads to the curvature correction in the Einstein-Hilbert action and vice versa Zhu . The corrected entropy takes the form Zhang $S_{h}=\frac{A}{4G}-\alpha\ln\frac{A}{4G}+\beta\frac{4G}{A},$ (1) where $\alpha$ and $\beta$ are positive dimensionless constants of order unity. The exact values of these constants are not yet determined and still an open issue in loop quantum cosmology. These corrections arise in the black hole entropy in LQG due to thermal equilibrium fluctuations and quantum fluctuations Rovelli . It is important to note that in the literature different kind of modification of entropy expression have studied in classical level for various modified gravity theories Nojiri1 ; Nojiri11 . The log correction to the area-entropy relation appears to have an almost universal status, having been derived from multiple different approaches to the calculation of entropy from counting microscopic states in different quantum gravity models. Let us stress here that although in the literature there is doubt about the second correction term in entropy-corrected relation, however, it is widely believed Zhang that the next quantum correction term to black hole entropy have the form ${4G}/{A}$, which leads to the reasonable correction terms to Newton’s law of gravitation shey0 and will also lead to the corrected modified Friedmann equation as we will show in this paper. Besides, if thermodynamical interpretation of gravity near apparent horizon is generic feature, one needs to verify whether the results may hold not only for more general spacetimes but also for the other principles of thermodynamics, especially for the generalized second law of thermodynamics. The generalized second law of thermodynamics is a universal principle governing the universe. The generalized second law of thermodynamics in the accelerating universe enveloped by the apparent horizon has been studied extensively in wang1 ; wang2 ; Shey3 . For other gravity theories, the generalized second law has also been considered in akbar . The aim of this paper is twofold. The first is to derive modified Friedmann equations by applying the first law of thermodynamics, $dE=T_{h}dS_{h}+WdV$, at apparent horizon of a FRW universe and assuming the apparent horizon has an entropy expression like (1). The other is to see whether the quantum corrected entropy-area relation together with the matter field entropy inside the apparent horizon will satisfy the generalized second law of thermodynamics. ## II Modified Friedmann Equation from the First law of thermodynamics We consider a homogenous and isotropic FRW universe which is described by the line element $ds^{2}={h}_{\mu\nu}dx^{\mu}dx^{\nu}+\tilde{r}^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),$ (2) where $\tilde{r}=a(t)r$, $x^{0}=t,x^{1}=r$, the two dimensional metric $h_{\mu\nu}$=diag $(-1,a^{2}/(1-kr^{2}))$. Here $k$ denotes the curvature of space with $k=0,1,-1$ corresponding to open, flat, and closed universes, respectively. The dynamical apparent horizon, a marginally trapped surface with vanishing expansion, is determined by the relation $h^{\mu\nu}\partial_{\mu}\tilde{r}\partial_{\nu}\tilde{r}=0$. Straightforward calculation gives the apparent horizon radius for the FRW universe $\tilde{r}_{A}=\frac{1}{\sqrt{H^{2}+k/a^{2}}}.$ (3) The associated temperature with the apparent horizon can be defined as $T=\kappa/2\pi$, where $\kappa$ is the surface gravity $\kappa=\frac{1}{\sqrt{-h}}\partial_{\mu}\left(\sqrt{-h}h^{\mu\nu}\partial_{\mu\nu}\tilde{r}\right).$ Then one can easily show that the surface gravity at the apparent horizon of FRW universe can be written as $\kappa=-\frac{1}{\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$ (4) When $\dot{\tilde{r}}_{A}\leq 2H\tilde{r}_{A}$, the surface gravity $\kappa\leq 0$, which leads the temperature $T\leq 0$ if one defines the temperature of the apparent horizon as $T=\kappa/2\pi$ . Physically it is not easy to accept the negative temperature, the temperature on the apparent horizon should be defined as $T=\|\kappa\|/2\pi$. Recently, the connection between temperature on the apparent horizon and the Hawking radiation has been considered in cao , which gives more solid physical implication of the temperature associated with the apparent horizon. Suppose the matter source in the FRW universe is a perfect fluid with stress- energy tensor $T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu},$ (5) where $\rho$ and $p$ are the energy density and pressure, respectively. The energy conservation law then leads to $\dot{\rho}+3H(\rho+p)=0,$ (6) where $H=\dot{a}/a$ is the Hubble parameter. Following Hay2 , we define the work density as $W=-\frac{1}{2}T^{\mu\nu}h_{\mu\nu}.$ (7) In our case it becomes $W=\frac{1}{2}(\rho-p).$ (8) The work density term is regarded as the work done by the change of the apparent horizon. Assuming the first law of thermodynamics on the apparent horizon is satisfied and has the form $dE=T_{h}dS_{h}+WdV,$ (9) where $S_{h}$ is the quantum corrected entropy associated with the apparent horizon which has the form (1). One can see that in Eq. (9), the work density is replaced with the negative pressure if we compare with the standard first law of thermodynamics, $dE=TdS-pdV$. For a pure de Sitter space, $\rho=-p$, then the work term reduces to the standard $-pdV$ and we obtain exactly the standard first law of thermodynamics. We also assume $E=\rho V$ is the total energy content of the universe inside a $3$-sphere of radius $\tilde{r}_{A}$, where $V=\frac{4\pi}{3}\tilde{r}_{A}^{3}$ is the volume enveloped by 3-dimensional sphere with the area of apparent horizon $A=4\pi\tilde{r}_{A}^{2}$. Taking differential form of the relation $E=\rho\frac{4\pi}{3}\tilde{r}_{A}^{3}$ for the total matter and energy inside the apparent horizon, we get $dE=4\pi\tilde{r}_{A}^{2}\rho d\tilde{r}_{A}+\frac{4\pi}{3}\tilde{r}_{A}^{3}\dot{\rho}dt.$ (10) Using the continuity equation (6), we obtain $dE=4\pi\tilde{r}_{A}^{2}\rho d\tilde{r}_{A}-4\pi H\tilde{r}_{A}^{3}(\rho+p)dt.$ (11) Taking differential form of the corrected entropy (1), we have $dS_{h}=\frac{2\pi\tilde{r}_{A}}{G}\left[1-\frac{\alpha G}{\pi{\tilde{r}_{A}}^{2}}-\frac{\beta G^{2}}{\pi^{2}{\tilde{r}_{A}}^{4}}\right]d\tilde{r}_{A}.$ (12) Inserting Eqs. (8), (11) and (12) in the first law (9) and using the relation between temperature and surface gravity, we can get the differential form of the modified Friedmann equation $\frac{1}{4\pi G}\frac{d\tilde{r}_{A}}{\tilde{r}_{A}^{3}}\left[1-\frac{\alpha G}{\pi{\tilde{r}_{A}}^{2}}-\frac{\beta G^{2}}{\pi^{2}{\tilde{r}_{A}}^{4}}\right]=H(\rho+p)dt.$ (13) Using the continuty equation (6), we can rewrite it as $\frac{-2d\tilde{r}_{A}}{\tilde{r}_{A}^{3}}\left[1-\frac{\alpha G}{\pi{\tilde{r}_{A}}^{2}}-\frac{\beta G^{2}}{\pi^{2}{\tilde{r}_{A}}^{4}}\right]=\frac{8\pi G}{3}d\rho.$ (14) Integrating (14) yields $\frac{1}{{\tilde{r}_{A}}^{2}}-\frac{\alpha G}{2\pi{\tilde{r}_{A}}^{4}}-\frac{\beta G^{2}}{3\pi^{2}{\tilde{r}_{A}}^{6}}=\frac{8\pi G}{3}\rho,$ (15) where an integration constant, which is just the cosmological constant, has been absorbed into the energy density $\rho$. Substituting $\tilde{r}_{A}$ from Eq.(3) we obtain entropy-corrected Friedmann equation $H^{2}+\frac{k}{a^{2}}-\frac{\alpha G}{2\pi}\left(H^{2}+\frac{k}{a^{2}}\right)^{2}-\frac{\beta G^{2}}{3\pi^{2}}\left(H^{2}+\frac{k}{a^{2}}\right)^{3}=\frac{8\pi G}{3}\rho.$ (16) In this way we derive the entropy-corrected Friedmann equation by starting from the first law of thermodynamics, $dE=T_{h}dS_{h}+WdV$, at apparent horizon of a FRW universe, and assuming that the associated entropy with apparent horizon has a quantum corrected relation (1). In the absence of the correction terms $(\alpha=0=\beta)$, one recovers the well-known Friedmann equation in standard cosmology. Since the last two terms in Eq. (16) can be comparable to the first term only when $a$ is very small, the corrections make sense only at early stage of the universe where $a\rightarrow 0$. When the universe becomes large, the entropy-corrected Friedmann equation reduces to the standard Friedmann equation. It is important to note that in the literature many different modifications of entropy and therefore of Friedmann equations are studied in classical modified gravity theories. For example, in Nojiri2 the modified gravity with $\ln R$ or $R^{-n}$ terms which grow at small curvature was discussed. It was shown Nojiri2 that such a model may eliminate the need for dark energy and may provide the current cosmic acceleration. It was also demonstrated that $R^{2}$ terms are important not only for early time inflation but also to avoid the instabilities and the linear growth of the gravitational force. Thus, modified gravity with $R^{2}$ term seems to be viable classical theory. It was also argued in Nojiri11 ; Nojiri3 that the modified gravity where some arbitrary function of Gauss-Bonnet term is added to Einstein action can explain the dark energy dominated universe. It was shown that such theory may pass solar system tests and can describe the most interesting features of late-time cosmology such as the transition from deceleration to acceleration, crossing the phantom divide, current acceleration with effective (cosmological constant, quintessence or phantom) equation of state of the universe. In Nojiri4 the modification of the Friedmann equations which may be caused by $f(R)$ gravity, string-inspired scalar-Gauss-Bonnet, modified Gauss-Bonnet theories, and ideal fluid with the inhomogeneous equation of state. It was demonstrated Nojiri4 that the history of the expansion of the universe can be reconstructed through such a universal formulation. Further investigations on the cosmological implications of the modified theory of gravity have been carried out in Nojiri5 . It is also worth mentioning that Eq. (16) is in complete agreement with the result of Cai5 . However, our derivation is quite different from Cai5 . Let us stress the difference between here and Cai5 . First of all, the authors of Cai5 have derived modified Friedmann equations by applying the first law of thermodynamics, $TdS=-dE$, to the apparent horizon of a FRW universe with the assumption that the apparent horizon has temperature $T=1/2\pi\tilde{r}_{A}$ and corrected-entropy like (1). It is worthy to note that the notation $dE$ in Cai5 is quite different from the same we used in the present work. In Cai5 , $-dE$ is actually just the heat flux $\delta Q$ in Jac crossing the apparent horizon within an infinitesimal internal of time $dt$. But, here $dE$ is change in the the matter energy inside the apparent horizon. Besides, in Cai5 the apparent horizon radius $\tilde{r}_{A}$ has been assumed to be fixed. Thus, the temperature of apparent horizon can be approximated to $T=1/2\pi\tilde{r}_{A}$ and there is no the term of volume change in it. But, here, we have used the matter energy $E$ inside the apparent horizon and the apparent horizon radius changes with time. This is the reason why we have included the term $WdV$ in the first law (9). Indeed, the term $4\pi\tilde{r}_{A}^{2}\rho d\tilde{r}_{A}$ in Eq. (11) contributes to the work term, while this term is absent in $dE$ of Cai5 . This is consistent with the fact that in thermodynamics the work is done when the volume of the system is changed. We have assumed that $d\tilde{r}_{A}$ is the infinitesimal change in the radius of the apparent horizon in a small time interval $dt$ which causes a small change $dV$ of volume inside the apparent horizon. Since the matter energy $E$ is directly related to the radius of the apparent horizon, therefore, the change of apparent horizon radius will change the energy $dE$ inside the apparent horizon. ## III Generalized Second law of thermodynamics In this section we turn to investigate the validity of the generalized second law of thermodynamics in a region enclosed by the apparent horizon. Differentiating Eq. (15) with respect to the cosmic time and using Eq. (6) we get $\frac{-2\dot{\tilde{r}_{A}}}{\tilde{r}_{A}^{3}}\left[1-\frac{\alpha G}{\pi{\tilde{r}_{A}}^{2}}-\frac{\beta G^{2}}{\pi^{2}{\tilde{r}_{A}}^{4}}\right]=-8\pi GH(\rho+p).$ (17) Solving for $\dot{\tilde{r}_{A}}$ we find $\dot{\tilde{r}_{A}}=4\pi GH\tilde{r}_{A}^{3}(\rho+p)\left[1-\frac{\alpha G}{\pi{\tilde{r}_{A}}^{2}}-\frac{\beta G^{2}}{\pi^{2}{\tilde{r}_{A}}^{4}}\right]^{-1}.$ (18) One can see from the above equation that $\dot{\tilde{r}}_{A}>0$ provided the dominant energy condition, $\rho+p>0$, holds. Let us now turn to find out $T_{h}\dot{S_{h}}$: $T_{h}\dot{S_{h}}=\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)\frac{d}{dt}\left(\frac{A}{4G}-\alpha\ln\frac{A}{4G}+\beta\frac{4G}{A}\right).$ (19) After some simplification and using Eq. (18) we obtain $T_{h}\dot{S_{h}}=4\pi H{\tilde{r}_{A}^{3}}(\rho+p)\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$ (20) In the accelerating universe the dominant energy condition may violate, $\rho+p<0$, indicating that the second law of thermodynamics ,$\dot{S_{h}}\geq 0$, does not hold. However, as we will see below the generalized second law of thermodynamics, $\dot{S_{h}}+\dot{S_{m}}\geq 0$, is still fulfilled throughout the history of the universe. From the Gibbs equation we have Pavon2 $T_{m}dS_{m}=d(\rho V)+pdV=Vd\rho+(\rho+p)dV,$ (21) where $T_{m}$ and $S_{m}$ are, respectively, the temperature and the entropy of the matter fields inside the apparent horizon. We limit ourselves to the assumption that the thermal system bounded by the apparent horizon remains in equilibrium so that the temperature of the system must be uniform and the same as the temperature of its boundary. This requires that the temperature $T_{m}$ of the energy inside the apparent horizon should be in equilibrium with the temperature $T_{h}$ associated with the apparent horizon, so we have $T_{m}=T_{h}$Pavon2 . This expression holds in the local equilibrium hypothesis. If the temperature of the fluid differs much from that of the horizon, there will be spontaneous heat flow between the horizon and the fluid and the local equilibrium hypothesis will no longer hold. Therefore from the Gibbs equation (21) we can obtain $T_{h}\dot{S_{m}}=4\pi{\tilde{r}_{A}^{2}}\dot{\tilde{r}}_{A}(\rho+p)-4\pi{\tilde{r}_{A}^{3}}H(\rho+p).$ (22) To check the generalized second law of thermodynamics, we have to examine the evolution of the total entropy $S_{h}+S_{m}$. Adding equations (20) and (22), we get $T_{h}(\dot{S_{h}}+\dot{S_{m}})=2\pi{\tilde{r}_{A}}^{2}(\rho+p)\dot{\tilde{r}}_{A}=\frac{A}{2}(\rho+p)\dot{\tilde{r}}_{A}.$ (23) where $A$ is the apparent horizon area. Substituting $\dot{\tilde{r}}_{A}$ from Eq. (18) into (23) we find $T_{h}(\dot{S_{h}}+\dot{S_{m}})=2\pi GAH{\tilde{r}_{A}}^{3}(\rho+p)^{2}\left[1-\frac{\alpha G}{\pi{\tilde{r}_{A}}^{2}}-\frac{\beta G^{2}}{\pi^{2}{\tilde{r}_{A}}^{4}}\right]^{-1}.$ (24) It is important to note that the expression in the bracket of Eq. (24) is positive at the present time, i.e., $\left[1-\frac{\alpha G}{\pi{\tilde{r}_{A}}^{2}}-\frac{\beta G^{2}}{\pi^{2}{\tilde{r}_{A}}^{4}}\right]>0.$ (25) This is due to the fact that at the present time ${\tilde{r}_{A}}\gg 1$ while $\alpha\sim O(1)$, $\beta\sim O(1)$ and $G\sim 10^{-11}$, thus $\frac{\alpha G}{\pi{\tilde{r}_{A}}^{2}}\ll 1$ and $\frac{\beta G^{2}}{\pi^{2}{\tilde{r}_{A}}^{4}}\ll 1$. At the early time where $\tilde{r}_{A}\rightarrow 0$ the generalized second law of thermodynamics may be violated but in that case the local equilibrium hypothesis is failed too. Besides, from the physical point of view, the effect of the correction terms on the entropy should be less than uncorrected term. Thus, the second and third terms on the right hand side of Eqs. (1) and (12) should be much smaller than the first term, otherwise these terms cannot be regarded as the correction terms. For all above reasons we can expand the right hand side of Eq. (24), up to the linear order of $\alpha$ and $\beta$, $T_{h}(\dot{S_{h}}+\dot{S_{m}})=2\pi GAH{\tilde{r}_{A}}^{3}(\rho+p)^{2}\left[1+\frac{\alpha G}{\pi{\tilde{r}_{A}}^{2}}+\frac{\beta G^{2}}{\pi^{2}{\tilde{r}_{A}}^{4}}\right].$ (26) The right hand side of the above equation cannot be negative throughout the history of the universe, which means that $\dot{S_{h}}+\dot{S_{m}}\geq 0$ always holds. This indicates that for a universe with any spacial curvature the generalized second law of thermodynamics is fulfilled in a region enclosed by the apparent horizon. ## IV Conclusions In summary, applying the first law of thermodynamics, $dE=T_{h}dS_{h}+WdV$, to apparent horizon of a FRW universe with any spatial curvature and assuming that the apparent horizon has temperature $T=\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)$, and a quantum corrected entropy-area relation, $S_{h}=\frac{A}{4G}-\alpha\ln\frac{A}{4G}+\beta\frac{4G}{A}$, we are able to derive modified Friedmann equations governing the dynamical evolution of the universe. We have also investigated the validity of the generalized second law of thermodynamics for the FRW universe with any spatial curvature. We have shown that, when thermal system bounded by the apparent horizon remains in equilibrium with its boundary such that $T_{m}=T_{h}$, the generalized second law of thermodynamics is fulfilled in a region enclosed by the apparent horizon. The validity of the generalized second law of thermodynamics for quantum corrected entropy area relation further supports the thermodynamical interpretation of gravity and provides more confidence on the profound connection between gravity and thermodynamics. It is worth noting that although we derived modified Friedmann equations corresponding to the corrected entropy-area relation (1) by applying the first law of thermodynamics to apparent horizon, it would be of great interest to see whether one is able to get modified Einstein field equation by following Jacobson argument Jac . 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1012.0513	# Physical measures and absolute continuity for one-dimensional center direction Marcelo Viana1 and Jiagang Yang2,1 viana@impa.br www.impa.br/ viana/ yangjg@impa.br ###### Abstract. For a class of partially hyperbolic $C^{k}$, $k>1$ diffeomorphisms with circle center leaves we prove existence and finiteness of physical (or Sinai-Ruelle- Bowen) measures, whose basins cover a full Lebesgue measure subset of the ambient manifold. Our conditions contain an open and dense subset of all $C^{k}$ partially hyperbolic skew-products on compact circle bundles. Our arguments blend ideas from the theory of Gibbs states for diffeomorphisms with mostly contracting center direction together with recent progress in the theory of cocycles over hyperbolic systems that call into play geometric properties of invariant foliations such as absolute continuity. Recent results show that absolute continuity of the center foliation is often a rigid property among volume preserving systems. We prove that this is not at all the case in the dissipative setting, where absolute continuity can even be robust. Partially supported by CNPq and PRONEX-Dynamical Systems. J.Y. was supported by scholarships from TWAS-CNPq and FAPERJ. 1IMPA, Est. D. Castorina 110, 22460-320 Rio de Janeiro, Brazil 2Instituto de Matemática, Universidade Federal Fluminense, Niterói, Brazil ###### Contents 1. 1 Introduction 2. 2 Statement of results 3. 3 Gibbs $u$-states 4. 4 Mostly contracting center 5. 5 Finiteness and stability of physical measures 6. 6 Absolute continuity for mostly contracting center 7. 7 Robust absolute continuity ## 1\. Introduction Let $f:N\to N$ be a diffeomorphism on some compact Riemannian manifold $N$. An invariant probability $\mu$ is a _physical (Sinai, Ruelle, Bowen) measure_ for $f$ if the set of points $z\in N$ for which (1) $\frac{1}{n}\sum_{j=0}^{n-1}\delta_{f^{i}(z)}\to\mu\quad\text{(in the weak${}^{}$ sense)}$ has positive volume. This set is denoted $B(\mu)$ and called the _basin_ of $\mu$. A program for investigating the physical measures of partially hyperbolic diffeomorphisms was initiated by Alves, Bonatti, Viana in [5, 19], who proved existence and finiteness when $f$ is either “mostly expanding” (asymptotic forward expansion) or “mostly contracting” (asymptotic forward contraction) along the center direction. In this paper we analyze the existence and finiteness problem without a priori conditions on the behavior along the center direction, in the case when the center bundle has dimension $1$. Our results are illustrated by the following example. Suppose $N=M\times S^{1}$, for some compact manifold $M$, and $f_{0}:N\to N$ is a partially hyperbolic _skew-product_ (2) $f_{0}:M\times S^{1}\to M\times S^{1},\quad f_{0}(x,\theta)=(g_{0}(x),h_{0}(x,\theta))$ with center bundle $E^{c}$ coinciding with the vertical direction $\\{0\\}\times TS^{1}$ at every point. This implies $g_{0}$ is an Anosov diffeomorphism, and we also take it to be transitive (all known Anosov diffeomorphisms being transitive). Assume $f_{0}$ is of class $C^{k}$ for some $k>1$, not necessarily an integer. ###### Theorem A. There exists a $C^{k}$ neighborhood ${\mathcal{U}}_{0}$ of $f_{0}$ such that for every $f\in{\mathcal{U}}_{0}$ which is accessible and whose center stable foliation is absolutely continuous there exists a finite number of physical measures. These measures are ergodic, the union of their basins has full volume in $N$, and the center Lyapunov exponents are either negative or zero. In the latter (zero) case the physical measure is unique. The subset of accessible diffeomorphisms is $C^{1}$ open and $C^{k}$ dense in the neighborhood of $f_{0}$ (Theorem 1.5 of Niţicǎ, Török [32]). Absolute continuity is also quite common in this context as we are going to see. That is surprising, since Avila, Viana, Wilkinson [14] have recently shown that absolute continuity of the center foliation is a rigid property for _volume preserving_ perturbations of skew-products. In contrast, here we prove ###### Theorem B. Suppose $f_{0}$ exhibits some periodic vertical leaf $\ell$ such that $f_{0}^{per(\ell)}\mid\ell$ is Morse-Smale with a unique periodic attractor and repeller. Then $f_{0}$ is in the closure of an open set ${\mathcal{V}}$ of $C^{k}$ diffeomorphisms such that for every $f\in{\mathcal{V}}$, • the center stable, the center unstable, and the center foliation are absolutely continuous * • both $f$ and its inverse have a unique physical measure, whose basin has full Lebesgue measure in $N$. Then the same is true for $f_{0}=g_{0}\times\operatorname{id}$, since it is $C^{k}$ approximated by diffeomorphisms as in the hypothesis of the theorem. Although we are primarily interested in general (dissipative) diffeomorphisms, our methods also shed some light on the issue of absolute continuity in the volume preserving context. Let $\lambda^{c}(f)$ denote the integrated center Lyapunov exponent of $f$ relative to the Lebesgue measure. ###### Theorem C. For any small $C^{1}$ neighborhood ${\mathcal{W}}$ of $f_{0}=g_{0}\times\operatorname{id}$ in the space of volume preserving diffeomorphisms of $N$, 1. (1) the subset ${\mathcal{W}}_{0}$ of diffeomorphisms $f\in{\mathcal{W}}$ such that $\lambda^{c}(f)\neq 0$ is $C^{1}$ open and dense in ${\mathcal{W}}$; 2. (2) if $f\in{\mathcal{W}}_{0}$ and $\lambda^{c}(f)>0$ then the center foliation and the center stable foliation are not (even upper leafwise) absolutely continuous; 3. (3) there exists a non-empty $C^{1}$ open set ${\mathcal{W}}_{1}\subset\\{f\in{\mathcal{W}}_{0}:\lambda^{c}(f)>0\\}$ such that the center unstable foliation of every $g\in{\mathcal{W}}_{1}$ is absolutely continuous. Claims (2) and (3) remain true when $\lambda^{c}(f)<0$, if one exchanges center stable with center unstable. Every $C^{k}$, $k>1$ diffeomorphism $f\in{\mathcal{W}}_{1}$ has a $C^{k}$ neighborhood ${\mathcal{W}}_{f}$ in the space of all (possibly dissipative) diffeomorphisms where the center unstable foliation remains absolutely continuous. Theorems A and B follow from more detailed statements that we present in the next section, where we also recall the main notions involved. A discussion of the volume preserving case is given in Section 7.3, including the proof of Theorem C and a (partly conjectural) scenario. ## 2\. Statement of results Let ${\mathcal{P}}_{}^{k}(N)$ be the space of partially hyperbolic, dynamically coherent, $C^{k}$ diffeomorphisms whose center leaves are compact, with any dimension, and form a fiber bundle. Unless otherwise stated, we always assume $k>1$. Most of our results concern the subspace ${\mathcal{P}}_{1}^{k}(N)$ of diffeomorphisms with $1$-dimensional center dimension. Let us begin by recalling the notions involved in these definitions. ### 2.1. Basic concepts A diffeomorphism $f:N\to N$ is _partially hyperbolic_ if there exists a continuous $Df$-invariant splitting $TN=E^{u}\oplus E^{c}\oplus E^{s}$ and there exist constants $C>0$ and $\lambda<1$ such that (a) $\\|Df_{x}^{-n}(v^{u})\\|\leq C\lambda^{n}$ and $\\|Df_{x}^{n}(v^{s})\\|\leq C\lambda^{n}$ * (b) $\\|Df_{x}^{-n}(v^{u})\\|\leq C\lambda^{n}\\|Df_{x}^{-n}(v_{c})\\|$ and $\\|Df_{x}^{n}(v^{s})\\|\leq C\lambda^{n}\\|Df_{x}^{n}(v_{c})\\|$ for all unit vectors $v^{u}\in E_{x}^{u}$, $v^{c}\in E_{x}^{c}$, $v^{s}\in E_{x}^{s}$, and all $x\in N$ and $n\geq 0$. Condition (a) means that the derivative $Df$ is uniformly expanding along $E^{u}$ and uniformly contracting along $E^{s}$. Condition (b) means that the behavior of $Df$ along the _center bundle_ $E^{c}$ is dominated by the behavior along the other two factors. Here all three bundles are assumed to have positive dimension. The bundles $E^{u}$ and $E^{s}$ are always integrable: there exist foliations ${\mathcal{W}}^{u}$ and ${\mathcal{W}}^{s}$ of $N$ tangent to $E^{u}$ and $E^{s}$, respectively, at every point. In fact these foliations are unique. Moreover, they are _absolutely continuous_ , meaning that the projections along the leaves between any two cross-sections preserve the class of sets with zero volume inside the cross-section. See [20, 28, 42]. A diffeomorphism $f:N\to N$ is _dynamically coherent_ if the bundles $E^{cu}=E^{c}\oplus E^{u}$ and $E^{cs}=E^{c}\oplus E^{s}$ also admit integral foliations, ${\mathcal{W}}^{cu}$ and ${\mathcal{W}}^{cs}$. Then, intersecting their leaves one obtains a _center foliation_ ${\mathcal{W}}^{c}$ tangent at every point to the center bundle $E^{c}$. We say that the center leaves _form a fiber bundle_ over the leaf space $N/{\mathcal{W}}^{c}$ if for any ${\mathcal{W}}^{c}(x)\in N/{\mathcal{W}}^{c}$ there is a neighborhood $V\subset N/{\mathcal{W}}^{c}$ of ${\mathcal{W}}^{c}(x)$ and a homeomorphism $h_{x}:V\times{\mathcal{W}}^{c}(x)\to\pi_{c}^{-1}(V)$ smooth along the verticals $\\{\ell\\}\times{\mathcal{W}}^{c}(x)$ and mapping each vertical onto the corresponding center leaf $\ell$. ###### Remark 2.1. The fiber bundle condition is probably not necessary. Indeed, when the diffeomorphisms are volume preserving, Avila, Viana, Wilkinson [14] prove that if $\dim E^{c}=1$ and the generic center leaves are circles then the center leaves form a fiber bundle _up to a finite cover_. In particular, all leaves are circles. Our arguments extend easily to this situation. A partially hyperbolic diffeomorphism $f:N\to N$ is _accessible_ if any points $z$, $w\in N$ can be joined by a piecewise smooth curve $\gamma$ such that every smooth leg of $\gamma$ is tangent to either $E^{u}$ or $E^{s}$ at every point. Equivalently, every smooth leg of the curve $\gamma$ is contained in a leaf of either ${\mathcal{W}}^{u}$ or ${\mathcal{W}}^{s}$. The _center Lyapunov exponent_ $\lambda^{c}(\mu)$ of an $f$-invariant probability measure $\mu$ is defined by (3) $\lambda^{c}(\mu)=\int\lambda^{c}(z)\,d\mu(z)\quad\text{where $\lambda^{c}(z)=\lim_{n\to\infty}\frac{1}{n}\log\|Df^{n}\mid E^{c}_{z}\|$.}$ By the ergodic theorem, this may be rewritten (4) $\lambda^{c}(\mu)=\int\log\|Df\mid E^{c}_{z}\|\,d\mu(z).$ If $\mu$ is ergodic then $\lambda^{c}(\mu)=\lambda^{c}(z)$ for $\mu$-almost every $z$. Finally, the center direction is _mostly contracting_ (Bonatti, Viana [19]) if (5) $\limsup_{n\to+\infty}\frac{1}{n}\log\\|Df^{n}\mid E^{c}_{x}\\|<0.$ for a positive volume measure subset of any disk inside a strong unstable leaf. It was shown by Andersson [9] that this is a $C^{k}$, $k>1$ open property. ### 2.2. The leaf space Let $d$ be the Riemannian distance on $N$. We endow the leaf space $N/{\mathcal{W}}^{c}$ with the distance defined by $d_{c}(\xi,\eta)=\sup_{x\in\xi}\inf_{y\in\eta}d(x,y)+\sup_{y\in\eta}\inf_{x\in\xi}d(x,y)\quad\text{for each }\xi,\eta\in N/{\mathcal{W}}^{c}.$ The quotient map $\pi_{c}:(N,d)\to(N/{\mathcal{W}}^{c},d_{c})$ is continuous and onto. In particular, the metric space $(N/{\mathcal{W}}^{c},d_{c})$ is compact. Let $f_{c}:N/{\mathcal{W}}^{c}\to N/{\mathcal{W}}^{c}$ be the map induced by $f$ on the quotient space $N/{\mathcal{W}}^{c}$. The stable set of a point $\xi\in N/{\mathcal{W}}^{c}$ for $f_{c}$ is defined by $W^{s}(\xi)=\\{\eta\in N/{\mathcal{W}}^{c}:d_{c}(f_{c}^{n}(\xi),f_{c}^{n}(\eta))\to 0\text{ when }n\to+\infty\\}$ and the local stable set of size $\varepsilon>0$ is defined by $W^{s}_{\varepsilon}(\xi)=\\{\eta\in N/{\mathcal{W}}^{c}:d_{c}(f_{c}^{n}(\xi),f_{c}^{n}(\eta))\leq\varepsilon\text{ for all }n\geq 0\\}.$ The unstable set and local unstable set of size $\varepsilon>0$ are defined in the same way, for backward iterates. It follows from the definitions that there exist constants $K$, $\tau$, $\varepsilon$, $\delta>0$ such that 1. (1) $d_{c}(f_{c}^{n}(\eta_{1}),f_{c}^{n}(\eta_{2}))\leq Ke^{-\tau n}d_{c}(\eta_{1},\eta_{2})$ for all $\eta_{1},\eta_{2}\in W^{s}_{\varepsilon}(\xi)$, $n\geq 0$; 2. (2) $d_{c}(f_{c}^{-n}(\zeta_{1}),f_{c}^{-n}(\zeta_{2}))\leq Ke^{-\tau n}d_{c}(\zeta_{1},\zeta_{2})$ for all $\zeta_{1},\zeta_{2}\in W^{u}_{\varepsilon}(\xi)$, $n\geq 0$; 3. (3) if $d_{c}(\xi_{1},\xi_{2})\leq\delta$ then $W^{s}_{\varepsilon}(\xi_{1})$ and $W^{u}_{\varepsilon}(\xi_{2})$ intersect at exactly one point, denoted $[\xi_{1}\,\xi_{2}]$ and this point depends continuously on $(\xi_{1},\xi_{2})$. This means that $f_{c}$ is a hyperbolic homeomorphism (in the sense of Viana [48]). We denote ${\mathcal{W}}^{c}(\Lambda)=\pi_{c}^{-1}(\Lambda)$, for any subset $\Lambda$ of $N/{\mathcal{W}}^{c}$. By Anosov’s closing lemma [10], periodic points are dense in the non-wandering set of $f_{c}$. Smale’s spectral decomposition theorem [45], the non-wandering set splits into a finite number of compact, invariant, transitive, pairwise disjoint subsets. Among these basic pieces of the non-wandering set, the _attractors_ $\Lambda_{i}$, $i=1,\dots,k$ of $f_{c}$ are characterized by the fact that $\Lambda_{i}=\bigcap_{n=0}^{\infty}f_{c}^{n}(U_{i})$ for some neighborhood $U_{i}$ of $\Lambda_{i}$ and it is transitive. The union of the stable sets $W^{s}(\Lambda_{i})$, $i=1,\dots,k$ is an open dense subset of $N/{\mathcal{W}}^{c}$. Every attractor $\Lambda_{i}$ consists of entire unstable sets, and so ${\mathcal{W}}^{c}(\Lambda_{i})$ is ${\mathcal{W}}^{u}$-saturated, that is, it consists of entire strong unstable leaves of $f$. Additionally, every $\Lambda_{i}$ has finitely many connected components $\Lambda_{i,j}$, $j=1,\dots,n_{i}$ that are mapped to one another cyclically. The unstable set $W^{u}(x)$ of every $x\in\Lambda_{i,j}$ is contained and dense in $\Lambda_{i,j}$. In particular, ${\mathcal{W}}^{c}(\Lambda_{i,j})$ is also ${\mathcal{W}}^{u}$-saturated. If $f_{c}$ is transitive, there is a unique attractor $\Lambda_{1}=N/{\mathcal{W}}^{c}$. We say $f$ is _accessible on $\Lambda_{i}$_ if, for every $j$, any points $z$, $w\in{\mathcal{W}}^{c}(\Lambda_{i,j})$ can be joined by a piecewise smooth curve $\gamma$ such that every smooth leg of $\gamma$ is tangent to either $E^{u}$ or $E^{s}$ at every point and the corner points belong to the same ${\mathcal{W}}^{c}(\Lambda_{i,j})$. The center direction of $f\mid{{\mathcal{W}}^{c}(\Lambda_{i})}$ is _mostly contracting_ if (5) holds for a positive volume measure subset of any disk inside a strong unstable leaf contained in ${\mathcal{W}}^{c}(\Lambda_{i})$. ### 2.3. Physical measures We are ready to state our main result on existence and finiteness of physical measures: ###### Theorem D. If $f\in{\mathcal{P}}_{1}^{k}(N)$, $k>1$ is accessible on every attractor and the center stable foliation is absolutely continuous then, for each attractor $\Lambda_{i}$, either * (a) there is a Lipschitz metric on each leaf of ${\mathcal{W}}^{c}(\Lambda_{i})$, depending continuously on the leaf and invariant under $f$; then $f$ admits a unique physical measure, which is ergodic, whose basin has full volume in the stable set of ${\mathcal{W}}^{c}(\Lambda_{i})$, and whose center Lyapunov exponent vanishes; * (b) or the center direction of $f\mid{{\mathcal{W}}^{c}(\Lambda_{i})}$ is mostly contracting; then $f\mid{{\mathcal{W}}^{c}(\Lambda_{i})}$ has finitely many physical measures, they are ergodic for $f$ and Bernoulli for some iterate, the union of their basins is a full volume subset of the stable set of ${\mathcal{W}}^{c}(\Lambda_{i})$, and their center Lyapunov exponents are negative. The union of the basins of these physical measures has full volume in $N$. To see that Theorem A is contained in Theorem D let us to note that, for every $k\geq 1$, any $C^{k}$ partially hyperbolic skew-product $f_{0}$ is in the interior of ${\mathcal{P}}_{1}^{k}(N)$. Indeed, partial hyperbolicity is well known to be a $C^{1}$ open property and the stability theorem for normally hyperbolic foliations (Hirsch, Pugh, Shub [28]) gives that every $f$ in a $C^{1}$ neighborhood of $f_{0}$ admits an invariant ${\mathcal{W}}^{}_{f}$ foliation, for each $\in\\{cu,cs,c\\}$, and there exists a homeomorphism mapping the leaves of ${\mathcal{W}}^{}_{f}$ diffeomorphically to the leaves of ${\mathcal{W}}^{}_{f_{0}}$. In particular, the center leaves of $f$ form a circle fiber bundle. ###### Remark 2.2. When the center fiber bundle is trivial, as happens near skew-products, part (a) of the Theorem D gives that $f\mid{{\mathcal{W}}^{c}(\Lambda_{i})}$ is topologically conjugate to a rotation extension $\Lambda_{i}\times{\mathbb{R}}/{\mathbb{Z}}\to\Lambda_{i}\times{\mathbb{R}}/{\mathbb{Z}},\quad(x,\theta)\mapsto(f_{c}(x),\theta+\omega(x)).$ To see this, fix some consistent orientation of the center leaves and any continuous section $\sigma:N/{\mathcal{W}}^{c}\to N$ of the center foliation, that is, any continuous map such that $\sigma(\ell)\in\ell$ for every $\ell\in N/{\mathcal{W}}^{c}$. Then define $h:{\mathcal{W}}^{c}(\Lambda_{i})\to\Lambda_{i}\times{\mathbb{R}}/{\mathbb{Z}},\quad h(z)=(\pi_{c}(z),\|\sigma(\pi_{c}(z)),z\|)$ where $\|\sigma(\pi_{c}(z)),z\|$ denotes the length, with respect to the $f$-invariant Lipschitz metric, of the (oriented) curve segment from $\sigma(\pi_{c}(z))$ to $z$ inside the center leaf. This map sends the center leaves of $f$ to verticals $\\{w\\}\times{\mathbb{R}}/{\mathbb{Z}}$, mapping the $f$-invariant Lipschitz metric on the center leaves to the standard metric on ${\mathbb{R}}/{\mathbb{Z}}$. Then $h\circ f\circ h^{-1}$ preserves the standard metric measure on the verticals, and so it is a rotation extension, as stated. Observe that, in addition, both $h$ and its inverse are Lipschitz on every leaf. Explicit bounds on the number of physical measures can be given in many cases. For instance, we will see in Theorem 5.3 that if $f$ admits some periodic center leaf $\ell$ restricted to which $f$ is Morse-Smale then the number of physical measures over the attractor containing $\pi_{c}(\ell)$ is bounded by the number of periodic orbits on $\ell$. Notice that we must have alternative (b) of Theorem D in this case, since alternative (a) is incompatible with the existence of hyperbolic periodic points. We also want to analyze the dependence of the physical measures on the dynamics. For this, we assume $N=M\times S^{1}$ and restrict ourselves to the subset ${\mathcal{S}}^{k}(N)\subset{\mathcal{P}}^{k}_{1}(N)$ of skew-product maps. We prove in Theorem 5.6 that there is an open and dense subset of diffeomorphisms $f\in{\mathcal{S}}^{k}(N)$ with mostly contracting center direction, such that the number of physical measures is locally constant and the physical measures vary continuously with the diffeomorphism. This property of _statistical stability_ has been studied in a number of recent works, including Alves, Viana [8], Vásquez [47], Andersson [9]. As mentioned before, existence and finiteness of physical measures for partially hyperbolic diffeomorphisms was proved by Alves, Bonatti, Viana [5, 19], under certain assumptions of weak hyperbolicity along the center direction. Substantial improvements followed, by Alves, Luzzatto, Pinheiro [6, 7], Alves, Araujo [4], Vasquez [47], Pinheiro [36], and Andersson [9], among others. Perturbations of certain skew-products over hyperbolic maps have been studied by Alves [2, 3], Buzzi, Sestier, Tsujii [23], and Gouezel [25]. In a remarkable recent paper, Tsujii [46] proved that generic (dense $G_{\delta}$) partially hyperbolic surface _endomorphisms_ do admit finitely many physical measures, such that the union of their basins has full Lebesgue measure. His approach is very different from the one in the present paper and it is not clear how it could be extended to diffeomorphisms in higher dimensions, even in the case of one-dimensional center bundle. ### 2.4. Absolute continuity It has been pointed out by Shub, Wilkinson [44] that foliations tangent to the center subbundle $E^{c}$ are often _not_ absolutely continuous. In fact, Ruelle, Wilkinson [41] showed that the disintegration of Lebesgue measure along the leaves is often atomic. Moreover, Avila, Viana, Wilkinson [14] observed recently that for certain classes of volume preserving diffeomorphisms, including perturbations of skew-products (2), absolute continuity of the center foliation is a rigid property: it implies that the center foliation is actually smooth, and the map is smoothly conjugate to a rigid model. However, we prove that this is not at all the case in our dissipative setting: ###### Theorem E. There is an open set ${\mathcal{U}}\subset{\mathcal{P}}_{1}^{k}(N)$, $k>1$, such that the center stable, the center unstable, and the center foliation are absolutely continuous for every $f\in{\mathcal{U}}$. Moreover, ${\mathcal{U}}$ may be chosen to accumulate on every skew-product map $f_{0}$ that admits a periodic vertical fiber restricted to which the map is Morse-Smale with a unique periodic attractor and repeller. Two weaker forms of absolute continuity are considered by Avila, Viana, Wilkinson [14]. Let $\operatorname{vol}$ denote Lebesgue measure in the ambient manifold and $\operatorname{vol}_{L}$ be Lebesgue measure restricted to some submanifold $L$. A foliation ${\mathcal{F}}$ on $N$ is (_lower_) _leafwise absolutely continuous_ if for every zero $\operatorname{vol}$-measure set $Y\subset N$ and $\operatorname{vol}$-almost every $z\in M$, the leaf $L$ through $z$ meets $Y$ in a zero $\operatorname{vol}_{L}$-measure set. Similarly, ${\mathcal{F}}$ is _upper leafwise absolutely continuous_ if $\operatorname{vol}_{L}(Y)=0$ for every leaf $L$ through a full measure subset of points $z\in M$ implies $\operatorname{vol}(Y)=0$. Absolute continuity implies both lower and upper leafwise absolute continuity (see [14, 21]); the converse is not true in general. We will see in Proposition 6.2 that the center stable foliation of a partially hyperbolic, dynamically coherent diffeomorphism with mostly contracting center direction is always upper leafwise absolutely continuous. This does not extend to lower leafwise absolutely continuity, in general: robust counter-examples will appear in [49]; see also Example 6.1 for a related construction. However, as stated before, full absolute continuity of the center foliation does hold on some open subsets of diffeomorphisms with mostly contracting center. ## 3\. Gibbs $u$-states Let $f:N\to N$ be a partially hyperbolic diffeomorphism. In what follows we denote $I_{r}=[-r,r]$ for $r>0$ and $d_{}=\dim E^{}$ for each $\in\\{u,cu,c,cs,s\\}$. We use $\operatorname{vol}^{}$ to represent the volume measure induced by the restriction of the Riemannian structure on the leaves of the foliation ${\mathcal{W}}^{}$ for each $\in\\{u,cu,c,cs,s\\}$. Following Pesin, Sinai [34] and Alves, Bonatti, Viana [5, 19] (see also [17, Chapter 11]), we call _Gibbs $u$-state_ any invariant probability measure $m$ whose conditional probabilities (Rokhlin [40]) along strong unstable leaves are absolutely continuous with respect to the volume measure $\operatorname{vol}^{u}$ on the leaf. More precisely, let $\Phi:I_{1}^{d_{u}}\times I_{1}^{d_{cs}}\to N$ be any _foliated box_ for the strong unstable foliation. By this we mean that $\Phi$ is a homeomorphism and maps every horizontal plaque $I_{1}^{d_{u}}\times\\{\eta\\}$ diffeomorphically to a disk inside some strong unstable leaf. Pulling $m$ back under $\Phi$ one obtains a measure $m_{\Phi}$ on $I_{1}^{d_{u}}\times I_{1}^{d_{cs}}$. The definition of Gibbs $u$-state means that there exists a measurable function $\alpha_{\Phi}(\cdot\,,\cdot)\geq 0$ and a measure $m_{\Phi}^{cs}$ on $I_{1}^{d_{cs}}$ such that (6) $m_{\Phi}(A)=\int_{A}\alpha_{\Phi}(\xi,\zeta)\,d\xi\,dm_{\Phi}^{cs}(\zeta)$ for every measurable set $A\subset I_{1}^{d_{u}}\times I_{1}^{d_{cs}}$. Proofs for the following basic properties of Gibbs $u$-states can be found in Section 11.2 of Bonatti, Díaz, Viana [17]: ###### Proposition 3.1. Let $f:N\to N$ be a partially hyperbolic diffeomorphism. 1. (1) The densities of a Gibbs $u$-state with respect to Lebesgue measure along strong unstable plaques are positive and bounded from zero and infinity. 2. (2) The support of every Gibbs $u$-state is ${\mathcal{W}}^{u}$-saturated, that is, it consists of entire strong unstable leaves. 3. (3) The set of Gibbs $u$-states is non-empty, weak∗ compact, and convex. Ergodic components of Gibbs $u$-states are Gibbs $u$-states. 4. (4) Every physical measure of $f$ is a Gibbs $u$-state and, conversely, every ergodic $u$-state whose center Lyapunov exponents are negative is a physical measure. Now let $f\in{\mathcal{P}}^{k}_{}(N)$. Recall that $\pi_{c}:N\to N/{\mathcal{W}}^{c}$ denotes the natural quotient map and $f_{c}:N/{\mathcal{W}}^{c}\to N/{\mathcal{W}}^{c}$ is the hyperbolic homeomorphism induced by $f$ in the leaf space. Given small neighborhoods $V^{s}_{\xi}\subset W_{\varepsilon}^{s}(\xi)$ and $V^{u}_{\xi}\subset W_{\varepsilon}^{u}(\xi)$ inside the corresponding stable and unstable sets, the map (7) $(\eta,\zeta)\mapsto[\eta,\zeta]$ defines a homeomorphism between $V^{u}_{\xi}\times V^{s}_{\xi}$ and some neighborhood $V_{\xi}$ of $\xi$. A probability measure $\mu$ on $N/{\mathcal{W}}^{c}$ has _local product structure_ if for $\mu$-almost every point $\xi$ and any such product neighborhood $V_{\xi}$ the restriction $\mu\mid V_{\xi}$ is equivalent to a product $\nu^{u}\times\nu^{s}$, where $\nu^{u}$ is a measure on $V_{\xi}^{u}$ and $\nu^{s}$ is a measure on $V_{\xi}^{s}$. In the sequel we prove three additional facts about Gibbs $u$-states that are important for our arguments. ###### Proposition 3.2. Take $f\in{\mathcal{P}}_{}^{k}(N)$, $k>1$ such that the center stable foliation is absolutely continuous. For every ergodic Gibbs $u$-state $m$ the support of the projection $(\pi_{c})_{}(m)$ coincides with some attractor of $f_{c}$. In particular, periodic points are dense in the support of $(\pi_{c})_{}(m)$. Moreover, any two such projections with the same support must coincide. In particular, the set of projections of all ergodic Gibbs $u$-states of $f$ down to $N/{\mathcal{W}}^{c}$ is finite. ###### Proposition 3.3. Take $f\in{\mathcal{P}}_{}^{k}(N)$, $k>1$ such that the center stable foliation is absolutely continuous. If $m$ is a Gibbs $u$-state for $f$ then $\mu=(\pi_{c})_{}(m)$ has local product structure. ###### Remark 3.4. Suppose $f$ is volume preserving. The Lebesgue measure $\operatorname{vol}$ is both an $s$-state and a $u$-state, because the strong stable foliation and the strong unstable foliation are both absolutely continuous. Thus, Proposition 3.3 implies that $(\pi_{c})_{}(m)$ has local product structure if _either_ ${\mathcal{W}}^{cu}$ _or_ ${\mathcal{W}}^{cs}$ is absolutely continuous. ###### Proposition 3.5. Let $f\in{\mathcal{P}}_{}^{k}(N)$, $k>1$ and $\Lambda$ be an attractor of $f_{c}$. Suppose the center stable foliation of $f$ is absolutely continuous and $f$ is accessible on $\Lambda$. Then every ergodic Gibbs $u$-state of $f$ supported in ${\mathcal{W}}^{c}(\Lambda)$ has some non-positive center Lyapunov exponent. As a special case, we get that if $f\in{\mathcal{P}}_{1}^{k}(N)$, $k>1$ is accessible on an attractor $\Lambda$ of $f_{c}$ and the center stable foliation is absolutely continuous, then the (unique) center Lyapunov exponent of every ergodic Gibbs $u$-state supported in ${\mathcal{W}}^{c}(\Lambda)$ is non-positive. The proofs of these propositions are given in Sections 3.1 through 3.3. ### 3.1. Finiteness in leaf space Here we prove Proposition 3.2. Let $m_{1}$ be any ergodic Gibbs $u$-state and $\mu_{1}=(\pi_{c})_{}(m_{1})$. Notice that $\mu_{1}$ is ergodic and so its support is a transitive set for $f_{c}$. Moreover, $\operatorname{supp}\mu_{1}=\pi_{c}(\operatorname{supp}m_{1})$ consists of entire unstable sets, because the support of $m_{1}$ is ${\mathcal{W}}^{u}$-saturated (Proposition 3.1). Thus, $\operatorname{supp}\mu_{1}$ is an attractor $\Lambda$ of $f_{c}$. As pointed out before, periodic points are dense in each attractor of $f_{c}$. Now we only have to show that if $\mu_{2}=(\pi_{c})_{}m_{2}$ for another ergodic Gibbs $u$-state $m_{2}$ and $\operatorname{supp}\mu_{2}=\Lambda=\operatorname{supp}\mu_{1}$ then $\mu_{1}=\mu_{2}$. For this, take $x_{c}\in\Lambda$, let $U_{c}$ be a neighborhood of $x_{c}$ in the quotient space $N/{\mathcal{W}}^{c}$, and let $U=\pi_{c}^{-1}(U_{c})$. Then $U$ has positive $m_{i}$-measure for $i=1,2$. So, since the $m_{i}$ are ergodic Gibbs $u$-states, there are disks $D_{i}\subset U$, $i=1,2$ contained in strong unstable leaves and such that Lebesgue almost every point in $D_{i}$ is in the basin $B(m_{i})$ of $m_{i}$. Moreover, these disks may be chosen such that the center stable foliation induces a holonomy map $h^{cs}:D_{1}\to D_{2}$. Since the center stable foliation is absolutely continuous, it follows that $h^{cs}$ maps some point $x_{1}\in D_{1}\cap B(m_{1})$ to a point $x_{2}\in D_{2}\cap B(m_{2})$ in the basin of $m_{2}$. Then $x_{1}$ and $x_{2}$ belong to the same center stable leaf of $f$, and so their projections $\pi_{c}(x_{1})$ and $\pi_{c}(x_{2})$ belong to the same stable set of $f_{c}$. Notice that $\pi_{c}(B(m_{i}))\subset B(\mu_{i})$ for $i=1,2$, and so each point $\pi(x_{i})\in B(\mu_{i})$. Since either basin consists of entire stable sets, this proves that $B(\mu_{1})$ and $B(\mu_{2})$ intersect each other, and so $\mu_{1}=\mu_{2}$. This completes the proof of Proposition 3.2. ### 3.2. Local product structure Here we prove Proposition 3.3. Let $m$ be any Gibbs $u$-state and $\ell_{0}$ be any center leaf. Since the center leaves form a fiber bundle, we may find a neighborhood $V\subset N/{\mathcal{W}}^{c}$ and a homeomorphism $\phi:V\times\ell_{0}\mapsto\pi_{c}^{-1}(V),\quad(\ell,\zeta)\mapsto\phi(\theta,\zeta)$ that maps each vertical $\\{\ell\\}\times\ell_{0}$ to the corresponding center leaf $\ell$. Clearly, we may choose $V$ to be the image of the bracket (recall Section 2.2) $W^{u}_{\varepsilon}(\ell_{0})\times W^{s}_{\varepsilon}(\ell_{0})\to V,\quad(\xi,\eta)\mapsto[\xi,\eta]$ for some small $\varepsilon>0$. Then, by dynamical coherence, the homeomorphism (8) $W^{u}_{\varepsilon}(\ell_{0})\times W^{s}_{\varepsilon}(\ell_{0})\times\ell_{0}\to\pi_{c}^{-1}(V),\quad(\xi,\eta,\zeta)\mapsto\phi([\xi,\eta],\zeta)$ maps each $\\{\xi\\}\times W_{\varepsilon}^{s}(\ell_{0})\times\ell_{0}$ onto a center stable leaf and each $W_{\varepsilon}^{u}(\ell_{0})\times\\{\eta\\}\times\ell_{0}$ onto a center unstable leaf. For each $x\in\pi_{c}^{-1}(V)$, let ${\mathcal{W}}^{u}_{loc}(x)$ denote the local strong unstable leaf over $V$, that is, the connected component of ${\mathcal{W}}^{u}(x)\cap\pi_{c}^{-1}(V)$ that contains $x$. Each ${\mathcal{W}}^{u}_{loc}(x)$ is a graph over the unstable set $W^{u}(\pi_{c}(x))$ and the center stable holonomy defines a homeomorphism $h_{x,y}^{cs}:{\mathcal{W}}^{u}_{loc}(x)\to{\mathcal{W}}^{u}_{loc}(y)$ between any two local strong unstable leaves. By assumption, all these homeomorphisms are absolutely continuous. Now let $m\mid\pi_{c}^{-1}(V)=\int m_{x}\,d\hat{m}$ be the disintegration of $m$ relative to the partition of $\pi^{-1}_{c}(V)$ into local strong unstable leaves. By definition of Gibbs $u$-states, each $m_{x}$ is equivalent to the Lebesgue measure along ${\mathcal{W}}^{u}_{loc}(x)$. It follows that the center stable holonomies are absolutely continuous relative to the conditional probabilities of $m$ along local strong unstable leaves: (9) $m_{x}(E)=0\text{ if and only if }m_{y}(h_{x.y}^{cs}(E))=0$ for $x$ and $y$ in some full $m$-measure subset of $\pi_{c}^{-1}(V)$ and for any measurable set $E\subset{\mathcal{W}}^{u}_{loc}(x)$. By the construction of (8), center stable holonomies preserve the coordinate $\xi$. Thus, identifying $\pi_{c}^{-1}(V)$ with the space $W_{\varepsilon}(\ell_{0})\times W^{s}_{\varepsilon}(\ell_{0})\times\ell_{0}$ through the homeomorphism (8), property (9) becomes (10) $m_{x}(A\times W_{\varepsilon}^{s}(\ell_{0})\times\ell_{0})=0\text{ if and only if }m_{y}(A\times W^{s}_{\varepsilon}(\ell_{0})\times\ell_{0})=0$ for any measurable set $A\subset W^{u}_{\varepsilon}(\ell_{0})$ and for $m$-almost every $x$ and $y$ in $\pi_{c}^{-1}(V)$. Let $\mu\mid V=\int\mu^{u}_{\eta}\,d\mu^{s}(\eta)$ be the disintegration of $\mu$ relative to the partition of $V$ into unstable slices $W^{u}(\ell_{0})\times\\{\eta\\}$; notice that $\mu^{s}$ is just the projection of $\mu\mid V$ to $W^{s}_{\varepsilon}(\ell_{0})$. Projecting $m\mid\pi_{c}^{-1}(V)$ down to $V\approx W^{u}_{\varepsilon}(\ell_{0})\times W^{s}_{\varepsilon}(\ell_{0})$, property (10) yields (11) $\mu_{\eta}(A\times W_{\varepsilon}^{s}(\ell_{0}))=0\text{ if and only if }\mu_{\eta^{\prime}}(A\times W^{s}_{\varepsilon}(\ell_{0}))=0$ for any measurable set $A\subset W^{u}_{\varepsilon}(\ell_{0})$ and for $\mu$-almost every $\eta$ and $\eta^{\prime}$ in $V$. This means that the conditional probabilities $\mu^{u}_{\eta}$ are (almost) all equivalent. Consequently, there is $\rho:W^{u}_{\varepsilon}(\ell_{0})\times W^{s}_{\varepsilon}(\ell_{0})\to(0,\infty)$ such that $\mu^{u}_{\eta}=\rho(\cdot,\eta)\mu^{u}$ at $\mu$-almost every point, where $\mu^{u}$ denotes the projection of $\mu\mid V$ to $W^{u}_{\varepsilon}(\ell_{0})$. Replacing in the disintegration of $\mu\mid V$, we get that $\mu\mid V=\rho\,\mu^{u}\times\mu^{s}$. This proves that $\mu$ has local product structure, as claimed. ### 3.3. Positive Gibbs $u$-states Here we prove Proposition 3.5. We begin by proving the following fact, which is interesting in itself: ###### Proposition 3.6. For $f\in{\mathcal{P}}_{}^{1}(N)$, given $c>0$ and $l\geq 1$ there is $n_{0}$ such that $\\#\big{(}S\cap\Gamma_{c,l}\big{)}<n_{0}$ for every center leaf $S$, where $\Gamma_{c,l}=\\{x\in N:\liminf\frac{1}{n}\sum_{i=1}^{n}\log\\|Df^{-l}\mid{E^{c}(f^{il}(x))}\\|^{-1}\geq c\\}.$ ###### Proof. Recall that $\operatorname{vol}^{c}$ denotes the Riemannian volume on center leaves. The main ingredient is ###### Lemma 3.7. Given $c>0$ and $l\geq 1$ there exists $\delta>0$ such that for any $x\in S\cap\Gamma_{c,l}$ and any neighborhood $U$ of $x$ inside the center leaf $S$ that contains $x$, one has $\liminf\frac{1}{n}\sum_{i=0}^{n-1}\operatorname{vol}^{c}(f^{il}(U))\geq\delta.$ ###### Proof. Let $x\in S\cap\Gamma_{c,l}$ be fixed. Fix $0<c_{1}<c_{2}<c$ and define $H(c_{2})$ to be the set of $c_{2}$-hyperbolic times for $x$, that is, the set of times $m\geq 1$ such that (12) $\frac{1}{k}\sum_{i=m-k+1}^{m}\log\\|Df^{-l}\mid E^{c}_{f^{il}(x)}\\|^{-1}\geq c_{2}\quad\text{for all $1\leq k\leq m$.}$ By the Pliss Lemma (see [2, 5]) , there exist $n_{1}\geq 1$ and $\delta_{1}>0$ such that $\\#\big{(}H(c_{2})\cap[1,n)\big{)}\geq n\delta_{1}\quad\text{for all $n\geq n_{1}$.}$ Notice that (12) implies $Df^{-kl}$ is an exponential contraction on $E^{c}_{f^{ml}(x)}$: $\\|Df^{-kl}\mid E^{c}_{f^{ml}(x)}\\|\leq\prod_{i=m-k+1}^{m}\\|Df^{-l}\mid E^{c}_{f^{il}(x)}\\|\leq e^{-c_{2}k}\quad\text{for all $1\leq k\leq m$.}$ It also follows from [5] that the points $f^{ml}(x)$ with $m\in H(c_{2})$ admit backward-contracting center disks with size uniformly bounded from below: there is $r>0$ depending only on $f$ and the constants $c_{1}$ and $c_{2}$ such that $f^{-kl}(B_{r}^{c}(f^{ml}(x)))\subset B^{c}_{e^{-c_{1}k}r}(f^{(m-k)l}(x))\quad\text{for all $1\leq k\leq m$.}$ where $B^{c}_{\rho}(y)$ denotes the ball inside ${\mathcal{W}}^{c}_{y}$ of radius $\rho$ around any point $y$. Let $a_{1}>0$ be a lower bound for $m^{c}(B^{c}_{r}(y))$ over all $y\in N$. Fix $n_{2}$ such that the ball of radius $e^{-c_{1}k}r$ around $x$ is contained in $U$ for every $k\geq n_{2}$. Then, in particular, $f^{ml}(U)\supset B^{c}_{r}(f^{ml}(x))\quad\text{and so}\quad m^{c}(f^{ml}(U))\geq a_{1}$ for every $m\in H(c_{2})$ with $m\geq n_{2}$. So, for $n\gg\max\\{n_{1},n_{2}\\}$, $\frac{1}{n}\sum_{i=0}^{n-1}m^{c}(f^{il}(U))\geq\frac{1}{n}a_{1}\big{[}\\#(H(c_{2})\cap[1,n))-n_{2}\big{]}\geq\frac{1}{n}a_{1}\big{[}n\delta_{1}-n_{2}\big{]}\geq\frac{\delta_{1}}{2}a_{1}\,.$ To finish the proof of Lemma 3.7 it suffices to take $\delta=a_{1}\delta_{2}/2$. ∎ To deduce Proposition 3.6 from Lemma 3.7, take any $n_{0}\geq V/\delta$ where $V$ is an upper bound for the volume of center leaves. Suppose $S\cap\Gamma_{c,l}$ contains $n_{0}$ distinct points $x_{j}$, $j=1,\ldots,n_{0}$. Let $U_{j}$, $j=1,\ldots,n_{0}$ be pairwise disjoint neighborhoods of the $x_{j}$ inside $S$. Take $n$ large enough that $\frac{1}{n}\sum_{i=0}^{n-1}m^{c}(f^{i}(U_{j}))>\delta\quad\text{for $1\leq j\leq n_{0}$.}$ Then $V\geq\frac{1}{n}\sum_{i=0}^{n-1}m^{c}(f^{i}(S))\geq\sum_{j=1}^{n_{0}}\frac{1}{n}\sum_{i=0}^{n-1}m^{c}(f^{i}(U_{j}))>n_{0}\delta>V.$ This contradiction proves Proposition 3.6. ∎ ###### Proof of Proposition 3.5. We argue by contradiction. Suppose there exists some ergodic Gibbs $u$-state $\nu$ supported in ${\mathcal{W}}^{c}(\Lambda)$ whose center Lyapunov exponents are all positive. ###### Lemma 3.8. There is $k_{0}\geq 1$ and some ergodic Gibbs $u$-state $\nu_{}$ of $f^{k_{0}}$ supported in ${\mathcal{W}}^{c}(\Lambda)$ such that (13) $\int\log\\|Df^{-k_{0}}\mid E^{c}_{x}\\|^{-1}d\nu_{}(x)>0.$ ###### Proof. Arguing as in [48, Section 2.1] one can find $k_{0}\geq 1$ such that $\int\log\\|Df^{-k_{0}}\mid E^{c}_{x}\\|^{-1}d\nu(x)>0$ The measure $\nu$ needs not be ergodic for $f^{k_{0}}$ but, since it is ergodic for $f$, it has a finite number $k$ of ergodic components $\nu_{i}$ ($k$ divides $k_{0}$). Moreover, $\int\log\\|Df^{-k_{0}}\mid E^{c}_{x}\\|^{-1}d\nu_{i}(x)>0$ for some ergodic component $\nu_{i}$. Since, by Proposition 3.1, each ergodic component $\nu_{i}$ is a Gibbs $u$-state, this completes the proof of the lemma. ∎ Let $k_{0}\geq 1$ be fixed from now on and $\lambda>0$ denote the expression on the left hand side of (13). Let $g=f^{k_{0}}$ and $\Gamma=\\{x\in N:\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}\log\\|Dg^{-1}\mid E^{c}_{g^{j}(x)}\\|^{-1}=\lambda\\}$ be the set of regular points of $\log\\|Dg^{-1}\mid E^{c}\\|$ for the transformation $g$. By ergodicity, $\nu_{}(\Gamma)=1$. A statement similar to the next corollary was proved by Ruelle, Wilkinson [41] when the diffeomorphism is $C^{1+\varepsilon}$ and the center is $1$-dimensional. ###### Corollary 3.9. There is $n_{0}\geq 1$ such that $\\#\big{(}{\mathcal{W}}^{c}(w)\cap\Gamma\big{)}<n_{0}$ for every $w\in N$. ###### Proof. Just use Proposition 3.6 with $c=\lambda/2$ and $l=k_{0}$. Clearly, $\Gamma\subset\Gamma_{c,l}$. ∎ Let $\ell_{0}$ be any periodic center leaf intersecting $\operatorname{supp}\nu_{}$ (periodic center leaves are dense in the support, by Proposition 3.2) and $\kappa\geq 1$ be minimal such that $g^{\kappa}(\ell_{0})=\ell_{0}$. Since $\nu_{}$ is a Gibbs $u$-state and $\Gamma$ has full measure, $\operatorname{vol}^{u}({\mathcal{W}}^{u}(x)\setminus\Gamma)=0$ for $\nu_{}$-almost every $x$, where $\operatorname{vol}^{u}$ denotes the Riemannian volume along strong unstable manifolds. In particular, the stable set ${\mathcal{W}}^{s}(\ell_{0})=\cup_{z\in\ell_{0}}{\mathcal{W}}^{s}(z)$ must intersect some strong unstable disk $D^{u}$ such that $\operatorname{vol}^{u}(D^{u}\setminus\Gamma)=0$. See Figure 1. Figure 1. ###### Lemma 3.10. Every point $x\in D^{u}\cap{\mathcal{W}}^{s}(\ell_{0})$ belongs to the strong stable manifold of some periodic point $y\in\ell_{0}$ of $f$ with period bounded by $k_{0}\kappa n_{0}$. ###### Proof. Let $y\in\ell_{0}$ be such that $x\in{\mathcal{W}}^{s}(y)$ and let $g_{0}=g^{\kappa}\mid\ell_{0}$. Suppose first that the orbit of $y$ under $g_{0}$ is infinite. We refer the reader to Figure 2. Fix $y^{}\in\omega(y)$ and let $(y_{j})_{j}$ be an injective sequence of iterates of $y$ converging to $y^{}$. Let $(x_{j})_{j}$ be a sequence of iterates of $x$ with $x_{j}\in{\mathcal{W}}^{s}(y_{j})$ and $d(x_{j},y_{j})\to 0$. Choose disks $D^{u}_{j}$ around the $x_{j}$ inside the forward iterates of $D^{u}$, small but with _uniform size_. Since $\Gamma$ is an invariant set, $m^{u}(D_{j}^{u}\setminus\Gamma)=0$ for every $j$. For every large $j$, the center leaves ${\mathcal{W}}^{c}(x_{j})$ is close to $\ell_{0}$ and so one can define a $cs$-holonomy map $\pi^{cs}$ from $D^{u}_{j}$ to the local strong unstable leaf through $y^{}$. Since ${\mathcal{W}}^{cs}$ is absolutely continuous, the image of every $D_{j}^{u}\cap\Gamma$ is a full volume measure subset of a neighborhood of $y^{}$ inside ${\mathcal{W}}^{u}(y^{})$, where these neighborhoods also have uniform size for all large $j$. Let $J=\\{j_{0},j_{0}+1,\ldots,j_{0}+n_{0}\\}$ where $j_{0}$ is some large integer and $n_{0}$ is as in Corollary 3.9. On the one hand, it follows from the previous considerations that $\Gamma^{}=\bigcap_{j\in J}\pi^{cs}(D_{j}^{u}\cap\Gamma)$ is a full volume measure subset of some neighborhood of $y^{}$ inside ${\mathcal{W}}^{u}(y^{})$. Fix some $w\in\Gamma^{}$ close to $y^{}$. For each $j\in J$, let $w_{j}\in D_{j}^{u}\cap\Gamma$ be such that $\pi^{cs}(w_{j})=w$. Moreover, let $z_{j}$ be the point where the local strong stable manifold of $w_{j}$ intersects ${\mathcal{W}}^{cu}(y^{})={\mathcal{W}}^{cu}(w)$. It is clear from the definition that $w_{j}\in{\mathcal{W}}^{cs}(w)$ and so $z_{j}\in{\mathcal{W}}^{c}(w)$ for all $j\in J$. Moreover, by choosing $w$ close enough to $y^{}$ we can ensure that $w_{j}$ is close to $x_{j}$ for every $j\in J$ and so $z_{j}$ is close to $y_{j}$ for all $j\in J$. The latter implies that the $z_{j}$ are all distinct. Observe also that $z_{j}\in\Gamma$ for all $j\in J$, because $\Gamma$ is (clearly) saturated by strong stable leaves. This proves that $\\#({\mathcal{W}}^{c}(w)\cap\Gamma)\geq\\#J>n_{0}$, in contradiction with Corollary 3.9. This contradiction proves that the $g_{0}$-orbit of $y$ can not be infinite. Figure 2. Similar arguments handle the case when $y$ is a periodic point for $g_{0}$. Let $k\geq 1$ be the (minimal) period of $y$ for $g_{0}$. Forward iterates of $D^{u}$ accumulate on the strong unstable manifolds of the iterates of $y$. Using, in much the same way as before, that the center stable foliation is absolutely continuous and $\Gamma$ is saturated by strong stable leaves, we find $w\in{\mathcal{W}}^{cu}(y)$ arbitrarily close to $y$ whose center leaf ${\mathcal{W}}^{c}_{w}$ intersects $\Gamma$ at points close to each of the $k$ iterates of $y$. In view of Corollary 3.9 this implies that $k<n_{0}$. This means that the period of $y$ for $f$ is less than $k_{0}\kappa n_{0}$ as stated. The proof of Lemma 3.10 is complete. ∎ ###### Lemma 3.11. Every point $z\in\ell_{0}$ is periodic for $f$, with period bounded by $k_{0}\kappa n_{0}$. ###### Proof. Let $y\in\ell_{0}$ be a periodic point as in Lemma 3.10 and let $z\in\ell_{0}$ be arbitrary. Choose $y^{\prime}\in{\mathcal{W}}^{u}(y)\cap{\mathcal{W}}^{c}(\Lambda_{i})\setminus\ell_{0}$ and $z^{\prime}\in{\mathcal{W}}^{s}(z)\cap{\mathcal{W}}^{c}(\Lambda_{i})\setminus\ell_{0}$. By accessibility, there exists some $su$-path connecting $y^{\prime}$ to $z^{\prime}$ or, in other words, there exist points $b_{0}=y,a_{1}=y^{\prime},b_{1},\ldots,a_{i},b_{i},\ldots,a_{s}=z^{\prime},b_{s}=z$ which belong to ${\mathcal{W}}^{c}(\Lambda_{i})$ such that $a_{j}$ and $b_{j}$ belong to the same strong stable manifold and $b_{j}$ and $a_{j+1}$ belong to the same strong unstable manifold. We are going to find an (arbitrarily) nearby $su$-path (14) $\tilde{b}_{0}=y,\tilde{a}_{1},\tilde{b}_{1},\ldots,a_{i},b_{i},\ldots,\tilde{a}_{s},\tilde{b}_{s}$ with $\tilde{b}_{s}\in\ell_{0}$ and such that every $\tilde{b}_{i}$ belongs to some periodic center leaf in ${\mathcal{W}}^{c}(\Lambda_{i})$. The first step is to observe that, since periodic leaves are dense, one may always find periodic leaves $\ell_{1},\ldots,\ell_{s-1}$ arbitrarily close to ${\mathcal{W}}^{c}({b_{1}}),\ldots,{\mathcal{W}}^{c}({b_{s-1}})$, respectively. Let $\ell_{s}=\ell_{0}$. Assume $\tilde{b}_{0},\tilde{a}_{1},\ldots,\tilde{b}_{k}$ have been defined, for some $0\leq k<s$. Since ${\mathcal{W}}^{u}(b_{k})$ intersects the stable set of ${\mathcal{W}}^{c}(b_{k+1})$ transversely at $a_{k+1}$, and stable and unstable sets vary continuously with the base point, we can find $\tilde{b}_{k+1}\in\ell_{k+1}$ close to $b_{k+1}$ such that ${\mathcal{W}}^{u}(\tilde{b}_{k})$ intersects ${\mathcal{W}}^{s}(\tilde{b}_{k+1})$ at some point $\tilde{a}_{k+1}$ close to $a_{k}$. Repeating this procedure $s$ times, we obtain an $su$-path as in (14). The next step is to prove that the points $\tilde{b}_{i}$ themselves are periodic. Recall that $\tilde{b}_{0}=y$ is taken to be periodic and $D^{u}$ intersects ${\mathcal{W}}^{s}(\tilde{b}_{0})$. So, the iterates accumulate on ${\mathcal{W}}^{u}(\tilde{b}_{0})$ and, in particular, on $\tilde{a}_{1}$. This implies there exist points $w\in\ell_{1}$ arbitrarily close to $\tilde{b}_{1}$ whose strong stable manifold intersects $f^{n}(D^{u})$ for some $n$. Since $\Gamma$ has full volume inside every $f^{n}(D^{u})$, we may use Lemma 3.10 to conclude that $w$ is periodic, with period uniformly bounded. Consequently, $\tilde{b}_{1}$ itself is periodic. It also follows that the iterates of $D^{u}$ accumulate on ${\mathcal{W}}^{u}(\tilde{b}_{1})$. This means we may now repeat the construction with $\tilde{b}_{1}$ in the place of $\tilde{b}_{0}$ and conclude that $\tilde{b}_{2}$ is periodic. After $s$ steps we conclude that $\tilde{z}=\tilde{b}_{s}$ is periodic. Since $\tilde{z}$ is arbitrarily close to $z$, and all the periods are bounded, we get that $z$ itself is periodic. This completes the proof of the lemma. ∎ In particular, Lemma 3.11 implies that no periodic point on the support of $\nu_{}$ is hyperbolic. This is a contradiction since, by a classical result of Katok [29], the support of any hyperbolic measure contains hyperbolic periodic points. This completes the proof of Proposition 3.5. ∎ ## 4\. Mostly contracting center In this section we prove some useful facts about partially hyperbolic diffeomorphisms with mostly contracting center direction. We call _${\mathcal{W}}^{u}$ -disk_ any image of a ball in $E^{u}$ embedded inside some strong unstable leaf. ###### Lemma 4.1. The center direction of $f$ is mostly contracting if and only if the center Lyapunov exponents of all ergodic Gibbs $u$-states are negative. If $f\in{\mathcal{P}}_{1}^{k}(N)$, $k>1$ and $\Lambda$ is an attractor of $f_{c}$, then the center direction of $f\mid{{\mathcal{W}}^{c}(\Lambda)}$ is mostly contracting if and only if the center Lyapunov exponent is negative for every ergodic Gibbs $u$-state supported in ${\mathcal{W}}^{c}(\Lambda)$. ###### Proof. Bonatti, Viana [19] show that if the center direction is mostly contracting then the center exponents of every ergodic Gibbs $u$-state are negative. To prove the converse, let $D$ be any disk inside a strong unstable leaf. By [17, Lemma 11.12] every Cesaro accumulation point of the iterates of Lebesgue measure on $D$ is a Gibbs $u$-state. By [17, Lemma 11.13] every ergodic component of a Gibbs $u$-state is again a Gibbs $u$-state. This implies that the iterates $f^{n}(D)$ accumulate on the support of some ergodic Gibbs $u$-state $\nu$. The hypothesis implies that $\nu$-almost every point has a Pesin (local) stable manifold which is an embedded disk of dimension $d_{cs}$. Using also the absolute continuity of the Pesin stable foliation (Pesin [35]), we conclude that a positive Lebesgue measure subset of points in some $f^{n}(D)$ belong to the union of these $d_{s}$-disks. This implies that (5) holds on a positive Lebesgue measure subset of $D$, as we wanted to show. The second part of the lemma follows from similar arguments. ∎ ### 4.1. Supports of Gibbs $u$-states ###### Lemma 4.2. If the center direction of $f$ is mostly contracting then the supports of the ergodic Gibbs $u$-states of $f$ are pairwise disjoint. ###### Proof. Let $m_{1}$ and $m_{2}$ be ergodic Gibbs $u$-states of $f$ and suppose $\operatorname{supp}m_{1}\cap\operatorname{supp}m_{2}$ contains some point $z$. Let $D$ be any ${\mathcal{W}}^{u}$-disk around $z$. Then $D\subset\operatorname{supp}m_{1}\cap\operatorname{supp}m_{2}$, since the supports are ${\mathcal{W}}^{u}$-saturated (Proposition 3.1). By Lemmas 11.12 and 11.13 in [17], every ergodic component $\nu$ of every Cesaro accumulation point of the iterates of Lebesgue measure on $D$ is an ergodic Gibbs $u$-state. Clearly, the support of $\nu$ is contained in $\operatorname{supp}m_{1}\cap\operatorname{supp}m_{2}$. By Pesin theory (see [19] for this particular setting) $\nu$-almost every point has a local stable manifold which is an embedded $d_{cs}$-disk. Recall (Proposition 3.1) that the density of Gibbs $u$-states along strong unstable leaves is positive and finite. Thus, we may find a ${\mathcal{W}}^{u}$-disk $D_{\nu}\subset\operatorname{supp}\nu$ such that every point $x$ in a full Lebesgue measure subset $D_{\nu}^{}$ has a Pesin stable manifold and belongs to the basin of $\nu$. Moreover, $D_{\nu}$ is accumulated by ${\mathcal{W}}^{u}$-disks $D_{i}\subset\operatorname{supp}m_{1}$ such that Lebesgue almost every point is in the basin of $m_{1}$. Assuming $D_{i}$ is close enough to $D_{\nu}$, it must intersect the union of the local stable manifolds through the points of $D_{\nu}^{}$ on some positive Lebesgue measure subset $D_{i}^{}$ (because the Pesin local stable lamination is absolutely continuous [35]). Then $D_{i}^{}$ is contained in the basin of $\nu$, and some full Lebesgue measure subset is contained in the basin of $m_{1}$. That implies $m_{1}=\nu$. Analogously, $m_{2}=\nu$, and so the ergodic Gibbs $u$-states $m_{1}$ and $m_{2}$ coincide. That completes the proof of the lemma. ∎ ###### Remark 4.3. It follows from Proposition 3.1 and Lemma 4.2 that if $f$ has mostly contracting center direction and minimal strong unstable foliation then it has a unique Gibbs $u$-state. This was first observed in [19]. ###### Proposition 4.4. Suppose the center direction of $f$ is mostly contracting, and let $m$ be an ergodic Gibbs $u$-state of $f$. Then the support of $m$ has a finite number of connected components. Moreover, each connected component $S$ is ${\mathcal{W}}^{u}$-saturated and ${\mathcal{W}}^{u}(x)$ is dense in $S$ for any $x\in S$. ###### Proof. Let $p$ be any periodic point in the support of $m$ with stable index equal to $d_{cs}$ (such periodic points do exist, by Katok [29]) and let $\kappa$ be its period. By Proposition 3.1, the unstable manifold of every $f^{j}(p)$ is contained in $\operatorname{supp}m$. We claim that $\cup_{j=1}^{\kappa}{\mathcal{W}}^{u}(f^{j}(p))$ is dense in $\operatorname{supp}m$. To see this, let $D$ be any disk inside ${\mathcal{W}}^{u}(p)$. Consider the forward iterates of Lebesgue measure on $D$. Using Lemmas 11.12 and Lemma 11.13 in [17], one gets that any ergodic component of any Cesaro accumulation point of these iterates is an ergodic Gibbs $u$-state $\nu$ supported inside the closure of $\cup_{j=1}^{\kappa}{\mathcal{W}}^{u}(f^{j}(p))$. By Lemma 4.2, the Gibbs $u$-states $m$ and $\nu$ must coincide. In particular, $\operatorname{supp}m$ is contained in the closure of $\cup_{j=1}^{\kappa}{\mathcal{W}}^{u}(f^{j}(p))$. That proves our claim. Since $m$ is ergodic for $f$, its ergodic decomposition relative to $f^{\kappa}$ has the form $m=l^{-1}\sum_{i=1}^{l}f^{i}_{}\tilde{m}$ where $l$ divides $\kappa$ and $\tilde{m}$ is $f^{\kappa}$-invariant and ergodic. Then $\operatorname{supp}m=\bigcup_{i=1}^{l}f^{i}(\operatorname{supp}\tilde{m}).$ We claim that the $f^{i}(\operatorname{supp}\tilde{m})$, $i=1,\dots,l$ are precisely the connected components of $\operatorname{supp}m$. On the one hand, the previous paragraph gives that $p\in f^{s}(\operatorname{supp}\tilde{m})$ for some $s$. Replacing either $p$ or $\tilde{m}$ by an iterate, we may suppose $s=0$. Then, by the argument in the previous paragraph applied to $f^{\kappa}$ (it is clear from the definition (5) that if $f$ has mostly contracting then so does any positive iterate), $\operatorname{supp}\tilde{m}$ coincides with the closure of ${\mathcal{W}}^{u}(p)$ and, in particular, it is connected. On the other hand, Lemma 4.2 gives that the $f^{i}(\operatorname{supp}\tilde{m})$, $i=1,\dots,l$ are pairwise disjoint. Since they are closed, it follows that they are also open in $\operatorname{supp}m$. This proves our claim. We are left to prove that the strong unstable foliation is minimal in each connected component $S_{i}=f^{i}(\operatorname{supp}\tilde{m})$. This will follow from an argument of Bonatti, Díaz, Ures [16]: ###### Lemma 4.5. There is a neighborhood $U^{s}_{i}$ of $f^{i}(p)$ inside $W^{s}(f^{i}(p))$ such that every unstable leaf in $S_{i}$ has some transverse intersection with $U^{s}_{i}$. ###### Proof. For any $x\in S_{i}$, let $D_{x}$ be a small ${\mathcal{W}}^{u}$-disk around $x$. Since $\tilde{m}_{j}$ is the unique ergodic $u$-state of $f^{\kappa}$ with support contained in $S_{j}$. It is also the unique Cesaro accumulation point of the iterates of $\operatorname{vol}_{D_{x}}$ under $f^{\kappa}$. In particular, there is $n_{x}\geq 1$ such that $f^{n_{x}\kappa}(D_{x})$ intersects the local stable manifold of $f^{i}(p)$ transversely. This implies that $D_{x}$ intersects the global stable manifold of $f^{i}(p)$ transversely. Then, by continuity of the strong unstable foliation, there is a neighborhood $V_{x}$ of $x$ and a bounded open set $U_{x}\subset W_{f^{\kappa}}^{s}(f^{i}(p))$ such that ${\mathcal{W}}^{u}(y)$ intersects $U_{x}$ transversely for every $y\in V_{x}$. The family $\\{V_{x}:x\in S_{i}\\}$ is an open cover of the compact set $S_{i}$. Let $\\{V_{x_{1}},\cdots,V_{x_{m}}\\}$ be a finite subcover. Choose $U^{s}_{j}$ a bounded neighborhood of $f^{i}(p)$ inside $W_{f^{\kappa}}^{s}(f^{i}(p))$ containing $U_{x_{j}}$ for all $j=1,\dots,m$. It follows from the construction that every strong unstable leaf contained in $S_{i}$ intersects $U_{i}^{s}$ transversely. This finishes the proof of the lemma. ∎ Let us go back to proving Proposition 4.4. The lemma gives that ${\mathcal{W}}^{u}(f^{-n\kappa}(x))$ intersects $U_{i}^{s}$ transversely, and so ${\mathcal{W}}^{u}(x)$ intersects $f^{n\kappa}(U_{i}^{s})$ transversely, for every $x\in S_{i}$ and every $n\geq 0$. Since $f^{n\kappa}(U_{i}^{s})$ converges to $f^{i}(p)$ when $n\to\infty$, it follows that $W^{u}(f^{i}(p))$ is contained in the closure of $W^{u}(x)$. Hence, $W^{u}(x)$ is dense in $S_{j}$, as claimed. The proof of the proposition is complete. ∎ ### 4.2. Bernoulli property An invariant ergodic measure $\eta$ of a transformation $g$ is called _Bernoulli_ if $(g,\eta)$ is ergodically conjugate to a Bernoulli shift. ###### Theorem 4.6. Suppose $f$ is a $C^{k}$, $k>1$ partially hyperbolic diffeomorphism with mostly contracting center direction. Then there is $l\geq 1$ and a $C^{k}$ neighborhood $\mathcal{U}$ of $f$ such that for any $g\in\mathcal{U}$, every ergodic $u$-state of $g^{l}$ is Bernoulli. ###### Proof. Let $m_{1},\dots,m_{u}$ be the ergodic Gibbs $u$-states of $f$. Proposition 4.4 gives that for each $j=1,\dots,u$ there exists $l_{j}\geq 1$ such that the support of $m_{j}$ has $l_{j}$ connected components $S_{j,i}$, $i=1,\dots,l_{j}$. Moreover, each connected component $S_{j,i}$ carries an ergodic component $m_{j,i}=f^{i}_{}\tilde{m}_{j}$ of the Gibbs $u$-state $m_{j}$ for the iterate $f^{l_{j}}$. Let $l$ be any common multiple of $l_{1},\dots,l_{u}$. Then every $S_{j,i}$ is fixed under $f^{l}$. Moreover, every Gibbs $u$-state $m_{j,i}$ is $f^{l}$-invariant and $f^{nl}$-ergodic for every $n\geq 1$: otherwise $S_{i}$ would break into more than one connected component (cf. the proof of Lemma 4.2). Then, by Ornstein, Weiss [33], every $m_{i,j}$ is a Bernoulli measure for $f^{l}$. We claim that $\\{m_{j,i}:1\leq j\leq u\text{ and }1\leq i\leq l_{j}\\}$ contains all the ergodic $u$-states of $f^{nl}$ for every $n\geq 1$. Indeed, let $m_{}$ be any ergodic $u$-state for $f^{nl}$. Then $m=\frac{1}{nl}\sum_{k=1}^{nl}f_{}^{k}m_{}$ is a $u$-state for $f$. Let $m=a_{1}m_{1}+\cdots+a_{u}m_{u}$ be its ergodic decomposition for $f$ and let $s$ be such that $a_{s}>0$. Then $\operatorname{supp}m_{s}\subset\operatorname{supp}m$. Since $\operatorname{supp}m_{s}$ is $f$-invariant, it must intersect $\operatorname{supp}m_{}$. Using Lemma 4.2 for $f^{nl}$ we conclude that $m_{}$ must coincide with some ergodic component of $m_{s}$ for the iterate $f^{nl}$. In other words, it must coincide with $m_{s,i}$ for some $i=1,\dots,l_{s}$, and this proves our claim. Now we extend these conclusions to any diffeomorphism $g$ in a $C^{k}$, $k>1$ neighborhood of $f$. By Andersson [9], any such $g$ has mostly contracting center direction, and so the previous argument applies to it. However, we must also prove that the integer $l$ can be taken uniform on a whole neighborhood of $f$. Notice that the only constraint on $l$ was that it should be a multiple of the periods $l_{j}$ of the ergodic components $m_{j}$. Observe that [9] also gives that the number of ergodic Gibbs $u$-states does not exceed the number of ergodic Gibbs $u$-states of $f$. So, we only need to check that the periods $l_{j}$ remain uniformly bounded for any $g$ in a neighborhood. We do this by arguing with periodic points, as follows. Let us fix, once and for all, $f$-periodic points $p_{j}$ with stable index $d_{cs}$ in the support of each $m_{j}$, $j=1,\dots,u$. The period of each $p_{j}$ is a (fixed) multiple of $l_{j}$. Let $p_{j}(g)$ be the continuation of these periodic points for some nearby diffeomorphism $g$, and let $\\{m_{1}(g),\dots,m_{s}(g)\\}$, with $s\leq u$ be the ergodic Gibbs $u$-states of $g$. We claim that every $\operatorname{supp}m_{j}(g)$, $1\leq j\leq s$ contains some $p_{i}(g)$, $1\leq i\leq u$. This can be seen as follows. As observed before, any accumulation point of Gibbs $u$-states of $g$ when $g\to f$ is a Gibbs $u$-state for $f$. We fix some small $\varepsilon>0$ and consider the $\varepsilon$-neighborhoods $B(p_{j},\varepsilon)$ of the periodic points $p_{j}$. Then, for any $g$ close enough to $f$ every ergodic Gibbs $u$-state $m_{j}(g)$ must give positive weight to some $B(p_{i},\varepsilon)$ and, consequently, also to $B(p_{i}(g),2\varepsilon)$. By continuous dependence of stable manifolds of periodic points on the dynamics, and the fact that the supports of Gibbs $u$-states are $u$-saturated, it follows that $\operatorname{supp}m_{j}(g)$ contains some ${\mathcal{W}}^{u}$-disk that intersects $W^{s}(p_{i}(g))$ transversely. Then, the support of $m_{j}(g)$ must contain $p_{i}(g)$. This proves our claim. It follows that the period $l_{j}(g)$ of each ergodic Gibbs $u$-state of $g$ divides the period of some $p_{i}(g)$ which, of course, coincides with the period of $p_{i}$. Since the latter have been fixed once and for all, this proves that the $l_{j}(g)$ are indeed uniformly bounded on a neighborhood of $f$. The proof of the theorem is complete. ∎ ### 4.3. Abundance of mostly contracting center We also give a family of new examples of diffeomorphisms with mostly contracting center. ###### Theorem 4.7. Suppose $\dim M=3$. The set of ergodic diffeomorphisms such that either $f$ or $f^{-1}$ has mostly contracting center direction is $C^{1}$ open and dense in the space of $C^{k}$, $k>1$ partially hyperbolic volume preserving diffeomorphisms with 1-dimensional center and some fixed compact center leaf. ###### Proof. Denote by ${\mathcal{V}}_{m}^{k}$ the set of $C^{k}$ volume preserving partially hyperbolic diffeomorphisms with 1-dimensional center and some fixed compact center leaf. This is a $C^{1}$ open set, cf. [28, Theorem 4.1]. Moreover, the diffeomorphisms such that both the strong stable foliation and the strong unstable foliation is minimal fill an open and dense subset $U_{1}$ of ${\mathcal{V}}_{m}^{1}$. This follows from a conservative version of the results of [16]: one only has to observe that blenders, that they use for the proof in the dissipative context, can be constructed also in the conservative setting, as shown by [26]. By [15], there is an open and dense subset $U_{2}$ for which the center Lyapunov exponent $\int\log\|Df\mid{E^{c}(x)}\|dm(x)\neq 0.$ Furthermore, by [24], there is an open and dense subset $U_{3}$ of ${\mathcal{V}}^{1}_{m}$ consisting of accessible diffeomorphisms. Let $U=U_{1}\cap U_{2}\cap U_{3}$. Before proceeding, let us recall that $C^{\infty}$ are $C^{1}$ dense in the space of volume preserving diffeomorphisms, by [11]. In particular, the $C^{1}$ open and dense subset $U$ has non-trivial intersection with the space of $C^{k}$ diffeomorphisms, for any $k>1$. We claim that for every $C^{k}$, $k>1$ diffeomorphism $f$ in $U$, either $f$ or its inverse has a unique ergodic Gibbs $u$-state and the corresponding center Lyapunov exponent is negative. In particular, by Lemma 4.1, either $f$ or its inverse has mostly contracting center direction. The first step is to note that $f$ is ergodic, since it is accessible (see [22, 27, 38]). Then the Lebesgue measure $\operatorname{vol}$ is an ergodic Gibbs $u$-state for both $f$ and $f^{-1}$. Since the strong stable and strong unstable foliations are minimal, the Gibbs $u$-state is unique; see Remark 4.3. This completes the proof of Theorem 4.7. ∎ ## 5\. Finiteness and stability of physical measures In this section we prove Theorem D. As remarked before, Theorem A is a particular case. We begin by recalling certain ideas from Bonatti, Gomez-Mont, Viana [18] and Avila, Viana [13] that we use for handling the case when the center Lyapunov exponent vanishes. ### 5.1. Smooth cocycles By assumption, the center leaves of $f$ define a fiber bundle $\pi_{c}:N\to N/{\mathcal{W}}^{c}$ over the leaf space. Then $f$ may be seen as a smooth cocycle (as defined in [13]) over $f_{c}$: $\begin{array}[]{rrcl}f:&N&\rightarrow&N\\\ &\downarrow&&\downarrow\\\ f_{c}:&N/{\mathcal{W}}^{c}&\rightarrow&N/{\mathcal{W}}^{c}\end{array}$ It follows from the form of our maps that the strong stable manifold ${\mathcal{W}}^{s}(x)$ of every point $x\in M$ is a graph over the stable set $W^{s}_{\pi_{c}(x)}$ of $\pi_{c}(x)\in N/{\mathcal{W}}^{c}$. For each $\eta\in W^{s}(\xi)$, the strong stable holonomy defines a homeomorphism $h^{s}_{\xi,\eta}:\xi\to\eta$ between the two center leaves. In fact (see [13, Proposition 4.1]), (15) $h^{s}_{\xi,\eta}(\theta)=\lim_{n\to\infty}(f^{n}\mid\eta)^{-1}\circ(f^{n}\mid\xi)(\theta),$ (for large $n$ one can identify $f^{n}(\xi)\approx f^{n}(\eta)$ via the fiber bundle structure) for each $\theta\in\xi$ and the limit is uniform on the set of all $(\xi,\eta,\theta)$ with $\theta\in\xi$ and $\xi$ and $\eta$ in the same local stable set. These _$s$ -holonomy_ maps satisfy • $h^{s}_{\eta,\zeta}\circ h^{s}_{\xi,\eta}=h^{s}_{\xi,\zeta}$ and $h^{s}_{\xi,\xi}=\operatorname{id}$ * • $f\circ h^{s}_{\xi,\eta}=h^{s}_{f_{c}(\xi),f_{c}(\eta)}\circ f$ * • $(\xi,\eta,\theta)\mapsto h^{s}_{\xi,\eta}(\theta)$ is continuous on the set of triples $(\xi,\eta,\theta)$ with $\xi$ and $\eta$ in the same local stable set and $\theta\in{\mathcal{W}}^{c}(\xi)$. Let $m$ be any $f$-invariant probability measure and $\mu=(\pi_{c})_{}(m)$. A _disintegration_ of $m$ into conditional probabilities along the center leaves is a measurable family $\\{m_{\xi}:\xi\in\operatorname{supp}\mu\\}$ of probability measures with $m_{\xi}(\xi)=1$ for $\mu$-almost every $\xi$ and (16) $m(E)=\int m_{\xi}(E)\,d\mu(\xi)$ for every measurable set $E\subset M$. By Rokhlin [40], such a family exists and is essentially unique. A disintegration is called _$s$ -invariant_ if $(h^{s}_{\xi,\eta})_{}m_{\xi}=m_{\eta}\quad\text{for every $\xi$, $\eta\in\operatorname{supp}\mu$ in the same stable set.}$ In a dual way one defines _$u$ -holonomy_ maps and _$u$ -invariance_. We call a disintegration _bi-invariant_ if it is both $s$-invariant and $u$-invariant, and we call it _continuous_ if $m_{\xi}$ varies continuously with $\xi$ on the support of $\mu$, relative to the weak∗ topology. ###### Proposition 5.1. Let $f\in{\mathcal{P}}_{}^{k}(N)$, $k>1$ be such that the center stable foliation is absolutely continuous. Let $m$ be an ergodic Gibbs $u$-state with vanishing center Lyapunov exponents. Then $m$ admits a disintegration $\\{m_{\xi}:\xi\in\operatorname{supp}\mu\\}$ into conditional probabilities along the center leaves which is continuous and bi-invariant. ###### Proof. Proposition 3.3 gives that $(\pi_{c})_{}m$ has local product structure. Thus, we are in a position to use Theorem D of Avila, Viana [13] to obtain the conclusion of the present proposition. ∎ ### 5.2. Zero Lyapunov exponent case The following result provides a characterization of the systems exhibiting ergodic Gibbs $u$-states with vanishing central exponent. ###### Proposition 5.2. Let $f\in{\mathcal{P}}_{1}^{k}(N)$, $k>1$ be such that the center stable foliation is absolutely continuous. Let $\Lambda$ be an attractor of $f_{c}$ such that $f$ is accessible on $\Lambda$, and let $m$ be an ergodic Gibbs $u$-state with vanishing center Lyapunov exponent. Then * (1) the conditional probabilities $\\{m_{x}:x\in\Lambda\\}$ along the center leaves are equivalent to the Lebesgue measure on the leaves, with densities uniformly bounded from zero and infinity; * (2) $\operatorname{supp}m=W^{c}(\Lambda)$ and this is the unique Gibbs $u$-state supported in ${\mathcal{W}}^{c}(\Lambda)$; * (3) the basin of $m$ covers a full Lebesgue measure subset of a neighborhood of ${\mathcal{W}}^{c}(\Lambda)$. ###### Proof. By Proposition 5.1, there is a disintegration $\\{m_{x}:x\in\Lambda\\}$ of $m$ along the center foliation which is continuous, $s$-invariant, and $u$-invariant. Let $\xi$ and $\eta$ be any two points in ${\mathcal{W}}^{c}(\Lambda)$. By accessibility on $\Lambda$, one can find an $su$-path $b_{0}=\xi,b_{1},\ldots,b_{s-1},b_{s}=\eta$ connecting $\xi$ to $\eta$. This $su$-path induces a holonomy map $h:{\mathcal{W}}^{c}(\xi)\to{\mathcal{W}}^{c}(\eta)$, defined as the composition of all strong stable/unstable holonomy maps $h_{i}:{\mathcal{W}}^{c}(b_{i-1})\to{\mathcal{W}}^{c}(b_{i})$. The fact that the disintegration is bi-invariant gives, in particular, that (17) $m_{\eta}(h(B^{c}_{\varepsilon}(\xi))=m_{\xi}(B^{c}_{\varepsilon}(\xi))$ It is a classical fact that the strong stable and strong unstable foliations are absolutely continuous in a strong sense: their holonomy maps have bounded Jacobians. See [20, 31, 1]. Those arguments extend directly to their restrictions to each center stable or center unstable leaf, respectively: the restricted strong stable and strong unstable foliations are also absolutely continuous with bounded Jacobians. By compactness, the $su$-path may be chosen such that the number $s$ of legs and the length of each leg are uniformly bounded, independent of $\xi$ and $\eta$ 111This may be deduced from [12] as follows. By Proposition 8.3 in [12], given any $x_{0}\in M$ there exists $w\in M$ such that $x_{0}$ is connected to every point in a neighborhood of $w$ by a uniformly bounded $su$-path. Then the same is true if one replaces $w$ by an arbitrary point $z\in M$: connect $w$ to $z$ by some $su$-path; the ”same” $su$-path determines a bijection between neighborhoods of $w$ and $z$; concatenating with $su$-paths from $x_{0}$ to the neighborhood of $w$ one obtains uniformly bounded $su$-paths from $x_{0}$ to any point near $z$. The claim now follows by compactness of the ambient manifold.. Then, we may fix a uniform upper bound constant $K>1$ on the Jacobians of all associated strong stable and strong unstable holonomies. Notice $\operatorname{vol}^{c}(B_{r}(\zeta))=2r$, since the center leaves are one- dimensional. Then (18) $K^{-1}\operatorname{vol}^{c}(B^{c}_{\varepsilon}(\xi))\leq\operatorname{vol}^{c}(h(B^{c}_{\varepsilon}(\xi)))\leq K\operatorname{vol}^{c}(B^{c}_{\varepsilon}(\xi)).$ From (17) and (18) we obtain $\frac{1}{K}\frac{m_{\xi}(B^{c}_{\varepsilon}(\xi))}{\operatorname{vol}^{c}(B^{c}_{\varepsilon}(\xi))}\leq\frac{m_{\eta}(h(B^{c}_{\varepsilon}(\xi)))}{\operatorname{vol}^{c}(h(B^{c}_{\varepsilon}(\xi)))}\leq K\frac{m_{\xi}(B^{c}_{\varepsilon}(\xi))}{\operatorname{vol}^{c}(B^{c}_{\varepsilon}(\xi))}.$ and, taking the limit as $\varepsilon\to 0$, $\frac{1}{K}\frac{dm_{\xi}}{d\operatorname{vol}^{c}}(\xi)\leq\frac{dm_{\eta}}{d\operatorname{vol}^{c}}(\eta)\leq K\frac{dm_{\xi}}{d\operatorname{vol}^{c}}(\xi).$ Since we can always find $\eta$ where the density is less or equal than $1$ (respectively, greater or equal than $1$), this implies that (19) $\frac{dm_{\xi}}{d\operatorname{vol}^{c}}(\xi)\in\big{[}K^{-1},K]$ for every $\xi$, and that proves claim (1). Now let $m^{\prime}$ be any other ergodic Gibbs $u$-state supported in ${\mathcal{W}}^{c}(\Lambda)$. The center Lyapunov exponent of $m^{\prime}$ must vanish: otherwise, by [29], there would be some hyperbolic periodic point in ${\mathcal{W}}^{c}(\Lambda)$, and that is incompatible with the conclusion in part (1) that there exist invariant conditional probabilities equivalent to Lebesgue measure along the center leaves. So, all the previous considerations apply to $m^{\prime}$ as well. In particular, it has a continuous disintegration $\\{m^{\prime}_{x}:x\in\Lambda\\}$ along the center foliation such that each $m^{\prime}_{x}$ is equivalent with $\operatorname{vol}^{c}$. Moreover, by Proposition 3.2, $(\pi_{c})_{}(m)=(\pi_{c})_{}(m^{\prime})$. Then, $\operatorname{vol}^{c}$-almost every point in almost every center leaf, relative to $(\pi_{c})_{}(m)=(\pi_{c})_{}(m^{\prime})$, belongs to the basin of both $m$ and $m^{\prime}$. In particular, the two basins intersect, and that implies $m=m^{\prime}$. That completes the proof of claim (2). From conclusion (1) we get that there exists a full $m$-measure set $\Delta$ consisting of leaves such that $\operatorname{vol}^{c}$-almost every point in the leaf belongs to the basin of $m$. Then, since $m$ is a Gibbs $u$-state, we can find a ${\mathcal{W}}^{u}$-disk $D^{u}_{0}$ such that $\Delta\cap D^{u}_{0}$ has full measure in $D^{u}_{0}$. Consider the $cu$-disk $D_{0}^{cu}=\bigcup_{\xi\in D_{0}^{u}}{\mathcal{W}}^{c}(\xi)$ Observe that the center foliation ${\mathcal{W}}^{c}$ is absolutely continuous on each center unstable leaf, because the corresponding holonomy maps between unstable leaves coincide with the corresponding holonomy maps for the center stable foliation, which we assume to be absolutely continuous. Using this fact, and a Fubini argument, we get that Lebesgue almost every point in $D_{0}^{cu}$ belongs to the basin of $m$. Next, consider the open set $D_{0}=\bigcup_{\zeta\in D_{0}^{cu}}{\mathcal{W}}^{s}_{loc}(\zeta).$ Since the strong stable foliation is absolutely continuous and the basin is ${\mathcal{W}}^{s}$-saturated, it follows that Lebesgue almost every point in $D_{0}$ belongs to the basin of $m$. It is clear that we can cover ${\mathcal{W}}^{c}(\Lambda)$ by finitely many such open sets $D_{0}$. This proves (3), and so the proof of the lemma is complete. ∎ ### 5.3. Construction of physical measures We are nearly done with the proof of Theorem D. By Proposition 3.5, all ergodic Gibbs $u$-states have non-negative center Lyapunov exponent. The case when the exponent vanishes for some Gibbs $u$-state is handled by Proposition 5.2: we get alternative (a) of the theorem in this case. Finally, if the center Lyapunov exponent is negative for all Gibbs $u$-states over some attractor $\Lambda_{i}$ of $f_{c}$ then, by Lemma 4.1, the center direction of $f$ is mostly contracting on that attractor $\Lambda_{i}$. Then, by Bonatti, Viana [19], there are finitely many ergodic Gibbs $u$-states supported in ${\mathcal{W}}^{c}(\Lambda_{i})$, these $u$-states are the physical measures of $f$, and the union of their basins covers a full volume measure subset of a neighborhood of ${\mathcal{W}}^{c}(\Lambda_{i})$. By Theorem 4.6, all these physical measures are Bernoulli for some iterate of $f$. Thus, we get alternative (b) of the theorem in this case. From now on, let $\\{m_{i,j}\\}_{j=1}^{J(i)}$ be the physical measures supported on each attractor $\Lambda_{i}$. As we have just seen, their basins cover a full Lebesgue measure subset of a neighborhood $U_{i}$ of ${\mathcal{W}}^{c}(\Lambda_{i})$. We want to prove that the union of all these basins contains a full Lebesgue measure subset of the ambient manifold. Suppose otherwise, that is, suppose the complement $C$ of this union has positive Lebesgue measure. Let $C_{0}\subset C$ be the set of Lebesgue density points of $C$. Notice that $C_{0}$ is $f$-invariant and $\operatorname{vol}(C_{0})=\operatorname{vol}(C)$. Since the unstable foliation is absolutely continuous, there is a ${\mathcal{W}}^{u}$-disk $D^{u}$ such that $\operatorname{vol}_{D^{u}}(D^{u}\cap C_{0})>0$. Denote $I^{u}=D^{u}\cap C_{0}$. Then every Cesaro accumulation point of the iterates of Lebesgue measure on $I^{u}$ is a Gibbs $u$-state (see [17], section 11.2), and so its ergodic components are ergodic Gibbs $u$-states. Let $m^{}$ be any such accumulation point and $m_{i,j}$ be an ergodic component of $m^{}$. The support of $m_{i,j}$ is contained in $U_{i}$, and so there is $n_{0}\geq 1$ such that $f^{n_{0}}(I^{u})$ intersects $U_{i}$. Recalling that $C_{0}$ is invariant, we get that $\operatorname{vol}(C_{0}\cap U_{i})>0$. This contradicts the definition of $C_{0}$, since Lebesgue almost every point in $U_{i}$ belongs to the basin of $m_{i,l}$ for some $l=1,\dots,J(i)$. This contradiction proves that the union of the basins does have full Lebesgue measure in $N$. That completes the proof of Theorem D. ### 5.4. Number of physical measures In this section, we give explicit upper bounds on the number of physical measures for some diffeomorphisms with mostly contracting center direction: ###### Theorem 5.3. Let $f\in{\mathcal{P}}_{1}^{k}(N)$, $k>1$ be accessible on some attractor $\Lambda$ and have absolutely continuous center stable foliation. Assume there exists some center leaf $\ell\subset{\mathcal{W}}^{c}(\Lambda)$ such that $f^{\kappa}(\ell)=\ell$ for some $\kappa\geq 1$ and $f^{\kappa}\mid\ell$ is Morse-Smale with periodic points $p_{1},\cdots,p_{s}$. Then the center direction is mostly contracting over $\Lambda$ and $f$ has at most $s$ physical measures supported in ${\mathcal{W}}^{c}(\Lambda)$. If ${\mathcal{W}}^{u}(p_{i})$ intersects $W^{s}(\ell)\setminus\cup_{j=1}^{s}{\mathcal{W}}^{s}(p_{j})$ for every $i$ then $f$ has at most $s/2$ physical measures supported in ${\mathcal{W}}^{c}(\Lambda)$. ###### Proof. Since $f$ has hyperbolic periodic point in ${\mathcal{W}}^{c}(\Lambda)$ the restriction of $f$ to ${\mathcal{W}}^{c}(\Lambda)$ can not be conjugate to a rotation extension over $\Lambda$. Thus, by Theorem D, $f$ has mostly contracting center direction over $\Lambda$. ###### Lemma 5.4. Suppose $f\in{\mathcal{P}}_{1}^{k}(N)$, $k>1$ has mostly contracting center direction on an attractor $\Lambda$ and let $p$ be any periodic point in ${\mathcal{W}}^{c}(\Lambda)$. Then any disk $D^{u}$ in unstable manifold of $p$ contains a positive measure subset $I^{u}$ such that any $\xi\in I^{u}$ belongs to the basin of some physical measure and has local stable manifold $W_{loc}^{s}(\xi)$. ###### Proof. As in the proof of Lemma 4.1, there is a positive measure subset $I^{u}$ of $D^{u}$ belonging to the basin of some physical measure $m$, and for $\xi\in I^{u}$, there is $n_{0}$ such that $f^{n_{0}}(\xi)$ belongs to the Pesin stable manifold of some point $\zeta$. Iterating backward we obtain a local stable manifold for $\xi$. ∎ Suppose $f$ has physical measures $\\{m_{j}\\}_{j=1}^{J}$ on ${\mathcal{W}}^{c}(\Lambda)$. Let $p_{t}$, $t=1,\dots,s$ be fixed as in Theorem 5.3. Since the support of each physical measure is a $u$-saturated compact set, the following fact is an immediate consequence of Lemma 4.2: ###### Corollary 5.5. For each $1\leq t\leq s$ there is at most one physical measure whose support intersects $W^{s}(p_{t})$. As observed before, the unstable foliation is minimal in every attractor in the quotient. So, the orbit of every strong unstable leaf intersects $W^{s}(\ell)=\cup_{t=1}^{s}W^{s}(p_{t})$. Since the supports of physical measures are ${\mathcal{W}}^{u}$-saturated and invariant, it follows that for every $1\leq j\leq J$ there exists some $1\leq t\leq s$ such that $\operatorname{supp}m_{j}$ intersects $W^{s}(p_{t})$. So, by Corollary 5.5, $J\leq s$. Let $\\{p_{s_{i}}\\}_{i=1}^{s/2}$ be periodic points in $\ell$ with stable index $d_{s}$ (i.e. repellers for $f\mid\ell$) and let $\\{p_{s_{i}}\\}_{i=s/2+1}^{s}$ be periodic points in $\ell$ with stable index $d_{cs}$ (i.e. attractors for $f\mid\ell$). We claim that if ${\mathcal{W}}^{u}(p_{i})$ intersects ${\mathcal{W}}^{s}(\ell)\setminus\cup_{j=1}^{s}{\mathcal{W}}^{s}(p_{j})$ for every $i$, then the support of every physical measure contains some $p_{i}$, $s/2+1\leq i\leq s$. Indeed, by the previous observations the support must intersect $W^{s}(p_{i})$ for some $i$, corresponding to either an attractor or a repeller of $f\mid\ell$. In the former case, the claim is proved; in the latter case, our assumption on $\ell$ implies that the support intersects the stable set of some other periodic point $p_{j}$ which is an attractor, and so the claim follows in just the same way. So, by the previous argument, the number of physical measures can not exceed $s/2$ in this case. The proof of Theorem 5.3 is complete. ∎ ### 5.5. Statistical stability We also want to analyze the dependence of the physical measures on the dynamics. For this, we assume $N=M\times S^{1}$ and restrict ourselves to the set ${\mathcal{S}}^{k}(N)\subset{\mathcal{P}}_{1}^{k}(N)$ of skew-product maps. Notice that every $f\in{\mathcal{S}}^{k}(N)$ is dynamically coherent, has compact one-dimensional center leaves, and absolutely continuous center stable foliation. As pointed out before, partially hyperbolicity is an open property and accessibility holds on an open and dense subset of ${\mathcal{S}}^{k}(N)$. ###### Theorem 5.6. For any $k>1$ there exists a $C^{1}$ open and $C^{k}$ dense subset ${\mathcal{B}}^{k}(N)$ of ${\mathcal{S}}^{k}(N)$ such that every $f\in{\mathcal{B}}^{k}(N)$ has mostly contracting center direction. Moreover, on a $C^{k}$ open and dense subset of ${\mathcal{B}}^{k}(N)$ the number of physical measures is locally constant and the physical measures depend continuously on the diffeomorphisms. ###### Proof. Notice that every $f\in{\mathcal{S}}^{k}(N)$ is dynamically coherent, has compact one-dimensional center leaves, and absolutely continuous center stable foliation. By a variation of an argument of Niţică, Török [32], one gets that the set of diffeomorphisms in ${\mathcal{S}}^{k}(N)$ which are accessible on all attractors are $C^{1}$ open and $C^{r}$ dense. Let us comment a bit on this, since our setting is not exactly the same. The heart of the proof is to show that the accessibility class of any point contains the corresponding center leaf. This is done by considering 4-leg $su$-paths linking the point to every nearby point in the center leaf; in this way one gets that every accessibility class is open in the center leaf; then, connectivity gives the conclusion. The only difference in our case is that we deal with accessibility over each of the attractors, not the whole ambient manifold. However, the arguments remains unchanged, just with the additional caution to choose the corners of the $4$-leg $su$-path to be points over the attractor. It is easy to see that the set of diffeomorphisms in ${\mathcal{S}}^{k}(N)$ which have a center leaf containing some hyperbolic periodic point is $C^{1}$ open and $C^{r}$ dense. Take ${\mathcal{B}}^{k}$ be the intersection of above two sets. Then by Theorem D, any $f\in{\mathcal{B}}^{k}$ has mostly contracting center bundle. By Andersson [9], for any partially hyperbolic diffeomorphism $f$ with mostly contracting center direction there is a $C^{k}$, $k>1$ neighborhood $\mathcal{U}$ of $f$ such that any $g\in\mathcal{U}$ has mostly contracting center direction also, and on a $C^{k}$ open and dense subset of $\mathcal{U}$, the number of physical measures is locally constant and the physical measures depend continuously on the diffeomorphism. This ends the proof of Theorem 5.6. ∎ ## 6\. Absolute continuity for mostly contracting center Throughout this section $f:N\to N$ is a partially hyperbolic, dynamically coherent, $C^{k}$, $k>1$ diffeomorphism with mostly contracting center direction. Recall the later is a robust (open) condition, by Andersson [9]. We develop certain criteria for proving absolute continuity of the center stable, center unstable, and center foliations and we apply these tools to exhibit several robust examples of absolute continuity. In particular, this yields a proof of Theorem E. The starting point for our criteria is the observation that for maps with mostly contracting center the Pesin stable manifolds are contained in, and have the same dimension as the center stable leaves. Since the Pesin stable lamination is absolutely continuous ([35, 37]), in this way one can get a local property of absolute continuity for the center stable foliation. This initial step of the construction is carried out in Section 6.2. Then one would like to propagate this behavior to the whole ambient manifold, in order to obtain actual absolute continuity. It is important to point out that this can not possibly work without additional conditions. Example 6.1 below illustrates some issues one encounters. A more detailed analysis, including explicit robust counter-examples will appear in [49]. Suitable assumptions are introduced in Section 6.1, where we also give the precise statements of our criteria. In Section 6.3 we present the main tool for propagating local to global behavior. The criteria are proved in Sections 6.4 through 6.6. Before proceeding, let us give a simple example of a map whose center foliation is leafwise absolutely continuous and locally absolutely continuous, but not globally absolutely continuous. This kind of construction explains why Pesin theory alone can not give (global) absolute continuity of center foliations, even when the center direction is mostly contracting. ###### Example 6.1. Let us start with $f_{0}:S^{1}\times[0,1]\to S^{1}\times[0,1]$, $f_{0}(x,t)=(2x,g(t))$ where $g:[0,1]\to[0,1]$ is a $C^{2}$ diffeomorphism such that $g(0)=0$, $g(1)=1$, $g(t)<t$ for all $t\in(0,1)$, and $0<g^{\prime}(t)<2$ for every $t\in[0,1]$. Then $f_{0}$ is a partially hyperbolic endomorphism of the cylinder, with the vertical segments as center leaves. Next, let $f:S^{1}\times[0,1]\to S^{1}\times[0,1]$ be a $C^{2}$-small perturbation, preserving the two boundary circles ${\mathcal{C}}_{i}=S^{1}\times\\{i\\}$, $i=0,1$ and the vertical line $\\{0\\}\times[0,1]$ through the fixed point $(0,0)$. Moreover, the horizontal derivatives of $f$ at the endpoints of this vertical line should be different: (20) $\frac{\partial f}{\partial x}(0,0)\neq\frac{\partial f}{\partial x}(0,1).$ By the stability of center foliations ([28], the new map $f$ has a center foliation whose leaves are curve segments with endpoints in the two boundary circles. Thus, they induce a holonomy map $h:{\mathcal{C}}_{0}\to{\mathcal{C}}_{1}$ that conjugates the two expanding maps $f\mid{\mathcal{C}}_{0}$ and $f\mid{\mathcal{C}}_{1}$. Condition (20) implies that the conjugacy can not be absolutely continuous (see [43]). This shows that the center foliation is not absolutely continuous. Yet, it is absolutely continuous restricted to $S^{1}\times[0,1)$, as we are going to explain. Notice that our assumptions imply that $g^{\prime}(0)<1<g^{\prime}(1)$ and so the lower boundary component ${\mathcal{C}}_{0}$ is an attractor for $f_{0}$, with $S^{1}\times[0,1)$ as its basin of attraction. Then the same is true for the perturbation $f$. Moreover, restricted to this basin, the center leaves coincide with the Pesin stable manifolds of the points in the attractor, and so they do form an absolutely continuous foliation. In particular, this also shows that the center foliation is leafwise absolutely continuous. ### 6.1. Criteria for absolute continuity We assume that some small cone field around the strong unstable bundle has been fixed. We call _$u$ -disk_ any embedded disk of dimension $d_{u}$ whose tangent space is contained in that unstable cone field at every point. Previously, we introduced the special case of ${\mathcal{W}}^{u}$-disks, which are contained in strong unstable leaves. To begin with, in Section 6.4 we prove that upper leafwise absolute continuity always holds in the present context: ###### Proposition 6.2. The center stable foliation of $f$ is upper leafwise absolutely continuous, if it exists. For the next criterion we assume the diffeomorphism is non-expanding along the center direction. This notion is defined as follows. Assume also $f$ is dynamically coherent. Given $r>0$ and $\in\\{s,cs,c,cu,u\\}$, we denote by ${\mathcal{W}}^{}_{r}(x)\subset{\mathcal{W}}^{}(x)$ the ball of radius $r$ around $x$, relative to the distance induced by the Riemannian metric of $N$ on the leaf ${\mathcal{W}}^{}(x)$. In what follows we always suppose $r$ is small enough so that ${\mathcal{W}}^{}_{r}(x)$ is an embedded disk of dimension $d_{}$ for all $x\in M$ and every choice of $$. We use $\widehat{W}^{s}(p)$ and $\widehat{W}^{u}(p)$ to denote the stable and unstable sets of a periodic point $p$. We say that $f$ is _non-expanding along the center direction_ if there exist $\rho>0$ and $\varepsilon>0$ such that • $f^{n}({\mathcal{W}}^{cs}_{\varepsilon}(x))\subset{\mathcal{W}}^{cs}_{\rho}(f^{n}(x))$ for every $n\geq 0$ and almost any $x$ in any $u$-disk. * • the support of every ergodic Gibbs $u$-state $m$ contains some periodic point $p$ such that $\widehat{W}^{s}(p)\supset{\mathcal{W}}^{cs}_{2\rho}(p)$ ###### Proposition 6.3. If $f$ is non-expanding along the center direction then the center stable foliation is absolutely continuous. The proof of this proposition is given in Sections 6.2 through 6.5. We will see that the hypothesis holds for a classical construction of partially hyperbolic, robustly transitive diffeomorphisms due to Mañé [30] (Section 7.1). It also holds for a more recent class of examples introduced by Bonatti, Viana [19], which are not even partially hyperbolic (though they do admit a dominated invariant splitting of the tangent bundle), but this fact will not be proved here. Let $f\in{\mathcal{P}}_{1}^{k}(N)$. Let $\ell$ be a periodic center leaf $\ell$, with period $\kappa\geq 1$. For $\in\\{s,u\\}$, we denote ${\mathcal{W}}^{}(\ell)=\cup_{\zeta\in\ell}{\mathcal{W}}^{}(\zeta)$. We call _homoclinic leaf_ associated to $\ell$ any center leaf $\ell^{\prime}$ contained in ${\mathcal{W}}^{s}(\ell)\cap{\mathcal{W}}^{u}(\ell)$. Then there exist strong stable and strong unstable holonomy maps (21) $h^{s}:\ell\to\ell^{\prime}\quad\text{and}\quad h^{u}:\ell\to\ell^{\prime}$ We say that $\ell$ is _in general position_ if (a) $f^{\kappa}\mid\ell$ is Morse-Smale with a single periodic attractor $a$ and a single periodic repeller $r$; * (b) $h^{s}(a\cup r)$ is disjoint from $h^{u}(a\cup r)$, for some homoclinic leaf associated to the center leaf $\ell$. Notice that ${\mathcal{W}}^{s}(\ell^{\prime})\setminus{\mathcal{W}}^{s}(h^{s}(r))$ is contained in the stable manifold $\widehat{W}^{s}(a)$ of the attractor. Thus, condition (b) implies that ${\mathcal{W}}^{u}(a)$ and ${\mathcal{W}}^{u}(r)$ intersect $\widehat{W}^{s}(a)$ transversely. Analogously, ${\mathcal{W}}^{s}(a)$ and ${\mathcal{W}}^{s}(r)$ intersect $\widehat{W}^{u}(r)$ transversely. ###### Proposition 6.4. Suppose $f\in{\mathcal{P}}^{k}_{1}(N)$ has some center leaf $\ell$ in general position and such that every strong unstable leaf intersects ${\mathcal{W}}^{s}(\ell)$. Then the center stable foliation of $f$ is absolutely continuous. This proposition is proved in Section 6.6. In Section 7.2 we use it to prove Theorems B and E, and in Section 7.3 we give an application to volume preserving systems. Noticing that, apart from dynamical coherence, all the hypotheses of Proposition 6.4 are robust, we get the following immediate consequence: ###### Corollary 6.5. Suppose $f\in{\mathcal{P}}_{1}^{k}(N)$ is robustly dynamically coherent and has some periodic center leaf $\ell$ in general position and such that every strong unstable leaf intersects ${\mathcal{W}}^{s}(\ell)$. Then the center stable foliation is robustly absolutely continuous. ### 6.2. Local absolute continuity The following lemma will allow us to obtain some property of local absolute continuity: ###### Lemma 6.6. For any ergodic $u$-state $m$ of $f$ and any disk $D$ contained in an unstable leaf inside $\operatorname{supp}m$, there is a positive measure subset $\Gamma$ such that the points in $\Gamma$ have (Pesin) stable manifolds with uniform size. Moreover, these stable manifolds form an absolutely continuous lamination, in the following sense: there is $K>0$ such that for any two $u$-disks $D_{1}$, $D_{2}$ sufficiently close to $D$, the stable manifolds of points in $\Gamma$ define a holonomy map between subsets of $D_{1}$ and $D_{2}$, and this is absolutely continuous, with Jacobian between $1/K$ and $K$. ###### Proof. Because $f$ has mostly contracting center direction, $m$ is a hyperbolic ergodic measure of $f$, by Pesin theory, there is a Pesin block $\Lambda$ with positive $m$ measure such that every point $x\in\Lambda$ has uniform size of stable manifold, and these stable manifolds on $\Lambda$ is uniformly absolutely continuous. Notice that the stable manifolds are contained in the center stable leaves. Since $m$ is a $u$-state, there is a disk $D_{0}$ contained in an unstable leaf inside the support and intersecting $\Lambda$ on a $m^{u}$-positive measure subset $D_{0}^{}$. Then the points in $D_{0}^{}$ have stable manifolds of size bounded below by some $\delta_{0}>0$. Denote $B_{0}=\cup_{x\in D_{0}^{}}W^{s}_{\delta_{0}}(x)$. Since $m$ is a $u$-state, $m(B_{0})=a_{0}>0$. We claim that there is $n_{0}>0$ such that $(f^{n_{0}})_{}\operatorname{vol}_{D}(B_{0})\neq 0$. Let us prove this claim. Let $D_{\varepsilon}^{}$ be the $\varepsilon$-neighborhood of $D_{0}^{}$ inside the corresponding unstable leaf. Denote by $B_{\varepsilon}=\cup_{x\in D_{\varepsilon}^{}}W^{cs}_{\delta_{0}}(x)$, it is an open set, and $m(B_{\varepsilon})\geq a_{0}>0$. Because every Cesaro accumulation point of the iterates of Lebesgue measure on $D$ is a Gibbs $u$-state with support contained in $\operatorname{supp}m$, and there is a unique ergodic $u$-state with support contained in $\operatorname{supp}m$, then $m$ is the unique Cesaro accumulation of the iterates of Lebesgue measure on $D$. Since $B_{\varepsilon}$ is open, one has $\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}(f^{i})_{}\operatorname{vol}_{D}(B_{\varepsilon})\geq m(B_{\varepsilon})=a_{0}$, so there is arbitrarily big $n$ such that $f_{}^{n}(\operatorname{vol}_{D})(B_{\varepsilon})>a_{0}/2$. For $\delta>0$ sufficiently small, denote by $D_{\delta}=\\{x\in D,d^{u}(x,\partial(D))\geq\delta\\}$, one has $m^{u}(D\setminus D_{\delta})<a_{0}/4$. Then there is $y\in D_{\delta}$ such that $f^{n}(y)\in B_{\varepsilon}$ and $f^{n}(D)$ contains a disk $D_{y}$ around $y$ and for any $x\in D_{\varepsilon}^{}$ one has $W^{cs}_{\delta_{0}}(x)\cap D_{y}\neq\emptyset$. Then the stable manifolds of $D_{0}^{}$ define a holonomy map between $D_{0}^{}$ and $B_{0}\cap D_{y}$, by the uniform absolute continuity of these stable manifolds, $\operatorname{vol}_{D_{y}}(D_{y}\cap B_{0})>0$, then $f_{}^{n_{0}}(\operatorname{vol}_{D})(B_{0})>0$. This proves the claim. This claim implies $\operatorname{vol}_{f^{n}(D)}(f^{n}(D)\cap B^{}_{0})>0$, let $\Gamma=D\cap f^{-n_{0}}(B^{}_{0})$, then $\operatorname{vol}_{D}(\Gamma)>0$, every point in $\Gamma$ has uniform size of stable manifold, and these stable manifolds are uniformly absolutely continuous. ∎ Suppose $f\in{\rm Diff}^{k}(N)$, $k>1$ admits dominated splitting $E^{cu}\oplus E^{cs}$, and it is dynamically coherent, that is, it has center stable and center unstable foliation. We call _$cs$ -block_ for $f$ the image ${\mathcal{B}}=h(\Sigma\times I^{d_{cs}})$ of any embedding $h:\Sigma\times I^{d_{cs}}\to N$, with $\Sigma\subset I^{d_{cu}}$, satisfying the following properties: 1. (1) $h(\\{a\\}\times I^{d_{cs}})$ is contained in ${\mathcal{W}}^{cs}(h(a,0))$, for every $a\in\Sigma$ 2. (2) $h(\\{a\\}\times I^{d_{cs}})$ is contained in the stable set of $h(a,0)$, for every $a\in\Sigma$ 3. (3) $h(\Sigma\times\\{0\\})$ is a positive measure subset of some disk $D$ transverse to ${\mathcal{W}}^{cs}$; 4. (4) there is $K>0$ such that for any $u$-disks $D_{1},D_{2}\subset N$ which cross $h(\Sigma\times I^{u})$, that is, $D_{i}$ intersects $h(a\times I^{cs})$ for every $a\in\Sigma$, there is a holonomy map $h^{cs}$ induced by ${\mathcal{W}}^{cs}$ from $D_{1}\cap h(a\times I^{cs})$ to $D_{2}\cap h(a\times I^{cs})$, the Jacobian of the holonomy map between $\operatorname{vol}_{D_{1}}$ and $\operatorname{vol}_{D_{2}}$ is bounded by $K$ from above and $1/K$ from below. We also say that ${\mathcal{B}}$ is a $cs$-block over the disk $D$ in (3). If $D$ is contained in the unstable manifold of an index $d_{cs}$ periodic point $p$, then we say the $cs$-block is associated with p. ###### Remark 6.7. If $D$ is in the support of some ergodic Gibbs $u$-state $m$ then $m({\mathcal{B}})>0$: this is a consequence of the absolute continuity property (4) and the fact that Gibbs $u$-states have positive densities along strong unstable leaves (Proposition 3.1). We say that the $cs$-block has size $r>0$ if the _plaque_ $h(\\{a\\}\times I^{d_{cs}})$ contains ${\mathcal{W}}^{cs}_{r}(h(a,0))$ for every $a\in\Sigma$. If a map $\tilde{h}:\Sigma\times I^{d_{cs}}\to N$ satisfies $\tilde{h}\mid{\Sigma\times\\{0\\}}\equiv h\mid{\Sigma\times\\{0\\}}\quad\text{and}\quad\tilde{h}(a\times I^{d_{cs}})\subset h(a\times I^{d_{cs}})$ for every $a\in\Sigma$ then $\tilde{\mathcal{B}}=\tilde{h}(\Sigma\times I^{d_{cs}})$ is called a sub-block of ${\mathcal{B}}$. ###### Lemma 6.8. Let $m$ be an ergodic $u$-state of $f$ and $p\in\operatorname{supp}m$ be a periodic point of stable index $d_{cs}$ whose stable manifold $\widehat{W}^{s}(p)$ has size $r$. Then there is a $cs$-block associated with $p$ with size $r$. ###### Proof. By Lemma 6.6, there is a $cs$-block over any $u$-disk $D\subset{\mathcal{W}}^{u}(p)$. Let $\kappa$ be the period of $p$. For every large $n$, the backward image $f^{-n\kappa}({\mathcal{B}})$ is a $cs$-block of size $r$ over the $u$-disk $f^{-n}(D)$. ∎ ### 6.3. Recurrence to $cs$-blocks The next proposition is a key ingredient in the proof of our criteria for absolute continuity. ###### Proposition 6.9. Let $m_{i}$, $i=1,\dots,s$ be the ergodic Gibbs $u$-states of $f$ and $B_{i}$, $i=1,\dots,n$ be $cs$-blocks over ${\mathcal{W}}^{u}$-disks $D_{i}\subset\operatorname{supp}m_{i}$. Then for any positive Lebesgue measure subset $D^{}$ of any ${\mathcal{W}}^{u}$-disk $D$, there exists $n>0$ arbitrarily large and there exists $1\leq i\leq s$ such that $\operatorname{vol}_{D}(D^{}\cap f^{-n}(B_{i}))>0$. ###### Proof. (For notational simplicity, we use $m^{u}$ to denote $\operatorname{vol}_{f^{n}(\Gamma)}$ for any $u$-disk $\Gamma$ and any $n>0$.) Let $D^{}_{\varepsilon}=\cup_{x\in D^{}}B_{\varepsilon}(x,D)$ where $B_{\varepsilon}(x,D)$ is the ball in $D$ with radius $\varepsilon$ and center in $x$, and $m$, $m_{\varepsilon}$ be Cesaro accumulation points of the iterates of Lebesgue measure on $D^{}$ and $D^{}_{\varepsilon}$ respectively, such that there is $\\{n_{j}\\}_{j=1}^{\infty}$ satisfying $\lim_{j\to\infty}\frac{1}{n_{j}m^{u}(D^{})}\sum_{i=0}^{n_{j}-1}(f^{i})_{}m^{u}\mid{D^{}}=m;$ $\lim_{j\to\infty}\frac{1}{n_{j}m^{u}(D^{}_{\varepsilon})}\sum_{i=0}^{n_{j}-1}(f^{i})_{}m^{u}\mid{D^{}_{\varepsilon}}=m_{\varepsilon}.$ Then they are $u$-states, denote by $m=a_{1}m_{1}+\cdots+a_{m}m_{s}$ the ergodic decomposition of $m$, suppose $a_{1}\neq 0$. For $\varepsilon$ sufficiently small, $m^{u}(D^{}_{\varepsilon})\approx m^{u}(D^{})$, and then $m_{\varepsilon}\approx m$, denote by $m_{\varepsilon}=a_{1,\varepsilon}m_{1}+\cdots+a_{s,\varepsilon}m_{m}$ the ergodic decomposition of $m_{\varepsilon}$, one has $a_{1,\varepsilon}\approx a_{1}$. Denote $D_{1}^{}=D_{1}\cap B_{1}$ and $D_{1,\delta}^{}=\cup_{x\in D_{1}^{}}B^{u}_{\delta}(x)$ and $B_{1,\delta}^{}=\\{z:z\in{\mathcal{W}}^{cs}_{loc}(x)\cap{\mathcal{W}}^{u}_{loc}(y)\text{ for }x\in D_{1,\delta}^{}\text{ and }y\in B_{1}\\}.$ Then there is $K_{1}>0$ such that for any $u$-disk $\Gamma$ which crosses $B_{1,\delta}^{}$, one has $\frac{m^{u}(\Gamma\cap B_{1})}{m^{u}(\Gamma\cap B_{1,\delta}^{})}>K_{1},$ that is because $m^{u}(\Gamma\cap B_{1})>\frac{1}{K}m^{u}(D_{1}\cap B_{1})>0$ and $m^{u}(\Gamma\cap B_{1,\delta}^{})$ is bounded above, where $K$ is the bound for the Jacobian of the center stable foliation in $B_{1}$. We can choose $\varepsilon$ properly such that $m^{u}(\partial(D^{}_{\varepsilon}))=0$, by Remark 6.7, suppose $m_{1}(B_{1})=b_{0}>0$. Because $B_{1,\delta}^{}$ is open, $\lim_{j\to\infty}\frac{1}{n_{j}m^{u}(D^{}_{\varepsilon})}\sum_{i=0}^{n_{j}-1}\big{(}f^{i}_{}m^{u}\mid D\big{)}\big{(}f^{i}(D^{}_{\varepsilon})\cap B_{1,\delta}^{}\big{)}\geq m_{\varepsilon}(B_{1,\delta})\gtrsim b_{0}a_{\varepsilon}>\frac{a_{1}b_{0}}{2}.$ So there is $n_{j}$ arbitrarily big such that $\big{(}f^{n_{j}}_{}m^{u}\mid D\big{)}\big{(}f^{n_{j}}(D^{}_{\varepsilon})\cap B_{1,\delta}^{}\big{)}\geq\frac{a_{1}b_{0}}{4}m^{u}(D_{\varepsilon}^{}).$ We claim that there is $b_{1}>0$ such that, for every $\varepsilon>0$ sufficiently small, $\big{(}f^{n_{j}}_{}m^{u}\mid D\big{)}\big{(}f^{n_{j}}({D^{}_{\varepsilon}})\cap B_{1}\big{)}\geq 2b_{1}m^{u}(D_{\varepsilon}^{}).$ Let us prove the claim. For $\varepsilon_{1}<\varepsilon$, denote $D^{}_{\varepsilon,\varepsilon_{1}}=\\{x\in D^{}_{\varepsilon};d_{D}(x,\partial(D^{}_{\varepsilon}))>\varepsilon_{1}\\}$. Then, for $\varepsilon_{1}$ sufficiently small, one has $m^{u}({D^{}_{\varepsilon,\varepsilon_{1}}})>m^{u}(D^{}_{\varepsilon})-a_{1}b_{0}/8$. It follows that $\big{(}f^{n_{j}}_{}m^{u}\mid D\big{)}\big{(}B_{1,\delta}^{}\big{)}\geq\frac{a_{1}b_{0}}{8}m^{u}(D^{}_{\varepsilon})$ and for any $x\in f^{n_{j}}(D^{}_{\varepsilon,\varepsilon_{1}})\cap B_{1,\delta}^{}$, there is a $u$-disk $D_{x}\subset f^{n_{j}}(D^{}_{\varepsilon})$ containing $x$ such that $D_{x}\cap{\mathcal{W}}^{cs}_{loc}(y)\neq\emptyset$ for any $y\in D_{1,\delta}^{}$. Since $\frac{m^{u}(D_{x}\cap B_{1})}{m^{u}(D_{x}\cap B_{1,\delta}^{})}>K_{1},$ and the distortion property on ${\mathcal{W}}^{u}$, there is $K_{2}>0$ such that $\frac{m^{u}(f^{-n_{j}}(D_{x}\cap B_{1}))}{m^{u}(f^{-n_{j}}(D_{x}\cap B_{1,\delta}^{}))}>K_{2}.$ Then, taking $b_{1}={K_{2}a_{1}b_{0}}/{16}$, we get our claim: $m^{u}(D^{}_{\varepsilon}\cap f^{-n_{j}}(B_{1}))>K_{2}m^{u}(D^{}_{\varepsilon,\varepsilon_{1}}\cap f^{-n_{j}}(B_{1,\delta}^{}))>2b_{1}m^{u}(D^{}_{\varepsilon}),$ Since $\lim_{\varepsilon\to 0}m^{u}(D^{}_{\varepsilon}\setminus D^{})=0$, this proves that $m^{u}(D^{}\cap f^{-n_{j}}(B_{1}))>b_{1}m^{u}(D^{})$. This completes the proof of the proposition. ∎ ###### Remark 6.10. Assuming there exists a unique Gibbs $u$-state, the arguments in the proof of Proposition 6.9 yield a slightly stronger conclusion that will be useful in the sequel: there exists $b_{1}>0$ such that for any positive Lebesgue measure subset $D^{}$ of any ${\mathcal{W}}^{u}$-disk $D$ and for any $1\leq i\leq s$ there exist arbitrarily large values of $n>0$ such that $\operatorname{vol}_{D}(D^{}\cap f^{-n}(B_{i}))\geq b_{1}\operatorname{vol}_{D}(D^{})$. ### 6.4. Upper leafwise absolute continuity Here we prove Proposition 6.2. Suppose there exists some measurable set $Y$ with $\operatorname{vol}(Y)>0$ that meets almost every center stable leaf ${\mathcal{W}}^{cs}(z)$ on a zero $\operatorname{vol}^{cs}$-measure subset. Up to replacing $Y$ by some full measure subset, we may suppose that every $x\in f^{n}(Y)$ is a Lebesgue density point of $f^{n}(Y)$ for every $n\geq 0$: (22) $\lim_{\rho\to 0}\frac{\operatorname{vol}(B_{\rho}(x)\cap f^{n}(Y))}{\operatorname{vol}(B_{\rho}(x))}=1.$ Since $f$ has finitely many ergodic $u$-states and their basins cover a full measure subset of $N$ (see [19]), it is no restriction to suppose that $Y$ is contained in the basin of some ergodic Gibbs $u$-state $m$. Let ${\mathcal{B}}$ be a $cs$-block over some $u$-disk contained in the support of $m$ (recall Proposition 3.1 and Section 6.2). Since the strong unstable foliation is absolutely continuous (see [20]), we can find a $u$-disk $D$ such that $D^{}=D\cap Y$ has positive $\operatorname{vol}_{D}$-measure. By Proposition 6.9, there exists $n>0$ such that $\operatorname{vol}_{f^{n}(D)}(f^{n}(D^{})\cap{\mathcal{B}})>0$. Take $y\in D^{}$ such that $f^{n}(y)\in{\mathcal{B}}$ and $f^{n}(y)$ is a Lebesgue density point for $f^{n}(D^{})\cap{\mathcal{B}}$ inside $f^{n}(D)$. Then, for every small $\rho>0$, $\frac{\operatorname{vol}_{f^{n}(D)}\big{(}B^{u}_{\rho}(f^{n}(y))\cap{\mathcal{B}}\big{)}}{\operatorname{vol}_{f^{n}(D)}\big{(}B^{u}_{\rho}(f^{n}(y)\big{)}}\geq\frac{\operatorname{vol}_{f^{n}(D)}\big{(}B^{u}_{\rho}(f^{n}(y))\cap f^{n}(D^{})\cap{\mathcal{B}}\big{)}}{\operatorname{vol}_{f^{n}(D)}\big{(}B^{u}_{\rho}(f^{n}(y)\big{)}}\approx 1,$ where $B^{u}_{\rho}(x)$ denotes the connected component of $B_{\rho}(x)\cap{\mathcal{W}}^{u}(x)$ that contains $x$. Then, since the center stable foliation is uniformly absolutely continuous on the $cs$-block, there exists $c>0$ such that $\frac{\operatorname{vol}\big{(}B_{\rho}(f^{n}(y))\cap{\mathcal{B}}\big{)}}{\operatorname{vol}\big{(}B_{\rho}(f^{n}(y)\big{)}}\geq c\quad\text{for all small }\rho>0.$ Together with (22), this implies that $\operatorname{vol}(f^{n}(Y)\cap{\mathcal{B}})>0$. On the other hand, the hypothesis implies that $f^{n}(Y)$ intersects almost every center stable leaf on a zero Lebesgue measure subset. Using, once more, that the center stable leaf is absolutely continuous on the $cs$-block, we get that $\operatorname{vol}(f^{n}(Y)\cap{\mathcal{B}})=0$. This contradicts the previous conclusion, and that contradiction completes the proof of Proposition 6.2. ### 6.5. Non-expansion along the center Now we prove Proposition 6.3. Let the ergodic $u$-states $\\{m_{i}\\}_{i=1}^{m}$, periodic points $\\{p_{i}\\}_{i=1}^{m},$ and $\rho,\varepsilon$ given in the definition of non-expansion along the center. By Lemma 6.8, we can choose $cs$-blocks $\\{{\mathcal{B}}_{i}\\}_{i=1}^{m}$ associated with $m_{i}$ with size $\rho$. In order to prove the center stable foliation is absolutely continuous, we just need show that for any two $u$-disks $D_{1},D_{2}$ which are $\varepsilon$ near, the holonomy map induced by ${\mathcal{W}}^{cs}$ between $D_{1}$ and $D_{2}$ maps Lebesgue positive measure subset to a Lebesgue positive measure subset, where two $u$-disks $D_{1},D_{2}$ are $\varepsilon$ near if for any $x\in D_{1}$, there is $y\in D_{2}$ belonging to ${\mathcal{W}}^{cs}_{\varepsilon}(x)$. Suppose $D_{1}^{}\subset D_{1}$ is a positive measure subset, denote by $D_{2}^{}\subset D_{2}$ the image of $D_{1}^{}$ under cs-holonomy map. Since $f$ is non-expanding along the center, we can assume that for any $x\in D_{1}^{}$, one has $f^{n}({\mathcal{W}}^{cs}_{\varepsilon}(x))\subset{\mathcal{W}}^{cs}_{\rho}(f^{n}(x))$ for $n>0$. Choose $\tilde{B}_{i}$ a sub-block of $B_{i}$ with arbitrarily small size, then by Proposition 6.9, there is $n$ and $j$ such that $m^{u}(f^{n}(D_{1}^{})\cap\tilde{B}_{j})>0$. Because $f^{n}(D_{1}^{})$ and $f^{n}(D_{2}^{})$ are $\rho$ near, and the cs-holonomy map in $B_{i}$ is absolutely continuous, one has that $m^{u}(f^{n}(D_{2}^{})\cap B_{j})>0$, this implies $m^{u}(D_{2}^{})>0$, so ${\mathcal{W}}^{cs}$ is absolutely continuous. ### 6.6. Center leaves in general position We are going to prove Proposition 6.4. Let us start by giving an overview of the argument. We need to compare a set on any $u$-disk with its projection to another $u$-disk under $cs$-holonomy. The idea is to consider appropriate iterates of both $u$-disks intersecting a given $cs$-block, and then take advantage of the uniform structure on the $cs$-block. The problem is that, because $cs$-blocks have gaps along the center direction, one can not immediately ensure that iterates of both disks intersect the same $cs$-block. To this end, we use the twisting property in the assumption of general position to find a pair of $cs$-blocks whose union covers the whole center direction, in the sense that it intersects any large iterate of any $u$-disk. Then, we show that some iterate of any of the disks intersects both $cs$-blocks, which gives the required property. Now we fill in the details in the proof. Let $f$ and $\ell$ be as in the statement of the proposition. For simplicity, consider the center leaf $\ell$ to be fixed (in other words, $\kappa=1$) and we also take the attractor $a$ and repeller $r$ of $f\mid\ell$ to be fixed. Extension to the general case is straightforward. ###### Lemma 6.11. The diffeomorphism $f$ has a unique ergodic $u$-state and its support contains the attractor $a$. ###### Proof. By Lemma 4.2, the supports of all ergodic Gibbs $u$-states are pairwise disjoint. Thus, it suffices to show that the support of any ergodic $u$-state contains $a$. By Proposition 3.1, the support of $m$ consists of entire unstable leaves. So, it suffices to prove that every strong unstable leaf intersects the stable manifold $\widehat{W}^{s}(a)$ of the attractor. By hypothesis, every strong unstable leaf intersects ${\mathcal{W}}^{s}(\ell)$. Moreover, ${\mathcal{W}}^{s}(\ell)$ is the union of $\widehat{W}^{s}(a)$ with the strong stable leaf through the repeller $r$. If a strong unstable leaf $L$ intersects ${\mathcal{W}}^{s}(r)$ then its forward orbit accumulates on ${\mathcal{W}}^{u}(r)$ and, in particular, on $h^{u}(r)$. Since $\ell$ is in general position, $h^{u}(r)\neq h^{s}(r)$ and so $h^{u}(r)$ belongs to the stable manifold of $a$. Hence, in any case, $L$ does intersect $\widehat{W}^{s}(a)$. This completes the argument. ∎ Consider the four points $a_{s}=h^{s}(a)$, $a_{u}=h^{u}(a)$, $r_{s}=h^{s}(r)$, $r_{u}=h^{u}(r)$ in $\ell^{\prime}$. For $\rho$, $\varepsilon>0$ small, and $\zeta\in\ell^{\prime}$, denote ${\mathcal{W}}_{\rho}^{s}(\ell^{\prime})=\cup_{\xi\in\ell^{\prime}}{\mathcal{W}}^{s}_{\rho}(\xi)\quad\text{and}\quad V^{cs}_{\varepsilon}(\zeta)=\cup_{\xi\in B^{c}_{\varepsilon}(\zeta)}{\mathcal{W}}^{s}_{\rho}(\xi).$ Let $\tilde{\mathcal{B}}$ be a $cs$-block over ${\mathcal{W}}^{u}_{loc}(a)$ (Lemma 6.8). Then for $n$ large, $f^{-n}(\tilde{\mathcal{B}})$ intersects ${\mathcal{W}}^{u}_{loc}(a_{s})$ in a set $\tilde{D}_{1}^{}$ with positive Lebesgue measure, and we may choose a $cs$-block ${\mathcal{B}}_{1}\subset f^{-n}(\tilde{\mathcal{B}})$ over a $u$-disk $\tilde{D}_{1}\supset\tilde{D}_{1}^{}$ such that ${\mathcal{W}}^{u}_{2\tau}(\zeta)\cap{\mathcal{B}}_{1}\neq\emptyset\quad\text{for all }\zeta\in{\mathcal{W}}^{s}_{\rho}(\ell^{\prime})\setminus V^{cs}_{\varepsilon}(r_{s}).$ We think of the union $W^{u}_{2\tau}(V^{cs}_{\varepsilon}(r_{s}))$ of the local unstable manifolds through the local center stable manifold of $r_{s}$ as the gap of ${\mathcal{B}}_{1}$ along the center direction. See Figure 3. Figure 3. Dually, consider a $cs$-block ${\mathcal{B}}_{2}\subset f^{-n}(\tilde{\mathcal{B}})$ over a $u$-disk $\tilde{D}_{2}\subset{\mathcal{W}}^{u}(a)$ such that ${\mathcal{W}}^{u}_{2\tau}(\zeta)\cap{\mathcal{B}}_{2}\neq\emptyset\quad\text{for all }\zeta\in{\mathcal{W}}^{s}_{\rho}(\ell^{\prime})\setminus V^{cs}_{\varepsilon}(r_{u}).$ Again, the union $W^{u}_{2\tau}(V^{cs}_{\varepsilon}(r_{u}))$ of the local unstable manifolds through the local center stable manifold of $r_{u}$ is the gap of ${\mathcal{B}}_{2}$ along the center direction. Moreover, we may fix $\delta_{0}>0$ such that, for any $\zeta\in{\mathcal{W}}^{s}_{\rho}(\ell^{\prime})$, either $m^{u}({\mathcal{W}}^{u}_{2\tau}(\zeta)\cap{\mathcal{B}}_{1})>\delta_{0}\quad\text{or}\quad m^{u}({\mathcal{W}}^{u}_{2\tau}(\zeta)\cap{\mathcal{B}}_{2})>\delta_{0}.$ This is, in precise terms, what we meant when we announced that the union ${\mathcal{B}}_{1}\cup{\mathcal{B}}_{2}$ of the two $cs$-blocks would cover the whole center direction. Now consider a new $cs$-block ${\mathcal{B}}$ defined as the union of $({\mathcal{W}}^{u}_{2\tau}(\xi)\cap{\mathcal{B}}_{2})\cup({\mathcal{W}}^{u}_{2\tau}(\xi)\cap{\mathcal{B}}_{1})$ over all $\xi\in W_{\rho}^{s}(\ell^{\prime})\setminus\big{(}V^{cs}_{\varepsilon}(r_{s})\cup V^{cs}_{\varepsilon}(r_{u})\big{)}$. In other words, ${\mathcal{B}}$ is obtained from ${\mathcal{B}}_{1}\cup{\mathcal{B}}_{2}$ by removing the two gaps. Thus, ${\mathcal{B}}={\mathcal{B}}^{1}\cup{\mathcal{B}}^{2}$ with ${\mathcal{B}}^{1}\subset{\mathcal{B}}_{1}$ and ${\mathcal{B}}^{2}\subset{\mathcal{B}}_{2}$. We are going to show that arbitrarily large iterates of any $u$-disk intersect both connected components of ${\mathcal{B}}$ on positive measure subsets. ###### Lemma 6.12. Given any $u$-disk $D$ and any positive $\operatorname{vol}_{D}$-measure subset $D^{}$ there exists $\zeta\in D^{}$ and $k$ arbitrarily large such that $\operatorname{vol}_{f^{k}(D)}\big{(}{\mathcal{W}}^{u}_{2\tau}(f^{n}(\zeta))\cap f^{k}(D^{})\cap{\mathcal{B}}^{i}\big{)}>0\quad\text{for both }i=1,2.$ ###### Proof. It is no restriction to suppose every point of $D^{}$ is a Lebesgue density point. Fix $\varepsilon>0$ small (the precise choice will be given later). Take any point $x\in D^{}$ and let $r>0$ small enough so that $\operatorname{vol}_{D}(D_{r}^{})>(1-\varepsilon)\operatorname{vol}_{D}(D_{r})$, where $D_{r}$ is the disk of radius $r$ around $x$ and $D_{r}^{}=D_{r}\cap D^{}$. By Proposition 6.9 and Remark 4.3 there exists $b_{1}>0$, independent of $x$ and $r$, such that $\operatorname{vol}_{D}(D_{r}^{}\cap f^{-n_{i}}({\mathcal{B}}^{1}))\geq b_{1}\operatorname{vol}_{D}(D_{r}^{})\geq b_{1}(1-\varepsilon)\operatorname{vol}_{D}(D_{r})$ for a sequence $n_{i}\to\infty$. Let $\rho>0$ be slightly smaller than $r$, so that $\operatorname{vol}_{D}(D_{\rho})>(1-\varepsilon)\operatorname{vol}_{D}(D_{r}).$ Then, for any $n_{i}$ sufficiently large and any $y\in D_{\rho}$, we have $f^{-n_{i}}(W^{u}_{loc}(f^{n_{i}}(y)))\subset D_{r}$. Since the local unstable manifold of $f^{n_{i}}(y)$ cuts across both ${\mathcal{B}}^{1}$ and ${\mathcal{B}}^{2}$, this means that we can associate to $y\in D^{}_{\rho}\cap f^{-n_{i}}({\mathcal{B}}_{1})$ the following subsets of $D_{r}$: $D^{1}_{i}(y)=f^{-n_{i}}(W^{u}_{loc}(f^{n_{i}}(y))\cap{\mathcal{B}}^{1})\quad\text{and}\quad D^{2}_{i}(y)=f^{-n_{i}}(W^{u}_{loc}(f^{n_{i}}(y))\cap{\mathcal{B}}^{2}).$ By bounded distortion, there exists $\kappa=\kappa(f)>0$ such that $\operatorname{vol}_{D}(D^{2}_{i}(y))\geq\kappa\operatorname{vol}_{D}(D^{1}_{i}(y))\quad\text{for every $y$ and every $i$.}$ We also denote by $D^{1}_{i}$ and $D^{2}_{i}$ the (disjoint) unions of $D^{1}_{i}(y)$ and $D^{2}_{i}(y)$, respectively, over all $y\in D^{}_{\rho}\cap f^{-n_{i}}({\mathcal{B}}_{1})$. Then, the previous inequality gives $\operatorname{vol}_{D}(D^{2}_{i})\geq\kappa\operatorname{vol}_{D}(D^{1}_{i})\quad\text{for every $i$.}$ By Proposition 6.9 and Remark 6.10, there exists a sequence $(n_{i})_{i}$ of positive integers and there exists $b_{1}>0$ such that $\operatorname{vol}_{D}(D_{\rho}^{}\cap f^{-n_{i}}({\mathcal{B}}^{1}))\geq b_{1}\operatorname{vol}_{D_{\rho}}(D_{\rho}^{})\geq b_{1}(1-\varepsilon)^{2}\operatorname{vol}_{D}(D_{\rho}).$ Consequently, $\operatorname{vol}_{D}(D^{1}_{i})\geq b_{1}(1-\varepsilon)^{2}\operatorname{vol}_{D}(D_{\rho})\geq b_{1}(1-\varepsilon)^{3}\operatorname{vol}_{D}(D_{r}).$ This implies that $\operatorname{vol}_{D}(D^{2}_{i})\geq b_{2}\operatorname{vol}_{D}(D_{r})$, where the constant $b_{2}>0$ is independent of $i$ and the choice of $r$. Now, suppose the lemma is false. Then $D^{2}_{i}(y)\cap D^{}$ is empty, for every $y\in D_{\rho}^{}\cap f^{-n_{i}}({\mathcal{B}}^{1})$, that is, $D^{2}_{i}\cap D^{}=\emptyset$. It follows that $\operatorname{vol}_{D}(D_{r}^{})\leq(1-b_{2})\operatorname{vol}_{D}(D_{r})$. This contradicts the choice of $D_{r}^{}$ at the beginning of the proof, as long as we fix $\varepsilon<b_{2}$. The proof of the lemma is complete. ∎ ###### Proof of Proposition 6.4. Let $h^{cs}:D^{1}\to D^{2}$ be a $cs$-holonomy between $u$-disks $D_{1}$ and $D_{2}$. Let $D_{1}^{}\subset D_{1}$ be a positive $\operatorname{vol}_{D_{1}}$-measure subset and $D_{2}^{}=h^{cs}(D_{1}^{})$. We want to prove that $\operatorname{vol}_{D_{2}}(D_{2}^{})$ is also positive. By Lemma 6.12, there exists $\zeta\in D^{}_{1}$ and $k\geq 1$ such that (23) $\operatorname{vol}_{f^{k}(D)}\big{(}{\mathcal{W}}^{u}_{2\tau}(f^{k}(\zeta))\cap f^{k}(D^{}_{1})\cap{\mathcal{B}}^{i}\big{)}>0\quad\text{for both }i=1,2.$ Notice that for $k$ big enough, ${\mathcal{W}}^{u}_{2\tau}(f^{k}(\zeta))$ and ${\mathcal{W}}^{u}_{2\tau}(f^{k}(h^{cs}(\zeta)))$ are contained in nearby $cu$-disks. That is because the stable foliation is uniformly contracting. Then ${\mathcal{W}}^{u}_{2\tau}(h^{cs}(\zeta))\cap{\mathcal{W}}^{s}_{\rho}(\ell^{\prime})\neq\emptyset$. This implies ${\mathcal{W}}^{u}_{2\tau}(h^{cs}(\zeta))\cap\tilde{{\mathcal{B}}}_{1}\neq\emptyset$ or ${\mathcal{W}}^{u}_{2\tau}(h^{cs}(\zeta))\cap{\mathcal{B}}_{2}\neq\emptyset$. Since $\tilde{{\mathcal{B}}}_{1},{{\mathcal{B}}}_{2}$ are $cs$-blocks, whose $cs$-foliations are uniformly absolutely continuous, from (23) one gets that $\operatorname{vol}_{f^{k}(D_{2})}\big{(}{\mathcal{W}}^{u}_{2\tau}(f^{k}(h^{cs}(\zeta)))\cap f^{k}(D_{2}^{})\cap{{\mathcal{B}}^{i}}\big{)}>0$ for either $i=1$ or $i=2$. This implies that $\operatorname{vol}_{D_{2}}(D_{2}^{})>0$. Thus, the center stable foliation is absolutely continuous, as claimed. ∎ ## 7\. Robust absolute continuity Here we use the results in the previous section to give examples of open sets of diffeomorphisms with absolutely continuous center stable/unstable foliations. ### 7.1. Mañé’s example Mañé [30] constructed a $C^{1}$ open set of diffeomorphisms ${\mathcal{U}}$ such that every $f\in{\mathcal{U}}$ is partially hyperbolic (but not hyperbolic), dynamically coherent, and transitive. From Proposition 6.3 one gets that every $C^{k}$, $k>1$ diffeomorphism $f$ in some non-empty $C^{1}$ open subset ${\mathcal{U}}^{\prime}$ has absolutely continuous center stable foliation. To explain this, let us recall some main features in Mañé’s construction. One starts from a convenient linear Anosov map $A:{\mathbb{T}}^{3}\to{\mathbb{T}}^{3}$ with eigenvalues $0<\lambda_{1}<\lambda_{2}<1<\lambda_{3}$. Let $p$ be a fixed point of $A$ and $\rho>0$ be small. One deforms $A$ inside the $\rho$-neighborhood of $p$, so as to create some fixed point with stable index $1$, while keeping the diffeomorphism unchanged outside $B_{\rho}(p)$. Mañé [30] shows that this can be done in such a way that the diffeomorphism $f_{0}:{\mathbb{T}}^{3}\to{\mathbb{T}}^{3}$ thus obtained is partially hyperbolic, with splitting $E^{s}\oplus E^{c}\oplus E^{s}$ where all factors have dimension $1$, and every diffeomorphism in some $C^{1}$ neighborhood ${\mathcal{U}}$ is dynamically coherent and transitive. The presence of periodic points with both stable indices $1$ and $2$ ensures that $f_{0}$ is not Anosov. Bonatti, Viana [19] observed that every $C^{k}$, $k>1$ diffeomorphism $f\in{\mathcal{U}}$ has mostly contracting center direction. Here, as well as in the steps that follow, one may have to reduce the neighborhood ${\mathcal{U}}$. Then Bonatti, Díaz, Ures [16] showed that the unstable foliation of every $f\in{\mathcal{U}}$ is minimal. According to [19], this implies that every $C^{k}$, $k>1$ diffeomorphism $f\in{\mathcal{U}}$ admits a unique physical measure, whose basin contains Lebesgue almost every point. The non-expansion condition in Proposition 6.3 can be checked as follows. A crucial observation is that the center stable bundle $E^{c}\oplus E^{s}$ is uniformly contracting outside $B_{\rho}(p)$, for all diffeomorphisms in a neighborhood, because $f_{0}=A$ outside $B_{\rho}(p)$. Let $q$ be another fixed or periodic point of $A$ and assume $\rho$ was chosen much smaller than the distance from $p$ to the orbit of $q$. Then $q$ remains a periodic point for $f_{0}$, with stable index $2$ and stable manifold of size $\geq 5\rho$. Let $q_{f}$ denote the hyperbolic continuation of $q$ for every $f$ in a neighborhood of $f_{0}$: $q_{f}$ is a periodic point with stable index $2$ and stable manifold of size $\geq 4\rho$. The fact that $E^{c}\oplus E^{s}$ is uniformly contracting outside $B_{\rho}(p)$ also implies that $f^{n}({\mathcal{W}}^{cs}_{\rho}(x))\subset{\mathcal{W}}^{cs}_{2\rho}(f^{n}(x))$ for all $x\in{\mathbb{T}}^{3}$ and $n\geq 0$. This proves that $f$ is non- expanding along the center direction, and so we may apply Proposition 6.3 to conclude that the center stable foliation of every $f$ near $f_{0}$ is absolutely continuous. We ignore whether the center unstable foliation and the center foliation are absolutely continuous or not in this case. However, in the next section, a different construction allows us to give examples where all three invariant foliations are robustly absolutely continuous. ### 7.2. Robust absolute continuity for all invariant foliations Here we prove Theorem E and use it to deduce Theorem B. We begin with an intermediate result: ###### Proposition 7.1. Let $f_{0}:N\to N$ be a $C^{k}$, $k>1$ skew-product $f_{0}(x,\theta)=(g_{0}(x),h_{0}(x,\theta))$, where $g_{0}$ is a transitive Anosov diffeomorphism. Assume that $f_{0}$ is accessible and has some periodic center leaf in general position. Then there exists a $C^{k}$ neighborhood ${\mathcal{V}}$ of $f_{0}$ such that for every $f\in{\mathcal{V}}$, the center stable, center unstable, and center foliation are absolutely continuous. ###### Proof. Every skew-product has absolutely continuous center stable and center unstable foliation and is robustly dynamically coherent (by [28]; the center foliation of a partially hyperbolic skew-product is always plaque expansive). In particular, $f_{0}$ satisfies all the hypotheses of Theorem D. The presence of a Morse-Smale center leaf prevents $f_{0}$ from being conjugate to a rotation extension. Thus, the center direction is mostly contracting in a whole neighborhood of $f_{0}$. The assumption that $g_{0}$ is transitive also ensures that every strong unstable leaf intersects ${\mathcal{W}}^{s}(\ell)$. So, we are in a position to apply Corollary 6.5 to conclude that the center stable foliation is robustly absolutely continuous. The same reasoning applied to the inverse of $f_{0}$ gives that the center unstable foliation is also robustly absolutely continuous. From the following general fact we get that the center foliation is also robustly absolutely continuous: ###### Lemma 7.2 (Pugh, Viana, Wilkinson [39]). Let ${\mathcal{F}}^{1}$, ${\mathcal{F}}^{2}$, ${\mathcal{F}}^{3}$ be foliation in some smooth manifold $N$ such that ${\mathcal{F}}^{1}$ and ${\mathcal{F}}^{2}$ are transverse at every point and the leaves of ${\mathcal{F}}^{3}$ are coincide with the intersections of leaves of ${\mathcal{F}}^{1}$ and ${\mathcal{F}}^{2}$: for every point $x\in N$, ${\mathcal{F}}^{3}(x)={\mathcal{F}}^{1}(x)\cap{\mathcal{F}}^{2}(x)$. If ${\mathcal{F}}_{1}$ and ${\mathcal{F}}_{2}$ are absolutely continuous then so is ${\mathcal{F}}_{3}$. ###### Proof. Suppose $D_{1},D_{2}$ are two disks transverse with ${\mathcal{F}}^{3}$, and $h^{3}:D_{1}\rightarrow D_{2}$ is the holonomy map induced by ${\mathcal{F}}^{3}$. Then ${\mathcal{F}}^{1}$ and ${\mathcal{F}}^{2}$ induce two foliations $\hat{{\mathcal{F}}}^{1}_{i}$ and $\hat{{\mathcal{F}}}^{2}_{i}$ on $D_{i}$, $i=1,2$, and these two foliations absolutely continuous in $D_{i}$. Fix $l_{1}\subset D_{1}$ a leaf of $\hat{{\mathcal{F}}}^{2}_{1}$, and denote by $l_{2}=h^{3}(l_{1})$, then $l_{2}$ is a leaf of $\hat{{\mathcal{F}}}^{2}_{2}$. Since the foliations $\hat{{\mathcal{F}}}^{1}_{i}$, $i=1,2$ are absolutely continuous, one has that the disintegration of the Lebesgue measure $\operatorname{vol}_{D_{1}}$ along the foliation $\hat{{\mathcal{F}}}^{1}_{1}$ is $\operatorname{vol}_{D_{1}}=\varphi_{x}(y)d\operatorname{vol}_{\hat{{\mathcal{F}}}^{1}_{1}(x)}(y)d\operatorname{vol}_{l_{1}}(x),\quad\text{where }\varphi_{x}(y)>0$ and the disintegration of the Lebesgue measure of $D_{2}$ along the foliation $\hat{{\mathcal{F}}}^{1}_{2}$ is $\operatorname{vol}_{D_{2}}=\phi_{x}(y)d\operatorname{vol}_{\hat{{\mathcal{F}}}^{1}_{2}(x)}(y)d\operatorname{vol}_{l_{2}}(x),\quad\text{where }\phi_{x}(y)>0.$ Now for any set $\Delta_{1}\subset D_{1}$ with $\operatorname{vol}_{D_{1}}(\Delta_{1})>0$, denote its image for $h^{3}$ by $\Delta_{2}$. By the above formulas for the disintegration, there is a positive $\operatorname{vol}_{l_{1}}$ measure subset $\Gamma_{1}\subset l_{1}$ such that for any $x\in\Gamma_{1}$, one has $\operatorname{vol}_{\hat{F}^{1}_{1}(x)}(\Delta_{1}\cap\hat{F}^{1}_{1}(x))>0.$ Denote $\Gamma_{2}=h^{3}(\Gamma_{1})\subset l_{2}$. By the absolute continuity of ${\mathcal{F}}^{1}$ and ${\mathcal{F}}^{2}$, $\operatorname{vol}_{l_{2}}(\Delta_{2})>0$ and $\operatorname{vol}_{\hat{F}^{1}_{2}(x)}(\Delta_{2}\cap\hat{F}^{1}_{2}(x))>0$ for any $x\in\Gamma_{2}$. This implies $\operatorname{vol}_{D_{2}}(\Delta_{2})>0$, and so the proof is complete. ∎ This completes the proof of Proposition 7.1. ∎ To complete the proof of Theorem E it suffices to note that any skew-product $f_{0}$ with a Morse-Smale center leaf, as in the statement of the theorem, is approximated by skew-products with center leaves in general position: all that is missing is property (b) in the definition of general position, and this can be achieved by a $C^{k}$ small perturbation inside the space of skew-products. Then Theorem E follows from Proposition 7.1. Now Theorem B is deduced as follows. For any skew-product $f_{0}$ as in the statement, Let ${\mathcal{U}}$ be an open set that accumulates on $f_{0}$ as given by Theorem E: for any $f\in{\mathcal{U}}$ the center stable, center unstable, and center foliations are absolutely continuous. Then let ${\mathcal{V}}\subset{\mathcal{U}}$ be an open subset such that every $f\in{\mathcal{V}}$ is accessible ([32]). Then every $f\in{\mathcal{V}}$ has finitely many physical measures, with basins containing almost every point. The Morse-Smale behavior on the center leaf $\ell$ prevents $f$ from being conjugate to a rotation extension. Thus, we are in case (b) of Theorem D. From the fact that $\ell$ contains a unique periodic attractor we also get that the physical measure is unique (see Theorem 5.6). The same argument applies for $f^{-1}$. This finishes the proof of Theorem B. ### 7.3. Volume preserving systems Here we prove Theorem C and a pair of related results. Based on these, we also describe a, partially conjectural, scenario for absolute continuity of foliations of conservative and dissipative systems. Part (1) of Theorem C is a direct consequence of the main result of Baraviera, Bonatti [15]. Part (2) is given by the following result: ###### Lemma 7.3. For any $f\in{\mathcal{W}}_{0}$ with $\lambda^{c}(f)>0$, the center foliation and the center stable foliation are not upper leafwise absolutely continuous. ###### Proof. Fix $c\in(0,\lambda^{c}(f))$. Then, by the Birkhoff ergodic theorem, the set $\Gamma_{c,1}=\\{x\in N:\lim\frac{1}{n}\sum_{i=1}^{n}\log\\|Df^{-1}\mid{E^{c}(f^{i}(x))}\\|^{-1}\geq c\\}$ has positive volume. Then, by Proposition 3.6, there is $n_{0}\geq 1$ such that the intersection of any center leaf with $\Gamma_{c,1}$ has at most $n_{0}$ points. In particular, the intersection has zero volume inside the center leaf. So, the center foliation of $f$ is not upper leafwise absolutely continuous. Next, observe that the set $\Gamma_{c,1}$ consists of entire strong stable leaves. So, the intersection of $\Gamma_{c,1}$ with any center stable leaf consists of no more than $n_{0}$ strong stable leaves. This implies that the intersection has zero volume inside the center stable leaf. Consequently, the center stable foliation is not upper leafwise absolutely continuous. In particular, we get that the center foliation and the center stable foliation are not absolutely continuous, as claimed. ∎ Now we prove part (3) of the theorem. Let $p\in M$ be a periodic point of $g_{0}$ and $a\in M$ be a homoclinic point associated to $p$. For simplicity, we take the periodic point to be fixed. Let us begin by constructing ${\mathcal{W}}_{1}$. The first step is to approximate $f_{0}$ by some diffeomorphism $f_{1}$ such that $\lambda^{c}(g)>0$ for any $g$ in a $C^{1}$ neighborhood. This can be done by the perturbation method in [15]; the perturbation may be chosen such that $f_{1}=f_{0}$ on a neighborhood of $\\{p\\}\times S^{1}$, and we assume that this is the case in what follows. The second step is to find $f_{2}$ arbitrarily close to $f_{1}$ such that, denoting by $\ell_{p}$ and $\ell_{a}$ the center leaves associated to the continuation of $p$ and $a$, * • every strong unstable leaf of $f_{2}$ intersects $W^{s}(\ell_{p})$; * • the restriction of $f_{2}$ to $\ell_{p}$ is a Morse-Smale diffeomorphism, with a single attractor $\xi$ and a single repeller $\eta$ * • and ${\mathcal{W}}^{u}(\eta)$ and $W^{s}(\xi)$ are in general position (we call this non-strong connection). These properties remain valid in a small neighborhood of $f_{2}$. As a final step, we use [16, 27, 26] to find a diffeomorphism $f_{3}$ arbitrarily close to $f_{2}$ and such that the strong stable and the strong unstable foliations are minimal in a whole $C^{1}$ neighborhood of $f_{3}$. We take ${\mathcal{W}}_{1}$ to be such a neighborhood. By [19], for every diffeomorphism $f\in{\mathcal{W}}$ the inverse $f^{-1}$ has mostly center direction. Then, by [9], the same is true in a whole $C^{k}$ neighborhood ${\mathcal{W}}_{f}$ in the space of all (possibly dissipative). diffeomorphisms. Hence, we are in a position to apply Corollary 6.5 to conclude that the center unstable foliation is absolutely continuous for every diffeomorphism in ${\mathcal{W}}_{f}$. This finishes the proof of Theorem C. The next proposition is a variation of results in [14] where center foliations are replaced by center stable or center unstable foliations. ###### Proposition 7.4. Let $f_{0}$ be as in Theorem A, where $M$ is a surface, and let $f$ be any $C^{1}$ nearby accessible, volume preserving diffeomorphism with $\lambda^{c}(f)=0$. If either the center stable foliation or the center unstable foliation is absolutely then $f$ is smoothly conjugate to a rotation extension and the center foliation is a smooth foliation. ###### Proof. Suppose ${\mathcal{W}}^{cs}$ is absolutely continuous. Then we may apply Theorem D. In this case Lebesgue measure is a Gibbs $u$-state with zero center exponent, and so we are in the elliptic case (a) of the theorem. In particular, the center foliation is leafwise absolutely continuous. Then we can apply [14] to conclude that the center foliation is smooth and $f$ is smoothly conjugate to a rigid model. In present case, where the center fiber bundle is trivial, we get that $f$ is topologically conjugate to a rotation extension (cf. Remark 4.3). ∎ ###### Remark 7.5. Suppose $f$ is partially hyperbolic, dynamically coherent, volume preserving, and all the center exponents are negative at almost every point. Then the center stable foliation of $f$ is upper leafwise absolutely continuous. This is a fairly direct consequence of Pesin theory. Indeed, if all the Lyapunov exponents are negative then the Pesin local stable manifold of almost every point is a neighborhood of the point inside its center stable leaf. Then the absolute continuity of Pesin laminations [35] implies that the center stable foliation is upper leafwise absolutely continuous. We close with a couple of conjectures on the issue of absolute continuity. The first one deals with dissipative systems. ###### Conjecture 7.6. Let $k>1$ and ${\mathcal{C}}_{k}$ be the space of partially hyperbolic, dynamically coherent $C^{k}$ diffeomorphisms with mostly contracting center direction. Then, for an open and dense subset, * • if there is a unique physical measure then the center stable foliation is absolutely continuous; * • if there is more than one physical measure then the center stable foliation is not upper leafwise absolutely continuous. Examples of the second situation will appear in a forthcoming paper [49]. Figure 4. ###### Conjecture 7.7. Let $k>1$ and ${\mathcal{V}}_{k}$ be the space of partially hyperbolic, dynamically coherent, volume preserving $C^{k}$ diffeomorphisms whose center Lyapunov exponents are negative at almost every point. Then, for an open and dense subset, the center stable foliation is absolutely continuous. Figure 4 outlines a scenario for these issues in a relevant special case, namely near the map $f_{0}=g_{0}\times\operatorname{id}$ as in Theorem A. Accessibility is assumed throughout (but is not needed for the negative results in $\lambda^{c}\neq 0$). Generically means for open and dense in $C^{k}$ topology, $k\geq 1$. Upper leafwise absolute continuity of the center unstable is known for $\lambda^{c}>0$, as we have seen, and we have also found an open subset with (full) absolute continuity of the center unstable. ## References * [1] F. Abdenur and M. Viana. Flavors of partial hyperbolicity. Preprint www.preprint.impa.br 2009. * [2] J. F. Alves. 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1012.0514	# The entropy conjecture for diffeomorphisms away from tangencies Gang Liao1, Marcelo Viana2, Jiagang Yang3 liaogang@math.pku.edu.cn viana@impa.br yangjg@impa.br (Date: September, 2010) ###### Abstract. We prove that every $C^{1}$ diffeomorphism away from homoclinic tangencies is entropy expansive, with locally uniform expansivity constant. Consequently, such diffeomorphisms satisfy Shub’s entropy conjecture: the entropy is bounded from below by the spectral radius in homology. Moreover, they admit principal symbolic extensions, and the topological entropy and metrical entropy vary continuously with the map. In contrast, generic diffeomorphisms with persistent tangencies are not entropy expansive and have no symbolic extensions. ###### Key words and phrases: Entropy conjecture, principal symbolic extensions, upper semi-continuity of the entropy, homoclinic tangencies ###### 1991 Mathematics Subject Classification: 37G25; 37B10; 37B40; 37C20 1 School of Mathematical Sciences, Peking University, Beijing 100871, China 2 IMPA, Estrada D. Castorina 110, Jardim Botânico, 22460-320 Rio de Janeiro, Brazil. 3 Departamento de Geometria, Instituto de Matemática, Universidade Federal Fluminense, Niterói, Brazil GL is supported by CSC-China. MV and JY are partially supported by CNPq, FAPERJ, and PRONEX Dynamical Systems. ## 1\. Introduction In this paper we prove that the dynamics of any diffeomorphism away from homoclinic tangencies admits a very precise description at the topological level. Let us begin by introducing the set-up of our results. For each $r\geq 1$, let $\operatorname{Diff}^{r}(M)$ denote the space of $C^{r}$ diffeomorphisms on some compact Riemannian manifold $M$, endowed with the $C^{r}$ topology. A periodic point $p$ of $f\in\operatorname{Diff}^{r}(M)$ is _hyperbolic_ if the derivative $Df^{\kappa}(p)$, $\kappa=\operatorname{per}(p)$ has no eigenvalues with norm $1$. Then there exist $C^{r}$ curves $W^{s}(p)$ and $W^{u}(q)$ \- the _stable_ and _unstable_ manifolds of $p$ \- that intersect transversely at $p$ and satisfy $f^{n\kappa}(q)\to p\ \text{for all }q\in W^{s}(p)\quad\text{and}\quad f^{-n\kappa}(q)\to p\ \text{for all }q\in W^{u}(p).$ A point $q\in W^{s}(p)\cap W^{u}(p)$ distinct from $p$ is a _homoclinic point_ associated to $p$. The homoclinic point $q$ is _transverse_ if $T_{q}M=T_{q}W^{u}(p)+T_{q}W^{s}(p).$ We say that $f$ has a _homoclinic tangency_ if there exists a non-transverse homoclinic point associated to some hyperbolic periodic point. The set of $C^{r}$ diffeomorphisms that have some homoclinic tangency will be denoted $\operatorname{HT}^{r}$. For notational simplicity, we also write $\operatorname{Diff}(M)=\operatorname{Diff}^{1}(M)$ and $\operatorname{HT}=\operatorname{HT}^{1}$. Our main results, that we are going to state in a while, hold for diffeomorphisms in $\operatorname{Diff}(M)\setminus\overline{\operatorname{HT}}$, that we call diffeomorphisms _away from tangencies_. ### 1.1. Entropy conjecture Let $m=\dim M$ and $f_{,k}:H_{k}(M,{\mathbb{R}})\to H_{k}(M,{\mathbb{R}})$, $0\leq k\leq m$ be the action induced by $f$ on the real homology groups of $M$. Let $\operatorname{sp}(f_{})=\max_{0\leq k\leq m}\operatorname{sp}(f_{,k}),$ where $\operatorname{sp}(f_{,k})$ denotes the spectral radius of $f_{,k}$. Shub [32] has conjectured (see also Shub, Sullivan [33]) that the logarithm of $\operatorname{sp}(f_{})$ is a lower bound for the topological entropy of $f$: (1) $\log\operatorname{sp}(f_{})\,\leq\,h(f)\quad\text{for every $f\in\operatorname{Diff}(M)$.}$ We prove that the conjecture does hold for diffeomorphisms away from tangencies: ###### Theorem A. The entropy conjecture (1) holds for every $f\in\operatorname{Diff}(M)\setminus\overline{\operatorname{HT}}$. This is the best result to date on the entropy conjecture in finite differentiability. We will also comment on the behavior of diffeomorphisms with tangencies. Before getting to that, let us briefly recall the history of this problem. The entropy conjecture is known to hold for an open and dense subset of the space $\operatorname{Homeo}(M)$ of homeomorphisms, when $\dim M\neq 4$. In fact, by Palis, Pugh, Shub, Sullivan [27], the conjecture always holds for an open and dense subset of any stable connected component of $\operatorname{Homeo}(M)$. When the dimension is different from $4$ all connected components are stable, by Kirby, Siebenmann [17], and that is how one gets the previous statement. Manning [19] proved that the weaker inequality $\log\operatorname{sp}(f_{,1})\leq h(f)$ always holds for homeomorphisms in any dimension. Using Poincaré duality, one deduces the full statement of the entropy conjecture for homeomorphisms on manifolds with $\dim M\leq 3$. The conjecture is also known to hold for homeomorphisms on any infra-nilmanifold, by Marzantowicz, Misiurewicz, Przytycki [22, 20]. Weaker versions of the conjecture, where one replaces the spectral radius of $f_{}$ by other topological invariants, have been proved in great generality. Bowen [4] showed that $\log\gamma_{1}\leq h(f)$ for every homeomorphism, where $\gamma_{1}$ is the growth rate of the fundamental group. This is a strengthening of Manning’s result mentioned previously. Ivanov [15] proved that the asymptotic Nielsen number is also a lower bound for the topological entropy, for every homeomorphism. Moreover, Misiurewicz, Przytycki [23] showed that the topological entropy of every homeomorphism is bounded from below by the logarithm of the degree. For local diffeomorphisms a proof can be given using the Perron-Fröbenius operator (see Oliveira, Viana [25]). On the other hand, Shub [32] exhibited a Lipschitz (piecewise affine) counterexample to the entropy conjecture: while the spectral radius is strictly positive, the topological entropy vanishes. Thus, some smoothness is necessary for a general (not just generic) statement. A major progress was the proof, by Yomdin [38], that the entropy conjecture is true for every $C^{\infty}$ diffeomorphism. The main ingredient is a relation between topological entropy $h(f)$ and the growth rate $v(f)$ of volume under iteration by a diffeomorphism. For $C^{\infty}$ diffeomorphisms the two numbers actually coincide (that is false in finite differentiability). The entropy conjecture is a consequence, because $\log\operatorname{sp}(f_{})\leq v(f)$ for any $C^{1}$ diffeomorphism $f$. The entropy conjecture has also been established for certain classes of systems with hyperbolicity properties: Anosov diffeomorphisms and, more generally, Axiom A diffeomorphisms with no cycles (Shub, Williams [34], Ruelle, Sullivan [30]), and partially hyperbolic systems with one-dimensional center bundle (Saghin, Xia [31]). All of these systems are away from tangencies, of course. ### 1.2. Entropy expansiveness and continuity of entropy Theorem A will be deduced from the following result: ###### Theorem B. Every diffeomorphism $f\in\operatorname{Diff}(M)\setminus\overline{\operatorname{HT}}$ is entropy expansive. ###### Remark 1.1. In contrast, there is a residual subset ${\mathcal{R}}$ of $\operatorname{Diff}(M)$ such that any $f\in{\mathcal{R}}\cap\overline{\operatorname{HT}}$ is _not_ entropy expansive. This is related to results of Downarwicz, Newhouse [12]. A proof will appear in Section 3.3. The notion of entropy expansiveness will be recalled in Section 2. It was first introduced by Bowen [2], who observed that for entropy expansive maps the metric entropy function (defined in the space of invariant probabilities) $\mu\mapsto h_{\mu}(f)$ is upper semi-continuous. In particular, for such maps there always exists some measure of maximum entropy. In view of these observations, Theorem B has the following direct consequence: ###### Corollary C. For any $f\in\operatorname{Diff}(M)\setminus\overline{\operatorname{HT}}$ the entropy function $\mu\to h_{\mu}(f)$ is upper semi-continuous and, thus, there is some invariant probability $\mu$ with $h_{\mu}(f)=h(f)$. The first examples of $C^{r}$ diffeomorphisms without measures of maximum entropy were given by Misiurewicz [21], for each $1\leq r<\infty$. He also introduced a weaker condition, called asymptotic entropy expansiveness, that suffices for upper semi-continuity of the metric entropy function. In addition, Misiurewicz [21] gave examples of $C^{r}$ diffeomorphisms, $1\leq r<\infty$ where the topological entropy function $f\mapsto h(f).$ fails to be upper semi-continuous. For $C^{\infty}$ diffeomorphisms, Newhouse [24] proved that the metric entropy function is always upper semi-continuous, and Yomdin [38] proved upper semi-continuity of the topological entropy function. Newhouse’s result has been improved by Buzzi [8], who showed that every $C^{\infty}$ diffeomorphism is asymptotically entropy expansive. Yomdin’s semi-continuity result also extends to every $C^{1}$ diffeomorphism away from tangencies: ###### Theorem D. The topological entropy is upper semi-continuous on $\operatorname{Diff}(M)\setminus\overline{\operatorname{HT}}$. Closing this section, let us observe that the metric entropy function is usually not lower semi-continuous. Indeed, by the ergodic closing lemma of Mañé [18], there is a residual subset ${\mathcal{R}}_{1}$ of $\operatorname{Diff}(M)$ such that for every $f\in{\mathcal{R}}_{1}$ every ergodic invariant measure is approximated by invariant measures supported on periodic orbits. Thus, for every $f\in{\mathcal{R}}_{1}$, either $h(f)=0$ or the metric entropy function fails to be lower semi-continuous. For maps on compact surfaces without boundary, it follows from Katok [16] that the topological entropy function is lower semi-continuous on $\operatorname{Diff}^{r}(M)$, for all $r>1$. By Gromov [13], this does not extend to surfaces with boundary. ### 1.3. Symbolic extensions A _symbolic extension_ of a map $f:M\to M$ is a subshift $\sigma:Y\to Y$ over a finite alphabet, together with a continuous surjective map $\pi:Y\to M$ such that $f\circ\pi=\pi\circ\sigma$. Markov partitions for uniformly hyperbolic systems (Bowen [3]) are the classical prototype. In general, a symbolic extension may carry a lot more dynamics than the original map $f$. We call a symbolic extension _principal_ if it is minimal in this regard: $h_{\mu}(f)=h_{ext}^{\pi}(\mu)$, where $h_{ext}^{\pi}(\mu)$ is the supremum of the entropy $h_{\nu}(\sigma)$ of the shift $\sigma$ over all invariant probabilities $\nu$ such that $\pi_{}\nu=\mu$. ###### Corollary E. Any $f\in\operatorname{Diff}(M)\setminus\overline{\operatorname{HT}}$ admits a principal symbolic extension. This follows directly from Theorem B together with the observation by Boyle, Fiebig, Fiebig [6] that every asymptotically entropy expansive diffeomorphism admits a principal symbolic extension. Let us also point out that Díaz, Fisher, Pacifico, Vieitez [11, 26] have, recently, constructed principal symbolic extensions for partially hyperbolic diffeomorphism admitting an invariant splitting into one dimensional subbundles. Indeed, they prove that such maps are entropy expansive. This is in contrast with previous work of Downarwicz, Newhouse [12], based on the theory developed by Boyle, Downarwicz [5], where it is shown that nonexistence of symbolic extensions is typical on the closure of the set of area preserving diffeomorphisms with homoclinic tangencies. Also very recently, Catalan, Tahzibi [9] proved non-existence of symbolic extensions for generic symplectic diffeomorphisms outside the Anosov domain. In this setting, they also find lower bounds for the topological entropy in terms of the eigenvalues at periodic points. ## 2\. Entropy theory Here we recall some basic facts about entropy. See Bowen [2] and Walters [35] for more information. Moreover, we propose an alternative definition of entropy expansiveness, in terms of invariant measures (almost entropy expansiveness), that will be useful in the sequel. ### 2.1. Definitions and statements Throughout, $f:M\to M$ is a continuous map on a compact metric space $M$. Let $K$ be a subset of $M$. For each $\varepsilon>0$ and $n\geq 1$, we consider the following objects. The _dynamical ball_ of radius $\varepsilon>0$ and length $n$ around $x\in M$ is the set $B_{n}(x,\varepsilon)=\\{y\in M:d(f^{j}(x),f^{j}(y))\leq\varepsilon\text{ for every }0\leq j<n\\}.$ A set $E\subset M$ is _$(n,\varepsilon)$ -spanning for $K$_ if for any $x\in K$ there is $y\in E$ such that $d(f^{i}x,f^{i}y)\leq\varepsilon$ for all $0\leq i<n$. In other words, the dynamical balls $B_{n}(y,\varepsilon)$, $y\in E$ cover $K$. Let $r_{n}(K,\varepsilon)$ denote the smallest cardinality of any $(n,\varepsilon)$-spanning set, and $r(K,\varepsilon)=\limsup_{n\to+\infty}\frac{1}{n}\log r_{n}(K,\varepsilon).$ A set $F\subset K$ is _$(n,\varepsilon)$ -separated_ if for any distinct points $x$ and $y$ in $F$ there is $0\leq i<n$ such that $d(f^{i}x,f^{i}y)>\varepsilon$. That is, no element of $F$ belongs to the dynamical ball $B_{n}(y,\varepsilon)$ of another. Let $s_{n}(K,\varepsilon)$ denote the largest cardinality of any $(n,\varepsilon)$-separated set, and $s(K,\varepsilon)=\limsup_{n\to+\infty}\frac{1}{n}\log s_{n}(K,\varepsilon).$ The _topological entropy of $f$ on $K$_ is defined by $h(f,K)=\lim_{\varepsilon\to 0}s(K,\varepsilon)=\lim_{\varepsilon\to 0}r(K,\varepsilon).$ The _topological entropy of $f$_ is defined by $h(f)=h(f,M)$. Given any finite open cover $\beta$ of $M$, let (2) $h(f,\beta)=\lim_{n\to\infty}\frac{1}{n}\log\|\beta^{n}\|=\inf_{n\geq 1}\frac{1}{n}\log\|\beta^{n}\|,$ where $\beta^{n}=\\{A_{0}\cap f^{-1}A_{1}\cap\cdots\cap f^{-n+1}A_{n-1)}:A_{i}\in\beta\text{ for }0\leq i\leq n-1\\}$ and $\|\beta^{n}\|$ is the smallest cardinality of a subcover of $\beta^{n}$. The topological entropy $h(f)$ coincides with the supremum of $h(f,\beta)$ over all finite open covers. ###### Remark 2.1. If $\operatorname{diam}(\beta)<\varepsilon$ then $r_{n}(M,\varepsilon)\leq s_{n}(M,\varepsilon)\leq\|\beta^{n}\|$ for every $n$. Hence, $r(M,\varepsilon)\leq s(M,\varepsilon)\leq h(f,\beta)$. ###### Lemma 2.2 (Bowen [2]). Let $0=t_{0}<t_{1}<\cdots<t_{r-1}<t_{r}=n$ and, for $0\leq i<r$, let $E_{i}$ be a $(t_{i+1}-t_{i},\varepsilon)$-spanning set for $f^{t_{i}}(F)$. Then $r_{n}(F,2\varepsilon)\leq\prod_{0\leq i<r}\\#(E_{i}).$ Now let $\mu$ be an $f$-invariant probability measure and $\xi=\\{A_{1},\cdots,A_{k}\\}$ be a finite partition of $M$ into measurable sets. The _entropy of $\xi$ with respect to $\mu$_ is $H_{\mu}(f,\,\xi)=-\sum_{i=1}^{k}\mu(A_{i})\log\mu(A_{i}).$ The _entropy of $f$ with respect to $\xi$ and $\mu$_ is given by $h_{\mu}(f,\,\xi)=\lim_{n\to+\infty}\frac{1}{n}\log H_{\mu}(f,\,\xi^{n}).$ Finally, the _entropy of $f$ with respect to $\mu$_ is given by $h_{\mu}(f)=\sup_{\xi}h_{\mu}(f,\xi),$ where $\xi$ ranges over all finite measurable partitions of $M$. For each $x\in M$ and $\varepsilon>0$, let $B_{\infty}(x,\varepsilon)=\\{y:d(f^{n}(x),f^{n}(y))\leq\varepsilon\text{ for }n\geq 0\\}$. The map $f$ is _entropy expansive_ if there exists $\varepsilon>0$ such that $\sup_{x\in M}h(f,B_{\infty}(x,\varepsilon))=0.$ Then we say that $f$ is _$\varepsilon$ -entropy expansive_. When $f$ is a homeomorphism, one may replace $B_{\infty}(x,\varepsilon)$ by $B^{\pm}_{\infty}(x,\varepsilon)=\\{y:d(f^{n}(x),f^{n}(y))\leq\varepsilon\text{ for }n\in{\mathbb{Z}}\\}$: indeed, Bowen [2, Corollary 2.3] gives that $\sup_{x}h(f,B_{\infty}(x,\varepsilon))=\sup_{x}h(f,B^{\pm}_{\infty}(x,\varepsilon))$ for every $\varepsilon>0$. ###### Lemma 2.3. Let ${\mathcal{W}}\subset\operatorname{Homeo}(M)$ and $\varepsilon>0$ be such that every $f\in{\mathcal{W}}$ is $\varepsilon$-entropy expansive. Then the topological entropy $f\mapsto h(f)$ is upper semi-continuous on ${\mathcal{W}}$. ###### Proof. Bowen [2, Theorem 2.4] asserts that $h(f)=r(M,\varepsilon)$ if $f$ is $\varepsilon$-entropy expansive. Then, by Remark 2.1, we have $h(f)=h(f,\beta)$ for every $f\in{\mathcal{W}}$ and every open covering $\beta$ of $M$ with $\operatorname{diam}\beta<\varepsilon$. Let $\beta$ be fixed. It is easy to see from the definition (2) that the map $f\mapsto h(f,\beta)$ is upper semi-continuous (because it is an infimum of upper semi- continuous functions). This gives the claim. ∎ Let $f$ be a homeomorphism and $\mu$ be any $f$-invariant probability measure. Given $\varepsilon>0$, we say that $f$ is _$(\mu,\varepsilon)$ -entropy expansive_ if (3) $h(f,B^{\pm}_{\infty}(x,\varepsilon))=0\quad\text{for $\mu$-almost every $x\in M$.}$ We say that $f$ is _$\varepsilon$ -almost entropy expansive_ if it is $(\mu,\varepsilon)$-entropy expansive for any invariant probability measure $\mu$. It is clear that $\varepsilon$-entropy expansiveness implies $\varepsilon$-almost entropy expansiveness. The converse is important for our purposes: ###### Proposition 2.4. If $f$ is $\varepsilon$-almost entropy expansive then $f$ is $\varepsilon$-entropy expansive. This follows from a stronger result, Proposition 2.5, that we present in the next section. The notion of almost entropy expansiveness extends to non- invertible maps, with $B_{\infty}(x,\varepsilon)$ instead of $B^{\pm}_{\infty}(x,\varepsilon)$ in the definition (3). Proposition 2.5 remains true, with the same change in the hypothesis, and so Proposition 2.4 also extends to the non-invertible case. ### 2.2. Entropy expansiveness from almost entropy expansiveness Let $f$ be a homeomorphism. We denote $B_{n}^{\pm}(x,\varepsilon)=\\{z\in M:d(f^{j}(x),f^{j}(z))\leq\delta\text{ for }\|j\|<n\\}$, for each $x\in M$ and $\varepsilon>0$. Proposition 2.4 is the particular case $a=0$ of ###### Proposition 2.5. Given $a\geq 0$, if $h(f,B^{\pm}_{\infty}(x,\varepsilon))\leq a$ for $\mu$-almost every $x\in M$ and every $f$-invariant probability $\mu$, then $h(f,B_{\infty}(x,\varepsilon))\leq a$ for every $x\in M$. ###### Proof. Suppose that $h(f,B_{\infty}(x_{0},\varepsilon))>a$ for some $x_{0}\in M$. Fix constants $a_{1}$ and $a_{2}$ such that $h(f,B_{\infty}(x_{0},\varepsilon))>a_{1}>a_{2}>a$. Then, there exists $\delta>0$, arbitrarily small, and a subsequence $(m_{i})_{i}\to\infty$ such that (4) $r_{m_{i}}(B_{\infty}(x_{0},\varepsilon),\delta)>e^{a_{1}m_{i}}\quad\text{for every $i$.}$ Write $\mu_{m_{i}}=({1}/{m_{i}})\sum_{j=0}^{m_{i}-1}\delta_{f^{j}(x_{0})}$. By compactness, $(\mu_{m_{i}})_{i}$ may be taken to converge, in the weak∗ topology, to some invariant measure $\mu$. For each $n\geq 1$, denote $\Gamma_{n}=\\{x\in M:r_{m}(B^{\pm}_{\infty}(x,\varepsilon),\delta/4)<e^{a_{2}m}\text{ for any }m\geq n\\}.$ These sets form an increasing sequence and, as long as $\delta$ is sufficiently small, the hypothesis implies that $\cup_{n}\Gamma_{n}$ has full $\mu$-measure. So, we may choose an increasing sequence of compact sets $\Lambda_{n}\subset\Gamma_{n}$ such that $\mu(\cup_{n}\Lambda_{n})=1$. For each $n\geq 1$ and $y\in\Lambda_{n}$, let $E_{n}(y)$ be an $(n,\delta/4)$-spanning set for $B^{\pm}_{\infty}(y,\varepsilon)$ with $\\#E_{n}(y)<e^{a_{2}n}$. Then $U_{n}(y)=\bigcup_{z\in E_{n}(y)}B_{n}(z,\delta/2)$ is a neighborhood of the compact set $B^{\pm}_{\infty}(y,\varepsilon)$. So, we may choose $N=N_{n}(y)$ and an open neighborhood $V_{n}(y)$ of $y\in\Lambda_{n}$ such that $B^{\pm}_{N}(u,\varepsilon)\subset U_{n}(y)$ for every $u\in V_{n}(y)$. Choose $y_{1},\dots,y_{s}\in\Lambda_{n}$ such that the $V_{n}(y_{i})$, $i=1,\dots,s$ cover the compact set $\Lambda_{n}$. Then let $W_{n}=\bigcup_{1\leq i\leq s}V_{n}(y_{i})$ and $L(n)=\max\\{n,N_{n}(y_{1}),\dots,N_{n}(y_{s})\\}$. The fact that $W_{n}$ is an open neighborhood of $\Lambda_{n}$ ensures that (5) $\lim_{i\to\infty}\mu_{m_{i}}(W_{n})\geq\mu(W_{n})\geq\mu(\Lambda_{n}).$ Consider the sequence of integers $0=t_{0}<t_{1}<\cdots<t_{r}=m_{i}$ defined as follows. Let $j\geq 0$ and suppose that $t_{0},\cdots,t_{j}$ have been defined. Then, take $t_{j+1}=\left\\{\begin{array}[]{ll}t_{j}+n&\text{if $f^{t_{j}}(x_{0})\in W_{n}$ and $L(n)\leq t_{j}<m_{i}-L(n)$}\\\ t_{j}+1&\text{otherwise.}\end{array}\right.$ Write $\\{t_{0},t_{1},\cdots,t_{r}\\}$ as a disjoint union $A\cup B$, where $t_{j}\in A$ if $f^{t_{j}}(x_{0})\in W_{n}$ and $L(n)\leq t_{j}<m_{i}-L(n)$ and $t_{j}\in B$ otherwise. For $t_{j}\in A$, choose $s_{j}\in\\{1,\dots,s\\}$ such that $f^{t_{j}}(x_{0})\in V_{n}(y_{s_{j}})$. Then $f^{t_{j}}(B_{m_{i}}(x_{0},\varepsilon))\subset B_{L(n)}^{\pm}(f^{t_{j}}(x_{0}),\varepsilon)\subset U_{n}(y_{s_{j}})$ and so $f^{t_{j}}(B_{m_{i}}(x_{0},\varepsilon))$ is $(n,\delta/2)$-spanned by $E(y_{s_{j}})$. Fix any $\delta/2$-dense subset $E_{}$ of the ambient space $M$. Then $f^{t_{j}}(B_{m_{i}}(x_{0},\varepsilon))$ is $(1,\delta/2)$-spanned by $E_{}$ for any $t_{j}\in B$. So, Lemma 2.2 applies to give $r_{m_{i}}(B_{m_{i}}(x_{0},\varepsilon),\delta)\leq\prod_{t_{j}\in A}\\#E(y_{s_{j}})\cdot(\\#E_{})^{\\#B}\leq e^{a_{2}n\\#A}\cdot\kappa^{\\#B},$ where $\kappa=\\#E_{}$. The definitions also imply that $n\\#A\leq m_{i}$ and $\\#B\leq\\#\\{0\leq j<m_{i}:f^{j}(x_{0})\notin W_{n}\\}+2L(n)=(1-\mu_{m_{i}}(W_{n}))m_{i}+2L(n).$ Replacing in the previous inequality, we find that $\displaystyle r_{m_{i}}(B_{m_{i}}(x_{0},\varepsilon),\delta)$ $\displaystyle\leq e^{a_{2}m_{i}}\cdot\kappa^{(1-\mu_{m_{i}}(W_{n}))m_{i}+2L(n)}$ $\displaystyle=\exp\Big{(}m_{i}\big{(}a_{2}+(1-\mu_{m_{i}}(W_{n}))\log\kappa+\frac{2L(n)}{m_{i}}\log\kappa\big{)}\Big{)}$ Fix $n$ large enough so that $1-\mu(\Lambda_{n})<(a_{1}-a_{2})/(2\log\kappa)$. Then, using (5), take $m_{i}$ to be large enough so that $1-\mu_{m_{i}}(W_{n})$ and $2L(n)/m_{i}$ are both smaller than $(a_{1}-a_{2})/(2\log\kappa)$. Then the previous inequality yields $r_{m_{i}}(B_{\infty}(x_{0},\varepsilon),\delta)\leq r_{m_{i}}(B_{m_{i}}(x_{0},\varepsilon),\delta)<e^{a_{1}m_{i}},$ contradicting (4). This contradiction completes the proof of the proposition. ∎ ## 3\. Almost entropy expansiveness Here we prove that every diffeomorphism away from tangencies is robustly almost entropy expansive: ###### Theorem 3.1. Every diffeomorphism away from tangencies admits a $C^{1}$ neighborhood ${\mathcal{U}}$ and some constant $\varepsilon>0$ such that $h(g,B^{\pm}_{\infty}(x,\varepsilon))=0$ for every $g\in{\mathcal{U}}$, every $g$-invariant probability $\mu$, and $\mu$-almost every $x\in M$. In view of Proposition 2.4, this implies that every such diffeomorphism is robustly entropy expansive, with locally uniform expansiveness constant: ###### Corollary 3.2. Every diffeomorphism away from tangencies admits a $C^{1}$ neighborhood ${\mathcal{U}}$ and some constant $\varepsilon>0$ such that every $g\in{\mathcal{U}}$ is $\varepsilon$-entropy expansive. ### 3.1. Preparatory remarks Let $\Lambda\subset M$ be a compact set invariant under $f$. Let $T_{\Lambda}M=E^{1}\oplus\cdots\oplus E^{k}$ be a splitting of the tangent bundle over $\Lambda$ into $Df$-invariant subbundles (some of the $E^{j}$ may reduce to $\\{0\\}$). Given an integer $L\geq 1$, the splitting is called _$L$ -dominated_ if for every $i<j$, every $x\in\Lambda$, and every pair of non- zero vectors $u\in E^{i}_{x}$ and $v\in E^{j}_{x}$, one has $\frac{\\|Df_{x}^{L}(u)\\|}{\\|u\\|}<\frac{1}{2}\frac{\\|Df_{x}^{L}(v)\\|}{\\|v\\|}.$ In the sequel we focus on the case of dominated splittings $T_{\Lambda}M=E^{1}\oplus E^{2}\oplus E^{3}$ into three subbundles. Write $E^{ij}=E^{i}\oplus E^{j}$ for $i\neq j$. Given a foliation ${\mathcal{F}}$ and a point $y$ in the domain, we denote by ${\mathcal{F}}(y)$ the leaf through $y$ and by ${\mathcal{F}}(y,\rho)$ the neighborhood of radius $\rho>0$ around $y$ inside the leaf. Following Burns, Wilkinson [7] we avoid assuming dynamical coherence by using locally invariant (“fake”) foliations, a construction that goes back to Hirsch, Pugh, Shub [14]. For any $L$-dominated splitting over any invariant set of a diffeomorphism in some small neighborhood of $f$, the angles between the invariant subbundles are bounded from zero by a constant that depends only on $L$. This simple observation allows us to get the Hirsch, Pugh, Shub statement in a somewhat more global form: ###### Lemma 3.3. For any $f\in\operatorname{Diff}(M)$, $L\geq 1$, and $\zeta>0$ there is a $C^{1}$ neighborhood ${\mathcal{U}}_{f}$ of $f$ and real numbers $\rho>r_{0}>0$ with the following properties. For any $g\in{\mathcal{U}}_{f}$ let $\Lambda_{g}$ be a $g$-invariant compact set such that the tangent space over $\Lambda_{g}$ admits an $L$-dominated splitting $T_{\Lambda_{g}}M=E^{1}_{g}\oplus E^{2}_{g}\oplus E^{3}_{g}$. Then, the neighborhood $B(x,\rho)$ of every $x\in\Lambda_{g}$ admits foliations ${\mathcal{F}}^{1}_{g,x}$, ${\mathcal{F}}^{2}_{g,x}$, ${\mathcal{F}}^{3}_{g,x}$, ${\mathcal{F}}^{12}_{g,x}$, ${\mathcal{F}}^{23}_{g,x}$ such that for every $y\in B(x,r_{0})$ and $\in\\{1,2,3,12,23\\}$: 1. (1) the leaf ${\mathcal{F}}^{}_{g,x}(y)$ is $C^{1}$ and $T_{y}\big{(}{\mathcal{F}}^{}_{g,x}(y)\big{)}$ lies in a cone of width $\zeta$ about $E_{x}^{}$; 2. (2) $g({\mathcal{F}}_{g,x}^{}(y,r_{0}))\subset{\mathcal{F}}_{g,x}^{}(g(y))$ and $g^{-1}({\mathcal{F}}_{g,x}^{}(y,r_{0}))\subset{\mathcal{F}}_{g,x}^{}(g^{-1}(y))$; 3. (3) ${\mathcal{F}}_{g,x}^{1}$ and ${\mathcal{F}}_{g,x}^{2}$ subfoliate ${\mathcal{F}}_{g,x}^{12}$ and ${\mathcal{F}}_{g,x}^{2}$ and ${\mathcal{F}}_{g,x}^{3}$ subfoliate ${\mathcal{F}}_{g,x}^{23}$. For simplicity, let us drop the reference to $g$ in the notations for the invariant subbundles and foliations. Lemma 3.3 allows us to define product structures on the $r$-neighborhood of every point $x\in\Lambda_{g}$, as follows. For $y$, $z\in B(x,\rho)$, write • $[y,z]_{1,2}=a$ if $z\in{\mathcal{F}}_{x}^{12}(y)$ and ${\mathcal{F}}_{x}^{1}(y)$ intersects ${\mathcal{F}}_{x}^{2}(z)$ at $a\in B(x,\rho)$; * • $[y,z]_{12,3}=a$ if ${\mathcal{F}}_{x}^{12}(y)$ intersects ${\mathcal{F}}_{x}^{3}(z)$ at $a\in B(x,\rho)$. Analogously, one defines $[y,z]_{2,3}$ and $[y,z]_{1,23}$. By transversality (Lemma 3.3(1)), in each case the intersection point $a$ is unique when it exists. Moreover, one can find $r_{1}\in(0,r_{0}]$, independent of $g$, $\Lambda_{g}$, and $x$, such that $[y,z]_{}$ is well defined whenever $y$ and $z$ belong to $B(x,r_{1})$. Moreover, for any $y\in B(x,r_{1})$ there are points $y_{}\in{\mathcal{F}}^{}_{x}(x)$, for each $\in\\{1,3,12,23\\}$, such that (6) $[y_{3},y_{12}]_{12,3}=y=[y_{23},y_{1}]_{1,23}.$ Part (1) of Lemma 3.3 ensures (for sufficiently small $\zeta$) that the locally invariant foliations ${\mathcal{F}}^{}_{x}$ are transverse, with angles uniformly bounded from below. Thus, there exists $l>0$, independent of $g$, $\Lambda_{g}$, and $x$, such that (7) $y_{}\in{\mathcal{F}}^{}_{x}(x,lr)\quad\text{for all $\in\\{1,3,12,23\\}$ and}$ (8) $\left\\{\begin{array}[]{l}y_{1}=x\Rightarrow y=y_{23}\in{\mathcal{F}}^{23}_{x}(x,lr)\\\ y_{3}=x\Rightarrow y=y_{12}\in{\mathcal{F}}^{12}_{x}(x,lr)\end{array}\right\\}\Rightarrow y\in{\mathcal{F}}^{12}_{x}(x,lr)\cap{\mathcal{F}}^{23}_{x}(x,lr)={\mathcal{F}}^{2}_{x}(x,lr)$ for any $y\in B^{\pm}(x,r)$ with $lr<r_{1}$. Moreover, $y\in B_{\infty}(x,r)$ implies (9) $(f^{j}(y))_{}\in{\mathcal{F}}^{}_{f^{j}(x)}(f^{j}(x),lr)\quad\text{and}\quad f^{j}(y_{})=(f^{j}(y))_{}$ for all $j\in{\mathbb{Z}}$ and $\in\\{1,3,12,23\\}$ (by local invariance of the foliations). The next proposition improves on a main result of Yang [37], see also Crovisier [10], and is the key step for Theorem 3.1. The proof is given in Section 4. ###### Proposition 3.4. Let $f:M\to M$ be a diffeomorphism away from tangencies. Then there exist $\lambda_{0}>0$, $L_{0}\geq 1$, and a $C^{1}$ neighborhood ${\mathcal{U}}_{0}$ of $f$, such that, given any $g\in{\mathcal{U}}_{0}$, the support of any ergodic $g$-invariant measure $\mu$ admits an $L_{0}$-dominated splitting $T_{\operatorname{supp}\mu}M=E^{1}\oplus E^{2}\oplus E^{3}$ with $\dim(E^{2})\leq 1$ and, for $\mu$-almost every point $x$, (10) $\displaystyle\lim_{n\to\infty}$ $\displaystyle\frac{1}{n}\sum_{i=1}^{n}\log\\|Dg^{L_{0}}\mid E^{1}_{g^{-iL_{0}}(x)}\\|\leq-\lambda_{0}\quad\text{and}\quad$ $\displaystyle\lim_{n\to\infty}$ $\displaystyle\frac{1}{n}\sum_{i=1}^{n}\log\\|Dg^{-L_{0}}\mid E^{3}_{g^{iL_{0}}(x)}\\|\leq-\lambda_{0}.$ ### 3.2. Proof of Theorem 3.1 Let $\lambda_{0}$, $L_{0}$, and ${\mathcal{U}}_{0}$ be as in Proposition 3.4. Fix $\delta>0$ with $2\delta<\lambda_{0}$ and then let $\zeta>0$ and $r_{}>0$ be sufficiently small so that, for any $g\in{\mathcal{U}}_{0}$, we have (11) $e^{-\delta}\leq\frac{\\|Dg^{L_{0}}(x)u\\|}{\\|Dg^{L_{0}}(y)v\\|}\leq e^{\delta}\quad\text{and}\quad e^{-\delta}\leq\frac{\\|Dg^{-L_{0}}(x)u\\|}{\\|Dg^{-L_{0}}(y)v\\|}\leq e^{\delta}$ whenever $d(x,y)\leq r_{}$ and $\angle(u,v)\leq\zeta$ (begin by choosing some local trivialization of the tangent bundle). Let ${\mathcal{U}}_{f}$, $r_{1}$, and $l$ be as in Lemma 3.3 and the comments following it. Take ${\mathcal{U}}={\mathcal{U}}_{0}\cap{\mathcal{U}}_{f}$ and $\varepsilon=\min\\{r_{1}/l,r_{}/l\\}$. We are going to prove that the conclusion of Theorem 3.1 holds for these choices. By ergodic decomposition, it is no restriction to suppose that the measure $\mu$ is ergodic. Given $x\in M$, denote $x_{i}=g^{iL_{0}}(x)$ for each $i\in{\mathbb{Z}}$. Let $\Gamma$ be the set of points $x\in\operatorname{supp}\mu$ such that $\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\log\\|Dg^{L_{0}}\mid E^{1}_{x_{-i}}\\|\leq-\lambda_{0}\quad\text{and}\quad\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\log\\|Dg^{-L_{0}}\mid E^{3}_{x_{i}}\\|\leq-\lambda_{0}.$ Proposition 3.4 asserts that $\mu(\Gamma)=1$. Take $x\in\Gamma$ and $y\in B(x,\varepsilon)$, and then let $y_{}\in{\mathcal{F}}_{x}^{}$, $\in\\{1,3,12,23\\}$ be as in (6). We claim that (12) $y_{1}=x=y_{3}\quad\text{for every }y\in B^{\pm}_{\infty}(x,\varepsilon).$ If $E^{3}=\\{0\\}$ the leaf ${\mathcal{F}}^{3}_{x}(x)$ reduces to $\\{x\\}$ and there is nothing to prove. So, let us assume that $E^{3}$ is non-trivial. ###### Lemma 3.5 (Pliss [29]). Given $a_{}\leq c_{2}<c_{1}$ there exists $\theta=(c_{1}-c_{2})/(c_{1}-a_{})$ such that, given any real numbers $a_{1},\cdots,a_{N}$ with $\sum_{i=1}^{N}a_{i}\leq c_{2}N\quad\text{and}\quad a_{i}\geq a_{}\text{ for every $i$,}$ there exist $l>N\theta$ and $1\leq n_{1}<\cdots<n_{l}\leq N$ such that $\sum_{i=n+1}^{n_{j}}a_{i}\leq c_{1}(n_{j}-n)\quad\text{for all}\quad 0\leq n<n_{j}\quad\text{and}\quad j=1,\cdots,l.$ Take $a_{}=\min\\{\log\\|Dg^{-L_{0}}(x)\\|:g\in{\mathcal{U}}\text{ and }x\in M\\}$ and note that $a_{}\leq-\lambda_{0}$. Let $-\lambda_{0}<c_{2}<c_{1}=-\lambda_{0}+\delta$. Applying Lemma 3.5 to $a_{i}=\log\\|Dg^{-L_{0}}\mid E^{3}_{x_{i}}\\|$ and large values of $N$, we find an infinite sequence $1\leq n_{1}<n_{2}<\cdots<n_{j}<\cdots$ such that $\sum_{t=n+1}^{n_{j}}\log\\|Dg^{-L_{0}}\mid E^{3}_{x_{i}}\\|\leq(-\lambda_{0}+\delta)(n_{j}-n)\quad\text{for every $0\leq n<n_{j}$.}$ By Lemma 3.3, the relation (11), and our choice of $\varepsilon$, $e^{-\delta}\leq\frac{\\|Dg^{-L_{0}}\mid T_{z}{\mathcal{F}}_{x_{i}}^{3}(x_{i})\\|}{\\|Dg^{-L_{0}}\mid T_{x}{\mathcal{F}}_{x_{i}}^{3}(x_{i})}\\|\leq e^{\delta}\quad\text{for every $z\in{\mathcal{F}}^{1}_{x}(x_{i},l\varepsilon)$ and $i\in{\mathbb{Z}}$.}$ From these two relations one gets that $g^{(n-n_{j})L_{0}}({\mathcal{F}}^{3}_{x_{n_{j}}}(x_{n_{j}},l\varepsilon))\subset{\mathcal{F}}^{3}_{x_{n}}(x_{n},e^{(n_{j}-n)(-\lambda_{0}+2\delta)}l\varepsilon).$ for every $0\leq n<n_{j}$ and, in particular, (13) $g^{-n_{j}L_{0}}({\mathcal{F}}^{3}_{x_{n_{j}}}(x_{n_{j}},l\varepsilon))\subset{\mathcal{F}}^{3}_{x}(x,e^{n_{j}(-\lambda_{0}+2\delta)}l\varepsilon).$ Let $y\in B^{\pm}_{\infty}(x,\varepsilon)$. By (9) and our choice of $\varepsilon$, the point $g^{iL_{0}}(y_{3})=(g^{iL_{0}}(y))_{3}$ belongs to ${\mathcal{F}}^{3}(x_{i},l\varepsilon)$ for every $i$. In particular, $y_{3}$ belongs to the intersection of all $g^{-n_{j}L_{0}}({\mathcal{F}}^{3}_{x_{n_{j}}}(x_{n_{j}},l\varepsilon))$ over all $j$. By (13), this intersection reduces to $\\{x\\}$. So, $y_{3}=x$ as claimed in (12). The proof that $y_{1}=x$ is entirely analogous, and so the proof of the claim is complete. Together with the relations (8) and (9), this gives that $g^{j}(B^{\pm}_{\infty}(x,\varepsilon))\subset{\mathcal{F}}^{2}_{g^{j}(x)}(g^{j}(x),r_{1})\quad\text{for any $j\in{\mathbb{Z}}$.}$ Observe that the ${\mathcal{F}}^{2}_{g^{j}(x)}(g^{j}(x),r_{1})$ are curves length bounded by some uniform constant $C$ if $\dim E^{2}=1$, and they reduce to points if $\dim E^{2}=0$. In the first case one can easily see that $r_{n}(B^{\pm}_{\infty}(x,\varepsilon),\beta)\leq Cn/\beta$ for every $n\geq 1$ and $\beta>0$, whereas, in the second case $r_{n}(B^{\pm}_{\infty}(x,\varepsilon),\beta)=1$. So, in either case, $r(B^{\pm}_{\infty}(x,\varepsilon),\beta)=0$ for every $\beta>0$. In this way, we have reduced the proof of Theorem 3.1 to proving Proposition 3.4. ### 3.3. Proof of the main results We are in a position to deduce all our main results. As mentioned before, Corollary E follows from Theorem B and a result in [6]. Theorem D is a direct consequence of Lemma 2.3 and Corollary 3.2. Corollary C follows immediately from Theorem B, as we also observed before. Theorem B is a corollary of Proposition 2.4 and Corollary 3.2. Finally, to prove Theorem A one can argue as follows. Given any $f\in\operatorname{Diff}(M)\setminus\overline{\operatorname{HT}}$, let $(f_{n})_{n}$ be a sequence of $C^{\infty}$ diffeomorphisms converging to $f$ in the $C^{1}$ topology. We may assume that every $f_{n}$ belongs to the isotopy class of $f$, so that $\operatorname{sp}(f_{n})=\operatorname{sp}(f)$. Then, by upper semi-continuity of the topological entropy (Theorem D) and the main result in Yomdin [38], $h(f)\geq\limsup_{n\to\infty}h(f_{n})\geq\limsup_{n\to\infty}\log\operatorname{sp}((f_{n})_{})=\log\operatorname{sp}(f_{}).$ Therefore, $f$ satisfies the entropy conjecture, as stated. This completes the proof. Closing this section, we prove Remark 1.1. If $\overline{\operatorname{HT}}$ has empty interior (in the $C^{1}$ topology) then we may take ${\mathcal{R}}=\operatorname{Diff}(M)\setminus\overline{\operatorname{HT}}$, and there is nothing to prove. From now on, assume that $\operatorname{int}(\overline{\operatorname{HT}})$ is non-empty. For each $k\geq 1$, define ${\mathcal{R}}_{k}$ to be the set of diffeomorphisms which either are away from tangencies, or admit a hyperbolic set of the form (14) $\Lambda\cup f(\Lambda)\cup\cdots\cup f^{m-1}(\Lambda)$ for some $m\geq 1$, with $f^{m}(\Lambda)=\Lambda$ and $\operatorname{diam}(f^{j}(\Lambda))<1/k$ for every $j$. Since hyperbolic sets are stable under small perturbations of the diffeomorphism, and the diameter remains essentially unchanged, ${\mathcal{R}}_{k}$ is a $C^{1}$ open set. Moreover, ${\mathcal{R}}_{k}$ is $C^{1}$ dense in $\operatorname{Diff}(M)$. Indeed, consider any $g\in\operatorname{Diff}(M)$. If $g$ is away from tangencies then, by definition, it belongs to ${\mathcal{R}}_{k}$. So, we may suppose that $g\in\overline{\operatorname{HT}}$. It follows from homoclinic bifurcation theory (see, for instance, [28, Chapter 6]) that, given any $\varepsilon>0$, there exist diffeomorphisms $f$ arbitrarily close to $g$ such that $f$ admits a hyperbolic set of the form (14) with $\max_{j}\operatorname{diam}(f^{j}(\Lambda))<\varepsilon$. This proves that ${\mathcal{R}}_{k}$ is indeed dense, for every $n$. Then ${\mathcal{R}}=\cap_{k}{\mathcal{R}}_{k}$. ${\mathcal{R}}$ is a $C^{1}$ generic subset. One can easily verify that each diffeomorphism $f\in{\mathcal{R}}\cap\overline{\operatorname{HT}}$ has a sequence of periodic horseshoes with periodic diameters converging to $0$. This implies that $f$ is not entropy expansive, as claimed. ## 4\. Proof of Proposition 3.4 Let $f:M\to M$ be any diffeomorphism away from tangencies. We denote by $\tau(p,f)$ the smallest period of a periodic point $p$. The logarithms of the norms of eigenvalues of $Df^{\tau(p,f)}(p)$ are called _exponents_ of $f$ at the periodic point $p$. ###### Proposition 4.1 (Wen [36]). There are constants $\lambda_{1}$, $\gamma_{1}>0$, $L_{1}\geq 1$, and a neighborhood ${\mathcal{U}}_{1}$ of $f$ such that, for any periodic point $p$ of any diffeomorphism $g\in{\mathcal{U}}_{1}$, 1. (1) there is at most one exponent in $[-\gamma_{1},\gamma_{1}]$; if such an exponent does exist, the corresponding eigenvalue is real and of multiplicity $1$; 2. (2) there is an $L_{1}$-dominated splitting $T_{\operatorname{Orb}(p,g)}M=E^{cs}\oplus E^{c}\oplus E^{cu}$ over the orbit of $p$, where $E^{cs}$, $E^{c}$, $E^{cu}$ correspond to the sums of the eigenspaces of $Dg_{p}^{\tau(p,g)}$ whose exponents fall in $(-\infty,-\gamma_{1})$ and $[-\gamma_{1},\gamma_{1}]$ and $(\gamma_{1},+\infty)$. 3. (3) if $\tau(p,g)\geq L_{1}$, then $\displaystyle\frac{1}{[\tau(p,g)/L_{1}]}$ $\displaystyle\sum_{i=0}^{[\tau(p,g)/L_{1}]-1}\log\\|Dg^{L_{1}}\mid E^{cs}_{g^{iL_{1}}(p)}\\|<-\lambda_{1}\quad\text{and}\quad$ $\displaystyle\frac{1}{[\tau(p,g)/L_{1}]}$ $\displaystyle\sum_{i=0}^{[\tau(p,g)/L_{1}]-1}\log\\|Dg^{-L_{1}}\mid E^{cu}_{g^{-iL_{1}}(p)}\\|<-\lambda_{1}.$ Take $\lambda_{1}$, $\gamma_{1}$, $L_{1}$, and the neighborhood ${\mathcal{U}}_{1}$ to be fixed once and for all. Moreover, denote $K_{1}=\max\\{\|\log\\|Dg^{m}(x)\\|\,\|:g\in{\mathcal{U}}_{1}\text{ and }x\in M\text{ and }\|m\|\leq L_{1}\\}$. Let $g\in{\mathcal{U}}_{1}$ and $\mu$ be any ergodic $g$-invariant probability measure. We are going to use Mañé’s ergodic closing lemma: ###### Proposition 4.2 (Mañé [18]). Let $\mu$ be an ergodic measure of a diffeomorphism $g$. Then there exist diffeomorphisms $g_{n}$, $n\geq 1$ and probability measures $\mu_{n}$, $n\geq 1$, where each $\mu_{n}$ is $g_{n}$-invariant and supported on a periodic orbit $\operatorname{Orb}(p_{n},g_{n})$, such that $(g_{n})_{n}\to g$ in the $C^{1}$ topology and $(\mu_{n})_{n}\to\mu$ in the weak∗ topology. Of course, we may assume that $g_{n}\in{\mathcal{U}}_{1}$ for all $n$. Then, by Proposition 4.1, the orbit of each $p_{n}$ admits an $L_{1}$-dominated splitting $T_{\operatorname{Orb}(p_{n},g_{n})}M=E^{1}_{n}\oplus E^{2}_{n}\oplus E^{3}_{n}$ such that $\dim(E^{2}_{n})\leq 1$. Restricting to a subsequence if necessary, we may assume that the dimensions of the subbundles $E^{i}_{n}$ are independent of $n$. The fact that $(\mu_{n})_{n}$ converges to $\mu$ in the weak∗ topology implies that any Hausdorff limit of the sequence $(\operatorname{Orb}(p_{n},g_{n}))_{n}$ contains the support of $\mu$. It follows, that the support admits an $L_{1}$-dominated splitting $T_{\operatorname{supp}\mu}M=E^{1}\oplus E^{2}\oplus E^{3}$ with $\dim(E^{2})\leq 1$ (see remark at the end of page 288 in [1]). This gives the first claim in Proposition 3.4. For the proof of (10) it is convenient to distinguish two cases. ### 4.1. Measures with large support Take $\lambda_{0}\in(0,\lambda_{1})$ and ${\mathcal{U}}_{0}={\mathcal{U}}_{1}$ and $L_{0}$ to be an appropriately large multiple of $L_{1}$ (to be chosen along the way). We are going to prove that (10) holds for every ergodic invariant probability measure $\mu$ whose support contains at least $L_{1}$ points. Let $(g_{n})_{n}$ and $(\mu_{n})_{n}$ be as in the ergodic closing lemma. The assumption $\\#\operatorname{supp}\mu\geq L_{1}$ implies that $\tau(p_{n},g_{n})\geq L_{1}$ for arbitrarily large $n$. Then, restricting to a subsequence if necessary, we may assume that $\tau(p_{n},g_{n})\geq L_{1}$ for every $n$. Thus, we are in a position to use part (3) of Proposition 4.1. ###### Lemma 4.3. There exists $\theta_{0}>0$, and for any $n\geq 1$ there exists $\Lambda_{n}\subset\operatorname{Orb}(p_{n},g_{n})$, such that $\mu_{n}(\Lambda_{n})\geq\theta_{0}$ and $\frac{1}{k}\sum_{i=1}^{k}\log\\|Dg^{-L_{1}}\mid E^{3,n}_{g_{n}^{iL_{1}}(q)}\\|\leq-\lambda_{0}\quad\text{for every $q\in\Lambda_{n}$ and $k\geq 1$.}$ ###### Proof. We are going to apply Lemma 3.5 to $a_{i}=\log\\|Dg^{-L_{1}}\mid E^{3,n}_{g^{(i-1)L_{1}}(p_{n})}\\|$ for $i=1,\dots,N$, where $N\geq 1$ is some large integer (precise conditions are stated along the way). Take $a_{}=-K_{1}$ and $c_{2}=-\lambda_{1}$ and $c_{1}=-\lambda_{0}$ and $\theta=(\lambda_{1}-\lambda_{0})/(K_{1}-\lambda_{0})$. The assumption of the lemma is a direct consequence of part (3) of Proposition 4.1, as long as we choose $N$ to be a multiple $[\tau(p_{n},g)/L_{1}]$. The conclusion of the lemma yields $1\leq n_{1}<\cdots<n_{l}\leq N$ with $l>\theta N$ such that, for every $j=1,\dots,l$, $\sum_{i=m}^{n_{j}-1}\log\\|Dg^{-L_{1}}\mid E^{3,n}_{g_{n}^{-iL_{1}}(q)}\\|\leq-(n_{j}-m)\lambda_{0}\quad\text{for all }0\leq m<n_{j}.$ Denoting $q_{n,j}=g^{-n_{j}L_{1}}(p_{n})$, this may be rewritten as (15) $\sum_{i=1}^{k}\log\\|Dg^{-L_{1}}\mid E^{3,n}_{g_{n}^{iL_{1}}(q_{n,j})}\\|\leq-k\lambda_{0}\quad\text{for all }1\leq k\leq n_{j}.$ Assume that $n_{j}\geq\tau(p_{n},g)$. Observing that $g^{\tau(p_{n},g)L_{1}}(q_{n,j})=q_{n,j}$, one easily deduces that the inequality (15) holds for every $1\leq k<\infty$. This means that the conclusion of the lemma holds for every point $q$ in $\Lambda_{n}=\\{g^{-n_{j}L_{1}}(p_{n}):\tau(p_{n},g)\leq n_{j}<N\\}.$ Observe that $\\#\\{j:\tau(p_{n},g)\leq n_{j}<N\\}>\theta N-\tau(p_{n},g)$, but different values of $n_{j}$ may yield the same point in $\Lambda_{n}$. Take $N$ to be some large multiple $\kappa\tau(p_{n},g)$ of the period. Then $N$ is also a multiple of the smallest period $\tau(p_{n},g^{L_{1}})=\tau(p_{n},g)/\gcd(L_{1},\tau(p_{n},g))$, of $p_{n}$ relative to the iterate $g^{L_{1}}$. Hence, $\displaystyle\\#\Lambda_{n}$ $\displaystyle\geq\frac{\theta N-\tau(p_{n},g)}{N/\tau(p_{n},g^{L_{1}})}=\frac{\theta\kappa-1}{\kappa\gcd(L_{1},\tau(p_{n},g))}\,\tau(p_{n},g)$ $\displaystyle\geq\frac{\theta\kappa-1}{\kappa L_{1}}\,\tau(p_{n},g)\geq\frac{\theta}{2L_{1}}\,\tau(p_{n},g),$ as long as $\kappa$ is large enough. Then $\mu_{n}(\Lambda_{n})={\\#\Lambda_{n}}/{\tau(p_{n},g)}\geq\theta/(2L_{1})$. The proof of the lemma is complete. ∎ Let us proceed with the proof of (10) in the case $\operatorname{supp}\mu\geq L_{1}$. Restricting to a subsequence if necessary, we may assume that $(\Lambda_{n})_{n}$ converges to some compact set $\Lambda$ in the Hausdorff topology. Since $(\mu_{n})_{n}$ converges to $\mu$ in the weak∗ topology, we have that $\mu(\Lambda)\geq\theta_{0}$. Moreover, (16) $\frac{1}{k}\sum_{i=1}^{k}\log\\|Dg^{-L_{1}}\mid E^{3}_{g^{iL_{1}}(y)}\\|\leq-\lambda_{0}\quad\text{for every }k\geq 1\text{ and }y\in\Lambda.$ By ergodicity, for $\mu$-almost every $x$, there exists $n(x)\geq 1$ such that $g^{n(x)}(x)\in\Lambda$. Take $L_{0}=\kappa L_{1}$ for some large $\kappa\geq 1$ and denote $j_{0}=[n(x)/L_{0}]$. Clearly (17) $\sum_{j=1}^{j_{0}}\log\\|Dg^{-L_{0}}\mid E^{3}_{g^{jL_{0}}(x)}\\|\leq j_{0}\kappa K_{1}.$ Let $j_{1}=[(n(x)-j_{0}L_{0})/L_{1}]$ and $l_{1}=n(x)-j_{0}L_{0}-j_{1}L_{1}$. By construction, $j_{1}\in[0,\kappa)$ and $l_{1}\in[0,L_{1})$. Let us write $g^{-L_{0}}=g^{-l_{1}}\circ\big{(}g^{-L_{1}}\big{)}^{\kappa}\circ g^{l_{1}}$. Then, for every $j>j_{0}$, the expression $\log\\|Dg^{-L_{0}}\mid E^{3}_{g^{jL_{0}}(x)}\\|$ is bounded by (18) $\displaystyle\sum_{i=1}^{\kappa}\log\\|$ $\displaystyle Dg^{-L_{1}}\mid E^{3}_{g^{(j-1)L_{0}+l_{1}+iL_{1}}(x)}\\|+2K_{1}$ $\displaystyle=\sum_{i=(j-1-j_{0})\kappa+(1-j_{1})}^{i=(j-1-j_{0})\kappa+(\kappa- j_{1})}\log\\|Dg^{-L_{1}}\mid E^{3}_{g^{iL_{1}}(y)}\\|+2K_{1},$ where $y=g^{n(x)}(x)$. Adding (17) to the sum of (18) over $j=j_{0}+1,\dots,n$, we find that $\sum_{j=1}^{n}\log\\|Dg^{-L_{2}}\mid E^{3}_{g^{jL_{2}}(x)}\\|$ is bounded by $\displaystyle j_{0}\kappa K_{1}+$ $\displaystyle\sum_{i=(1-j_{1})}^{(n-j_{0})\kappa+(\kappa- j_{1})}\log\\|Dg^{-L_{1}}\mid E^{3}_{g^{iL_{1}}(y)}\\|+2K_{1}n$ $\displaystyle\leq\big{(}j_{0}\kappa+j_{1})K_{1}+\sum_{i=1}^{(n-j_{0})\kappa- j_{1}}\log\\|Dg^{-L_{1}}\mid E^{3}_{g^{iL_{1}}(y)}\\|+2K_{1}n.$ Consequently, $\displaystyle\limsup_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}\log\\|$ $\displaystyle Dg^{-L_{2}}\mid E^{3}_{g^{jL_{2}}(x)}\\|$ $\displaystyle\leq\kappa\limsup_{k\to\infty}\frac{1}{k}\sum_{i=1}^{k}\log\\|Dg^{-L_{1}}\mid E^{3}_{g^{iL_{1}}(y)}\\|+2K_{1}.$ According to (16), the right hand side is bounded by $-\kappa\lambda_{0}+2K_{1}\leq-\lambda_{0}$, as long as we choose $\kappa$ sufficiently large. This completes the proof of (10) in this case. ### 4.2. Measures with small support Finally, we extend the claims in (10) to ergodic measures supported on periodic orbits with period smaller than $L_{1}$. We need slightly more precise choices of $\lambda_{0}$, $L_{0}$, and ${\mathcal{U}}_{0}$, than in the previous section. These are made precise along the way. Let $\operatorname{Per}(f,L_{1})$ be the (compact) set of periodic points $p$ of $f$ such that $\tau(p,f)<L_{1}$. ###### Lemma 4.4. There is a positive integer $m>0$, such that for any $p\in\operatorname{Per}(f,L_{1})$ there exist $m_{\pm}(p)\in\\{1,\dots,m\\}$ satisfying $\log\\|Df^{m_{+}(p)\tau(p,f)}\mid E^{1}_{p}\\|<0\quad\text{and}\quad\log\\|Df^{-m_{-}(p)\tau(p,f)}\mid E^{3}_{p}\\|<0.$ ###### Proof. We explain how to find $m_{+}$ satisfying the first claim; the argument for the second claim is analogous. Suppose that for every $m\geq 1$ there is $p_{m}\in\operatorname{Per}(f,L_{1})$ such that $\log\\|Df^{n\tau(p_{m},f)}\mid E^{1}_{p_{m}}\\|\geq 0$ for all $1\leq n\leq m$. Restricting to a subsequence if necessary, we may suppose that the $L_{1}$-dominated splittings $T_{\operatorname{Orb}(p_{m},f)}M=E_{m}^{1}\oplus E_{m}^{2}\oplus E_{m}^{3}$ are such that the dimensions of the subbundles $E^{j}_{m}$ are independent of $m$. Analogously, we may suppose that the periods $\tau(p_{m},f)$ are independent of $m$ and $(p_{m})_{n}$ converges to some $p\in M$. Then $p$ is periodic, with $\tau(p,f)=\tau(p_{m},f)$, and there is an $L_{1}$-dominated splitting $T_{\operatorname{Orb}(p,f)}M=E^{1}\oplus E^{2}\oplus E^{3}$ with $\dim E^{j}=\dim E_{m}^{j}$. On the one hand, by continuity, (19) $\log\\|Df^{nL_{1}}\mid E^{1}_{p}\\|\geq 0\quad\text{for any $n\geq 1$.}$ On the other hand, all the exponents of $Df^{\tau(p_{m},f)}\mid E^{1}_{p_{m}}$ are bounded above by $-\gamma_{1}$ and so the same is true for the exponents of $Df^{\tau(p,f)}\mid E^{1}_{p}$. It follows that $\lim_{n\to\infty}\log\\|Df^{n\tau(p,f)}\mid E^{1}_{p}\\|=-\infty,$ which contradicts (19). This contradiction proves the claim. ∎ Lemma 4.4 implies that if $L_{0}\geq 1$ is chosen to be a multiple of $m!L_{1}!$ then $\log\\|Df^{L_{0}}\mid E^{1}_{x}\\|<0\quad\text{and}\quad\log\\|Df^{-L_{0}}\mid E^{3}_{x}\\|<0$ for every $x\in\operatorname{Per}(f,L_{1})$. Define $\lambda_{}=-\max\\{\log\\|Df^{L_{0}}\mid E^{1}_{x}\\|,\log\\|Df^{-L_{0}}\mid E^{3}_{x}\\|:p\in\operatorname{Per}(f,L_{1})\\}.$ Notice that $\lambda_{}>0$, since $\operatorname{Per}(f,L_{1})$ is compact. Moreover, by definition (20) $\log\\|Df^{L_{0}}\mid E^{1}_{x}\\|\leq-\lambda_{}\quad\text{and}\quad\log\\|Df^{-L_{0}}\mid E^{3}_{x}\\|\leq-\lambda_{}$ for all $x\in\operatorname{Per}(f,L_{1})$. Clearly, the map $g\mapsto\operatorname{Per}(g,L_{1})$ is upper semi-continuous: for any neighborhood $U_{0}$ of $\operatorname{Per}(f,L_{1})$, we have $\operatorname{Per}(g,L_{1})\subset U_{0}$ for every $g$ in a neighborhood of $f$. Reducing ${\mathcal{U}}_{0}$ if necessary, we may assume that this holds for every $g\in{\mathcal{U}}_{0}$. Choose $\lambda_{0}\in(0,\lambda_{})$. Taking some small $\delta>0$ and shrinking ${\mathcal{U}}_{0}$ and $U_{0}$ if necessary, * (a) for any $g\in{\mathcal{U}}_{0}$ and $x$, $y\in M$ with $d(x,y)<\delta$, we have $\displaystyle\|\log\\|Df^{L_{0}}\mid E^{1}_{x}\\|$ $\displaystyle-\log\\|Dg^{L_{0}}\mid E^{1}_{y}\\|\|<\lambda_{}-\lambda_{0}\quad\text{and}\quad$ $\displaystyle\|\log\\|Df^{-L_{0}}\mid E^{3}_{x}\\|$ $\displaystyle-\log\\|Dg^{-L_{0}}\mid E^{3}_{y}\\|\|<\lambda_{}-\lambda_{0}.$ * (b) for any $g\in{\mathcal{U}}_{0}$ and $y\in U_{0}$, there exists $x\in\operatorname{Per}(f,L_{1})$ such that $d(f^{jL_{0}}(x),g^{jL_{0}}(y))<\delta\quad\text{for all $\|j\|\leq L_{1}!$.}$ Fix $g\in{\mathcal{U}}$ and $q\in\operatorname{Per}(g,L_{1})\subset U_{0}$. By (b), there exists $p\in\operatorname{Per}(f,L_{1})$ such that $d(f^{jL_{0}}(p),g^{jL_{0}}(q))<\varepsilon\quad\text{whenever $\|j\|\leq L_{1}!$.}$ The periods $\tau(p,f)$ and $\tau(q,g)$ need not be the same. Combining (a)-(b) with (20), we get that (21) $\frac{1}{n}\sum_{i=1}^{n}\log\\|Df^{L_{0}}\mid E^{1}_{f^{-iL_{0}}(q)}\\|\leq-\lambda_{0}.$ for any $1\leq n\leq L_{1}!$. Since $\tau(q,g)<L_{1}!$, it follows that (21) holds for every $n\geq 1$. The proof of the claim about $\log\\|Df^{L_{0}}\mid E^{3}\\|$ is analogous. This finishes the proof of Proposition 3.4. ## References * [1] C. Bonatti, L. J. Díaz, and M. Viana. Dynamics beyond uniform hyperbolicity, volume 102 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, 2005. * [2] R. Bowen. Entropy expansive maps. Trans. Am. Math. Soc., 164:323–331, 1972. * [3] R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms, volume 470 of Lect. Notes in Math. Springer Verlag, 1975. * [4] R. Bowen. 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Topology, 14:329–338, 1975. * [35] P. Walters. An introduction to ergodic theory. Springer Verlag, 1982. * [36] L. Wen. Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles. Bull. Braz. Math. Soc., 35:419–452, 2004. * [37] J. Yang. $C^{1}$ dynamics far from tangencies. PhD thesis, IMPA¡ Rio de Janeiro. www.preprint.impa.br. * [38] Y. Yomdin. Volume growth and entropy. Israel J. Math., 57:285–300, 1987.	arxiv-papers	2010-12-02T17:33:30	2024-09-04T02:49:15.425487	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Liao Gang, Marcelo Viana, Jiagang Yang", "submitter": "Jiagang Yang", "url": "https://arxiv.org/abs/1012.0514" }
1012.0761	11institutetext: Max–Planck–Institut für Sonnensystemforschung, 37191 Katlenburg-Lindau, Germany 22institutetext: Geophysical Institute, University of Alaska, Fairbanks, AK 99775, USA 22email: ao@how.gi.alaska.edu # About the relative importance of compressional heating and current dissipation for the formation of coronal X-ray Bright Points S. Javadi , J. Büchner A. Otto 11 javadi@mps.mpg.de 1122 J.C. Santos 11 ###### Abstract Context. The solar corona is heated to high temperatures of the order of $10^{6}K$. The coronal energy budget and specifically possible mechanisms of coronal heating (wave, DC-electric fields, ..) are poorly understood. This is particularly true as far as the formation of X-ray bright points (BPs) is concerned. Aims. Investigation of the energy budget with emphasis on the relative role and contribution of adiabatic compression versus current dissipation to the formation of coronal BPs. Methods. Three-dimensional resistive MHD simulation starts with the extrapolation of the observed magnetic field from SOHO/MDI magnetograms, which are associated with a BP observed on 19 December 2006 by Hinode. The initial radially non-uniform plasma density and temperature distribution is in accordance with an equilibrium model of chromosphere and corona. The plasma motion is included in the model as a source of energy for coronal heating. Results. Investigation of the energy conversion due to Lorentz force, pressure gradient force and Ohmic current dissipation for this bright point shows the minor effect of Joule heating in comparison to the work done by pressure gradient force in increasing the thermal energy by adiabatic compression. Especially at the time when the temperature enhancement above the bright point starts to form, compressional effects are quite dominant over the direct Joule heating. Conclusions. Choosing non-realistic high resistivity in compressible MHD models for simulation of solar corona can lead to unphysical consequences for the energy balance analysis, especially when local thermal energy enhancements are being considered. ###### Key Words.: Sun: atmosphere – Sun: magnetic topology – Magnetohydrodynamics (MHD) – Methods: numerical – Sun: corona ††offprints: S.Javadi ## 1 Introduction The mechanisms of coronal heating are not well understood. A particular object for studying heating processes are coronal bright points (further abbreviated BPs). Due to the increasing accuracy of observations our knowledge about BP has greatly advanced from the time of their discovery in soft X-ray images Vaiana et al. (1970). According to X-ray and EUV observations the linear size of BPs is on average about 30-40 arcsec with, typically, an embedded bright core of about 5-10 arcsec Madjarska et al. (2003). The average lifetime of X-ray BPs is about 8 hours Golub et al. (1974) and 20 hours for EUV BPs Zhang et al. (2001). For a long time it has been known that BPs are associated with small bipolar magnetic features in the photosphere Krieger et al. (1971); Brown et al. (2001). About one third of BPs lie over emerging regions of magnetic flux, while the rest of them lie above moving magnetic features. This was a base for the ”cancelling magnetic feature” (CMF) model Priest et al. (1994). Lifetime and energy release of BPs are known to be closely related to the different phases of the motion of this photospheric magnetic feature Brown et al. (2001). First theories were mainly addressing the topology of the magnetic field below BPs by e.g., Parnell et al. (1994); Longcope, (1998). Using higher resolution and cadence observations of BP’s intensity and taking into account a more comprehensive patterns of motion in particular in regions with highly divergent magnetic field, (Brown et al., 2001) could associate different patterns of motion of the solar photospheric magnetic features to different stages of a BP evolution. The plasma motion in the regions of strong magnetic field was first included by Büchner (2004a,b) in their three- dimensional numerical resistive MHD model using their 3D numerical simulation model, LINMOD3d. The latter considers dissipation of currents generated by plasma motion in photosphere on time scales longer than an Alfvén time as a one of the heating processes in the solar corona Parker, (1972). In their model they took into account current dissipation due to anomalous resistivity (Büchner & Elkina, 2005, 2006) that causes Joule heating. Since LINMOD3d considers the compressibility of the plasma, the resulting heating could be due also to compressional effects. Later on two-dimensional MHD simulation studies were carried out by von Rekowski B. et al. (2006a, b); von Rekowski et al. (2008). These authors used an analytical initial equilibrium and imposed a magnetic flux footpoint motion to model coronal bright point heating as being due to canceling magnetic features. To obtain the desired heating rate they used an enhanced resistivity for which the values were above the theoretically justifiable resistivity. This raises the general question of the energy budget and energy conversion in solar flux tubes. Even with low resistivity, current simulations are unable to resolve the diffusion regions of reconnection and thus overestimate Joule heating. It is also unresolved how much heating is caused by pressure gradient forces. To clarify this question we continued the work of Büchner et al. (2004a, b, c); Büchner (2006, 2007); Santos & Büchner (2007); Santos et al. (2008). These authors demonstrated the formation of localized current sheets in and above the transition region at the position of a EUV BPs as a result of photospheric plasma motion. This study is extending their results through a systematic study of the energy conversion and budget in magnetic flux tubes. The investigation uses the 3D simulation model LINMOD3d to simulate the solar atmosphere in the region of an X-ray BP observed by the Hinode spacecraft on 19 December 2006 between 22.17 UT and 22.22 UT. In section 2 we briefly review the main features of the numerical simulation model LINMOD3d. In section 3 we describe the specific simulation setup used in our study and section 4 provides some simulation results for the chosen BP data. In section 5 we present results of energy budget analysis by investigating the role of different forces and in section 6 we summarize and discuss our results. ## 2 Simulation model Our simulation model uses the approach of the LINMOD3d code (Büchner et al., 2004a, b, c). This means that the initial magnetic field is obtained by extrapolating the observed photospheric line-of sight (LOS) magnetic fields. The initial plasma distribution is non-uniform containing a dense and cool chromosphere as well as the transition to a rarefied and hot corona. The photospheric driving is switched on by coupling the chromospheric plasma with a moving background neutral gas. Some details of our code have been given briefly in the following subsection. ### 2.1 Equations In our study we solve the following set of MHD equations: $\displaystyle\frac{\partial\rho}{\partial t}$ $\displaystyle=-\@vec{\nabla}\cdot\rho\@vec{u}$ (1) $\displaystyle\frac{\partial\rho\@vec{u}}{\partial t}$ $\displaystyle=-\@vec{\nabla}\cdot\rho\@vec{u}\@vec{u}-\@vec{\nabla}p+\@vec{j}\times\@vec{B}-\nu\rho(\@vec{u}-\@vec{u}_{0})$ (2) $\displaystyle\frac{\partial\@vec{B}}{\partial t}$ $\displaystyle=\@vec{\nabla}\times(\@vec{u}\times\@vec{B}-\eta\@vec{j})$ (3) $\displaystyle\frac{\partial p}{\partial t}$ $\displaystyle=-\@vec{\nabla}\cdot p\@vec{u}-(\gamma-1)p\@vec{\nabla}\cdot\@vec{u}+(\gamma-1)\eta j^{2}$ (4) where $\rho$ and $\@vec{u}$ are plasma density and velocity, $\@vec{B}$ is the magnetic field and P is the thermal pressure. A plasma-neutral gas coupling in photosphere and chromosphere is included through the collision term in the momentum equation, where $\@vec{u}_{0}$ denotes the neutral gas velocity. The neutral gas serves as a frictional background to communication photospheric footpoint motion to the plasma and magnetic field through frictional interaction. It also leads to a reflection of coronal Alfvén waves back to the corona from the transition region, so that the influence of coronal Alfvén waves can be neglected at the photospheric boundary. In order to set the plasma in motion a number of incompressible flow eddies is used according to observed horizontal drifts in the photosphere $\nabla\cdot\textbf{u}_{0}=0$ is imposed via the neutral gas, where $\textbf{u}_{0}$ is dependent in x and y. It is constant along z and derived from a potential using $\textbf{u}_{0}=\nabla\times(U\textbf{e}_{z})$, with $U=u_{00}/\cosh\left(\frac{x-y+c_{0}}{L_{0}}\right)/\cosh\left(\frac{x+y+c_{1}}{L_{1}}\right)$ (5) Note that the contour lines of this function are streamlines of the flow. The magnitudes of velocity scale with $u_{00}/L_{0}$ and $u_{00}/L_{1}$, chosen in accordance with the observed plasma motion in the photosphere. In our simulation we approximated the observed motion by three vortices with amplitudes of the velocity $u_{00}$ equal to 5.5, 5 and 2 $km/s$, respectively. The values of $c_{0}$, $L_{0}$, $c_{1}$ and $L_{1}$ are 9, 6, 51 and 6 Mm for the first vortex 5, 6, 28 and 6 Mm for the second and 19, 7, 38 and 7 Mm for the third vortex. The height-dependent collision frequency $\nu$ is chosen to be sufficiently large only below the transition region. This way the plasma is forced to move, dragged by the neutral gas, in the model chromosphere but not above the transition region. This way the horizontal motion generates a Poynting flux into the corona. On the other hand the collision frequency is chosen in a way that coronal Alfvén waves are properly reflected while wave perturbations in the chromosphere are heavily damped by the frictional interaction with the neutral background. Our choice of equations means that in this study we do not consider energy losses due to radiation and heat conduction and we also excluded the action of the solar gravitation in this study. The system of equations is closed by Ohm’s and Ampère’s laws and the temperature is defined via the ideal gas law for a fully ionized plasma: $\displaystyle\@vec{E}$ $\displaystyle=-\@vec{u}\times\@vec{B}+\eta\@vec{j}$ (6) $\displaystyle\@vec{\nabla}\times\@vec{B}$ $\displaystyle=\mu_{0}\@vec{j}$ (7) $\displaystyle p$ $\displaystyle=2n\kappa_{B}T$ (8) The value of the resistivity $\eta$ is varied in accordance with three models described in subsection 2.3. The MHD equations are discretized by means of a second order weakly dissipative Leapfrog scheme. Due to stability reasons the induction equation is discretized using Dufort-Frankel scheme, Potter (1973). ### 2.2 Simulation box and normalization The lower boundary of the simulation box is a horizontal square in the photosphere sized $46.4\times 46.4\,Mm^{2}$. The simulation box extends 15.45 Mm toward the corona. A nonuniform grid in the z direction supplies the proper resolution of the transition layer, where the grid distance $\Delta z$ corresponds to 160 km, Büchner et al. (2004a). This corresponds to 64 grid points in z direction, while in the x, y plane a $128\times 128$ grid are used. We solve for dimensionless variables that are normalized to natural scales as listed in table. 1. Note that the maximum imposed velocity of the neutral gas is smaller than 5 km/s while the typical (normalizing) electron thermal velocity is $v_{the}=1470$ km/s and the Alfvén speed is $v_{A}=50$ km/s. Hence, one can be certain that the inserted neutral gas motion is gentle, sub-Alfvénic and sub-slow velocities. ### 2.3 Resistivity models In order to verify the influence of different resistivity models on the BP plasma heating we solved the equations for the same initial and boundary conditions but varying the resistivity model. The resistivity $\eta$ can be expressed via an effective collision frequency $\mu$ as $\eta=\frac{\mu}{\epsilon_{0}\ \omega^{2}_{pe}}$, where $\omega_{pe}$ is the electron plasma frequency ($\omega_{pe}=\sqrt{ne^{2}/\epsilon_{0}m_{e}}$). In our model we always apply a constant physically justified background resistivity $\eta_{0}$ which exceeds exceeds the numerical resistivity. It is appropriate to chose for effective collision frequency of the background resistivity the Spitzer, (1962) value $\mu=(ne^{4}Ln\Lambda T^{-\frac{3}{2}}/16\pi\epsilon^{2}_{0}m^{\frac{1}{2}}_{e}K^{\frac{3}{2}}_{B}$). Based on the typical plasma parameters of our model we chose for the collision-driven background resistivity $\eta_{0}=10^{-4}$ (in normalized units). In two models we switched on additional, anomalous, resistivity in places where either the current density of the current carrier velocity ($u_{ccv}$ determined as the current density divided by the charge density) exceeds a physically justified thresholds of micro-instabilities. In the first resistivity model anomalous resistivity is switched on when the current carrier velocity ($u_{ccv}$ exceeds a critical velocity (Roussev et al., 2002; Büchner & Elkina, 2005): $\displaystyle\eta=\eta_{0}+\begin{cases}0,&\text{if}\,\|u_{ccv}\|<u_{crit}\\\ \eta_{eff}\left({\frac{\|u_{ccv}\|}{u_{crit}}-1}\right),&\text{if}\,\|u_{ccv}\|\geq u_{crit}\\\ \end{cases}$ (9) A natural choice for the threshold velocity is the electron thermal velocity $v_{the}$, in our for the normalizing quantities 1470 km/s or to $5.8.10^{-4}$ in normalized units. In the first resistivity model we chose $5.10^{-2}$ to follow the ideal evolution of the plasma as long as possible. The additional term for resistivity can be estimated e.g., for a nonlinear ion- acoustic instability (Büchner & Elkina, 2006) as $\eta_{eff}=\frac{\mu_{eff}}{\epsilon_{0}\ \omega^{2}_{pe}}=\frac{\omega_{pi}}{\epsilon_{0}\ \omega^{2}_{pe}}$ (10) Here $\omega_{pi}$ denotes plasma ion frequency ($\omega_{pi}=\sqrt{ne^{2}/\epsilon_{0}m_{i}}$). For the typical parameters of our simulation this estimate would reveal $\eta=2.5$, i.e. a magnetic Reynolds number of less than unity. In this case many current sheets would immediately diffuse away. On the the other hand, since the plasma $\beta$ is relatively large for our simulation parameters obliquely propagating waves would be present in the spectrum of the micro-turbulence. In this case the estimate of the effective collision frequency has to take into account lower-hybrid waves (Silin & Büchner, 2005). For our normalizing values this results in $\eta_{eff}=0.03$. In a second model calculation we considered a current density dependent resistivity used before, e.g., by Neukirch et al. (1997), in which the resistivity increases even stronger (quadratic dependence) after the current density exceeds a critical value $j_{crit}$: $\displaystyle\eta=\eta_{0}+\begin{cases}0,&if\|j\|<j_{crit}\cr\eta_{eff}(\frac{\|j\|}{j_{crit}}-1)^{2}&if\|j\|\geq j_{crit}\cr\end{cases}$ (11) The critical current density is related to critical velocity via $j_{crit}=e\ n_{e}\ u_{crit}$. Here we will report the results of our simulations obtained according to the second model for which we chose a threshold as low as $j_{crit}=0.69$ in order to discuss the consequences of an early switch on of additional, anomalous resistivity. For comparison we solved the problem also by assuming for a third model a constant enhanced uniform resistivity as usually done in global MHD simulations. Concerning the values of the chosen $\eta_{eff}$ one should note that the width of the actual current sheets in which turbulence effectively operates is of the order of the ion inertial scale $d_{i}={c\over\omega_{pi}}$. This scale cannot be resolved in any realistic 3D MHD simulation of the solar corona. In order to introduce micro-turbulent anomalous resistivity the threshold velocity (- current density) has to up-scaled to the actual resolution of the simulation by a factor of $5.10^{4}$. By the same reason the resistive electric field builds up in very (perhaps $d_{i}$-) thin current sheets. To consider the correct values of the electric field on the much coarser MHD- simulation grid anomalous resistivity used in the simulation has also to be scaled up by the above scaling factor. This approach allows to consider the correct amount of Joule heating. Table 1: Normalization values. $\begin{array}[]{p{0.5\linewidth}l}\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr Variable&\ Normalization\ value\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\\\ density&N_{0}=2.10^{15}m^{-3}\\\ \\\ lenght&L_{0}=500\ km\\\ \\\ magnetic field&B_{0}=1G=10^{-4}\ T\\\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\\ pressure&P_{0}=\frac{B_{0}^{2}}{2\mu_{0}}=4\times 10^{-3}\ J/m^{2}\\\ \\\ temperature&T_{0}=\frac{P_{0}}{2\ n_{0}\ \kappa_{B}}=7.2\times 10^{4}\ K\\\ \\\ Alfv\'{e}n velocity&v_{A0}=\frac{B_{0}}{\sqrt{\mu_{0}\ m_{i}\ N_{0}}}=50\ km/s\\\ \\\ time&\tau_{0}=L_{0}/\ v_{A0}=10\ s\\\ \hline\cr\end{array}$ ### 2.4 Initial and boundary conditions We first carried out a potential field extrapolation to the Fourier decomposed normal field component of the magnetic field taken from the MDI-observation. The resulting 3D magnetic configuration is used as the initial condition of the simulation code. In the potential field approximation the normal field component is related to $(B_{x},B_{y})$ through $\nabla\cdot B=0$ and $\nabla\times B=0$. The initial density and temperature height profiles for the plasma is taken in accordance with the VAL model that assumes pressure being in a hydrostatic equilibrium. The simulation box has 6 boundaries: 4 lateral, 1 top and 1 bottom boundaries. For the side boundaries a line symmetric boundary condition is used with the line symmetry with respect to the centers of the sides of the simulation box. For the upper boundary the derivatives in the normal direction are put to zero. At the lower boundary the normal velocity is set to be zero, while the tangential velocity is taken from the neutral motion. ## 3 Simulation setup Our study is based on an X-ray BP observed by the XRT X-ray telescope on board of the Hinode spacecraft on 19 December 2006\. The corresponding X-ray image is shown in Fig. 1. For the initial magnetic field we used the observed $line- of-sight$ (LOS) component of the photospheric magnetic field taken by the Michelson Doppler Interferometer MDI onboard the Soho spacecraft at 22:17 UT. For that sake data from a field of view with the horizontal size of 64 $\times$ 64 $arcsec^{2}$ was chosen that properly covers the magnetic features associated to this BP (insert in Fig. 1). Note that we use the LOS component as the initial normal field component at the lower boundary of our simulation box, the photosphere, since the BP observation was made close to the center of the solar disc. Figure 1: X-ray Image taken from XRT/Hinode on 19 December 2006 at 22:08 UT. Insert: the LOS component of the photospheric magnetic field in a 64 $\times$ 64 $arcsec^{2}$ horizontal plane taken from MDI/Soho, where white (black) spots correspond to upward (downward) directed Line-of-sight components of the photospheric magnetic field. The BP and the related magnetic field feature are indicated in the images. Fourier filtering was applied to the LOS component of the magnetic field. By taking into account only the first eight Fourier modes, details of magnetic field structure smaller that 6 Mm are neglected. The extension of structures arising from smaller scale magnetic features would not extend higher up into the corona, they are dissipated at an early stage of the evolution in the highly collisional chromospheric plasma. Figure 2: Potential magnetic field extrapolated from the filtered MDI magnetograms and used as initial configuration for the magnetic field in our simulation. The blue lines show the magnetic field lines. The color code depicts the LOS component of the magnetic field. Note that axes here are in terms of grid points, 64 in z direction and $128\times 128$ in x,y plane. Fig. 2 shows a three-dimensional view of the magnetic field extrapolated from the photospheric boundary for the magnetic field observed at 22:17 UT on December 19, 2006. The blue lines show the magnetic field lines. The color code depicts the LOS component of the photospheric magnetic field. Magnetic fields directed upward from the photosphere are colored in red, downward directed in blue. With the chosen normalization length of $L_{0}=500$ km, the box size in x and y direction correspond to 92.8 $L_{0}$ and the z direction extend to 30.9 $L_{0}$. The photospheric plasma velocities are obtained by applying the local-correlation-tracking(LCT) method November & Simon (1988) to the Fourier filtered LOS magnetic component of the photospheric magnetic field observed between 22:17 UT and 22:22 UT. The left panel of Fig. 3 shows the velocity pattern obtained by the LCT method. For the simulation we used incompressible velocity vortices to mimic the observed velocity pattern, as shown in the right panel of Fig. 3. Note that the interval chosen for the simulation starts a few hours after the time the BP first appeared in the X-ray images and that the bright point continues to glow a few more hours afterwards. During the whole simulation time interval the relative shear motion of the two main magnetic flux concentrations of opposite polarity is negligible. Figure 3: Horizontal plasma velocities in the photosphere. Left panel: velocities obtained by applying LCT technique to MDI magnetograms between 22:17 and 22:22 UT. The right panel shows the vortices that are used to approximate this motion in the simulation. ## 4 Simulation results The simulation results are first shown in a plane at x = 45.7 (Fig. 2), which crossed through the center of the two main magnetic polarities. The vertical profile of the temperature is shown in Fig. 4 for t = 0 (top panel), 80 (middle panel) and 160 s (bottom panel). In $t=0$ we have a height dependent temperature as defined by the initial condition. At $t=80$ s the effects of plasma compression and expansion, together with Joule heating, shape the temperature profile. An arc of hot plasma is formed above the two opposite magnetic polarities. The increase in temperature in this layer is approximately 0.5 in normalized units, what corresponds to 36000 K. The region that is located just below it, however, experiences some drop in temperature. At $t=160$ s the arc of hot plasma leaves the simulation box and we are left with a corona in which the differences in temperature can reach one orders of magnitudes. Figure 4: Temperature distribution in the vertical plane at x = 45.7 at the beginning (upper panel) and at t = 80 s and t = 160 s. Note the temperature in color bar in presented in terms of the normalization value. Fig. 5 shows the parallel and perpendicular components of the current with respect to the magnetic field direction at t = 80 and t = 160s. It can be seen that enhanced current flows coincide well with the temperature increase. This would lead to an interpretation of the heating as being due to current dissipation only. However, as shown later, adiabatic heating can have an important contribution to temperature increase. Figure 5: Parallel and perpendicular components of electrical current at t = 80 s (upper two panels) and t = 160 s (lower two panels) in the same diagnostic plane as in Fig.4. Note the enhancement in perpendicular current is located at the same place of the temperature maximum. ## 5 Energy balance Let us now diagnose the different contributions to plasma heating in the BP region. First, in subsection 5.1, we discuss the overall global heating. In subsection 5.2 the dependence on the resistivity model is presented. Finally, the flux-tube heating is analyzed in subsection 5.3. ### 5.1 Global effect of current dissipation and compression In order to understand the relative contribution of current dissipation and plasma compression to the coronal plasma heating in the BP region it is appropriate to analyze the pressure changes rewriting the Eq. 4 in terms of a continuity equation for the Temperature evolution. This leaves two source terms on the right hand side of the equation: $\frac{\partial T}{\partial t}+\@vec{\nabla}\cdot T\@vec{u}=-(\gamma-1)\ T\ \@vec{\nabla}\cdot\@vec{u}+(\gamma-1)\ \eta j^{2}/\rho$ (12) Let us first analyze the first case, where an anomalous resistivity is used when the current carrier velocity exceeds a critical value. Fig. 6 shows the resulting distribution of $-(\gamma-1)\ T\ \@vec{\nabla}\cdot\@vec{u}$ in the vertical diagnostic plane. This way we have a proxy for temperature changes associated to pressure compression and expansion. Adiabatic heating has an important role on the formation of the high temperature arc that propagates upward towards the top boundary. It is also due to expansion that temperature decreases below this hot arc. Figure 6: Adiabatic cooling/heating rate after 80 s and 160 s in the plane x = 45.7, according to the first term in the r.h.s of Eq. 11 over density, ($-(\gamma-1)T\nabla\cdot u<0$). The values of second term in the right hand side of the Eq. 11, $(\gamma-1)\ \eta j^{2}/\ \rho$, is shown in Fig. 7 in the plane x = 45.7 at t = 80s and t = 160s. By comparison with the compressional part the contribution of the Joule heating appears to be negligible. For a better comparison the contribution of the two terms in the right hand side of the Eq. 11 in the temperature evaluation, the horizontal view is shown at the height of transition region in two different instance of time, t = 80 s in left and t = 160 s in the right panel of Fig. 8. Figure 7: Joule heating rate, (second term in the r.h.s of Eq. 12, divided by density), after 80 s and 160 s in the plane x = 45.7. Figure 8: Temperature, first and second terms of Eq. 12 over density, in the left, middle and right panels, respectively. The result is shown at a horizontal plane in transition region at two instance of time, t = 80s in top and t = 160s in the bottom panels. In the following, we will analyze in some more detail to what degree compression and Joule heating contribute to the evolution of the temperature. For this sake and in order to study the role of the forces involved in the energy conversion process, we performed a volume integration of the time rates of change of kinetic, magnetic and thermal energies in the simulation box above the chosen Bright Point region. Our approach is similar to Birn et al. (2009) when they used energy transport equations to analyze the properties of energy conversions associated with a reconnection process. The contribution of different terms in the energy transport process can be studied from the following equations: $\displaystyle\frac{d\varepsilon_{kin}}{dt}=-\frac{1}{2}\int_{S_{V}}\rho u^{2}\@vec{u}\cdot d\@vec{s}+\int_{V}(-\@vec{u}\cdot\nabla p+\@vec{u}\cdot\@vec{j}\times\@vec{B})d^{3}v$ (13) $\displaystyle\frac{d\varepsilon_{mag}}{dt}=-\frac{1}{\mu_{0}}\int_{S_{V}}(-\@vec{u}\@vec{B}^{2}+(\@vec{u}\cdot\@vec{B})\@vec{B}-\eta\@vec{j}\times\@vec{B})\cdot d\@vec{s}$ (14) $\displaystyle+\int_{V}(-\@vec{u}\cdot\@vec{j}\times\@vec{B}-\eta\@vec{j}^{2})d^{3}v$ $\displaystyle\frac{d\varepsilon_{th}}{dt}=-\frac{\gamma}{\gamma-1}\int_{S_{V}}p\@vec{u}\cdot d\@vec{s}+\int_{V}(\@vec{u}\cdot\nabla p+\eta\@vec{j}^{2})d^{3}v$ (15) Where $\varepsilon_{kin}$, $\varepsilon_{mag}$ and $\varepsilon_{th}$ denote kinetic, $\rho u^{2}/2$ , magnetic, $\@vec{B}^{2}/2\mu_{0}$, and thermal, $P/(\gamma-1)$, energies, respectively. The volume integrals (second term on the right-hand side) in these equations represent the energy conversion from one form into the another. This energy conversion are explicitly written in terms of the work done by Lorentz force, pressure gradient force and Joule dissipation, (left panel of Fig. 9). Note that the initial spike in the Lorentz force is in part caused by numerical discretization errors and in part by the onset of photospheric footpoint motion. The initial oscillations are damped substantially during, approximately, two Alfvén times, followed by a state of an approximate force balance. This effect was found to be smaller in a run where footpoint motion was not included. The initial perturbation has a minor effect on the initial extrapolated magnetic field, it does not affect the currents and Lorentz forces at a later times. The surface integrals are also needed to obtain the energy rates, when they indicate the transport of each of the three form of energies. With the chosen boundary condition however, the values of these surface integrals are zero at the lower boundary. They compensate each other through the side boundaries of the simulation box as well. At the upper boundary however, one needs to consider the contribution of this surface integrals in the rate of energy transfer. This means $\textbf{E}\times\textbf{B}$, $P\textbf{u}$ and $\rho u^{2}\textbf{u}$ for the transport of magnetic, kinetic and thermal energies, respectively. The values of these terms at the upper boundary are shown in Fig. 10. One can see that the contribution due to these terms is insignificant, so it would be a good approximation to consider only the volume integrals for the change in the energy rates. Figure 9: Rates of the contribution to the total energy changes, (bottom panel). Different contributions to the changes of the energy, (top panel). The changes in energy rates are shown in the right panel of Fig. 9, the forces responsible for these changes are depicted in the left panel of the Figure. As one can see by comparing the two panels the magnetic energy is transferred to kinetic energy almost completely via the work done by the Lorentz force that accelerates the plasma. It is an intermediate step however, followed by the work done by pressure gradient force that converts kinetic energy into thermal energy. This decelerates the plasma motion until, finally, the Lorentz force is balanced. The direct transformation of magnetic energy to thermal energy (Joule heating) is via Ohmic current dissipation, $\eta J^{2}$. A comparison of the energy conversions rates (see Fig. 9, right panel) however shows that Joule dissipation plays a minor role in the energy exchange process while the other contributions are orders of magnitudes larger. The minor role of Joule heating in comparison to adiabatic process in the increase of thermal energy was also found for the case of a solar flare by Birn et al. (2009), where they explained the compressional heating in two almost simultaneously steps: acceleration by Lorentz force and deceleration by pressure gradients. Figure 10: First term in the right hand side of the Eqs.12-14, (surface integrals) is shown in the upper boundary of the simulation box after 160 s. ### 5.2 Influence of different resistivity models The previous calculation was based on an anomalous resistivity model with the current carrier velocity as a critical value for a local switch-on of additional resistivity. In order to better understand the influence of the resistivity we performed the simulation also with two other resistivity models, one that uses a current density dependent resistivity and another with constant resistivity respectively. Figure 11: Energy change rates for three different resistivity models, (top panel). The bottom panel shows the work done by the Lorentz force, pressure gradient force and the Joule heating power. different lines correspond to anomalous current carrier dependent (dashed), anomalous current dependent (dotted) and constant (solid line) resistivity models. Fig. 11 depicts the resulting energy conversion rates and the work done by the involved forces v $\cdot\ J\times B$, v $\cdot\ \nabla P$ and by $\eta\ J^{2}$, for all the three resistivity models by using different line styles for the results obtained by using the different resistivity model. The results obtained for the three cases show that the resistivity model influences the dynamics of the system and the thermal energy rate mainly through the pressure gradient force. While magnetic and kinetic energy rates of change depend only weakly on the resistivity model, the rate of temperature change is significantly influenced. Nevertheless, independent on the used resistivity model the heating is due mainly to the work done by the pressure gradient force. At the same time the contribution of the Joule heating is about two orders of magnitude smaller (Note the scale of the plots in the top row.) We conclude that the adiabatic compression is the dominant effect in increasing temperature in the BP region in all three cases. ### 5.3 Flux tube heating In order to locate the heating effect better it is appropriate to determine it for individual flux tubes, integrating along the magnetic field lines instead of taking values averaged over the whole simulation box as reported in the previous sections. In this integration one has to take into account the changing cross-section of flux tubes. This can be done by applying the concept of the differential flux tube volume $V=\int{B}^{-1}ds$, where ds indicates the step size along the field line. This way the flux conservation in a flux tube ($\Phi=A\times B=const.$) is taken into account by the proportionally of the cross-section to $B^{-1}$. Note that large flux tube volumes correspond to field line rising high into the corona or hitting regions of vanishing magnetic field. The energy is transported in accordance with the upward directed Poynting flux $E\times B$, enhanced magnetic tension is carried away by wave propagation. Figure 12: Integration along the field lines using the differential flux tube volume concept for the work done by Lorentz force, pressure gradient force and Joule heating, (top panel, from left to right), and the changes in rates of magnetic field change, the temperature and the kinetic energy, (bottom panel, from left to right) at t = 160s. For the quantities described in section 5.1 the resulting flux tube integrated values are shown in Fig. 12 in the horizontal reference plane just above the transition region. The values reached indicate once more the negligible role of Joule heating by current dissipation for the thermal energy change in the bright point region compared to the dominant role of the pressure gradient force. Please note the different range of the plots in Figure 12 as indicated by the color bar. It also can be seen that the locations at which this force and also maximum rates of energy changes appear coincide. Furthermore, the same pattern has formed in the integration result of v $\cdot\ \nabla P$, v $\cdot\ J\times B$ and the rate of change of the different kinds of energy. This pattern can clearly be seen in the integration of total energy along the field lines (Fig. 13), which is the sum of the kinetic, magnetic and thermal energies: $\varepsilon=\varepsilon_{kin}+\varepsilon_{mag}+\varepsilon_{th}=\int_{V}\\{\frac{1}{2}\rho u^{2}+\frac{1}{2\mu_{0}}B^{2}+\frac{p}{\gamma-1}\\}d^{3}v$ Figure 13: Results of integration along the magnetic field lines using the differential flux tube volume concept for total energy(top, left panel) and magnetic(top, right panel), kinetic(bottom, left panel) and thermal energies(bottom, right panel), at t = 160s. The left panel of Fig. 14 shows the result of this integration for temperature and flux tube volume. The coincidence of the temperature enhancement with the maxima obtained in the flux tube integrated energy change rates and forces shows that the heat is provided by the plasma compression due to the Lorentz force. Figure 14: Temperature and flux tube volume, integrated along the magnetic field lines, at t = 160s. Enhanced flux tube integrated values follow the same pattern as the BP. This indicates that the regions of enhanced temperatures correspond to the foot points of field lines leading to higher altitudes or to regions where the magnetic field vanishes. The plasma motion across these regions supplies the magnetic energy that is converted to thermal energy. ## 6 Summary and discussion We have presented the results of heating processes in the region of an observed X-ray coronal bright point. In particular we have investigated the importance of the work done by adiabatic compression in comparison with Joule heating in the course of the dynamic evolution and heat production near the bright point. The simulation shows that an arc-shaped structure of enhanced temperature forms that is 2-4 times hotter than the background plasma. This structure is located above the two main opposite photospheric magnetic flux concentration. It coincides with the location where the electrical current densities are maximum. The structures of temperature and current density enhancements, indeed, coincide. We further examined the contribution of the Lorentz force, pressure gradient force and Joule heating performing volume integrals in the simulation box that determine the magnetic, kinetic and thermal energy change rates for three different resistivity models. We found that independent on the resistivity model magnetic energy was transformed to kinetic energy through the work done by Lorentz force. Kinetic energy in turn is converted to thermal energy due to pressure gradients that balance the Lorentz force. A comparison of the effect of the three energy conversion through v $\cdot\ J\times B$, v $\cdot\ \nabla P$ and $\eta\ J^{2}$ show that adiabatic compression has an important role in temperature increase in the upper corona. This is not dependent on the resistivity model used in the simulation. For a better understanding of the heating processes we utilized the concept of differential flux tube integration of the different contributions along the magnetic field lines. A quantitative comparison in the horizontal plane, from where the integration starts, shows that energy conversion rate, total energies and work done by Lorentz and pressure gradient forces are located in the same flux tubes, also temperature and flux tube volume are maximum at the same place. We conclude that the conversion of magnetic energy to kinetic energy via the work done by the Lorentz force and from kinetic to thermal energy due to the work done against the pressure gradient force determine the heating of this bright point. We could show that plasma compression dominates the heating of the bright point. In contrast, the role of Joule dissipation appeared to be negligibly small. The temperature enhancement follows the same pattern. The fact that the pattern obtained by calculating flux volume integrals coincides with the one of temperature and energy change rates bring us to the conclusion that plasma motion at the footpoints of the flux tubes carries the energy upward and makes the flux tubes rise to the higher corona. The magnetic energy is converted to thermal energy until the plasma compression is balanced by the Lorentz force. In the local, flux-tube oriented consideration we also could see that the role of the Joule heating in these energy conversion processes was negligible and the heating of plasma in the bright point region is basically due to pressure gradient force. First, the fact that Joule heating is weak in the corona was not entirely unexpected but it is quantitatively confirmed here. It is worth to remember that the necessary up-scaling of the resistivity and of the onset condition of micro-turbulent anomalous resistivity to the resolved by the MHD simulation grid scales does even overestimate the actual Joule heating. As a result Joule heating cannot be considered a viable process unless there is a convincing argument that the dissipation regions are volume filling to a much larger extend than the already large one used in the present model. Second, the results demonstrate very clearly that compression is an important processes in the energy budget. It is not clear in how far compression can contribute to the overall coronal heating but it is certainly important for the local heating of BPs. Third, in this context the nature and the consequences of plasma compression are worth some consideration. In ideal MHD adiabatic compression is reversible. But the consequent flux tube heating is, however, irreversible due to magnetic reconnection and other mixing processes. 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1012.0821	# Interactive proofs with competing teams of no-signaling provers Gus Gutoski Institute for Quantum Computing and School of Computer Science University of Waterloo, Waterloo, Ontario, Canada (December 3, 2010 (Minor revisions: October 14, 2011)) ###### Abstract This paper studies a generalization of multi-prover interactive proofs in which a verifier interacts with two competing teams of provers: one team attempts to convince the verifier to accept while the other attempts to convince the verifier to reject. Each team consists of two provers who jointly implement a _no-signaling_ strategy. No-signaling strategies are a curious class of joint strategy that cannot in general be implemented without communication between the provers, yet cannot be used as a black box to establish communication between them. Attention is restricted in this paper to _two-turn_ interactions in which the verifier asks questions of each of the four provers and decides whether to accept or reject based on their responses. We prove that the complexity class of decision problems that admit two-turn interactive proofs with competing teams of no-signaling provers is a subset of $\mathrm{PSPACE}$. This upper bound matches existing $\mathrm{PSPACE}$ lower bounds on the following two disparate and weaker classes of interactive proof: 1. 1. Two-turn multi-prover interactive proofs with only one team of no-signaling provers. 2. 2. Two-turn competing-prover interactive proofs with only one prover per team. Our result implies that the complexity of these two models is unchanged by the addition of a second competing team of no-signaling provers in the first case and by the addition of a second no-signaling prover to each team in the second case. Moreover, our result unifies and subsumes prior $\mathrm{PSPACE}$ upper bounds on these classes. ## 1 Introduction Interactive proofs were introduced in the mid-1980’s as a generalization of the concept of efficient proof verification and the complexity class $\mathrm{NP}$ [Bab85, BM88, GMR89]. Informally speaking, an _interactive proof_ is a conversation between a randomized polynomial-time _verifier_ and a computationally unbounded _prover_ regarding some common input string $x$. A decision problem $L$ is said to admit an interactive proof if there exists a verifier such that (i) if $x$ is a yes-instance of $L$ then there is a prover who can convince the verifier to accept $x$ with high probability, and (ii) if $x$ is a no-instance of $L$ then no prover can convince the verifier to accept $x$ except with small probability. In a dramatic testament to the surprising power of randomization and interaction, it was soon discovered that every problem in $\mathrm{PSPACE}$ admits an interactive proof, yielding the well- known identity $\mathrm{IP}=\mathrm{PSPACE}$ [LFKN92, Sha92]. #### Multi-prover interactive proofs, no-signaling provers The fruitful study of interactive proofs has prompted further generalization of the model. One such generalization is the _multi-prover_ interactive proof model of Ben-Or _et al._ [BOGKW88] wherein several provers cooperate in their attempt to convince the verifier to accept the input string $x$. The key aspect that sets this model apart from single-prover interactive proofs is the fact that the provers cannot communicate with one another during the protocol. Amazingly, this small distinction is enough to increase the power of the model from $\mathrm{PSPACE}$ all the way up to $\mathrm{NEXP}$ [BFL91, FRS94], even when the interaction is restricted to only two turns with only two provers [FL92]. In terms of complexity classes, the corresponding identity is $\mathrm{MIP}=\mathrm{NEXP}$. Intermediate classes of multi-prover interactive proofs are obtained by tinkering with the set of strategies available to the provers. Consider, for example, a joint strategy where the distribution of answers from one prover is independent of the question asked of the other prover—these are the _no- signaling_ strategies. Clearly, such a strategy cannot be used in a black-box fashion by the provers to establish communication. At first glance it may seem that the no-signaling condition is equivalent to the standard definition of a multi-prover interactive proof. However, there exist no-signaling strategies that cannot be implemented without communication between the provers, suggesting that this model might be a nontrivial intermediary between single- and multi-prover interactive proofs. Indeed, it was established by Ito, Kobayashi, and Matsumoto [IKM09] that the two-turn, two-prover protocol for $\mathrm{PSPACE}$ of Cai, Condon, and Lipton [CCL94] is sound even against no-signaling provers. By contrast, $\mathrm{PSPACE}$ is known not to admit two-turn _single_ -prover interactive proofs unless the polynomial hierarchy collapses and $\mathrm{PSPACE}=\mathrm{AM}$ [Bab85, GS89]. A converse result was proven by Ito, who showed that every problem that admits a two-turn interactive proof with two no-signaling provers is also in $\mathrm{PSPACE}$ [Ito10]. Thus, the interactive proof model is even _more_ sensitive to change than suggested by the difference between single- and multi-prover interactive proofs, as even the smaller difference between no-signaling and standard multi-prover interactive proofs is sufficient to make the jump from $\mathrm{PSPACE}$ up to $\mathrm{NEXP}$ (at least in the case of two turns and two provers). In addition to this prior work, parallel repetition results for multi-prover interactive proofs with no-signaling provers were established in Refs. [Hol09, KR10]. The reader is referred to Ito [Ito10] for more detailed history and references. #### Inspiration from quantum information Though the present paper contains no formal discussion of quantum information, it is proper to acknowledge its role in motivating the study of no-signaling provers. Interest in this model was originally drawn from the study of multi- prover _quantum_ interactive proofs, in which the provers (and possibly the verifier) are permitted to exchange and manipulate quantum information. It is easy to see that interactive proofs with ordinary, “classical” provers are not affected by the ability of the provers to sample from a common source of randomness. Quantum provers, on the other hand, might use shared pieces of some entangled quantum state to implement a _nonlocal_ strategy that correlates their messages in ways that cannot otherwise be achieved [Bel64]. (The phenomenon of nonlocality was famously branded by Einstein as “spooky action at a distance.”) Indeed, some classical protocols which are sound against classical provers are known to become unsound when the provers share entanglement [CHTW04, CGJ09]. Whereas the set of strategies that admit shared entanglement is highly complex, the set of no-signaling strategies is relatively simple and it includes entanglement-sharing strategies as a proper subset. So, for example, any protocol that is sound against no-signaling provers is also sound against quantum provers who share entanglement. It is also interesting to find differences between no-signaling strategies and entanglement-sharing strategies, as this difference sheds light on the extent to which no-signaling can be used as a proxy for shared entanglement. In some protocols the allowance of arbitrary no-signaling strategies leads to implausible consequences [vD05, BBL+06]. Such protocols can be viewed as mathematical evidence against physical theories that admit so-called “super-strong” nonlocality such as that found in no-signaling strategies but not entanglement-sharing strategies. The present paper establishes a scenario in which two no-signalling provers are equivalent to two signaling provers. #### Interactive proofs with competing provers Another generalization of the single-prover model is an interactive proof with _competing provers_ , in which one prover tries to convince the verifier to accept the input string $x$ while the other prover tries to convince the verifier to reject $x$. One may consider proofs in which all messages are known to all provers (_complete_ information) or in which each prover sees only the messages he exchanges with the verifier (_incomplete_ information). These two forms of competing-prover interactive proofs were studied by several authors in the 1990’s [FST90, FS92, FKS95, FK97]. But for our purpose in this paper it only makes sense to consider protocols with incomplete information. In the jargon of game theory, interactive proofs with competing provers are _zero-sum games_ , about which there exists a vast body of literature in computer science, economics, and other disciplines. For instance, fast algorithms for zero-sum games of incomplete information in _extensive form_ imply that the complexity class $\mathrm{RG}$ of problems that admit interactive proofs with competing provers is a subset of $\mathrm{EXP}$ [KM92, KMvS94]. Feige and Kilian proved the reverse containment [FK97], yielding the competing-prover analogy $\mathrm{RG}=\mathrm{EXP}$ of the aforementioned identity $\mathrm{IP}=\mathrm{PSPACE}$ for single-prover interactive proofs. Feige and Kilian also studied _two-turn_ interactive proofs with competing provers, providing a matching upper and lower bound of $\mathrm{PSPACE}$ on the complexity of this model [FK97]. The complexity of $k$-turn interactive proofs with competing provers for constants $k\geq 3$ is an open question of interest to both complexity theorists and game theorists alike. #### Interactive proofs with competing teams of provers, our result Multi-prover interactive proofs and interactive proofs with competing provers are two distinct generalizations of the single-prover model. The next logical step is to unify these two generalizations in the obvious way via interactive proofs with competing _teams_ of provers. Combining established naming conventions for complexity classes based on interactive proofs, we let $\mathrm{MRG}$ denote the class of decision problems that admit interactive proofs with competing teams of provers. To the author’s knowledge, this model was considered prior to the present work only by Feigenbaum, Koller, and Shor [FKS95]. Those authors studied this class under the game-theoretic guise of zero-sum games of _imperfect recall_ and proved the containments $\mathrm{EXP^{NP}}\subseteq\mathrm{MRG}\subseteq\mathrm{\Sigma^{EXP}_{2}}\cap\mathrm{\Pi^{EXP}_{2}}$ where $\mathrm{\Sigma^{EXP}_{2}}$ and $\mathrm{\Pi^{EXP}_{2}}$ are classes in the second level of the exponential hierarchy, which is the exponential-time version of the familiar polynomial hierarchy. In this paper, we consider interactive proofs with competing teams of _no- signaling_ provers. Our main result is as follows. ###### Theorem 1. Every decision problem that admits a two-turn interactive proof with competing teams of two no-signaling provers per team is also in $\mathrm{PSPACE}$. This upper bound matches the aforementioned $\mathrm{PSPACE}$ lower bounds on the following two disparate and weaker classes of interactive proof: 1. 1. Two-turn multi-prover interactive proofs with only one team of no-signaling provers [CCL94, IKM09]. 2. 2. Two-turn competing-prover interactive proofs with only one prover per team [FK97]. Our result implies that the complexity of these two models is unchanged by the addition of a second competing team of no-signaling provers in the first case and by the addition of a second no-signaling prover to each team in the second case. Moreover, our result unifies and subsumes prior $\mathrm{PSPACE}$ upper bounds on these classes [Ito10, FK97]. #### Limitations of the present approach Attention is restricted in this paper to interactions with no more than two no-signaling provers per team and no more than two messages exchanged with each prover. The purpose for this restriction, quite simply, is that this class of interactions appears to be the largest to which our techniques apply. For all we know, interactions with three messages for a prover or three provers on a team could be sufficiently powerful to capture all of $\mathrm{EXP}$. Indeed, it is consistent with current knowledge that a three- message protocol for $\mathrm{EXP}$ might require only _one_ prover per team, or that a three-prover no-signaling protocol for $\mathrm{EXP}$ might require only _one_ team of provers. Given this paucity of upper bounds for similar, seemingly weaker models it is hoped that any reservation at the restrictions in our model is more than compensated by the fact that we are able to say anything at all about it. Let us list some natural extensions of the two-prover, two-turn model and point out exactly where our metchod fails for these extensions. More than two turns, only one prover per team. Perhaps the most important open problem related to our work is the complexity of $k$-turn interactive proofs with competing provers for constants $k\geq 3$. This problem, which dates back at least to 1997 [FK97], is still open even in the special case of only one prover per team. With only one prover per team, the question is really a game-theoretic question with a much wider application than just interactive proofs. Our method fails for this case because we do not have a bound on the verifier matrix of the form $V\leq e_{\mathcal{A}_{01}\mathcal{B}_{01}}p^{}$ such as that appearing in Proposition 3. Thus, we do not obtain a good enough bound on the loss vectors appearing in our variant of the multiplicative weights update method. More than two turns, only one team of no-signaling provers. The complexity of $k$-turn multi-prover interactive proofs with two no- signaling provers is still open for $k\geq 3$, even with only one team of provers [Ito10]. For ordinary multi-prover interactive proofs—in which the provers are not allowed to implement arbitrary no-signaling strategies—it is known that a multi-turn protocol with any number of provers can be simulated by another protocol with only two turns and two provers [FL92]. Our method fails here for the same reason as above—that we cannot bound the loss vectors in the multiplicative weights update method for a multi-turn verifier. More than two provers, only one team of no-signaling provers. Similarly, the complexity of two-turn multi-prover interactive proofs with more than two no-signaling provers is still open, even with only one team of provers [Ito10]. As mentioned above, ordinary multi-prover interactive proofs require only two provers [FL92]. Our method does not extend to this case either, as there is no known analogue of Lemma 6 for more than two provers. Quantum verifier and/or provers. Even with two no-signaling provers, two turns of interaction, and only one team of provers, it is still not known that the $\mathrm{PSPACE}$ upper bound holds when either the verifier or provers can send quantum messages [Ito10]. Here the problem is that Lemma 6 does not hold for quantum states. #### Techniques Theorem 1 is proven by means of an efficient parallel algorithm that, given an explicit description of a verifier and an accuracy parameter $\delta$, finds no-signaling strategies for the teams that are within $\delta$ of optimal. Containment in $\mathrm{PSPACE}$ then follows in the usual way by observing that the description of the verifier has size exponential in the length of the input string $x$ and then employing the fact that a parallel algorithm with succinct input can be simulated in polynomial space [Bor77]. Our algorithm is an example of the _multiplicative weights update method (MWUM)_ as discussed in the survey paper [AHK05] and in the PhD thesis of Kale [Kal07]. (See also Ref. [WK06].) In its simplest form, the MWUM solves a min- max optimization problem on probability distributions. In the present paper we use the MWUM to optimize not just a _single_ distribution, but many distributions _simultaneously_ in the form of a stochastic matrix that represents a strategy for one of the teams. This trick seems to work only for two-turn protocols, as otherwise it is not clear how to ensure sufficient accuracy. Let us compare our algorithm to the two previous algorithms it subsumes: • The polynomial-space algorithm of Feige and Kilian for two-turn interactive proofs with competing provers [FK97] is a complicated and highly specialized precursor to the MWUM that, like our algorithm, optimizes over stochastic matrices that represent strategies for the provers. Their algorithm works by nondeterministically guessing the entries of the matrix and scanning them in a read-once fashion. This approach cannot be extended to optimize over no-signaling strategies, as the read-once model does not allow verification of the no-signaling condition. * • The parallel algorithm of Ito for two-turn, two-prover interactive proofs with no-signaling provers [Ito10] is essentially a reduction to the _mixed packing and covering problem_ , which is a special type of linear program that is known to admit an efficient parallel algorithm [You01]. This approach, too, cannot be extended to competing teams of no-signaling provers, as any linear programming formulation of the protocol is unlikely to be a mixed packing and covering problem. Our study has benefitted from the valuable experience of recent applications of the MWUM to parallel algorithms for quantum complexity classes [JW09, JUW09, JJUW10, Wu10, GW11]. Indeed, we follow the same high-level approach as the recent proof of $\mathrm{DQIP}=\mathrm{DIP}=\mathrm{PSPACE}$ [GW11]. Namely, * • The domain of admissible (no-signaling) strategies is a strict subset of the “natural” domain (stochastic matrices) for the MWUM. * • To get around this problem, the strategy domain is extended to _all_ the stochastic matrices and a _penalty term_ is introduced so as to remove any incentive for a team to use an inadmissible strategy. (See Section 3). * • Finally, one must prove a “rounding” theorem (Corollary 4.1), which establishes that near-optimal, fully-admissible strategies can be obtained from near-optimal strategies in the extended domain with penalty term. ## 2 Preliminaries ### 2.1 Definition of two-turn interactive proofs with competing teams of provers In this paper we are concerned with decision problems that admit two-turn interactive proofs with competing teams of no-signaling provers. Let us clarify this concept. A _two-turn verifier_ is a randomized polynomial-time algorithm that, given an input string $x$, produces questions $i,j$ for the two teams of provers. The teams select their answers $k,l$ (possibly using randomness to do so) and then the verifier accepts or rejects the input $x$ according to some boolean function of $i,j,k,l$. For convenience, the teams shall be called _Team Alice_ and _Team Bob_. It is the goal of Team Alice to convince the verifier to accept the input string $x$, while Team Bob’s goal is to convince the verifier to reject $x$. In the protocols we consider each team consists of two provers. The provers of Team Alice shall be called _Alice 0_ and _Alice 1_, while the provers of Team Bob shall be called _Bob 0_ and _Bob 1_. Each individual prover on each team receives his or her own private question and supplies his or her own separate answer to the verifier. In particular, the question $i$ asked of Team Alice is actually a pair $i=(i_{0},i_{1})$ with question $i_{c}$ going to prover Alicec for both values of the bit $c\in\\{0,1\\}$. Similarly, the question $j$ asked of Team Bob is also a pair $j=(j_{0},j_{1})$ with question $j_{c}$ going to prover Bobc. The answers $k,l$ received from the two teams are also pairs $k=(k_{0},k_{1})$ and $l=(l_{0},l_{1})$ with answers $k_{c}$ and $l_{c}$ coming from Alicec and Bobc, respectively. The entire interaction is illustrated in Figure 1. Figure 1: A two-turn interactive proof with competing teams of two no- signaling provers per team. Each team may jointly implement any no-signaling strategy in order to produce its answers. Briefly, a strategy for, say, Team Alice is _no-signaling_ if the marginal distribution on answers $k_{0}$ from Alice0 does not depend upon the question $i_{1}$ asked of Alice1 and _vice versa_. No-signaling strategies are discussed in greater detail in Section 2.5. A decision problem $L$ is said to admit a two-turn interactive proof with competing teams of no-signaling provers with _completeness_ $c$ and _soundness_ $s$ if there exists a fixed two-turn verifier with the following properties: Completeness. If the input string $x$ is a yes-instance of $L$ then there exists a no- signaling strategy for Team Alice that convinces the verifier to accept $x$ with probability at least $c$, regardless of the no-signaling strategy employed by Team Bob. Soundness. If the input string $x$ is a no-instance of $L$ then there exists a no- signaling strategy for Team Bob that convinces the verifier to reject $x$ with probability at least $1-s$, regardless of the no-signaling strategy employed by Team Alice. The completeness and soundness parameters need not be fixed constants. Rather, they may vary as a function of the input string $x$. The complexity class $\mathrm{MRG_{\mathrm{ns}}}(2,2)$ consists of all decision problems that admit two-turn interactive proofs with competing teams of two no-signaling provers per team with completeness $c$ and soundness $s$ such that there exists a fixed polynomial-bounded function $p$ on strings with $c-s\geq 1/p$. (The first parameter of the class $\mathrm{MRG}_{\mathrm{ns}}(2,2)$ denotes the number of provers per team, the second denotes the number of turns in the protocol. It is also common to parameterize interactive proof classes according to the number of _rounds_ of communication, rather than the number of _turns_. Under this scheme, the class $\mathrm{MRG_{\mathrm{ns}}}(2,2)$ might be called $\mathrm{MRG_{\mathrm{ns}}}(2,1)$ by some authors.) In this paper we prove $\mathrm{MRG_{\mathrm{ns}}}(2,2)\subseteq\mathrm{PSPACE}$. It then follows from existing lower bounds on weaker classes [IKM09, FK97] that $\mathrm{MRG_{\mathrm{ns}}}(2,2)=\mathrm{PSPACE}.$ ### 2.2 Notation, the Kronecker product To each interactive proof with input $x$ we associate eight distinct finite- dimensional real Euclidean spaces—four _question_ spaces and four _answer_ spaces. These spaces are denoted as follows for both $c\in\\{0,1\\}$: $\displaystyle\mathcal{S}_{c}\quad$ The question space for prover Alicec $\displaystyle\mathcal{A}_{c}\quad$ The answer space for prover Alicec $\displaystyle\mathcal{T}_{c}\quad$ The question space for prover Bobc $\displaystyle\mathcal{B}_{c}\quad$ The answer space for prover Bobc The dimension of each space is the number of distinct questions or answers available to that prover. (For example, prover Alice0 can be asked any of $\dim(\mathcal{S}_{0})$ distinct questions and may respond with any of $\dim(\mathcal{A}_{0})$ distinct answers.) Individual questions or answers are indexed by positive integers denoted for both $c\in\\{0,1\\}$ as follows: $\displaystyle\textrm{Questions for Alice${}_{c}$}:\quad$ $\displaystyle i_{c}=1,\dots,\dim(\mathcal{S}_{c})$ $\displaystyle\textrm{Questions for Bob${}_{c}$}:\quad$ $\displaystyle j_{c}=1,\dots,\dim(\mathcal{T}_{c})$ $\displaystyle\textrm{Answers from Alice${}_{c}$}:\quad$ $\displaystyle k_{c}=1,\dots,\dim(\mathcal{A}_{c})$ $\displaystyle\textrm{Answers from Bob${}_{c}$}:\quad$ $\displaystyle l_{c}=1,\dots,\dim(\mathcal{B}_{c})$ Since the verifier acts in polynomial time, the bit length of the questions and answers is at most a polynomial in the bit length $\|x\|$ of the input string $x$. Since $n$ bits suffice to encode $2^{n}$ distinct questions or answers, the dimension of the spaces $\mathcal{S}_{c},\mathcal{T}_{c},\mathcal{A}_{c},\mathcal{B}_{c}$ can be exponential in $\|x\|$. The _Kronecker product_ (or _tensor product_) of two spaces $\mathcal{X},\mathcal{Y}$ is another space with dimension $\dim(\mathcal{X})\dim(\mathcal{Y})$. This product space is typically denoted by $\mathcal{X}\otimes\mathcal{Y}$, which we abbreviate to $\mathcal{X}\mathcal{Y}$. Kronecker products involving the eight spaces $\mathcal{S}_{c},\mathcal{T}_{c},\mathcal{A}_{c},\mathcal{B}_{c}$ are further abbreviated so that $\mathcal{S}_{01}=\mathcal{S}_{0}\mathcal{S}_{1}=\mathcal{S}_{0}\otimes\mathcal{S}_{1}$ and so on. The Kronecker product extends in a natural way to vectors and linear operators. In this paper each vector or linear operator is implicitly associated with its representation as a column or a matrix, for which the Kronecker product is given by a straightforward formula. For example, if $A,B$ are $2\times 2$ matrices given by $A=\left[\begin{array}[]{cc}a&b\\\ c&d\end{array}\right],\qquad B=\left[\begin{array}[]{cc}p&q\\\ r&s\end{array}\right]$ then the Kronecker product $A\otimes B$ is given by $A\otimes B=\left[\begin{array}[]{cc}aB&bB\\\ cB&dB\end{array}\right]=\left[\begin{array}[]{cc}a\left[\begin{array}[]{cc}p&q\\\ r&s\end{array}\right]&b\left[\begin{array}[]{cc}p&q\\\ r&s\end{array}\right]\\\ c\left[\begin{array}[]{cc}p&q\\\ r&s\end{array}\right]&d\left[\begin{array}[]{cc}p&q\\\ r&s\end{array}\right]\end{array}\right]=\left[\begin{array}[]{cccc}ap&aq&bp&bq\\\ ar&as&br&bs\\\ cp&cq&dp&dq\\\ cr&cs&dr&ds\end{array}\right].$ This definition extends in the obvious way to arbitrary matrices of any dimension, including column vectors and other non-square matrices. We also make use of the following symbols: $e_{\mathcal{X}}$ \| The all-ones vector of dimension $\dim(\mathcal{X})$. ---\|--- $I_{\mathcal{X}}$ \| The identity matrix acting on $\mathcal{X}$. $M^{}$ \| The _adjoint_ of a linear mapping $M$. If $M$ is a matrix or column vector then $M^{}$ is simply the transpose of $M$. $\langle A,B\rangle$ \| The _matrix inner product_ , defined as $\operatorname{Tr}(A^{}B)$. This inner product is defined only when the dimensions of $A,B$ are equal. If $A,B$ are vectors then $\langle A,B\rangle$ is called the _vector inner product_. $\leq,\geq$ \| Matrix inequalities are entrywise. ${\overline{c}}$ \| Given a bit $c\in\\{0,1\\}$, the compliment ${\overline{c}}$ is given by ${\overline{c}}=1$ if $c=0$, otherwise ${\overline{c}}=0$. ### 2.3 Min-max formalism for interactive proofs with competing provers Given a fixed two-turn verifier and a fixed input string $x$, let $\pi_{i,j}$ denote the probability with which the verifier asks questions $i=(i_{0},i_{1})$ to Team Alice and $j=(j_{0},j_{1})$ to Team Bob. For each 4-tuple $(i,j)$ of questions to the provers let $v_{i,j}\in\mathcal{A}_{01}\mathcal{B}_{01}$ denote the 0-1 vector of _payouts_ to Team Bob. That is, for each $k=(k_{0},k_{1})$ and each $l=(l_{0},l_{1})$ the $(k,l)$th entry of $v_{i,j}$ is either zero or one according to whether the verifier accepts or rejects $x$ in the event that the verifier asks questions $(i,j)$ to the teams and they respond with answers $(k,l)$.111 One could consider a more general referee in which the payouts are awarded probabilistically so that each entry of $v_{i,j}$ lies in the interval $[0,1]$. But it is easily seen that this model is equivalent to the one we have just described.222 The payout vector $v_{i,j}$ is defined so that 0 indicates acceptance of $x$ while 1 indicates rejection. This arbitrary choice is opposite of convention, but it better facilitates the forthcoming presentation of our multiplicative weights update algorithm. Consider the entrywise nonnegative matrix $V:\mathcal{S}_{01}\mathcal{T}_{01}\to\mathcal{A}_{01}\mathcal{B}_{01}$ whose $(i,j)$th column is $\pi_{i,j}v_{i,j}$. This matrix uniquely specifies the actions of the verifier. Strategies for the teams are specified as follows. For each pair $i$ of questions let $a_{i}\in\mathcal{A}_{01}$ denote the probability vector of Team Alice’s responses to $i$. That is, for each pair $k$ of answers the $k$th entry of $a_{i}$ denotes the probability with which Team Alice replies with answers $k$ given that questions $i$ were asked. Thus, the actions of Team Alice are uniquely specified by the stochastic matrix $A:\mathcal{S}_{01}\to\mathcal{A}_{01}$ whose $i$th column is $a_{i}$. Similarly, for each pair $j$ of questions let $b_{j}\in\mathcal{B}_{01}$ denote the probability vector of Team Bob’s responses to $j$. The actions of Team Bob are uniquely specified by the stochastic matrix $B:\mathcal{T}_{01}\to\mathcal{B}_{01}$ whose $j$th column is $b_{j}$. Not every stochastic matrix denotes a valid no- signaling strategy for the teams. Criteria for no-signaling strategies are discussed in Section 2.5. For now, it suffices to note that the set of all strategies available to each team is a compact convex subset of stochastic matrices. Conditioned on the verifier asking questions $(i,j)$, it is clear that the probability of rejection is given by the vector inner product $\left\langle v_{i,j},a_{i}\otimes b_{j}\right\rangle.$ It follows that the probability of rejection—taken over all questions $(i,j)$—given strategies $A$ for Team Alice and $B$ for Team Bob is given by the matrix inner product $\Pr[\textrm{$V$ rejects $x$}\mid A,B]=\langle V,A\otimes B\rangle=\sum_{i,j}\pi_{i,j}\left\langle v_{i,j},a_{i}\otimes b_{j}\right\rangle.$ Of course, Team Bob wishes to maximize this quantity while Team Alice wishes to minimize this quantity. Given that the above inner product is bilinear in $(A,B)$ and that the sets of admissible strategies for the two teams are compact and convex, it follows from standard min-max theorems [Vil38, Fan53] that every interactive proof with verifier $V$ has an _equilibrium value_ , which we denote by $\lambda(V)$, given by $\lambda(V)=\min_{A}\max_{B}\langle V,A\otimes B\rangle=\max_{B}\min_{A}\langle V,A\otimes B\rangle$ where the minimum is over all no-signaling matrices $A:\mathcal{S}_{01}\to\mathcal{A}_{01}$ and the maximum is over all no- signaling matrices $B:\mathcal{T}_{01}\to\mathcal{B}_{01}$. In particular, for every protocol there exists at least one _equilibrium point_ $(A^{\star},B^{\star})$ with the property that $\displaystyle\langle V,A^{\star}\otimes B\rangle$ $\displaystyle\leq\lambda(V)\quad\textrm{for all $B$},$ $\displaystyle\langle V,A\otimes B^{\star}\rangle$ $\displaystyle\geq\lambda(V)\quad\textrm{for all $A$}.$ Thus, the strategy $B^{\star}$ always ensures maximum likelihood of rejection, while $A^{\star}$ always ensures minimum likelihood of rejection. This min-max theorem applies to every min-max expression considered throughout this paper. Henceforth we do not bother to explicitly remark upon this fact. Here and throughout the paper we adopt the convention that for any min-max problem of the form $\nu(g)=\min_{a\in\mathbf{A}}\max_{b\in\mathbf{B}}g(a,b)$ elements $\tilde{a}\in\mathbf{A}$ and $\tilde{b}\in\mathbf{B}$ are _$\delta$ -optimal_ if $\displaystyle g(\tilde{a},b)$ $\displaystyle\leq\nu(g)+\delta\quad\textrm{for all $b\in\mathbf{B}$},$ $\displaystyle g(a,\tilde{b})$ $\displaystyle\geq\nu(g)-\delta\quad\textrm{for all $a\in\mathbf{A}$}.$ Elements that are $0$-optimal—such as $A^{\star},B^{\star}$ above—are simply called _optimal_. ### 2.4 Notation for marginal distributions Before we discuss no-signaling strategies in detail it is beneficial to introduce notation for marginal probability distributions that will be used throughout the remainder of this paper. Suppose, for instance, that $a\in\mathcal{A}_{01}$ is a probability vector of answers from Team Alice to some question from the verifier. We let $\operatorname{mar}_{\mathcal{A}_{1}}(a)\in\mathcal{A}_{0}$ denote the probability vector for the marginal distribution on answers from the prover Alice0. Basic probability theory dictates that the mapping $\operatorname{mar}_{\mathcal{A}_{1}}$ satisfy $\textrm{$k_{0}$th entry of $\operatorname{mar}_{\mathcal{A}_{1}}(a)$}\equiv\sum_{k_{1}=1}^{\dim(\mathcal{A}_{1})}\textrm{$(k_{0},k_{1})$th entry of $a$}.$ Of course, this mapping may be extended to arbitrary real vectors. For arbitrary spaces $\mathcal{X},\mathcal{Y}$ the linear mapping $\operatorname{mar}_{\mathcal{Y}}$ is defined by $\operatorname{mar}_{\mathcal{Y}}:\mathcal{X}\mathcal{Y}\to\mathcal{X}:x\otimes y\mapsto\langle e_{\mathcal{Y}},y\rangle x.$ (The matrix representation of $\operatorname{mar}_{\mathcal{Y}}$ is $e_{\mathcal{Y}}^{}\otimes I_{\mathcal{X}}$.) While this mapping is primarily intended to denote marginal probability distributions, we will have occasion to use it on non-probability vectors in this paper. The mapping $\operatorname{mar}_{\mathcal{Y}}$ is to vectors as the _partial trace_ is to square matrices. Readers familiar with quantum information know that the state of a quantum register can be computed from a joint state of several registers via the partial trace. So too with probability distributions: the distribution on states of a classical register can be computed from a joint distribution on states of several registers via $\operatorname{mar}_{\mathcal{Y}}$. The mapping $\operatorname{mar}_{\mathcal{Y}}$ extends naturally from vectors to matrices by applying $\operatorname{mar}_{\mathcal{Y}}$ to each column: $\textrm{$i$th column of $\operatorname{mar}_{\mathcal{Y}}(A)$}\equiv\operatorname{mar}_{\mathcal{Y}}\left(\textrm{$i$th column of $A$}\right).$ So, for example, if Team Alice acts according to the stochastic matrix $A$ then the stochastic matrix $\operatorname{mar}_{\mathcal{A}_{1}}(A):\mathcal{S}_{01}\to\mathcal{A}_{0}$ describes the “marginal” strategy for prover Alice0. That is, the $(i_{0},i_{1})$th column of $\operatorname{mar}_{\mathcal{A}_{1}}(A)$ is the distribution on answers $k_{0}$ from Alice0 given questions $(i_{0},i_{1})$ from the verifier. ### 2.5 Characterization of no-signaling strategies Recall that a strategy for Team Alice is _no-signaling_ if for both values of the bit $c\in\\{0,1\\}$ the marginal distribution on answers $k_{c}$ from Alicec does not depend on the question $i_{\overline{c}}$ asked of Alice${}_{\overline{c}}$. In terms of Team Alice’s stochastic matrix $A$, this condition means that for each $i_{c}$ the $(i_{0},i_{1})$th column of $\operatorname{mar}_{\mathcal{A}_{\overline{c}}}(A)$ is identical for all subindices $i_{\overline{c}}$. Letting $a_{i_{c}}$ denote this fixed probability vector and letting $A_{c}:\mathcal{S}_{c}\to\mathcal{A}_{c}$ denote the stochastic matrix whose columns are $a_{i_{c}}$, the above condition can be written as $\operatorname{mar}_{\mathcal{A}_{c}}(A)=A_{c}\otimes e_{\mathcal{S}_{\overline{c}}}^{}.$ We have just proven the following simple proposition. ###### Proposition 2 (Characterization of no-signaling strategies). A stochastic matrix $A:\mathcal{S}_{01}\to\mathcal{A}_{01}$ denotes a no- signaling strategy for Team Alice if and only if for both values of the bit $c\in\\{0,1\\}$ there exists a stochastic matrix $A_{c}:\mathcal{S}_{c}\to\mathcal{A}_{c}$ such that $\operatorname{mar}_{\mathcal{A}_{\overline{c}}}(A)=A_{c}\otimes e_{\mathcal{S}_{\overline{c}}}^{}.$ A similar characterization holds for Team Bob. Stochastic matrices $A$ meeting this condition are called _no-signaling matrices_. The matrices $A_{c}$ are said to _witness_ the fact that $A$ is a no-signaling matrix. It follows immediately from Proposition 2 that the set of all no-signaling strategies available to each team is compact and convex—a fact already used in Section 2.3 to assert the existence of optimal strategies for the teams. ## 3 A relaxed min-max problem with penalties As mentioned in the introduction, the MWUM in its simplest form solves min-max optimization problems over probability vectors. We optimize over stochastic matrices for the teams by using the MWUM simultaneously on each column of these matrices—a trick that works only for two-turn protocols, as we shall soon see. We noted in Section 2.5 that the no-signaling matrices available to the teams form a strict subset of the stochastic matrices. In order to optimize only over no-signaling matrices, in this section we specify a new min-max optimization problem $\mu(V)$ in which the teams may use _arbitrary_ strategies but pay a _penalty_ for strategies that violate the no-signaling condition. By a careful choice of penalty, we remove the incentive of the teams to select inadmissible strategies without ruining the precarious convergence properties of the MWUM. Some preliminary observations are given in Section 3.1 before the formal definition of the new min-max problem $\mu(V)$ in Section 3.2. Equivalence of $\mu(V)$ and $\lambda(V)$ is proven in Section 3.3 with proofs of some lemmas in Section 3.4. ### 3.1 Bounds on two-turn verifiers First, for ease of notation we let $\Phi_{V}$ denote the unique linear transformation satisfying $\left\langle V,A\otimes B\right\rangle=\left\langle\Phi_{V}(A),B\right\rangle=\left\langle A,\Phi_{V}^{}(B)\right\rangle$ for all matrices $A,B$. Though a precise formula for $\Phi_{V}$ is of little use in this paper, for completeness we note that $\displaystyle\Phi_{V}(A)$ $\displaystyle=\operatorname{Tr}_{\mathcal{S}_{01}}\left(\left(A^{}\otimes I_{\mathcal{B}_{01}}\right)V\right)$ $\displaystyle\Phi_{V}^{}(B)$ $\displaystyle=\operatorname{Tr}_{\mathcal{T}_{01}}\left(\left(I_{\mathcal{A}_{01}}\otimes B^{}\right)V\right)$ where $\operatorname{Tr}_{\mathcal{S}_{01}}$ and $\operatorname{Tr}_{\mathcal{T}_{01}}$ denote _partial trace_ transformations. At the risk of hijacking terminology from functional analysis, the matrix $\Phi_{V}(A)$ can be viewed as a _partial inner product_ between $V$ and $A$. This matrix can also be viewed as a new two-turn verifier for Team Bob obtained by “hard-wiring” Team Alice’s strategy $A$ into the original verifier $V$. Next, let $p\in\mathcal{S}_{01}\mathcal{T}_{01}$ denote the probability vector for the distribution on questions asked by the verifier. In the notation of Section 2.3, the $(i,j)$th entry of $p$ is $\pi_{i,j}$—the probability with which the verifier asks questions $i$ to Team Alice and $j$ to Team Bob. Let $p_{\textrm{Alice}}\in\mathcal{S}_{01}$ denote the marginal distribution $p_{\textrm{Alice}}=\operatorname{mar}_{\mathcal{T}_{01}}(p)$ on questions to Team Alice, so that the $i$th entry of $p_{\textrm{Alice}}$ is $\sum_{j}\pi_{i,j}$. It is not hard to see that $V\leq e_{\mathcal{A}_{01}\mathcal{B}_{01}}p^{}$ with equality achieved in the extreme case that each of the verifier’s payout vectors $v_{i,j}$ is equal to the all-ones vector $e_{\mathcal{A}_{01}\mathcal{B}_{01}}$. (Recall that matrix inequalities are entrywise.) Similarly, it is easy to prove analogous inequalities for $\Phi_{V}(A),\Phi_{V}^{}(B)$. For example: ###### Proposition 3. For any stochastic matrix $B:\mathcal{T}_{01}\to\mathcal{B}_{01}$ it holds that $\Phi_{V}^{}(B)\leq e_{\mathcal{A}_{01}}p_{\textrm{Alice}}^{}.$ ###### Proof. Let $A:\mathcal{S}_{01}\to\mathcal{A}_{01}$ be any nonnegative matrix and let $a_{i},b_{j}$ denote the columns of $A,B$, respectively. Then $\left\langle A,\Phi_{V}^{}(B)\right\rangle=\left\langle V,A\otimes B\right\rangle\leq\left\langle e_{\mathcal{A}_{01}\mathcal{B}_{01}}p^{},A\otimes B\right\rangle=\sum_{i,j}\pi_{i,j}\left\langle e_{\mathcal{A}_{01}},a_{i}\right\rangle\left\langle e_{\mathcal{B}_{01}},b_{j}\right\rangle$ As $B$ is stochastic it must be that $\left\langle e_{\mathcal{B}_{01}},b_{j}\right\rangle=1$ for each $j$. The above expression then simplifies to $\sum_{i}\left(\sum_{j}\pi_{i,j}\right)\left\langle e_{\mathcal{A}_{01}},a_{i}\right\rangle=\left\langle e_{\mathcal{A}_{01}}p_{\textrm{Alice}}^{},A\right\rangle.$ As this inequality holds for all nonnegative matrices $A$ it must be that $\Phi_{V}^{}(B)\leq e_{\mathcal{A}_{01}}p_{\textrm{Alice}}^{}$ as claimed. ∎ ### 3.2 Definition of the relaxed min-max problem The relaxation $\mu(V)$ of $\lambda(V)$ is defined by $\mu(V)=\min_{(A,A_{0},A_{1})}\max_{(B,\Pi_{0},\Pi_{1})}\left\langle f_{V}(A,A_{0},A_{1}),(B,\Pi_{0},\Pi_{1})\right\rangle$ where the triples $(A,A_{0},A_{1})$ and $(B,\Pi_{0},\Pi_{1})$ have the form $\displaystyle A$ $\displaystyle:\mathcal{S}_{01}\to\mathcal{A}_{01}$ any stochastic $\displaystyle A_{c}$ $\displaystyle:\mathcal{S}_{c}\to\mathcal{A}_{c}$ any stochastic $\displaystyle\qquad c\in\\{0,1\\}$ $\displaystyle B$ $\displaystyle:\mathcal{T}_{01}\to\mathcal{B}_{01}$ no-signaling only $\displaystyle\Pi_{c}$ $\displaystyle:\mathcal{S}_{01}\to\mathcal{A}_{c}\qquad\qquad$ $\displaystyle 0\leq\Pi_{c}\leq e_{\mathcal{A}_{c}}p_{\textrm{Alice}}^{}$ $\displaystyle\qquad c\in\\{0,1\\}.$ The linear mapping $f_{V}$ appearing in the inner product (and its adjoint) is defined by $\displaystyle f_{V}$ $\displaystyle:(A,A_{0},A_{1})\mapsto\left(\Phi_{V}(A)\,,\,\operatorname{mar}_{\mathcal{A}_{1}}(A)-A_{0}\otimes e_{\mathcal{S}_{1}}^{}\,,\,\operatorname{mar}_{\mathcal{A}_{0}}(A)-A_{1}\otimes e_{\mathcal{S}_{0}}^{}\right)$ $\displaystyle f_{V}^{}$ $\displaystyle:(B,\Pi_{0},\Pi_{1})\mapsto\left(\Phi_{V}^{}(B)+e_{\mathcal{A}_{1}}\otimes\Pi_{0}+e_{\mathcal{A}_{0}}\otimes\Pi_{1}\,,\,-\Pi_{0}\left(I_{\mathcal{S}_{0}}\otimes e_{\mathcal{S}_{1}}\right)\,,\,-\Pi_{1}\left(I_{\mathcal{S}_{1}}\otimes e_{\mathcal{S}_{0}}\right)\right)$ so that $\left\langle f_{V}(A,A_{0},A_{1}),(B,\Pi_{0},\Pi_{1})\right\rangle=\left\langle V,A\otimes B\right\rangle+\sum_{c\in\\{0,1\\}}\left\langle\operatorname{mar}_{\mathcal{A}_{\overline{c}}}(A)-A_{c}\otimes e_{\mathcal{S}_{\overline{c}}}^{},\Pi_{c}\right\rangle$ for all $(A,A_{0},A_{1})$ and all $(B,\Pi_{0},\Pi_{1})$. (The adjoint mapping $f_{V}^{}$ is not used until the algorithm of Figure 2 and its proof of correctness in Proposition 8.) #### Intuition Some explanation is in order. As with the original min-max problem $\lambda(V)$, the matrices $A$ and $B$ represent the strategies employed by the teams. Note, however, that in the definition of $\mu(V)$ Team Alice is now free to choose among arbitrary stochastic matrices for its strategy. The matrices $A_{0},A_{1}$ for Team Alice are purported witnesses to the claim that $A$ is a valid no-signaling matrix. For the moment, we are concerned with relaxing the domain only of Team Alice’s strategies, so Bob’s strategy $B$ must still be no-signaling. Bob’s strategies will be addressed in Section 4.2. The matrices $\Pi_{0},\Pi_{1}$ for Team Bob are _penalty matrices_ —they are the means by which Team Bob penalizes Team Alice according to the extent that $A_{0},A_{1}$ are false witnesses to the claim that $A$ is no-signaling. The new objective function $\left\langle f_{V}(A,A_{0},A_{1}),(B,\Pi_{0},\Pi_{1})\right\rangle$ equals the old objective function $\langle V,A\otimes B\rangle$ plus two _penalty terms_. If $A$ is not a no-signaling matrix then the difference matrix $\Delta_{c}\equiv\operatorname{mar}_{\mathcal{A}_{\overline{c}}}(A)-A_{c}\otimes e_{\mathcal{S}_{\overline{c}}}^{}$ must be nonzero for at least one $c$. In this case, Bob selects $\Pi_{c}$ to pick out the positive entries of $\Delta_{c}$, which are then added the verifier’s probability of rejection. Let us informally explain why the restriction $0\leq\Pi_{c}\leq e_{\mathcal{A}_{c}}p_{\textrm{Alice}}^{}$ on penalty matrices is sufficient to remove Team Alice’s incentive to cheat. Suppose the $k_{c}$th entry of the $i$th column of the difference matrix $\Delta_{c}$ is a positive real number $\delta>0$ and suppose that $A^{\prime}$ is a valid no-signaling matrix witnessed by $A_{0},A_{1}$. Since the verifier asks questions $i$ of Team Alice with probability $\pi_{i}$, it must be that, when selecting the probability with which to answer $k_{c}$, the advantage gained by Team Alice from using the inadmissible strategy $A$ instead of the no-signaling strategy $A^{\prime}$ is at most $\delta\pi_{i}$. By selecting a penalty matrix $\Pi_{c}$ so that the $k_{c}$th entry of the $i$th column of $\Pi_{c}$ is equal to $\pi_{i}$, Team Bob adds precisely the quantity $\delta\pi_{i}$ to the verifier’s probability of rejection, thus eliminating the advantage obtained by Team Alice in acting according to $A$ instead of $A^{\prime}$ for this particular choice of questions $i$ and answer $k_{c}$ from Alicec. Repeating this logic for all entries $(i,k_{c})$ of $\Delta_{c}$, we find that Team Bob should select the penalty matrix $\Pi_{c}$ so that the $(i,k_{c})$th entry is either zero or $\pi_{i}$ according to whether the corresponding entry of $\Delta_{c}$ is nonpositive or positive. A penalty matrix of this form is called _optimal for $(A,A_{0},A_{1})$_ and satisfies $\left\langle\Delta_{c},\Pi_{c}\right\rangle=\left\langle\Delta_{c}^{+},e_{\mathcal{A}_{c}}p_{\textrm{Alice}}^{}\right\rangle$ where $\Delta_{c}^{+}$ is the positive part of $\Delta_{c}$. (Here the _positive part_ of a real matrix $X$ is the matrix $X^{+}$ with the property that if $x$ is any entry of $X$ then the corresponding entry of $X^{+}$ is $\max\\{0,x\\}$.) ### 3.3 Equivalence of the two min-max problems We are now ready to prove the desired “rounding theorem” mentioned in the introduction, a corollary of which is the equivalence of the min-max problems $\mu(V)$ and $\lambda(V)$ (Corollary 4.1). The theorem employs two lemmas and their corollaries, the proofs of which appear below in Section 3.4. ###### Theorem 4 (Rounding theorem). Let $(A,A_{0},A_{1})$ be a feasible solution for $\mu(V)$ and let $\Pi_{0}^{A},\Pi_{1}^{A}$ be optimal penalties for $(A,A_{0},A_{1})$. There exists a no-signaling matrix $A_{\mathrm{ns}}$ witnessed by $A_{0},A_{1}$ such that for all stochastic matrices $B$ it holds that $\left\langle V,A_{\mathrm{ns}}\otimes B\right\rangle\leq\left\langle f_{V}(A,A_{0},A_{1}),(B,\Pi_{0}^{A},\Pi_{1}^{A})\right\rangle.$ Moreover, $A_{\mathrm{ns}}$ can be computed efficiently in parallel given $(A,A_{0},A_{1})$. ###### Proof. For both $c\in\\{0,1\\}$ let $\Delta_{c}^{+}$ be the positive part of $\operatorname{mar}_{\mathcal{A}_{\overline{c}}}(A)-A_{c}\otimes e_{\mathcal{S}_{\overline{c}}}^{}$ and observe that $\Delta_{c}^{+}\leq\operatorname{mar}_{\mathcal{A}_{\overline{c}}}(A).$ By Corollary 5.1 below there exists a preimage $D_{0}^{+}\geq 0$ of $\Delta_{0}^{+}$ with $\displaystyle A-D_{0}^{+}$ $\displaystyle\geq 0$ $\displaystyle\operatorname{mar}_{\mathcal{A}_{1}}(D_{0}^{+})$ $\displaystyle=\Delta_{0}^{+}.$ Let $\Gamma_{1}^{+}$ be the positive part of $\operatorname{mar}_{\mathcal{A}_{0}}\left(A-D_{0}^{+}\right)-A_{1}\otimes e_{\mathcal{S}_{0}}^{}.$ As with $\Delta_{c}$ above, observe that $\Gamma_{1}^{+}\leq\operatorname{mar}_{\mathcal{A}_{0}}\left(A-D_{0}^{+}\right).$ (Moreover, it is easy to see that $\Gamma_{1}^{+}\leq\Delta_{1}^{+}$—a fact we employ later in this proof.) Apply Corollary 5.1 again to obtain a preimage $C_{1}^{+}\geq 0$ of $\Gamma_{1}^{+}$ with $\displaystyle A-D_{0}^{+}-C_{1}^{+}$ $\displaystyle\geq 0$ $\displaystyle\operatorname{mar}_{\mathcal{A}_{0}}(C_{1}^{+})$ $\displaystyle=\Gamma_{1}^{+}.$ Thus, we have a matrix $A-D_{0}^{+}-C_{1}^{+}\geq 0$ such that for both $c\in\\{0,1\\}$ it holds that $\operatorname{mar}_{\mathcal{A}_{\overline{c}}}\left(A-D_{0}^{+}-C_{1}^{+}\right)\leq A_{c}\otimes e_{\mathcal{S}_{\overline{c}}}^{}.$ Hence there exist nonnegative matrices $T_{c}:\mathcal{S}_{01}\to\mathcal{A}_{c}$ with $\operatorname{mar}_{\mathcal{A}_{\overline{c}}}\left(A-D_{0}^{+}-C_{1}^{+}\right)+T_{c}=A_{c}\otimes e_{\mathcal{S}_{\overline{c}}}^{}.$ Applying $\operatorname{mar}_{A_{c}}$ to both sides of this equation we see that $\operatorname{mar}_{\mathcal{A}_{0}}(T_{0})=\operatorname{mar}_{\mathcal{A}_{1}}(T_{1}).$ By Corollary 6.1 below there exists a nonnegative matrix $T:\mathcal{S}_{01}\to\mathcal{A}_{01}$ with $\operatorname{mar}_{\mathcal{A}_{\overline{c}}}(T)=T_{c}$ for both $c\in\\{0,1\\}$. The desired no-signaling matrix $A_{\mathrm{ns}}$ is given by $A_{\mathrm{ns}}=A-D_{0}^{+}-C_{1}^{+}+T.$ As $D_{0}^{+}$, $C_{1}^{+}$, and $T$ can be computed efficiently in parallel, so too can $A_{\mathrm{ns}}$. To see that $A_{\mathrm{ns}}$ is a no-signaling matrix witnessed by $A_{0},A_{1}$ it suffices to observe that $\operatorname{mar}_{\mathcal{A}_{\overline{c}}}(A_{\mathrm{ns}})=\operatorname{mar}_{\mathcal{A}_{\overline{c}}}\left(A-D_{0}^{+}-C_{1}^{+}\right)+T_{c}=A_{c}\otimes e_{\mathcal{S}_{\overline{c}}}^{}.$ It remains only to verify the stated inequality. To this end, we have $\displaystyle\left\langle V,A_{\mathrm{ns}}\otimes B\right\rangle$ $\displaystyle=\left\langle A,\Phi_{V}^{}(B)\right\rangle-\left\langle D_{0}^{+}+C_{1}^{+},\Phi_{V}^{}(B)\right\rangle+\left\langle T,\Phi_{V}^{}(B)\right\rangle$ $\displaystyle\leq\left\langle A,\Phi_{V}^{}(B)\right\rangle+\left\langle T,\Phi_{V}^{}(B)\right\rangle$ $\displaystyle\leq\left\langle A,\Phi_{V}^{}(B)\right\rangle+\left\langle T,e_{\mathcal{A}_{01}}p_{\textrm{Alice}}^{}\right\rangle$ As $A_{\mathrm{ns}}$ and $A$ are both stochastic matrices, it must be that $D_{0}^{+}+C_{1}^{+}$ and $T$ have the same column sums. As $\langle T,e_{\mathcal{A}_{01}}p_{\textrm{Alice}}^{}\rangle$ equals the sum of the column sums of $T$ weighted according to $p_{\textrm{Alice}}$, the matrix $T$ can be replaced by $D_{0}^{+}+C_{1}^{+}$ without affecting this inner product. That is $\langle T,e_{\mathcal{A}_{01}}p_{\textrm{Alice}}^{}\rangle=\langle D_{0}^{+}+C_{1}^{+},e_{\mathcal{A}_{01}}p_{\textrm{Alice}}^{}\rangle.$ Expanding the right side of this equality we obtain $\left\langle\operatorname{mar}_{\mathcal{A}_{1}}(D_{0}^{+}),e_{\mathcal{A}_{0}}p_{\textrm{Alice}}^{}\right\rangle+\left\langle\operatorname{mar}_{\mathcal{A}_{0}}(C_{1}^{+}),e_{\mathcal{A}_{1}}p_{\textrm{Alice}}^{}\right\rangle=\left\langle\Delta_{0}^{+},e_{\mathcal{A}_{0}}p_{\textrm{Alice}}^{}\right\rangle+\left\langle\Gamma_{1}^{+},e_{\mathcal{A}_{1}}p_{\textrm{Alice}}^{}\right\rangle.$ As $\Gamma_{1}^{+}\leq\Delta_{1}^{+}$ this quantity is at most $\left\langle\Delta_{0}^{+},e_{\mathcal{A}_{0}}p_{\textrm{Alice}}^{}\right\rangle+\left\langle\Delta_{1}^{+},e_{\mathcal{A}_{1}}p_{\textrm{Alice}}^{}\right\rangle.$ Putting everything together, we have $\displaystyle\langle V,A_{\mathrm{ns}}\otimes B\rangle$ $\displaystyle\leq\langle A,\Phi_{V}^{}(B)\rangle+\left\langle\Delta_{0}^{+},e_{\mathcal{A}_{0}}p_{\textrm{Alice}}^{}\right\rangle+\left\langle\Delta_{1}^{+},e_{\mathcal{A}_{1}}p_{\textrm{Alice}}^{}\right\rangle$ $\displaystyle=\langle A,\Phi_{V}^{}(B)\rangle+\left\langle\operatorname{mar}_{\mathcal{A}_{1}}(A)-A_{0}\otimes e_{\mathcal{S}_{1}}^{},\Pi_{0}^{A}\right\rangle+\left\langle\operatorname{mar}_{\mathcal{A}_{0}}(A)-A_{1}\otimes e_{\mathcal{S}_{0}}^{},\Pi_{1}^{A}\right\rangle$ $\displaystyle=\left\langle f_{V}(A,A_{0},A_{1}),(B,\Pi_{0}^{A},\Pi_{1}^{A})\right\rangle.$ as desired. ∎ ###### Corollary 4.1 (Equivalence of min-max problems). The following hold for any verifier $V$ and any $\delta\geq 0$: 1. 1. $\mu(V)=\lambda(V).$ 2. 2. If $(B^{\mu},\Pi_{0}^{\mu},\Pi_{1}^{\mu})$ is $\delta$-optimal for $\mu(V)$ then $B^{\mu}$ is $\delta$-optimal for $\lambda(V)$. 3. 3. If $(A^{\mu},A_{0}^{\mu},A_{1}^{\mu})$ is $\delta$-optimal for $\mu(V)$ then there exists $A_{\mathrm{ns}}$ such that $A_{\mathrm{ns}}$ is $\delta$-optimal for $\lambda(V)$ and $A_{\mathrm{ns}}$ can be computed efficiently in parallel given $(A^{\mu},A_{0}^{\mu},A_{1}^{\mu})$. ###### Proof. We begin with item 1. It is easy to prove $\lambda(V)\geq\mu(V)$: let $A^{\lambda}$ be optimal for $\lambda(V)$, let $A_{0},A_{1}$ witness the fact that $A^{\lambda}$ is no-signaling, and let $(B^{\mu},\Pi_{0}^{\mu},\Pi_{1}^{\mu})$ be optimal for $\mu(V)$. Then $\lambda(V)\geq\left\langle V,A^{\lambda}\otimes B^{\mu}\right\rangle=\left\langle f_{V}(A^{\lambda},A_{0},A_{1}),(B^{\mu},\Pi_{0}^{\mu},\Pi_{1}^{\mu})\right\rangle\geq\mu(V).$ For the reverse inequality, let $(A^{\mu},A_{0}^{\mu},A_{1}^{\mu})$ be optimal for $\mu(V)$, let $\Pi_{0}^{A^{\mu}},\Pi_{1}^{A^{\mu}}$ be optimal penalties for $(A^{\mu},A_{0}^{\mu},A_{1}^{\mu})$, and let $B^{\lambda}$ be optimal for $\lambda(V)$. By Theorem 4 there exists a no-signaling matrix $A_{\mathrm{ns}}$ witnessed by $A_{0}^{\mu},A_{1}^{\mu}$ such that $\left\langle V,A_{\mathrm{ns}}\otimes B^{\lambda}\right\rangle\leq\left\langle f_{V}\left(A^{\mu},A_{0}^{\mu},A_{1}^{\mu}\right),\left(B^{\lambda},\Pi_{0}^{A^{\mu}},\Pi_{1}^{A^{\mu}}\right)\right\rangle.$ The desired inequality $\lambda(V)\leq\mu(V)$ follows from the fact that the left side is at least $\lambda(V)$ and the right side is at most $\mu(V)$. The proof of item 1 is complete. Item 2 follows easily from item 1. Let $A$ be a no-signaling matrix and let $A_{0},A_{1}$ witness this fact. Then $\lambda(V)-\delta=\mu(V)-\delta\leq\left\langle f_{V}(A,A_{0},A_{1}),(B^{\mu},\Pi_{0}^{\mu},\Pi_{1}^{\mu})\right\rangle=\left\langle V,A\otimes B^{\mu}\right\rangle.$ As $A$ was chosen arbitrarily, it follows that $B^{\mu}$ is $\delta$-optimal for $\lambda(V)$. For item 3, let $B$ be any no-signaling matrix and let $\Pi_{0}^{A^{\mu}},\Pi_{1}^{A^{\mu}}$ be optimal penalties for the given $\delta$-optimal solution $(A^{\mu},A_{0}^{\mu},A_{1}^{\mu})$. By Theorem 4 there exists a no-signaling matrix $A_{\mathrm{ns}}$ witnessed by $A_{0}^{\mu},A_{1}^{\mu}$ such that $\left\langle V,A_{\mathrm{ns}}\otimes B\right\rangle\leq\left\langle f_{V}\left(A^{\mu},A_{0}^{\mu},A_{1}^{\mu}\right),\left(B,\Pi_{0}^{A^{\mu}},\Pi_{1}^{A^{\mu}}\right)\right\rangle\leq\mu(V)+\delta=\lambda(V)+\delta.$ As $B$ was chosen arbitrarily, it follows that $A_{\mathrm{ns}}$ is $\delta$-optimal for $\lambda(V)$. ∎ ### 3.4 Lemmas used in the rounding theorem The lemmas used in the proof of Theorem 4 are not difficult. It is quite likely that some form of these lemmas is part of computer science “folklore,” though our notation may be nonstandard. ###### Lemma 5 (Small marginals have small preimages). Let $a\in\mathcal{A}_{01}$ and $\vec{\delta}\in\mathcal{A}_{0}$ be nonnegative vectors with $\vec{\delta}\leq\operatorname{mar}_{\mathcal{A}_{1}}(a).$ There exists a nonnegative vector $d\in\mathcal{A}_{01}$ with $d\leq a$ and $\operatorname{mar}_{\mathcal{A}_{1}}(d)=\vec{\delta}$. Moreover, $d$ can be computed efficiently in parallel given $a,\vec{\delta}$. ###### Proof. Let $a_{(k_{0},k_{1})}$ and $\vec{\delta}_{k_{0}}$ denote the nonnegative entries of $a$ and $\vec{\delta}$, respectively. Let $s_{k_{0}}$ denote the $k_{0}$th entry of $\operatorname{mar}_{\mathcal{A}_{1}}(a)$ so that $s_{k_{0}}=\sum_{k_{1}=1}^{\dim(\mathcal{A}_{1})}a_{(k_{0},k_{1})}.$ The desired vector $d$ has entries $d_{(k_{0},k_{1})}$ given by $d_{(k_{0},k_{1})}=\left\\{\begin{array}[]{ll}\displaystyle\vec{\delta}_{k_{0}}\frac{a_{(k_{0},k_{1})}}{s_{k_{0}}}&\textrm{when $s_{k_{0}}\neq 0$}\\\ 0&\textrm{otherwise}\end{array}\right.$ (Intuitively, the weight $\vec{\delta}_{k_{0}}$ required of $\sum_{k_{1}}d_{(k_{0},k_{1})}$ is “spread out” over each $d_{(k_{0},k_{1})}$ proportionately according to $a_{(k_{0},k_{1})}$.) It is clear that this construction can be implemented efficiently in parallel. Let us verify that $d\leq a$. Observe that for the case $s_{k_{0}}\neq 0$ the ratio $\vec{\delta}_{k_{0}}/s_{k_{0}}$ is at most one because $\vec{\delta}\leq\operatorname{mar}_{\mathcal{A}_{1}}(a)$. Then $d_{(k_{0},k_{1})}=a_{(k_{0},k_{1})}\frac{\vec{\delta}_{k_{0}}}{s_{k_{0}}}\leq a_{(k_{0},k_{1})}$ as desired. Of course, if $s_{k_{0}}=0$ then $d_{(k_{0},k_{1})}=0$ by definition and hence $d_{(k_{0},k_{1})}\leq a_{(k_{0},k_{1})}$ because $a\geq 0$. Let us verify that $\operatorname{mar}_{\mathcal{A}_{1}}(d)=\vec{\delta}$. For the case $s_{k_{0}}\neq 0$ the $k_{0}$th entry of $\operatorname{mar}_{\mathcal{A}_{1}}(d)$ is given by $\sum_{k_{1}=1}^{\dim(A_{1})}d_{(k_{0},k_{1})}=\frac{\vec{\delta}_{k_{0}}}{s_{k_{0}}}\sum_{k_{1}=1}^{\dim(A_{1})}a_{(k_{0},k_{1})}=\vec{\delta}_{k_{0}}$ as desired. As above, if $s_{k_{0}}=0$ then by definition $d_{(k_{0},k_{1})}=0$ for each $k_{1}$ and hence $\sum_{k_{1}}d_{(k_{0},k_{1})}=0$. As $0\leq\vec{\delta}_{k_{0}}\leq s_{k_{0}}$ it must be that $\vec{\delta}_{k_{0}}=0$, too. ∎ ###### Corollary 5.1. Let $A:\mathcal{S}_{01}\to\mathcal{A}_{01}$ and $\Delta:\mathcal{S}_{01}\to\mathcal{A}_{0}$ be nonnegative matrices with $\Delta\leq\operatorname{mar}_{\mathcal{A}_{1}}(A).$ There exists a nonnegative matrix $D:\mathcal{S}_{01}\to\mathcal{A}_{01}$ with $D\leq A$ and $\operatorname{mar}_{\mathcal{A}_{1}}(D)=\Delta$. Moreover, $D$ can be computed efficiently in parallel given $A,\Delta$. ###### Proof. Apply Lemma 5 to each of the columns of $A,\Delta$. ∎ ###### Lemma 6 (Disjoint marginals are always consistent). For both $c\in\\{0,1\\}$ let $t_{c}\in\mathcal{A}_{c}$ be nonnegative vectors whose entries sum to the same value. There exists a nonnegative vector $t\in\mathcal{A}_{01}$ with $\operatorname{mar}_{\mathcal{A}_{\overline{c}}}(t)=t_{c}$ for both $c\in\\{0,1\\}$. Moreover, $t$ can be computed efficiently in parallel given $t_{0},t_{1}$. ###### Proof. Let $p_{k_{0}}$ and $q_{k_{1}}$ be the nonnegative entries of $t_{0}$ and $t_{1}$, respectively. Let $s$ denote the sum of the entries of $t_{0},t_{1}$ so that $s=\sum_{k_{0}=1}^{\dim(\mathcal{A}_{0})}p_{k_{0}}=\sum_{k_{1}=1}^{\dim(\mathcal{A}_{1})}q_{k_{1}}.$ If $s=0$ then it is clear that the desired vector $t$ is the zero vector. For the remainder of the proof assume that $s\neq 0$. The desired vector $t$ has entries $t_{(k_{0},k_{1})}$ given by $t_{(k_{0},k_{1})}=\frac{p_{k_{0}}q_{k_{1}}}{s}$ It is clear that this construction can be implemented efficiently in parallel. Let us verify that $\operatorname{mar}_{\mathcal{A}_{\overline{c}}}(t)=t_{c}$ for both $c\in\\{0,1\\}$. For the case $c=0$ the $k_{0}$th entry of $\operatorname{mar}_{\mathcal{A}_{1}}(t)$ is given by $\sum_{k_{1}=1}^{\dim(\mathcal{A}_{1})}\frac{p_{k_{0}}q_{k_{1}}}{s}=\frac{p_{i,j}s}{s}=p_{k_{0}}$ as desired. The case $c=1$ is handled similarly. ∎ ###### Corollary 6.1. For both $c\in\\{0,1\\}$ let $T_{c}:\mathcal{S}_{01}\to\mathcal{A}_{c}$ be nonnegative matrices with $\operatorname{mar}_{\mathcal{A}_{0}}(T_{0})=\operatorname{mar}_{\mathcal{A}_{1}}(T_{1})$. There exists a nonnegative matrix $T:\mathcal{S}_{01}\to\mathcal{A}_{01}$ with $\operatorname{mar}_{\mathcal{A}_{\overline{c}}}(T)=T_{c}$ for both $c\in\\{0,1\\}$. Moreover, $T$ can be computed efficiently in parallel given $T_{0},T_{1}$. ###### Proof. Apply Lemma 6 to each of the columns of $T_{0},T_{1}$. ∎ ## 4 A parallel multiplicative weights algorithm In this section we complete the proof of our main result—that every decision problem that admits a two-turn interactive proof with competing teams of no- signaling provers is also in $\mathrm{PSPACE}$. Most of the detail appears in Section 4.1 wherein we present an efficient parallel oracle-algorithm based on the MWUM that produces $\delta$-optimal no-signaling strategies for the teams, given an oracle for “best responses” for Team Bob to a given candidate strategy for Alice. We describe an efficient parallel implementation of the required oracle in Section 4.2, from which the unconditional efficiency of our algorithm immediately follows. The ensuing inclusion of $\mathrm{MRG}_{\mathrm{ns}}(2,2)$ inside $\mathrm{PSPACE}$ is discussed in Section 4.3. ### 4.1 The parallel algorithm Precise statements of the problem solved by our algorithm and the oracle it requires are given below. All input numbers are written as rational numbers in binary. For matrix inputs, each entry is written explicitly. ###### Problem 1 (Weak no-signaling equilibrium). _Input:_ \| A verifier matrix $V:\mathcal{S}_{01}\mathcal{T}_{01}\to\mathcal{A}_{01}\mathcal{B}_{01}$ and an accuracy parameter $\delta>0$. ---\|--- _Oracle:_ \| Weak no-signaling optimization. (See Problem 2 below.) _Output:_ \| $\delta$-optimal no-signaling strategies $\tilde{A},\tilde{B}$ for the min-max problem $\lambda(V)$. ###### Problem 2 (Weak no-signaling optimization). _Input:_ \| A verifier-Alice matrix $S:\mathcal{T}_{01}\to\mathcal{B}_{01}$ and an accuracy parameter $\delta>0$. ---\|--- _Output:_ \| A $\delta$-optimal no-signaling strategy $\tilde{B}$ for Team Bob. (That is, a no-signaling matrix $\tilde{B}$ such that $\langle S,\tilde{B}\rangle\geq\langle S,B\rangle-\delta$ for all no-signaling matrices $B$.) Given Corollary 4.1, it suffices to find $\delta$-optimal solutions $(\tilde{A},\tilde{A}_{0},\tilde{A}_{1})$ and $(\tilde{B},\tilde{\Pi}_{0},\tilde{\Pi}_{1})$ for $\mu(V)$ and then convert these solutions into $\delta$-optimal strategies for $\lambda(V)$. This method is codified in the algorithm of Figure 2. This algorithm is a straightforward modification of the standard multiplicative weights update method for equilibrium problems. The precise formulation of the MWUM used in this paper is stated as Theorem 7. Our statement of this theorem is somewhat nonstandard: the result is usually presented in the form of an algorithm, whereas our presentation is purely mathematical. However, a cursory examination of the literature—say, Kale’s thesis [Kal07, Chapter 2]—reveals that our mathematical formulation is equivalent to the more conventional algorithmic form. ###### Theorem 7 (Multiplicative weights update method—see Ref. [Kal07, Theorem 2]). Fix an $\varepsilon\in(0,1/2)$. Let $m^{1},\dots,m^{T}$ be arbitrary $D$-dimensional “loss” vectors whose entries $m^{t}_{i}$ lay in the interval $[-\alpha,\alpha]$. Let $w^{1},\dots,w^{T}$ be $D$-dimensional nonnegative “weight” vectors whose entries $w^{t}_{i}$ are given recursively via $\displaystyle w^{1}_{i}$ $\displaystyle=1$ $\displaystyle w^{t+1}_{i}$ $\displaystyle=w^{t}_{i}\left(1-\varepsilon m^{t}_{i}\right).$ Let $p^{1},\dots,p^{T}$ be probability vectors obtained by normalizing each $w^{1},\dots,w^{T}$. For all probability vectors $p$ it holds that $\frac{1}{T}\sum_{t=1}^{T}\left\langle p^{t},m^{t}\right\rangle\leq\left\langle p,\frac{1}{T}\sum_{t=1}^{T}m^{t}\right\rangle+\alpha\left(\varepsilon+\frac{\ln D}{\varepsilon T}\right).$ Note that Theorem 7 holds for _all_ choices of loss vectors $m^{1},\dots,m^{T}$, including the case in which each $m^{t}$ is chosen adversarially based upon $w^{t}$. This adaptive selection of loss vectors is typical in implementations of the MWUM. 1. 1. Let $\varepsilon=\delta/10$ and let $T=\left\lceil\frac{\ln(\dim(\mathcal{A}_{01}))}{\varepsilon^{2}}\right\rceil$. Let $\left(W^{1},W_{0}^{1},W_{1}^{1}\right)$ denote the triple of all-ones matrices and let $\left(A^{1},A_{0}^{1},A_{1}^{1}\right)$ denote the uniformly random strategy for Alice obtained by normalizing the columns of $\left(W^{1},W_{0}^{1},W_{1}^{1}\right)$. 2. 2. Repeat for each $t=1,\dots,T$: 1. (a) Compute optimal penalties $\Pi_{0}^{t},\Pi_{1}^{t}$ for $(A^{t},A_{0}^{t},A_{1}^{t})$ as described in Section 3.2. Use the oracle for Problem 2 to obtain a $\delta/2$-best response $B^{t}$ to the verifier-Alice matrix $\Phi_{V}(A^{t})$. 2. (b) Compute the loss matrices $\left(M^{t},M_{0}^{t},M_{1}^{t}\right)=f_{V}^{}\left(B^{t},\Pi_{0}^{t},\Pi_{1}^{t}\right)$. Exit the loop now if $t=T$. 3. (c) Update the weight matrices according to the standard multiplicative weights update rule: $\left(W^{t+1},W_{0}^{t+1},W_{1}^{t+1}\right)=\left(W^{t},W_{0}^{t},W_{1}^{t}\right)\boxtimes\left(\underbrace{\left(W^{1},W_{0}^{1},W_{1}^{1}\right)}_{\textrm{all- ones matrices}}-\varepsilon\left(M^{t},M_{0}^{t},M_{1}^{t}\right)\right)$ where $\boxtimes$ denotes the (entrywise) matrix Schur product. (See Theorem 7.) 4. (d) Compute the updated triple $(A^{t+1},A_{0}^{t+1},A_{1}^{t+1})$ of stochastic matrices for Team Alice by normalizing the columns of $(W^{t+1},W_{0}^{t+1},W_{1}^{t+1})$. 3. 3. Compute $(\tilde{A},\tilde{A}_{0},\tilde{A}_{1})=\frac{1}{T}\sum_{t=1}^{T}(A^{t},A_{0}^{t},A_{1}^{t})\qquad\textrm{and}\qquad(\tilde{B},\tilde{\Pi}_{0},\tilde{\Pi}_{1})=\frac{1}{T}\sum_{t=1}^{T}(B^{t},\Pi_{0}^{t},\Pi_{1}^{t})$ both of which are $\delta$-optimal for $\mu(V)$. Compute the no-signaling matrix $\tilde{A}_{\mathrm{ns}}$ from $(\tilde{A},\tilde{A}_{0},\tilde{A}_{1})$ as described in Corollary 4.1. 4. 4. Return $(\tilde{A}_{\mathrm{ns}},\tilde{B})$ as the $\delta$-optimal strategies of Team Alice and Team Bob for $\lambda(V)$. Figure 2: Algorithm that finds $\delta$-optimal solutions to the equilibrium problem $\lambda(V)$ for two-turn interactive proofs with competing teams of no-signaling provers (Problem 1). ###### Proposition 8. The oracle-algorithm presented in Figure 2 solves the weak no-signaling equilibrium problem (Problem 1). Assuming unit cost for the oracle, this algorithm can be implemented in parallel with run time bounded by a polynomial in $1/\delta$ and $\log(\dim(\mathcal{S}_{01}\mathcal{T}_{01}\mathcal{A}_{01}\mathcal{B}_{01}))$. ###### Proof. For each pair $i=(i_{0},i_{1})$ of questions let $\pi_{i}$ denote the probability with which the verifier asks questions $i$ to Team Alice. Let $m^{t}$ denote the $i$th column of $M^{t}$ for each $t=1,\dots,T$. We argue that the entries of $m^{t}$ lay in the interval $[0,3\pi_{i}]$. To this end, observe that the loss matrix $M^{t}$ is defined in Figure 2 via the adjoint mapping $f_{V}^{}$ as $M^{t}=\Phi_{V}^{}(B^{t})+e_{\mathcal{A}_{1}}\otimes\Pi_{0}^{t}+e_{\mathcal{A}_{0}}\otimes\Pi_{1}^{t}\leq 3e_{\mathcal{A}_{01}}p_{\textrm{Alice}}^{}$ where the inequality follows immediately from the bound $\Phi_{V}^{}(B)\leq e_{\mathcal{A}_{01}}p_{\textrm{Alice}}^{}$ of Proposition 3 and the restriction $\Pi_{c}\leq e_{\mathcal{A}_{c}}p_{\textrm{Alice}}^{}$ on penalty matrices. The desired bound on the entries of $m^{t}$ follows from the observation that the $i$th column of $3e_{\mathcal{A}_{01}}p_{\textrm{Alice}}^{}$ is the vector whose entries are all equal to $3\pi_{i}$. Let $a^{t}$ denote the $i$th column of $A^{t}$ for $t=1,\dots,T$. It is clear that the construction of the probability vectors $a^{t}$ in terms of the loss vectors $m^{t}$ presented in Figure 2 obeys the condition of Theorem 7. It therefore follows that for any probability vector $a\in\mathcal{A}_{01}$ we have $\frac{1}{T}\sum_{t=1}^{T}\left\langle a^{t},m^{t}\right\rangle\leq\left\langle a,\frac{1}{T}\sum_{t=1}^{T}m^{t}\right\rangle+3\pi_{i}\left(\varepsilon+\frac{\ln(\dim(\mathcal{A}_{01}))}{\varepsilon T}\right).$ Summing these inequalities over all columns $i$ we find that for any stochastic matrix $A$ it holds that $\frac{1}{T}\sum_{t=1}^{T}\left\langle A^{t},M^{t}\right\rangle\leq\left\langle A,\frac{1}{T}\sum_{t=1}^{T}M^{t}\right\rangle+3\left(\varepsilon+\frac{\ln(\dim(\mathcal{A}_{01}))}{\varepsilon T}\right).$ A similar bound on the stochastic matrices $A_{0}^{t},A_{1}^{t}$ in terms of the loss matrices $M_{0}^{t},M_{1}^{t}$ can be derived in much the same way. For completeness, let us make this argument explicit. For both $c\in\\{0,1\\}$ and for each question $i_{c}$ let $\pi_{i_{c}}$ denote the probability with which the referee asks question $i_{c}$ to Alicec. Let $m_{c}^{t}$ denote the $i_{c}$th column of $M_{c}^{t}$ for each $t=1,\dots,T$. We argue that the entries of $m_{c}^{t}$ lay in the interval $[-\pi_{i_{c}},0]$. Recall the loss matrix $M_{c}^{t}$ is defined in Figure 2 via the adjoint mapping $f_{V}^{}$ as $M_{c}^{t}=-\Pi_{c}^{t}\left(I_{\mathcal{S}_{c}}\otimes e_{\mathcal{S}_{\overline{c}}}\right)\geq- e_{\mathcal{A}_{c}}\operatorname{mar}_{\mathcal{S}_{\overline{c}}}(p_{\textrm{Alice}})^{}$ where the inequality follows immediately from the restriction $\Pi_{c}\leq e_{\mathcal{A}_{c}}p_{\textrm{Alice}}^{}$ on penalty matrices. The desired bound on the entries of $m_{c}^{t}$ follows from the observation that the $i_{c}$th column of $e_{\mathcal{A}_{c}}\operatorname{mar}_{\mathcal{S}_{\overline{c}}}(p_{\textrm{Alice}})^{}$ is the vector whose entries are all equal to $\pi_{i_{c}}$. As above, let $a_{c}^{t}$ denote the $i_{c}$th column of $A_{c}^{t}$ for $t=1,\dots,T$. It is clear that the construction of the probability vectors $a_{c}^{t}$ in terms of the loss vectors $m_{c}^{t}$ presented in Figure 2 obeys the condition of Theorem 7. It therefore follows that for any probability vector $a_{c}\in\mathcal{A}_{c}$ we have $\frac{1}{T}\sum_{t=1}^{T}\left\langle a_{c}^{t},m^{t}\right\rangle\leq\left\langle a_{c},\frac{1}{T}\sum_{t=1}^{T}m_{c}^{t}\right\rangle+\pi_{i_{c}}\left(\varepsilon+\frac{\ln(\dim(\mathcal{A}_{c}))}{\varepsilon T}\right).$ Summing these inequalities over all columns $i_{c}$ we find that for any stochastic matrix $A_{c}$ it holds that $\frac{1}{T}\sum_{t=1}^{T}\left\langle A_{c}^{t},M_{c}^{t}\right\rangle\leq\left\langle A_{c},\frac{1}{T}\sum_{t=1}^{T}M_{c}^{t}\right\rangle+\varepsilon+\frac{\ln(\dim(\mathcal{A}_{c}))}{\varepsilon T}.$ At this point we have derived three inequalities for three arbitrary stochastic matrices $A,A_{0},A_{1}$. Summing these inequalities and substituting $(M^{t},M_{0}^{t},M_{1}^{t})=f_{V}^{}(B^{t},\Pi_{0}^{t},\Pi_{1}^{t})$ and the choices of $\varepsilon,T$ listed in Figure 2 we find that for any triple $(A,A_{0},A_{1})$ of stochastic matrices it holds that $\frac{1}{T}\sum_{t=1}^{T}\left\langle f_{V}(A^{t},A_{0}^{t},A_{1}^{t}),(B^{t},\Pi_{0}^{t},\Pi_{1}^{t})\right\rangle\leq\left\langle f_{V}(A,A_{0},A_{1}),\frac{1}{T}\sum_{t=1}^{T}(B^{t},\Pi_{0}^{t},\Pi_{1}^{t})\right\rangle+\delta/2.$ (1) The remainder of this proof is a straightforward adaptation of Kale’s analysis for the much simpler class of two-player zero-sum games in normal form [Kal07, Section 2.3.1]. We argue that the triples $(\tilde{A},\tilde{A}_{0},\tilde{A}_{1})$ and $(\tilde{B},\tilde{\Pi}_{0},\tilde{\Pi}_{1})$ appearing Figure 2 are $\delta$-optimal for $\mu(V)$. Let us begin with the triple $(\tilde{A},\tilde{A}_{0},\tilde{A}_{1})$. Choose any $(B,\Pi_{0},\Pi_{1})$ and let $(A^{\star},A_{0}^{\star},A_{1}^{\star})$ be optimal for $\mu(V)$. We have $\displaystyle\left\langle\frac{1}{T}\sum_{t=1}^{T}f_{V}(A^{t},A_{0}^{t},A_{1}^{t}),(B,\Pi_{0},\Pi_{1})\right\rangle$ $\displaystyle\leq\frac{1}{T}\sum_{t=1}^{T}\left\langle f_{V}(A^{t},A_{0}^{t},A_{1}^{t}),(B^{t},\Pi_{0}^{t},\Pi_{1}^{t})\right\rangle+\delta/2$ $\displaystyle\leq\left\langle f_{V}(A^{\star},A_{0}^{\star},A_{1}^{\star}),\frac{1}{T}\sum_{t=1}^{T}(B^{t},\Pi_{0}^{t},\Pi_{1}^{t})\right\rangle+\delta\leq\mu(V)+\delta$ as desired. (The first inequality is because each $(B^{t},\Pi_{0}^{t},\Pi_{1}^{t})$ is a $\delta/2$-best response to $(A^{t},A_{0}^{t},A_{1}^{t})$; the second is Eq. (1).) To see that $(\tilde{B},\tilde{\Pi}_{0},\tilde{\Pi}_{1})$ is $\delta$-optimal for $\mu(V)$, let $(A,A_{0},A_{1})$ be any triple of stochastic matrices. We have $\left\langle f_{V}(A,A_{0},A_{1}),\frac{1}{T}\sum_{t=1}^{T}(B^{t},\Pi_{0}^{t},\Pi_{1}^{t})\right\rangle\geq\frac{1}{T}\sum_{t=1}^{T}\left\langle f_{V}(A^{t},A_{0}^{t},A_{1}^{t}),(B^{t},\Pi_{0}^{t},\Pi_{1}^{t})\right\rangle-\delta/2\geq\mu(V)-\delta$ as desired. (The first inequality is Eq. (1); the second is because each $(B^{t},\Pi_{0}^{t},\Pi_{1}^{t})$ is a $\delta/2$-best response to $(A^{t},A_{0}^{t},A_{1}^{t})$.) Finally, it follows from Corollary 4.1 that $\tilde{A}_{\mathrm{ns}}$ and $\tilde{B}$ are $\delta$-optimal strategies for $\lambda(V)$. That the algorithm admits an efficient parallel implementation is straightforward. In each iteration computations of optimal penalties, the loss matrices (via $f_{V}^{}$), the multiplicative weights update rule, and normalization are all simple operations involving only addition and multiplication of individual rational entries of matrices that can easily be implemented in parallel. Efficiency follows from the fact that the total number of iterations is bounded by a polynomial in $1/\delta$ and the logarithm of $\dim(\mathcal{S}_{01}\mathcal{T}_{01}\mathcal{A}_{01}\mathcal{B}_{01})$, the size of the verifier matrix. ∎ ### 4.2 Implementations of the best-response oracle for Team Bob In order for the algorithm of Figure 2 to be unconditionally efficient, we require a parallel implementation of the oracle for weak no-signaling optimization (Problem 2). Fortunately, all the work is already done: Problem 2 is the optimization problem that arises naturally from two-turn, two-prover interactive proofs with no-signaling provers. Thus, the parallel algorithm of Ito [Ito10] can be re-used to implement the oracle in our algorithm without complication. In Ito’s terminology, the verifier-Alice matrix $\Phi_{V}(A)$ specifies a _game_ and the two no-signaling provers comprising Team Bob are the _players_. Ito does not claim that an explicit strategy for the players can be found efficiently in parallel. Rather, he claims only that the task of distinguishing high success probability from low success probability admits a parallel algorithm, as this simpler task is sufficient to put $\mathrm{MIP}_{\mathrm{ns}}(2,2)$ inside $\mathrm{PSPACE}$. However, a cursory glance at the details of Ito’s proof reveals a parallel construction of near- optimal no-signaling strategies for the players as required by Problem 2. Alternatively, the oracle for weak no-signaling optimization (Problem 2) can be implemented by re-using the algorithm for weak no-signaling equilibrium (Problem 1) listed in Figure 2 of the present paper. Indeed, Problem 2 is a special case of Problem 1 in which one team has a trivial strategy space. In this special case the required “oracle” demands only weak no-signaling optimization over a trivial strategy space, which of course admits a trivial parallel implementation. In other words, the algorithm of Figure 2 can be used in a two-level recursive fashion to give an unconditionally efficient parallel algorithm for Problem 1. ### 4.3 Containment in PSPACE The desired containment of $\mathrm{MRG}_{\mathrm{ns}}(2,2)$ inside $\mathrm{PSPACE}$ now follows in the usual way: ###### Theorem 1. Every decision problem that admits a two-turn interactive proof with competing teams of two no-signaling provers per team is also in $\mathrm{PSPACE}$. Thus, we obtain the identity $\mathrm{MRG}_{\mathrm{ns}}(2,2)=\mathrm{PSPACE}$. ###### Proof. Let $L$ be a decision problem in $\mathrm{MRG}_{\mathrm{ns}}(2,2)$ with completeness $c$ and soundness $s$ and let $x$ be any input string. Each entry of the exponential-size verifier matrix $V:\mathcal{S}_{01}\mathcal{T}_{01}\to\mathcal{A}_{01}\mathcal{B}_{01}$ induced by the verifier on input $x$ can be computed in space polynomial in $\|x\|$ by simulating every choice of randomness for the verifier. In order to decide whether $x$ is a yes-instance or no-instance of $L$ it suffices to find $\delta$-optimal strategies for the teams for $\delta=(c-s)/3$, which permits us to distinguish $\lambda(V)\geq c$ from $\lambda(V)\leq s$. It follows from Proposition 8 and the discussion in Section 4.2 that the algorithm of Figure 2 can be used to find $\delta$-optimal strategies for the teams and can be implemented in parallel with run time bounded by a polynomial in $1/\delta$ and the logarithm of the dimensions of $V$. As the dimensions of $V$ scale exponentially with $\|x\|$ and $\delta$ scales as an inverse polynomial in $\|x\|$ the total run time of this parallel algorithm scales polynomially with $\|x\|$ and can therefore be simulated in polynomial space in the usual way [Bor77]. ∎ ## Acknowledgements The author is grateful to Tsuyoshi Ito, Sarvagya Upadhyay, John Watrous, and Xiaodi Wu for helpful discussions. 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1012.0923	# Analysis of the radiative thermal transfer in planar multi-layer systems with various emissivity and transmissivity properties S. Spanulescu1 1Department of Physics, Hyperion University of Bucharest, 030615, Romania ###### Abstract Abstract The paper analyzes the radiative thermal transfer in a liquid helium cryostat with liquid nitrogen shielding. A infinite plane walls model is used for demonstrating a method for lowering the radiative heat transfer and the numerical results for two such systems are presented. Some advantages concerning the opportunity of using semi-transparent walls are analytically and numerically demonstrated. ###### pacs: 44.40.+a Keywords: Radiative heat transfer, emissivity, thermal shield, cryostat ## I Introduction In a previous paper ssarxiv4 the thermal conductive flux in a cryostat was analyzed, and some methods for its reduction were suggested. In this paper we analyze the other major process of heat transfer which occurs in the same system, the radiative heat transfer. The classical method for lowering the values of this flux involves a number of reflective screens interposed in the vacuum chamber between the hot and the cold wall sieg , as in figure 1. Figure 1: The system of two infinite planes in vacuum The net radiative thermal flux from the warm wall to the cold one is strongly dependent on the properties of the walls and shields, especially their emissivity hprs , emis . Although these properties are not entirely known for cryogenic temperatures and have to be experimentally determined, some theoretical consideration concerning the general relationships describing the radiative thermal transfer may be drawn and they allow the proper design for improved cryostats. We analyze in the following the basic relationships for calculating this process, and a method for significantly reducing it is presented. ## II Radiative heat transfer between reflective infinite planes in vacuum Let us consider the system composed by two plane surfaces ${S_{a}}$ and ${S_{b}}$ placed in vacuum as in figure 2 in steady state conditions at temperatures ${T_{a}}$ and ${T_{b}}$. Figure 2: The system of two infinite planes in vacuum We analyze the radiative thermal flux between the two surfaces, from ${S_{a}}$ to ${S_{b}}$ if the first one is in contact with liquid nitrogen at the temperature ${T_{a}}=77.36\rm K$ and the second one is in contact with liquid helium at the temperature ${T_{b}}=4.22\rm K$. The thermal flux emitted by the surface unit by ${S_{a}}$, if it presents the emissivity ${e_{1}}$ considered constant with the temperature is (figure 2): ${E_{{a_{1}}}}={e_{1}}\sigma T_{a}^{4}$ (1) conforming to the Stephan-Boltzmann law, with the constant landau $\sigma=\frac{{2{\pi^{2}}k_{B}^{4}}}{{15{h^{3}}{c^{2}}}}=5,6692\cdot{10^{-8}}\rm W/({m^{2}}{K^{4}})$ This thermal flux is partially reflected by the surface ${S_{b}}$, so that a fraction of it is returned to the surface ${S_{a}}$. According to Kirchhoff law, the reflexivity $r$ of the surface is $r=1-e$, so that this fraction is chis ${E_{b1}}=(1-{e_{2}}){E_{a1}}={e_{1}}(1-{e_{2}})\sigma T_{a}^{4}$ (2) This flux is again reflected to ${S_{a}}$,as a secondary flux ${E_{{a_{2}}}}$. ${E_{{a_{2}}}}=(1-{e_{1}}){E_{{b_{1}}}}={e_{1}}(1-{e_{1}})(1-{e_{2}})\sigma T_{a}^{4}$ (3) After other two reflections the surface ${S_{a}}$ transmits also the flux: ${E_{{a_{3}}}}={e_{1}}{(1-{e_{1}})^{2}}{(1-{e_{2}})^{2}}\sigma T_{a}^{4}$ (4) and after $k$ double reflections the flux is: ${E_{{a_{k}}}}={e_{1}}{(1-{e_{1}})^{(k-1)}}{(1-{e_{2}})^{(k-1)}}T_{a}^{4}$ (5) It follows that the total flux ${E_{0}}$ emitted by ${S_{a}}$ and incident on ${S_{b}}$ is: $\begin{gathered}{E_{a}}={E_{a1}}+{E_{a2}}+...+{E_{ak}}+...={e_{1}}\sigma T_{a}^{4}+{e_{1}}(1-{e_{1}})(1-{e_{2}})\sigma T_{a}^{4}\hfill\\\ +{e_{1}}{(1-{e_{1}})^{2}}{(1-{e_{2}})^{2}}\sigma T_{a}^{4}+...+{e_{1}}{(1-{e_{1}})^{k-1}}{(1-{e_{2}})^{k-1}}\sigma T_{a}^{4}+...\hfill\\\ \end{gathered}$ (6) ${E_{a}}={e_{1}}\sigma T_{a}^{4}\frac{1}{{1-(1-{e_{1}})(1-{e_{2}})}}$ (7) The total flux ${E_{ab}}$ emitted by ${S_{a}}$ and absorbed by ${S_{b}}$ is: ${E_{ab}}={e_{2}}{E_{a}}={e_{2}}{e_{1}}\sigma T_{a}^{4}\frac{1}{{1-(1-{e_{1}})(1-{e_{2}})}}$ (8) For calculating the net flux flowing from the wall ${S_{a}}$ to the wall ${S_{b}}$ we must subtract the flux emitted by ${S_{b}}$ and absorbed by ${S_{a}}$ (as in figure 3). At the temperature ${T_{b}}$ this flux has an expression similar with (8): ${E_{ba}}=\frac{{{e_{1}}{e_{2}}\sigma T_{b}^{4}}}{{1-(1-{e_{1}})(1-{e_{2}})}}$ (9) If ${T_{a}}>{T_{b}}$, the thermal flux emitted by ${S_{a}}$ is greater than that emitted by ${S_{b}}$, and the total net thermal flux from ${S_{a}}$ to ${S_{b}}$ is: $E={E_{ab}}-{E_{ba}}$ (10) $E=\frac{1}{{\frac{1}{{{e_{1}}}}+\frac{1}{{{e_{2}}}}-1}}\sigma(T_{a}^{4}-T_{b}^{4})$ (11) In the case that the walls have the same emissivity ${e_{1}}={e_{2}}=e$: $E=\frac{e}{{2-e}}\sigma(T_{a}^{4}-T_{b}^{4})$ (12) For calculating the net thermal flux the emissivity coefficients ${e_{1}}$ and ${e_{2}}$ and the temperatures are necessary. If the temperature ${T_{b}}$ is not given, it may be calculated taking into consideration that if ${S_{b}}$ is placed in vacuum in steady state conditions the thermal flux absorbed is equal with the emitted one. So, imposing a value for the total net flux through the surface ${S_{b}}$,its temperature is: ${T_{b}}={\left[{T_{a}^{4}-\frac{E}{\sigma}\left({\frac{1}{{{e_{1}}}}+\frac{1}{{{e_{2}}}}-1}\right)}\right]^{\frac{1}{4}}}$ (13) In the case that between the two walls there is a third wall with the left side emissivity $e_{2s}$ and the right side emissivity $e_{2d}$ (as in figure 3), for calculating the thermal flux we have to consider the condition that in steady state regime the thermal flux emitted by ${S_{a}}$ and absorbed by ${S_{b}}$ is equal to the flux emitted by ${S_{b}}$ and absorbed by ${S_{c}}$. We obtain the equations $\left\\{\begin{gathered}E=\frac{1}{{\frac{1}{{{e_{1d}}}}+\frac{1}{{{e_{2s}}}}-1}}\sigma(T_{a}^{4}-T_{b}^{4})\hfill\\\ E=\frac{1}{{\frac{1}{{{e_{2d}}}}+\frac{1}{{{e_{3s}}}}-1}}\sigma(T_{b}^{4}-T_{c}^{4})\hfill\\\ \end{gathered}\right.$ (14) Figure 3: The system of two infinite planes in vacuum If the exterior walls temperatures ${T_{a}}$ and ${T_{c}}$ are known, by solving this system we obtain the unknowns $E$ and ${T_{b}}$ $E=\frac{1}{{\frac{1}{{{e_{1d}}}}+\frac{1}{{{e_{2s}}}}+\frac{1}{{{e_{2d}}}}+\frac{1}{{{e_{3s}}}}-2}}\sigma(T_{a}^{4}-T_{c}^{4})$ (15) ${T_{b}}={\left[{\frac{{\left({\frac{1}{{{e_{3s}}}}+\frac{1}{{{e_{2d}}}}-1}\right)T_{a}^{4}+\left({\frac{1}{{e2}}+\frac{1}{{{e_{1d}}}}-1}\right)T_{c}^{4}}}{{\frac{1}{{{e_{3s}}}}+\frac{1}{{{e_{2s}}}}+\frac{1}{{{e_{1d}}}}+\frac{1}{{{e_{2d}}}}-2}}}\right]^{\frac{1}{4}}}$ (16) In the case that all the surfaces have the same emissivity ${e_{1d}}={e_{2s}}={e_{2d}}={e_{3s}}=e$, the expressions simplify as $E=\frac{e}{{2(2-e)}}\sigma(T_{a}^{4}-T_{c}^{4})$ (17) ${T_{b}}={\left({\frac{{T_{a}^{4}+T_{c}^{4}}}{2}}\right)^{\frac{1}{4}}}$ (18) One may notice that the thermal flux is twice smaller that in the case of two walls, and the fourth power of middle wall temperature is the average value of the exterior temperature at the fourth power. Generalizing, for the case of $n$ walls with the same emissivity $e$, and with the extreme temperatures $T_{1}$ and $T_{2}$, the total net flux transmitted from the warmest wall and absorbed by the coldest one is $E=\frac{e}{{(n-1)(2-e)}}\sigma(T_{1}^{4}-T_{n}^{4})$ (19) and the intermediate walls temperatures $T_{k}$ are ${T_{k}}={\left\\{{\frac{1}{n-1}[(n-k)T_{1}^{4}+(k-1)T_{n}^{4}]}\right\\}^{\frac{1}{4}}},\quad k=2,3,...,n-1$ (20) In the table 1 we present numerical results for the radiative heat transfer between two plane infinite walls with different emissivity coefficients. One may notice that the minimum radiative heat transfer is obtained for the lowest values of the emissivity of both walls, as expected. These numerical results may be used as a reference for more sophisticated methods for reducing the thermal flux, as it will be presented further. For a system of $n$ walls with the same emissivity, the corresponding value on the diagonal should be divided by $n-1$. Table 1: The heat transfer power reported to the surface between two plane walls, for various values of the emissivity of the right side and the left side ($\rm W/m^{2}$). ${}_{e_{l}}\backslash^{e_{r}}$ \| 0.1 \| 0.2 \| 0.3 \| 0.4 \| 0.5 \| 0.6 \| 0.7 \| 0.8 \| 0.9 ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- 0.1 \| 0.105 \| 0.142 \| 0.162 \| 0.173 \| 0.181 \| 0.187 \| 0.191 \| 0.194 \| 0.197 0.2 \| 0.142 \| 0.221 \| 0.272 \| 0.307 \| 0.332 \| 0.352 \| 0.367 \| 0.38 \| 0.39 0.3 \| 0.162 \| 0.272 \| 0.352 \| 0.412 \| 0.46 \| 0.498 \| 0.53 \| 0.556 \| 0.579 0.4 \| 0.173 \| 0.307 \| 0.412 \| 0.498 \| 0.569 \| 0.629 \| 0.68 \| 0.725 \| 0.763 0.5 \| 0.181 \| 0.332 \| 0.46 \| 0.569 \| 0.664 \| 0.747 \| 0.821 \| 0.886 \| 0.944 0.6 \| 0.187 \| 0.352 \| 0.498 \| 0.629 \| 0.747 \| 0.854 \| 0.951 \| 1.04 \| 1.121 0.7 \| 0.191 \| 0.367 \| 0.53 \| 0.68 \| 0.821 \| 0.951 \| 1.073 \| 1.187 \| 1.294 0.8 \| 0.194 \| 0.38 \| 0.556 \| 0.725 \| 0.886 \| 1.04 \| 1.187 \| 1.329 \| 1.464 0.9 \| 0.197 \| 0.39 \| 0.579 \| 0.763 \| 0.944 \| 1.121 \| 1.294 \| 1.464 \| 1.631 Figure 4: The amount of heat transfer by radiation in the system of two infinite planes in vacuum ## III Radiative heat transfer in the case of semi-transmissive walls If a part $T$ of the incident thermal radiation $I$ passes through the wall (supposed thermic ”thin”) and another part $R$ is returned to the region where it came, it may be possible that, in certain condition, the total net heat flux to be diminished. In steady state conditions we may write $I=T+R+A$ (21) where $A$ is the radiation amount that is absorbed by the wall. If we denote by $t=\frac{T}{I},\;r=\frac{r}{I},\;e=\frac{A}{I}$ (22) the transmissivity, reflexivity and emissivity of the wall, taking into account the Kirchhoff low of radiation which states that the emitted radiation is equal with the absorbed one, we obtain $t=1-e-t$ (23) Therefore the equations (8) and (9) have to be written with this reflection coefficient instead $1-e$, so that the net flux emitted by the semi- transparent wall $S_{a}$ and absorbed by $S_{b}$ is ${E_{st}}={e_{2}}{e_{1}}\sigma\frac{T_{a}^{4}-T_{b}^{4}}{{1-(1-{e_{1}}-t_{1})(1-{e_{2}}-t_{2})}}$ (24) If we use a wall with a transparency close to $t=1-e$ (25) (an ideal case of semi-transparency) then the denominator in the relationship (11) becomes unity, and the thermal flux emitted by this wall and absorbed by the other becomes ${E_{ab}}={e_{2}}{e_{1}}\sigma\left(T_{a}^{4}-T_{a}^{4}\right)$ (26) which reduces the heat transfer by the ratio $\frac{E}{E_{st}}=\frac{1}{{1-(1-{e_{1}})(1-{e_{2}})}}$ (27) For example, with a typical high reflective surface with $e=0.1$, if the transmissivity is $t=0.9$, the heat transfer reduction ratio is almost five. However, for heat isolation purposes one may take into account the necessity that the transmitted part of the flux may still be absorbed after the final, coldest wall. That is why the walls must have an unidirectional transparency, from the colder wall to the warmer one. This way, the thermal radiation is reflected to the exterior of the system and dissipated there. Hence, the right formula for calculating the heat transfer by radiation is obtained considering $t_{2}=0$ in the expression (24) ${E_{st}}={e_{2}}{e_{1}}\sigma T_{a}^{4}\frac{1}{{1-(1-{e_{1}}-t_{1})(1-{e_{2}})}}$ (28) In the ideal case given by the condition (25), the net thermal flux is given by equation (26), being considerably lower than in the opaque wall case. If there are $n$ walls with the same emissivity $e$ and one side transparency $t=1-e$, the net thermal flux is obtained with an expression similar to (29) $E_{st}=\frac{e}{n-1}\sigma(T_{1}^{4}-T_{n}^{4})$ (29) ## IV Numerical results and conclusions The numerical results obtained for a system of two walls with ideal transparency (the warmer one) are presented in table 2 and figure 5. One may notice an important reduction of the radiation flux especially for the common case of low emissivity values. Table 2: The heat transfer power reported to the surface between two plane ideally semi-transparent walls, for various values of the emissivity of the right side and the left side ($\rm W/m^{2}$). ${}_{e_{l}}\backslash^{e_{r}}$ \| 0.1 \| 0.2 \| 0.3 \| 0.4 \| 0.5 \| 0.6 \| 0.7 \| 0.8 \| 0.9 ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- 0.1 \| 0.02 \| 0.04 \| 0.06 \| 0.08 \| 0.1 \| 0.12 \| 0.14 \| 0.159 \| 0.179 0.2 \| 0.04 \| 0.08 \| 0.12 \| 0.159 \| 0.199 \| 0.239 \| 0.279 \| 0.319 \| 0.359 0.3 \| 0.06 \| 0.12 \| 0.179 \| 0.239 \| 0.299 \| 0.359 \| 0.419 \| 0.478 \| 0.538 0.4 \| 0.08 \| 0.159 \| 0.239 \| 0.319 \| 0.399 \| 0.478 \| 0.558 \| 0.638 \| 0.717 0.5 \| 0.1 \| 0.199 \| 0.299 \| 0.399 \| 0.498 \| 0.598 \| 0.698 \| 0.797 \| 0.897 0.6 \| 0.12 \| 0.239 \| 0.359 \| 0.478 \| 0.598 \| 0.717 \| 0.837 \| 0.957 \| 1.076 0.7 \| 0.14 \| 0.279 \| 0.419 \| 0.558 \| 0.698 \| 0.837 \| 0.977 \| 1.116 \| 1.256 0.8 \| 0.159 \| 0.319 \| 0.478 \| 0.638 \| 0.797 \| 0.957 \| 1.116 \| 1.275 \| 1.435 0.9 \| 0.179 \| 0.359 \| 0.538 \| 0.717 \| 0.897 \| 1.076 \| 1.256 \| 1.435 \| 1.614 Figure 5: The amount of heat transfer by radiation in the system of two infinite planes in vacuum Of course, there are technological difficulties concerning the obtaining of the right condition of transparency, but it may be achieved using thin film deposition of materials with the proper index of refraction. One may use the Fresnel formulae for the reflective coefficient calculus, but it has to be considered the temperature and especially the wavelength dependence of the index of refraction of the involved materials, and some other special effects mentioned in the literature Robi . The presented formulae may be used as a first step in evaluating the performances of cryogenic shields, but surely some experimental steps have to be included for a final cryostat design. ## Acknowledgements This work was supported by the Romanian National Research Authority (ANCS) under Grant 22-139/2008. ## REFERENCES ## References * (1) S. Spanulescu,arXiv:0902.4144v1 (2009) * (2) R.Siegel, J. R. Howell, Thermal Radiation Heat Transfer, Hemisphere, Washington DC, 1992. * (3) A. Schirrmacher, HSPS, 33, 2, 299-335 (2003) * (4) http://www.monarchserver.com/TableofEmissivity.pdf * (5) L. D. Landau and E.M. Lifshitz, Statistical Physics, Elsevier, Amsterdam (2005). * (6) M. Marinescu, A. Chisacof, P. Raducanu, A. O. Motorcea, Bazele Termodinamicii tehnice, vol. 2, Politehnica Press, Bucharest, 2009 * (7) P-M. L. Robitaille. An analysis of universality in blackbody radiation. Progress in Physics, 2, 22 23,(2006)	arxiv-papers	2010-12-04T14:11:10	2024-09-04T02:49:15.465364	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Sever Spanulescu", "submitter": "Sever Spanulescu Prof", "url": "https://arxiv.org/abs/1012.0923" }
1012.0963	# Tricyclic graphs with exactly two main eigenvalues ††thanks: This research was partially supported by the NSF of China(No.10971086). Xiaoxia Fan, Yanfeng Luo Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PR China fanxx06@lzu.cn ###### Abstract An eigenvalue of a graph $G$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected tricyclic graphs with exactly two main eigenvalues are determined. 2000 Mathematics Subject Classification: 05C50 Keywords: Main eigenvalues; Tricyclic graphs; 2-walk $(a,b)$-linear graphs ## 1 Introduction All graphs considered in this paper are finite, undirected and simple. Let $G=(V,E)$ be a graph with vertex set $V(G)$ and edge set $E(G)$. Denote by $A(G)$ the adjacency matrix of $G$. The eigenvalues of $G$ are those of $A(G)$. An eigenvalue of a graph $G$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. It is well known that a graph is regular if and only if it has exactly one main eigenvalue. A long-standing problem posed by Cvetkovic ([2]) is that of how to characterize graphs with exactly $k(k\geq 2)$ main eigenvalues. Hagos [3] gave an alternative characterization of graphs with exactly two main eigenvalues. Recently, Hou and Zhou [4] characterized the trees with exactly two main eigenvalues. A vertex of a graph $G$ is said to be pendant if it has degree one. Denote by $C_{n}$ and $P_{n}$ the cycle and path of order $n$, respectively. A connected graph is said to be tricyclic (resp., unicyclic and bicyclic), if $\|E(G)\|=\|V(G)\|+2$ (resp., $\|E(G)\|=\|V(G)\|$ and $\|E(G)\|=\|V(G)\|+1$). Hou and Tian [5] showed that the graphs $C_{r}^{k}$ for some positive integers $k,r$ with $r\geq 3$, where $C_{r}^{k}$ is the graph obtained from $C_{r}$ by attaching $k>0$ pendant vertices to every vertex of $C_{r}$, are the only connected unicyclic graphs with exactly two main eigenvalues. Hu et al. [6] and Shi [7] characterized all connected bicyclic graphs with exactly two main eigenvalues independently. This paper will continue the line of this research and determine all connected tricyclic graphs with exactly two main eigenvalues. For any tricyclic graph $G$, the base of $G$, denoted by $G_{B}$ is the minimal tricyclic subgraph of $G$. Clearly, $G_{B}$ is the unique tricyclic subgraph of $G$ containing no pendent vertex, and $G$ can be obtained from $G_{B}$ by attaching trees to some vertices of $G_{B}$. It follows from [8] that there are 8 types of bases for tricyclic graphs, say, ${\mathcal{T}}_{i},i=1,\dots,8$, which are depicted in Fig. 1. Figure 1: The 8 types of bases for tricyclic graphs. ## 2 Preliminaries In this section, we will present some notations and known results which will be used in the next section. The reader is referred to [1] for any undefined notation and terminology on graphs in this paper. Let $G$ be a graph. As usual, we denote by $d(v)=d_{G}(v)$ and $N(v)=N_{G}(v)$ the degree of vertex $v$ and the set of all neighbors of $v$ in $G$. Let $S(v)=\sum\limits_{u\in N(v)}d(u).$ (2.1) A graph $G$ is called 2-walk $(a,b)$-linear if there exist unique rational numbers $a,b$ such that $S(v)=ad(v)+b$ (2.2) holds for every vertex $v\in V(G)$. An internal path of $G$ is a walk $v_{0}v_{1}\dots v_{s}$ such that the vertices $v_{0},v_{1},\dots,v_{s}$ are distinct, $d(v_{0})>2,d(v_{s})>2$, and $d(v_{i})=2$ for $0<i<s$. An internal path is called an internal cycle if $v_{0}=v_{s}$. If $R$ is a path or a cycle of $G$, the length of $R$, denoted by $l(R)$, is defined as the number of edges of $R$. ###### Lemma 2.1 ([3]). A graph $G$ has exactly two main eigenvalues if and only if $G$ is 2-walk $(a,b)$-linear. ###### Lemma 2.2 ([5]). Let $G$ be a 2-walk (a,b)-linear graph. Then both $a$ and $b$ are integers. ###### Lemma 2.3 ([6]). Let $G$ be a 2-walk $(a,b)$-linear graph and $v,u$ be two vertices of $G$ with unequal degree $d(v),d(u)$, respectively. Then $a=\frac{S(v)-S(u)}{d(v)-d(u)},\ b=\frac{d(u)S(v)-d(v)S(u)}{d(v)-d(u)}.$ (2.3) In the following, for convenience, we always assume that ${\mathcal{G}}$ is the set of tricyclic graphs with exactly two main eigenvalues, $x$ is a pendent vertex of $G$ (if exist). For each $G\in{\mathcal{G}}$, let $G_{0}$ be the graph obtained from $G$ by deleting all pendent vertices. From the proof Lemmas 3.1-3.7 in [6], we know that those Lemmas are also hold for tricyclic graphs. Hence we have the following two Lemmas. ###### Lemma 2.4 Let $G\in{\mathcal{G}}$ and $R=x_{1}x_{2}\dots x_{t}$ be an internal path of length at least $2$ in $G$. Then $l(R)\leq 3$. In particular, if $l(R)=3$, then there exists no path $Q=y_{1}y_{2}y_{3}$ in $G$ such that $d(y_{1})=d(y_{3})=d(x_{1})$ and $d(y_{2})=2$. ###### Lemma 2.5 Let $G\in{\mathcal{G}}$ and $v\in V(G_{0})$. Then (i) $G_{0}\in{\mathcal{T}}_{i},i=1,\dots,8$ (see Fig. 1); (ii) $d(v)=d_{G_{0}}(v)$ or $a+b$; (iii) if $G$ has at least one pendent vertex, then $S(x)=a+b\geq 3$ and $a\geq 2$; (iv) for a cycle $C=x_{1}x_{2}\dots x_{t}x_{1}$ of $G$ with $d_{G_{0}}(x_{1})\geq 3$, $d_{G_{0}}(x_{2})=2$, if $G$ has at least one pendent vertex, then there is an integer $i\in\\{1,2,\dots,t\\}$ such that $d(x_{i})\neq a+b$. ## 3 Tricyclic graphs with exactly two main eigenvalues In this section, we will determine all tricyclic graphs with exactly two main eigenvalues. By Lemma 2.1, it is sufficient to determine all 2-walk $(a,b)$-linear tricyclic graphs. ###### Lemma 3.1 Let $G\in{\mathcal{G}}$ has at least one pendent vertex and let $R=x_{1}x_{2}\dots x_{t}$ be an internal path or an internal cycle of length at least $3$ in $G_{0}$ with $d_{G_{0}}(x_{1})=d_{G_{0}}(x_{t})\in\\{3,4,6\\}$ or $d_{G_{0}}(x_{1})=3,d_{G_{0}}(x_{t})=5$. Then (i) $d(x_{2})=d(x_{3})=\dots=d(x_{t-1})\in\\{2,a+b\\}$ and $d(x_{1})=d(x_{t})$; (ii) if $d(x_{2})=2$, then $l(R)=3$; (iii) if $R$ is a cycle with $d_{G_{0}}(x_{1})\in\\{3,4,6\\}$, then $l(R)=3$ and $d(x_{2})=d(x_{3})=2$. In particular, if $d_{G_{0}}(x_{1})=3$, then $a=2$; (iv) if $R$ is a cycle with $d_{G_{0}}(x_{1})=5$, then $l(R)=3$, $d(x_{2})=d(x_{3})\in\\{2,3\\}$. In particular, if $d(x_{2})=d(x_{3})=3$, then $d(x_{1})=5$ and $a=3,b=0$. Proof. (i) By way of contradiction, assume that there is an integer $i\in\\{2,3,\dots,t-2\\}$ such that $d(x_{i})\neq d(x_{i+1})$. Without loss of generality, suppose that $i$ is the smallest integer such that $d(x_{i})\neq d(x_{i+1})$. By Lemma 2.5 (ii), we may assume that $d(x_{i})=2$ and $d(x_{i+1})=a+b$. Hence $d(x_{2})=d(x_{3})=\dots=d(x_{i})=2$. Applying (2.3) with $(v,u)=(x_{i+1},x)$, we have $a=\frac{S(x_{i+1})-S(x)}{d(x_{i+1})-d(x)}=\frac{a+b-2+2+d(x_{i+2})-(a+b))}{a+b-1}=\frac{d(x_{i+2})}{a+b-1}.$ This together with Lemma 2.5 (iii) implies that $d(x_{i+2})=a(a+b-1)\geq 2(a+b-1)\geq a+b+1>max\\{a+b,3\\}.$ (3.1) By Lemma 2.5 (ii), we have $d(x_{j})\in\\{a+b,2\\}$ for $2\leq j\leq t-1$. Thus $x_{i+2}\in\\{x_{1},x_{t}\\}$. If $d_{G_{0}}(x_{1})=d_{G_{0}}(x_{t})=3$, then $d(x_{i+2})\in\\{d(x_{1}),d(x_{t})\\}\subseteq\\{3,a+b\\}$, contrary to (3.1). If $d_{G_{0}}(x_{1})=d_{G_{0}}(x_{t})=4$, then $d(x_{1}),d(x_{t})\in\\{4,a+b\\}$. It follows from (3.1) that $d(x_{i+2})=a(a+b-1)=4$. This together with Lemma 2.5 (iii) implies that $a=2,b=1$. Note that $d(x_{2})=2$. Then $S(x_{2})=5$ by (2.2). On the other hand, $d_{G_{0}}(x_{1})=4>a+b$, so $d(x_{1})=4$ by Lemma 2.5 (ii). Thus by (2.1), $S(x_{2})=d(x_{1})+d(x_{3})=4+d(x_{3})>5$, a contradiction. If $d_{G_{0}}(x_{1})=d_{G_{0}}(x_{t})=6$, with a similar argument of the case $d_{G_{0}}(x_{1})=d_{G_{0}}(x_{t})=4$, we will get a contradiction again. If $d_{G_{0}}(x_{1})=3,d_{G_{0}}(x_{t})=5$, then by (3.1) $d(x_{i+2})=d(x_{t})=d_{G_{0}}(x_{t})=5$ and $a(a+b-1)=5$. It contradicts the fact that $a\geq 2,a+b\geq 3$. Hence $d(x_{2})=d(x_{3})=\dots=d(x_{t-1})\in\\{2,a+b\\}$. Therefore $S(x_{2})=S(x_{t-1})$ by (2.2). On the other hand, by (2.1), $S(x_{2})=d(x_{2})-2+d(x_{1})+d(x_{3}),S(x_{t-1})=d(x_{t-1})-2+d(x_{t})+d(x_{t-3})$. It follows that $d(x_{1})=d(x_{t})$. (ii) Suppose contrary that $l(R)\geq 4$. Then $d(x_{3})=d(x_{4})=2$ by (i). Applying (2.3) with $(v,u)=(x_{3},x)$, we have $a=4-(a+b)$, which is impossible by Lemma 2.5 (iii). (iii) By Lemma 2.5 (ii), we have $d(x_{2})\in\\{2,a+b\\}$. If $d(x_{2})=a+b$, then $d(x_{i})=a+b$ for $2\leq i\leq t-1$ by (i). Thus $d(x_{1})=d_{G_{0}}(x_{1})\neq a+b$ by Lemma 2.5 (iv). Applying (2.3) with $(v,u)=(x_{2},x)$, we have $a=\frac{a+b-2+d(x_{1})+a+b-(a+b)}{a+b-1}=1+\frac{d(x_{1})-1}{a+b-1}=1+\frac{d_{G_{0}}(x_{1})-1}{a+b-1}.$ (3.2) Note that $d_{G_{0}}(x_{1})=3,4$ or $6$ and $d_{G_{0}}(x_{1})\neq a+b\geq 3$. It follows from (3.2) that $a$ can not be an integer. It contradicts Lemma 2.2. Hence $d(x_{2})=2$. Therefore $l(R)=3$ and $d(x_{3})=d(x_{2})=2$. In particular, if $d_{G_{0}}(x_{1})=3$, then $a=2+d(x_{1})-(a+b)$ by applying (2.3) with $(v,u)=(x_{2},x)$. If $d(x_{1})=a+b$, then $a=2$. If $d(x_{1})=d_{G_{0}}(x_{1})=3$, then $a=5-(a+b)$. It follows from Lemmas 2.2 and 2.5 (iii) that $a=2$. (iv) By Lemma 2.5 (ii), $d(x_{2})\in\\{2,a+b\\}$. If $d(x_{2})=2$, then $l(R)=3$ and $d(x_{3})=2$ by (i) and (ii). If $d(x_{2})=a+b$, then $d(x_{i})=a+b$ for $2\leq i\leq t-1$ by (i). So $d(x_{1})=d_{G_{0}}(x_{1})=5\neq a+b$ by Lemma 2.5 (iv). Applying (2.3) with $(v,u)=(x_{2},x)$, we have $a=1+\frac{4}{a+b-1}$. Thus $a+b=3$ and $a=3$ by Lemmas 2.2 and 2.5 (iii). Suppose that $l(R)\geq 4$. Then $d(x_{2})=d(x_{3})=d(x_{4})=a+b=3$ by (ii). Applying (2.3) with $(v,u)=(x_{3},x)$, we have $a=2$. It is a contradiction. Therefore $l(R)=3$. $\square$ Figure 2: The graphs $T_{i}$ for $i=1,\dots,8$. Figure 3: The graphs $H_{i}$ for i=1,…,30. ###### Lemma 3.2 Let $G\in{\mathcal{G}}$ with $G_{0}\in{\mathcal{T}}_{1}$ (see Fig. 1). Then $G=H_{i}$ for $i=1,2$ (see Fig. 3). Proof. If $G_{0}\in{\mathcal{T}}_{1}$, then $d_{G_{0}}(u_{1}),d_{G_{0}}(v_{1}),d_{G_{0}}(w_{1})\in\\{3,4,5,6\\}$ and so each cycle of $G_{0}$ has the length of $3$ by Lemmas 2.4 and 3.1. Hence $G_{0}=T_{1}$ (see Fig. 2), where $n,m,k\geq 1$. For convenience, we set $w_{k}=v_{m}=u_{n}$. Case 1. $n=m=k=1$. If $G$ has no pendent vertex, then $G=H_{1}$ (see Fig. 3). By (2.3), $H_{1}$ is 2-walk $(1,6)$-linear. If $G$ has at least one pendent vertex, say $x$, then $x\in N(u_{1})$ since $d(x_{i})=2$ for $i=1,\dots,6$ by Lemma 3.1 (iii). It follows from Lemma 2.5 (ii) that $d(u_{1})=a+b>6$. Applying (2.3) with $(v,u)=(u_{1},x)$, we have $a=\frac{a+b-6+12-(a+b)}{a+b-1}<2$. This contradicts Lemma 2.5 (iii). Case 2. $n\geq 2$. We consider the following two cases: Subcase 1. $G$ has no pendent vertex. Then $S(x_{1})=2+d(w_{1})=S(x_{5})=5$ by (2.1) and (2.2). Hence $d(w_{1})=3$. It implies that $k\geq 2$. Similarly, we have $d(v_{1})=3$ and $m\geq 2$. By Lemma 2.4, we have $n,m,k\in\\{2,4\\}$. Without loss of generality, suppose that $n\geq m\geq k$. If $n=2$, then $m=k=2$. Hence $S(u_{1})=7,S(u_{2})=9$ by (2.1). On the other hand, $d(u_{1})=d(u_{2})=3$, so $S(u_{1})=S(u_{2})$ by (2.2), a contradiction. Hence $n=4$. Similarly, we have $m=k=4$. Therefore $G=H_{2}$ (see Fig. 3). By (2.3), $H_{2}$ is 2-walk $(1,3)$-linear. Subcase 2. $G$ has at least one pendent vertex. In this case, we show that there is no such graph with exactly two main eigenvalues. Since $d_{G_{0}}(u_{1})=3$, we have $a=2$ by Lemma 3.1 (iii). So $d(x_{i})=2$ for $i=1,\dots,6$ by Lemma 3.1 (iii) and (iv). We claim that $m,k\geq 2$. Otherwise, let $k=1$. Then $S(x_{1})=2+d(w_{1})=S(x_{5})=2+d(u_{1})$ by (2.1) and (2.2). It follows from Lemma 2.5 (ii) and the fact that $d_{G_{0}}(u_{1})\neq d_{G_{0}}(w_{1})$ that $d(w_{1})=d(u_{1})=a+b$. Hence $S(u_{1})=S(w_{1})$ by (2.2). By (2.1), $S(u_{1})=a+b-3+4+d(u_{2}),S(w_{1})=\Big{\\{}\begin{array}[]{ll}a+b-5+8+d(u_{n-1}),&\mbox{if}~{}m=1,\\\ a+b-4+4+d(v_{m-1})+d(u_{n-1}),&\mbox{if}~{}m\geq 2.\\\ \end{array}$ If $m=1$, then $d(u_{2})=d(u_{n-1})+2$. This is impossible by Lemma 3.1 (i). If $m\geq 2$, then $d(u_{2})+1=d(v_{m-1})+d(u_{n-1})$. Obviously, $u_{2}=u_{n-1}$ when $n=3$ and $d(u_{2})=d(u_{n-1})$ when $n\geq 4$ by Lemma 3.1 (i). Hence $d(v_{m-1})=1$, it contradicts the fact that $d(v_{m-1})\geq d_{G_{0}}(v_{m-1})=2$. Therefore $k\geq 2$. Dually, we have $m\geq 2$. Note that $d(x_{1})=d(x_{3})=d(x_{5})$, we have $S(x_{1})=2+d(w_{1})=S(x_{3})=2+d(v_{1})=S(x_{5})=2+d(u_{1})$ by (2.1) and (2.2). It implies that $d(w_{1})=d(v_{1})=d(u_{1})=3$ or $a+b$. If $d(u_{1})=3$. Applying (2.3) with $(v,u)=(x_{5},x)$, we have $a=5-(a+b)$. So $a=2,b=1$. Furthermore, $S(u_{1})=4+d(u_{2})=7$ by (2.1) and (2.2). Thus $d(u_{2})=3$. It follows from Lemma 3.1 (i) that $d(u_{i})=3$ for $2\leq i\leq n-1$. Similarly, we have $d(v_{i})=d(w_{j})=3$ for $2\leq i\leq m-1$ and $2\leq j\leq k-1$. Note that $d_{G_{0}}(u_{n})=a+b=3$. We have $d(u_{n})=3$ and $S(u_{n})=7$ by (2.2). On the other hand, $S(u_{n})=9$ by (2.1), a contradiction. If $d(u_{1})=a+b\neq 3$, then $a+b\geq 4$ by Lemma 2.5 (iii). Applying (2.3) with $(v,u)=(u_{1},x)$, we have $2=\frac{a+b-3+4+d(u_{2})-(a+b)}{a+b-1}$. Note that $d_{G_{0}}(u_{2})=2$ or $3$, we have $d(u_{2})=2(a+b)-3>max\\{a+b,d_{G_{0}}(u_{2})\\}$. It contradicts Lemma 2.5 (ii). $\square$ Figure 4: The graphs ${\mathcal{G}}_{i}$ for i=1,…,8, where $l_{1},b\geq 1$ and $max\\{k_{1},k_{2}\\}\geq 1$ ###### Lemma 3.3 Let $G\in{\mathcal{G}}$ with $G_{0}\in{\mathcal{T}}_{2}$ (see Fig. 1). Then $G=H_{i}$ for $i=3,4,5,6$ (see Fig. 3) or $G\in{\mathcal{G}}_{j}$ for $j=1,2$ (see Fig. 4). Proof. If $G_{0}\in{\mathcal{T}}_{2}$, then $d_{G_{0}}(w_{1}),d_{G_{0}}(r_{1})\in\\{3,4\\}$ and so each cycle of $G_{0}$ has the length of $3$ by Lemmas 2.4 and 3.1 (iii). Hence $G_{0}=T_{2}$ and $d(x_{i})=2$ for $i=1,\dots,4$ (see Fig. 2), where $k,l\geq 1,n,m\geq 2$. For convenience, we set $w_{k}=v_{1}=u_{1}$ and $v_{m}=u_{n}=r_{l}$. By (2.1) and (2.2), $S(x_{1})=2+d(w_{1})=S(x_{3})=2+d(r_{1})$. Hence $d(w_{1})=d(r_{1})$. If $G$ has no pendent vertex, then $k,l\in\\{1,2,4\\}$ and $m,n\in\\{2,4\\}$ by Lemma 2.4. First, let $k=1$. Then $d(w_{1})=d(r_{1})=4$. So $l=1$. Therefore $G=H_{i}$ for $i=3,4$ (see Fig. 3). By (2.3), $H_{3}$ and $H_{4}$ are 2-walk $(2,2)$-linear and 2-walk $(1,4)$-linear, respectively. Next, let $k=2$. Then $d(w_{1})=d(r_{1})=3$. By (2.1) and (2.2), $S(r_{1})=4+d(r_{2})=S(w_{1})=7$. So $d(r_{2})=3$ and $l=2$. Similarly, $S(u_{1})=3+d(u_{2})+d(v_{2})=S(w_{1})=7$. So $d(u_{2})=d(v_{2})=2$. Note that $n,m=2$ or $4$, we have $n=m=4$ and $d(u_{3})=d(v_{3})=2$. Therefore, $G=H_{5}$ (see Fig. 3). By (2.3), $H_{5}$ is 2-walk (2,1)-linear. Finally, let $k=4$. With a similar argument of the case $k=2$, we have $G=H_{6}$ is 2-walk (1,3)-linear (see Fig. 3). If $G$ has at least one pendent vertex. We consider the following two cases: Case 1. $k=l=1$. Then $d(w_{1})=d(r_{1})=4$ or $a+b$ by Lemma 2.5 (ii). If $d(w_{1})=4$. Applying (2.3) with $(v,u)=(x_{1},x)$, we have $a=6-(a+b)$. It follows from Lemmas 2.2 and 2.5 (iii) that $a+b=3$ or $4$. If $a+b=3$, then $a=3,b=0$. So $S(w_{1})=4+d(u_{2})+d(v_{2})=12$ by (2.1) and (2.2). By Lemma 2.5 (ii), $d(u_{2}),d(v_{2})\in\\{2,4,a+b\\}$. Thus $d(u_{2})=d(v_{2})=4$. It implies that $n=m=2$, which is impossible since $G$ is simple. If $a+b=4$, then $a=b=2$. So $S(w_{1})=4+d(u_{2})+d(v_{2})=10$ by (2.1) and (2.2). By Lemma 2.5 (ii), $d(u_{2}),d(v_{2})\in\\{2,4\\}$. Without loss of generality, suppose that $d(u_{2})=2,d(v_{2})=4$. Then $d(v_{i})=4$ for $2\leq i\leq m-1$ by Lemma 3.1 (i). For the vertex $u_{2}$, $S(u_{2})=4+d(u_{3})=6$ by (2.1) and (2.2). So $d(u_{3})=2$. Thus $n\geq 4$. Hence $n=4$ by Lemma 3.1 (ii). Therefore $G\in{\mathcal{G}}_{1}$ (see Fig. 4), where $l_{1}\geq 1$. It is easy to see that any graph $G\in{\mathcal{G}}_{1}$ is 2-walk $(2,2)$-linear. If $d(w_{1})=a+b>d_{G_{0}}(u_{1})=4$. In this case, we show that there is no such graph with exactly two main eigenvalues. Applying (2.3) with $(v,u)=(x_{1},x)$ and $(v,u)=(w_{1},x)$, respectively. We have $a=2$ and $a=\frac{a+b-4+4+d(u_{2})+d(v_{2})-(a+b)}{a+b-1}$, respectively. Hence $d(u_{2})+d(v_{2})=2(a+b)-2$. By Lemma 2.5 (ii) and the fact that $d(u_{n})=d(r_{1})=d(w_{1})=a+b$, we have $d(u_{2}),d(v_{2})\in\\{2,a+b\\}$. If $d(u_{2})=2,d(v_{2})\in\\{2,a+b\\}$ or $d(u_{2})=a+b,d(u_{2})=2$, then $a+b=3$ or $4$. It contradicts the fact that $a+b>4$. If $d(u_{2})=d(v_{2})=a+b$, then $2(a+b)-2=2(a+b)$, a contradiction. Case 2. $k\geq 2$. Then $a=2$ by Lemma 3.1 (iii). By Lemmas 2.5 (ii), $d(w_{1})=d(r_{1})=3$ or $a+b$. We first show that $d(w_{1})=d(r_{1})=3$. Otherwise, let $d(w_{1})=d(r_{1})=a+b\neq 3$. Then $a+b\geq 4$ by Lemma 2.5 (iii). Applying (2.3) with $(v,u)=(w_{1},x)$, we have $2=\frac{a+b-3+4+d(w_{2})-(a+b)}{a+b-1}$. Note that $a+b\geq 4$ and $d_{G_{0}}(w_{2})=2$ or $3$. We have $d(w_{2})=2(a+b)-3>max\\{a+b,d_{G_{0}}(w_{2})\\}$. It contradicts Lemma 2.5 (ii). Hence $d(w_{1})=d(r_{1})=3$. It implies that $l\geq 2$. Next, applying (2.3) with $(v,u)=(x_{1},x)$, we have $a=5-(a+b)$. So $a+b=3$ and $a=2,b=1$. For the vertex $w_{1}$, $S(w_{1})=4+d(w_{2})=7$ by (2.1) and (2.2). So $d(w_{2})=3$. Thus $d(w_{i})=3$ for $2\leq i\leq k-1$ by Lemma 3.1 (i). Note that $d_{G_{0}}(w_{k})=3$. We have $d(w_{k})=3$ by Lemma 2.5 (ii). Hence $d(w_{i})=3$ for $1\leq i\leq k$. Similarly, we have $d(r_{j})=3$ for $1\leq j\leq l$. For the vertex $w_{k}$, $S(w_{k})=3+d(u_{2})+d(v_{2})=7$. Note that $d(u_{2}),d(v_{2})\in\\{2,3\\}$ by Lemma 2.5 (ii). We have $d(u_{2})=d(v_{2})=2$. It follows from Lemmas 2.4 and 3.1 that $m=n=4$ and $d(u_{3})=d(v_{3})=2$. Therefore $G\in{\mathcal{G}}_{2}$ (see Fig. 4), where $max\\{k_{1},k_{2}\\}\geq 1$. It is easy to see that any graph $G\in{\mathcal{G}}_{2}$ is 2-walk (2,1)-linear. $\square$ ###### Lemma 3.4 Let $G\in{\mathcal{G}}$ with $G_{0}\in{\mathcal{T}}_{3}$ (see Fig. 1). Then $G=H_{7}$ (see Fig. 3). Proof. If $G_{0}\in{\mathcal{T}}_{3}$, then $d_{G_{0}}(r_{1})\in\\{3,5\\}$ and so the cycle of $G_{0}$ has the length of $3$ by Lemmas 2.4 and 3.1. Hence $G_{0}=T_{3}$ (see Fig. 2), where $n,m,k\geq 2,l\geq 1$. For convenience, we set $u_{1}=v_{1}=w_{1}$ and $u_{n}=v_{m}=w_{k}=r_{l}$. We first show that $G$ contains at least one pendent vertex. On the contrary, suppose that $G$ has no pendent vertex. Then $n,m,k,l\leq 4$ by Lemma 2.4. Without loss of generality, suppose that $n\geq m\geq k$. If $l=1$, then $S(x_{1})=7$ and $S(u_{2})=5$ or $8$ by (2.1). So $S(u_{2})\neq S(x_{1})$. On the other hand, $S(u_{2})=S(x_{1})$ by (2.2), a contradiction. If $l\geq 2$, then $d(u_{n})=4,d(r_{1})=3$. So $S(x_{1})=5$ and $S(u_{n-1})=6$ or $7$ by (2.1). On the other hand, $S(x_{1})=S(u_{n-1})$ by (2.2), a contradiction. Therefore, $G$ has at least one pendent vertex. We consider the following two cases: Case 1. $l=1$. By Lemma 3.1 (iv), $d(x_{1})=d(x_{2})=3$ or $d(x_{1})=d(x_{2})=2$. If $d(x_{1})=d(x_{2})=3$, then $a=3,b=0,d(u_{n})=5$ by Lemma 3.1 (iv). So $d(u_{1})=3\neq d(u_{n})$ by Lemma 2.5 (ii). It follows from Lemma 3.1 (i) that $n,m,k\leq 3$. Without loss of generality, suppose that $n=m=3,k=2$ or $3$. Then $S(u_{3})=6+d(u_{2})+d(v_{2})+d(w_{k-1})=15$ by (2.1) and (2.2). Note that $d(u_{2}),d(v_{2}),d(w_{k-1})=2$ or $3$ by Lemma 2.5 (ii). We have $d(u_{2})=d(v_{2})=d(w_{k-1})=3$. So $S(u_{1})=6+d(w_{2})=9$ by (2.1) and (2.2). Thus $d(w_{2})=3$. It implies that $k=3$. Therefore $G=H_{7}$ (see Fig. 3). By (2.3), $H_{7}$ is 2-walk $(3,0)$-linear. If $d(x_{1})=d(x_{2})=2$. We show that in this case there is no such graph with exactly two main eigenvalues. We consider the following two cases: Subcase 1. $max\\{n,m,k\\}\geq 4$. Without loss of generality, suppose that $n\geq 4$. Then $d(u_{1})=d(u_{n})$ and $d(u_{i})=d(u_{2})$ for $2\leq i\leq n-1$ by Lemma 3.1 (i). Note that $d_{G_{0}}(u_{1})\neq d_{G_{0}}(u_{n})$. We have $d(u_{1})=d(u_{n})=a+b\geq d_{G_{0}}(u_{n})=5$ by Lemma 2.5 (ii). Hence $S(u_{1})=a+b-3+d(u_{2})+d(v_{2})+d(w_{2}),\ S(u_{n})=a+b-5+4+d(u_{n-1})+d(v_{m-1})+d(w_{k-1}).$ We next show that $d(v_{2})=d(v_{m-1})$. If $m=2$, then $v_{2}=u_{n}$, $v_{m-1}=u_{1}$. So $d(v_{2})=d(u_{n})=d(u_{1})=d(v_{m-1})$. If $m=3$, clearly, $d(v_{2})=d(v_{m-1})$. If $m\geq 4$, then $d(v_{2})=d(v_{m-1})$ by Lemma 3.1 (i). Therefore $d(v_{2})=d(v_{m-1})$ for all $m\geq 2$. Similarly, we have $d(w_{2})=d(w_{k-1})$ for all $k\geq 2$. Hence $S(u_{1})\neq S(u_{n})$. On the other hand, $S(u_{1})=S(u_{n})$ by (2.2), a contradiction. Subcase 2. $max\\{n,m,k\\}\leq 3$. Without loss of generality, suppose that $n=m=3$ and $k=2$ or $3$. We claim that $d(u_{2})=a+b$. Otherwise, let $d(u_{2})=d_{G_{0}}(u_{2})=2$. Then $d(u_{1})+d(u_{3})=S(u_{2})=S(x_{1})=2+d(u_{3})$ by (2.1) and (2.2), which is impossible since $d(u_{1})\geq d_{G_{0}}(u_{1})=3$. Hence $d(u_{2})=a+b$. Similarly, we have $d(v_{2})=a+b$. Note that $d(u_{3})\in\\{a+b,5\\}$ by Lemma 2.5 (ii). We consider the following two cases: If $d(u_{3})=a+b\geq d_{G_{0}}(u_{3})=5$. Applying Lemma 2.5 (iv) with $C=u_{1}u_{2}u_{3}v_{2}u_{1}$, we have $d(u_{1})\neq a+b$. So $d(u_{1})=3$. Applying (2.3) with $(v,u)=(u_{1},x)$ and $(v,u)=(x_{1},x)$, respectively. We have $a=\frac{a+b+d(w_{2})}{2}$ and $a=2$, respectively. Hence $d(w_{2})=4-(a+b)<0$, a contradiction. If $d(u_{3})=d_{G_{0}}(u_{3})=5\neq a+b$, then $a=7-(a+b)$ by applying (2.3) with $(v,u)=(x_{1},x)$. It follows from Lemma 2.5 (iii) that $a+b=3$ or $4$. If $a+b=3$, then $a=4,b=-1$. Hence $S(u_{2})=d(u_{1})+6=11$ by (2.1) and (2.2). This is impossible since $d(u_{1})=3$ by Lemma 2.5 (ii). If $a+b=4$, then $a=3,b=1$. Thus $S(u_{2})=d(u_{1})+7=13$ by (2.1) and (2.2). This is also impossible since $d(u_{1})=3$ or $4$. Case 2. $l\geq 2$. We show that in this case there is no such graph with exactly two main eigenvalues. By Lemma 3.1 (iii), we have $a=2$ and $d(x_{1})=d(x_{2})=2$. Applying (2.3) with $(v,u)=(x_{1},x)$, we have $2=2+d(r_{1})-(a+b)$. So $d(r_{1})=a+b$. Applying (2.3) with $(v,u)=(r_{1},x)$, we have $2=\frac{a+b-3+4+d(r_{2})-(a+b)}{a+b-1}$. Hence $d(r_{2})=2(a+b)-3$. By Lemma 2.5 (ii), $d(r_{2})\in\\{2,4,a+b\\}$. If $d(r_{2})=2$ or $4$, then $a+b$ is not an integer, a contradiction. If $d(r_{2})=a+b$, then $a+b=3$. So $a=2,b=1$. By Lemmas 2.5 (ii) and 3.1 (i), we have $d(r_{l})=4$ and $d(r_{l-1})=d(r_{2})=3$. Hence $S(r_{l})=3+d(u_{n-1})+d(w_{k-1})+d(v_{m-1})=9$ by (2.1) and (2.2). It follows that $d(u_{n-1})=d(w_{k-1})=d(v_{m-1})=2$. Thus $S(u_{n-1})=5$ by (2.2). On the other hand, $S(u_{n-1})=4+d(u_{n-2})\geq 6$ by (2.1), a contradiction. $\square$ ###### Lemma 3.5 Let $G\in{\mathcal{G}}$ with $G_{0}\in{\mathcal{T}}_{4}$ (see Fig. 1). Then $G=H_{i}$ for $i=8,9,10$ (see Fig. 3) or $G\in{\mathcal{G}}_{3}$ (see Fig. 4). Proof. If $G_{0}\in{\mathcal{T}}_{4}$, then $d_{G_{0}}(w_{1})\in\\{3,4\\}$ and so the cycle of $G_{0}$ has the length of $3$ by Lemmas 2.4 and 3.1 (iii). Hence $G_{0}=T_{4}$ (see Fig. 2), where $k\geq 1$ and $p,q,n,m\geq 2$. For convenience, we set $u_{1}=v_{1}=s_{1},u_{n}=v_{m}=t_{1}$ and $s_{p}=t_{q}=w_{k}$. If $G$ has no pendent vertex. Without loss of generality, we may assume that $n\geq m,p\geq q$ and consider the following two cases: Case 1. $k=1$. Then $S(x_{1})=6$ by (2.1). We claim that $p=2$. Otherwise, let $p\geq 3$. Then $d(s_{2})=2$ and $S(s_{2})=5$ or $7$ by (2.1). So $S(x_{1})\neq S(s_{2})$. On the other hand, $S(x_{1})=S(s_{2})$ by (2.2), a contradiction. Hence $p=2$. Similarly, we have $q=2,n=3,m=2$ or $3$. If $m=2$, then applying (2.3) with $(v,u)=(w_{1},u_{2})$ and $(v,u)=(w_{1},u_{1})$, respectively, we have $a=2$ and $a=1$, respectively. A contradiction. If $m=3$, then $G=H_{8}$ (see Fig. 3). By (2.3), $H_{8}$ is 2-walk $(2,2)$-linear. Case 2. $k>1$. By Lemma 2.4, we have $k,p,q,m,n=2$ or $4$. If $k=2$, then $S(w_{1})=7=S(w_{2})=3+d(s_{p-1})+d(t_{q-1})$ by (2.1) and (2.2). So $d(s_{2})=d(t_{2})=2$. It implies that $p=q=4$. Hence $n=4,m=2$. Otherwise, let $m=n=4$. Then $S(u_{1})=6\neq S(w_{1})=7$ by (2.1). On the other hand, $S(u_{1})=S(w_{1})$ by (2.2), a contradiction. Therefore $n=4,m=2$ and $G=H_{9}$ (see Fig. 3). By (2.3), $H_{9}$ is 2-walk $(2,1)$-linear. If $k=4$, with a similar argument, we have $G=H_{10}$ is 2-walk $(1,3)$-linear (see Fig. 3). If $G$ has at least one pendent vertex, then $k>1$. Otherwise, suppose that $k=1$. Then $d(w_{1})=4$ or $a+b$ by Lemma 2.5 (ii). If $d(w_{1})=4\neq a+b$. Applying (2.3) with $(v,u)=(x_{1},x)$, we have $a=6-(a+b)$. It follows from Lemma 2.5 (iii) that $a=3,b=0$. So $S(w_{1})=4+d(s_{p-1})+d(t_{q-1})=12$ by (2.1) and (2.2). This is impossible since $d(s_{p-1}),d(t_{q-1})=2$ or $3$ by Lemma 2.5 (ii). If $d(w_{1})=a+b\geq 4$. Applying (2.3) with $(v,u)=(x_{1},x)$ and $(v,u)=(w_{1},x)$, respectively, we have $a=2$ and $a=\frac{a+b-4+4+d(t_{q-1})+d(s_{p-1})-(a+b)}{a+b-1}$, respectively. Thus $d(t_{q-1})+d(s_{p-1})=2(a+b)-2.$ (3.3) Note that $d(t_{q-1}),d(s_{p-1})\in\\{2,3,a+b\\}$. We consider the following five cases by symmetry: If $d(t_{q-1})=d(s_{p-1})=2$, then $a+b=3$. This contradicts the fact that $a+b\geq 4$. If $d(t_{q-1})=2,d(s_{p-1})=3$, then $a+b=3.5$. This contradicts Lemma 2.2. If $d(t_{q-1})=2,d(s_{p-1})=a+b$, then $a+b=4$ by (3.3). Note that $a=2$. We have $b=2$ and $d(s_{p-1})=d(w_{1})=4$. By Lemma 3.1, $d(s_{i})=4$ for $2\leq i\leq p-1$. So $S(s_{2})=6+d(u_{1})=10$ by (2.1) and (2.2). Thus $d(u_{1})=4$. Similarly, we have $d(u_{n})=4$. By (2.1) and (2.2), $S(u_{1})=5+d(u_{2})+d(v_{2})=10$. This is impossible since $d(u_{2}),d(v_{2})=2$ or $4$. If $d(t_{q-1})=3,d(s_{p-1})=a+b$, then $a+b=5$ by (3.3). Recall that $a=2$, we have $b=3$ and $d(w_{1})=5$. Note that $d(t_{q-1})=3\neq a+b$. We have $q=2$. Thus $S(t_{q-1})=9=5+d(u_{n-1})+d(v_{n-1})$ by (2.1) and (2.2). By Lemma 2.5 (ii), $d(u_{n-1}),d(v_{n-1})\in\\{2,3,5\\}$. Hence $d(u_{n-1})=d(v_{n-1})=2$. Therefore $S(u_{n-1})=3+d(u_{n-2})=7$, which is impossible since $d(u_{n-2})\in\\{2,3,5\\}$. If $d(t_{q-1})=d(s_{p-1})=a+b$, then $2(a+b)=2(a+b)-2$ by (3.3), a contradiction. Hence $k>1$. It follows from Lemma 3.1 (iii) that $a=2$ and $d(x_{1})=d(x_{2})=2$. Applying (2.3) with $(v,u)=(x_{1},x)$, we have $2=2+d(w_{1})-(a+b)$. So $d(w_{1})=a+b$. Applying (2.3) with $(v,u)=(w_{1},x)$, we have $2=\frac{a+b-3+4+d(w_{2})-(a+b)}{a+b-1}$. Thus $d(w_{2})=2(a+b)-3\geq 3$. Note that $d(w_{2})\in\\{2,3,a+b\\}$ by Lemma 2.5 (ii). We have $d(w_{2})=3$ or $a+b$. It follows that $a+b=3$ and $a=2,b=1$. By Lemma 3.1 (i), $d(w_{i})=d(w_{2})=3$ for $2\leq i\leq k-1$. By Lemma 2.5 (ii), $d(w_{1})=d(w_{k})=d(u_{1})=d(u_{n})=3$. For the vertex $w_{k}$, we have $S(w_{k})=3+d(s_{p-1})+d(t_{q-1})=7$. It follows from Lemma 2.5 (ii) that $d(s_{p-1})=d(t_{q-1})=2$ and $p,q\geq 3$. Hence $p=q=4$ and $d(s_{2})=d(t_{2})=2$ by Lemmas 2.4 and 3.1 (ii). For the vertex $u_{1}$, we have $S(u_{1})=2+d(v_{2})+d(u_{2})=7$. Note that $d(u_{2}),d(v_{2})\in\\{2,3\\}$. We may assume that $d(v_{2})=2,d(u_{2})=3$ by symmetry. It follows from Lemmas 2.4 and 3.1 (ii) that $m=4,d(v_{3})=2$ and $d(u_{i})=3$ for $2\leq i\leq n-1$. Therefore $G\in{\mathcal{G}}_{3}$ (see Fig. 4), where $max\\{k_{1},k_{2}\\}\geq 1$. It is easy to see that any graph $G\in{\mathcal{G}}_{3}$ is 2-walk $(2,1)$-linear. Up to now, we complete the proof of the Lemma. $\square$ ###### Lemma 3.6 There is no graph $G\in{\mathcal{G}}$ with $G_{0}\in{\mathcal{T}}_{5}$ (see Fig. 1). Proof. By way of contradiction, suppose that $G\in{\mathcal{G}}$ with $G_{0}\in{\mathcal{T}}_{5}$. Let $G_{0}=T_{5}$ (see Fig. 2), where $n,m,k,p,q\geq 2$. For convenience, we set $u_{1}=v_{1}=s_{1},u_{n}=w_{k}=t_{q},v_{m}=s_{p}=w_{1}=t_{1}$ and consider the following two cases: Case 1. There is a vertex $v\in\\{v_{m-1},s_{p-1},w_{2},t_{2}\\}$ such that $d_{G_{0}}(v)=2$ and $d(v)=a+b$. Without loss of generality, let $v=v_{m-1}$. Then $d(v_{i})=a+b$ for $2\leq i\leq m-1$ by Lemma 3.1 (i). In particular, $a\geq 2,a+b\geq 3$ by Lemma 2.5 (iii). We first show that $d(u_{1})=d(w_{1})=a+b$. If $m\geq 4$, then $d(u_{1})=d(w_{1})$ by Lemma 3.1 (i). Note that $d_{G_{0}}(u_{1})\neq d_{G_{0}}(w_{1})$. We have $d(u_{1})=d(w_{1})=a+b$ by Lemma 2.5 (ii). If $m=3$. Applying (2.3) with $(v,u)=(v_{2},x)$, we have $a=\frac{a+b-2+d(u_{1})+d(w_{1})-(a+b)}{a+b-1}=\frac{d(u_{1})+d(w_{1})-2}{a+b-1}.$ By Lemma 2.5 (ii), $d(u_{1})=3$ or $a+b$ and $d(w_{1})=4$ or $a+b$. If $d(u_{1})=3,d(w_{1})=4$, then $a=\frac{5}{a+b-1}$. Note that $a\geq 2$ is an integer. We have $a+b=2$. It contradicts the fact that $a+b\geq 3$. If $d(u_{1})=3,d(w_{1})=a+b\geq 4$, then $a=1+\frac{2}{a+b-1}<2$, a contradiction. If $d(u_{1})=a+b\geq 3,d(w_{1})=4\neq a+b$, then $a=1+\frac{3}{a+b-1}$ is not an integer, a contradiction. Hence $d(u_{1})=d(w_{1})=a+b$. Applying Lemma 2.5 (iv) with $C=u_{1}v_{2}\dots v_{m-1}w_{1}s_{p-1}\dots s_{2}u_{1}$, we have $p\geq 3$ and $d(s_{i})\neq a+b$ for some $2\leq i\leq p-1$. It follows from Lemmas 2.5 (ii) and 3.1 (i) that $d(s_{i})=2$ for $2\leq i\leq p-1$. Applying (2.3) with $(v,u)=(v_{2},x)$ and $(v,u)=(u_{1},x)$, respectively. We have $a=2$ and $a=1+\frac{d(u_{2})}{a+b-1}$, respectively. This together with Lemma 2.5 (ii) and the fact that $a+b\geq d_{G_{0}}(w_{1})=4$ implies that $d(u_{2})=a+b-1=d_{G_{0}}(u_{2})\geq 3$. Hence $n=2,d(u_{2})=d_{G_{0}}(u_{2})=3$. Therefore $a+b=4$ and $a=b=2$. By (2.1) and (2.2), $S(u_{2})=4+d(w_{k-1})+d(t_{q-1})=8$. By Lemma 2.5 (ii), $d(w_{k-1}),d(t_{q-1})=2$ or $4$. Thus $d(w_{k-1})=d(t_{q-1})=2$. Hence $S(w_{k-1})=3+d(w_{k-2})=6$ by (2.1) and (2.2), which is impossible since $d(w_{k-2})=2$ or $4$. Case 2. For any vertex $v\in\\{v_{m-1},s_{p-1},w_{2},t_{2}\\}$, we have $d_{G_{0}}(v)=3$ or $d(v)=2$. It follows from Lemma 2.4 that $m,k,p,q\leq 4$. Without loss of generality, suppose that $m\geq p,k\geq q$. Hence $m,k=3$ or $4$. Subcase 1. $max\\{m,k\\}=4$. Without loss of generality, suppose that $m=4$. Then $d(u_{1})=d(w_{1})$ by Lemma 3.1 (i). Note that $d_{G_{0}}(u_{1})\neq d_{G_{0}}(w_{1})$. We have $d(u_{1})=d(w_{1})=a+b\geq d_{G_{0}}(w_{1})=4$ by Lemma 2.5 (ii). Thus $G$ has at least one pendent vertex. Applying (2.3) with $(v,u)=(v_{2},x)$ and $(v,u)=(u_{1},x)$, respectively. We have $a=2$ and $a=\frac{a+b-3+2+d(s_{2})+d(u_{2})-(a+b)}{a+b-1}$, respectively. Hence $d(s_{2})+d(u_{2})=2(a+b)-1$. If $p>2$, then $d(s_{2})=2$. It follows from the fact that $d_{G_{0}}(u_{2})=2$ or $3$ and $a+b\geq 4$ that $d(u_{2})=2(a+b)-3>max\\{a+b,d_{G_{0}}(u_{2})\\}$, which is impossible by Lemma 2.5 (ii). If $p=2$, then $d(s_{2})=d(w_{1})=a+b$. So $d(u_{2})=a+b-1\geq 3$. It follows that $n=2,d(u_{2})=d_{G_{0}}(u_{2})=3$ and $a+b=4$. Hence $a=b=2$. By (2.2), $S(w_{k-1})=6$. On the other hand, $S(w_{k-1})=d(w_{k-2})+d(u_{2})=5$ or $7$ by (2.1), a contradiction. Subcase 2. $m=k=3$. Then $d(w_{1})+d(u_{1})=S(v_{2})=S(w_{2})=d(w_{1})+d(u_{n})$ by (2.1) and (2.2). So $d(u_{1})=d(u_{n})$. Thus $S(u_{1})=S(u_{n})$ by (2.2). We claim that $p=q=2$ or $3$. Otherwise, let $p\neq q$. Without loss of generality, suppose that $p=3,q=2$. Then $S(u_{1})=d(u_{1})-3+4+d(u_{2})$ and $S(u_{n})=d(u_{n})-3+2+d(w_{1})+d(u_{n-1})$. Note that $d(u_{2})=d(u_{n-1}),d(w_{1})>2$. We have $S(u_{1})\neq S(u_{n})$, a contradiction. Hence $p=q=2$ or $3$. We consider the following two cases: Subcase 2.1. $G$ has no pendent vertex. Then $n=2$. Otherwise, suppose that $n>2$. Then $d(u_{2})=d(v_{2})=2$. By (2.1) and (2.2), $3+d(u_{3})=S(u_{2})=S(v_{2})=7$. This is impossible since $d(u_{3})=2$ or $3$. Hence $n=2$. If $p=q=2$. Applying (2.3) with $(v,u)=(u_{1},v_{2})$ and $(v,u)=(w_{1},u_{1})$, respectively. We have $a=2$ and $a=1$, respectively. A contradiction. If $p=q=3$, also applying (2.3) with $(v,u)=(u_{1},v_{2})$ and $(v,u)=(w_{1},u_{1})$, respectively, we have $a=0$ and $a=1$, respectively. Also a contradiction. Subcase 2.2. $G$ has at least one pendent vertex. Then $a\geq 2,a+b\geq 3$. First, let $p=q=3$. Applying (2.3) with $(v,u)=(w_{1},x)$, we have $a=\frac{d(w_{1})-4+8-(a+b)}{d(w_{1})-1}$. By Lemma 2.5 (ii), $d(w_{1})=a+b$ or $4$. It follows that $a<2$, a contradiction. Next, suppose that $p=q=2$. By Lemma 2.5 (ii), $d(w_{1})=a+b$ or $4$. If $d(w_{1})=a+b\geq 4$. Applying (2.3) with $(v,u)=(w_{1},x)$ and note that $d(u_{1})=d(u_{n})$, we have $a=\frac{2d(u_{1})}{a+b-1}$. If $d(u_{1})=a+b$, then $a=2+\frac{2}{a+b-1}$ is not an integer, which is impossible by Lemma 2.2. If $d(u_{1})=d_{G_{0}}(u_{1})=3\neq a+b$, then $a+b=4$ and $a=b=2$. So $S(v_{2})=6$ by (2.2). On the other hand, $S(v_{2})=d(u_{1})+d(w_{1})=7$ by (2.1), a contradiction. If $d(w_{1})=4\neq a+b$. Applying (2.3) with $(v,u)=(v_{2},x)$, we have $a=4+d(u_{1})-(a+b)$. If $d(u_{1})=a+b$, then $a=4$. For the vertex $u_{1}$, we have $S(u_{1})=4(a+b)+b=a+b-3+6+d(u_{2})$. It follows that $d(u_{2})=4(a+b)-7>max\\{a+b,d_{G_{0}}(u_{2})\\}$, which is impossible by Lemma 2.5 (ii). If $d(u_{1})=3\neq a+b$, then $a=7-(a+b)$. Note that $a+b\neq 3,4$. We have $a+b=5$ and $a=2,b=3$ by Lemma 2.5 (iii). By (2.1) and (2.2), $S(w_{1})=11=4+d(u_{1})+d(u_{n})$. This is impossible since $d(u_{1})=d(u_{n})$. Up to now, we have completed the proof of the Lemma. $\square$. ###### Lemma 3.7 Let $G\in{\mathcal{G}}$ with $G_{0}\in{\mathcal{T}}_{6}$. Then $G=H_{i}$ for $i=11,\dots,15$ (see Fig. 3) or $G\in{\mathcal{G}}_{4}$ (see Fig. 4). Proof. Let $G_{0}=T_{6}$ (see Fig. 2), where $n,m,p,k\geq 2$. For convenience, we set $u_{1}=v_{1}=w_{1}=s_{1}$ and $u_{n}=v_{m}=w_{k}=s_{p}$. Case 1. There is a vertex $v\in\\{u_{2},v_{2},w_{2},s_{2}\\}$ such that $d_{G_{0}}(v)=2$ and $d(v)=a+b$. Without loss of generality, suppose that $v=u_{2}$. Applying (2.3) with $(v,u)=(u_{2},x)$, we have $a=\frac{a+b-2+d(u_{1})+d(u_{3})-(a+b)}{a+b-1}=\frac{d(u_{1})+d(u_{3})-2}{a+b-1}.$ (3.4) By Lemmas 2.5 (ii), $d(u_{1}),d(u_{3})=4$ or $a+b$. We consider the following three cases: Subcase 1. $d(u_{1})=d(u_{3})=a+b$, then $a=2$ by (3.4). We now show that $d(u_{i})=a+b$ for $1\leq i\leq n$. If $n=3$, then obviously, $d(u_{i})=a+b$ for $1\leq i\leq n$. If $n\geq 4$, then $d(u_{i})=d(u_{2})=a+b$ for $1<i<n$ and $d(u_{n})=d(u_{1})=a+b$ by Lemma 3.1 (i). Hence $d(u_{i})=a+b$ for $1\leq i\leq n$. Applying Lemma 2.5 (iv) with $C=u_{1}u_{2}\dots u_{n}v_{m-1}\dots v_{2}u_{1}$, we have $d(v_{i})\neq a+b$ for some $2\leq i\leq m-1$. It follows from Lemmas 2.5 (ii) and 3.1 (i) that $m\geq 3$ and $d(v_{i})=2$ for all $2\leq i\leq m-1$. Thus $S(v_{2})=2a+b=a+b+d(v_{3})$. Note that $a=2$. We have $d(v_{3})=2$. So $m\geq 4$. It follows from Lemma 3.1 (ii) that $m=4$. Similarly, we have $k=p=4$ and $d(w_{2})=d(w_{3})=d(s_{2})=d(s_{3})=2$. For the vertex $u_{1}$, $S(u_{1})=2(a+b)+b=a+b-4+6+a+b$ by (2.1) and (2.2). Hence $b=2$. It follows that $d(u_{i})=a+b=4$ for $1\leq i\leq n$. Therefore $G\in{\mathcal{G}}_{4}$ (see Fig. 4), where $l_{1}\geq 1$. It is easy to see that any graph $G\in{\mathcal{G}}_{4}$ is 2-walk $(2,2)$-linear. Subcase 2. $d(u_{1})=a+b,d(u_{3})=4$ or $d(u_{1})=4,d(u_{3})=a+b$ and $a+b\neq 4$. Then $a=1+\frac{3}{a+b-1}$ is not an integer by (3.4), a contradiction. Subcase 3. $d(u_{1})=d(u_{3})=4\neq a+b$. Then $n=3$ and $a=\frac{6}{a+b-1}$. It follows from Lemma 2.5 (iii) that $a+b=3$ and $a=3,b=0$. Thus $d(u_{2})=a+b=3$. For the vertex $u_{1}$, $S(u_{1})=12=3+d(v_{2})+d(w_{2})+d(s_{2})$ by (2.1) and (2.2). Note that $d(v_{2}),d(w_{2}),d(s_{2})\in\\{2,4,a+b\\}$. We have $d(v_{2})=d(w_{2})=d(s_{2})=a+b=3$. So $S(v_{2})=9=5+d(v_{3})$. Thus $d(v_{3})=4\neq a+b$. It implies that $m=3$. Similarly, we have $k=p=3$. Therefore $G=H_{11}$ (see Fig. 3). By (2.3), $H_{11}$ is 2-walk $(3,0)$-linear. Case 2. For any vertex $v\in\\{u_{2},v_{2},w_{2},s_{2}\\}$, $d_{G_{0}}(v)=4$ or $d(v)=2$. By Lemma 2.4, $n,m,p,k\leq 4$. Without loss of generality, suppose that $n\geq m\geq p\geq k$. Then $n\geq m\geq p\geq 3$ and $d(u_{2})=d(v_{2})=d(s_{2})=2$. By (2.1) and (2.2), $S(u_{2})=d(u_{1})+d(u_{3})=S(v_{2})=d(u_{1})+d(v_{3})$. Thus $d(v_{3})=d(u_{3})$. It implies that $m=n$. Similarly, we have $p=n$ and $k=n$ or $2$. Hence $k=p=m=n\in\\{3,4\\}$ or $k=2,p=m=n\in\\{3,4\\}$. If $G$ has no pendent vertex, then $G=H_{i}$ for $i=12,\dots,15$ (see Fig. 3). By (2.3), $H_{12}$ is 2-walk $(1,6)$-linear, $H_{13}$ is 2-walk $(0,8)$-linear, $H_{14}$ is 2-walk $(2,2)$-linear and $H_{15}$ is 2-walk $(1,4)$-linear. If $G$ has at least one pendent vertex $x$, then $x\in N(u_{1})$ or $x\in N(u_{n})$. Without loss of generality, suppose that $x\in N(u_{1})$. Then $d(u_{1})=a+b\geq 5$. Applying (2.3) with $(v,u)=(u_{1},x)$, we have $a=\frac{d(w_{2})+2}{a+b-1}$. By Lemma 2.5 (ii), $d(w_{2})=2,4$ or $a+b$. It implies that $a<2$. This is impossible by Lemma 2.5 (iii). Up to now, we have competed the proof of the Lemma. $\square$ ###### Lemma 3.8 Let $G\in{\mathcal{G}}$ with $G_{0}\in{\mathcal{T}}_{7}$(see Fig. 1). Then $G=H_{i}$ for $i=16,\dots,23$ (see Fig. 3) or $G\in{\mathcal{G}}_{j}$ for $j=5,6$ (see Fig. 4). Proof. Let $G_{0}=T_{7}$ (see Fig. 2), where $n,m,k,l,p,q\geq 2$. For convenience, we set $u_{1}=w_{1}=s_{1},u_{n}=w_{k}=t_{1},v_{1}=r_{1}=s_{p}$ and $v_{m}=r_{l}=t_{q}$. If $G$ has no pendent vertex, then $n,m,k,l,p,q\leq 4$ by Lemma 2.4. Without loss of generality, suppose that $n\geq k,m\geq l,p\geq q$. Then $n=3$ or $4$. By (2.1) and (2.2), $S(u_{1})=d(u_{2})+d(w_{2})+d(s_{2})=S(u_{n})=d(u_{n-1})+d(w_{k-1})+d(t_{2})$. Note that $d(u_{2})=d(u_{n-1}),d(w_{2})=d(w_{k-1})$. We have $d(s_{2})=d(t_{2})$. It implies that $p=q$. Similarly, we have $k=l$. If $n=3$, then $m=3,k,p=2$ or $3$ by Lemma 2.4. Hence $G=H_{i}$ for $i=16,17,18,19$ (see Fig. 3). If $n=4$, then $m=4,k,p=2$ or $4$ by Lemma 2.4. Hence $G=H_{j}$ for $j=20,21,22,23$ (see Fig. 3). By (2.3), $H_{16}$ is 2-walk $(2,2)$-linear, $H_{17}$ and $H_{18}$ are 2-walk $(1,4)$-linear, $H_{19}$ is 2-walk $(0,6)$-linear, $H_{20}$ is 2-walk $(3,-1)$-linear, $H_{21}$ and $H_{22}$ are 2-walk $(2,1)$-linear, $H_{23}$ is 2-walk $(1,3)$-linear. If $G$ has at least one pendent vertex. Then $a\geq 2,a+b\geq 3$. We first show that $d(v_{1})=d(v_{m})$. If $max\\{m,l\\}\geq 4$, then $d(v_{1})=d(v_{m})$ by Lemma 3.1 (i). If $m,l\leq 3$. By way of contradiction, suppose that $d(v_{1})\neq d(v_{m})$. By Lemma 2.5 (ii), we may assume that $d(v_{1})=a+b>3,d(v_{m})=3$ without loss of generality. Applying (2.3) with $(v,u)=(v_{1},v_{3})$, we have $a=\begin{cases}\frac{d(s_{p-1})-d(t_{q-1})}{a+b-3}&\mbox{if}\ m=3,l=2\ \mbox{or}\ m=2,l=3,\\\ 1+\frac{d(s_{p-1})-d(t_{q-1})}{a+b-3}\quad&\mbox{if}\ m=3,l=3.\end{cases}$ By Lemma 2.5 (ii), $d(s_{p-1}),d(t_{q-1})=2,3$ or $a+b$. If $m=3,l=2$ or $m=2,l=3$, then $d(s_{p-1})=a+b,d(t_{q-1})=2$ since $a\geq 2,a+b\geq 3$. So $a=1+\frac{1}{a+b-3}$. It implies that $a+b=4$ and $a=b=2$. By (2.1) and (2.2), $S(v_{3})=6+d(v_{2})=8$. So $d(v_{2})=2$ and $S(v_{2})=6$ by (2.2). On the other hand, $S(v_{2})=7$ by (2.1), a contradiction. If $m=l=3$, then $d(s_{p-1})=a+b\neq 3,d(t_{q-1})=3$ or $d(s_{p-1})=a+b,d(t_{q-1})=2$ since $a\geq 2,a+b\geq 3$. If $d(s_{p-1})=a+b\neq 3,d(t_{q-1})=3$, then $a+b\geq 4$ and $a=2$. Thus $b\geq 2$. By (2.1) and (2.2), $S(v_{3})=6+b=d(v_{2})+d(r_{2})+3$, which is impossible since $d(v_{2}),d(r_{2})=2$ or $2+b$ and $b\geq 2$. If $d(s_{p-1})=a+b,d(t_{q-1})=2$, then $a=2+\frac{1}{a+b-3}$. It implies that $a+b=4$ and $a=3,b=1$. By (2.1) and (2.2), $S(v_{3})=d(v_{2})+d(r_{2})+2=10$. Note that $d(v_{2}),d(r_{2})=2$ or $4$ by Lemma 2.5 (ii), we have $d(v_{2})=d(r_{2})=4$. Thus $S(v_{2})=13$ by (2.2). On the other hand, $S(v_{2})=9$ by (2.1), a contradiction. Hence $d(v_{1})=d(v_{m})$. Next, we show that $d(v_{1})=d(v_{m})=a+b$. On the contrary, suppose that $d(v_{1})=d(v_{m})=3\neq a+b$ by Lemma 2.5 (ii). Then $a+b\geq 4$. Note that $max\\{m,l\\}\geq 3$. Without loss of generality, suppose that $m\geq 3$. By Lemma 2.5 (ii), we have $d(v_{2})=a+b$ or $2$. If $d(v_{2})=a+b$. Applying (2.3) with $(v,u)=(v_{2},x)$, we have $a=\frac{d(v_{3})+1}{a+b-1}$. By Lemma 2.5 (ii), $d(v_{3})=2,3$ or $a+b$. This together with $a+b\geq 4$ implies that $a<2$. It contradicts the fact that $a\geq 2$. If $d(v_{2})=2$. Applying (2.3) with $(v,u)=(v_{2},x)$, we have $a=3+d(v_{3})-(a+b)$. If $m\geq 4$, then $d(v_{3})=2$ by Lemma 3.1 (i). Note that $a+b\geq 4$. We have $a=5-(a+b)\leq 1$. It contradicts the fact $a\geq 2$. Thus $m=3$. Hence $d(v_{3})=3$ and $a=6-(a+b)$. Note that $a+b\geq 4,a\geq 2$. We have $a+b=4$ and $a=b=2$. By (2.1) and (2.2), $S(v_{1})=2+d(r_{2})+d(s_{p-1})=8$. So $d(r_{2})+d(s_{p-1})=6$. If $l=2$, then $d(r_{2})=d(v_{m})=3$. So $d(s_{p-1})=3\neq a+b$. It implies that $p=2$ and $d(u_{1})=d(s_{p-1})=3$. Similarly, we get $q=2$ and $d(u_{n})=3$. By (2.1) and (2.2), $S(u_{1})=3+d(w_{2})+d(u_{2})=8$. Note that $d(w_{2}),d(u_{2})=2,3$ or $4$ by Lemma 2.5 (ii). It follows that $d(u_{2})=2,d(w_{2})=3$ or $d(u_{2})=3,d(w_{2})=2$. We may suppose that $d(u_{2})=2,d(w_{2})=3$ without loss of generality. Then $k=3$ and $d(u_{i})=2$ for $2\leq i\leq n-1$ by Lemma 3.1 (i). Hence $G$ has no pendent vertex. This is a contradiction. If $l\geq 3$, then $d(r_{2})=2$ or $4$. If $d(r_{2})=2$, then $d(s_{p-1})=6-d(r_{2})=4$. Thus $S(s_{p-1})=10$ by (2.2). This is impossible since $S(s_{p-1})=4-2+3+d(s_{p-2})\leq 9$ by (2.1). If $d(r_{2})=4$, then $S(r_{2})=10$ by (2.2). This is also impossible since $S(r_{2})=4-2+3+d(r_{3})\leq 9$ by (2.1). Hence $d(v_{1})=d(v_{m})=a+b$. Dually, we have $d(u_{1})=d(u_{n})=a+b$. Applying Lemma 2.5 (iv) with $C=v_{1}v_{2}\dots v_{m}r_{l-1}\dots r_{2}v_{1}$, we have, without loss of generality, $m\geq 3$ and $d(v_{i})\neq a+b$ for some $2\leq i\leq m-1$. So $d(v_{i})=2$ for all $2\leq i\leq m-1$ by Lemmas 2.5 (ii) and 3.1. Applying (2.3) with $(v,u)=(v_{1},x)$ and $(v,u)=(v_{2},x)$, respectively. We have $a=\frac{d(r_{2})+d(s_{p-1})-1}{a+b-1}\ \mbox{and}\ a=d(v_{3}),$ (3.5) respectively. We claim that $m\geq 4$. Otherwise, suppose that $m=3$. Then $a=d(v_{3})=a+b$ and $d(r_{2})+d(s_{p-1})=a(a+b-1)+1\geq 3(a+b)-2>2(a+b)$, which is impossible since $d(r_{2}),d(s_{p-1})=2$ or $a+b$ by Lemma 2.5 (ii). Hence $m\geq 4$. Therefore $m=4$ by Lemma 3.1 (ii). Thus $a=d(v_{3})=2$. It follows from (3.5) that $d(r_{2})+d(s_{p-1})=2(a+b)-1$. By Lemma 2.5 (iii), $d(r_{2}),d(s_{p-1})=2$ or $a+b$. If $d(r_{2})=d(s_{p-1})=2$, then $a+b=2.5$, a contradiction. If $d(r_{2})=d(s_{p-1})=a+b$, then $2(a+b)=2(a+b)-1$, a contradiction. If $d(r_{2})=2,d(s_{p-1})=a+b$, then $a+b=3$. So $a=2,b=1$. By Lemma 3.1 (i), $d(s_{i})=d(s_{p-1})=3$ for $2\leq i\leq p-1$. For the vertex $u_{1}$, $S(u_{1})=3+d(u_{2})+d(w_{2})=7$ by (2.1) and (2.2). By Lemma 2.5 (ii), $d(u_{2}),d(w_{2})=2$ or $3$. It follows that $d(u_{2})=d(w_{2})=2$. So $S(u_{2})=3+d(u_{3})=5$. Thus $d(u_{3})=2$ and $n\geq 4$. Hence $n=4$ by Lemma 3.1 (ii). Similarly, we have $d(w_{3})=d(r_{2})=2$ and $k=l=4$. By (2.1) and (2.2), $S(u_{4})=4+d(t_{2})=7$. So $d(t_{2})=3$. Hence $d(t_{i})=3$ for $1\leq i\leq q-1$ by Lemma 3.1 (i). Therefore $G\in{\mathcal{G}}_{5}$ (see Fig. 4), where $max\\{k_{1},k_{2}\\}\geq 1$. It is easy to see that any graph $G\in{\mathcal{G}}_{5}$ is $2$-walk $(2,1)$-linear. If $d(r_{2})=a+b,d(s_{p-1})=2$, then with a similar argument of the case $d(r_{2})=2,d(s_{p-1})=a+b$, we have $G\in{\mathcal{G}}_{6}$ is $2$-walk $(2,1)$-linear (see Fig. 4), where $max\\{k_{1},k_{2}\\}\geq 1$. Up to now, we have complete the proof of the Lemma. $\square$ ###### Lemma 3.9 Let $G\in{\mathcal{G}}$ with $G_{0}\in{\mathcal{T}}_{8}$. Then $G\cong H_{i}$ for $i=24,\dots,30$ (see Fig. 3) or $G\in{\mathcal{G}}_{j}$ for $j=7,8$ (see Fig. 4). Proof. Let $G_{0}=T_{8}$ (see Fig. 2), where $n,m,k,l,p,q\geq 2$. For convenience, we set $u_{1}=v_{1}=w_{1}$, $u_{n}=t_{1}=s_{p}$, $v_{m}=r_{1}=t_{q}$ and $w_{k}=s_{1}=r_{l}$. Case 1. There is a vertex $v\in\\{w_{2},v_{2},u_{2}\\}$ with $d_{G_{0}}(v)=2$ and $d(v)=a+b$. Without loss of generality, suppose that $v=w_{2}$. Then $k\geq 3$ and $d(w_{i})=a+b$ for $2\leq i\leq k-1$ by Lemma 3.1 (i). In particular, $,a\geq 2,a+b\geq 3$ by Lemma 2.5 (iii). Applying Lemma 2.3 with $(v,u)=(w_{2},x)$, we have $a=\frac{d(w_{1})+d(w_{3})-2}{a+b-1}$. By Lemma 2.5 (ii), we have $d(w_{1}),d(w_{3})=3$ or $a+b$. If $d(w_{1})=d(w_{3})=3\neq a+b$, then $a+b\geq 4$ and $a=\frac{4}{a+b-1}<2$, a contradiction. If $d(w_{1})=a+b,d(w_{3})=3$ or $d(w_{3})=a+b,d(w_{1})=3$ and $a+b\neq 3$, then $a=\frac{a+b+1}{a+b-1}=1+\frac{2}{a+b-1}<2$, a contradiction. If $d(w_{1})=d(w_{3})=a+b$, then $a=2$. We claim that $a+b=3$. Otherwise, let $a+b>3$. For the vertex $w_{1}$, $S(w_{1})=2(a+b)+b=a+b-3+a+b+d(u_{2})+d(v_{2})$. Thus $d(u_{2})+d(v_{2})=a+b+1$. By Lemma 2.5 (ii), $d(u_{2}),d(v_{2})=2,3$ or $a+b$. Note that $a+b>3$. We have $d(u_{2})=2,d(v_{2})=3$ or $d(u_{2})=3,d(v_{2})=2$ or $d(u_{2})=d(v_{2})=3$. Without loss of generality, we consider the following two cases: If $d(u_{2})=2,d(v_{2})=3$, then $a+b=4$ and $m=2$. Note that $a=2$. We have $b=2$. If $k\geq 4$, then $d(w_{k})=d(w_{1})=4$ by Lemma 3.1 (i). If $k=3$, then $d(w_{k})=d(w_{3})=4$. So $d(r_{3})=2$ or $4$ by Lemma 2.5 (ii). On the other hand, $S(v_{2})=4+d(r_{2})+d(t_{q-1})=8$ by (2.1) and (2.2). So $d(r_{2})=d(t_{q-1})=2$. By (2.1) and (2.2), $S(r_{2})=6=3+d(r_{3})$. Thus $d(r_{3})=3$, a contradiction. If $d(u_{2})=d(v_{2})=3$, then $a+b=5$ and $m=n=2$. Note that $a=2$. We have $b=3$. By (2.1) and (2.2), $S(v_{2})=5+d(r_{2})+d(t_{q-1})=9$. Thus $d(r_{2})=d(t_{q-1})=2$. Hence $S(r_{2})=7=3+d(r_{3})$. This is impossible since $d(r_{3})\in\\{2,3,5\\}$. Therefore $a+b=3$ and $a=2,b=1$. By (2.1) and (2.2), $S(w_{1})=7=3+d(v_{2})+d(u_{2})$. So $d(v_{2})=d(u_{2})=2$. Hence $S(v_{2})=5=3+d(v_{3})$. It follows that $d(v_{3})=2$ and $m\geq 4$. Therefore $m=4$ by Lemma 3.1 (ii). Similarly, we have $n=l=p=4,d(u_{2})=d(u_{3})=d(r_{2})=d(r_{3})=d(s_{2})=d(s_{3})=2$ and $d(t_{i})=3$ for $2\leq i\leq q-1$. Therefore $G\in{\mathcal{G}}_{8}$ (see Fig. 4), where $max\\{k_{1},k_{2}\\}\geq 1$. It is easy to see that any graph $G\in{\mathcal{G}}_{8}$ is 2-walk $(2,1)$-linear. Case 2. For any vertex $v\in\\{w_{2},v_{2},u_{2}\\}$, we have $d_{G_{0}}(v)=3$ or $d(v)=2$. If $G$ has no pendent vertex, then $n,m,k,l,p,q\leq 4$ by Lemma 2.4. If $n=m=k=l=p=q=2$, then $G$ is regular. It is well known that a graph is regular if and only if it has exactly one main eigenvalues. Thus $max\\{n,m,k,p,q\\}\geq 3$. Without loss of generality, suppose that $n\geq 3$. We consider the following two cases: Subcase 1. $n=3$. Then $S(u_{2})=6$ and $m,k,l,p,q=2$ or $3$ by Lemma 2.4. If $m=k=2$, then $S(u_{1})=8=S(u_{3})=2+d(t_{2})+d(s_{p-1})$. So $d(t_{2})=d(s_{p-1})=3$ and hence $p=q=2$. Similarly, $S(u_{1})=8=S(v_{2})=6+d(r_{2})$. Thus $d(r_{2})=2$ and $l=3$. Therefore $G=H_{24}$ (see Fig. 3). By (2.3), $H_{24}$ is 2-walk $(2,2)$-linear. With a similar argument, we have: If $m=2,k=3$ or $m=3,k=2$, then $G=H_{25}$ is 2-walk $(1,4)$-linear (see Fig. 3). If $m=k=3$, then $G=H_{26}$ is 2-walk $(0,6)$-linear (see Fig. 3). Subcase 2. $n=4$. Then $m,k,l,p,q=2$ or $4$ by Lemma 2.4. With a similar argument of Subcase 1, we have the following cases: If $m=k=2$, then $G=H_{27}$ is 2-walk $(3,-1)$-linear (see Fig. 3). If $m=2,k=4$ or $m=4,k=2$, then $G=H_{28}$ is 2-walk $(2,1)$-linear (see Fig. 3). If $m=k=4$, then $G=H_{29}$ is 2-walk $(1,3)$-linear (see Fig. 3). If $G$ has at least one pendent vertex $x$, then $x\in N(v)$ for $v\in\\{w_{1},v_{m},u_{n},w_{k},r_{i_{1}},t_{i_{2}},s_{i_{3}}\\}$, where $2\leq i_{1}\leq l-1,2\leq i_{2}\leq q-1,2\leq i_{3}\leq p-1$. If $v\in\\{r_{i_{1}},t_{i_{2}},s_{i_{3}}\\}$, then with a similar argument of Case 1, we have $G\in\mathcal{G}_{8}$. Hence we may suppose that $v\in\\{w_{1},v_{m},u_{n}\\}$. In particular, assume that $x\in N(w_{1})$ without loss of generality. Then $d(w_{1})=a+b\geq 4$ and $d(u)=d_{G_{0}}(u)$ for $u\in\\{r_{i_{1}},t_{i_{2}},s_{i_{3}}\\}$, where $2\leq i_{1}\leq l-1,2\leq i_{2}\leq q-1,2\leq i_{3}\leq p-1$. Applying Lemma (2.3) with $(v,u)=(w_{1},x)$, we have $a=\frac{a+b-3+d(u_{2})+d(v_{2})+d(w_{2})-(a+b)}{a+b-1}=\frac{d(u_{2})+d(v_{2})+d(w_{2})-3}{a+b-1}.$ Note that $d(u_{2}),d(v_{2}),d(w_{2})=2,3$ or $a+b$ and $a+b\geq 4$. We consider the following cases without loss of generality: If $d(u_{2})=d(v_{2})=d(w_{2})=2$, then $a=\frac{3}{a+b-1}<2$, a contradiction. If $d(u_{2})=d(v_{2})=2,d(w_{2})=3$, then $a=\frac{4}{a+b-1}<2$, a contradiction. If $d(u_{2})=d(v_{2})=2,d(w_{2})=a+b$, then $a=1+\frac{2}{a+b-1}<2$, a contradiction. If $d(u_{2})=2,d(v_{2})=d(w_{2})=3$, then $a=\frac{5}{a+b-1}<2$, a contradiction. If $d(u_{2})=2,d(v_{2})=3,d(w_{2})=a+b$, then $m=2$ and $a=1+\frac{3}{a+b-1}$. By Lemmas 2.2 and 2.5 (ii), we have $a+b=4$ and $a=2$. Hence $d(w_{i})=d(w_{2})=4$ for $2\leq i\leq k-1$. By (2.1) and (2.2), $S(w_{k-1})=4+2+d(w_{k})=10$. Thus $d(w_{k})=4$. Similarly, $S(v_{2})=4+d(r_{2})+d(t_{q-1})=8$. It implies that $d(r_{2})=d(t_{q-1})=2$. Thus $S(r_{2})=6=3+d(r_{3})$. Hence $d(r_{3})=3$. On the other hand, $d(r_{3})\in\\{2,4\\}$ by Lemma 2.5 (ii) and the fact that $d(w_{k})=4$. This is a contradiction. If $d(u_{2})=2,d(v_{2})=d(w_{2})=a+b$, then $a=2+\frac{1}{a+b-1}$ is not an integer, a contradiction. If $d(u_{2})=d(v_{2})=d(w_{2})=3$, then $n=m=k=2$ and $a=\frac{6}{a+b-1}$. It follows from Lemma 2.5 (iii) that $a+b=4$ and $a=b=2$. By (2.1) and (2.2), $S(v_{2})=4+d(r_{2})+d(t_{q-1})=8$. It implies that $d(r_{2})=d(t_{q-1})=2$. Hence $S(r_{2})=3+d(r_{3})=6$. It follows that $d(r_{3})=3$ and $l=3$. Similarly, we have $p=q=3$. Therefore $G=H_{30}$ (see Fig. 3). By (2.3), $H_{30}$ is 2-walk $(2,2)$-linear. If $d(u_{2})=d(v_{2})=3,d(w_{2})=a+b$, then $n=m=2$ and $a=1+\frac{4}{a+b-1}$. Thus $a+b=5$ and $a=2$ by Lemma 2.5 (iii). Hence $d(w_{1})=d(w_{2})=a+b=5$. For the vertex $v_{2}$, $S(v_{2})=5+d(r_{2})+d(t_{q-1})=9$. It implies that $d(r_{2})=d(t_{q-1})=2$. Thus $S(r_{2})=3+d(r_{3})=7$, which is impossible since $d(r_{3})\in\\{2,3,5\\}$ by Lemma 2.5 (ii). If $d(u_{2})=3,d(v_{2})=d(w_{2})=a+b$, then $a=2+\frac{2}{a+b-1}$ is not an integer, a contradiction. If $d(u_{2})=d(v_{2})=d(w_{2})=a+b$, then $n=m=k=2$ and $a=3$. We claim that $p=l=2$. Otherwise, let $p,l>2$. By (2.1), $S(w_{1})=a+b-3+3(a+b),S(w_{2})=a+b-3+a+b+d(s_{2})+d(r_{l-1})$. Note that $d(s_{2}),d(r_{l-1})=2$ by assumption. We have $S(w_{1})>S(w_{2})$. On the other hand, $S(w_{1})=S(w_{2})$ by (2.2), a contradiction. Hence $p=l=2$. Similarly, we have $q=2$. Thus $G\in{\mathcal{G}}_{7}$ (see Fig. 4), where $b\geq 1$. It is easy to see that any graph $G\in{\mathcal{G}}_{7}$ is 2-walk $(3,b)$-linear. Up to now, we have complete the proof of the Lemma. $\square$ ###### Theorem 3.10 The graphs $H_{i}$ for $i=1,\dots,30$ and those in ${\mathcal{G}}_{j}$ for $j=1,\dots,8$ are all connected tricyclic graphs with exactly two main eigenvalues. Proof. It follows directly from Lemmas 3.2–3.9. $\square$ ## References * [1] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, Macmillan, New York, 1976. * [2] D. Cvetkovic, P. Rowlinson, S. Simic, Eigenspaces of Graphs, Cambridge University Press, Cambridge, 1997. * [3] E.M. Hagos, Some results on graph spectra. Linear Algebra Appl. 356 (2002) 103–111. * [4] Y.P. Hou, H.Q. Zhou, Trees with exactly two main eigenvalues, Acta of Hunan Normal University. 28(2) (2005) 1–3 (in Chinese). * [5] Y.P. Hou, F. Tian, Unicyclic graphs with exactly two main eigenvalues, Appl. Math. Lett. 19 (2006) 1143–1147. * [6] Z.Q. Hu, S.C. Li, C.F. Zhu, Bicyclic graphs with exactly two main eigenvalues, Linear Algebra Appl. 431 (2009) 1848–1857. * [7] L.S. Shi, On graphs with given main eigenvalues, Appl. Math. Lett. 22 (2009) 1870–1874. * [8] X.Y. Geng, S.C. Li, The spectral radius of tricyclic graphs with $n$ vertices and $k$ pendant vertices, Linear Algebra Appl. 428 (2008) 2639–2653.	arxiv-papers	2010-12-05T00:58:41	2024-09-04T02:49:15.471690	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Xiaoxia Fan and Yanfeng Luo", "submitter": "Xiaoxia Fan", "url": "https://arxiv.org/abs/1012.0963" }
1012.1097	# New static spheroidal solution in Jordan-Brands-Dicke theory. S.M.KOZYREV Scientific center gravity wave studies ”Dulkyn”, PB 595, Kazan, 420111, Russia ,Kazan, Russian Federation Email address: Sergey@tnpko.ru ###### Abstract The static spheroidal solutions of Jordan-Brands-Dicke theory (JBD) are studied. We consider the effect of the anisotropic stresses of scalar field on the shape of JBD self-graviting objects. It is shown that scalar fields can have significant effect on the structure and properties of self-graviting objects. In contrast with general relativity in JBD theory there are nonflat static spheroidal solutions. PACS: 04.20.Jb, 04.40.Dg, 95.35.+d ## 1 Introduction A wide spread assumption in the study of stellar structure is that the shape of star can be modeled as a spherical symmetry object. This approach has been used extensively in the study of star, star system and galaxies [1]. However, in many systems, deviation from spherical symmetry may play an important role in determining of them properties. Physical situation where unspherical shape may be relevant are very diverse. Scalar self-graviting objects resulting from the non minimal coupling scalar fields to gravity are a system where anisotropic pressure occurs naturally [2]. A model for the Universe where the dark matter and energy are the scalar nature can be realistic and could explain most of the observed structures [3]. The self-interaction of the scalar field could explain the behavior of galaxy rotation curves all along the background. Anisotropy appears as an extra assumption on the behavior of scalar fields and on the shape of equilibrium configuration. Since we still do not have a formulation of the possible anisotropic stresses is emerging in these or other contexts, we take the approach of finding several exact solutions representing physical situations, modelled by ellipsoid of revolution. Our goals hear is to find exact spheroidal solution, offering an analysis of the change in the physical properties of the stellar and galaxy models due to presence of non minimally coupled scalar fields. In this context, particularly interesting is the case of JBD theory [4], [5] where pressure anisotropies come in action. ## 2 Static spheroidal vacuum solutions of Jordan-Brans-Dicke theory. JBD theory can be thought of as a minimal extension of general relativity designed to properly accommodate both Mach’s principle and Dirac’s large number hypothesis. The progress in the understanding of scalar-tensor theories of gravity is closely connected with finding and investigation of exact solutions. Shortly after JBD theory was proposed, Heckmann obtained parametric form of the exact static vacuum solution to the JBD equations [7]. Later Brans [5] find the static, spherically symmetric, vacuum solution of the JBD equations in isotropic coordinates. In the Jordan conformal frame, the JBD action takes the form [4] (we use geometrized units such that G = c = 1 and we follow the signature +,-,-,-). $\displaystyle S$ $\displaystyle=$ $\displaystyle\int dx\sqrt{-g}(\phi R-\omega g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi)+S_{m}.$ (1) Here, $\omega$ coupling constant, R is the Ricci scalar curvature with respect to the space-time metric gμν and Sm denotes action of matter fields. (We use units in which gravitational constant G=1 and speed of light c=1.) Variation of (1) with respect to gμν and $\phi$ gives, respectively, the field equations: $R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\frac{1}{\phi\ }T_{\mu\nu}^{M}+T_{\mu\nu}^{JBD},$ (2) where $\displaystyle T_{\mu\nu}^{JBD}$ $\displaystyle=$ $\displaystyle[\frac{\omega}{\phi^{2}}\left(\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\nabla_{\alpha}\phi\nabla^{\alpha}\phi\right)+$ (3) $\displaystyle\ \ +\frac{1}{\phi}\left(\nabla_{\mu}\nabla_{\nu}\phi- g_{\mu\nu}\nabla_{\alpha}\nabla^{\alpha}\phi\right)],$ and $\nabla_{\alpha}\nabla^{\alpha}\phi=\frac{T_{\lambda}^{M\ \lambda}}{3+2\omega},$ (4) and $T_{\lambda}^{M\ \lambda}$ is the energy momentum tensor of ordinary matter which obeys the conservation equation $T_{\mu\nu;\lambda}^{M\ }$ gνλ= 0. One can derive the energy density and pressure of JBD field from (3) $\rho^{JBD}=T_{00}$, $P^{JBD}=T_{ii}$ $(i=1,2,3)$. As we have already mentioned we consider standard static spheroidal space- time. Solutions to the equation in spheroidal coordinates have application to a wide range of problems in physics [6]. The two-dimensional elliptic coordinate system is defined from the set of all ellipses and all hyperbolas with a common set of two focal points. Oblate spheroidal coordinates are derived from elliptic coordinates by rotating the elliptical coordinate system about the perpendicular bisector of the focal points. Similarly, one can obtain the prolate spheroidal coordinates by rotating it about the parallel bisector. We adopt coordinates that allow us to write spheroidal geometry in prolate form $\displaystyle ds^{2}=-B\left(\xi\right)dt^{2}+A\left(\xi\right)\left(c\sqrt{\frac{\xi^{2}-\eta^{2}}{\xi^{2}-1}}d\xi^{2}+c\sqrt{\frac{\xi^{2}-\eta^{2}}{1-\eta^{2}}}d\eta^{2}+c\sqrt{(\xi^{2}-1)(1-\eta^{2})}d\varphi^{2}\right),$ (5) and in oblate form $\displaystyle ds^{2}=-B\left(\xi\right)dt^{2}+A\left(\xi\right)\left(c\sqrt{\frac{\xi^{2}-\eta^{2}}{\xi^{2}-1}}d\xi^{2}+c\sqrt{\frac{\xi^{2}-\eta^{2}}{1-\eta^{2}}}d\eta^{2}+c\xi\eta d\varphi^{2}\right),$ (6) where A, B are function of $\xi$ and $\xi\geq 1$, $-1\leq\eta\leq 1$, $0\leq\varphi\leq 2\pi$. We denote the separation of the two focal points by $c$. Then the solutions of the gravitational field equations in the vacuum $T_{\mu\nu}^{M\ }$= 0 take in oblate case the form $A=a_{0}\left(\frac{1+\sqrt{1-\xi^{2}}}{-1+\sqrt{1-\xi^{2}}}\right)^{\beta\left(1+\frac{1}{\sqrt{-3-2\omega},}\right)},$ $B=b_{0}\left(\frac{1+\sqrt{1-\xi^{2}}}{-1+\sqrt{1-\xi^{2}}}\right)^{\beta\left(-1+\frac{1}{\sqrt{-3-2\omega},}\right)},\\\ $ (7) $\phi=\phi_{0}\left(\frac{1+\sqrt{1-\xi^{2}}}{-1+\sqrt{1-\xi^{2}}}\right)^{-\frac{\beta}{\sqrt{-3-2\omega},}},$ where $\beta,a_{0},b_{0},\phi_{0}$ arbitrary constants. For the prolate case $A=a_{0}\left(\frac{1+\xi}{-1+\xi}\right)^{\beta\left(\frac{1-\sqrt{-3-2\omega}}{1+\sqrt{-3-2\omega}}\right)},$ $B=b_{0}\left(\frac{1+\xi}{-1+\xi}\right)^{\beta},\\\ $ (8) $\phi=\phi_{0}\left(\frac{1+\xi}{-1+\xi}\right)^{\left(\frac{-\beta}{1+\sqrt{-3-2\omega}}\right)}.$ These solutions can take some possible forms, depending on the values of arbitrary constants appearing in the solution. Now choose the value $\omega=-2$ and redefine the arbitrary constant. Then the metric and scalar function become $A=1,$ $B=b_{0}\left(\frac{1+\sqrt{1-\xi^{2}}}{-1+\sqrt{1-\xi^{2}}}\right)^{\beta},\\\ $ (9) $\phi=\phi_{0}\left(\frac{1+\sqrt{1-\xi^{2}}}{-1+\sqrt{1-\xi^{2}}}\right)^{-\frac{\beta}{2,}},$ and $A=1,$ $B=b_{0}\left(\frac{1+\xi}{-1+\xi}\right)^{\beta},\\\ $ (10) $\phi=\phi_{0}\left(\frac{1+\xi}{-1+\xi}\right)^{-\frac{\beta}{2}}.$ To see that all these metrics is asymptotically flat it is enough to show that the metric components behave in an appropriate way at large $\xi$-coordinate values, e.g., $g_{\mu\nu}=\eta_{\mu\nu}+O(1/\xi)$ as $\xi\rightarrow\infty$. By inspection of the coefficients, we verify that this is so. It is certainly true that any vacuum solution of Einstein’s equations is also a solution of JBD equations (2) with $\phi$ strictly constant, and that $\phi=const$ is the solution of the equation(4) for $\omega\rightarrow\infty$. However, this by no means implies that all JBD solutions satisfy Einstein’s equations in the limit $\omega\rightarrow\infty$ or in the limit $\phi\rightarrow const$. In fact, it is easy to show that Einstein’s field equations yield only the flat space $A=1$, $B=1$. We now deduce the results obtained in the oblate and prolate spheroidal coordinates. Firstly, in both cases, the JBD scalar field appears to play the role of dark matter component. Secondly, the directional components of equation of state of JBD field are anisotropic and in both cases indicate that the JBD scalar field appears to be an ”exotic” type of dark matter. ## 3 Discussion In this article we delineated the qualitative features one would expect from spheroidal scalar field object. It is demonstrated that our model can successfully predict the spheroidal configuration in terms of a self- gravitating spacetime solution to the JBD field equations and reproduce the not spherically-symmetric shape in terms of the non-trivial energy density and anisotropic pressure of the JBD scalar field which was absent in the context of general relativity. We believe that following this hypotheses the shape of galaxy and rotation curve may be explained by action of scalar fields. The solution presented here could be a first approximation at the galactic space- time provided the presence of scalar field. Therefore, it is necessary to study how these results modify the standard method of interpretation rotation data. Further investigation into the nature solutions with view to separating the real rotational effects from the scalar fields anisotropy might be rewarding. ## References * [1] M.R.Finch, J.E.F. Skea, A review of Relativistic Static Fluid Sphere, http://www.dft.if.uerj.br/users/JimSkea/papers/pfrev.ps * [2] S. Lee, Anisotropy due to Brans-Dicke Theory, arXiv:0811.1643 (2008). * [3] Clifford M.Will,Theory and Experiment in Gravitational Physics, Cambridge University Press, Cambridge, England, 1993 * [4] P. Jordan, Schwerkraft und Weltall, Vieweg (Braunschweig) 1955. * [5] C. Brans and R. H. Dicke, Mach’s Principle and a Relativistic Theory of Gravitation, Phys. Rev. 124, 925-935, (1961). * [6] C. Flammer. Spheroidal Wave Functions. Stanford University Press, 1957. * [7] O. Heckmann, P. Jordan, R.Fricke, Astroph., Zur erweiterten Gravitationstheorie, Z. 28, 113-149, (1951).	arxiv-papers	2010-12-06T08:56:16	2024-09-04T02:49:15.486729	{ "license": "Public Domain", "authors": "S.M.Kozyrev", "submitter": "Sergey Kozyrev", "url": "https://arxiv.org/abs/1012.1097" }
1012.1190	# An improvement upon unmixed decomposition of an algebraic variety ††thanks: Partially supported by a NKBRPC (2004CB318000) Zhenyi Ji, Yongbin Li. _School of Applied Mathematics,_ _University of Electronic Science and Technology of China,_ _Chengdu, Sichuan, 610054, China_ E-mail: jizhenyi0010@163.com (Z.Y.Ji)E-mail: yongbinli@uestc.edu.cn (Y.B.Li) ###### Abstract Decomposing an algebraic variety into irreducible or equidimensional components is a fundamental task in classical algebraic geometry and has various applications in modern geometry engineering. Several researchers studied the problem and developed efficient algorithms using $Gr$ö$bner$ basis method. In this paper, we try to modify the computation of unmixed decomposition of an algebraic variety based on improving the computation of $Zero(sat(\mathbb{T}))$, where $\mathbb{T}$ is a triangular set in K[X]. Keywords:unmixed decomposition,weakly non-degenerate conditions, $Wu^{\prime}s$ characteristic set,U-set. ## 1 Introduction Let K be a field of characteristic 0 and $\textbf{K}[x_{1},x_{2},\ldots,x_{n}]$ (or K[X] for short) the ring of polynomials in the variables $(x_{1},x_{2},\ldots,x_{n})$ with coefficients in K. A polynomial set is a finite set $\mathbb{P}$ of nonzero polynomials in K[X]. The ideal of K[X] generated by all elements of $\mathbb{P}$ is denoted by $Ideal(\mathbb{P})$ and the algebraic variety of $\mathbb{P}$ is denoted by $Zero(\mathbb{P})$. The method of $Gr$ö$bner$ bases introduced by $Buchberge$[1,2]provides a powerful device for computing a basis of $Ideal(\mathbb{P})$. It is well known that $Wu$[22] provided an efficient method for constructing a $Wu^{\prime}s$ characteristic set of every polynomial set to compute the variety of $\mathbb{P}$ in 1978. Therefrom, various algorithms for computing triangular decomposition of polynomial sets and $systems$ are developed by some researchers[3,4,10,11,17,19,23,24,25]. Based on various triangular decompositions for polynomial systems, including the famous $Wu^{\prime}s$ characteristic set method, and the $Gr$ö$bner$ Basis method we can get the unmixed decomposition of an algebraic variety. According to the analytic method established by $Zhang$ et.al[28], we get the modified method to compute characteristic series. Furthermore, we try to improve the computation of $Zero(sat(\mathbb{T}))$, where $\mathbb{T}$ is a triangular set. Some examples can illustrate our improvement. ## 2 Preliminaries ### 2.1 $Wu^{\prime}s$ characteristic set For any polynomial $p\notin\textbf{K}$, the biggest index $k$ such that $deg(p,x_{k})>0$ is called the _class_ , $x_{k}$ the _leading variable_ , $deg(p,x_{k})$ the _leading degree_ of $p$, and $lcoeff(p,x_{k})$ the _leading coefficient_ of $p$, denoted by $cls(p)$, $lv(p)$, $ldeg(p)$, $ini(p)$, respectively. Definition 2.1.1. A finite nonempty ordered set $\mathbb{T}=[f_{1},f_{2},\ldots,f_{s}]$ of polynomials in $\textbf{K[X]}\backslash\textbf{K}$ is called a _triangular set_ if $cls(f_{1})<cls(f_{2})<\cdots<cls(f_{s})$. Triangular set $\mathbb{T}$ is written as the following form $\mathbb{T}=[f_{1}(u_{1},\ldots,u_{r},y_{1}),\ldots,f_{s}(u_{1},\ldots,u_{r},y_{1},\ldots,y_{s})]$ (1) where $(u_{1},\ldots,u_{r},y_{1},\ldots,y_{s})$ is a permutation of $(x_{1},\ldots,x_{n})$. Let $f\neq 0$ be a polynomial and $g$ any polynomial in K[X],the $\emph{pesudo-}\emph{remainde}r$ of $g$ with respect to $f$ in $lv(f)$ is denoted by $prem(g,f,lv(f))$. One can find a formal definition of $pesudo- remainder$[7,18]or two alternative ones[18,19]. For any polynomial $p$ and triangular set $\mathbb{T}$ $prem(p,\mathbb{T})$ stands for the _pesudo- remainder_ of $p$ with respect to $\mathbb{T}$ is defined by $\displaystyle prem(p,\mathbb{T})=prem(\ldots prem(p,f_{s},y_{s}),\ldots,f_{1},y_{1}).$ (2) One can easily deduce the following pesudo-remainder formula $\displaystyle\prod\limits_{i=1}^{s}{ini(f_{i})}^{d_{i}}p=\sum\limits_{i=1}^{s}{q_{i}f_{i}}+prem(p,\mathbb{T}),$ (3) where each $d_{i}$ is a nonnegative integer and $q_{i}\in\textbf{K[X]}$ for $1\leq i\leq s$. For any polynomial set $\mathbb{P}\subset\textbf{K[X]}$, we write $prem(\mathbb{P},\mathbb{T})\triangleq\\{prem(p,\mathbb{T})\|p\in\mathbb{P}\\}$. Given two polynomials $f$, $g$ $\in\textbf{K[X]}$, the Sylvester resultant of $f$ and $g$ with respect to some $x_{k}$ $(1\leq k\leq n)$ is denoted by $res(f,g,x_{k})$. Let $p$ be any polynomial and $\mathbb{T}=[f_{1},f_{2},\ldots,f_{s}]$ a triangular set in K[X] as (1). The polynomial $res(p,\mathbb{T})\triangleq res(\ldots res(p,f_{s},y_{s}),\ldots,f_{1},y_{1})$ is called the _resultant_ of $p$ with respect to $\mathbb{T}$. Let $\mathbb{T}$ is a triangular set as $(1)$ and $p$ any polynomial, $p$ is said to be reduced with respect to $\mathbb{T}$ if $deg(p,y_{i})<deg(f_{i},y_{i})$ for all $i$. $\mathbb{T}$ is said to be _noncontradictory ascending set_ if every $f\in\mathbb{T}\cup(ini(\mathbb{T}))$ is reduced to $\mathbb{T}\setminus\\{f\\}$. Definition 2.1.2. A triangular set $\mathbb{T}=[f_{1},f_{2},\ldots,f_{s}]$ is called perfect, if $Zero(\mathbb{T},ini(\mathbb{T}))\neq\emptyset$ . Definition 2.1.3.[22] A noncontradictory ascending set $\mathbb{T}$ is called a $Wu^{\prime}s$ _characteristic set_ of polynomial set $\mathbb{P}\subset\textbf{K[X]}$ if $\mathbb{T}\subset Ideal(\mathbb{P})$, $prem(\mathbb{P},\mathbb{T})=\\{0\\}$. Definition 2.1.4. A finite set $\mathbb{T}_{1},\mathbb{T}_{2},\ldots,\mathbb{T}_{s}$ is called a _characteristic series_ of polynomial set $\mathbb{P}$ in K[X] if the following zero decomposition holds $Zero(\mathbb{P})=\bigcup\limits_{i=1}^{s}{Zero(\mathbb{T}_{i}/ini(\mathbb{T}_{i}))}$ (4) and $prem(\mathbb{P},\mathbb{T}_{i})=\\{0\\}$ for every $i$. Defnition 2.1.5.[10,25] A triangular set $\mathbb{T}=[f_{1},f_{2},\ldots,f_{s}]$ is called a _regular set_ if $res(I,\mathbb{T})\neq 0$ for all $I\in ini(\mathbb{T}).$ Definition 2.1.6.[4,19] A triangular set $\mathbb{T}=[f_{1},f_{2},\ldots,f_{s}]$ as (1) is called a _normal set_ if $ini(\mathbb{T})\in\textbf{K[U]}$. Definition 2.1.7. Let $\mathbb{T}$ be a triangular set in K[X]. The _saturation ideal_ of $\mathbb{T}$ $sat(\mathbb{T})\triangleq Ideal(\mathbb{T}):J^{\infty}=\\{g\in\textbf{K[X]}\mid J^{q}g\in Ideal(\mathbb{T})$ for some $q>0$$\\}$, where $J=\prod_{f\in\mathbb{T}}ini(f)$. One can compute a basis of $sat(\mathbb{T})$ by the following Lemma. Lemma 2.1.8.[5,8,20] Let $\mathbb{T}=[f_{1},f_{2},\ldots,f_{s}]$ be a triangular set in K[X], $z$ is a new variable, $\mathbb{H}=\mathbb{T}\cup\\{zJ-1\\}=\\{f_{1},f_{2},\ldots,f_{s},zJ-1\\}$, $Gb$ be the $Gr$ö$bner$ basis of $\mathbb{H}$ with respect to a _lexicographic ordering_ where $z$ is greater than every $x_{i}$. Then $sat(\mathbb{T})=Ideal(\mathbb{H})\cap\textbf{K[X]}=Ideal(Gb\cap\textbf{K[X]}).$ (5) ### 2.2 The theory of weakly non-degenerate conditions and its application Let $\mathbb{T}$ be as $(1)$, we denote ${\mathbb{C}_{f}}_{i}$ the set of all the nonzero coefficients of $f_{i}$ in $y_{i}$, $\mathbb{R}_{f_{i}}=\\{res(c,\mathbb{T})\neq 0:c\in\mathbb{C}_{f_{i}}\\}$ for any $f_{i}\in\mathbb{T}$. For any $\overline{\textbf{z}}=({\overline{\textbf{u}}},\overline{y}_{1},\ldots,\overline{y}_{s})\in Zero(\mathbb{T})$, we write ${\overline{\textbf{z}}}^{\\{{j}\\}}$ for ${\overline{\textbf{u}}}$,$\overline{y}_{1}$,$\ldots$,$\overline{y}_{j}$ or $(\overline{\textbf{u}},\overline{y}_{1},\ldots,\overline{y}_{j})$ with $0\leq j\leq s.$ Definiton 2.2.1.[28] Let $\mathbb{T}$ as $(1)$ be a regular set in K[X]. A zero $\textbf{z}_{0}\in Zero(\mathbb{T})$ is called a quasi-normal zero if $\textbf{z}_{0}^{\\{i-1\\}}\notin Zero(\mathbb{C}_{f_{i}})$ for any $1\leq i\leq s,$ also said to satisfying the _weakly non-degenerate conditions_. The following definition is an extension of the concept of _quasi-normal zero_ of regular set to triangular set. Definiton 2.2.2.[13] Let $\mathbb{T}=[f_{1},f_{2},\ldots,f_{s}]$ as (1) be a triangular set in K[X]. A zero $z_{0}$ $\in Zero(\mathbb{T})$ is called a _quasi-normal zero_ of $\mathbb{T}$ if for any $1\leq i\leq s$, _either conditions holds_ : a. $I_{i}({\bf z}^{\\{i-1\\}}_{0})\neq 0$ if ${{res}}(I_{i},\mathbb{T})=0$; b. ${\bf z}^{\\{i-1\\}}_{0}\notin{\rm Zero}({\mathbb{C}}_{f_{i}})$ if ${res}(I_{i},{\mathbb{T}})\neq 0$. For any triangular set $\mathbb{T}$ in K[X], we denote $QnZero(\mathbb{T})$ the set of all _quasi-normal zeros_ of $\mathbb{T}$ and $\overline{QnZero(\mathbb{T})}^{E}$ the closure of $QnZero(\mathbb{T})$ in topological space $\textbf{K}^{n}$, where $\textbf{K}^{n}$ is induced by follow metric $\|{\bf z}-{\bf z}^{}\|={\rm max}\\{\|x_{1}-x^{}_{1}\|,\|x_{2}-x^{}_{2}\|,\ldots,\|x_{n}-x^{}_{n}\|\\}$ for any ${\bf z},{\bf z}^{}\in\tilde{{\bf K}}^{n}$, then we have the following theorem. Theorem 2.2.3.[13] For any triangular set $\mathbb{T}=[f_{1}(\textbf{u},y_{1}),\ldots,f_{s}(\textbf{u},y_{1},\ldots,y_{s})]$, we have $Zero(\mathbb{T}/ini(\mathbb{T}))\subseteq\overline{QnZero(\mathbb{T})}^{E}\subseteq Zero(sat(\mathbb{T}))$. The following definition plays an important role in this paper. Definition 2.2.4.[9,13] Let $\mathbb{T}$ be a triangular set in K[X], We establish $\mathbb{U}_{\mathbb{T}}\triangleq\\{c\in\mathbb{C}_{f}:res(ini(f),\mathbb{T})\neq 0$ and $\mathbb{R}_{f}\cap K=\emptyset,f\in\mathbb{T}\\}\cup$ $\\{c:res(c,\mathbb{T})=0,c\in ini(\mathbb{T})\\}.$ Remark: This definition has slightly difference with the notion in[9,13]. Example 2.2.5. Let $\mathbb{T}=[f_{1},f_{2},f_{3}]$, under $x_{1}\prec x_{2}\prec x_{3}\prec x_{4}$, where $f_{1}=x_{1}x_{2}^{2}+x_{2}+2x_{1}^{2},$ $f_{2}=x_{2}x_{3}+x_{1}x_{2}^{2}+x_{2}x_{1}+2x_{1},$ $f_{3}=x_{2}x_{4}^{2}-x_{4}-x_{2}-x_{3}.$ By above notation, we know $\mathbb{C}_{f_{1}}=\\{x_{1},1,2x_{1}^{2}\\}$, $\mathbb{C}_{f_{2}}=\\{x_{2},x_{1}x_{2}^{2}+x_{1}x_{2}+2x_{1}\\}$, $\mathbb{C}_{f_{3}}=\\{x_{3},-1,-x_{2}-x_{3}\\}$. It is easy to see that $\mathbb{R}_{f_{1}}=\mathbb{C}_{f_{1}}=\\{x_{1},1,2x_{1}^{2}\\}$, $\mathbb{R}_{f_{2}}=\\{2x_{1}^{2},{x_{{1}}}^{2}\left(4\,{x_{{1}}}^{4}-6\,{x_{{1}}}^{3}+2\,{x_{{1}}}^{2}-2\,x_{{1}}+2\right)\\}$, $\mathbb{R}_{f_{3}}=\\{2x_{1},-1,4\,{x_{{1}}}^{6}-14\,{x_{{1}}}^{5}+14\,{x_{{1}}}^{4}+2\,{x_{{1}}}^{2}-2\,x_{{1}}\\}$. Thus $\mathbb{U}_{\mathbb{T}}=\\{x_{2}\\}$. Similarly, one can compute that $\mathbb{U}_{\mathbb{T}^{}}=\emptyset$ where $\mathbb{T}^{}=[g_{1},g_{2},g_{3}]$ $g_{1}=-x_{{2}}{x_{{3}}}^{2}-x_{{3}}+x_{{1}}{x_{{2}}}^{2}-x_{{2}}x_{{1}},$ $g_{2}=x_{{1}}{x_{{4}}}^{2}+x_{{3}}{x_{{4}}}^{2}+x_{{4}}+x_{{3}}-2\,x_{{2}}+x_{{2}}x_{{1}}+x_{{1}}{x_{{2}}}^{2},$ $g_{3}=x_{{3}}{x_{{5}}}^{2}+2\,x_{{5}}+2\,x_{{1}}{x_{{2}}}^{2}+x_{{2}}x_{{1}}+2\,x_{{3}},$ under $x_{1}\prec x_{2}\prec x_{3}\prec x_{4}\prec x_{5}$. Theorem 2.2.6.[13] For any triangular set $\mathbb{T}$, we have $Zero(\mathbb{T}/\mathbb{U}_{\mathbb{T}})\subseteq\overline{QnZero(\mathbb{T})}^{E}\subseteq Zero(sat(\mathbb{T}))$. Corollary 2.2.7.[13] Let $\mathbb{T}$ be a triangular set in K[X] with $\mathbb{U}_{\mathbb{T}}=\emptyset$. Then $Zero(\mathbb{T})=Zero(sat(\mathbb{T}))$. Based on the the theory of weakly non-degenerate conditions we get the following modified algorithm $CharserA$[9,13] to compute characteristic series. Algorithm CharserA: $\Psi\leftarrow CharserA(\mathbb{P})$. Given a nonempty polynomial set $\mathbb{P}$ in K[X], this algorithm computes a finite sets $\Psi$ such that $Zero(\mathbb{P})=\bigcup\limits_{\mathbb{T}\in\Psi}{Zero(\mathbb{T}/\mathbb{U}_{\mathbb{T}})}$ $C1:$ Set $\Phi\leftarrow\\{\mathbb{P}\\}$, $\Psi\leftarrow\emptyset$. $C2:$ While $\Phi\neq\emptyset$ do: $C2.1$. Let $\mathbb{F}$ be an element of $\Psi$ and set $\Psi\leftarrow\Psi\backslash{\mathbb{F}}$. $C2.2$ Compute $\mathbb{T}\leftarrow Charset(\mathbb{F})$. $C2.3$ If $\mathbb{T}$ is noncontradictory, then compute $\mathbb{U}_{\mathbb{T}}$. $C2.4$ If $\mathbb{U}_{\mathbb{T}}=\emptyset$, then set $\Psi\leftarrow\Psi\cup{\mathbb{T}}$. $C2.5$ If $\mathbb{U}_{\mathbb{T}}\neq\emptyset$, then set $\Psi\leftarrow\Psi\cup{\mathbb{T}}$, $\Phi\leftarrow\Phi\\{\mathbb{F}\cup\mathbb{T}\cup\\{I\\}:I\in\mathbb{U}_{\mathbb{T}}\\}$. Example 2.2.8. Let $\mathbb{P}=\\{p_{1},p_{2},p_{3}\\}$ in $\textbf{K}[x_{1},x_{2},x_{3},x_{4}]$, where $p_{1}={x_{{3}}}^{2}+2\,x_{{3}}x_{{2}}+x_{{1}}x_{{2}}+2$, $p_{2}={x_{{1}}}^{3}+2-4\,x_{{3}}{x_{{2}}}^{2}-2\,x_{{2}}{x_{{3}}}^{2}+2\,{x_{{1}}}^{2}-2\,x_{{2}}$, $p_{3}=x_{{2}}x_{{3}}{x_{{4}}}^{2}+x_{{4}}+{x_{{1}}}^{2}+2\,x_{{3}}+x_{{2}}$. Under the variable ordering $x_{1}\prec x_{2}\prec x_{3}\prec x_{4}$. By the above description, one can easily get $CharserA=\\{\mathbb{T}\\}$, where $\mathbb{T}=[2\,x_{{1}}{x_{{2}}}^{2}+2\,x_{{2}}+{x_{{1}}}^{3}+2\,{x_{{1}}}^{2}+2,{x_{{3}}}^{2}+2\,x_{{2}}x_{{3}}+x_{{1}}x_{{2}}+2,x_{{2}}x_{{3}}{x_{{4}}}^{2}+x_{{4}}+$ ${x_{{1}}}^{2}+2\,x_{{3}}+x_{{2}}]$. It is easy to see that $\mathbb{U}_{T}=\emptyset$, this implies that $Zero(\mathbb{P})=Zero(\mathbb{T})$ Compared with $MMP$, $epsilon$, $Regular$, we get the following zero decomposition directly. $Zero(\mathbb{P})=Zero(\mathbb{T}_{1}/\\{x_{1},x_{2}x_{3}\\})\bigcup\limits_{i=2}^{5}{Zero(\mathbb{T}_{i})}$. ($MMP$) where $\mathbb{T}_{1}=[2\,x_{{1}}{x_{{2}}}^{2}+2\,x_{{2}}+2+2\,{x_{{1}}}^{2}+{x_{{1}}}^{3},{x_{{3}}}^{2}+2\,x_{{2}}x_{{3}}+x_{{1}}x_{{2}}+2,x_{{2}}x_{{3}}{x_{{4}}}^{2}+x_{{4}}+$ ${x_{{1}}}^{2}+2\,x_{{3}}+x_{{2}}].$ $\mathbb{T}_{2}=[-{x_{{1}}}^{7}-4\,{x_{{1}}}^{6}-4\,{x_{{1}}}^{5}-4\,{x_{{1}}}^{4}-12\,{x_{{1}}}^{3}-8\,{x_{{1}}}^{2}-4\,x_{{1}}-8,-2\,x_{{2}}+{x_{{1}}}^{3}+2\,{x_{{1}}}^{2}$ $+2,-{x_{{1}}}^{3}x_{{3}}-2\,{x_{{1}}}^{2}x_{{3}}-2\,x_{{3}},{x_{{1}}}^{4}x_{{4}}+2\,{x_{{1}}}^{3}x_{{4}}+2\,x_{{1}}x_{{4}}+{x_{{1}}}^{6}+2\,{x_{{1}}}^{5}-4\,{x_{{1}}}^{2}-4].$ $\mathbb{T}_{3}=[-2\,{x_{{1}}}^{3}-4\,{x_{{1}}}^{2}-4,4\,x_{{2}},4\,{x_{{3}}}^{2}+8,4\,x_{{4}}+8\,x_{{3}}+4\,{x_{{1}}}^{2}].$ $\mathbb{T}_{4}=[-{x_{{1}}}^{3}-2\,{x_{{1}}}^{2}-2,-2\,x_{{2}},-2\,{x_{{3}}}^{2}-4,-2\,x_{{4}}-4\,x_{{3}}-2\,{x_{{1}}}^{2}].$ $\mathbb{T}_{5}=[2\,x_{{1}},4\,x_{{2}}+4,2\,{x_{{3}}}^{2}-4\,x_{{3}}+4,-4\,x_{{3}}{x_{{4}}}^{2}+4\,x_{{4}}+8\,x_{{3}}-4]$. $Zero(\mathbb{P})=Zero(\mathbb{T}_{1}/\\{x_{1},x_{2}x_{3}\\})\bigcup\limits_{i=2}^{4}{Zero(\mathbb{T}_{i})}$. ($epsilon$) where $\mathbb{T}_{1}=[2\,x_{{1}}{x_{{2}}}^{2}+2\,x_{{2}}+2+2\,{x_{{1}}}^{2}+{x_{{1}}}^{3},{x_{{3}}}^{2}+2\,x_{{2}}x_{{3}}+x_{{1}}x_{{2}}+2,x_{{2}}x_{{3}}{x_{{4}}}^{2}+x_{{4}}$ $+{x_{{1}}}^{2}+2\,x_{{3}}+x_{{2}}].$ $\mathbb{T}_{2}=[{x_{{1}}}^{4}+2\,{x_{{1}}}^{3}+2\,x_{{1}}+4,2\,x_{{2}}-{x_{{1}}}^{3}-2-2\,{x_{{1}}}^{2},x_{{3}},2\,x_{{4}}+4\,{x_{{1}}}^{2}+{x_{{1}}}^{3}+2].$ $\mathbb{T}_{3}=[x_{{1}},1+x_{{2}},{x_{{3}}}^{2}+2-2\,x_{{3}},x_{{3}}{x_{{4}}}^{2}-x_{{4}}-2\,x_{{3}}+1].$ $\mathbb{T}_{4}=[{x_{{1}}}^{3}+2+2\,{x_{{1}}}^{2},x_{{2}},{x_{{3}}}^{2}+2,x_{{4}}+{x_{{1}}}^{2}+2\,x_{{3}}].$ $Zero(\mathbb{T})=Zero(\mathbb{T}/\\{x_{1},x_{2}x_{3}\\})$. $(Regular)$ where $\mathbb{T}=[2\,x_{{1}}{x_{{2}}}^{2}+2\,x_{{2}}+2+2\,{x_{{1}}}^{2}+{x_{{1}}}^{3},{x_{{3}}}^{2}+2\,x_{{2}}x_{{3}}+x_{{1}}x_{{2}}+2,x_{{2}}x_{{3}}{x_{{4}}}^{2}$ $+x_{{4}}+{x_{{1}}}^{2}+2\,x_{{3}}+x_{{2}}];$ Other experiment comparison see table 1. Table 1: Number of triangular sets for nine test examples. Polynomial set \| $MMP^{[7]}$ \| $epsilon^{[21]}$ \| $Regular^{[16]}$ \| $CharserA^{[9,13]}$ ---\|---\|---\|---\|--- \| Branches \| Branches \| Branches \| Branches $\mathbb{P}_{1}$ \| 1 \| 1 \| 2 \| 1 $\mathbb{P}_{2}$ \| 2 \| 3 \| 5 \| 2 $\mathbb{P}_{3}$ \| 3 \| 3 \| 6 \| 1 $\mathbb{P}_{4}$ \| 3 \| 2 \| 1 \| 2 $\mathbb{P}_{5}$ \| 4 \| 3 \| 1 \| 1 $\mathbb{P}_{6}$ \| 6 \| ? \| 2 \| 4 $\mathbb{P}_{7}$ \| 9 \| 5 \| 7 \| 1 $\mathbb{P}_{8}$ \| 9 \| 6 \| 1 \| 2 $\mathbb{P}_{9}$ \| 6 \| 4 \| 4 \| 3 ## 3 Traditional method for unmixed decomposition An algebraic variety is said to be unmixed or equidimensional if all irredundant irreducible components have the same dimension. Refer to the zero decomposition (4) which provides a representation of the variety $Zero(\mathbb{P})$ in terms of its subvarieties determined by $\mathbb{T}_{i}$. However, each $Zero(\mathbb{T}_{i}/ini(\mathbb{T}_{i}))$ is not necessarily an algebraic variety, it is a _quasi-algebraic variety_. In what follows, we shall see how a corresponding variety decomposition may be obtained by determining, from each $\mathbb{T}_{i}$, a finite set of polynomials. Theorem 3.1 Let $\mathbb{P}$ be a polynomial set in K[X], $\mathbb{T}_{1},\mathbb{T}_{2},\ldots,\mathbb{T}_{s}$ is a characteristic series of $\mathbb{P}$, then we have $Zero(\mathbb{P})=\bigcup_{i=1}^{s}Zero(sat(\mathbb{T}_{i})).$ (6) The following result provides a useful criterion for removing some redundant subvarieties in the decomposition (4) without computing their defining sets. Lemma 3.2[3] Let $\mathbb{P}$ and $\mathbb{T}_{i}$ be as in Theorem $3.1$, if $\|\mathbb{T}_{i}\|>\|\mathbb{P}\|$, then we have $Zero(sat(\mathbb{T}_{i}))\subset\mathop{\bigcup}\limits_{\mathop{1\leq i\leq s}\limits_{i\neq j}}Zero(sat(\mathbb{T}_{j})).$ (7) The next theorem can make sure this decomposition is unmixed. Theorem 3.3[6] Let $\mathbb{T}=[f_{1},f_{2},\ldots,f_{s}]$ be a triangular set in K[X], if $\mathbb{T}$ is not perfect, then $sat(\mathbb{T})=\textbf{K[X]}$. If $\mathbb{T}$ is perfect, then $Zero(sat(\mathbb{T}))$ is an unmixed variety of dimension $n-\|\mathbb{T}\|$. For every $i$ let $\mathbb{G}_{i}$ be the finite basis of $sat(\mathbb{T}_{i})$, which can computed by Lemma 2.1.6. If $sat(\mathbb{T}_{i})=\textbf{K[X]}$ then $Zero(sat(\mathbb{T}_{i}))=\emptyset$, hence, we can remove it. Thus a variety decomposition of the following form is obtained: $Zero(\mathbb{P})=\bigcup_{i=1}^{s}Zero(\mathbb{G}_{i})$ (8) By theorem 3.3, each $\mathbb{G}_{i}$ defines an unmixed algebraic variety. ## 4 Improvement Based upon the theory of weakly non-degenerate conditions, we have the following result. Theorem 4.1 Let $\mathbb{T}=[f_{1},f_{2},\ldots,f_{s}]$ as $(1)$ be a triangular set in K[X], then $Zero(sat(\mathbb{T}))=Zero(Ideal({\mathbb{T}}):U^{\infty})$ (9) where $U=\prod\limits_{u\in\mathbb{U}_{\mathbb{T}}}u.$ Proof: It is easy to see that $Zero(sat(\mathbb{T}))\subset Zero(Ideal({\mathbb{T}}):U^{\infty}).$ Next, to establish containment in the opposite direction, let $\textbf{X}\in Zero(Ideal({\mathbb{T}}):U^{\infty}).$ Equivalently, if $fU^{m}\in Ideal(\mathbb{T})$ for some $m>0$, then $f(\textbf{X})=0$. Now, let $f\in Ideal(Zero(\mathbb{T}/U)$, then $fU$ vanishes on $Zero(\mathbb{T})$. Thus, by the $Nullstellensatz$, $fU\in\sqrt{Ideal(\mathbb{T})}$, so $(fU)^{l}\in Ideal(\mathbb{T})$ for some $l>0$. Hence, $f^{l}\in Ideal(\mathbb{T}):U^{\infty}$. we have $f(\textbf{X})=0$, Thus $\textbf{X}\in Zero(Ideal(Zero(\mathbb{T}/U))$. This establish that $Zero(Ideal(\mathbb{T}):U^{\infty})\subset Zero(Ideal(Zero(\mathbb{T}/U))$. We also know $Zero(Ideal(\mathbb{T}):U^{\infty})\supset Zero(Ideal(Zero(\mathbb{T}/U))$, then $Zero(Ideal(\mathbb{T}):U^{\infty})=Zero(Ideal(Zero(\mathbb{T}/U))$ $=\overline{Zero(\mathbb{T}/U)}.$ where $\overline{Zero(\mathbb{T}/U)}$ denotes the Zariski _closure_ of $Zero(\mathbb{T}/U)$. From the definition of Zariski _closure_ and $Zero(\mathbb{T}/U)\subset Zero(sat(\mathbb{T}))$, then $Zero(sat(T))\supset Zero(Ideal({\mathbb{T}}):U^{\infty})$. This completes the proof. Applying $CharserA^{[9,13]}$ and above theorem we get the modified algorithm for unmixed decomposition of an algebraic variety. Algorithm 4.2: $\Psi\leftarrow UnmVarDec(\mathbb{P})$. Given a nonempty set $\mathbb{P}$, this algorithm computes finite set $\Psi$ of polynomial sets $\mathbb{G}_{1},\ldots,\mathbb{G}_{s}$ such that the decomposition (8) holds and each $\mathbb{G}_{i}$ defines an unmixed algebraic variety. U1: Compute $\Phi\leftarrow CharserA(\mathbb{P})$, and set $\Psi\leftarrow\varnothing$: U2: While $\Phi\neq\emptyset$, do: U2.1: Let $\mathbb{T}$ be an element in $\Phi$, and $\Phi\leftarrow\Phi\setminus\\{\mathbb{T}\\}$. if $\|\mathbb{T}\|>\|\mathbb{P}\|$, then go to $U2$: U2.2: Compute $Gr$ö$bner$ basis $\mathbb{G}$ of $Ideal(\mathbb{T}):U^{\infty}$ according to Lemma 2.1.8, and set $\Psi\leftarrow\Psi\cup\\{\mathbb{G}\\}$: U3: While $\exists$ $\mathbb{G},\mathbb{G}^{}$ such that $rem(\mathbb{G},\mathbb{G}^{})=\\{0\\}$,do: set $\Psi\leftarrow\Psi\ \backslash\\{\mathbb{G}^{}\\}$. where $rem(\mathbb{G},\mathbb{G}^{})\triangleq\\{rem(p,\mathbb{G}^{})\|p\in\mathbb{G}\\}$ and $rem(p,\mathbb{G}^{})$ see[1,2] for details. Example 4.3. Let $\mathbb{P}=\\{f_{1},f_{2},f_{3},f_{4}\\}$ be a polynomial set in $\textbf{K}[x_{1},x_{2},x_{3},x_{4}]$ where $g_{1}=2\,{x_{{3}}}^{2}x_{{1}}+{x_{{3}}}^{2}x_{{2}}+x_{{3}}+x_{{1}},$ $g_{2}=-x_{{2}}x_{{3}}x_{{4}}+2\,x_{{1}}{x_{{2}}}^{2}-x_{{3}}{x_{{4}}}^{2}+x_{{2}}+2\,x_{{1}}-x_{{4}},$ $g_{3}={x_{{5}}}^{2}{x_{{1}}}^{2}-x_{{2}}{x_{{5}}}^{2}+x_{{5}}-x_{{3}}x_{{4}}+x_{{2}}+x_{{1}},$ $g_{4}=2\,{x_{{2}}}^{3}{x_{{3}}}^{2}+2\,x_{{2}}{x_{{3}}}^{3}x_{{4}}+2\,{x_{{4}}}^{2}{x_{{3}}}^{3}+2\,{x_{{2}}}^{2}x_{{3}}+x_{{2}}x_{{3}}x_{{4}}+2\,x_{{4}}{x_{{3}}}^{2}+x_{{3}}{x_{{4}}}^{2}-x_{{2}}+$ $2\,x_{{3}}+x_{{4}}.$ Under variable ordering $x_{1}\prec x_{2}\prec x_{3}\prec x_{4}\prec x_{5}$ $\mathbb{P}$ is decomposed into six characteristic sets $\mathbb{T}_{i}^{}$ such that $Zero(\mathbb{P})=\bigcup\limits_{i=1}^{6}{Zero(\mathbb{T}_{i}^{}/ini(\mathbb{T}_{i}^{}))}$ where $\mathbb{T}^{}_{1}=[2\,x_{{1}}{x_{{3}}}^{2}+x_{{2}}{x_{{3}}}^{2}+x_{{3}}+x_{{1}},-x_{{3}}{x_{{4}}}^{2}-x_{{2}}x_{{3}}x_{{4}}-x_{{4}}+x_{{2}}+2\,x_{{1}}+2\,x_{{1}}{x_{{2}}}^{2},-{x_{{1}}}^{2}{x_{{5}}}^{2}$ $+x_{{2}}{x_{{5}}}^{2}-x_{{5}}+x_{{3}}x_{{4}}-x_{{2}}-x_{{1}}],$ $\mathbb{T}^{}_{2}=[x_{{2}}-{x_{{1}}}^{2},2\,x_{{1}}{x_{{3}}}^{2}+{x_{{1}}}^{2}{x_{{3}}}^{2}+x_{{3}}+x_{{1}},x_{{3}}{x_{{4}}}^{2}+x_{{4}}+x_{{3}}{x_{{1}}}^{2}x_{{4}}-2\,x_{{1}}-{x_{{1}}}^{2}-$ $2\,{x_{{1}}}^{5},-x_{{5}}+x_{{3}}x_{{4}}-x_{{1}}-{x_{{1}}}^{2}],$ $\mathbb{T}^{}_{3}=[x_{{2}}+2\,x_{{1}},x_{{3}}+x_{{1}},x_{{1}}{x_{{4}}}^{2}-x_{{4}}-2\,{x_{{1}}}^{2}x_{{4}}+8\,{x_{{1}}}^{3},{x_{{1}}}^{2}{x_{{5}}}^{2}+2\,x_{{1}}{x_{{5}}}^{2}+x_{{5}}-$ $x_{{1}}+x_{{1}}x_{{4}}],$ $\mathbb{T}^{}_{4}=[x_{{1}},x_{{3}},-x_{{4}}+x_{{2}},x_{{2}}{x_{{5}}}^{2}-x_{{5}}-x_{{2}}],$ $\mathbb{T}^{}_{5}=[x_{{1}}+2,x_{{2}}-4,x_{{3}}-2,2\,{x_{{4}}}^{2}+9\,x_{{4}}+64,x_{{5}}+2-2\,x_{{4}}],$ $\mathbb{T}^{}_{6}=[x_{{1}},x_{{2}},x_{{3}},x_{{4}},x_{{5}}].$ $\mathbb{T}^{}_{5}$ and $\mathbb{T}^{}_{6}$ contains five polynomials and thus need not be considered for the variety decomposition by Lemma 3.2. In order to obtain an unmixed decomposition of $Zero(\mathbb{P})$, It remains to determine $sat(\mathbb{T}^{}_{1})$, $sat(\mathbb{T}^{}_{2})$, $sat(\mathbb{T}^{}_{3})$, $sat(\mathbb{T}^{}_{4})$ by computing the respectively $Gr$ö$bner$ basis $\mathbb{G}_{1}$, $\mathbb{G}_{2}$, $\mathbb{G}_{3}$, $\mathbb{G}_{4}$ of $\mathbb{T}^{}_{1}\cup\\{1+zx_{3}(2x_{1}+x_{2})(x_{2}-x_{1}^{2})\\}$, $\mathbb{T}_{2}^{}\cup\\{1-zx_{3}(2x_{1}+x_{1}^{2})\\}$, $\mathbb{T}_{3}^{}\cup\\{1-zx_{1}(x_{1}+2)\\}$, $\mathbb{T}_{4}^{}\cup\\{1-zx_{2}\\}$ according to Lemma 2.1.8. The $Gr$ö$bner$ base may be found to consist of 15, 9, 10 and 6 polynomials respectively. Let $sat(\mathbb{T}^{}_{i})=\mathbb{G}_{i}\cap\textbf{K}[x_{1},\ldots,x_{5}]$ for $i=1,2,3,4$. We have $sat(\mathbb{T}^{}_{1})=$ $\left[{\begin{array}[]{{20}c}{2\,{x_{{3}}}^{2}x_{{1}}+x_{{2}}{x_{{3}}}^{2}+x_{{3}}+x_{{1}},\hfill}\\\ {2\,x_{{3}}{x_{{2}}}^{3}x_{{1}}+4\,x_{{3}}{x_{{2}}}^{2}{x_{{1}}}^{2}-x_{{4}}x_{{3}}x_{{2}}+x_{{3}}{x_{{2}}}^{2}+x_{{1}}{x_{{4}}}^{2}-2\,x_{{1}}x_{{4}}x_{{3}}+x_{{1}}x_{{4}}x_{{2}}+\hfill}\\\ {4\,x_{{1}}x_{{3}}x_{{2}}+2\,{x_{{2}}}^{2}x_{{1}}+4\,x_{{3}}{x_{{1}}}^{2}-x_{{4}}+x_{{2}}+2\,x_{{1}},\hfill}\\\ {x_{{4}}x_{{3}}x_{{2}}-2\,{x_{{2}}}^{2}x_{{1}}+{x_{{4}}}^{2}x_{{3}}-x_{{2}}-2\,x_{{1}}+x_{{4}},\hfill}\\\ {-{x_{{5}}}^{2}{x_{{1}}}^{2}+x_{{2}}{x_{{5}}}^{2}-x_{{5}}+x_{{4}}x_{{3}}-x_{{2}}-x_{{1}},\hfill}\\\ {{x_{{5}}}^{2}x_{{3}}-{x_{{3}}}^{2}x_{{1}}-x_{{3}}-x_{{1}}+{x_{{5}}}^{2}{x_{{3}}}^{2}{x_{{1}}}^{2}+x_{{5}}{x_{{3}}}^{2}+x_{{1}}{x_{{5}}}^{2}-{x_{{3}}}^{3}x_{{4}}+\hfill}\\\ {2\,{x_{{5}}}^{2}{x_{{3}}}^{2}x_{{1}}.\hfill}\\\ \end{array}}\right]$ $sat(\mathbb{T}_{2}^{})=$ $\left[{\begin{array}[]{{20}c}{-x_{1}+x_{2}^{2},\hfill}\\\ {2\,{x_{{3}}}^{2}x_{{1}}+x_{{3}}+x_{{1}}+{x_{{1}}}^{2}{x_{{3}}}^{2},\hfill}\\\ {2\,x_{{3}}{x_{{2}}}^{3}x_{{1}}+4\,x_{{3}}{x_{{2}}}^{2}{x_{{1}}}^{2}-x_{{3}}x_{{2}}x_{{4}}+x_{{3}}{x_{{2}}}^{2}+{x_{{4}}}^{2}x_{{1}}-2\,x_{{4}}x_{{3}}x_{{1}}+x_{{4}}x_{{2}}x_{{1}}+\hfill}\\\ {4\,x_{{3}}x_{{2}}x_{{1}}+2\,x_{{1}}{x_{{2}}}^{2}+4\,x_{{3}}{x_{{1}}}^{2}-x_{{4}}+x_{{2}}+2\,x_{{1}},\hfill}\\\ {x_{{3}}x_{{2}}x_{{4}}-2\,x_{{1}}{x_{{2}}}^{2}+x_{{3}}{x_{{4}}}^{2}-x_{{2}}-2\,x_{{1}}+x_{{4}},\hfill}\\\ {-{x_{{5}}}^{2}{x_{{1}}}^{2}+x_{{2}}{x_{{5}}}^{2}-x_{{5}}+x_{{3}}x_{{4}}-x_{{2}}-x_{{1}},\hfill}\\\ {-{x_{{3}}}^{2}x_{{1}}-x_{{3}}-x_{{1}}+{x_{{5}}}^{2}x_{{1}}+{x_{{1}}}^{2}{x_{{5}}}^{2}{x_{{3}}}^{2}+{x_{{3}}}^{2}x_{{5}}-{x_{{3}}}^{3}x_{{4}}+x_{{3}}{x_{{5}}}^{2}\hfill}\\\ {+2\,{x_{{5}}}^{2}{x_{{3}}}^{2}x_{{1}}.\hfill}\\\ \end{array}}\right]$ $sat(\mathbb{T}_{3}^{})=$ $\left[{\begin{array}[]{{20}c}{2x_{1}+x_{2},\hfill}\\\ {x_{3}+x_{1},\hfill}\\\ {2\,{x_{{3}}}^{2}x_{{1}}+x_{{3}}+x_{{1}}+{x_{{1}}}^{2}{x_{{3}}}^{2},\hfill}\\\ {{x_{{5}}}^{2}{x_{{1}}}^{2}-x_{{1}}+x_{{5}}+2\,{x_{{5}}}^{2}x_{{1}}+x_{{4}}x_{{1}},\hfill}\\\ {{x_{{5}}}^{2}x_{{4}}x_{{1}}+2\,{x_{{5}}}^{2}x_{{4}}+x_{{5}}{x_{{4}}}^{2}-2\,x_{{5}}x_{{4}}x_{{1}}+8\,x_{{5}}{x_{{1}}}^{2}+{x_{{4}}}^{2}-x_{{4}},\hfill}\\\ {2\,{x_{{4}}}^{2}{x_{{5}}}^{2}+{x_{{4}}}^{3}x_{{5}}+4\,x_{{5}}x_{{4}}{x_{{1}}}^{2}+16\,x_{{5}}{x_{{1}}}^{3}+9\,{x_{{5}}}^{2}x_{{4}}+4\,x_{{5}}{x_{{4}}}^{2}+32\,x_{{5}}{x_{{1}}}^{2}-\hfill}\\\ {32\,{x_{{5}}}^{2}x_{{1}}-8\,x_{{5}}x_{{4}}x_{{1}}+{x_{{4}}}^{3}+4\,x_{{4}}{x_{{1}}}^{2}+16\,{x_{{1}}}^{3}-4\,x_{{5}}x_{{4}}+8\,x_{{5}}x_{{1}}-6\,x_{{4}}\hfill}\\\ {+3\,{x_{{4}}}^{2}-14\,x_{{4}}x_{{1}}-8\,{x_{{1}}}^{2}-16\,x_{{5}}+16\,x_{{1}}.\hfill}\\\ \end{array}}\right]$ $sat(\mathbb{T}_{4}^{})=\mathbb{T}_{4}^{}.$ It is easy to verify that $Zero(sat(\mathbb{T}_{2}^{}))$, $Zero(sat(\mathbb{T}_{3}^{}))$ and $Zero(sat(\mathbb{T}_{4}^{}))$ are subvarieties of $Zero(sat(\mathbb{T}_{1}^{}))$. Therefore, $Zero(\mathbb{P})=Zero(sat(\mathbb{T}^{}_{1}))$ is an unmixed decomposition. By our improvement, $\mathbb{P}$ is decomposition into two characteristic sets such that $Zero(\mathbb{P})=Zero(\mathbb{T}_{1}^{}/{x_{3}})\cup Zero(\mathbb{T}_{4}^{})$ since $\mathbb{U}_{\mathbb{T}_{1}^{}}=\\{x_{3}\\}$ and $\mathbb{U}_{\mathbb{T}_{4}^{}}=\emptyset$, where $\mathbb{T}_{1}^{}$ and $\mathbb{T}_{4}^{}$ as above. In order to determine $Zero(sa(\mathbb{T}_{1}^{}))$, we only compute the $Gr$ö$bner$ base $\mathbb{G}_{1}$ of $\mathbb{T}_{1}^{}\cup\\{1-zx_{3}\\}$ according to Theorem 4.1. The $Gr$ö$bner$ base may be found to consist of 8 polynomials. Let $Ideal(\mathbb{T}^{}_{1}):x_{3}^{\infty}=\mathbb{G}_{1}\cap\textbf{K}[x_{1},\ldots,x_{5}]$. We have $Ideal(\mathbb{T}^{}_{1}):x_{3}^{\infty}=sat(\mathbb{T}_{1}^{})$, and remove $\mathbb{T}_{4}^{}$ according to $U3$, then we get the result as above. Example 4.4. Let $\mathbb{P}=\\{f_{1},f_{2},f_{3},f_{4}\\}$ be a polynomial set in $\textbf{K}[x_{1},x_{2},x_{3},x_{4},x_{5}]$ where $f_{1}=2\,{x_{{2}}}^{2}x_{{1}}+x_{{1}}x_{{2}}+{x_{{5}}}^{2}x_{{3}}+2\,x_{{5}}+2\,x_{{3}},$ $f_{2}={x_{{3}}}^{2}x_{{2}}+x_{{3}}+2\,x_{{1}}x_{{2}}+3\,{x_{{4}}}^{2}{x_{{1}}}^{2}+{x_{{4}}}^{2}x_{{2}}+2\,x_{{4}},$ $f_{3}=x_{{3}}{x_{{5}}}^{2}+2\,x_{{5}}+2\,x_{{1}}{x_{{2}}}^{2}+3\,x_{{3}}+{x_{{3}}}^{2}x_{{2}}+2\,x_{{1}}x_{{2}}+{x_{{2}}}^{3}x_{{1}},$ $f_{4}=2\,x_{{5}}+3\,x_{{1}}x_{{2}}+x_{{3}}{x_{{5}}}^{2}+4\,x_{{3}}+2\,x_{{2}}{x_{{3}}}^{2}.$ Under variable ordering $x_{1}\prec x_{2}\prec x_{3}\prec x_{4}\prec x_{5}$ $\mathbb{P}$ is decomposed into eight characteristic sets $\mathbb{T}_{i}$ such that $Zero(\mathbb{P})=\bigcup\limits_{i=1}^{8}{Zero(\mathbb{T}/ini(\mathbb{T}_{i}))}$ where $\mathbb{T}_{1}=[-x_{{2}}{x_{{3}}}^{2}-x_{{3}}+{x_{{2}}}^{2}x_{{1}}-x_{{1}}x_{{2}},-x_{{1}}{x_{{4}}}^{2}-x_{{3}}{x_{{4}}}^{2}-x_{{4}}-x_{{3}}+2\,x_{{2}}-x_{{1}}x_{{2}}$ $-{x_{{2}}}^{2}x_{{1}},x_{{3}}{x_{{5}}}^{2}+2\,x_{{5}}+2\,{x_{{2}}}^{2}x_{{1}}+x_{{1}}x_{{2}}+2\,x_{{3}}],$ $\mathbb{T}_{2}=[x_{{1}}{x_{{2}}}^{2}-x_{{1}}x_{{2}},x_{{3}},x_{{1}}{x_{{4}}}^{2}+x_{{4}}-2\,x_{{2}}+2\,x_{{1}}x_{{2}},2\,x_{{5}}+3\,x_{{1}}x_{{2}}],$ $\mathbb{T}_{3}=[x_{1},-x_{3},-x_{4}+2x_{2},2x_{5}],$ $\mathbb{T}_{4}=[x_{{1}}{x_{{2}}}^{2}-x_{{1}}x_{{2}}-{x_{{1}}}^{2}x_{{2}}+x_{{1}},-x_{3}-x_{1},x_{{4}}-2\,x_{{2}}+2\,x_{{1}}x_{{2}}-2\,x_{{1}}+{x_{{1}}}^{2}x_{{2}},-x_{{1}}{x_{{5}}}^{2}+$ $2\,x_{{5}}+3\,x_{{1}}x_{{2}}-4\,x_{{1}}+2\,{x_{{1}}}^{2}x_{{2}}],$ $\mathbb{T}_{5}=[-x_{1},-x_{3},x_{4}-2x_{2},2x_{5}],$ $\mathbb{T}_{6}=[x_{1},-x_{3},x_{4}-2x_{2},2x_{5}],$ $\mathbb{T}_{7}=[-x_{2},-x_{3},-x_{1}x_{4}^{2}-x_{4},2x_{5}],$ $\mathbb{T}_{8}=[-x_{1},-x_{2},-x_{3},-x_{4},2x_{5}].$ We can remove $\mathbb{T}_{8}$ according to Lemma 3.2, and remove $\mathbb{T}_{i}(i=2,\ldots,7)$ by compute $sat(\mathbb{T}_{i})(i=2,\ldots,7)$ and $U.3$, where $sat(\mathbb{T}_{1})=$ $\left[{\begin{array}[]{{20}c}{{-{x_{{2}}}^{2}x_{{1}}+x_{{3}}+x_{{2}}{x_{{3}}}^{2}+x_{{1}}x_{{2}}}},\hfill\\\ {{-2\,{x_{{2}}}^{2}x_{{3}}+4\,{x_{{2}}}^{2}x_{{1}}-x_{{1}}x_{{2}}x_{{3}}+x_{{3}}x_{{1}}{x_{{2}}}^{2}+x_{{2}}x_{{3}}x_{{4}}-{x_{{1}}}^{2}{x_{{2}}}^{2}-{x_{{1}}}^{2}{x_{{2}}}^{3}+\hfill}}\\\ {{x_{{3}}x_{{1}}{x_{{2}}}^{3}-{x_{{4}}}^{2}{x_{{1}}}^{2}x_{{2}}+{x_{{4}}}^{2}x_{{1}}{x_{{2}}}^{2}-{x_{{4}}}^{2}x_{{1}}x_{{2}}-x_{{4}}x_{{1}}x_{{2}}+x_{{1}}{x_{{4}}}^{2}+x_{{4}}-2\,x_{{2}},\hfill}}\\\ {{x_{{1}}{x_{{4}}}^{2}+{x_{{4}}}^{2}x_{{3}}+x_{{4}}+x_{{3}}-2\,x_{{2}}+x_{{1}}x_{{2}}+{x_{{2}}}^{2}x_{{1}},\hfill}}\\\ {{2\,x_{{5}}+{x_{{5}}}^{2}{x_{{2}}}^{2}x_{{1}}+2\,x_{{3}}x_{{1}}{x_{{2}}}^{3}-{x_{{5}}}^{2}x_{{1}}x_{{2}}+x_{{3}}x_{{1}}{x_{{2}}}^{2}+2\,x_{{2}}x_{{3}}x_{{5}}+4\,{x_{{2}}}^{2}x_{{1}}\hfill}}\\\ {{-x_{{1}}x_{{2}}}},\hfill\\\ {{2\,{x_{{2}}}^{2}x_{{1}}+x_{{1}}x_{{2}}+{x_{{5}}}^{2}x_{{3}}+2\,x_{{5}}+2\,x_{{3}},\hfill}}\\\ {{-2\,{x_{{4}}}^{2}x_{{5}}-x_{{2}}x_{{4}}+2\,{x_{{2}}}^{2}-8\,{x_{{2}}}^{3}-2\,{x_{{2}}}^{3}x_{{3}}+4\,x_{{4}}{x_{{2}}}^{2}-4\,{x_{{2}}}^{4}x_{{3}}-8\,x_{{5}}{x_{{2}}}^{2}\hfill}}\\\ {{-2\,x_{{5}}{x_{{2}}}^{3}x_{{3}}-2\,x_{{5}}{x_{{2}}}^{2}x_{{3}}-2\,{x_{{5}}}^{2}{x_{{2}}}^{3}+2\,{x_{{5}}}^{2}{x_{{2}}}^{2}+{x_{{5}}}^{2}x_{{4}}{x_{{2}}}^{2}+2\,x_{{4}}{x_{{2}}}^{3}x_{{3}}\hfill}}\\\ {{-{x_{{5}}}^{2}x_{{4}}x_{{2}}+{x_{{2}}}^{2}x_{{4}}x_{{3}}-2\,x_{{5}}{x_{{2}}}^{2}{x_{{4}}}^{2}+2\,{x_{{4}}}^{2}x_{{1}}x_{{5}}x_{{2}}+2\,x_{{5}}x_{{4}}x_{{2}}+2\,x_{{2}}x_{{3}}x_{{5}}\hfill}}\\\ {{+2\,x_{{5}}x_{{1}}{x_{{2}}}^{2}+2\,x_{{5}}x_{{1}}{x_{{2}}}^{3}+2\,x_{{5}}{x_{{4}}}^{2}x_{{2}},\hfill}}\\\ {{-8\,x_{{2}}+4\,x_{{4}}-4\,x_{{5}}+2\,x_{{1}}x_{{2}}+4\,x_{{1}}{x_{{4}}}^{2}+{x_{{5}}}^{2}x_{{4}}-4\,{x_{{2}}}^{2}x_{{3}}-2\,{x_{{4}}}^{2}x_{{5}}+\hfill}}\\\ {{4\,{x_{{2}}}^{2}x_{{1}}-2\,x_{{1}}x_{{2}}x_{{3}}+x_{{3}}x_{{1}}{x_{{2}}}^{2}+2\,x_{{2}}x_{{3}}x_{{4}}-2\,{x_{{1}}}^{2}{x_{{2}}}^{2}-2\,{x_{{1}}}^{2}{x_{{2}}}^{3}+{x_{{4}}}^{2}{x_{{5}}}^{2}x_{{1}}\hfill}}\\\ {{-2\,x_{{2}}x_{{3}}x_{{5}}-2\,{x_{{4}}}^{2}{x_{{1}}}^{2}x_{{2}}-3\,{x_{{4}}}^{2}x_{{1}}x_{{2}}-2\,x_{{4}}x_{{1}}x_{{2}}+2\,{x_{{5}}}^{2}x_{{1}}x_{{2}}-2\,{x_{{5}}}^{2}x_{{2}}}}.\hfill\\\ \end{array}}\right]$ then $Zero(\mathbb{P})=Zero(sat(\mathbb{T}_{1}))$. By our improvement we get $CharserA(\mathbb{P})=\\{\mathbb{T}^{}\\}$, where $\mathbb{T}^{}=\left[{\begin{array}[]{{20}c}{-x_{{2}}{x_{{3}}}^{2}-x_{{3}}+x_{{1}}{x_{{2}}}^{2}-x_{{1}}x_{{2}},\hfill}\\\ {x_{{1}}{x_{{4}}}^{2}+x_{{3}}{x_{{4}}}^{2}+x_{{4}}+x_{{3}}-2\,x_{{2}}+x_{{1}}x_{{2}}+x_{{1}}{x_{{2}}}^{2},\hfill}\\\ {x_{{3}}{x_{{5}}}^{2}+2\,x_{{5}}+2\,x_{{1}}{x_{{2}}}^{2}+x_{{1}}x_{{2}}+2\,x_{{3}}.\hfill}\\\ \end{array}}\right]$ It is easy to see that $\mathbb{U}_{\mathbb{T}^{}}=\emptyset$, then $Zero(sat(\mathbb{T}^{}))=Zero(\mathbb{T}^{})$ according to Lemma 4.1, so we get $Zero(\mathbb{P})=Zero(\mathbb{T}^{})$ directly. ## References * [1] Buchberger,B. Ein Algorithmus Zum Auffinden der Basiselement des Restklassentinges nach einem nulldimensionalen Polynomial. Ph.D. thesis, Univesitat Innsbruck, Austria, 1965. * [2] Buchberger,B. Groebner bases: An gorithmic method in polynomial ideal theory. 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Education Publ.House, Shanghai,1996(in Chinese). * [27] Yang, L.,Hou,X.-R. Gather-and-Shift. A symbolic method for solving polnomials. In:_Proceedings ATCM’95_ , Singapore, 1995,771-780. * [28] Zhang, J.-Z, Yang, L., Hou, X.-R. A note on Wu Wen-Tsün’s nondegenerate condition. _Technical Report ICTP/91/160_ ,International Center for Theoretical Physics,International Atomic Energy Agency, Miramare, Trieste,1991; Also in Chises Science Bullentin, 1993,38:1,86-87.	arxiv-papers	2010-12-06T14:52:02	2024-09-04T02:49:15.493875	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Zhenyi Ji, Yongbin Li", "submitter": "Yongbin Li", "url": "https://arxiv.org/abs/1012.1190" }
1012.1197		arxiv-papers	2010-12-06T15:08:27	2024-09-04T02:49:15.500812	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Yongbin Li", "submitter": "Yongbin Li", "url": "https://arxiv.org/abs/1012.1197" }
1012.1218	# BeppoSAX observations of the X–ray pulsar MAXI J1409$-$619 in low state: discovery of cyclotron resonance features Mauro Orlandini11affiliation: INAF/Istituto di Astrofisica Spaziale e Fisica Cosmica (IASF) Bologna, via Gobetti 101, 40129 Bologna, Italy , Filippo Frontera11affiliation: INAF/Istituto di Astrofisica Spaziale e Fisica Cosmica (IASF) Bologna, via Gobetti 101, 40129 Bologna, Italy 22affiliation: Dipartimento di Fisica, Università degli Studi di Ferrara, via Saragat 1, 44122 Ferrara, Italy , Nicola Masetti11affiliation: INAF/Istituto di Astrofisica Spaziale e Fisica Cosmica (IASF) Bologna, via Gobetti 101, 40129 Bologna, Italy , Vito Sguera11affiliation: INAF/Istituto di Astrofisica Spaziale e Fisica Cosmica (IASF) Bologna, via Gobetti 101, 40129 Bologna, Italy , Lara Sidoli33affiliation: INAF/Istituto di Astrofisica Spaziale e Fisica Cosmica (IASF) Milano, via Bassini 15, 20133 Milano, Italy ###### Abstract The transient 500 s X–ray pulsar MAXI J1409$-$619 was discovered by the slit cameras aboard MAXI on October 17, 2010, and soon after accurately localized by Swift. We found that the source position was serendipitously observed in 2000 during BeppoSAX observations of the Galactic plane. Two sources are clearly detected in the MECS: one is consistent with the position of IGR J14043$-$6148 and the other one with that of MAXI J1409$-$619\. We report on the analysis of this archival BeppoSAX/MECS observation integrated with newly analyzed observation from ASCA and a set of high-energy observations obtained from the offset fields of the BeppoSAX/PDS instrument. For the ON-source observation, the 1.8–100 keV spectrum is fit by an absorbed power law with a photon index $\Gamma=0.87_{-0.19}^{+0.29}$, corresponding to 2–10 and 15–100 keV unabsorbed fluxes of $2.7\times 10^{-12}$ and $4\times 10^{-11}$ erg cm-2 s-1, respectively, and a 2–10 keV luminosity of $7\\!\times\\!10^{34}$ erg s-1 for a 15 kpc distance. For a PDS offset field observation, performed about one year later and showing a 15–100 keV flux of $7\times 10^{-11}$ erg cm-2 s-1, we clearly pinpoint three spectral absorption features at 44, 73, and 128 keV, resolved both in the spectral fit and in the Crab ratio. We interpret these not harmonically spaced features as due to cyclotron resonances. The fundamental energy of 44$\pm$3 keV corresponds to a magnetic field strength at the neutron star surface of $3.8\times 10^{12}(1+z)$ G, where $z$ is the gravitational redshift. We discuss the nature of the source in the light of its possible counterpart. ###### Subject headings: X–rays: binaries — Stars: individual: MAXI J1409$-$619 — Stars: individual: IGR J14043$-$6148 ††slugcomment: The Astrophysical Journal, 747:, 2012 March 1 ## 1\. Introduction On October 17, 2010 Yamaoka et al. (2010b) reported on an outburst from a new source, named MAXI J1409$-$619, revealed by the Gas Slit Camera (GSC; Mihara et al. 2002) of the Monitor of All-sky X-ray Image (MAXI; Matsuoka et al. 2009) experiment aboard the International Space Station. The 4–10 keV flux decreased from its initial value of $\sim$40 mCrab on October 17 to $\sim$30 mCrab on the following day. On October 20 Swift (Gehrels et al. 2004) began a target of opportunity observation of the MAXI J1409$-$619 error circle (0$\fdg$2 radius). Kennea et al. (2010b) used the X–Ray Telescope (XRT; Burrows et al. 2005) to pinpoint the position of the new source with an estimated 90% confidence level uncertainty of 1$\farcs$9 and coordinates RA(J2000): 14h 08m 02$\fs$56, Dec(J2000): $-$61° 59′ 00$\farcs$3\. Its spectrum is fit by an absorbed power law, with photon index $\Gamma=-0.5^{+0.1}_{-0.6}$. A follow-up XRT observation on November 30 found the source $\sim$7 times brighter than on October 20, with an average 0.3–10 keV flux of $7\times 10^{-10}$ erg cm-2 s-1 (Kennea et al. 2010a). Only in this latter observation a $\sim$500 s pulsation was detected, with a 42% sinusoidal rms modulation. The source was detected since October 18 also in the 15–50 keV energy band by the Swift Burst Alert Telescope (BAT; Barthelmy et al. 2005) at the level of $\sim$30 mCrab (Kennea et al. 2011). Neither catalogued radio nor X–ray sources were present in the Swift error circle, while a 2MASS IR star, the likely IR counterpart of the X–ray transient, lays 2$\farcs$1 from the XRT position (Kennea et al. 2010b). The source was also observed on October 22 by the Proportional Counter Array (PCA; Jahoda et al. 1996) aboard RXTE (Bradt et al. 1993). The X–ray spectrum shows a strong 6.5 keV Iron line (E.W. 200 eV), and a continuum modeled by reflection or partially covering absorption (Yamaoka et al. 2010a). The 2–10 keV flux was about $10^{-10}$ erg cm-2 s-1, and was strongly variable on time- scales of hundreds of seconds. An observation performed on December 4 found MAXI J1409$-$619 at a flux level 6–7 times higher than the October observation (Yamamoto et al. 2010), and confirmed the 500 s pulse period observed by Swift. From observations performed on December 2 and 3 by the GLAST Burst Monitor (GBM; Meegan et al. 2007) aboard the Fermi satellite, the presence of a 500 s double peaked pulsating signal was confirmed (Camero-Arranz et al. 2010). Following the MAXI J1409$-$619 localization by Swift, we searched for BeppoSAX (Boella et al. 1997a) archival observations with the transient position within the Narrow Field Instruments (NFIs) Field of View (FoV). We found that an observation performed in the framework of a Galactic plane survey contained the MAXI J1409$-$619 position (Sguera et al. 2010). In this Paper we report on the spectral analysis of this BeppoSAX observation, integrated with high-energy data obtained from the offset fields of the PDS (Frontera et al. 1997) instrument. We further present the analysis of a newly analyzed ASCA observation performed during a Galactic plane survey. ## 2\. Observations ### 2.1. BeppoSAX/PDS Observations We searched our BeppoSAX archive for observations covering the MAXI J1409$-$619 position. From our local archive we are able to extract not only products from the NFI nominal pointings (also available from ASDC – the ASI Scientific data Center111http://www.asdc.asi.it/bepposax), but also net spectra from the PDS offset fields. Table 1List of PDS pointings covering the MAXI J1409$-$619 error box Seq \| OPn \| Date \| Duration \| Source Name \| Dist \| RA \| DEC \| Type \| PDS rate \| S/N ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| \| \| (ks) \| \| $(\arcmin)$ \| (J2000) \| (J2000) \| \| (15–100 keV c/s) \| 1 \| 08317 \| 29/01/2000 \| 42 \| Gal Plane Sur 2 \| 19.6 \| 211.3474 \| $-$61.88437 \| ON \| $0.247\pm 0.069$ \| 3.6 2 \| 10480 \| 07/01/2001 \| 220 \| Circinus Galaxy \| 27.3 \| 212.8952 \| $-$61.80250 \| MOFF \| $0.450\pm 0.032$ \| 14.1 3 \| 08386 \| 07/02/2000 \| 52 \| Gal Plane Sur 16 \| 49.5 \| 210.2841 \| $-$61.83828 \| ON \| $0.086\pm 0.049$ \| 1.7 4 \| 01652 \| 23/02/1997 \| 90 \| Alpha Cen \| 53.6 \| 213.5273 \| $-$62.53019 \| POFF \| $0.086\pm 0.052$ \| 1.6 5 \| 08484 \| 19/02/2000 \| 47 \| Gal Plane Sur 23 \| 70.0 \| 213.5936 \| $-$61.09281 \| POFF \| $0.144\pm 0.072$ \| 2.0 Note. — The “Source Name” refers, in case of offset fields, to the nominal ON pointing. The distance listed in column 6 refers to the angular distance of the PDS pointing from the MAXI J1409$-$619 position. Indeed, because of the rocking technique used to derive the PDS background, two positions offset by 3$\fdg$5 with respect to the NFI direction were alternatively observed every 96 s. For fields free of sources, we expect that differences between the count spectra in the “plus OFF” (thereafter POFF) and “minus OFF” (thereafter MOFF) positions will be consistent with zero. If, on the other hand, a contaminating source is present in one of the two offset fields, then the spectral difference corresponds to a net, background subtracted PDS spectrum of the offset field. We found three sets of observations containing the MAXI J1409$-$619 position in their FoVs. In Figure 1 we show a finding chart centered on the MAXI J1409$-$619 position and showing both the ON-source and OFF-source pointings (the PDS FoV is 1$\fdg$3 FWHM). In Table 1 we detail the whole set of BeppoSAX observations used in our analysis. All errors are given at the 90% confidence level for a single parameter. Figure 1.— Region centered on the MAXI J1409$-$619 position (the small black circle) with the PDS pointings: ON pointings (red), POFF pointings (green), and MOFF pointings (cyan). The PDS FoV is 1$\fdg$3 (FWHM). Note the two PDS observations (one ON and one MOFF) fully covering the source position, and three (one ON and two POFF) partially covering the position (with the term “partially” we intend that the source is beyond the collimator FWHM but before its FWZI). The bold dotted line corresponds to the Galactic plane. #### 2.1.1 ON-source observations (OP08317 and OP08386) The first ON-source BeppoSAX pointing with MAXI J1409$-$619 in the NFI FoV was performed on January 29, 2000 within a program aimed at performing a systematic study of part of the Galactic plane (OP08317 – Galactic Plane Survey Field 02). Another ON-source observation (OP08386 – Galactic Plane Survey Field 16), was performed a week later. This observation partially overlaps the OP08317 field (see Fig. 1). The Galactic plane Field 02 net exposures of the two imaging instruments Low- Energy Concentrator Spectrometer (LECS; 0.1–10 keV; 37′ FoV; Parmar et al. 1997) and Medium-Energy Concentrator Spectrometer (MECS; 1.8–10 keV; 56′ FoV; Boella et al. 1997b) are 6812 and 22060 s, respectively. The difference is due to the constraint that the LECS can be operated only when in the Earth shadow. Good data were selected when the instrument configurations were nominal, and with an elevation angle above the Earth limb $>$4°. Table 2Sources detected in the MECS field for OP08317 # \| RateaaIn units of $10^{-3}$ Counts/s \| pixel \| \| R.A. \| Dec \| \| S/N \| Counterpart ---\|---\|---\|---\|---\|---\|---\|---\|--- \| (2–10 keV) \| x \| y \| \| (2000) \| \| \| 1 \| 45.5$\,\pm\,$ \| 3.6 \| 373.2 \| 317.1 \| \| 14 04 31.3 \| $-$61 47 10.8 \| \| 12.6 \| IGR J14043$-$6148 2 \| 10.6$\,\pm\,$ \| 1.3 \| 188.2 \| 233.2 \| \| 14 08 00.4 \| $-$61 58 24.1 \| \| 8.15 \| MAXI J1409$-$619 3 \| 3.77$\,\pm\,$ \| 0.79 \| 293.9 \| 250.1 \| \| 14 06 00.5 \| $-$61 56 09.6 \| \| 4.77 \| Figure 2.— 2–10 keV MECS image of a mosaic of the two, partially overlapping, ON-source BeppoSAX observations OP08317 and OP08386, smoothed with a Gaussian filter with a $\sigma$ of 24″. Two sources are clearly detected. The one present in the MAXI J1409$-$619 error box is listed as #2 in Table 2. The two “cuts-out” are due to the removal of the calibration source events. A mosaic of the two, partially overlapping, ON-source OP08317 and OP08386 MECS images is shown in Figure 2, where two relatively bright sources are clearly present (see Table 2). Source #2 position is consistent with MAXI J1409$-$619, while the source #1 position is consistent with the X–ray source IGR J14043$-$6148 (Bird et al. 2010). The third detection (source #3) corresponds to extended low-energy emission observed in the LECS image, and may be associated with a radio source at approximately the same position (Cohen & Green 2001). The uncertainty in the MECS positions is 30″ (Perri & Capalbi 2002). We extracted LECS and MECS data from a 4′ (30 pixel) radius circular region centered on the MAXI J1409$-$619 position. Concerning LECS data, there are not enough counts to allow a spectral reconstruction. On the other hand, we are able to extract a 1.8–10 keV MECS spectrum, that we rebinned to a minimum of 30 counts per bin to allow the use of $\chi^{2}$ statistics. We used a response matrix appropriate for the off-axis position of the source. Because the position of MAXI J1409$-$619 is close to the Galactic plane, we checked for possible contamination from the Galactic ridge emission. To this end, from the MECS image we extracted background spectra from two source-free regions, offset by about the same angle as the source (to take into account vignetting), and with the same extraction radii. These backgrounds are consistent with the mean background used routinely for MECS observations. For this reason the mean background was used in the spectral analysis. Given that the PDS is not an imaging instrument, before performing joint MECS/PDS spectral fits it is necessary to address the problem of determining whether the high energy emission observed by the PDS is due to MAXI J1409$-$619 or to the INTEGRAL source. Bear in mind that, in computing fluxes, it is necessary to take into account the triangular angular response of the PDS collimators (see Fig. 2 in Frontera et al. 2007). First, we analyzed the ON-source OP08386 MECS observation, that contains IGR J14043$-$6148 (detected at $>$15$\sigma$) but not MAXI J1409$-$619\. For this observation we have only a marginal PDS detection (see Table 1), therefore it seems plausible to associate the high-energy emission observed in the OP08317 observation with MAXI J1409$-$619\. To further confirm this association, we verified that the PDS spectrum extrapolated at lower energies is consistent with the MAXI J1409$-$619 MECS spectrum. Figure 3.— 1.8–100 keV joint MECS/PDS count rate spectrum (plus signs) and the power law best fit model (histogram) of MAXI J1409$-$619 in the OP08317 field. The fit residuals are shown in the bottom panel. In Figure 3 we show the joint MECS/PDS 1.8–100 keV MAXI J1409$-$619 count rate spectrum, together with its power law best fit (normalized $\chi^{2}_{\nu}$ of 0.18 for 15 degrees of freedom – dof). It is characterized by a hard spectrum, with power law photon index $0.87_{-0.19}^{+0.29}$, $N_{H}=2.8_{-2.2}^{+3.4}\times 10^{22}$ cm-2 (consistent with galactic absorption, Dickey & Lockman 1990) and unabsorbed 2–10 and 15–100 keV fluxes of $2.7\times 10^{-12}$ and $4\times 10^{-11}$ erg cm-2 s-1, respectively. #### 2.1.2 MOFF offset field observation (OP10480) From Figure 1 we can see that we have three sets of PDS offset observations covering the MAXI J1409$-$619 position. For two of them (OP01652 and OP08484) we do not have sufficient statistics to perform a useful analysis (see the observed count rates in Table 1). On the other hand, for the MOFF offset field of the Circinus Galaxy observation, OP10480, data have enough statistics for a thorough spectral analysis. First, we checked for the presence of any catalogued X–ray sources in the MOFF observation FoV. The only source present is the EGRET (Hartman et al. 1999) source 3EG J1410$-$6147, that was recently associated with the 50 ms radio pulsar PSR J1410$-$6132 (O’Brien et al. 2008). No significant X–ray emission has been detected around the EGRET source (Doherty et al. 2003); therefore we associate the PDS detection with MAXI J1409$-$619. We started by extracting a 15–100 keV coarser binned PDS spectrum of the MOFF field (assuming as its background the corresponding POFF spectrum). The fit with a power law gives a photon index of $2.2\pm 0.3$, with a normalized $\chi^{2}_{\nu}$ of 1.31 for 11 dof. From the shape of the residuals and from the quite different photon index with respect to the ON-source observation, we suspected the presence of a change of slope in the spectrum between 20 and 30 keV. This break is not detected in the ON-source observation because of its poor statistics. We tried both a broken and a cutoff power law. By fixing the cutoff energy at 25 keV (we were only able to constrain the break/cutoff energy between 20 and 40 keV), a cutoff power law fit gives a power law index $\Gamma=0.9_{-0.3}^{+0.4}$, with a normalized $\chi^{2}_{\nu}$ of 1.06 for 11 dof. While from a statistical point of view the two fits are equivalent (an F-test gives a 36% probability of chance improvement (PCI) of the $\chi^{2}$ – see Appendix A for the discussion on the correctness of the use of the F-test), the cutoff fit nicely matches the spectral index observed at lower energies. The corresponding 15–100 keV flux, corrected for the triangular response of the collimator, is $7.0\times 10^{-11}$ erg cm-2 s-1. Intrigued by the shape of the residuals, we tried a smaller rebinning factor and a wider (15–200 keV) energy range. We were surprised to find the spectrum shown in Figure 4. Absorption features at $\sim$40, $\sim$70, and $\sim$120 keV are quite evident. We also noted that the points that trail these features at $\sim$50, $\sim$90, and $\sim$150 keV seems to be in agreement with a hard continuum up to 200 keV. To test this hypothesis, we performed a fit to the data below 30 keV together with these trailing points. No cutoff (the break energy is not constrained) or broken power law (the two photon indices are equal) model are able to fit the data, while a power law with photon index $1.0\pm 0.2$ does ($\chi^{2}_{\nu}=0.56$ for 20 dof). Therefore the change of slope in 20–40 keV derived from the analysis of the coarser binned energy spectrum is really due to the presence of the absorption features. The first absorption feature drives the position of the break, while the higher energy ones are spread by the rebinning, resulting in low-count, high energy bins that are fit by a change of slope. Figure 4.— 15–200 keV MOFF PDS count rate spectrum (plus signs) and the best fit power law plus 3 Gaussians in absorption model (histogram). The fit residuals are shown in the bottom panel. The Gaussian energies are left free and are $44\pm 3$, $73_{-5}^{+4}$, and $128_{-8}^{+5}$ keV. The line widths were constrained to 4, 4, and 7 keV for the three lines. To further support this interpretation, we performed a joint fit with the MECS spectrum (this analysis is detailed in Appendix B). We find that a cutoff or a broken power law models are not able to fit this broad band spectrum with the constraint $\Gamma\sim 0.9$ below 10 keV. This implies that the origin of the change of slope is not due to the presence of a cutoff but to something else, likely absorption features. For this reasons as the continuum we will use a power law with $\Gamma$ fixed at $0.87$, as derived by the analysis on the broad band spectrum of the first observation. As throughly discussed in Appendix A, the assessment of the statistical significance of these absorption features has to be done with great care. Our strategy was to evaluate the significance of each feature one at a time. We started with a power law fit in the energy range 15–55 keV, and we obtained an unacceptable $\chi^{2}_{\nu}$ of 1.4 for 25 dof. The inclusion of a Gaussian in absorption (gabs in Xspec) with centroid energy $E_{\rm cyc}=44\pm 3$ keV, $\tau=16_{-7}^{+14}$, and width fixed at 4 keV significantly improved the fit ($\chi^{2}_{\nu}$ equal to 0.48 for 23 dof). Two comments on this measurement are in order: first, the $\sim$44 keV line is statistically significant (PCI is $6\times 10^{-3}$). Second, the difficulties in determining the other line parameters (the line width had to be fixed, while the line normalization is affected by large errors) are due to the fact that the lines are seen as “holes” in the continuum: we have not enough statistics to clearly reconstruct the line profile. The line width is therefore computed from the “hole” width, while for the line depth we can infer only an upper limit (this will be more evident for the lines at higher energies). Following our detection strategy, we performed a second fit on the 15–80 keV energy range. The fit with only one Gaussian in absorption was not acceptable: $\chi^{2}_{\nu}$ was 3.6 for 32 dof. By adding another Gaussian in absorption we obtained a significantly better fit, with $\chi^{2}_{\nu}$ of 0.48 for 30 dof (PCI is $2\times 10^{-7}$). The second line parameters are $E_{\rm cyc}=73_{-5}^{+4}$ keV, $\tau=91_{-42}^{+\infty}$, while the the line width was fixed at 7 keV. Note how the two line energies are not in an harmonic ratio. Then we extended the energy range to the whole 15–200 keV band. Again the fit with two lines was not acceptable ($\chi^{2}_{\nu}$ of 4.5 for 46 dof), so we added another absorption line, the width of which was fixed at 7 keV. The fit significantly improved ($\chi^{2}_{\nu}$ of 0.58 for 44 dof, PCI of $10^{-10}$), with the best fit $E_{\rm cyc}=128_{-8}^{+5}$ keV. $\tau$ was not constrained (this was not a surprise because, as discussed above, we have no signal in the “hole”, and therefore we are not able to put a limit to the line normalization). Finally, we constrained the line energies to be in an harmonic ratio. In this case we obtain a $\chi^{2}_{\nu}$ of 1.22 for 46 dof. The best fit parameters of the fundamental222Thereafter we will indicate the fundamental resonance as (1:1), meaning with this the ordinal number of the resonance with respect to the fundamental. With this notation the first harmonic will be indicated as (1:2), the third (1:3), and so on. were $E_{\rm cyc}=41\pm 1$ keV, $\tau=11_{-7}^{+9}$. All the line widths were fixed, as in the previous case with the line energies left free, but we were not able to constrain the (1:2) and (1:3) line normalizations. In order to verify that the observed features were not of instrumental origin, and at the same time to better characterize the cyclotron resonance features (thereafter CRFs), we performed a normalized Crab ratio analysis (Orlandini 2004) on the MOFF spectrum, as we successfully employed for the detection of CRFs in numerous X–ray pulsars (see, e.g. Orlandini et al. 1998). As it is evident from the top panel of Figure 5, the ratio between the MAXI J1409$-$619 and the Crab count rate spectra shows a “hole” in the 38–44 keV range, due to the fundamental CRF. To enhance the feature, we multiplied the MAXI J1409$-$619/Crab ratio by the functional form of the Crab spectrum, i.e. a simple power law with photon index equal to 2.1. The result is shown in the second panel of Figure 5. Finally, in the lower panel, we show the ratio between the previous function and the MAXI J1409$-$619 best fit continuum power law model, together with a Gaussian fit to the fundamental CRF. The Gaussian width was fixed at 4 keV, while the Gaussian centroid energy is $43_{-1}^{+2}$ keV. Please note that this value is a lower limit, because it does not take into account the intrinsic energy resolution of the PDS instrument. Figure 5.— Top: Ratio between MAXI J1409$-$619 MOFF and Crab count rate spectra. Middle: the MAXI J1409$-$619/Crab ratio multiplied by E-2.1, the functional form of the Crab spectrum. The dotted line corresponds to the best fit power law continuum. Bottom: ratio between the former expression and the best fit power law continuum, together with a Gaussian fit to the $\sim$43 keV CRF. The statistical assessment of the CRF has been evaluated with a run test, finding that we can reject the line being random at the 99% confidence level (see text and Appendix A for details). In order to give a quantitative evaluation of the CRF in the normalized Crab Ratio we performed a run test (see detailed discussion in Appendix A). Up to $\sim$34 keV the residuals are consistent with random fluctuations ($N_{+}=12$; $N_{-}=9$; $N_{r}=13$; consistent with random fluctuation at the 84%). On the other hand, in the 34–50 keV band we see a clear structure in the residuals. In this case from a run test with $N_{+}=2$, $N_{-}=14$, $N_{r}=2$ we we can reject the null hypothesis of randomness at the 99% confidence level. ### 2.2. ASCA observation The region around MAXI J1409$-$619 was observed by ASCA on March 2, 1998 for a net exposure time of about 18 ks. The source is clearly detected, and its spectrum is confirmed to be quite hard, with a best fit power law photon index $\Gamma=0.1_{-0.6}^{+0.8}$ and an unabsorbed 1–10 keV flux of $3\times 10^{-12}$ erg cm-2 s-1. The absorption in the direction of the source is not well constrained by the fit, with an upper limit of $3\times 10^{22}$ cm-2, consistent with the Swift and BeppoSAX/MECS results. ## 3\. A very red(dened) counterpart for MAXI J1409$-$619 As stressed by Kennea et al. (2010b), a NIR source belonging to the 2MASS catalogue (Skrutskie et al. 2006) is found within 2$\farcs$1 of the MAXI J1409$-$619 position as determined by the XRT. According to that catalogue, this source (labeled as 2MASS J14080271$-$6159020) has NIR magnitudes $J$ = 15.874$\pm$0.086, $H$ = 13.620$\pm$0.022 and $K$ = 12.560$\pm$0.021. No catalogued optical counterpart is present at the position of this NIR source, and inspection of the DSS-II-Red digitized archival plates333http://archive.eso.org/dss/dss does not show any evident optical object at that location. Following Monet et al. (2003) we can thus place a conservative upper limit, $R\\!\\!>$20, to the $R$-band magnitude of the optical counterpart of this source. The above magnitudes thus indicate that this is an extremely red object. Indeed, assuming the Milky Way extinction law (Cardelli et al. 1989), we find that the NIR color indices of this source are consistent with those of a late O/early B-type star (Wegner 1994) with a reddening of $A_{V}\\!\\!\approx\\!\\!20$ mag. This, using the formula of Predehl & Schmitt (1995), implies a column density of NH $\sim$ 3.6$\times$1022 cm-2, which is consistent with that inferred from the Swift and BeppoSAX spectral analysis results. This large extinction is also supported by the non-detection of the optical counterpart of the source. All this points to a likely HMXB nature of this source, hosting a heavily absorbed early-type star, similar to several cases of INTEGRAL hard X–ray sources identified as HMXBs at optical and/or NIR wavelengths (see, e.g., Masetti et al. 2010). Assuming then $A_{V}\\!\\!\approx\\!\\!20$ mag along the MAXI J1409$-$619 line of sight and a B0 spectral type for the companion star in this system, we can infer its distance, depending on the luminosity class (main sequence, giant or supergiant) of the star. Using the tabulated absolute magnitudes for this type of star (Lang 1992) we find, for these three cases, respective distances of $\sim$4.6, $\sim$7.9 and $\sim$14.5 kpc. Given that the large absorption found along the line of sight of this source at both NIR and X–ray wavelengths is consistent with the Galactic one, we consider most likely that the correct distance is the largest one. Thus, MAXI J1409$-$619 is quite likely located in the far side of the Galaxy, i.e. in the farthest parts of the Sagittarius- Carina arm, and its mass donor star is likely an early-type supergiant. ## 4\. Discussion We found five sets of observations containing the position of MAXI J1409$-$619 in our BeppoSAX/PDS archive, performed in 1997, 2000, and 2001, and one in the ASCA archive (March 1998). In all our observations the source was in a low state, with 15–100 keV fluxes in the range $\sim$2–8 mCrab, and no spectral variability during the observations. For comparison, an integrated exposure (over 5 years) of 2.4 Ms by INTEGRAL/IBIS provides 2$\sigma$ upper limits on the persistent quiescent emission of 0.2 and 0.4 mCrab in the 20–40 and 40–100 keV energy bands, respectively (Sguera et al. 2010). When assuming the source fluxes in outburst as measured by Swift (Kennea et al. 2010b, 2011) and RXTE (Yamamoto et al. 2010), from our low state measurement we can infer a dynamic range of 400 in 15–50 keV, and of 300 in 2–10 keV. The discovery of CRFs in the low state spectrum of MAXI J1409$-$619, together with its $\sim$500 s pulsations, unambiguously identify the source as an accreting X–ray binary pulsar. Only few sources show multiple CRFs, and only two show resonances above the (1:2). Three CRFs were observed during the 2004–2005 outburst of the X–ray pulsar V0332+53 (Coburn et al. 2005; Kreykenbohm et al. 2005; Tsygankov et al. 2006), and five (possible six) CRFs were discovered in the spectrum of 4U 0115+63 during its 1999 giant outburst (Santangelo et al. 1999; Heindl et al. 1999; Ferrigno et al. 2009). In both cases, deviations from a pure harmonic ratio among the CRFs were observed, and were explained in terms of departures from a classical dipolar structure of the magnetic field in the line-forming region (Nishimura 2005, 2008). The magnetic field strength at the neutron star surface corresponding to $E_{\rm cyc}\\!=\\!44$ keV is $3.8\times 10^{12}(1+z)$ G (Canuto & Ventura 1977), where $z$, the gravitational redshift, for a typical neutron star of mass 1.4 M⊙ and radius 10 Km, is about 0.3. When left as free parameters in the fit, we found that the best fit CRF line energies do not follow the harmonic relation $E_{n}=n\,E_{1}$. A slightly non-harmonicity is expected when relativistic effects are taken into account (see, e.g., Meszaros 1992) $E_{n}=m_{e}c^{2}\,\frac{\displaystyle\sqrt{1+2n(B/B_{\rm crit})\sin^{2}\theta}-1}{\sin^{2}\theta}\,\frac{1}{1+z}$ (1) where $m_{e}$ is the electron rest mass, $c$ the speed of light, $\theta$ the angle between the photon and the magnetic field direction, and $B_{\rm crit}=4.414\times 10^{13}$ G is the critical magnetic field strength where the cyclotron energy equals the electron rest mass. The observed (1:2) and (1:3) ratios of the line energies with respect to the fundamental, 1.7$\pm$0.2 and 2.9$\pm$0.3, cannot be explained in terms of Eq. (1), as it is evident from Figure 6. A fit to the harmonic relation, shown as the dashed line, gives $E_{1}=41\pm 3$ keV (in agreement with the best fit value $E_{\rm cyc}=41\pm 1$ keV found when imposing the harmonic relation in the spectral fit), but with poor significance ($\chi^{2}_{\nu}=2.27$ for 2 dof). In the same figure we also show the harmonic relation from Eq. (1) for different values of the magnetic field and the angle $\theta$. Taking into account relativistic effects, we found that the magnetic field responsible for the (1:1) and (1:3) CRFs is about 20% higher than that responsible for the (1:2) CRF. Figure 6.— Harmonicity of the MAXI J1409$-$619 CRF line energies. The dashed black line corresponds to a linear fit (that is, $E_{n}=n\,E_{1}$) to the data, while the colored strips take into account relativistic effects, as detailed by Eq. (1). Each strip corresponds to a fixed value of the magnetic field strength $B_{12}=B\times 10^{12}$ G, and to a range 0.0001–1 for $\sin^{2}\theta$, where $\theta$ is the angle between the photon direction and that of the magnetic field. While $E_{1}$ and $E_{3}$ are consistent with $B_{12}\sim 3.8$, $E_{2}$ is consistent with a magnetic field about 20% lower. Two points are worth noticing: it is always the (1:2) resonance that shows the larger disagreement with the harmonic relation, and this could be due to the fact that this CRF is due to pure absorption (Nishimura 2003), while for the other resonances other effects, like multiple scattering and photon spawning, enter into play (see, e.g., Schönherr et al. 2007, and references therein). Unfortunately, because we are not able to reconstruct the CRF profiles, we cannot extract more information, like the electron temperature and the geometry of the emitting region. Second, at variance with the V0332+53 and 4U 0115+63 observations, both performed during giant outbursts, our BeppoSAX observations were performed while MAXI J1409$-$619 was in a low state. This did not allow the study of the dependence of the CRF parameters as a function of luminosity, an important tool for study of the physical conditions in the line-forming region (Mihara et al. 2004; Nakajima et al. 2006; Klochkov et al. 2011). The likely early-type optical counterpart and the 500 s pulsation makes MAXI J1409$-$619 a HMXB pulsar. According to the nature of the secondary star we have two possibilities: the source is a supergiant fast X–ray transient (SFXT; Sguera et al. 2005; Negueruela et al. 2006) or a Be/HMXB. In favor of the former interpretation is the highly reddened supergiant as possible counterpart, the typical outburst X–ray luminosity of $2\times 10^{37}$ erg s-1 (assuming a distance of 14.5 kpc), and the dynamic range of more than two orders of magnitude ($\sim$300) that is typical of the so-called “intermediate” SFXT (Sguera et al. 2007; Clark et al. 2010). Against we have that the source active phase, about two months long (Ueno et al. 2010; Kennea et al. 2011), is significantly longer that that typical of SFXT (see, e.g., Sidoli 2009). If this were the case, then our magnetic field measurement would rule out the magnetar nature for SFXTs (Bozzo et al. 2008). The observed properties of MAXI J1409$-$619 are also in agreement with those observed in other Be/HMXBs, like 1A 1118$-$615, a 400 s X–ray pulsar with a hard X–ray spectrum ($\Gamma\\!\sim\\!1$), a CRF at $\sim$55 keV, and long (tens of years) periods of quiescence interrupted by giant (Type-II) outbursts lasting weeks to months, in which the X–ray luminosity increases by a factor $\sim$200 (see, e.g., Rutledge et al. 2007). This would put the source a factor $\sim$2–3 closer. Figure 7.— Error box of AGL J1410$-$6143 (big red circle) superimposed on the 20–100 keV INTEGRAL/IBIS deep mosaic image ($\sim$2.3 Ms exposure). The BeppoSAX/MECS position of MAXI J1409$-$619 is marked by the green square. The two small yellow ellipses represent the Fermi $\gamma$–ray sources 2FGL J1409.9$-$6129 and 2FGL J1413.4$-$6204, respectively. White contours (from 50% to 99%) refer to the EGRET source 3EG J1410$-$6147. Finally, we note that MAXI J1409$-$619 is located within the $0\fdg 5$ error box of the unidentified transient MeV source AGL J1410$-$6143 (see Fig. 7) discovered on February 21, 2008 by the $\gamma$–ray satellite AGILE (Tavani et al. 2009a) during a bright MeV flare lasting only about one day (Pittori et al. 2008; Orlandini et al. 2008). The AGILE large position uncertainty makes very difficult the identification of its lower energy counterpart responsible for the $\gamma$–ray emission. Despite this drawback, it is intriguing to note that the flaring source MAXI J1409$-$619 is the only catalogued hard X–ray source above 20 keV (20–100 keV) to be located inside the AGILE error box, according to all the available catalogs in the HEASARC database. This spatial correlation is further supported by a similar transient nature for both MAXI J1409$-$619 and AGL J1410$-$6143\. We also point out that this is not a unique case. To date, a few HMXBs have been unambiguously detected as flaring MeV sources lasting only a few days (Sabatini et al. 2010; Tavani et al. 2009b; Abdo et al. 2009). In addition, there are several other HMXBs proposed as best candidate counterparts of unidentified transient MeV sources located on the Galactic plane (Sguera et al. 2009, 2011; Sguera 2009). For the sake of completeness, we note that within the large AGILE error box there are a number of high energy MeV sources (see Fig. 7): i) 2FGL J1409.9$-$6129 and 2FGL J1413.4$-$6204 have been reported in the second Fermi source catalog (Abdo et al. 2011) as firmly identified $\gamma$–ray pulsars and this unambiguously excludes their association with the transient AGL J1410$-$6143, ii) 3EG J1410$-$6147 is still unidentified although it has been likely associated with the $\gamma$–ray pulsar 2FGL J1409.9$-$6129 (O’Brien et al. 2008). However, Wallace et al. (2000) reported a possible MeV flare from 3EG J1410$-$6147 lasting a few days on November 1991\. This behavior is at variance with the proposed association with the $\gamma$–ray pulsar 2FGL J1409.9$-$6129 while it is more compatible with the flaring nature of AGL J1410$-$6143\. Further multi-wavelength studies (radio, NIR, X–ray and $\gamma$–ray) of the sky region are strongly needed and encouraged to shed more light on the nature of such high energy emitters. ## Acknowledgements We thank an anonymous referee for helpful comments and suggestions that greatly improved the paper. The BeppoSAX satellite was a joint Italian-Dutch programme. Part of this work is based on archival data, software or online services provided by the ASI Scientific Data center (ASDC), and the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA’s Goddard Space Flight Center. This publication makes use of data products from the 2MASS archive. 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(2010b) Yamaoka, K., Nakahira, S., Negoro, H., et al. 2010b, ATel, 2959, 1 ## Appendix A On the statistical significance of absorption features The analysis of the significance of spectral absorption features, and in particular CRFs, has historically been fraught with serious misunderstanding, that we attribute to the application of techniques made for emission features to absorption ones. This has a very profound impact on the statistical analysis, as detailed in the following. First, when searching for spectral features, and especially absorption features, the binning of data is crucial: indeed a narrow feature can be lost if the data binning is too high (see, e.g., Eadie et al. 1971, page 259), as we showed in the data analysis detailed in Section 2.1.2. But before searching for the best binning factor, it has become customary to rebin data in order to achieve a minimum number of counts in each bin. The reason of this procedure is that we must avoid bins with few counts if we want to use the $\chi^{2}$ statistics to test for the goodness of our fit (in other words, errors must be normally distributed). A minimum 20 counts per bin filter is sufficient to achieve this goal (Cash 1979). The application of this criterion to background subtracted spectra is correct for instruments that are not background dominated (that is, they have an intrinsically low number of counts per bin), like a detector on the focal plane of X–ray optics. On the other hand, when the net source spectrum is obtained by subtracting two high counts spectra this correction must not be applied (instruments like BeppoSAX/PDS or RXTE/HEXTE are background dominated, with a very high number of counts per energy channel). Although the net source counts can be very low, they are still Gaussian distributed, because resulting from the difference of two normally distributed counts. Second, if features are present in the source spectrum, they (clearly) show up in the fit residuals, and the standard goodness of fit estimator, the minimization of the $\chi^{2}$ statistics (Lampton et al. 1976), should (clearly) indicate that the fit is not “good” and an additional component is necessary. The inclusion of a new component to the continuum should result in a reduction of the reduced $\chi^{2}$. In order to assess whether the improvement of the $\chi^{2}$ is due to chance or it is because the new component is significant it is customary to use the F-test (see, e.g., Barlow 1989, page 160). A very important point has to be raised here, because we have two completely different approaches whether we are dealing with emission or absorption features. In the former case we are dealing, using a Xspec terminology, with an additive component, and we test the null hypothesis that the coefficient of the new term is zero (Bevington 1969, page 200). This is the F-test routine that is incorporated in the recent versions of Xspec. On the other hand, when dealing with an absorption feature, the component is not additive but multiplicative (see routines gabs or cyclabs). It is therefore obvious that we cannot use the F-test for an additive component, because we cannot vary to zero the new component coefficient without modifying the parameters of the original model. The two components (the multiplicative model and the original continuum) are strongly coupled, and variations in one will affect also the other. Therefore we must use a different F-test, as described in Press et al. (2007, page 730), that tests the null hypothesis that the observed variances comes from the same sample (and this routine is not incorporated in Xspec). This same test must be used to assess if two different models are statistically equivalent or not (and indeed we used it in Section 2.1.2 to compare the power law and the cutoff power law fits). It is worth noting that the application of the F-test to the statistical assessment of CRFs is not affected by the concerns raised by its use with emission lines (Protassov et al. 2002). Third, CRFs are very shallow features, with their count rates often at the instrument sensitivity level. We therefore expect errors to be dominant, and consequently the $\chi^{2}$ could not be the best estimator of the CRF significance. We need a test that takes into account the structure of the line (an information which is lost in the $\chi^{2}$, because we compute the square of the difference between the data and the model). In other words, we need to compute what is the probability that a particular structure visible in the fit residuals occurs by chance. The run test (also known as Wald-Wolfowitz test; Barlow 1989; Eadie et al. 1971) works on the signs of the deviations, that is on the form of the residuals. To better clarify how the run test works give a look at Figure 8, adapted from Barlow (1989): when fitting the 12 data points with a straight line the normalized $\chi^{2}$ is exactly 1 (likely due to error overestimation). But it is evident by eye that the fit is not good (indeed the data come from a parabolic model). The reason is that if the fit were good we should expect that the number of points “above” the fitting line should not group together, but should be intermixed with points “below” the fitting line (and this should be more true as the number of data points increases). If, on the other hand, we observe only small groups of data with the same “sign” (called runs), this means that our data are not randomly distributed with respect to the fitting model, but there is an underlying trend. Figure 8.— Example on how the goodness of fit estimation performed by the $\chi^{2}$ statistics not always is able to find the best fit model to the data. In this case the linear fit to the 12 data points yields a normalized $\chi^{2}$ of 1, but the deviation with respect to the straight line is evident. A run test shows that these deviations have only 1.3% probability to be due to random fluctuations (see text for details). As we can see from Figure 8, we have 12 points, 6 points “above” the fitting line (let us call them $N_{+}$) and 6 points “below” the fitting line ($N_{-}$). The number of runs $N_{r}$ is only 3, suspiciously small. Indeed the probability of obtaining $N_{r}\leq 3$ is 1.3%, telling us that the structure observed in the residuals is not due to random fluctuations but to a wrong modelization (the linear fit). From this example, and from the real-case analysis performed on the residuals shown in Figures 5 and 9, should be clear that the goodness of fit in the case of data structures in the residuals should not be addressed only with the $\chi^{2}$ estimator, but it has to be supported by the run test. In conclusion, this is a sort of cookbook for a correct evaluation of the statistical significance of absorption (CRFs) features in the spectra of X–ray sources: 1. 1. Be careful with the binning: although CRFs are usually broad features, a too small binning can hide structures. When dealing with background dominated data (like, for example, BeppoSAX/PDS or RXTE/HEXTE net spectra) never rebin data in order to have a minimum number of counts per bin, otherwise we risk to lose the absorption feature; 2. 2. In order to assess the significance of a CRF the F-test is perfectly suitable, but we must use the correct routine: the F-test routine included in Xspec is applicable only to an additive component (in particular, an emission line), while both gabs and cyclabs are multiplicative; 3. 3. The evaluation of the statistical significance of a CRF must be supported by other tests, especially in the presence of structures in the fit residuals: the run test is able to discriminate whether the structure is due to random fluctuations or not. ## Appendix B The continuum spectral model of MAXI J1409$-$619 The analysis performed on the broad band spectrum of the pointed MAXI J1409$-$619 observation yields as best fit continuum a power law with index $\sim$0.9 (see section 2.1.1). On the other hand, the analysis performed on a coarser binned spectrum of the offset observation indicates the presence of a change of slope around 20–40 keV (see section 2.1.2). Because a finer rebinning of this second observation revealed the presence of features that can be explained as CRSFs, it is important to understand whether this change of slope is due to a cutoff in the continuum or is a combined effect of the binning and the presence of the absorption features. From the pointed observation we know that data below 10 keV must be described by a power law with $\Gamma\sim 0.9$. To take into account this constraint we performed a joint fit of the high energy PDS spectrum from the second observation with the low energy MECS spectrum from the first observation. Although the two spectra come from different observations, X–ray pulsars are known to not show any spectral variability besides that at low energy due to reprocessing of X–rays by the circumstellar material (see, for example, the case of Vela X–1 or GX301–2). Because our source does not show any intrinsic absorption or emission lines, signatures of reprocessing, we are confident that our MECS spectrum did not change between the two observations. First we tried a fit with a power law, modified at low energy by photoelectric absorption. All the parameters are left free but the $N_{H}$ (fixed at the $2.8\times 10^{22}$ cm-2 value obtained from the ON-source observation), and the result is shown in panel a. of Figure 9. The fit parameters are listed in Table 3. Figure 9.— 1.8–200 keV joint MECS (from the pointed observation) / PDS (from the offset observation) spectrum. a. Fit with a power law continuum. b. Fit with a cutoff continuum. c. Fit with a broken power law continuum. d. Fit with a power law continuum and three CRSFs. The only model able to fit simultaneously the low (below 10 keV) and high energy data is the power law plus three CRSFs. The best fit parameters are listed in Table 3. The fit is not statistically acceptable, and the slope is driven by the low energy part of the spectrum. To fit also the PDS data we need something that is able to change the slope at about 40 keV. Now let us test whether we can explain this change of slope in terms of a cutoff in the continuum. We first tried a cutoff power law (see panel b. in Figure 9): the fit is statistically acceptable but the power law index does not match with that of the low energy part, below 10 keV. The systematic deviations from the model (information lost by the $\chi^{2}$ test, as discussed in Appendix A) can be quantified by means of the run test on the MECS residuals ($N_{+}=8$, $N_{-}=6$, $N_{r}=3$). The probability of obtaining $N_{r}\leq 3$ is 0.5%, therefore the structure observed in the residuals is not due to random fluctuations but to the wrong modelization of the continuum. The same result is obtained with the broken power law fit (see panel c. in Figure 9): there is no match to the MECS data. These results demonstrate that the change of slope is not in the continuum but is due to something else: we made the hypothesis that its origin are absorption features. Therefore the true continuum model is a power law together with CRSF features (see panel d. in Figure 9). Because we are not able to disentangle the continuum and the CRSFs, due to the broadness of the features and their low statistics, we need to fix the power law index to the value obtained by the broad band fit. Just as a test, we performed all the analysis on the CRSFs by fixing the power law index at 1.0, 1.1, and 1.2, corresponding to the $1.0\pm 0.2$ value obtained from the fit to the PDS data below 30 keV and the points trailing the CRSFs. The CRSF parameters are all consistent with each other, demonstrating that they do not depend on the particular value of the power law index. Table 3Best fit parameters for the joint MECS/PDS spectral fits \| Power Law \| Cutoff \| Broken PL \| Power Law ---\|---\|---\|---\|--- \| \| \| \| \+ 3 gabs $\Gamma$ \| $0.87\pm 0.07$ \| $-0.75_{-0.5}^{+0.3}$ \| $0.06_{-0.2}^{+0.2}$ \| $1.0_{-0.2}^{+0.3}$ $E_{\rm cutoff}$ \| $\cdots$ \| $14\pm 4$ \| $27_{-4}^{+7}$ \| $\cdots$ $\Gamma_{2}$ \| $\cdots$ \| $\cdots$ \| $2.93_{-0.6}^{+1.9}$ \| $\cdots$ $\chi^{2}_{\nu}$ (dof) \| 2.48 (66) \| 0.91 (65) \| 0.78 (64) \| 0.54 (59)	arxiv-papers	2010-12-06T16:16:10	2024-09-04T02:49:15.505311	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Mauro Orlandini (1), Filippo Frontera (2,1), Nicola Masetti (1), Vito\n Sguera (1) and Lara Sidoli (3) ((1) INAF/IASF Bologna, (2) Physics Dept,\n Ferrara Un, (3) INAF/IASF Milano)", "submitter": "Mauro Orlandini", "url": "https://arxiv.org/abs/1012.1218" }
1012.1358	# Trust transitivity in social networks Oliver Richters Tiago P. Peixoto tiago@fkp.tu-darmstadt.de Institut für Festkörperphysik, TU Darmstadt, Hochschulstrasse 6, 64289 Darmstadt, Germany ###### Abstract Non-centralized recommendation-based decision making is a central feature of several social and technological processes, such as market dynamics, peer-to- peer file-sharing and the web of trust of digital certification. We investigate the properties of trust propagation on networks, based on a simple metric of trust transitivity. We investigate analytically the percolation properties of trust transitivity in random networks with arbitrary degree distribution, and compare with numerical realizations. We find that the existence of a non-zero fraction of _absolute trust_ (i.e. entirely confident trust) is a requirement for the viability of global trust propagation in large systems: The average pair-wise trust is marked by a discontinuous transition at a specific fraction of absolute trust, below which it vanishes. Furthermore, we perform an extensive analysis of the Pretty Good Privacy (PGP) web of trust, in view of the concepts introduced. We compare different scenarios of trust distribution: community- and authority-centered. We find that these scenarios lead to sharply different patterns of trust propagation, due to the segregation of authority hubs and densely-connected communities. While the authority-centered scenario is more efficient, and leads to higher average trust values, it favours weakly-connected “fringe” nodes, which are directly trusted by authorities. The community-centered scheme, on the other hand, favours nodes with intermediate degrees, in detriment of the authorities and its “fringe” peers. ## I Introduction Several social and technological systems rely on the notion of trust, or recommendation, where agents must make their decision based on the trustworthiness of other agents, with which they interact. One example are buyers in markets Vriend (1994); Bornholdt and Schuster (2003), who may share among themselves their experiences with different sellers, or lenders which may share a belief that a given borrower will not be able to pay back Anand et al. (2009). Another example are peer-to-peer file-sharing programs Kamvar et al. (2003); Bornholdt and Schuster (2003), which often must know, without relying on a central authority, which other programs act in a fair manner, and which act selfishly. In the same line, an even more direct example is the web of trust of digital certification, such as the Pretty Good Privacy (PGP) system Guardiola et al. (2002); Boguñá et al. (2004), where regular individuals must certify the authenticity of other individuals with digital signatures. In all these systems, the agents lack global information, and must infer the reliability of other agents, based solely on the opinion of trusted peers, thus forming a network of trust. In this paper, we present an analysis of trust propagation based on the notion of _transitivity_ : If agent $a$ trusts agent $b$, and agent $b$ trusts agent $c$, then, to some extent, agent $a$ will also trust agent $c$. Based on this simple concept, we define a trust metric with which the reliability of any reachable agent may be inferred. Instead of concentrating on the minutiae of trust propagation semantics, we focus on the topological aspect of trust networks, using concepts from network theory Newman (2010). Using random networks as a simple model, we investigate the necessary conditions for trust to “percolate” through an entire system. We then apply the concepts introduced to investigate in detail the PGP web of trust, possibly the best “real” example of a trust propagation system, which is completely accessible for investigation. We focus on the role of the strongly connected nodes in the network — the so called _trust authorities_ — which represent a different paradigm of trust delegation, in comparison to the decentralized community-based approach, which is also heavily present in the network. This paper is divided as follows. In section II we define the trust metric used; in section III we consider the problem of trust percolation in random networks with different trust weight distributions. In in section IV we turn to the analysis of the PGP network, and provide an extensive analysis of topology of the PGP network, and of trust propagation according to different trust distribution scenarios. Finally, in section V we provide some final remarks and a conclusion. ## II Trust metric Trust is the measure of belief that a given entity will act as one expects. It is often associated with positive, desirable attributes, but it may not always be the case (e.g. one may have trust that someone will act undesirably). Humans use trust to make decisions when more direct information is unavailable. In general, humans will decide their level of trust based on arbitrary, heuristic rules, since there is no formal consensus on how to evaluate trust. We will deliberately avoid the detailed formalization of these rules, and instead rely on two simplifications: 1. We will treat trust simply as a probability that a given assessment about an agent is true or false (e.g. fair/reliable or not); 2. We further assume that this belief is _transitive_ , i.e. if agent $a$ trust agent $b$, which in turn trusts agent $c$, then $a$ will also trust $c$, to some extent. This makes trust propagation easier to analyse, while retaining the most intuitive properties of trust propagation. Figure 1: Examples of trust networks: Left: A directed tree. Right: A more realistic example. The edges in blue are the ones which contribute to the value of trust from Bob to Alice, according to Eq. 6. We will consider a system of $N$ agents which form a directed trust network: Each agent $v$ (represented by a vertex, or node) has a number of interactions (represented by directed edges, or links) with other agents $\\{u_{i}\\}$ for which a value $c_{v,u_{i}}\in[0,1]$ of _direct trust_ is defined a priori, and which can be interpreted as a probability. This value represents a direct experience agent $v$ had with $u_{i}$, which is not inferred from any other agent. We then define the _inferred trust_ $t_{ij}\in[0,1]$ from agent $i$ to any agent $j$, which is somehow based on the values of $c_{v,u_{i}}$. In a simple situation where there is only one possible path between any two given nodes (i.e. the network is a directed tree, as the example on the left in Fig. 1), one could simply multiply the values of $c$ along the single path to obtain $t$, e.g. $t_{\text{Alice},\text{Bob}}=c_{1}c_{3}$, in the example of Fig. 1. In general, however, the situation may be more complicated, as in the example on the right of Fig. 1, where there is a variety of possible (often “contradictory”) transitive paths between most pairs of nodes. Perhaps the simplest way of defining a trust metric would be to consider only the _best_ transitivity path between two nodes, i.e., the one where the trust transitivity is maximum, $s_{u,v}=\max\left\\{\prod_{\\{e_{i}\\}}c_{e_{i}}\right\\},\qquad\forall\,\\{e_{i}\\}\in P_{u\rightsquigarrow v},$ (1) where $P_{u\rightsquigarrow v}$ is the set of all paths from $u$ to $v$, $\\{e_{i}\\}$ is the set of edges in a given path, and $c_{e}$ is the direct trust associated with a given edge. This definition is an attractive one, since it corresponds directly to the concept of minimum distance on weighted graphs, which is defined as the sum of weights along the path with the smallest sum. This is easily seen by noticing that $\prod_{\\{e_{i}\\}}c_{e_{i}}=\exp\\{\sum_{\\{e_{i}\\}}\omega_{e_{i}}\\}$, with $\omega_{e_{i}}=-\ln c_{e_{i}}\geq 0$ being the edge weights (with the special value of $\omega_{e_{i}}=\infty$ if $c_{e_{i}}=0$). However, it is clear that this approach leads to an optimistic bias, since the best path obviously favors large values of trust, and uses only a small portion of the information available in the network. As an illustration consider the network on the right of Fig. 1, where the value of $s_{\text{Alice},\text{Bob}}$ is $1\times 0.9\times 0.6=0.54$, via Dave and Chuck. However, if Chuck is directly consulted, the transitivity drops to $0.3\times 0.6=0.18$. In principle, there is no reason to prefer any of the two assessments over the other. One may attempt to rectify this by considering instead _all_ possible paths between two nodes, $\tilde{t}_{u,v}=\frac{\displaystyle\sum_{u\rightsquigarrow v}\omega_{u\rightsquigarrow v}\prod_{e\in u\rightsquigarrow v}c_{e}}{\displaystyle\sum_{u\rightsquigarrow v}\omega_{u\rightsquigarrow v}},$ (2) where $\omega_{u\rightsquigarrow v}$ is a weight associated with a given path $u\rightsquigarrow v$. It should be chosen to minimize the effect of a very large number of paths with very low values of trust, without introducing an optimistic bias on the final trust value. One apparently good choice is to consider the transitivity value of the path itself, but not including the last edge, $\omega_{u\rightsquigarrow v}=\prod_{e\in u\rightsquigarrow v}c_{e}+(1-c_{e})\delta(e,e_{\to v}),$ (3) where $e_{\to v}$ is the last edge in the path, and $\delta$ is the Kronecker delta. Not only this avoids a bias in the final value of $\tilde{t}_{u,v}$, but also $\omega_{u\rightsquigarrow v}$ has a simple interpretation as being the value of trust on the _final_ recommendation, which is completed by the last edge. While this may seem reasonable, and uses all available information in the network, it has two major drawbacks: 1. It is very computationally costly to consider all possible paths between two nodes, even in moderately sized networks. It would represent an unreasonable effort on part of the agents to use all this information. 2. Computed as in Eq. 2, the value of $\tilde{t}_{u,v}$ has the unsettling behaviour of tending to zero, whenever the number of paths become large (as they often are), even when paths are differently weighted. Consider a simple scenario where the network is a complete graph, i.e. all possible edges in the network exist, and all of them have the same direct trust value $c$. Since there are ${N-2\choose l}l!$ paths of length $l+1$ between any two vertices, the value of inferred trust between any two nodes can be calculated as $\displaystyle\tilde{t}_{u,v}$ $\displaystyle=\frac{\sum_{l=0}^{N-2}{N-2\choose l}l!c^{2l+1}}{\sum_{l=0}^{N-2}{N-2\choose l}l!c^{l}}$ (4) $\displaystyle\leq c^{N-1}\exp\left(\frac{1}{c^{2}}-\frac{1}{c}\right),$ (5) from which it is easy to see that $\lim_{N\to\infty}\tilde{t}_{u,v}=0$ for $c<1$. This is an undesired behavior, since one would wish that such highly connected topologies (which often occur as subgraphs of social networks, known as _cliques_) would result in _higher_ values of trust. In order to compensate for this one would have to use a more aggressive weighting of the possible paths. We propose the following modification, which combines some features of both previous approaches: Instead of considering all possible paths, we consider only those with the largest weights to all the in-neighbours of the target vertex, as shown in Fig. 2. This leads to a trust metric defined as $t_{u,v}=\frac{\sum_{w}A_{w,v}\left(s^{G\setminus\\{v\\}}_{u,w}\right)^{2}c_{w,v}}{\sum_{w}A_{w,v}s^{G\setminus\\{v\\}}_{u,w}},$ (6) where the path weights are the best trust transitivity to the in-neighbours, $s^{G\setminus\\{v\\}}_{u,w}$, which are calculated after removing the target vertex from the graph (so that it cannot influence its own trust). Figure 2: Illustration of the paths used to calculate $t_{u,v}$ according to Eq. 6. The vertices $w_{i}$ are the in-neighbours of $v$, and the values $s_{i}=s^{G\setminus\\{v\\}}_{u,w_{i}}$ are the values of best trust (Eq. 1) from $u$ to $w_{i}$, with vertex $v$ removed from the graph. We call this trust metric _pervasive trust_ , and it corresponds to the intuitive strategy of searching for the nodes with a direct interaction with the target node (the final arbitrators), and weighting their opinions according to the best possible trust transitivity leading to them. It can be seen that this definition does not suffer form the same problems of Eq. 2, again by considering the same complete graph example, with uniform direct trust $c$. Since in this situation every target vertex has $N-2$ in-neighbours different from the source, and the shortest path to each of these in- neighbours is of length one, the value of pervasive trust can be easily calculated as $t_{u,v}=\frac{(N-2)c^{3}+c}{(N-2)c+1},$ (7) which converges to $t_{u,v}\approx c^{2}$ for $N\gg 1$. Thus the indirect opinions with value $c^{2}$ dominate the direct trust value $c$, but the inferred value does not vanish, as with the definition of Eq. 2. Considering again the example on the right of Fig. 1, we obtain the value $t_{\text{Alice,Bob}}=(0.9^{2}\times 0.6+(0.9\times 0.7)^{2}\times 0.3)/(0.9+0.9\times 0.7)\approx 0.4$, from the edges outlined in blue in the figure. Additionally, the definition of pervasive trust works as one would expect in the trivial example on the left of Fig. 1, where $s_{u,v}$ and $t_{u,v}$ have the same values. We note that the numerical computation of $s_{u,v}$ can be done by using Dijkstra’s shortest path algorithm Dijkstra (1959); Brandes and Erlebach (2005), which has a complexity of $O(N\log N)$. Thus the entire matrix $s_{u,v}$ can be calculated in $O(N^{2}\log N)$ time. The same algorithm can be used to calculate $t_{u,v}$, but since each target vertex needs to be removed from the graph, and thus a new search needs to be made for each different target, this results in $O(N^{3}\log N)$ time. It is possible to improve this by performing searches in the _reversed_ graph, i.e., for each target vertex $v$, the contribution to $t_{u,v}$ from all sources $u$ can be calculated simultaneously, after $v$ is removed, by performing a single reversed search from each of the in-neighbours of $v$ to each source $u$. This way, the entire $t_{u,v}$ matrix can be computed in $O(kN^{2}\log N)$ time (where $k=E/N$ is the average degree of the network), which is comparable to the computation time of $s_{u,v}$ for sparse graphs. ### II.1 Comparison with other trust metrics Other trust metrics have been proposed in the literature, mainly by computer scientists, seeking to formalize the notion of trust in peer-to-peer computer systems. Some are quite detailed, like the usage of subjective logic by Jøsang et al Jøsang et al. (2006), and others are comparable with the simplistic approach taken in this work, such as Eigentrust Kamvar et al. (2003) and more recently TrustWebRank Walter et al. (2009). These last metrics are based on the notion of _feedback centrality_ Brandes and Erlebach (2005), which are calculated by solving some linear system. The Eigentrust metric requires the trust network to be a stochastic matrix (i.e. the sum of the trust values of the out-edges of all vertices must sum to unity) and the inferred trust values are given by the steady state distribution of the corresponding Markov chain (i.e. the left eigenvector of the stochastic matrix with unity eigenvalue, hence the name of the metric). Thus the inferred trust values are _global_ properties, independent of any source vertex (i.e. non-personalized), which is non-intuitive. Additionally, the requirement that the trust network is stochastic means that only _relative_ values of trust are measured, and the absolute information is lost. Furthermore, such an approach is strongly affected by the presence of loops in the network, which get counted multiple times, which is also non-intuitive as far as trust transitivity is concerned. The metric TrustWebRank Walter et al. (2009) tries to fix some of these problems by borrowing ideas from the PageRank Page et al. (1999) algorithm, resulting in a metric which also requires a stochastic matrix, but is personalised. However, in order for the algorithm to converge, it depends on the introduction of an _damping factor_ which eliminates the contribution of longer paths in the network, independently of its trust value. This is an a priori assumption that these paths are not relevant, and may not correspond to reality. Additionally, the strange role of loops in the network is the same as in the Eigentrust metric. However, since there is no consensus on how a trust propagates, and the notion of trust lacks a formal, universally accepted definition, in the end there is no “correct” or “wrong” metric. We only emphasize that our approach is derived directly from the simple notion of trust transitivity, is easy to interpret, propagates _absolute_ values of trust, and makes no assumption whatsoever about the network topology, and direct trust distribution. ## III Trust percolation Trust transitivity is based on the multiplication of direct trust values, which may tend to be low if the paths become long. Therefore, it is a central problem to determine if the trust transitivity between two randomly chosen vertices of a large network vanishes if the system becomes very large. This provides important information about the viability of trust transitivity on large systems. As a simple network model, we will consider random directed networks with arbitrary degree distributions Newman et al. (2001). We will also suppose that the direct trust values in the range between $c$ and $c+dc$ will be independently distributed with probability $\rho_{c}(c)dc$, where $\rho_{c}(c)$ is an arbitrary distribution. The objective of this section is to calculate the average best trust transitivity $\left<s\right>$, given by Eq. 1, and the average pervasive trust $\left<t\right>$, Eq. 6, between randomly chosen pairs of source and target vertices. In random networks, the value of average pervasive trust will be given simply as $\left<t\right>=\left<s\right>\left<c\right>$, since the best paths to the in- neighbours of a given vertex are uncorrelated, and the probability that they pass through the node itself tend to zero, in the limit of large network size. Therefore we need only to concern ourselves with the average best trust transitivity $\left<s\right>$. Networks are composed of components of different types and sizes: For each vertex there will be an _out-component_ , which is the set of vertices reachable from it, and an _in-component_ , which is the set of vertices for which it is reachable. A maximal set of vertices which are mutually reachable is called a _strongly connected component_. Random graphs often display a phase transition in the size and number of these components: If the number of edges is large enough, there will be the sudden formation of a giant (in-, out-, strongly connected) component, which spans a non-vanishing fraction of the network Newman (2010); Newman et al. (2001). The existence of these giant components is obviously necessary for a non-vanishing value of trust to exist between most vertices, but it is not sufficient, since it is still necessary that the multiplication of direct trust values along most shortest paths do not become vanishingly small. As an illustration, consider a sparse graph (i.e. with finite average degree), with an arbitrary degree distribution. In the situation where there is a sufficiently large giant out-component in the graph, the average shortest path from a randomly chosen root vertex to the rest of the network is given approximately Newman et al. (2001) by $l\approx\frac{\ln(N/\left<k\right>)}{\ln(\left<k_{2}\right>/\left<k\right>)},$ (8) independently of the out-degree distribution (as long as $\left<k\right>$ and $\left<k_{2}\right>$ are finite positive), where $N$ is the number of vertices, $\left<k\right>$ is the average out-degree and $\left<k_{2}\right>$ is the average number of second out-neighbours, and it is assumed that $N\gg\left<k\right>$ and $\left<k_{2}\right>\gg\left<k\right>$ 111An analogous expression for the distance from the entire network to a randomly chosen _target_ can be obtained by replacing $\left<k\right>$ and $\left<k_{2}\right>$ with the average in-degree and second in-neighbours, $\left<j\right>$ and $\left<j_{2}\right>$ respectively.. Since the edges are weighted, the average length of the best paths can differ from $l$, but can never be smaller. Thus, an upper bound on the average best trust is given by $\left<s\right>=o(\max{\\{c_{i}\\}}^{l})$, where $\max{\\{c_{i}\\}}$ is the maximum value of direct trust in the network. In the situation where $\max{\\{c_{i}\\}}<1$, we have that $\lim_{N\to\infty}\left<s\right>=o(0)$, since $\lim_{N\to\infty}l=\infty$. Therefore, if there are no values of $c=1$ in the network, the average trust will always be zero in sparse networks. The only possible strategies for non-vanishing values of average trust is either to have a non-zero fraction of $c=1$ (which we will call _absolute trust_), or for the network to be dense, such that $l$ remains finite for $N\to\infty$. With the above consideration in mind, we now move to calculate the average trust transitivity values. For that we modify the generating function method used in Newman et al. (2001) to obtain the distribution of component sizes. The objective is to obtain a self-consistency condition for the distribution of best trust transitivity values by describing the direct neighbourhood of a single vertex, which is based on the following observation: A randomly chosen vertex $u$ with out-neighbours $w_{i}$, each with direct trust from $u$ given by $c_{i}$, will trust another randomly chosen vertex $v$ with a best trust transitivity value of $s^{+}$ only if $\max(c_{i}s_{i}^{+})=s^{+}$, where $s_{i}^{+}$ is the best trust transitivity from $w_{i}$ to $v$. In a random network which is sufficiently large, $s^{+}$ and $s_{i}^{+}$ should both be drawn from the same distribution $\rho^{+}_{s}(s)$. This gives a self- consistency condition for $\rho^{+}_{s}(s)$ which is given schematically as follows, where each term corresponds to the probability of the vertex having a given number of out-neighbours, and the maximum best trust transitivity being equal the desired value. Note that we have explicitly multiplied every instance of $s^{+}$ with a arbitrary free variable $u^{+}$, which cannot be determined by the above self-consistency alone, and has to be described separately. Each term on the right is weighted by the out-degree probability $p_{k}$. In terms of the cumulative distribution $\tilde{\rho}^{+}_{s}(s)=\int_{0}^{s}du\rho^{+}_{s}(u)$, this self-consistency can be expressed as $\tilde{\rho}^{+}_{s}(s)=\sum_{k}p_{k}[\tilde{\beta}^{+}(s)]^{k},$ (9) where $\tilde{\beta}^{+}(x)$ is the cumulative probability that $s^{+}c<x$, with $c$ distributed by $\rho_{c}(c)$, given by $\tilde{\beta}^{+}(s)=\int_{0}^{1}dx\rho_{c}(x)\tilde{\rho}^{+}_{s}(s/x).$ (10) The distribution $\rho^{+}_{s}(s)$ above does not equal $\rho_{s}(s)$, due to the remaining variable $u^{+}$, for which one must still find an appropriate distribution. This last piece is obtained by realizing that $\rho_{s}(s)$ must also be subject to a complementary self-consistency condition in the _opposite_ direction, following the in-neighbours: A randomly chosen vertex $u$ with in-neighbours $w_{i}$, each with direct trust to $u$ given by $c_{i}$, will _be trusted_ by another randomly chosen vertex $u$ with a best trust value of $s^{-}$ only if $\max(c_{i}s_{i}^{-})=s^{-}$, where $s_{i}^{-}$ is the best trust from $w_{i}$ to $u$. This results in an entirely analogous self-consistency condition for $\rho_{s}^{-}(s)$, where the out-degree distribution $p_{k}$ is replaced by the in-degree distribution $p_{j}$. Since this last self-consistency is also complete up to a free variable $u^{-}$, we can formulate the ansatz that $u^{+}=s^{-}$ and $u^{-}=s^{+}$, such that $s=s^{+}s^{-}.$ (11) With this connection it is possible to obtain $\rho_{s}(s)$ as $\displaystyle\rho_{s}(s)=$ $\displaystyle\int_{0}^{1}du\rho^{+}_{s}(u)\rho^{-}_{s}(s/u)/u,\;\;\text{or}$ (12) $\displaystyle=$ $\displaystyle\int_{0}^{1}du\rho^{+}_{s}(s/u)\rho^{-}_{s}(u)/u,$ (13) and the average $\left<s\right>$ more directly as $\displaystyle\left<s\right>=$ $\displaystyle\int_{0}^{1}\int_{0}^{1}ds^{-}ds^{+}s^{-}s^{+}\rho^{-}_{s}(s^{-})\rho^{+}_{s}(s^{+})$ (14) $\displaystyle=$ $\displaystyle\left<s^{-}\right>\left<s^{+}\right>.$ (15) By rewriting Eq. 9 in terms of the generating functions of the in- and out- degree distributions, $G(z)=\sum_{j}p_{j}z^{j}\qquad F(z)=\sum_{k}p_{k}z^{k},$ (16) one obtains the self-consistency equations in a more compact form, $\displaystyle\tilde{\rho}^{-}_{s}(s)$ $\displaystyle=F(\tilde{\beta}^{-}(s))$ (17) $\displaystyle\tilde{\rho}^{+}_{s}(s)$ $\displaystyle=G(\tilde{\beta}^{+}(s)).$ (18) These are integral equations, for which there are probably no general closed form solutions. However, it is possible to solve them numerically by successive iterations from an initial distribution, which we chose as $\tilde{\rho}^{0}(s)=\Theta(s-1)$, where $\Theta(x)$ is the Heaviside step function. From the numerical solutions the average values can be obtained as $\left<s^{-}\right>=\int_{0}^{1}ds\rho^{-}_{s}(s)s=1-\int_{0}^{1}ds\tilde{\rho}^{-}_{s}(s)$ (where the last expression is obtained by integration by parts), and in analogous fashion for $\left<s^{+}\right>$. The average value of best trust transitivity $\left<s\right>$ is then given by Eq. 15. We turn now to the conditions necessary for non-vanishing average trust transitivity. Both Eqs. 17 and 18 accept the trivial solution $\tilde{\rho}^{-/+}_{s}(s)=\Theta(s)$, which corresponds to $\rho^{-/+}_{s}(s)=\delta(s)$, i.e. the average best trust is zero. As discussed previously, for other solutions to be possible, we need to consider a non-vanishing fraction of edges with absolute trust $c=1$ in the network. Here we will consider direct trust distributions of the form, $\rho_{c}(c)=\gamma\delta(c-1)+(1-\gamma)\rho^{\prime}_{c}(c),$ (19) which correspond to a fraction $\gamma$ of edges with $c=1$, and a complementary fraction $(1-\gamma)$ with $c$ given with probability density $\rho^{\prime}_{c}(c)$. We will consider two different versions of $\rho^{\prime}_{c}(c)$: A uniform distribution $\rho^{\prime}_{c}(c)=1$, and a single-valued distribution $\rho^{\prime}_{c}(c)=\delta(c-\eta)$, with $\eta=1/2$. We will use two different degree distributions, the Poisson and Zipf 222Many real networks have broad degree distributions with a power-law tail, $p_{k}\sim k^{-\tau}$ for large $k$. These networks are called _scale- free_. The PGP network considered below is also an example of this. , and their respective generating functions, $\displaystyle p_{j}=\frac{\left<j\right>^{j}e^{-\left<j\right>}}{j!}$ $\displaystyle\qquad G(z)=e^{\left<j\right>(z-1)}$ (20) $\displaystyle p_{j}=\frac{j^{-\tau}}{\zeta(\tau)}$ $\displaystyle\qquad G(z)=\frac{\text{Li}_{\tau}(z)}{\zeta(\tau)},$ (21) where $\zeta(\tau)$ is the Riemann $\zeta$ function, and $\text{Li}_{n}(x)$ is the $n$th polylogarithm of $x$. For simplicity, we will consider only the situation where $p_{j}=p_{k}$, and both $j$ and $k$ are independently distributed. Figure 3: Average values of best trust $\left<s\right>$ and pervasive trust $\left<t\right>$ as a function of the fraction of edges with absolute trust $\gamma$. Top left: Networks with Poisson in- and out-degree distributions, and uniform trust distribution. Top right and bottom right: Poisson distribution, and single-valued trust distribution. Bottom left: Zipf distribution, and single-valued trust distribution. Solid lines correspond to analytical solutions, and symbols to numerical realizations of several networks of different sizes: $10^{4}$ (red), $10^{5}$ (green) and $10^{6}$ (blue) nodes. The dashed line shows the average direct trust $\left<c\right>=(\gamma+1)/2$. In Fig. 3 are plotted the values of $\left<s\right>$ and $\left<t\right>$, as a function of $\gamma$, for the different distributions. It is also compared with numerical computations on actual network realizations of different sizes. The main feature observed is a first-order transition from vanishing trust to positive trust, at specific values of $\gamma$. The transition values $\gamma^{}$ correspond exactly to the critical values of the formation of a giant component of the induced subgraph composed only of edges with $c=1$, which has average degree $\gamma\left<k\right>$ Newman et al. (2001). For graphs with Poisson degree distribution, this corresponds to $\gamma^{}=1/\left<k\right>$. It is worth observing that on finite graphs, the average trust does not vanish very rapidly, and is still non-zero for relatively large networks with $N=10^{6}$ nodes, even when $\gamma=0$. This is attributed to the so-called small-world effect where the average shortest path scales slowly as $l\sim\ln N$, as in Eq. 8. Therefore in practical situations where networks are large but finite, $\gamma>\gamma^{}$ it is not a strictly necessary condition for system-wide trust propagation. Another interesting feature is the behaviour of the average trust in graphs with Zipf degree distribution. There, the transition to positive trust is of second order, and the critical points are also $\gamma=1/\left<k\right>$. Additionally, the values of average trust are smaller than in networks with Poisson degree distribution and the same average degree, for intermediary values of $\gamma$ after the transitions. This is due to the smaller path multiplicity of graphs with scale-free distribution: Even though the average shortest path length is smaller in such graphs, the number of alternative paths is also smaller, due to the dominance of vertices with smaller degree. Thus, if the shortest path happens to have a small trust value, there will be a higher probability there will not be an alternative path. In Fig. 3 it is shown also the average best trust for $1<\tau<2$, for which the average degree diverges. For such dense networks, the values of $\left<s\right>$ are above zero for all values of $\gamma>0$, which is simply due to the fact that the average shortest path length does not diverge in this case. ## IV The Pretty Good Privacy (PGP) Network In this section we investigate trust propagation on the Pretty Good Privacy (PGP) network. In a broad manner PGP (or more precisely the OpenPGP standard Thayer et al. (2007)) refers to a family of computer programs for encryption and decryption of files, as well as data authentication, i.e. generation and verification of digital signatures. It is often used to sign, encrypt and decrypt email. It implements a scheme of public-key cryptography Menezes et al. (1996), where the keys used for encryption/decryption are split in two parts, one private and one public. Both parts are related in way, such that the private key is used exclusively for decryption and creation of signatures, and the public key only for encryption and signature verification. Thus any user is capable of sending encrypted messages and verifying the signature of a specific user with her public key, but only this user can decrypt these messages and generate signatures, using her private key, which she should never disclose. The public keys are usually published in so-called key servers, which mutually synchronize their databases, and thus become global non-centralized repositories of public keys. However, the mere existence of public key in a key server, associated with a given identity (usually a name and an email address) is no guarantee that this key really belongs to the respective person, since there is no inherent verification in the submission process. This problem is solved by the implementation of the so-called _web of trust_ of PGP keys, whereby a user can attach a signature to the public key of another user, indicating she trusts that this key belongs to its alleged owner. The validity of a given key can then be inferred by transitivity, in a self-organized manner, without the required presence of a central trust authority. As such, this system represents an almost perfect example of a trust propagation through transitivity. As a rule, key signatures should only be made after careful verification, which usually requires the two parties to physically meet. Such a requirement transforms the web of trust into a snapshot of a global social network of acquaintances, since the vast majority of keys correspond to human users, which tend to sign keys of people with which they normally interact. There is also a tendency to sign keys (upon verification) from people which do not belong to a close circle of acquaintances, with the sole purpose of strengthening the web of trust with more connections. This tendency is well reflected by the so-called “key signing parties”, where participants meet (usually after a large technological conference) to massively sign each other’s keys Brennen (2008). Thus the structure of the PGP network reflects the global dynamics of self-organization of human peers in a social context. This section is divided in two parts. In the first part we present some aspects of the topology and temporal organization of the network. In the second part we analyze the trust transitivity in the network, in view of the trust metric we discussed previously. ### IV.1 Network topology The PGP network used in this work was obtained from a snapshot of the globally synchronized SKS key servers 333Available at http://key-server.de/dump/ in November 2009. It is composed of $N\approx 2.5\times 10^{6}$ keys and $E\approx 7\times 10^{5}$ signatures with a very low average degree of $\left<j\right>=0.28$. This means that many keys are isolated and contain no signatures. Therefore we will concentrate on the largest _strongly connected component_ , i.e. a maximal set of vertices for which there is a path between any pair of vertices in the set. The number of vertices $N\approx 4\times 10^{4}$ in this component is much smaller, but the network is much denser, with on average $\left<j\right>\approx 7.58$ signatures per key (see summarized data in table 1). It represents the _de facto_ web of trust, since the rest of the network is so sparsely connected that no trust transitivity can be inferred from it. We note that keys may have multiple “subkeys” which correspond to different identities (usually different email addresses from the same person) and which can individually sign other subkeys. For simplicity, in this work we have collapsed subkeys into single keys, and possible multiple signatures into a single signature. We have also discarded invalid, and revoked keys and signatures. $N$ \| $E$ \| $\left<j\right>$ \| $r$ \| $a$ \| $c$ ---\|---\|---\|---\|---\|--- 2513677 \| 703142 \| $\approx 0.28$ \| $0.45$ \| $-0.02152(12)$ \| 0.02321(9) 39796 \| 301498 \| $\approx 7.58$ \| $0.69$ \| $0.0332(3)$ \| $0.461(2)$ Table 1: Summary of statistics for the whole PGP network (above) and the largest strongly connected component (below). $N$ is the number of vertices (keys), and $E$ is the number of edges (signatures), $\left<j\right>$ is the average in-degree, $r$ is the average reciprocity, $a$ is the assortativity coefficient and $c$ is the average clustering coefficient. The number of keys and signatures in the strongly connected component has been increasing over time, as shown in Fig. 4. The number of keys (which are now valid) was approximately the same for some time and then slightly decreased for a period up to around 2002, and has been increasing with an approximately constant rate since then. We note that the number of keys may decrease since keys can expire or be revoked. The number of signatures, on the other hand, seems to be increasing with an accelerated rate, which is approximately constant, and similar to the rate of growth of the number of keys. This means that the average degree of the network is increasing with time, as can be seen in Fig. 4. Keys and signatures grow in an organized manner, as shown by the waiting time distribution between the creation of two subsequent keys or signatures, as shown in Fig. 4. These distributions are broad for several orders of magnitude, from the order of seconds to days, approximately following a power-law in this region. The fact that keys and signatures are often created only seconds apart, and the waiting time distribution lacks any discernible characteristic scale, except for a cut-off at large times ($\sim 1$ day), shows that the network does not grow in a purely random fashion (which would generate exponentially-distributed waiting times), and serves as a signature of an underlying organized growth process. Figure 4: Number of keys and signatures as a function of time for the strongly connected component of the PGP network, and waiting time distribution between new keys and signatures. The straight lines are power-laws $\Delta t^{-\xi}$, with $\xi=1.3$ (top) and $\xi=0.18$ (bottom). We will characterize the topology of the network by its degree distribution and nearest-neighbours degree correlations, as well as other standard network measures such as clustering Newman (2003a), reciprocity Zamora-López et al. (2008) and community structure Newman and Girvan (2004). We will pay special attention to the most highly connected vertices, some of which correspond to so-called _certificate authorities_ and display a distinct connectivity pattern, which has a special meaning for trust propagation. The network has very heterogeneous degree distributions, as can be seen in Fig. 5, with some keys having on the order of $10^{3}$ signatures. They are possibly compatible with a power-law with exponent $\sim 2.5$ for large degrees, but the distributions are not broad enough for a precise identification. The number of signatures on a given key (the in-degree) and the number of signatures made by a the same key (the out-degree) are strongly correlated, as can be seen in Fig. 6, which shows the average out-degree ${\left<k\right>}$ as a function of the in-degree $j$. Figure 5: Several statistical properties for the PGP Network. Top left: In- and out-degree distributions, $p_{j}$ and $p_{k}$ respectively. The solid line corresponds to a power-law with exponent $2.5$. Top right: Average in- and out-degree of the nearest out-neighbours, as a function of the in- and out- degree. Bottom left: Average lustering coefficient as a function of in- and out-degree. Bottom right: Distribution of community sizes, for the unmodified and shuffled versions of the network. The solid lines correspond to power-laws with exponent $2.3$ (top) and $3.8$ (bottom). This is explained by the high reciprocity of the edges in the network, i.e. if a key $a$ signs a key $b$, there is a very high probability that key $b$ signs key $a$ as well. This is easy to understand, since key verification usually requires physical presence, and both parties take the opportunity to mutually verify each other keys in the same encounter. The edge reciprocity Zamora- López et al. (2008) is quantified as the fraction $r=n^{\leftrightarrow}_{e}/E$, where $n^{\leftrightarrow}_{e}$ is the number of reciprocal edges and $E$ is the total number of edges in the network. The PGP network has a high value of $r=0.69$. The reciprocity is distributed in a slightly heterogeneous fashion across the network, as is shown in Fig. 6, where is plotted the average reciprocity of the edges as a function of the in- and out-degrees of the source vertex. It can be seen that the keys with very few signatures tend to act in a very reciprocal manner, whereas the more prolific signers receive less signatures back. This heterogeneity is further amplified when one considers the degree correlation between nearest- neighbours, as shown in Fig. 5, where it is plotted the average in- and out- degree, ${\left<j\right>}_{\text{nn}}$ and ${\left<k\right>}_{\text{nn}}$, of the nearest out-neighbours of the vertices in the network, as a function of the in- and out-degree of the source vertex, $j$ and $k$. Figure 6: Left: Average out-degree as a function of the in-degree of the same vertex. Right: Average edge reciprocity, as a function of the in or out-degree of the source vertex. The degree correlation shows an _assortative_ regime for intermediary degree values ($\sim 10$ – $40$), meaning that vertices with higher degrees are connected preferentially with other vertices with high degree, but also some _dissortative_ features for vertices with very high and very low degrees, where vertices with low degree are connected preferentially with vertices with high degree, and _vice versa_. This mixed connectivity pattern leads to a very low scalar assortativity coefficient 444The scalar assortativity coefficient Newman (2003b) is defined for an undirected graph as $a=\frac{1}{\sigma_{q}^{2}}\sum_{ij}ij\left(e_{ij}-q_{i}q_{j}\right)$ where $e_{ij}$ is the fraction of edges that connect vertices of degrees $i$ and $j$, $q_{i}=\sum_{j}e_{ji}$ and $\sigma_{q}$ is the standard deviation of the distribution $q_{i}$. This definition yields values in the range $a\in[-1,1]$, with $a=-1$ for networks which are maximally dissortative, and $a=1$ for maximally assortative. For the PGP network, the direction of the edges was ignored in the calculation of $a$. of $a=0.0332(3)$, which is unusual for social networks Newman and Park (2003). These differences become more clear when one investigates more closely the keys with the largest degree in the network, as it is shown in table 2. Key ID \| Name \| $j$ \| $k$ \| $\left<j\right>_{\text{out}}$ \| $c$ \| Date ---\|---\|---\|---\|---\|---\|--- D2BB0D0165D0FD58 \| CA Cert Signing Authority (Root CA) <gpg@cacert.org> \| $965$ \| $1507$ \| $17.5(8)$ \| $0.0031$ \| 2003-07-11 2F951508AAE6022E \| Karlheinz Geyer (TUD) <geyerk.fv.tu@nds.tu-darmstadt.de> \| $661$ \| $744$ \| $59(2)$ \| $0.0660$ \| 2004-12-07 DBD245FCB3B2A12C \| ct magazine CERTIFICATE <pgpCA@ct.heise.de> \| $597$ \| $1348$ \| $18.3(12)$ \| $0.0033$ \| 1999-05-11 69D2A61DE263FCD4 \| Kurt Gramlich <kurt@skolelinux.de> \| $406$ \| $644$ \| $71(3))$ \| $0.0807$ \| 2002-10-17 948FD6A0E10F502E \| Marcus Frings <protagonist@gmx.net> \| $387$ \| $381$ \| $82(5)$ \| $0.1110$ \| 2002-03-22 29BE5D2268FD549F \| Martin Michlmayr <tbm@cyrius.com> \| $385$ \| $436$ \| $56(4)$ \| $0.0499$ \| 1999-08-04 566D362CEE0977E8 \| Jens Kubieziel <jens@kubieziel.de> \| $369$ \| $414$ \| $73(4)$ \| $0.1098$ \| 2002-08-23 3F101691D98502C5 \| Elmar Hoffmann <elho@elho.net> \| $352$ \| $1$ \| $348$ \| $0.1122$ \| 2005-02-17 957952D7CF3401A9 \| Elmar Hoffmann <elho@elho.net> \| $348$ \| $311$ \| $84(5)$ \| $0.1086$ \| 2005-02-17 CE8A79D798016DC7 \| Josef Spillner <josef@coolprojects.org> \| $344$ \| $429$ \| $71(4)$ \| $0.1007$ \| 2001-05-22 89CD4B21607559E6 \| Benjamin Hill (Mako) <mako@atdot.cc> \| $325$ \| $319$ \| $70(5)$ \| $0.0801$ \| 2000-07-13 Table 2: The eleven keys with the largest number of signatures in the network, their respective in-degree $i$, out-degree $j$, average in-degree of the nearest out-neighbours $\left<j\right>_{\text{out}}$, clustering coefficient $c$, and date of creation. As with the rest of the network, most of the largest keys belong to individuals, with the exception of the first and third keys with the most signatures, which belong to entities. These entities are known as _certificate authorities_ and are created by organizations with the intent of centralizing certification. The largest authority is the community-driven CAcert.org which issues digital certificates of various kinds to the public, free of charge 555See the CAcert.org website: http://cacert.org. The second largest authority is the German magazine c’t, which initiated a PGP certification campaign in 1997 666A second, older c’t key is also still among the largest hubs, with 289 signatures. See http://www.heise.de/security/dienste/ Krypto- Kampagne-2111.html for more details. . These authorities interact with individuals in a different manner, acting as a central mediator between loosely connected peers. This is evident by the low clustering coefficient ($c\approx 0.003$), which is one order of magnitude lower than the other (human) hubs ($c\sim 0.05$ – $0.11$), and the average in-degree of their out- neighbours, which is also significantly smaller than their human counterparts ($\sim 17$ vs. $60$ – $80$, respectively). These different patterns represent distinct paradigms of trust organization: Authority vs. Community-based; each with its set of advantages and disadvantages. An authority-based scenario relies on few universally trusted vertices which mediate all trust propagation. In this way, the responsibility of key verification is concentrated heavily on these vertices, which reduces the total amount of verification necessary, and is thus more efficient. The most obvious disadvantage is that the authorities represent central points of failure: if an authority itself is not trusted, neither will be the keys it certifies. Additionally, this approach may increase the probability of forgery, since only one party needs to be deceived in order for global trust to be achieved. The complementary scenario is the community-based approach, where densely- connected clusters of vertices provide certification for each other. This obviously requires more diligence from the participants, but has the advantage of larger resilience against errors, since the multiplicity of different paths between vertices is much larger. In the PGP network both these paradigms seem to be present simultaneously, as can be observed in detail by extracting its community structure Newman and Girvan (2004). This is done by obtaining the community partition of the network which maximizes the _modularity_ $Q$ of the network, defined as $Q=\frac{1}{2E}\sum_{i\neq j}\left[A_{ij}-\frac{k_{i}k_{j}}{2E}\right]\delta(s_{i},s_{j}),$ (22) where $E$ is the total number of edges, $A_{ij}$ is the adjacency matrix of the network, $k_{i}$ is the degree of vertex $i$, $s_{i}$ is the community label of vertex $i$ and $\delta$ is the Kronecker delta. According to this definition, a partition with high values of $Q$ is possible for networks with densely-connected groups of vertices, with fewer connections between different groups. The maximum value of $Q=1$ is achieved only for "perfect" partitions of extremely segregated communities. We note that the above definition is meaningful only for _undirected_ graphs, and thus we apply it to the undirected version of PGP network, where the direction of the edges is ignored. We used the method of Reichardt et al Reichardt and Bornholdt (2006) to obtain the best partition, which resulted in modularity value of $Q\approx 0.73$. As a comparison, we computed the modularity for a shuffled version of the network, where the edges were randomly placed, but the degrees of the vertices were preserved, which resulted in the significantly smaller value $Q\approx 0.03$. The distribution of community sizes seems to have a power-law tail with exponent $\sim 2.3$ ($\sim 3.8$ for the shuffled network), characterizing a scale-free structure. By isolating the individual communities, one can clearly see strong differences between those in the vicinity of the certificate authorities and “regular” communities. In Fig. 7 is shown two representative examples of these two types of communities: On top is the community around the CAcert.org certificate authority, and is composed of $677$ keys, with an average $6.9$ signatures per key. Its degree distributions are shown on the side, from which the large discrepancy between the most central vertex and the rest of the community can be observed. The colors on the vertices correspond to the Top-Level Domain (TLD) of the email addresses associated with each key, and serve as an indication of the geographical proximity of the individuals. For the community containing CAcert.org, a high degree of geographical heterogeneity is present. This is corroborated also by the fact that there are fewer direct edges between individuals. On the bottom of Fig. 7 it is shown a community composed almost exclusively of keys with Austrian email addresses (.at TLD) which show a completely different pattern, lacking any central authority. It is smaller, with $287$ keys, but denser, with $10$ signatures per key. This pattern is repeated for most of the largest communities in the graph. Some non- centralized communities have a broader degree distribution than the Austrian community, but only those associated with certificate authorities display a centralized pattern such as in the top of Fig. 7. We now turn to the trust propagation on the PGP network. Figure 7: Two example communities of the PGP network, and their in- and out- degree distributions. The colors on the vertices correspond to the top-level domain (TLD) of the email addresses. Top: Community containing the CACert.org certificate authority. Bottom: Community composed mostly of Austrian email addresses (.at TLD). ### IV.2 Trust transitivity In order to properly investigate trust transitivity in the PGP network, it is necessary to know the direct trust values associated with each signature, which indicate the level of scrutiny in the key verification process. The OpenPGP standard Thayer et al. (2007) defines four trust “classes” for signatures, according to the degree of verification made. Unfortunately, these classes are universally ignored, and most signatures fall into the “generic” class, from which no assertion can be made. Since the actual level of verification of the keys is in fact unknown, we will investigate hypothetical situations which represent different strategies the PGP users may use to verify keys. In the last section we have shown that the network is composed of different connection patterns: community clusters and centralized trust authorities. Depending on how these connection patterns are judged more trustworthy, the values of transitive trust will be different. Here we will consider three possible scenarios: 1. Random distribution, 2\. Authority- centered trust, and 3. Community-centered trust. In all situations we will consider that all signatures have the same trust value of $c=1/2$, except for a fraction $\gamma$ of edges which have absolute trust $c=1$, which is selected as follows for each situation: 1. 1. _Random:_ The $\gamma E$ edges are chosen randomly among all $E$ edges. 2. 2. _Authority-centered:_ The $\gamma E$ edges with the largest _betweenness_ Freeman (1977) $b_{e}$ are chosen, which is defined as $b_{e}=\sum_{i\neq j}\frac{\sigma_{ij}(e)}{\sigma_{ij}},$ (23) where $\sigma_{i,j}$ is the number of shortest paths from vertex $i$ to $j$, and $\sigma_{ij}(e)$ is the number of these paths which contain the edge $e$. This distribution favours edges adjacent to nodes with high degree, and also edges which bridge different communities. 3. 3. _Community-centered:_ The $\gamma E$ edges with the largest _edge clustering_ $\tau_{e}$ are chosen, which is defined as $\tau_{e}=\frac{\sum_{i}A_{s(e),i}A_{i,t(e)}}{\sqrt{k_{s(e)}j_{t(e)}}},$ (24) where $s(e)$ and $t(e)$ are the source and target vertices of edge $e$, $A_{i,j}$ is the adjacency matrix, and $j_{i}$ and $k_{i}$ are the in- and out-degrees of vertex $i$, respectively. This quantity measures the density of out-neighbours of the $s(e)$ which are also in-neighbours of $t(e)$, and simultaneously the density of in-neighbours of $t(e)$ which are out-neighbours of $s(e)$. This distribution favours edges with belong to densely-connected communities. For instance, the edges of a _clique_ (i.e. a complete subgraph) will all have the value $\tau_{e}=1-1/(n-1)$, where $n$ is the size of the clique, which will approach the maximum value $\tau_{e}\to 1$ for a sufficiently large clique size. Figure 8: Average best trust ${\left<s\right>}$ and pervasive trust ${\left<t\right>}$, as a function of the fraction of edges with absolute trust $\gamma$, for the PGP network. The different curves correspond to the different trust distribution scenarios described in the text. Figure 9: Average best trust ${\left<s\right>}$ and pervasive trust ${\left<t\right>}$, as a function of the in-degree $j$ and the fraction of edges with absolute trust $\gamma$, for the PGP network. The different plots correspond to the different trust distribution scenarios described in the text: (a) Random distribution, (b) authority-centered distribution and (c) community-centered distribution. The plots (d) correspond to a community-centered distribution, done on a shuffled version of network, with the same degree sequence. In Fig. 8 it is shown the average best trust transitivity, Eq. 1 and average pervasive trust Eq. 6 for the PGP network, as a function of $\gamma$ according to the different approaches. We note that, due to the relatively small size of the network, no discontinuous transition is seen. The authority-centered trust leads to significantly higher values of ${\left<s\right>}$ and ${\left<t\right>}$, and the community-based distribution to the lowest values. This is expected, since distributing trust according to the edge betweenness essentially _optimizes_ trust transitivity, putting the highest values along the shortest paths between vertices. The community-centered approach does exactly the opposite, favoring intra-community connections, and results in the lowest values of average trust. Thus, favoring the hubs and authorities is clearly more _efficient_ , if the objective is solely to increase the average trust in the network. However, pure efficiency may not be what is desired, since it relies in the opinion of a much smaller set of vertices, which eases the job of dishonest parties, which need only to convince these vertices in order to be trusted by a large portion of the network. Some of these issues become more clear by observing how nodes with different degrees receive trust with each of these strategies, as show in Fig. 9. For random distribution of trust, the vertices with higher degree receive a natural bias in the values of average best in-trust, ${\left<s\right>}$, since the shortest paths leading to them tend to be smaller. But the fair nature of the definition of $t$ compensates for this, and the values of ${\left<t\right>}$ are almost independent of the in-degree of the vertices. The highly connected nodes become more trusted only with the authority-centred approach. Interestingly, in this situation the nodes with the _smallest_ degrees also receive a large value of trust, since most of them are “fringe” nodes connected only with the hubs (see Fig. 5). The vertices with intermediary degrees are thus left in the limbo, and are in effect _penalized_ for their community pattern. The almost symmetrically opposite situation is obtained with the community-centered trust distribution, where both the vertices with smallest and largest degrees receive the smallest trust values, and the intermediary nodes are judged more trustworthy due to their strong communities. We note that this effect is not due simply to the way the values of trust are distributed, but depend strongly on the existence of communities in the network. This is evident when the same trust distribution is applied to a shuffled version of the network, with the same degree sequence, as is shown in Fig. 9. For such a network, the community structure disappears, and the highly connected nodes come again in the lead. ## V Conclusion We investigated properties of trust propagation on network based on the notion of trust transitivity. We defined a trust metric, called _pervasive trust_ which provides inferred trust values for pairs of nodes, based on a network of direct trust values. The metric extends trust transitivity to the situation where multiple paths between source and target exist, by combining the best trust transitivity to the in-neighbours of a given target node, and their direct trust to the target. The trust values so-obtained are unbiased, personalized and well defined for any possible network topology. Equipped with this metric we analyzed the conditions necessary for global trust propagation in large systems, using random networks with arbitrary degree distributions as a simple model. We analytically obtained the average best trust transitivity (as well as pervasive trust) as a function of the fraction $\gamma$ of edges with _absolute_ trust $c=1$. We found that there is a specific value of $\gamma=\gamma^{}$, below which the average trust is always zero. For $\gamma\geq\gamma^{}$ the average value jumps discontinuously to a positive value. Using the defined trust metric, we investigated trust propagation in the Pretty Good Privacy (PGP) network Guardiola et al. (2002); Boguñá et al. (2004). We gave an overview of the most important topological and dynamical features of the PGP network, and identified mixed connectivity patters which are relevant for trust propagation: namely the existence of trust authorities and of densely-connected non-centralized communities. Based on these distinct patterns, we formulated different scenarios of direct trust distribution, and compared the average inferred trust which results from them. We found that an authority-centered approach, where direct trust is given preferentially to nodes which are more central, leads to a much larger average trust, but at the same time benefits nodes at the fringe of the network, which are only connected to the authority hubs, and for which no other information is available. Symmetrically, a community-centered approach, where edges belonging to densely-connected communities are favoured with more trust, results in less overall trust, but both the fringe nodes and the authorities receive significantly less trust than average. These differences are not simply due to the different ways the direct is distributed, but rather to the fact that the dense communities and the trust authorities are somewhat segregated. These differences illustrate the advantages and disadvantages of both paradigms of trust propagation, which seem to be coexist in the PGP network. It also serves as an insightful example of how dramatically the direct trust distribution can influence the inferred trust, even when the underlying topology remains the same. In this work, we have concentrated on static properties of trust propagation. However most trust-based systems are dynamic, and change according to some rules which are influenced by the trust propagation itself. One particularly good example is market dynamics Vriend (1994); Anand et al. (2009); Bornholdt and Schuster (2003) where sellers (or borrowers) do not perform well if they have a poor track record, which will be partially influenced by trust. Thus, it remains to be seen how trust transitivity can be carried over to such types of models, and what role it plays in shaping their dynamics. ## Acknowledgments We thank Alexandre Hannud Abdo for carefully reading the manuscript. This work has been supported by the DFG under Contract No. Dr300/5-1. ## References Vriend (1994) N. J. Vriend, Santa Fe Institute Working Paper 94-03-013 (1994). * Bornholdt and Schuster (2003) S. Bornholdt and H. G. Schuster, _Handbook of graphs and networks_ (Wiley-VCH Weinheim, 2003). * Anand et al. (2009) K. Anand, P. Gai, and M. Marsili, arXiv:0911.3099 (2009). * Kamvar et al. (2003) S. D. Kamvar, M. T. Schlosser, and H. Garcia-Molina, in _Proceedings of the 12th international conference on World Wide Web_ (ACM, Budapest, Hungary, 2003), pp. 640–651. * Guardiola et al. (2002) X. Guardiola, R. Guimera, A. Arenas, A. Diaz-Guilera, D. Streib, and L. A. N. Amaral, arXiv:cond-mat/0206240 (2002). * Boguñá et al. (2004) M. Boguñá, R. Pastor-Satorras, A. Díaz-Guilera, and A. Arenas, Physical Review E 70, 056122 (2004). * Newman (2010) M. Newman, _Networks: An Introduction_ (Oxford University Press, 2010). * Dijkstra (1959) E. W. Dijkstra, Numerische mathematik 1, 269–271 (1959). * Brandes and Erlebach (2005) U. Brandes and T. Erlebach, _Network Analysis: Methodological Foundations_ (Springer, 2005), 1st ed. * Jøsang et al. (2006) A. Jøsang, R. Hayward, and S. Pope, in _Proceedings of the 29th Australasian Computer Science Conference-Volume 48_ (2006), p. 94. * Walter et al. (2009) F. E. Walter, S. Battiston, and F. Schweitzer, in _Proceedings of the third ACM conference on Recommender systems - RecSys ’09_ (New York, New York, USA, 2009), p. 197. * Page et al. (1999) L. Page, S. Brin, R. Motwani, and T. Winograd, Stanford Infolab p. 17 (1999). * Newman et al. (2001) M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Physical Review E 64, 026118 (2001). * Thayer et al. (2007) R. Thayer, L. Donnerhacke, D. Shaw, H. Finney, and J. Callas, _OpenPGP message format_ , http://tools.ietf.org/html/rfc4880 (2007). * Menezes et al. (1996) A. Menezes, P. van Oorschot, and S. Vanstone, _Handbook of Applied Cryptography_ (CRC Press, 1996), 1st ed. * Brennen (2008) V. A. Brennen, _The keysigning party HOWTO_ (2008), URL http://cryptnet.net/fdp/crypto/keysigning_party/en/keysigning_party.html. * Newman (2003a) M. E. J. Newman, SIAM Review 45, 167 (2003a). * Zamora-López et al. (2008) G. Zamora-López, V. Zlatić, C. Zhou, H. Štefančić, and J. Kurths, Physical Review E 77, 016106 (2008). * Newman and Girvan (2004) M. E. J. Newman and M. Girvan, Physical Review E 69, 026113 (2004). * Newman and Park (2003) M. E. J. Newman and J. Park, Physical Review E 68, 036122 (2003). * Reichardt and Bornholdt (2006) J. Reichardt and S. Bornholdt, Phys. Rev. E 74, 016110 (2006). * Freeman (1977) L. C. Freeman, Sociometry 40, 35 (1977). * Newman (2003b) M. E. J. Newman, Phys. Rev. E 67, 026126 (2003b).	arxiv-papers	2010-12-06T22:25:27	2024-09-04T02:49:15.517631	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Oliver Richters, Tiago P. Peixoto", "submitter": "Tiago Peixoto", "url": "https://arxiv.org/abs/1012.1358" }
1012.1514	# Modified DGLAP Evolution for Fragmentation Functions in Nuclei and QGP Wei-Tian Deng Ning-Bo Chang Xin-Nian Wang Frankfurt Institute for Advanced Studies (FIAS) Ruth-Moufang-Strasse 1, D-60438 Frankfurt am Main, Germany School of Physics, Shandong University, Jinan, Shandong 250100, China Nuclear Science Division, MS 70R0319, Lawrence Berkeley National Laboratory, Berkeley, California 94720 ###### Abstract Within the framework of generalized factorization of higher-twist contributions, including modification to splitting functions of both quark and gluon, we get and numerically resolve the medium-modified DGLAP (mDGLAP) evolution equations. With Woods-Saxon nuclear geometry and Hirano 3D ideal hydrodynamic simulations of hot medium, we study the medium modified fragmentation functions (mFF) in DIS and Au+Au collisions in RHIC. Our calculation imply that the parton density in hot medium produced in RHIC is about 30 times larger than cold nucleon. ###### keywords: Modified splitting functions; Modified Fragmentation Functions; nuclear modification factor ## 1 Introduction Since jet quenching phenomenons are observed in RHIC, many phenomenological studies of it indicate a scenario of strong interaction between energetic partons and the hot medium with an extremely high initial parton density [3]. The same phenomena are also predicted in deeply inelastic scattering (DIS) off large nuclei when the struck quark propagates through the target nuclei [4]. In the presence of nuclear or hot QCD medium, the initially produced energetic partons will have to go through multiple scattering and induced gluon bremsstrahlung before hadronization. The induced gluon bremsstrahlung effectively reduces the leading parton’s energy and softens the final hadron spectra or parton fragmentation functions. To take into account multiple induced gluon emissions, one can follow the resummation of gluon bremsstrahlung in vacuum and assume that multiple medium induced bremsstrahlung can be resummed in the same way to obtain the mDGLAP evolution equations for the modified fragmentation functions [1, 2, 13], $\displaystyle\frac{\partial\tilde{D}_{q}^{h}(z_{h},\mu^{2})}{\partial\ln\mu^{2}}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}(\mu^{2})}{2\pi}\int_{z_{h}}^{1}\frac{dz}{z}\left[\tilde{\gamma}_{q\rightarrow qg}(z,\mu^{2})\tilde{D}_{q}^{h}(\frac{z_{h}}{z},\mu^{2})\right.+\left.\tilde{\gamma}_{q\rightarrow gq}(z,\mu^{2})\tilde{D}_{g}^{h}(\frac{z_{h}}{z},\mu^{2})\right],$ (1) $\displaystyle\frac{\partial\tilde{D}_{g}^{h}(z_{h},\mu^{2})}{\partial\ln\mu^{2}}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}(\mu^{2})}{2\pi}\int_{z_{h}}^{1}\frac{dz}{z}\left[\sum_{q=1}^{2n_{f}}\tilde{\gamma}_{g\rightarrow q\bar{q}}(z,\mu^{2})\tilde{D}_{q}^{h}(\frac{z_{h}}{z},\mu^{2})\right.+\left.\tilde{\gamma}_{g\rightarrow gg}(z,\mu^{2})\tilde{D}_{g}^{h}(\frac{z_{h}}{z},\mu^{2})\right],$ (2) where the modified splitting functions are given by the sum of the vacuum ones and the medium modification $\tilde{\gamma}_{a\rightarrow bc}(z,l_{T}^{2})=\gamma_{a\rightarrow bc}(z)+\Delta\gamma_{a\rightarrow bc}(z,l_{T}^{2})$, The first mDGALP evolution equation for quark in Eq. (1) is derived in Refs. [1, 2, 14] in DIS off nuclei. The modification to the splitting functions for quark $\displaystyle\Delta\gamma_{q\rightarrow qg}(z,x_{B},x_{L},l_{T}^{2})=\frac{1}{l_{T}^{2}}\left[\left(C_{A}\frac{1+z^{2}}{(1-z)_{+}}+C_{F}(1-z)(1+z^{2})\right)T_{qg}^{A}(x_{B},x_{L})+\delta(1-z)\Delta^{(q)}T_{qg}^{A}(x_{B},l_{T}^{2})\right]\times\frac{2\pi\alpha_{s}}{N_{c}f_{q}^{A}(x_{B})}$ (3) are obtained from the induced gluon bremsstrahlung spectra and therefore are related to the twist-four nuclear quark-gluon correlation distribution $T^{A}_{qg}$ which essentially describe the amplitude of the second hard parton scattering and the induced gluon radiation. The matrix element $\Delta^{(q)}T_{qg}^{A}$ in the second term in Eq.(3) comes from virtual corrections to the multiple parton scattering cross section. This term can be constructed from the momentum sum rule (or momentum conservation) for mFF [1, 2, 5], $\int dzz\Delta D_{q}^{h}(z,Q^{2})=0$. In order to get the complete coupled mDGLAP evolution equations, we also consider multiple scattering and induced gluon bremsstrahlung for a gluon jet to get the mDGLAP evolution equation for gluon in Eq. (2). From gluon-gluon scattering matrix elements, one can obtain the medium modification to the splitting functions for gluon $\Delta\gamma_{g\rightarrow gg}$ and $\Delta\gamma_{g\rightarrow q\bar{q}}$ (see details in Ref. [14]). In terms of the generalized jet transport parameter $\hat{q}$ in nuclear medium, we can express the quark-gluon correlation function $T^{A}_{qg}$ approximately as the integration over the parton’s trajectory through the medium [6, 8, 7] $\frac{2\pi\alpha_{s}}{N_{c}}\frac{T_{qg}^{A}(x_{B},x_{L})}{f_{q}^{A}(x_{B})}\approx\int dy^{-}\hat{q}_{F}(y)4\sin^{2}(x_{L}p^{+}y^{-}/2)\,.$ (5) Given the initial conditions of fragmentation functions at initial energy scale $\tilde{D}_{a}(Q_{0}^{2})$, We can numerically solve the coupled mDGLAP evolution equations using modified HOPPET [10] Fortran 95 package in LO. Such initial conditions in principle should be different in medium and vacuum. To take into account medium modification to the fragmentation functions at the initial scale $Q_{0}^{2}$, we will assume in this study [13]$\tilde{D}_{a}(Q_{0}^{2})=D_{a}(Q_{0}^{2})+\Delta D_{a}(Q_{0}^{2})$, where $D_{a}(Q_{0}^{2})$ is the vacuum fragmentation function and $\Delta D_{a}(Q_{0}^{2})$ is generated purely from medium via the mDGLAP starting at $\mu^{2}=0$. ## 2 Modified Fragmentation Function in DIS To calculate the mFF in semi-inclusive DIS off a nucleus, we employ the Woods- Saxon nuclear geometry. Considering the initial quark jet produced at $y_{0}$ that travels along a direction with impact parameter $b$, we assume that the jet transport parameter along the quark jet trajectory is proportional to the nuclear density $\hat{q}(y,b)=\hat{q}_{0}\rho_{A}(y,b)/\rho_{A}(0,0)$. If we neglect the nuclear and impact parameter dependence of the nuclear quark distribution function, the photon-nucleon cross section that produces a quark at $(y_{0},b)$ is proportional to the nuclear density distribution $\rho_{A}(y_{0},b)$. Then the averaged mFF should be $\tilde{D}(z)=\langle\tilde{D}(z,y_{0},b)\rangle=\frac{\pi}{A}\int_{0}^{\infty}db^{2}\int_{-\infty}^{\infty}dy_{0}\tilde{D}(z,y_{0},b)\rho_{A}(y_{0},b).$ (6) In order to calculate the $\tilde{D}(z,y_{0},b)$ for a quark produced at location $(y_{0},b)$, the path integral in the modified splitting functions should be replaced by the following $\int 4dy^{-}\hat{q}(y)\sin^{2}(x_{L}p^{+}y^{-}/2)=\frac{\hat{q}_{0}}{\rho_{A}(0,0)}\int_{y0}^{\infty}4dy^{-}\rho_{A}(y,b)\sin^{2}\left[\frac{l^{2}_{T}(y-y_{0})}{4q^{-}z(1-z)}\right]$ (7) Shown in Fig. (1) are the calculated nuclear modification factors of $\pi^{\pm}$, $K^{\pm}$ and $p(\bar{p})$ with different values of jet transport parameter $\hat{q}_{0}=0.015\pm 0.005$ GeV2 as compared to the HERMES experimental data [11]: $\displaystyle R_{M}^{h}(z,\nu)=\left(\frac{N^{h}(z,\nu)}{N^{e}(\nu)}\|_{A}\right)/\left(\frac{N^{h}(z,\nu)}{N^{e}(\nu)}\|_{D}\right)=\left(\frac{\Sigma e_{f}^{2}q_{f}(x)D_{f}^{h}(z)}{\Sigma e_{f}^{2}q_{f}(x)}\|_{A}\right)/\left(\frac{\Sigma e_{f}^{2}q_{f}(x)D_{f}^{h}(z)}{\Sigma e_{f}^{2}q_{f}(x)}\|_{D}\right).$ (8) Figure 1: The energy dependency of the nuclear modification factors with different values of the jet transport parameter $\hat{q}_{0}$ compared with the HERMES [11] data for $Ne$, $Kr$ and $Xe$ targets. For clear presentation the modification factors for different targets have been shifted vertically by some value($Kr$ by -0.05 and $Xe$ by -0.2). Illustrated in Fig. (1), the medium modification of the mFF gradually disappears as the initial jet energy $E$ increases. The agreement between our theory calculations and experimental data is generally good except at lower energy where hadronic absorption might become important. We can see significant different modification at HERMES for $p$ and $\bar{p}$, this may be because of the process of quark-antiquark annihilation in twist-4 double scattering and the asymmetry of $q$ and $\bar{q}$ in nuclei [14, 15], which we have not put into mDGLAP yet. As we have discussed, the initial condition for mFF in the medium at $Q_{0}^{2}=1$ GeV2 is different from which in vacuum. Therefore, most of the medium modification to the mFF come from mDGLAP evolution at low $Q^{2}$ while contribution from high $Q^{2}$ region is power-suppressed. This will lead to a very weak $Q^{2}$ dependence as shown in the left panel of Fig. 2. The calculated suppression factors are almost independent of $Q^{2}$, consistent with the experimental data [11]. If one has chosen the initial condition at $Q_{0}^{2}$ as the same as the vacuum one, one would obtain a too strong $Q^{2}$ dependence of modification factor. ## 3 Modified Fragmentation Function in Au+Au Collisions One can extend the calculation for medium modified parton fragmentation functions in DIS to hot medium [8, 16] like QGP or hot hadronic matter created in high-energy heavy-ion collisions. To take into account both the the longitudinal and transverse expansion of the hot matter we use a 3D ideal hydrodynamic simulations [18, 19] which give us information of the temprature, energy density, the fraction of the hadron phase and so on, on each step of the hot matter evolution. The jet transport parameter in hot medium at given time $\tau$ and local position $r$ is assumed to include the contribution from both the QGP phase and hadronic phase [17] $\hat{q}(\tau,\textbf{r})=\hat{q}_{0}\frac{\rho^{QGP}(\tau,\textbf{r})}{\rho^{QGP}(\tau_{0},0)}(1-f)+\hat{q}_{N}\frac{\rho_{h}(\tau,\textbf{r})}{\rho_{N}}f\,,$ (9) where f is the fraction of the hadronic phase, $\hat{q}_{0}$ is the jet transport parameter at the center of the bulk medium in the QGP phase at its formation time $\tau_{0}$. $\rho_{h}(\tau,\textbf{r})$ is the number density of the hadron resonance gas, $\rho_{N}=n_{0}\approx 0.17$ fm-3 is the nucleon density in the center of a large nucleus and $\hat{q}_{N}=0.015$ GeV${}^{2}/$fm is the jet transport parameter in cold nuclear matter we got in last section. Then we can calculate the mFF $\tilde{D}_{p}^{h}(z_{h},Q^{2},E,\textbf{r},\phi,\textbf{b})$ in hot medium, and average it over the initial parton production position and the out going directions. Assume the parton production cross section is proportional to the overlap function, we can get $\displaystyle\langle\tilde{D}_{p}^{h}(z_{h},Q^{2},E,\textbf{b})\rangle=\frac{\int d\phi d^{2}{\textbf{r}}t_{A}(\|\textbf{r}+\textbf{b}\|)t_{A}(\|\textbf{r}-\textbf{b}\|)\tilde{D}_{p}^{h}(z_{h},Q^{2},E,\textbf{r},\phi,\textbf{b})}{2\pi\int d^{2}{\textbf{r}}t_{A}(\|\textbf{r}+\textbf{b}\|)t_{A}(\|\textbf{r}-\textbf{b}\|)}.$ (10) If we neglect nuclear effect such as the shadowing effect on the initial parton distribution function, we can get the nuclear modification factor $R_{AA}$ for a fixed impact parameter b $\displaystyle R_{AA}(\textbf{b})=\frac{d\sigma_{AB}^{h}/dyd^{2}p_{T}d^{2}\textbf{b}}{T_{AA}(\textbf{b})d\sigma_{pp}^{h}/dyd^{2}p_{T}}=\frac{f^{p}(x_{1},x_{2})\otimes\mathrm{d}\sigma\otimes\langle\tilde{D}_{p}^{h}(z_{h},Q^{2},E,\textbf{b})\rangle}{f^{p}(x_{1},x_{2})\otimes\mathrm{d}\sigma\otimes D(z_{h},Q^{2})}.$ (11) Shown in the right panel of Fig. 2 is the comparison of our result about the medium modification factor for top $5\%$ centrality Au+Au collisions to the PHENIX data [20]. The jet transport parameter at the center of the bulk medium in the QGP phase at its formation time $\hat{q}_{0}=0.5\pm 0.1$ GeV${}^{2}/$fm can fit the experiment data in error bar. Figure 2: left: Comparison of the modified multiplicity ratios as a function of $Q^{2}$ at fixed value of $z$ and jet energy $E$ with the HERMES [11] data for $Ne$, $Kr$ and $Xe$ targets. For clear presentation the modification factors for different targets have been shifted vertically by some value($Kr$ by -0.05 and $Xe$ by -0.2). right: comparison of the nucleon modification factor $R_{AA}$ for $\pi^{0}$ in $0-5\%$ biased events. ## 4 Conclusions We have got the coupled mDGLAP evolution equations within the framework of generalized factorization of higher-twist contributions. By solving the mDGLAP equations numerically, we study the nuclear modified fragmentation functions both in cold nucleon and hot medium produced in RHIC. From our results, one is suggested that the gluon density in QGP is about 30 times larger than which in cold nucleon. ## Acknowledgements We would like to thank Xiao-Fang Chen and Zhe Xu for their helpful discuss. This work is supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Divisions of Nuclear Physics, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. W.-T. Deng was financialy supported by Helmholtz International Center for FAIR within the framework of the LOEWE program launched by the State of Hesse. N.-B. Chang was financialy supported by National Natural Science Foundation of China under Project Nos. 10975092. ## References * [1] X. F. Guo and X. N. Wang, Phys. Rev. Lett. 85, 3591 (2000) [arXiv:hep-ph/0005044]. * [2] X. N. Wang and X. F. Guo, Nucl. Phys. A 696, 788 (2001) [arXiv:hep-ph/0102230]. * [3] X. N. Wang, Phys. Lett. B 595, 165 (2004) [arXiv:nucl-th/0305010]. * [4] E. Wang and X.-N. Wang, Phys. Rev. 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G 34, S809 (2007). * [16] A. Majumder, C. Nonaka and S. A. Bass, Phys. Rev. C 76, 041902 (2007) [arXiv:nucl-th/0703019]. * [17] X. F. Chen, C. Greiner, E. Wang, X. N. Wang and Z. Xu, Phys. Rev. C 81, 064908 (2010) [arXiv:1002.1165 [nucl-th]]. * [18] T. Hirano, U. W. Heinz, D. Kharzeev, R. Lacey and Y. Nara, Phys. Lett. B 636, 299 (2006) [arXiv:nucl-th/0511046]. * [19] T. Hirano, U. W. Heinz, D. Kharzeev, R. Lacey and Y. Nara, Phys. Rev. C 77, 044909 (2008) [arXiv:0710.5795 [nucl-th]]. * [20] A. Adare et al. [PHENIX Collaboration], Phys. Rev. Lett. 101, 232301 (2008) [arXiv:0801.4020 [nucl-ex]].	arxiv-papers	2010-12-07T14:14:03	2024-09-04T02:49:15.530955	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Wei-Tian Deng, Ning-Bo Chang, Xin-Nian Wang", "submitter": "Wei-Tian Deng", "url": "https://arxiv.org/abs/1012.1514" }
1012.1893	# PHENIX photons and dileptons Takao Sakaguchi, for the PHENIX Collaboration Brookhaven National Laboratory, Physics Department ###### Abstract Electro-magnetic probes such as dileptons and photons are strong probes to investigate the thermodynamical state of the early stages of collisions since they leave the system unscathed. The PHENIX experiment has measured both photons and dileptons in p+p, d+Au and Au+Au collisions. An excess of dilepton yield over the expected hadronic contribution is seen in 0.2-0.8 GeV/$c^{2}$ in Au+Au collisions, which is prominent in lower $p_{T}$ and most central. Direct photons are measured through their internal conversion to electron pairs. We saw a large enhancement in Au+Au collisions over p+p yield scaled by the number of binary collisions. It turned out from the latest results on d+Au collisions that this enhancement is not explainable by a nuclear effect. ###### keywords: ## 1 Introduction Many intriguing phenomena have been observed at RHIC where a hot and dense matter is expected to be created. The large suppression of the yield of the single hadrons at high transverse momentum ($p_{T}$) suggested that the matter is so dense to stem fast partons with large $Q^{2}$ emerged from the very early stage in the medium [1]. The large elliptic flow of particles and its scaling in terms of kinetic energy of the particles suggests that the system is locally in equilibrium at as early as 0.3 fm/c. Although the hadronic probes already exhibited many interesting phenomena, since they are suffered from strong interactions in later stages to some extent, observation of more direct and penetrating probes have been desired. Electro-magnetic probes such as photons or dileptons are ideal in this sense, since they interact with medium or other particles only electro-magnetically, once produced [2]. Therefore, these probes are of interest already from the beginning of the history of the heavy ion collisions. At the leading order, the production processes of photons are annihilation ($q\bar{q}\rightarrow\gamma g$) and Compton scattering ($qg\rightarrow\gamma q$). Their yields are proportional to $\alpha\alpha_{s}$, which are $\sim$40 times lower than the ones from strong interaction. The dilepton production is mainly from annihilation of quark and anti-quark ($q\bar{q}\rightarrow\gamma^{}$), and the yield is proportional to $\alpha^{2}$. Except for some difference in production processes, photons and dileptons can be treated as same observables. A theoretical calculation tells that photons and dileptons share similar emission sources [3]. Fig 1 shows a mapping of photons and dileptons in one coordinate. Figure 1: Photons and dileptons in one coordinate with three degrees of freedom. PHENIX has measured dileptons and photons since the RHIC has started its running in 2000. In this paper, we show the most recent results on dileptons and photons in p+p, d+Au and Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV. ## 2 Dilepton analysis The PHENIX detector has a capability of measuring momentum of charged particles with high accuracy and of identifying electrons with high efficiency and strong rejection power of hadrons [4]. In measuring dileptons, we have a huge combinatorial background $e^{+}e^{-}$ arising from random combination of electrons from photons converted at beam pipe, Dalitz decays of $\pi^{0}$ or $\eta$. We constructed combinatorial background by combining electrons or positrons from different events, and subtracted them from foreground mass distributions. The subtracted distributions still include unphysical correlated pairs from back-to-back jets. We estimated the contribution using PYTHIA event generator followed by a detector simulation, and subtracted it off from the invariant mass distribution. The signal to background ratio is $\sim 10^{-2}$ in total. The resulting distribution is corrected for efficiencies of single electrons, and compared to various hadronic contributions. ## 3 Low mass and low $p_{T}$ dileptons in Au+Au collisions Fig 2(a) shows the invariant mass distribution of electron pairs measured in Au+Au collisions at $\sqrt{s_{NN}}$=200GeV [5]. The data is compared with electron pairs from various hardonic sources calculated using a Monte Carlo and filtered by the PHENIX acceptance. A large excess is seen in 0.2-0.8 GeV/$c^{2}$ in Minimum bias Au+Au collisions. In order to study this excess in detail, we divided the data set into various $p_{T}$, mass and centrality bins. Fig. 2(b) and (c) show the mass spectra in various $p_{T}$ and centrality bins, respectively. Figure 2: (a, left) Invariant mass distributions in Minimum bias Au+Au events with hadronic cocktail calculation, and the ones in (b, middle) various $p_{T}$ regions and in (c, right) centrality classes. The excess dies out at higher $p_{T}$ and lower centralities, which suggests that the excess is contributed from a thermal source. We made summary plots for the detail studies as shown in Fig. 3(a) and (b). Figure 3: (a, left) Yield and (b, right) slope parameters of dileptons in low mass and low $p_{T}$ region. For slope parameters, solid points are calculated from truncated means of the mass distributions, while open points are from exponential fits to the distributions. As we already saw in the invariant mass distributions, the yield of excess region increases as collisions become more central. The slope parameters are low for low $m_{T}-m_{0}$ and high for high $m_{T}-m_{0}$, implying that the local slopes are roughly proportional to the energy of photons emitted. This is another evidence of that they are from thermal sources. ## 4 Low mass and high $p_{T}$ dileptons in Au+Au and p+p Turning the region of interest to higher $p_{T}$, where $p_{T}>>M$, the yield of dileptons are considered to be dominated by internal conversion of real photons as depicted in Fig 4(a). Figure 4: (a, left ) Diagram of real photons and internal conversion of photons. (b, right) Fit to data with direct photon and hadronic cocktail distributions. Taking this advantage, we performed direct photon measurement through dilepton measurement [6]. The invariant mass distribution of Dalitz decay of $\pi^{0}$, $\eta$ and direct photons can be calculated using Kroll-Wada formula as shown below. $\frac{1}{N_{\gamma}}\frac{dN_{ee}}{dm_{ee}}=\frac{2\alpha}{3\pi}\sqrt{1-\frac{4m_{e}^{2}}{m_{ee}^{2}}}\left(1+\frac{2m_{e}^{2}}{m_{ee}^{2}}\right)\frac{1}{m_{ee}}\|F(m_{ee}^{2})\|^{2}\left(1-\frac{m_{ee}^{2}}{M^{2}}\right)^{3}$ Then, the real invariant mass distribution is fitted with a function of: $F=(1-r)f_{c}+rf_{d}$ where $f_{c}$ is the cocktail calculation, $f_{d}$ is the mass distribution for direct photons, and $r$ is the free parameter in the fit. Using Kroll-Wada formula, $r$ is converted to direct photon to inclusive photon ratio as follows: $r=\frac{\gamma^{}_{dir}(m_{ee}>0.15)}{\gamma^{}_{inc}(m_{ee}>0.15)}\propto\ \frac{\gamma^{}_{dir}(m_{ee}\approx 0)}{\gamma^{}_{inc}(m_{ee}\approx 0)}\ =\frac{\gamma_{dir}}{\gamma_{inc}}\equiv r_{\gamma}$ Once $r_{\gamma}$ is obtained, the invariant yield of direct photons are calculated as $\gamma_{inc}\times r_{\gamma}$. The procedure demonstrated in Fig 4(b) is for 1.0$<p_{T}<$1.5 GeV/c. The dotted lines show the contributions from various hadrons, the solid blue is the sum of these contributions, and the solid red line show the distribution by direct photons converted internally. The $r$ value is determined by the fit to the data. The error of the fit corresponds to the statistical error. We applied the procedure for various $p_{T}$s and centralities in p+p and Au+Au collisions, and obtained the $p_{T}$ spectra as shown in Fig 5(a). Figure 5: (a, left) Direct photon yields in p+p and Au+Au collisions. (b, right) Comparison of the yield in 0-20 % Au+Au collisions with various theoretical models. The result is obtained from 810 million Minimum bias Au+Au events collected in RHIC Year-4 run, and 1.5 billion minimum bias events and 270 million high $p_{T}$ single electron trigger events collected in RHIC Year-5 p+p run. The distributions are for 0-20 %, 20-40 % centrality and Minimum Bias collisions in case of Au+Au collisions. The distributions were then fitted with the p+p fit plus exponential function. We obtain slopes and dN/dy for three centralities as shown in Table 1. Table 1: dN/dy and slopes of direct photon $p_{T}$ distributions obtained from the fit to the data with the p+p fit plus exponential function. Centrality \| dN/dy ($p_{T}>$1 GeV/c) \| Slope (MeV) \| $\chi^{2}$/DOF ---\|---\|---\|--- 0-20 % \| 1.50$\pm$0.23$\pm$0.35 \| 221$\pm$19$\pm$19 \| 4.7/4 20-40 % \| 0.65$\pm$0.08$\pm$0.15 \| 217$\pm$18$\pm$16 \| 5.0/4 MinBias \| 0.49$\pm$0.05$\pm$0.11 \| 233$\pm$14$\pm$19 \| 3.2/4 There are many models in the market that have a wide range of initial temperatures and thermalization times. Fig 5(b) shows the comparison of the data with the models. Since the initial temperature and thermalization time are highly correlated, the comparison of data and models does not help constraining two parameters to definite numbers. ## 5 Low mass and high $p_{T}$ dileptons in d+Au It has been a question whether or not the excess is purely from the source that only exists in Au+Au collisions. A possible effect that increases the the yield in nucleus collisions is $k_{T}$ broadening, or so-called Cronin effect. In order to quantify the effect, we analyzed d+Au data from RHIC Year-8 run with the same procedure. Fig. 6(a) shows the invariant mass distribution in d+Au collisions for the lowest $p_{T}$ where the direct photon yield is significant. After repeating over $p_{T}$ bins, we obtained the $r_{\gamma}$ as a function of $p_{T}$. Fig 6(b) shows the $r_{\gamma}$ in various collision systems. The lines show the ratios calculated using the direct photon contributions estimated by NLO pQCD prediction scaled by the number of binary collisions ($N_{coll}$). Three lines show the predictions with three mass scales ($\mu=p_{T}/2,p_{T},2p_{T}$). The p+p ratio is very close to the NLO pQCD prediction, while Au+Au data largely deviates in low $p_{T}$ from the prediction. The d+Au data shows a moderate excess, but the excess is not as much as the one in Au+Au collisions. Figure 6: (a, left) Invariant mass distribution together with the calculation for hadronic cocktail and direct photons. The procedure to obtain $r^{{}^{\prime}}$ is described in analysis section. (b, right) Ratios of $\gamma_{dir}$ to $\gamma_{inc}$ for p+p, d+Au and Au+Au collisions. From the ratios obtained, we calculated the direct photon spectra, again following the procedure explained previously. Fig 7(a) shows the direct photon spectra for d+Au Minimum bias events, together with the real photon result from RHIC Year-3 run. Figure 7: (a, left) Direct photon yield in d+Au collisions from this analysis together with Run3 real photon result and p+p yield scaled by $N_{coll}$. (b, right) The same data with NLO pQCD calculation, instead of p+p yield. A comparison is made with direct photon spectra from p+p collisions scaled by $N_{coll}$. Fig. 7(b) shows a comparison of data with NLO pQCD calculation scaled by $N_{coll}$. In both cases, the yield in d+Au collisions are higher than the one expected from p+p yield, suggesting that the nuclear effect is seen in d+Au collisions. The ultimate interest in the series of the direct photon measurements is whether or not the excess seen in Au+Au collisions can be explained by cold nuclear effect (Cronin effect). In order to evaluate this, we compared the Au+Au yield with the d+Au yield scaled by the difference of $N_{coll}$ as shown in Fig 8(a). Figure 8: (a, left) Direct photon yields in Au+Au and d+Au collisions scaled by the difference of $N_{coll}$ in both collision systems. (b, right) Ratio of Au+Au yield to d+Au yield scaled by $N_{coll}$. As seen in the plot, the Au+Au yield is higher than the one in d+Au in $p_{T}<$2.5 GeV/c, which is close to what is expected in a literature [3]. When we divide Au+Au by $N_{coll}$-scaled d+Au data, we clearly see the existence of an additional effect in Au+Au collisions (Fig. 8(b)). Since both statistical and systematic uncertainties are still very large in d+Au measurement, it is hard to exhibit a concrete quantitative conclusion. However, in order to visually improve the result, we tried to parameterize the d+Au points by using a fit function. Fig 9(a) shows a fit to the d+Au points and Fig 9(b) shows the fit scaled by $N_{coll}$ and Au+Au points. Figure 9: (a, left) Fit to the d+Au invariant yield, and (b, right) comparison of the fit and Au+Au yield. The excess in Au+Au for $p_{T}<$2.5GeV/c still looks to be definite compared to the yield in d+Au, implying that the excess is caused by a source created in hot dense medium. ## 6 Toward better measurement of dileptons The statistical uncertainty and a part of systematic uncertainty in dilepton measurement are governed by the a background arising from random combinations of electrons from photons converted at beam pipe and Dalitz decay of $\pi^{0}$ and $\eta$. If we could tag and remove such electrons from foreground electron pair measurement, the errors would be reduced significantly. We developed a hadron blind detector (HBD) to realize this tagging. The detector is windowless Čerenkov type detector with CF4 gas in its radiator. We operate the detector in magnetic field free region in order to look for an opening angle of electron pairs. The principle of operation is following; if we see a cluster that has charge corresponding to the number of photo- electrons for single electrons, we keep the track associated with the cluster. If we see a cluster with double of the number of the photon-electrons, we reject the tract associated with the cluster, assuming that this is contributed by two electrons, which are likely by photon conversion or Dalitz decay. The detail of the detector description and its performance can be found in [7]. Fig. 10 shows the invariant mass distributions with and without using HBD information from Year-9 p+p run at $\sqrt{s}$=500GeV. Figure 10: Dilepton invariant mass spectra (a, left) without HBD matching and (b, right) with HBD matching and a charge cut. The signal to background ratios in $\phi$, $\omega$ and $\rho$ meson region are significantly improved after applying a charge cut in HBD. We are planning to also apply the charge cut in HBD in Au+Au collisions in RHIC Year-10 run data. It is essential in Au+Au collisions because the particle multiplicity is much higher than that in d+Au or p+p. ## 7 Summary Electro-magnetic probes such as dileptons and photons are strong probes to investigate the thermodynamical state of the early stages of collisions since they leave the system unscathed. The PHENIX experiment has measured both photons and dileptons in p+p, d+Au and Au+Au collisions. An excess of dilepton yield over the expected hadronic contribution is seen in 0.2-0.8 GeV/$c^{2}$ in Au+Au collisions, which is prominent in lower $p_{T}$ and most central. The invariant slopes are low for low $m_{T}-m_{0}$ and high for high $m_{T}-m_{0}$, implying that the local slopes are roughly proportional to the energy of photons emitted. Direct photons are measured through their internal conversion to electron pairs. We saw a large enhancement in Au+Au collisions over p+p yield scaled by the number of binary collisions. It turned out from the latest results on d+Au collisions that this enhancement is not explainable by a nuclear effect. ## References [1] K. Adcox et al. [PHENIX Collaboration], Nucl. Phys. A 757, 184 (2005) [arXiv:nucl-ex/0410003]. * [2] P. Stankus, Ann. Rev. Nucl. Part. Sci. 55, 517 (2005). * [3] S. Turbide, R. Rapp and C. Gale, Phys. Rev. C 69, 014903 (2004) [arXiv:hep-ph/0308085]. * [4] K. Adcox et al. [PHENIX Collaboration], Nucl. Instrum. Meth. A 499, 469 (2003). * [5] A. Adare et al. [PHENIX Collaboration], Phys. Rev. C 81, 034911 (2010) [arXiv:0912.0244 [nucl-ex]]. * [6] A. Adare et al. [PHENIX Collaboration], Phys. Rev. Lett. 104, 132301 (2010) [arXiv:0804.4168 [nucl-ex]]. * [7] M. Makek (PHENIX coll.), this proceedings.	arxiv-papers	2010-12-08T23:51:21	2024-09-04T02:49:15.543285	{ "license": "Public Domain", "authors": "Takao Sakaguchi (for the PHENIX Collaboration)", "submitter": "Takao Sakaguchi", "url": "https://arxiv.org/abs/1012.1893" }
1012.1926	# Forming Close-in Earth-like Planets via a Collision-Merger Mechanism in Late-stage Planet Formation Jianghui Ji11affiliation: Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China; jijh@pmo.ac.cn 22affiliation: Department of Astrophysics, School of Physics, University of New South Wales, NSW 2052, Australia , Sheng Jin11affiliation: Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China; jijh@pmo.ac.cn 33affiliation: Graduate School of Chinese Academy of Sciences, Beijing 100049, China; qingxiaojin@gmail.com , C. G. Tinney22affiliation: Department of Astrophysics, School of Physics, University of New South Wales, NSW 2052, Australia ###### Abstract The large number of exoplanets found to orbit their host stars in very close orbits have significantly advanced our understanding of the planetary formation process. It is now widely accepted that such short-period planets cannot have formed in situ, but rather must have migrated to their current orbits from a formation location much farther from their host star. In the late stages of planetary formation, once the gas in the proto-planetary disk has dissipated and migration has halted, gas-giants orbiting in the inner disk regions will excite planetesimals and planetary embryos, resulting in an increased rate of orbital crossings and large impacts. We present the results of dynamical simulations for planetesimal evolution in this later stage of planet formation. We find that a mechanism is revealed by which the collision- merger of planetary embryos can kick terrestrial planets directly into orbits extremely close to their parent stars. celestial mechanics – methods:numerical – planets and satellites:formation – stars:individual (OGLE-06-109L, 47 Ursae Majoris) ## 1 INTRODUCTION To date over 490 extrasolar planets have been discovered, revealing a wide diversity of planetary systems (http://exoplanet.eu). One of more unusual phenomena so revealed has been the population of “Hot Jupiters” – gas-giants found in very small orbits (periods $<$ 8d) about their parent stars – of which the prototype was the very first gas-giant exoplanet discovered, 51 Peg (Mayor & Queloz, 1995). It is believed that such short-period gas-giants cannot have formed this close to their parent stars, and so must have migrated in, or been scattered in, from a more distant formation region (Lin et al., 1996; Weidenschilling & Marzari, 1996; Ida & Lin, 2004; Chambers, 2009). The measurement precisions that make the detection of such short-period exoplanets possible have over recent years continually improved for both Doppler (e.g. Gl 876 d (Rivera et al., 2005), Gl 581 c (Udry et al., 2007), 61 Vir b (Vogt et al., 2010)) and transit (e.g. Kepler-4b (Borucki et al., 2010)) detection. What then are the possible formation mechanisms that can produce such close-in terrestrial and super-terrestrial planets? Several models have been proposed for the formation of close-in terrestrial planets. Raymond et al. (2006) have shown that super-Earths could form interior to a migrating Jovian planet. As they migrate inward, such gas-giants can shepherd planetary embryos interior to their orbits, which can then further collide and merge to generate Earth-like planets (Zhou et al., 2005). It has also been suggested that orbital migration and planet-planet scattering could potentially produce short-period super-Earths (Brunini & Cionco, 2005; Terquem & Papaloizou, 2007; Raymond et al., 2008). Whatever the mechanism for their formation, it is likely that such planets are common around at least low-mass stars (Kennedy & Kenyon, 2008). In all these scenarios, the formation of short-period Earth-like planets is associated with the migration of gas-giant planets. According to the core accretion paradigm for planetary formation, the isolation cores in the terrestrial planet formation region, and the solid cores of gas-giants, are both formed within $\sim 1$ Myr from kilometer-sized planetesimals (Safronov, 1969; Wetherill, 1980). Subsequently massive solid cores accrete disk gas to form giant planets (Kokubo & Ida, 2002; Ida & Lin, 2004) at $\sim 3-6$ Myr, before the disk disperses (Haisch et al., 2001). In the late stage of planet formation, when giant planets have ceased migration after the gas disk clears, the disk of countless planetesimals and planetary embryos will become turbulent due to stirring by gas-giants over hundreds of million years (or potentially even longer). In the meantime, it is expected that orbital crossings and giant impacts will frequently occur, which could lead to the formation of terrestrial planets (Chambers, 2001; Raymond et al., 2004; Zhang & Ji, 2009) and short-period Earth-like planets. In this Letter, we present a potential new formation mechanism for short- period Earth-like planets in the late stage of planet formation through a collision-merger scenario. In this mechanism, a planetary embryo is directly kicked to a close-in orbit after a collision with another embryo, and then the larger merged body is seized by the central star as a hot Earth-like planet. ## 2 SIMULATION SETUP Extrasolar planetary systems that harbor pairs of Jupiter-to-Saturn-mass companions are of particular interest to researchers (Gozdziewski, 2002; Ji et al., 2005; Zhang et al., 2010), e.g., OGLE-06-109L bc (Gaudi et al., 2008), 47 Uma bc (Butler & Marcy, 1996; Fischer et al., 2002), Gl 876 bc (Marcy et al., 2001). It is interesting to consider whether it is likely that such systems might host additional hot terrestrial planets (as, for example, the Gl 876 system does in the form of Gl 876 d – Rivera et al. (2005)), and further how such planets might form and evolve. We have therefore carried out simulations that explore such a system architecture. In total, 30 runs were performed using a hybrid symplectic algorithm in the MERCURY package (Chambers, 1999) for following two such systems. The initial conditions of the two systems simulated were: * Simulation 1 \- two giant planets are simulated with initial orbital parameters ($M_{P}$, $a$, $e_{p}$) $=$ (0.71 MJup, 2.3 AU, 0.001) and (0.27 MJup, 4.6 AU, 0.11), to emulate the OGLE-2006-BLG-109L system (Gaudi et al., 2008). 500 planetary embryos and planetesimals 111Herein the masses of embryos range from several lunar-mass to Mar-mass, and those of smaller ”planetesimals” have approximately a lunar mass, rather than a planetesimal mass. with total mass 10 M⊕ were distributed between 0.3 AU $<a<$ 5.2 AU and with $e<0.02$. Each of the 26 runs carried out over 400 Myr. * Simulation 2 \- two giant planets are simulated with initial orbital parameters ($M_{P}$, $a$, $e_{p}$) $=$ (2.9 MJup, 2.08 AU, 0.05) and (1.1 MJup, 3.97 AU, 0.001), to emulate the 47 Uma system (Fischer et al., 2002). 648 planetary objects with total mass of 5.14 M⊕were distributed in the region 0.3 AU $<a<$ 1.6 AU with $e<0.02$. Each of the four runs evolved over 100 Myr. The other initial orbital elements of each planetary embryo (or planetesimal) are randomly generated – the arguments of periastron, longitudes of the ascending node, and mean anomalies range from $0^{\circ}$ to $360^{\circ}$, and inclinations are from $0^{\circ}$ to $1^{\circ}$. In addition, the hybrid integrator parameters are adopted as a stepsize of $3$ days ($\sim$ a twentieth of a period for the innermost possible body at $0.3$ AU), and a Bulirsch-Stoer tolerance of $10^{-12}$. At the end of integration, the changes of energy and angular momenta are $10^{-3}$ and $10^{-11}$, respectively. In these runs, the gravitational interactions of all bodies are taken into account. Two bodies are assumed to collide whenever the distance between them is less than the sum of their physical radii (Chambers, 1999). If two objects collide, they are merged into a single body, without fragmentation, after the collision. ## 3 RESULTS ### 3.1 Simulation results In our simulations, we find that the collision-merger mechanism produces close-in terrestrial planets in 20% of the runs carried out (5 of 26 Simulation 1 runs, and 1 of 4 Simulation 2 runs). The simulations exhibit a classical planetary accretion scenario in their late stage formation (Chambers, 2001; Raymond et al., 2004). Figure 1 shows snapshots at various evolution times for a representative run of Simulation 1. Initially, the embryos and planetesimals reside in a cold disk, which is quickly stirred by the two gas-giants and excited to highly eccentric orbits within 0.1 Myr. We also see that three small bodies are involved in a 1:1 resonance with the inner giant by that time. By the end of 1 Myr, most of the initial objects have been removed by ejection or collision due to frequent orbital crossings in this chaotic stage. In addition, we see that a close-in planet has formed at $\sim$ 1 Myr which subsequently remains very stable. At the conclusion of the run (400 Myr), three planetesimals survive, of which one has been seized as a Trojan body by the inner giant, and the other two move at $\sim 1$ AU in eccentric orbits. Figure 1: A snapshot of planet formation in the late stage for Simulation 1. The panels show the orbital eccentricity versus semi-major axis for each surviving body at simulation times of 0, 0.1, 1.0, 10, 50 & 400 Myr. The radii and the color of the embryos and planetesimals are related to their mass, with radius proportional to $m^{1/3}$. The two giants are, respectively, at 2.3 and 4.6 AU. A close-in terrestrial planet forms at $\sim$ 1 Myr and it remains stable over secular evolution. Figure 2 shows the time evolution of the mass, semi-major axis, and eccentricity of the short-period terrestrial planet formed in the Simulation 1 run shown in Fig. 1. At 0.0356 Myr, two bodies that may be excited by secular resonance of gas-giants, collide at very high eccentricities ($e$ = 0.91 and 0.80, shown by the red and black lines in Fig. 2, respectively) and are then assumed to merge into a single planetary embryo. That merged body (the remaining black line in Fig. 2) is captured by the parent star as a short- period planet, and its orbit dramatically shrinks from $\sim 0.4$ AU at the time of the collision, down to 0.077 AU. Subsequently, three additional collisions take place over the further late-stage evolution of that merged object. Fig. 2 shows that the embryo moves slightly inward at each collision, and that its mass also increases. Moreover, we note that it finally becomes a 3.3 Mercury-mass planet with a close-in orbit about 0.056 AU, and its eccentricity drops down to $e$=0.13 after the last collision. The orbit may then, of course, be further circularized by tidal interaction with the star over even longer timescales. Figure 2: Mass, semi-major axis and eccentricity evolution of the short-period terrestrial planet the emerges from the Simulation 1 run shown in Fig. 1. The black and red lines in the lower two panels show the semi-major axis and eccentricity evolution of the two bodies that collide to form a merged planetary embryo, which is kicked from 0.4 AU to 0.077 AU at 0.0356 Myr. Subsequently that merged embryo (shown as a single black line after 0.0356 Myr) is subject to further collision-mergers, with the epoch of each collision shown by the solid triangles. The resultant mass evolution of this body is shown in the upper panel. Figure 3: Mass, semi-major axis and eccentricity evolution of a short-period terrestrial planet that emerges from a run of Simulation 2 – layout is the same as for Fig. 2. In this case the first collision-merger event occurs at 2.2 Myr, and the embryo is thrown from its location of $\sim 0.8$ AU at the time of the collision to 0.06 AU. Figure 3 shows the formation and evolution of a similar terrestrial planet that emerges in one of the runs for Simulation 2. At 2.2 Myr, the semi-major axis of a $\sim 0.9$ Mercury-mass embryo drops down from $\sim 0.8$ AU to 0.06 AU as a result of a collision with a highly-eccentric planetesimal excited more than a million years earlier. The merged body has an eccentricity that drops from 0.90 to 0.50 immediately after the impact. The enlarged, merged body subsequently undergoes additional collisions, and its eccentricity further evolves to $e$=0.33 (after its last collision) with a final mass of 1.3 Mercury-mass. Here, the collision-merger scenario may provide some clues of the origins of the moderate eccentricities seen in super-Earths detected to date (e.g., HD 181433 b (Bouchy et al., 2009)). The major difference in the evolution of these two examples is that the short- period planet that evolved in Simulation 1 was moved into an inner orbit at a very early stage, and subsequently accreted a majority of the mass available in nearby orbits; while the Simulation 2 planet had almost completed accretion into a Mercury-mass embryo before it moved closer to the star. In all simulations, we notice that terrestrial planets and bodies formed at short periods via collision-merger events come into being within 10-30 Myr, which agrees with the estimated timescale of terrestrial core formation (Yin et al., 2002), as derived from the chronometry of meteorites and numerical simulations of terrestrial planet formation (Chambers, 2001; Raymond et al., 2004; Zhang & Ji, 2009). In addition, we find that all survivors remain stable in their final configurations. These results indicate that a collision-merger mechanism could indeed produce short-period, terrestrial planets in two systems that host two gas-giants. Similarly, we also find the above outcomes in other 4 runs. However, a natural question then arises – do the bodies that take part in these collisions really merge? Or will they become fragmented? ### 3.2 Merger versus fragmentation In the accretion model of MERCURY, a collision-merger scenario occurs whenever the distance between two bodies is less than the sum of their physical radii (Chambers, 1999), and the package models the two bodies merging inelastically to form a single new body that conserves mass and total momentum. The collisions in the runs, therefore, are considered to be perfect gravitational aggregations, which assumes that enough energy is dissipated in the collision for the two bodies to remain gravitationally bound. However, actual collisions could have a result that ranges anywhere from this result (complete merger), through partial fragmentation, to the complete shattering and disintegration of both impactors. (Wetherill & Stewart, 1993). Whether these bodies either fragment or cohere in a collision will obviously depend on the – currently poorly understood – physical properties of the colliding bodies (Wetherill, 1980). What can be said is that the outcome will be extremely complicated. To assess the likely state of the merger vs fragmentation for two bodies in a collision, we can, though, make order-of- magnitude estimates. Consider two bodies of the same mass $m$, with relative velocity at infinity $\sigma$, and the sum of the physical radii $R_{s}$. The collision velocity for a head-on collision between them is (Safronov, 1969; Wetherill, 1980; Armitage, 2007), $v_{c}=(\sigma^{2}+v^{2}_{esc})^{1/2}$ (1) where $v_{esc}$=$\sqrt{4Gm/R_{s}}$ is the escape velocity at the point of collision, a parameter used to evaluate whether they will physically collide. Take the coefficient of restitution as $\epsilon$, then accretion will result if $\epsilon v_{c}<v_{esc}$, even if the initial impact results in fragmentation into two bodies. Conversely, the bodies will be unbound if $\epsilon v_{c}>v_{esc}$. Thus, the threshold value of the coefficient of restitution for these outcomes is (Armitage, 2007), $\epsilon=\bigg{(}1+\frac{\sigma^{2}}{v^{2}_{esc}}\bigg{)}^{-1/2}$ (2) This shows that if $\sigma\ll v_{esc}$, merger and growth is likely unless collision is totally elastic; whereas $\sigma\gg v_{esc}$ leads to fragmentation. For the Simulation 1 run shown in Fig. 2 , we have used the above equations to assess the outcome of the first collision as it happens between two identical Mercury-like embryos, which allows us to make a rough evaluation of the likely outcome by calculating the instantaneous velocities of the impactors at the epoch just before the collision. Now we notice that at the first collision the body was impacted onto a close-in orbit. The $v_{esc}$ of the two impactors are nearly the same – 3.12 km s-1 (assuming equal bulk density); the velocities of the impactors at the collision epoch near the pericenter are estimated to be 43.21 km s-1 and 35.50 km s-1, respectively, thus we have an approximate relative velocity projected to the relative position of two colliding bodies of 13.49 km s-1. In this case, a merger requires $\epsilon\leq 0.23$, which is close to the accretion condition ($\epsilon\leq 0.34$) in realistic accretion model for head-on collision (Kokubo & Genda, 2010). On the basis of above analysis, a merger seem to be possible for two eccentric objects when the collision occurs in the nearby region of central star, subsequently the merged body is seized by the star at close-in orbit. In the collision-merger process, moderate energy should be released, and they could be converted into the internal heat of the merger in the collision between embryos, e.g., simulations of a supposed Moon-forming impact show that the collision can deliver prodigious energy to the Earth, which could lead the proto-Earth to a mixed solid-melt state (Canup, 2008).222At the very time before/after the collision, the fractional energy change due to integrator was about 9 part in $10^{4}$. Additional energy loss may arise from the ejection of other embryos or transfer to the envelope and core of giant planets (Li et al., 2010). We also obtain similar results for the Simulation 2 run shown in Fig. 3. In addition, Leinhardt & Richardson (2002) showed that a large mass ratio between two impactors will tend to lead to merger and aggregation – the accretion probability is $\sim 60$% (averaged over all impact parameters) for average mass ratio of 1:5. This suggests that the first collision seen in this run, where the mass ratio of 1:3.43, is likely to result in a merger. ## 4 DISCUSSION and CONCLUSION We have uncovered a new mechanism for producing short-period terrestrial planets via collisions-mergers in the late stages of planetary formation. In this mechanism, two highly-eccentric bodies first undergo a severe orbital crossing and then form a short-period planet via collision-merger. In the set of simulations performed to date, this mechanism produces a short-period, terrestrial planet in 20$\%$ of runs. As mentioned previously, the formation rate for short-period terrestrial planets via a collision-merger process is only a moderate 20%. However, this low rate may be a result of the limits imposed on our simulations by current computational capabilities, which restrict our adopted population of embryos and planetesimals to a few hundred objects with a total mass of only several times that of the Earth. The resultant planetesimal disk in our simulations is much smaller than that of the Minimum Mass Solar Nebula ($\sim 0.01$ $M_{\odot}$ within 30 AU (Weidenschilling, 1977; Hayashi, 1981)) – which would also contain billions of small bodies. Increasing the number of bodies and the total mass of the proto-planetary disk would likely increase the efficiency with which this mechanism produces short-period terrestrial planets. In addition, it is worth noting that close-in planets emerge from our simulations within a few million years. This is a significantly shorter timescale than the billion years over which the Solar System is thought to have undergone significant evolution. So, while near-infrared observations of young cluster samples, indicate an overall dust disk lifetime of $\sim 6$ Myr (Haisch et al., 2001), the planetary system will actually continue to evolve over much longer timescales following the clearing of the dust and gas disk. During this late stage of planetary formation, frequent orbital crossings and huge impacts will occur, which are likely to significantly boost the feasibility of collision-merger events producing short-period terrestrial bodies. The collision-merger scenario for the formation of short-period planets does not require perfect accretion. Rather it relies on the collisions pushing the resultant body inward, so that the central star can capture it as a short- period planet. In this sense, such a mechanism could play a key role in throwing the largest fragments resulting from severe impacts into short-period orbits. On the other hand, given the diversity in the architectures of currently known systems, exoplanets are likely to form through a variety of mechanisms rather than through a uniform dominant process (D. Lin 2009, private communication). Our simulation results show one potential mechanism for the origin of short-period terrestrial planets in a compact disk with two gas-giants, and may predict an abundance of close-in bodies for this family. We thank the anonymous referee for useful comments and suggestions that helped to improve the contents. We also thank J.E. Chambers, D.N.C. Lin and E. Kokubo for informative discussions and insightful comments. J.H.J. is very grateful to the Australian Academy of Sciences for the support of his stay at UNSW, and to Chris Tinney and UNSW Astrophysics Department for their hospitality. This work is financially supported by the National Natural Science Foundation of China (Grants 10973044, 10833001, 10573040, 10673006, 10233020), the Natural Science Foundation of Jiangsu Province, and the Foundation of Minor Planets of Purple Mountain Observatory. ## References * Armitage (2007) Armitage, P. 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N., et al. 2006, Science, 313, 1413 * Raymond et al. (2008) Raymond, S. N., et al. 2008, MNRAS, 384, 663 * Rivera et al. (2005) Rivera, E. J., et al. 2005, ApJ, 634, 625 * Safronov (1969) Safronov,V.S. 1969, Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets (Moscow:Nauka) * Terquem & Papaloizou (2007) Terquem, C., & Papaloizou, J. C. B. 2007, ApJ, 654,1110 * Udry et al. (2007) Udry, S., et al. 2007, A&A, 469, L43 * Vogt et al. (2010) Vogt, S. S., et al. 2010, ApJ, 708, 1366 * Weidenschilling (1977) Weidenschilling, S. J. 1977, MNRAS, 180,57 * Weidenschilling & Marzari (1996) Weidenschilling, S. J., & Marzari, F. 1996, Nature, 384,619 * Wetherill (1980) Wetherill, G. W. 1980, ARA&A, 18, 77 * Wetherill & Stewart (1993) Wetherill, G. W., & Stewart, G.R. 1993, Icarus, 106, 190 * Yin et al. (2002) Yin, Q., et al. 2002, Nature, 418, 949 * Zhang & Ji (2009) Zhang, N., & Ji, J. H. 2009, Science in China Series G , 52(5), 794 * Zhang et al. (2010) Zhang, N., Ji, J. H., & Sun, Z. 2010, MNRAS, 405, 2016 * Zhou et al. (2005) Zhou, J.-L., Aarseth, S. J., et al. 2005, ApJ, 631,L85	arxiv-papers	2010-12-09T06:09:27	2024-09-04T02:49:15.549269	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jianghui Ji (1,2), Sheng Jin (1,3), C. G. Tinney (2) ((1) Purple\n Mountain Observatory, CAS (2) Department of Astrophysics, School of Physics,\n University of New South Wales, Australia (3) Graduate School of Chinese\n Academy of Sciences)", "submitter": "Jianghui Ji", "url": "https://arxiv.org/abs/1012.1926" }
1012.2172	# Higgs searches at the Tevatron Fermilab E-mail on behalf of the CDF and D0 collaborations ###### Abstract: We present combined CDF and D0 searches for the Standard Model (SM) and Minimal Supersymmetric Standard Model (MSSM) Higgs boson using up to 6.7 fb-1 of integrated luminosity from 1.96 TeV proton-antiproton collisions at the Tevatron. Specialized searches for Higgs bosons produced via gluon fusion, associated boson production, and vector boson fusion, decaying to $b\bar{b}$, $W^{+}W^{-}$, $\tau^{+}\tau^{-}$, and $\gamma\gamma$ are combined to produce 95% CL upper limits on SM Higgs production as a function of mass. Current Tevatron limits are shown, including a new exclusion for SM Higgs masses between 158 and 175 GeV$/c^{2}$. We also present prospects for future sensitivity. ## 1 Introduction The discovery of the Higgs boson will resolve the longstanding question of how the electroweak symmetry is broken, and how fermions and bosons acquire mass. The standard model of particle physics hypothesizes that a non-zero scalar field with four degrees of freedom permeates the universe, such that the $W^{+}$, $W^{-}$, and $Z$ boson gain mass through three of these degrees of freedom, while the fourth degree of freedom becomes a new scalar boson. The BEHHGK (pronounced ”beck”) mechanism and boson, named after the authors (Breit, Englert, Higgs, Hagen, Guralnik, Kibble), was proposed in 1964 [1], and is often referred to as the Higgs mechanism and boson. Finding the Higgs boson confirms that the Higgs field exists, and is the subject of an intense search at the Tevatron collider. Relationships between measurable electroweak parameters within the standard model, in conjunction with direct searches [3], constrain the Higgs boson mass to be between 114 GeV111c is set to 1 in this proceeding and 185 GeV at the 95% CL [2]. With enough data, this entire mass range is accessible to the 1.96 TeV center of mass Tevatron proton-antiproton collider, for either observation or exclusion of the standard model Higgs boson. The two multipurpose detectors at the Tevatron, CDF and D0, are able to reconstruct all of the final state particles and topologies resulting from SM Higgs boson production and decay. The Tevatron has delivered 9 fb-1 of luminosity each to CDF and D0. Data collection efficiencies were 85 - 90 % for this data, and an integrated luminosity of up to 6.7 fb-1 have been analyzed for the Higgs boson searches covered in these proceedings. The 350 nb-1 delivered to the 7 TeV proton- proton LHC is not sufficient for Higgs searches, meaning the search for the Higgs boson is currently unique to the Tevatron. The CDF and D0 experiments at the Tevatron have been producing combined Higgs searches since 2006. Each iteration, efforts are made to improve signal acceptance, such as by loosening lepton and $b$ hadron identification requirements, adding backup triggers which select events online using different aspects of the signal signature, and relaxing kinematic selection. As the signal acceptance is improved, the backgrounds increase and become more difficult to model, and are separated into categories with similar S $/\sqrt{B}$, such that high S $/\sqrt{B}$ categories provide the most signal sensitivity, whereas low S $/\sqrt{B}$ categories provide the best background constraints. The primary Higgs channels of $H\to W^{+}W^{-}$, $WH\to l\nu b\bar{b}$, $ZH\to\nu\nu b\bar{b}$, and $ZH\to l^{+}l^{-}b\bar{b}$ from CDF and D0 correspond to about 500 Higgs bosons produced at each mass, 114 $<m_{H}<$ 185 GeV. ## 2 Low mass SM Higgs searches Searches for a ”low mass” (m${}_{H}<$ 135 GeV) standard model Higgs boson are presented elsewhere in these conference proceedings for $H\to b\bar{b}$ [4], $H\to\tau^{+}\tau^{-}$ [5], and $H\to\gamma\gamma$ [6]. Here we report on the features of the most sensitive low mass Higgs search channels. A low mass Higgs boson preferentially decays $H\to b\bar{b}$. It is most easily identified in events produced via associated Higgs production, $WH$ and $ZH$, when the $W$ and $Z$ decay leptonically, into final states of $WH\to l\nu b\bar{b}$, $ZH\to\nu\nu b\bar{b}$, and $ZH\to l^{+}l^{-}b\bar{b}$ where $\ell=e$ or $\mu$. In events with a reconstructed $W$ or $Z$ boson and two or more additional jets, the di-jet invariant mass is used to search for a resonance originating from $H\to b\bar{b}$. To reduce the background from $W$ and $Z$ production in association with jets, $b$-jets are ”tagged” by identifying a secondary vertex or tracks with high impact parameter from $B$ hadron decay. The dijet mass is shown before and after one and two $b$-tags for $WH\to\ell\nu b\bar{b}$ candidates in Figure 1. Expected sensitivity is improved by categorizing events according to the number of $b$-tags and the purity of the applied $b$-tagging algorithms. Figure 1: The dijet mass for $WH\to\ell\nu b\bar{b}$ candidates with no $b$-tags, one $b$-tag, and two $b$-tags (left to right), demonstrating how the $W$+jets background is reduced while $WH$ signal, $W+b\bar{b}/c\bar{c}$, and $t\bar{t}$ heavy flavor backgrounds become more prominent. ## 3 High mass SM Higgs searches Searches for a ”high mass” Higgs boson ($m_{H}>$ 135 GeV) are reported in detail elsewhere in these proceedings for CDF and D0 [7]. The most sensitive mode at the Tevatron is $gg\to H\to WW\to\ell\nu\ell\nu$ due to the high cross section and well identified final state. The high mass analysis benefits by separating events into categories according to the number of jets and the number of leptons because of the difference in the kinematics of the signal production mechanisms and primary background processes. The signal cross section and uncertainties for the dominant signal process $gg\to H$ use state of the art NNLL and NNLO calculations as explained in Ref. [12]. Since there are neutrinos in the final state, the invariant mass of the Higgs boson cannot be reconstructed. The best variable for distinguishing $H\to WW$ from the background is the separation between the charged leptons, dR $=\sqrt{d\eta^{2}+d\phi^{2}}$, which is different for spin-zero $H\to WW$ decay than for spin-one $Z\to WW$ decay as demonstrated in Figure 2. Also shown is the equivalent plot for same sign leptons, one of the control regions used to test the background modeling. Figure 2: The $\Delta$R for opposite sign (OS) leptons with no accompanying jets (left) and for the same-sign-lepton control region (right). ## 4 Validation of multivariate analysis techniques Multivariate techniques are used in all the Tevatron SM Higgs analyses. The main algorithms used are Neural Networks, Matrix Element probabilities, and Boosted Decision Trees. By taking advantage of the different correlations for signal and background among the multiple input variables, these techniques typically improve analyses by about 20% with respect to simply fitting the leading two kinematically distinct variables. To help validate these multivariate techniques we use them to measure higher statistics standard model processes. Shown in Figure 3 are two measurements using multivariate techniques. We measure $\sigma_{WW+WZ}=$ 16.6${}^{+3.5}_{-3.0}$ pb in a $W$+2-jet final state compared to the SM expected cross section of 15.1 $\pm$ 0.8 pb [8], and also verify the $W$+jets background process rate and shape by analyzing a $WH\to\ell\nu b\bar{b}$ sample of events before the $b$-tagging requirements. Figure 3: A matrix element event probability discriminant from a diboson search for $WW/WZ\to\ell\nu jj$ (left) and a random forest (RF) of boosted decision trees tested in the pre b-tagged WH control region (right). ## 5 Combined SM Higgs search The CDF combined search and D0 combined search for the SM Higgs boson was presented elsewhere in these proceedings [9, 10] and the final results are shown in Figure 4. The 95% CL upper limit on the cross section divided by the SM value of the cross section is shown on the y-axis and is equal to one when standard model sensitivity is achieved. The dotted line is the expected upper limit on cross section in pseudo-experiments with no signal events, allowing systematic uncertainties to be fit within each pseudo-experiment, and fitting for the maximum signal that can be accommodated at the 95% CL. The green and yellow bands represent the one and two $\sigma$ variations of the expected limits. The solid line is the upper limit observed in actual data. CDF achieves expected sensitivity to the Higgs boson for m${}_{H}=$ 165 GeV, while D0 almost achieves observed sensitivity for m${}_{H}=$ 165 GeV. CDF is able to exclude 100 ¡ mH ¡ 102 GeV. CDF and D0 perform a joint search for the Higgs boson where shared systematic uncertainties are kept correlated (bottom of Fig. 4). The observed exclusion is 158 $<m_{H}<$ 175 GeV which is consistent with the expected exclusion of 156 $<m_{H}<$ 175 GeV. The combined search is also able to exclude Higgs masses below 109 GeV. At 115 GeV, just above the LEP limit, the expected (observed) exclusion is 1.45SM (1.56SM). For $m_{H}=$ 165 GeV, where the high mass exclusion is strongest, $H\to W^{+}W^{-}$ dominates, but for $m_{H}=$ 115 GeV, the sensitivity comes from a combination of analyses (see Table 1). We can collectively view the dozens of discriminant outputs from the multiple search channels by gathering all histogram bins from all sub-channels, summing up those bins with the same $S/B$ into the same bins, and then sorting the bins from lowest to highest $S/B$. The distribution is shown for m${}_{H}=$ 165 GeV and m${}_{H}=$ 115 GeV in Figure 5. For 165 GeV, the data clearly prefer the background-only model, while for m${}_{H}=$ 115 GeV an excess of events in bins with very high $S/B$ and a deficit of events with slightly lower $S/B$ result in the observed limit being approximately equal to the expected limit. Figure 4: SM Higgs combinations for CDF (left), D0 (right), and the Tevatron (bottom). Table 1: Tevatron analyses ordered by the expected upper limit at 95% CL for a Higgs boson search mass of 115 GeV. Analysis \| Experiment \| Expected \| Integrated ---\|---\|---\|--- channel \| \| limit $\sigma$/$\sigma_{SM}$ @ 115 GeV \| Luminosity (fb-1) $WH\to\ell\nu b\bar{b}$ \| CDF \| 3.4 \| 5.7 $ZH/WH\to\not\\!\\!E_{T}b\bar{b}$ \| CDF \| 4.0 \| 5.7 $ZH/WH\to\not\\!\\!E_{T}b\bar{b}$ \| D0 \| 4.2 \| 6.4 $WH\to\ell\nu b\bar{b}$ \| D0 \| 4.8 \| 5.3 $ZH\to\ell\ell b\bar{b}$ \| CDF \| 5.5 \| 5.7 $ZH\to\ell\ell b\bar{b}$ \| D0 \| 5.7 \| 6.2 $H\to WW$ \| CDF \| 10.6 \| 5.9 $H\to WW$ \| D0 \| 12 \| 6.7 $H\to\tau\tau$ \| D0 \| 16 \| 4.9 $ZH/WH\to qqb\bar{b}$ \| CDF \| 18 \| 4 $H\to\gamma\gamma$ \| D0 \| 18.5 \| 4.9 $H\to\gamma\gamma$ \| CDF \| 21 \| 5.4 $H\to\tau\tau$ \| CDF \| 25 \| 2.3 $ttH$ \| D0 \| 45 \| 2.1 Total \| CDF + D0 \| 1.45 \| 5.8 Figure 5: On left, for m${}_{H}=$ 165 GeV, the data from all search channels sorted into bins of varying $S/B$. On right, same for m${}_{H}=$ 115 GeV . ## 6 Beyond SM Higgs searches Beyond standard model Higgs bosons searches are presented in detail elsewhere in these conference proceedings [11]. The MSSM predicts enhanced Higgs production cross sections and decays to down-type fermions for high values of tan $\beta$, the ratio of the vacuum expectation value for up-type and down- type fermions, thereby enhanacing rates for $H\to b\bar{b}$ and $H\to\tau\tau$ final states. The Tevatron has searched for such signatures primarily using the invariant mass of the $b$-jets or the invariant mass of the visible $\tau$ decay products as discriminants and set limits close to the theoretically interesting value of tan $\beta=$ 30, which is approximately the ratio of mtop to mb. A small excess of 2 $\sigma$ (including a trials factor) is seen in the CDF $bb$ search (Fig. 6). Figure 6: Search for the MSSM in $H+b\to bb+b$ search channel at CDF shown with best fit to signal plus background hypothesis. ## 7 Outlook Figure 7 shows the expected a priori signal sensitivity as a function of mass and analyzed integrated luminosity per experiment. This includes a factor of 1.5 improvement in the expected limit based on a range of improvements from lepton identification efficiency, b-tagging efficiency, triggering, and jet energy resolution. The full dataset for Run II of the Tevatron will be collected by September 2011, and is expected to be about 10 fb-1 of analyzed luminosity per experiment. The projected sensitivity with this dataset is 3 $\sigma$ for m${}_{H}=$ 115 GeV. The sensitivity would be expected to be at least 2.4 $\sigma$ across the full Higgs boson mass range of 100 to 185 GeV. The possibility of running the Tevatron for another 3 years would allow for about 17 fb-1 of data to be analyzed per experiment. This would increase the expected sensitivity to 3 $\sigma$ across the full mass range, 4 $\sigma$ at 115 GeV. Figure 7: Projections of expected Higgs sensitivity, assuming a factor of 1.5 improvement in the expected limits at each mass point, and assuming D0 and CDF have equal sensitivity and equal analyzed integrated luminosity. Note that much of this factor of 1.5 has already been achieved at high mass. ## 8 Summary The CDF and D0 experiments at the Tevatron have a comprehensive and aggressive search program for the Higgs boson which can be divided into three main categories : high mass, low mass, and beyond standard model searches. The newest high mass combination from the Tevatron excludes 158 $<m_{H}<$ 175 GeV using a dataset of up to 6.7 fb-1. The low mass primary channels achieved 1.45SM (1.56SM) expected (observed) exclusion for $m_{H}=$ 115 GeV and a 95% CL exclusion for masses between 100 GeV and 109 GeV. Tevatron MSSM Higgs searches are setting limits at theoretically motivated values of tan $\beta$, and a $H\to b\bar{b}+b$ search with 2.2 fb${-1}$ has a 2 $\sigma$ excess at m${}_{H}=$ 140 GeV. The Tevatron is scheduled to deliver up to 12 fb-1 by its conclusion in September 2010, providing about 10 fb${-1}$ of analyzable data for Higgs analysis. With this dataset, a full combination of Higgs searches will have at least 2.4 $\sigma$ level sensitivity for 100 $<m_{H}<$ 185 GeV. This would increase to 3 $\sigma$ were the Tevatron to run an additional three additional years as is the Tevatron Run III proposal. ## References * [1] F. Englert and R. Brout, Broken symmetry and the mass of gauge vector mesons, Phys. Rev. Lett. 13 321 (1964); P. W. Higgs, Broken Symmetries and the mases of gauge bosons, Phys. Rev. Lett. 13 508 (1964); G. S. Guralnik, C. R. Hagen and T. W. B. Kibble, Global Conservation Laws and Massless particles, Phys. Rev. Lett. 13, 585 (1964). * [2] J. Alcaraz, Precision Electroweak Measurements and Constraints on the Standard Model, arXiv:0911.2604 [hep-ex]. * [3] R. Barate et al. [LEP Working Group for Higgs boson searches], Search for the standard model Higgs boson at LEP, Phys. Lett. B 565, 61 (2003) * [4] Y. Nagai, Standard Model low mass Higgs searches at CDF, PoS (ICHEP 2010) 067; Y. Enari, Standard Model low mass Higgs searches at D0, Pos (ICHEP 2010) 053. * [5] P. Totaro, Search for Standard Model Higgs boson in di-tau final state at the Tevatron, PoS (ICHEP 2010) 080. * [6] K. Peters, Search for Standard Model Higgs boson in gamma gamma final state at the Tevatron, PoS (ICHEP 2010) 071. * [7] D. Lucchesi, Standard Model high mass Higgs search at CDF, PoS (ICHEP 2010) 081; B. Tuchming, Standard Model high mass Higgs search at D0, PoS (ICHEP 2010) 061. * [8] T. Aaltonen et al. [CDF Collaboration], Measurement of the WW+WZ Production Cross Section Using the Lepton+Jets Final State at CDF II, Phys. Rev. Lett. 104, 101801 (2010). * [9] K. Potamianos, Combination of Standard Model Higgs searches at CDF, Pos (ICHEP 2010) 073; Conference note http://www-cdf.fnal.gov/physics/new/hdg//Results_files/results/cdfcombichep2010/ . * [10] M. Mulhearn, Combination of Standard Model Higgs searches at D0, Pos (ICHEP 2010) 066; Conference note http://www-d0.fnal.gov/Run2Physics/WWW/results/prelim/HIGGS/H96 . * [11] A. Patwa, Beyond Standard Model Higgs boson searches at the Tevatron, Pos (ICHEP 2010) 070. * [12] T. Aaltonen et al., [CDF and D0 Collaborations], Combined CDF and D0 Upper Limits on Standard Model Higgs-Boson Production with up to 6.7 fb-1 of Data, FERMILAB-CONF-10-257-E, arXiv:1007.4587 [hep-ex].	arxiv-papers	2010-12-10T03:53:08	2024-09-04T02:49:15.560371	{ "license": "Public Domain", "authors": "Ben Kilminster", "submitter": "Benjamin Kilminster", "url": "https://arxiv.org/abs/1012.2172" }
1012.2267	# PHOTON MASS AND VERY LONG BASELINE INTERFEROMETRY111This article is an extended version of the essay ‘COMBINING GENERAL RELATIVITY, MASSIVE QED AND VERY LONG BASELINE INTERFEROMETRY TO GRAVITATIONALLY CONSTRAIN THE PHOTON MASS’ (A. Accioly, J. Helayël-Neto and E. Scatena, Phys. Lett. A 374 (2010) 3806) which was awarded an “honorable mention” in the 2010 Essay Competition of the Gravity Research Foundation. ANTONIO ACCIOLY${}^{\dagger,\;\star,\;\amalg}$ JOSÉ HELAYËL- NETO${}^{\dagger,\;\diamond}$ and ESLLEY SCATENA${}^{\star,\;\ddagger}$ †Laboratório de Física Experimental (LAFEX), Centro Brasileiro de Pesquisas Físicas (CBPF), Rua Dr. Xavier Sigaud 150, Urca, 22290-180, Rio de Janeiro, RJ, Brazil ⋆Instituto de Física Teórica (IFT), São Paulo State University (UNESP), R. Dr. Bento Teobaldo Ferraz 271, Bl. II - Barra Funda, 01140-070, São Paulo, SP, Brazil ∐accioly@cbpf.br ⋄helayel@cbpf.br ‡scatena@ift.unesp.br (Day Month Year; Day Month Year) ###### Abstract A relation between the photon mass, its frequency, $\nu$, and the deflection parameter, $\gamma$, determined by experimentalists (which characterizes the contribution of space curvature to gravitational deflection) is found. This amazing result allows us to conclude that the knowledge of the parameters $\nu$ and $\gamma$ is all we need to set up gravitational bounds on the photon mass. By considering as inputs the most recent measurements of the solar gravitational deflection of radio waves obtained via the Very Long Baseline Interferometry, upper bounds on the photon mass are estimated. ###### keywords: Gravitational deflection; massive QED; photon mass, Very Long Baseline Interferometry. Managing Editor ## 1 Introduction As a consequence of the substantial improvement in the interferometry techniques that took place over the last few decades, a new light was shed on the physical phenomena in which gravitational and quantum effects are interwoven. Consider, for instance, the quantum mechanical phase shift of neutrons caused by their interaction with Earth’s gravitational field which was observed in the Colella-Overhauser-Werner[1] and Bonse-Wrobleski[2] experiments performed with neutron interferometry. These seminal and pioneering scientific tests have helped dramatically to convince physicists that there exist some experiments whose outcome necessarily depends upon both the gravitational constant and the Planck’s constant. The fact that in the great majority of phenomena of interest in terrestrial physics, gravity and quantum mechanics do not simultaneously play an important role, is certainly responsible for the wrong idea that gravity and quantum mechanics cannot be closely intertwined in some special circumstances. In the last four decades the COW experiments have become more sophisticated. The latest neutron interferometry experiments[3] report a statistically significant discrepancy between the theoretically predicted and experimentally measured values of the neutron phase shift due to gravity. Supposing the equality of inertial and gravitational masses for the neutron the experimenters found the neutron phase factor to be $1\%$ lower than predicted,[3] which clearly signals a possible violation of the classical equivalence principle. At first sight, it seems that these latest neutron interferometry experiments are in conflict with the more precise tests of the classical equivalence principle conducted via atomic interferometry, and with those based on torsion pendulum. Adunas, Rodriguez- Milla and Ahluwalia,[4] showed, however, that each of the aforementioned experiments probes a different aspect of gravity; and that current experiment techniques, when coupled to solar-neutrino data, may be able to explore quantum mechanically induced violations of the classical equivalence principle. They also predicted a quantum violation of the classical equivalence principle for the next generation of atomic interferometry experiments. Actually, from an operational point of view one cannot claim, even in principle, that there exists, for certain quantum systems, an exact equality of gravitational and inertial masses.[4, 5] Therefore, we come to the conclusion that quantum mechanics and the classical equivalence principle cannot coexist peacefully.[6, 7] Recently, an interesting experiment, not directly related to interferometry techniques, but that attests to the fact that neutrons can reside in quantum stationary states formed in the gravitational field of the Earth, was carried out. In this experiment, the lowest quantum state of neutrons in the Earth’s gravitational field was identified in the measurement of neutron transmission between a horizontal mirror on the bottom and an absorber/scatter on top.[8, 9] This result motivate Ernest to do a careful and through research on gravitational eigenstates in weak gravity.[10, 11] Actually, despite the almost universal study of quantum theory applied to atomic and molecular states, very little work has been done to investigate the properties of the hypothetical stationary states that should exist in similar types of gravitational central potentials wells, particularly those with large quantum numbers. The preceding experiments show clearly that physical phenomena in which gravity (specifically, Earth’s gravitational field) and quantum effects are merged together, are no more beyond our reach. We remark that in all these investigations the gravitational field of the Earth is described by Newton’s gravity. On the other hand, since its original publication in 1915, Einstein’s general theory of relativity continues to be an active area of both theoretical and experimental research. Presently, the theory successfully accounts for all data gathered to date.[12] In addition, among its so-called classical tests there is one, namely, light bending, that has been confirmed with an accuracy that increases as time goes by. In reality, it is expected that a series of improved designed experiments with the Very Long Baseline Interferometry (VLBI) will increase the present accuracy of the deflection parameter $\gamma$ by at least a factor of 4.[13] By the way, the current value for $\gamma$ is 0.9998$\pm$0.0003 (68$\%$ confidence level),[13] in agreement with general relativity. Besides, it is also a generally acknowledged fact that the gravitational deflection of light by the sun can be measured more accurately at radio wavelengths with interferometry techniques than at visible wavelength with available optical techniques.14-17 Indeed, at present the VLBI is the most accurate technique we have at our disposal for measuring radio-wave gravitational deflection.[13] Now, taking into account that the search for upper bounds on the photon mass222In all these researches, the photon is described by massive QED, which is nothing but the most straightforward extension of standard QED. Its Lagrangian can be written as $\mathcal{L}=-\frac{1}{4}F^{2}_{\mu\nu}+\frac{1}{2}m^{2}A^{2}_{\mu}-J^{\mu}A_{\mu},$ (1) where $F_{\mu\nu}$($=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$) is the field strength, $J^{\mu}$ is the conserved (electric) current and $m$ is the photon mass. Here, indices are raised and lowered with $\eta^{\mu\nu}$ and $\eta_{\mu\nu}$, respectively. In reality, massive QED is theoretically simpler than the standard theory,[18] besides being renormalizable.[19] has increased over the past several decades,20-22 it would be interesting to estimate gravitational bounds for the photon mass by considering the most recent measurements of the solar gravitational deflection of radio waves obtained by means of the VLBI. This is precisely the goal of this work. The article is organized as follows. In Sec. 2 we show that there exists a constraint on the photon mass, its frequency and the deflection parameter determined by experimenters (which characterizes the contribution of space curvature to gravitational deflection). From this amazing result, upper limits on the photon mass are found in Sec. 3, by considering as inputs the most recent measurements of the solar gravitational deflection of radio waves obtained via the VLBI.[13] To conclude, we discuss in Sec. 4, whether or not the bounds we have estimated can be improved. In our convention $\hbar=c=1$, and the signature is (+ - - -). ## 2 Combining General Relativity, Massive QED and Very Long Baseline Interferometry to Gravitationally Constrain the Photon Mass To start off, we recall that the Lagrangian for the gravitational minimally coupled massive photon field is $\displaystyle{\cal{L}}=\sqrt{-g}\left[-\frac{1}{4}g^{\mu\alpha}g^{\nu\beta}F_{\mu\nu}F_{\alpha\beta}+\frac{m^{2}}{2}g^{\mu\nu}A_{\mu}A_{\nu}\right].$ (2) On the other hand, for small fluctuations around the Minkowski metric $\eta$, the full metric can be written as file=foton-graviton.eps,width=5.9cm Figure 1: Feynman graph for the interaction between a massive photon and an external gravitational field. $\displaystyle g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu},$ (3) where $\kappa^{2}=32\pi G$. Here $G$ is Newton’s constant. Supposing then that the massive photon is scattered by an external weak gravitational field (See Fig. 1), we promptly obtain from Eqs. (2) and (3) that the Lagrangian for the interaction under discussion is $\displaystyle{\cal{L}}_{\mathrm{int}}=\frac{\kappa}{2}h_{\mathrm{ext}}^{\mu\nu}\left[\eta^{\alpha\beta}F_{\alpha\mu}F_{\beta\nu}-\frac{1}{4}F_{\alpha\beta}^{2}\eta_{\mu\nu}+\frac{m^{2}}{2}A_{\alpha}^{2}\eta_{\mu\nu}-m^{2}A_{\mu}A_{\nu}\right].$ (4) Note that now the indices are raised (lowered) with $\eta^{\mu\nu}(\eta_{\mu\nu})$. Accordingly, the vertex function for the alluded process is, in momentum space, given by $\displaystyle V_{\alpha\beta}(p,p^{\prime})$ $\displaystyle=$ $\displaystyle\frac{\kappa}{2}h^{\mu\nu}_{\mathrm{ext}}(\mathbf{k})\Big{[}(m^{2}-p\cdot p^{\prime})(\eta_{\mu\nu}\eta_{\alpha\beta}-2\eta_{\mu\alpha}\eta_{\nu\beta})+p^{\prime}_{\alpha}p_{\beta}\eta_{\mu\nu}$ (5) $\displaystyle+\;2(-p^{\prime}_{\alpha}p_{\nu}\eta_{\mu\beta}-p^{\prime}_{\mu}p_{\beta}\eta_{\nu\alpha}+p^{\prime}_{\mu}p_{\nu}\eta_{\alpha\beta})\Big{]}.$ Now, assuming that the external weak gravitational field is generated by a point mass $M$ at $\bf{r}$=$\bf{0}$, we immediately find the expression for the external gravitational field by solving Einstein’s linearized field equations in the de Donder gauge. The resulting expression is $h^{\mu\nu}_{\mathrm{ext}}(\mathbf{r})=\frac{\kappa M}{16\pi r}(\eta^{\mu\nu}-2\eta^{\mu 0}\eta^{\nu 0}).$ (6) Consequently, the momentum space gravitational field, $h_{\mathrm{ext}}^{\mu\nu}({\bf k})$, is given by $\displaystyle h^{\mu\nu}_{\mathrm{ext}}(\mathbf{k})$ $\displaystyle=$ $\displaystyle\int{d^{3}\mathbf{r}e^{-i\mathbf{k}\cdot\mathbf{r}}h^{\mu\nu}_{\mathrm{ext}}(\mathbf{r})}$ (7) $\displaystyle=$ $\displaystyle\frac{\kappa M}{2\mathbf{k}^{2}}\Big{(}\frac{\eta^{\mu\nu}}{2}-\eta^{\mu 0}\eta^{\nu 0}\Big{)}.$ We are now ready to compute the unpolarized differential cross section for the process displayed in Fig. 1. To do that we recall that the expression for the mentioned cross section is $\frac{d\sigma}{d\Omega}=\frac{1}{(4\pi)^{2}}\frac{1}{3}\sum^{3}_{r=1}\sum^{3}_{r^{\prime}=1}\mathcal{M}^{2}_{rr^{\prime}},$ (8) with ${\cal{M}}_{rr^{\prime}}=\epsilon_{r}^{\alpha}({\bf p})\epsilon_{r^{\prime}}^{\beta}({\bf p^{\prime}})V_{\alpha\beta}(p,p^{\prime}),$ (9) where $\epsilon_{r}^{\alpha}({\bf p})$ and $\epsilon_{r^{\prime}}^{\beta}({\bf p^{\prime}})$ are the polarizations vectors for the ingoing and outgoing vectorial bosons, respectively. These vectors, in turn, satisfy the relation $\sum_{r=1}^{3}\epsilon_{r}^{\mu}(\mathbf{p})\epsilon^{\nu}_{r}(\mathbf{p})=-\eta^{\mu\nu}+\frac{p^{\mu}p^{\nu}}{m^{2}}.$ (10) After algebraic manipulations, we find that the expression for the unpolarized differential cross section we are searching for reads $\frac{d\sigma}{d\Omega}=\frac{1}{6}\frac{M^{2}G^{2}}{\Big{(}\sin^{2}{\frac{\theta}{2}}\Big{)}^{2}}\Bigg{[}3\mathbf{p}^{4}+\frac{3}{2}m^{4}+2\mathbf{p}^{2}m^{2}+2\mathbf{p}^{2}(\mathbf{p}^{2}+2m^{2})\cos{\theta}+\mathbf{p}^{4}\cos^{2}{\theta}\Bigg{]},$ (11) where $\theta$ is the scattering angle. For small angles, it reduces to $\frac{d\sigma}{d\Omega}=\frac{16M^{2}G^{2}}{\theta^{4}}\left[\frac{1-\frac{m^{2}}{2E^{2}}}{1-\frac{m^{2}}{E^{2}}}\right]^{2},$ (12) where $E$ is the energy of the ingoing photon. The above differential cross section can be related to a classical trajectory with impact parameter $b$ via the relation $\displaystyle bdb=-\frac{d\sigma}{d\Omega}\theta d\theta.$ (13) From Eqs. (12) and (13), we arrive at the conclusion that $\displaystyle\theta=\frac{4MG}{b}\left(\frac{1-\frac{m^{2}}{2E^{2}}}{1-\frac{m^{2}}{E^{2}}}\right),$ (14) which, in the ultrarelativistic limit, i.e., $E\gg m$, leads to $\displaystyle\theta$ $\displaystyle=$ $\displaystyle\theta_{\mathrm{E}}\left(1+\frac{m^{2}}{2E^{2}}\right)$ (15) $\displaystyle=$ $\displaystyle\theta_{\mathrm{E}}\left(1+\frac{m^{2}}{8\pi^{2}\nu^{2}}\right),$ (16) where $\nu$ is the frequency of the massive photon and $\theta_{\mathrm{E}}\equiv\frac{4MG}{b}$. Before going on, some comments are in order. Recently, it was shown that the unpolarized differential cross sections for the scattering of different quantum particles are spin dependent, which is in disagreement with the classical equivalence principle[7] (See Table 1). This result raises an important question: Why the gravitational field perceives the spin? Because there is the presence of a nonzero momentum transfer (k) in the scattering, responsible for probing the internal structure (spin) of the particle. Nevertheless, if we choose any two expressions from those listed in Table 1, we get that the difference between them is always extremely small for typical deflection angles. To show this for the massless particles, for instance, we study the behavior of $\displaystyle\frac{\Delta(\frac{d\sigma}{d\Omega})}{(\frac{d\sigma}{d\Omega})_{s=0}}\equiv\frac{(\frac{d\sigma}{d\Omega})_{s}-(\frac{d\sigma}{d\Omega})_{s=0}}{(\frac{d\sigma}{d\Omega})_{s=0}},$ (17) as a function of the scattering angle $\theta$ (See Fig. 2). It is trivial to show that for small angles the preceding expression reduces to $\displaystyle\frac{\Delta(\frac{d\sigma}{d\Omega})}{(\frac{d\sigma}{d\Omega})_{s=0}}\approx-\frac{s\theta^{2}}{2}.$ (18) For a typical deflection angle, say $\theta\sim 10^{-6}$, we found $\frac{\Delta(\frac{d\sigma}{d\Omega})}{(\frac{d\sigma}{d\Omega})_{s=0}}\sim 10^{-12}$. The detection of so small an effect is, of course, beyond todays technology. Consequently, for these tiny deflection angles, the cross sections will be unaffected by the spin of the particle. Unpolarized differential cross-sections for the scattering of different quantum particles by an external weak gravitational field generated by a static point particle of mass $M$. Here $m$ is the particle mass, $s$ the spin, $\theta$ the scattering angle, and $\lambda\equiv\frac{m^{2}}{\mathbf{p}^{2}}=\frac{1-\mathbf{v}^{2}}{\mathbf{v}^{2}}$, with $\mathbf{v}$ and $\mathbf{p}$ being the velocity and three-momentum, in this order, of the incident particle. $m$ $s$ $\frac{d\sigma}{d\Omega}$ 0 0 $\Big{(}\frac{GM}{\sin^{2}{\frac{\theta}{2}}}\Big{)}^{2}$ $\neq 0$ 0 $\Big{(}\frac{GM}{\sin^{2}{\frac{\theta}{2}}}\Big{)}^{2}\Big{(}1+\frac{\lambda}{2}\Big{)}^{2}$ 0 $\frac{1}{2}$ $\Big{(}\frac{GM}{\sin^{2}{\frac{\theta}{2}}}\Big{)}^{2}\cos^{2}{\frac{\theta}{2}}$ $\neq 0$ $\frac{1}{2}$ $\Big{(}\frac{GM}{\sin^{2}{\frac{\theta}{2}}}\Big{)}^{2}\Big{[}\cos^{2}{\frac{\theta}{2}}+\frac{\lambda}{4}\Big{(}1+\lambda+3\cos^{2}{\frac{\theta}{2}}\Big{)}\Big{]}$ 0 1 $\Big{(}\frac{GM}{\sin^{2}{\frac{\theta}{2}}}\Big{)}^{2}\cos^{4}{\frac{\theta}{2}}$ $\neq 0$ 1 $\Big{(}\frac{GM}{\sin^{2}{\frac{\theta}{2}}}\Big{)}^{2}\Big{[}\frac{1}{3}+\frac{2}{3}\cos^{4}{\frac{\theta}{2}}-\frac{\lambda}{3}\Big{(}1-\frac{3\lambda}{4}-4\cos^{2}{\frac{\theta}{2}}\Big{)}\Big{]}$ 0 2 $\Big{(}\frac{GM}{\sin^{2}{\frac{\theta}{2}}}\Big{)}^{2}\Big{(}\sin^{8}{\frac{\theta}{2}}+\cos^{8}{\frac{\theta}{2}}\Big{)}$ In order to recover Einstein’s geometrical results from Table 1, we must have ${\bf k}\rightarrow{\bf 0}$; in other words, in the nontrivial limit of small momentum transfer, which corresponds to a nontrivial small angle limit since $\left\|{\bf k}\right\|=2\left\|{\bf p}\right\|\sin\frac{\theta}{2}$, the massive (massless) particles behave in the same way, regardless the spin. In fact, when the spin is ‘switched off’, i.e, for small angles, we obtain from Table 1 that for $m=0$, $\displaystyle\frac{d\sigma}{d\Omega}\approx\frac{16G^{2}M^{2}}{\theta^{4}},$ (19) while for $m\neq 0$, $\displaystyle\frac{d\sigma}{d\Omega}\approx\frac{16G^{2}M^{2}}{\theta^{4}}\left(1+\frac{\lambda}{2}\right)^{2}.$ (20) Using Eq. (12) we conclude that for $m=0$, $\displaystyle\theta\approx\frac{4GM}{b},$ (21) while for $m\neq 0$, $\displaystyle\theta\approx\frac{4GM}{b}\left(1+\frac{\lambda}{2}\right).$ (22) The former equation gives the gravitational deflection angle for a massless particle — a result foreseen by Einstein a long time ago; whereas the latter coincides with the prediction of general relativity for the deflection of a massive classical test particle by an external weak gravitational field.[23] It is worth noticing that Eqs. (15) and (22) are exactly the same. In short, for small angles the results of Table 1 not only reduce to those predicted by Einstein’s geometrical theory, they are also in agreement with the classical equivalence principle. file=spin3.eps,width=10cm Figure 2: $\frac{\Delta(\frac{d\sigma}{d\Omega})}{(\frac{d\sigma}{d\Omega})_{s=0}}$ as a function of the scattering angle $\theta$. At first glance it seems that Eq. (16) predicts a dispersive deflection angle for the massive photons. Actually, this is a false impression; indeed, it is straightforward to show that Eq. (16) can be rewritten as $\displaystyle\theta=\theta_{\mathrm{E}}\left(\frac{3-{\bf v^{2}}}{2}\right).$ (23) After these important digressions, we return to the analysis of Eq. (16). The first term in the expression (16) coincides with that obtained by Einstein in 1916, by solving the equation of light propagation in the field of a static body,[24] while the second one is the most important correction to the mass $m$ of the massive photon. On the other hand, the angle of gravitational bending determined by the experimental groups is in general expressed through the relation[25] $\displaystyle\theta_{\mathrm{exp}}=\frac{1+\gamma}{2}\theta_{\mathrm{E}},$ (24) where $\gamma$ is the deflection parameter, precisely and unambiguously determined by experimenters by measuring the deflection of electromagnetic radiation in the gravitational field of the sun. From Eqs. (16) and (24), we then get $\displaystyle m<2\pi\nu\sqrt{\left\|1-\gamma\right\|}.$ (25) This amazing result clearly shows that there exists a constraint between the photon mass and the parameters $\gamma$ and $\nu$. In addition, it tells us that the knowledge of these parameters is all we need to set up upper limits on the photon mass. ## 3 Finding Gravitational Upper Bounds on the Photon Mass We are now finally ready to find gravitational bounds on the photon mass. To accomplish this goal, we make use of the recent measurements of the solar gravitational deflection of radio waves found by Fomalont _et al._[13] using the VLBI. The solutions for $\gamma$ obtained from these measurements, as well as the corresponding bounds on the photon mass we have estimated using Eq. (25), are displayed in Table 2. In order to determine the optimum value of $(\gamma-1)$ from the experimental results, the aforementioned authors minimized the (normalized) chi-squared expression $\chi_{k}^{2}=\frac{1}{k}\sum_{d,i}\left(\frac{P_{d}(i)-0.5(\gamma-1)D_{d}(i)}{\sigma_{d}(i)}\right)^{2},$ (26) where $P_{d}(i)$ and $\sigma_{d}(i)$ are the measured position offset and error estimate, in this order. The term $D_{d}(i)$ is the differential general relativity gravitational bending prediction averaged over the session. The $\sigma_{\gamma}$’s and $\chi_{k}^{2}$’s directly related to the the solutions for $\gamma$ are shown in Table 2. Uppers bounds on the photon mass estimated using the solutions for $\gamma$ found by Fomalont _et al._ 13 Solution Type $(\gamma-1)\times 10^{-4}$ $\sigma_{\gamma}\times 10^{-4}$ $\chi^{2}_{k}$ $m\times 10^{-11}$($MeV$) $43\;GHz$ data (corona-free) -2.4 3.2 0.9 1.7 $43\;GHz$ data only -1.0 2.6 2.2 1.1 $43\;GHz$ data only - Oct05 -3.2 2.8 1.1 2.0 $23\;GHz$ data only - Oct05 -2.0 2.4 4.7 0.8 In Table 2, the $43\;GHz$ corona-free fit is the most accurate since it has the lowest $\chi^{2}$. This is due to the lessening of some coronal effects, and the increase of the position errors. The two $43\;GHz$ only solutions (with no removal of the ionosphere reduction) show the effect of the Oct05 session that was made relatively close to the sun; the first solution was found using data obtained during sessions that lasted several days, while the second one is based on data that were got within the space of one day only. Finally, the $23\;GHz$ only solution suggests that coronal refraction, which is four times larger than that at $43\;GHz$, is dominating the sensitivity of the experiment at $23\;GHz$. It is worth mentioning that Fomalont $et\;al.$[13] have used for the results in their paper an average of the four solutions exhibited in Table 2 to obtain $\gamma=0.9998\pm 0.0003$. From this result and assuming that the massive photon passing near the solar limb has a frequency $\nu=43GHz$, which is perfectly justifiable since their data came mainly from $43GHz$ observations where the refraction effects of the solar corona were negligible beyond 3 degrees from the sun, we obtain another gravitational bound on the photon mass, namely, $m\sim 3.4\times 10^{-11}MeV$. For reasons already discussed, we come to the conclusion that among the gravitational bounds on the photon mass we have found, the most reliable is $m\sim 1.7\times 10^{-11}MeV$. In fact, the associated $\gamma$ was determined, on the one hand, using one single frequency; on the other, it has the best chi-squared. Furthermore, the data used for computing $\gamma$ came mainly from observations where the refraction effects of the solar corona were negligible. ## 4 Discussion Certainly, the bounds we have found on the photon mass are considerably higher than the recently recommended limit published by the Particle Data Group ($m<1\times 10^{-18}eV$).[26] They are nevertheless comparable to other existing bounds.20-22 Let us then discuss whether or not a better limit on the photon mass might be obtained using Eq. (25). First, if the deflections measured using the VLBI could be made with greater accuracy the value of $\sqrt{\left\|1-\gamma\right\|}$ would be reduced giving, as a result, a better gravitational estimate. According to Fomalont et al.,[13] a series of designed experiments with the VLBI could increase the accuracy of the future experiments by at least a factor of 4. Second, if deflection measurements can be obtained at lower frequencies, while maintaining the value of the deflection parameter $\gamma$, the gravitational bound will be improved in direct proportion to the frequency. This point, however, is very delicate. In fact, as we have already mentioned, up till now the best results obtained for the gravitational deflection via the VLBI are those that come mainly from $43GHz$ where the refraction effects of the solar corona were negligible beyond 3 degrees from the sun. Incidentally, the lowest frequency employed by the radio astronomers was $2GHz$. However, the measurements made at this frequency are less reliable because of the refraction effects of the solar corona. Actually, the radio astronomers use in their experiments a mixing of different frequencies but the most significant contributions come in general from $\sim 43GHz$. The possibility of improving the gravitational limit on the photon mass in this case is thence very limited. We remark that up to now only two attempts to constrain the photon mass using gravitational deflection measurements were made.[27, 28] Unfortunately, these estimates are not very reliable because in both of them the values of the deflection parameter $\gamma$ are overestimated, while the values of the frequency are underestimated. In reality, the $\nu$-values used in the mentioned estimates are in the neighborhood of the the lowest frequency employed by radio astronomers ($\approx 2GHz$). We call attention to the fact that the gravitational estimates on the photon mass we have obtained, like the great majority of estimates made with the purpose of limiting the photon mass and which are available in the literature, are essentially order-of-magnitude arguments. Nonetheless, our efforts in this work are based on a new conceptual approach to the subject; besides, in the calculation of the bounds, the most accurate experimental data currently available have been taken as inputs. To conclude, we remark that recently we have found a quantum bound on the photon mass ($m\sim 1.6\times 10^{-10}MeV$) based on the computation of the anomalous electron magnetic moment in the framework of Proca electrodynamics.29,30 To the best of our knowledge, this is the firs time a quantum bound on the photon mass is estimated.333Note that the bound on the photon mass obtained by Boulware and Deser31 via the Aharonov-Bohm effect (which is present in massive (finite-range) electrodynamics) is a semiclassical limit; our bound, nonetheless, unlike the Bolware-Deser limit, is based on truly (loop) quantum effects. It is worth noticing that this quantum bound is one order of magnitude higher than the bound derived from the gravitational scattering. ## Acknowledgments The authors are very grateful to FAPERJ, CNPq, and CAPES (Brazilian funding agencies) for the financial support. ## References * [1] R. Colella, A. Overhauser and S. Werner, Phys. Rev. Lett. 34 (1975) 1472. * [2] U. Bonse and T. Wroblewski, Phys. Rev. Lett. 51 (1983) 1401. * [3] K. Littrell, B. Allman and S. Werner, Phys. Rev. A 56 (1997) 1767. * [4] G. Adunas, E. Rodriguez-Milla and D. V. Ahluwalia-Khalilova, Gen. Relativ. Gravit. 33 (2001) 183. * [5] G. Adunas, E. Rodriguez-Milla and D. V. Ahluwalia-Khalilova, Phys. Lett. B 485 (2000) 215. * [6] A. Accioly and R. Paszko, Phys. Rev. D 78 (2008) 064002. * [7] A. Accioly and R. Paszko, Adv. Studies Theor. Phys. 3 (2009) 65. * [8] V. Nesvizhevsky et al., Nature 415 (2002) 297. * [9] V. Nesvizhevsky et al., Phys. Rev. D 67 (2003) 102002. * [10] A. Ernest, J. Phys. A: Math. Theor. 42 (2009) 115207. * [11] A. Ernest, J. Phys. A: Math. Theor. 42 (2009) 115208. * [12] S. Turyshev, Usp. Fiz. Nauk 179 (2009) 3. * [13] E. Fomalont et al., Astrophys. J. 699 (2009) 1395. * [14] D. Robertson, W.Carter and W. Dillenger, Nature 349 (1991) 768. * [15] B. Bertotti, L. Iess, and P. Tortora, Nature 425 (2003) 374. * [16] S. Shapiro et al., Phys. Rev. Lett. 92 (2005) 121101. * [17] S. Lambert and C. Le Poncin-Lafitte, Astron. Astrophys. 499 (2009) 331. * [18] A. Ignatiev and G. Joshi, Phys. Rev. D 53 (1996) 984. * [19] D. Boulware, Ann. Phys. 56 (1970) 140. * [20] A. Goldhaber and M. Nieto, Rev. Mod. Phys. 43 (1971) 277. * [21] Liang-Cheng Tu, J. Luo, and G. Gilles, Rep. Prog. Phys. 68 (2005) 77. * [22] A. Goldhaber and M. Nieto, Rev. Mod. Phys. 82 (2010) 939. * [23] A. Accioly and S. Ragusa, Class. Quant. Grav. 19 (2002) 5429 [Erratum Class. Quant. Grav. 20 (2003) 4963]. * [24] A. Einstein, Ann. Phys. 49 (1916) 769. * [25] C. Will, Theory and Experiment in Gravitational Physics (Cambridge: Cambridge Univ. Press, 1993). * [26] C. Amsler et al. (Particle Data Group), Phys. Lett. B 667 (2008) 1. * [27] D. Lowenthal, Phys. Rev. D 8 (1973) 2349. * [28] A. Accioly and R. Paszko, Phys. Rev. D 69 (2004) 107501. * [29] A. Accioly, J. Helayël-Neto and E. Scatena, Phys. Rev. D 82 (2010) 065026. * [30] A. Accioly, J. Helayël-Neto, R. Turcati, J. Morais and E. Scatena, Classical Quantum Gravity 27 (2010) 205010. * [31] D. Boulware and S. Deser, Phys. Rev. Lett. 63 (1989) 2319.	arxiv-papers	2010-12-10T13:51:50	2024-09-04T02:49:15.568422	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "A. Accioly, J. Helay\\\"el-Neto and E. Scatena", "submitter": "Antonio Accioly", "url": "https://arxiv.org/abs/1012.2267" }
1012.2313	# Millimeter Imaging of the $\beta$ Pictoris Debris Disk: Evidence for a Planetesimal Belt David J. Wilner11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 , Sean M. Andrews11affiliation: Harvard- Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 , A. Meredith Hughes22affiliation: Department of Astronomy, 601 Campbell Hall, University of California, Berkeley, CA 94720 33affiliation: Miller Fellow ###### Abstract We present observations at 1.3 millimeters wavelength of the $\beta$ Pictoris debris disk with beam size $4\farcs 3\times 2\farcs 6$ ($83\times 50$ AU) from the Submillimeter Array. The emission shows two peaks separated by $\sim 7^{\prime\prime}$ along the disk plane, which we interpret as a highly inclined dust ring or belt. A simple model constrains the belt center to $94\pm 8$ AU, close to the prominent break in slope of the optical scattered light. We identify this region as the location as the main reservoir of dust producing planetesimals in the disk. circumstellar matter — planetary systems: planet-disk interactions— stars: individual ($\beta$ Pictoris) ††slugcomment: accepted by ApJL: December 9, 2010 ## 1 Introduction The discovery of excess far-infrared emission from the nearby ($19.44\pm 0.05$ pc; van Leeuwen, 2007) A6V-type main-sequence star $\beta$ Pictoris (Aumann, 1985) together with optical imaging of scattered light from circumstellar dust (Smith & Terrile, 1984) established the “debris disk” paradigm where dust grains orbiting the star originate from an eroding reservoir of larger bodies (see reviews by e.g. Artymowicz, 1997; Backman & Paresce, 1993; Wyatt, 2008). The nearly edge-on disk surrounding this young star ($12^{+8}_{-4}$ Myr; Zuckerman et al., 2001) is relatively luminous ($F_{disk}/F_{}=2.5\times 10^{-3}$; Lagrange et al., 2000) and has been studied in great detail with a panoply of observational techniques. High resolution images in the optical (Kalas & Jewitt, 1995; Heap et al., 2000; Golimowski et al., 2006), near- infrared (Mouillet et al., 1997a; Tamura et al., 2006; Boccaletti et al., 2009) and mid-infrared (Wahhaj et al., 2003; Weinberger et al., 2003; Okamoto et al., 2004; Telesco et al., 2005) all show a wealth of structure, including density concentrations, an inner cavity, and asymmetries such as warps. These features, including a secondary disk of scattered light inclined by about $5{\arcdeg}$ (Ahmic et al., 2009), have been variously ascribed to the gravitational influence of a giant planet or planets (e.g. Mouillet et al., 1997b; Augereau et al., 2001; Freistetter et al., 2007; Kennedy & Wyatt, 2010). Indeed, a planetary mass companion at a projected distance of 8 AU from the star now has been directly imaged (Lagrange et al., 2009, 2010). The emerging view of debris disks like $\beta$ Pictoris postulates the presence of a planetesimal belt that produces dust with a range of sizes through collisional cascades (Strubbe & Chiang, 2006; Wyatt, 2008; Kuchner & Stark, 2010). The stirring of the planetesimals may be due to the gravity of $\sim 1000$ km-sized objects formed within the belt (Kenyon & Bromley, 2004), or to the presence of planets located closer to the star (Moro-Martin et al., 2007). In either case, the dynamical effects of stellar radiation create a distribution of grain sizes that depends on distance from the star, e.g. the blow-out of the smallest “$\beta$-meteoroid” grains. An important consequence is that images of debris disks at different wavelengths are dominated by different grain sizes and can appear remarkably different (Wyatt, 2006). Observations at millimeter wavelengths are most sensitive to large grains that are minimally affected by radiative forces and thus have the potential to trace best the location of the dust producing parent planetesimals. The debris disk around Vega, for example, shows a clumpy ring confined to radii $<200$ AU at wavelengths of 350 $\mu$m and longward (Holland et al., 1998; Wilner et al., 2002; Marsh et al., 2006), while it appears smooth and featureless and extends to radii $\sim 800$ AU in mid-infrared light that arises predominantly from small grains expelled by radiation (Su et al., 2005). The debris disk around HR 8799, an A-type star that harbors three directly imaged planets, shows similar morphological changes with wavelength (Su et al., 2009). For the $\beta$ Pictoris debris disk, the angular resolutions of (sub-)millimeter images have been too coarse to reveal much structure. Images from several different telescopes generally show dust emission extended along a position angle of $\sim 30{\arcdeg}$, consistent with the optical disk: JCMT/SCUBA at 850 $\mu$m with a $14^{\prime\prime}$ beam (Holland et al., 1998), APEX/LABOCA at 870 $\mu$m with an $18^{\prime\prime}$ beam (Nilsson et al., 2009), SEST/SIMBA at 1200 $\mu$m with a $24^{\prime\prime}$ beam (Liseau et al., 2003), and Herschel/SPIRE at 250, 350 and 500 $\mu$m with $18$, $25$, and $37^{\prime\prime}$ beams, respectively (Vandenbussche et al., 2010). A separate peak or blob of dust emission is also found $\sim 30^{\prime\prime}$ to the southwest of the star, but the relationship of this peak to the main disk is unclear; Dent et al. (2000) and Vandenbussche et al. (2010) have suggested it is a background galaxy with a coincidental alignment with the disk plane. Millimeter interferometry offers a way to obtain higher angular resolution and more information on the largest detectable grain populations within the debris disk. In this Letter, we present imaging observations of $\beta$ Pictoris at 1.3 millimeters wavelength from the Submillimeter Array (SMA)111The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academica Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academica Sinica. that reveal a belt of emission around the star centered near a radius of $\sim 95$ AU that likely marks a reservoir of planetesimals. ## 2 Observations We used the eight element SMA (Ho et al., 2004) on Mauna Kea, Hawaii to observe $\beta$ Pictoris in the compact-north configuration (baselines 6-97 m) on 2010 August 9, and in the extended configuration (baselines 12-178 m) on 2010 September 1. The phase center was chosen to be $\alpha=5^{h}47^{m}17\fs 09$, $\delta=-51\arcdeg 03\arcmin 59\farcs 5$ (J2000), about $0\farcs 9$ from the stellar position at the epoch of the observations. The $\beta$ Pictoris system is a challenging target for the SMA as it never rises above $20{\arcdeg}$ elevation. Nonetheless, usable data were obtained in both configurations over the hour angle range $\pm 1.7$. The weather conditions were good on both days, with 225 GHz atmospheric opacities 0.07–0.09 and stable atmospheric phase. The correlator was configured to provide the maximum 4 GHz of bandwidth in each of two sidebands centered $\pm 6$ GHz from a central LO frequency of 235.6 GHz (wavelength of 1.3 millimeters), with a uniform spectral resolution of 0.8125 MHz. At this frequency, the primary beam size is $\sim 54^{\prime\prime}$ (FWHM). Observations of the strong source 3C454.3 obtained at the start of each track were used to calibrate the passband response. Observations of the quasars J0538-440 and J0522-364 were interleaved with $\beta$ Pictoris in order to calibrate time dependent gain variations. The astrometric uncertainty is $\lesssim 0\farcs 3$. The absolute flux scale was set with reference to observations of the standard calibrator Callisto in each track and should be accurate to better than 15%. The calibration procedure was performed using the IDL based MIR software. Subsequent imaging and deconvolution were done within the MIRIAD package. ## 3 Results and Analysis ### 3.1 1.3 Millimeter Emission Figure 3.1 shows a contour image of the 1.3 millimeter emission overlaying a Hubble Space Telescope/STIS coronographic image of optical scattered light from Heap et al. (2000). The 1.3 millimeter image was made using natural weighting and a modest taper in the east-west direction to avoid extreme ellipticity of the synthesized beam, which is $4\farcs 3\times 2\farcs 6$ ($83\times 50$ AU) oriented nearly north-south (position angle $2{\arcdeg}$). The maximum sidelobes of the dirty beam obtained with this weighting scheme are located about $13^{\prime\prime}$ to the east and west, with amplitude 15% of the central peak. The rms noise in this image is 0.6 mJy beam-1. The star symbol is plotted offset by $(0\farcs 35,0\farcs 70)$ from the phase center, within the uncertainties of the stellar position corrected for proper motion. The 1.3 millimeter emission shows two peaks at positions symmetrically offset from the stellar position by $\sim 3\farcs 5$ to the northeast and southwest, respectively. This basic morphology suggests a highly inclined ring or belt, where the peaks are due to limb brightening at the ansae (where the column density is highest). While the southwest peak appears slightly brighter, the difference lies within the noise and cannot be considered significant. SMA image of the 1.3 millimeter continuum emission from $\beta$ Pictoris overlayed on an image of optical scattered light from Heap et al. (2000). The contour levels are $-2,2,4,6,...\times 0.6$ mJy (the rms noise level), Negative contours are dotted. The ellipse in the lower left corner represents the $4\farcs 3\times 2\farcs 6$ (FWHM) synthesized beam size. The star symbol indicates the location of the stellar photosphere. Because the SMA observations are not sensitive enough to detect the stellar photosphere, the alignment of the images from the SMA and Hubble Space Telescope is limited by the absolute astrometry. Even taking account of this uncertainty, it seems that the two millimeter emission peaks do not align perfectly along the $30{\arcdeg}$ position angle of the primary optical disk (Kalas & Jewitt, 1995). Instead, examination of Figure 3.1 suggests that the peaks align more closely with the $34{\arcdeg}$ position angle of the scattered light secondary disk described by Golimowski et al. (2006). Observations with better resolution and sensitivity are needed to confirm this suggestion; the non-circular beam makes it difficult to assess small differences in orientation, and the millimeter emission structure itself may prove to be warped or more complex. The 1.3 millimeter flux in the detected structure is $13\pm 1.4$ mJy, estimated by integrating over the emission in the image. This value is only approximately half of the $24.3\pm 3.0$ mJy measured in the $24^{\prime\prime}$ SEST beam at 1.2 millimeters (Liseau et al., 2003), a discrepancy significantly larger than expected from the mutual absolute calibration uncertainties and the spectral slope. The difference suggests the presence of an additional, extended 1.3 millimeter emission component, missed in these observations by the spatial filtering properties of the interferometer. Given the shortest SMA baselines, the peak brightness is diminished already by 50% for a $20^{\prime\prime}$ (FWHM) Gaussian source (Wilner & Welch, 1994), a size scale smaller than the SEST beam. Judging from the partially resolved images from far-infrared and submillimeter filled aperture telescopes, this missing component is likely elongated along the disk, which extends beyond the SMA field of view. The separate dust peak to the southwest is detected at $\sim 5\sigma$, offset by ($-21\farcs 4\pm 0\farcs 4,-22\farcs 4\pm 0\farcs 6$) from the center of the field (not shown). The corresponding absolute position is $\alpha=5^{h}47^{m}14\fs 82$, $\delta=-51\arcdeg 04\arcmin 21\farcs 9$ (J2000). Because the primary beam correction is large and uncertain at this location beyond the half power point, it is difficult to provide an accurate estimate of the flux. The position is well determined, however, and shows that this peak does not lie along an extension of the optical disk. This supports previous suggestions that this feature is a background source, presumably a dusty galaxy. ### 3.2 Belt Location and Width We constrain the basic properties of the 1.3 millimeter emission with a simple model that assumes the structure is characterized by a flat, axisymmetric belt of emission. We take the radial profile of the emission to be $r^{-0.5}$, which is physically motivated by optically thin emission for constant surface density and a temperature profile of $r^{-0.5}$ due to stellar irradiation. We fix the inclination and orientation of the belt on the sky to $87\arcdeg$ and $33\arcdeg$, respectively; due to the limited resolution, small variations in these geometric parameters do not have significant effects on the results. This simple model has three parameters: the belt center $R$, belt width $\Delta R$, and flux $F$. We calculate a grid of models over the parameter ranges $60<R<130$ AU and $2<\Delta R<110$ AU in steps of $2$ AU, and $10<F<17$ mJy in steps of 0.5 mJy, and calculate $\chi^{2}$ values for each model using all of the SMA visibilities. The right panel of Figure 3.2 shows the resulting $\chi^{2}$ surface (marginalized over the parameter $F$). The cross marks the best fit at $R=94\pm 8$ AU, $\Delta R=34^{+44}_{-32}$ AU, $F=15\pm 2$ mJy; the uncertainties represent the formal $1\sigma$ errors. The data strongly constrain the belt center location and allow for widths up to sizes comparable to the resolution of the observations. The three left panels of Figure 3.2 show the 1.3 millimeter image from Figure 3.1 together with images of the best-fit model and the residuals, all made in the same way. The model reproduces the main features of the data, and the residuals are consistent with noise. If a steeper radial emissivity were assumed, then the outer edge of the emission could extend further. However, a belt with a width that encroaches much closer to the star than the best fit may be difficult to reconcile with the mid-infrared emission from the system. A proper model that considers the constraints of the full spectral energy distribution requires many more assumptions than made here, in particular about the grain composition, grain size distribution, collisional behaviors and dynamics. Left panels: SMA image of the 1.3 millimeter emission from $\beta$ Pictoris together with the image of the best-fit axisymmetric belt model and the residuals. The contour levels and beam size are the same as in Figure 3.1. The dashed line indicates a position angle of $34{\arcdeg}$. Right panel: The $\chi^{2}$ surface for the belt center and width model parameters, with contours at $1,2,3\sigma$. The cross marks the best-fit model. ## 4 Discussion The new millimeter observations improve substantially on previous single dish images and start to resolve fine structure in the $\beta$ Pictoris disk. Since large grains cannot travel far from their place of origin due to short grain- grain collisional timescales and negligible radiation effects, the emission at this long wavelength should trace the dust producing planetesimals. Inspection and analysis of the resolved millimeter emission suggest a highly inclined ring or belt centered at a radius within or near $\sim 95$ AU. Aside from the nearly edge-on viewing geometry, the millimeter morphology is strikingly similar to other well-studied A-type stars with substantial circumstellar dust, in particular Vega (e.g. Marsh et al., 2006) and Fomalhaut (e.g. Holland et al., 2003). The region interior to this belt in $\beta$ Pictoris is clearly not empty, as evidenced by mid-infrared imaging and spectroscopy (Telesco et al., 2005; Chen et al., 2007, e.g.), but it must be relatively deficient in dust mass or millimeter-sized grains or both. The location of the millimeter emission belt corresponds closely to a prominent break in the slope of the optical scattered light, as well as a change in the optical color gradient (Golimowski et al., 2006). These properties are plausibly explained in a model with dust producing planetesimals located just interior the break, with stellar radiation (and possibly also a stellar wind) creating a radial gradient in grain size (Augereau et al., 2001; Strubbe & Chiang, 2006). In this scenario, the extended halo of emission along the disk plane would be dominated by a population of small grains blown out onto highly elliptical or hyperbolic orbits, possibly with temperatures above the local blackbody values, that cover a large area on the sky and give rise to the fraction of millimeter emission missed by the interferometer. One implication of the multi-component emission structure is that the far- infrared to millimeter spectral index of $2.34\pm 0.07$ indicated by the integrated spectrum (Vandenbussche et al., 2010) may not be representative of any of the individual components. Depending on the details, it is possible, for example, that the belt component could show a steeper spectral index that would be closer to expectations for a steady-state collisional cascade, without resorting to unusual fragmentation prescriptions or wavy grain size distributions (Thébault & Augereau, 2007). Until multi-wavelength observations are available in this regime that clearly resolve the relevant structures, it will be problematic to use the integrated spectral index to make conclusive inferences about the grain properties and size distributions. The 1.3 millimeter image of the dust belt around $\beta$ Pictoris sets the stage for much improved future observations with the Atacama Large Millimeter/Submillimeter Array (ALMA), now under construction in Chile (and much better placed for aperture synthesis observations of this southern source). More detailed millimeter images have the potential to determine, e.g. if the dust belt center is offset from the star, or if the emission exhibits pericenter glow or other asymmetries that could point to dynamical perturbations from additional planets in this remarkable system. We thank the SMA staff for scheduling and executing the two filler tracks that provided the data used in this paper. 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1012.2419	, , # Multiple quantum collapse of the inflaton field and its implications on the birth of cosmic structure Gabriel León1, Adolfo De Unánue2 and Daniel Sudarsky3111On sabbatical leave from: Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, México D.F. 04510, México. 1 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, México D.F. 04510, México 2 C3 Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México, Torre de Ingeniería, Circuito Exterior S/N Ciudad Universitaria, México D.F. 04510 3 Instituto de Astronomía y Física del Espacio (UBA-CONICET), Casilla de Correos 67, Sucursal 28, 1428 Buenos Aires, Argentina gabriel.leon@nucleares.unam.mx adolfo@nucleares.unam.mx sudarsky@nucleares.unam.mx ###### Abstract The standard inflationary account for the origin of cosmic structure is, without a doubt, extremely successful. However, it is not fully satisfactory as has been argued in [A. Perez, H. Sahlmann, and D. Sudarsky, Class. Quantum Grav., 23, 2317, (2006)]. The central point is that, in the standard accounts, the inhomogeneity and anisotropy of our universe seems to emerge, unexplained, from an exactly homogeneous and isotropic initial state through processes that do not break those symmetries. The proposal made there to address this shortcoming calls for a dynamical and self-induced quantum collapse of the original homogeneous and isotropic state of the inflaton. In this article, we consider the possibility of a multiplicity of collapses in each one of the modes of the Quantum Field. As we will see, the results are sensitive to a more detailed characterization of the collapse than those studied in the previous works, and in this regard two simple options will be studied. We find important constraints on the model, most remarkably on the number of possible collapses for each mode. ###### pacs: 98.80.Cq, 98.80.Bp, 03.65.Ta ††: Class. Quantum Grav. ## 1 Introduction Modern cosmology has become a very successful field of research in recent years. One of the major ideas, incorporated in the cosmological model, is the existence of a period of accelerating expansion early in the Universe’s history, called Inflation. One of the major successes of inflationary cosmology is its ability to ‘account for’ the spectrum of the temperature anisotropies in the Cosmic Microwave Background (CMB), which is understood as the earliest observational data about the primordial density fluctuations that seed the growth of structure in our Universe. However, when considering this account in more detail, one immediately notes that there is something odd about it. Namely, that out of an initial situation, which is taken to be perfectly isotropic and homogeneous, and based on a dynamics that supposedly preserves those symmetries, one ends with a non- homogeneous and non-isotropic situation. The problem described above, has been acknowledged by some cosmologists222Sometimes this problem is formulated as the Quantum-to- Classical transition. [1] and even by some authors in recent textbooks [2, 3, 4]. Nevertheless, several researchers in the field continue to hold the belief that the issues have been successfully resolved [5, 6, 7, 8]. For an extensive discussion about why the standard explanations doesn’t solve this problem, we invite the reader to consult the reference [9]. In a recent series of works [10, 11, 12, 13, 14, 9, 15, 16] the problem has been analyzed leading to the conclusion that we need some new physics to be able to fully address the problem. The essential idea (as exposed in [10, 11, 12, 13, 14, 9, 15, 16]) is to introduce a new ingredient to the inflationary paradigm: _the self-induced collapse hypothesis:_ a phenomenological model incorporating the description of the effects of a dynamical collapse of the wave function of the inflaton on the subsequent cosmological evolution. The idea is inspired by L. Diósi [17, 18, 19] and R. Penrose’s arguments [20, 21, 22, 23] in the sense that the unification of quantum theory and the theory of gravitation would likely involve modifications in both theories, rather than only the latter as is more frequently assumed. Moreover, Penrose’s idea is that the resulting modifications of the former should involve something akin to a self-induced collapse of the wave-function occurring when the matter fields are in a quantum superposition corresponding to space-time geometries which are ‘too different from each other’. This sort of self-induced collapse would, in fact, be occurring in rather common situations, and would ultimately resolve the long standing ‘measurement problem’ in quantum mechanics. The collapse hypothesis in this context was originally inspired by Penrose’s ideas, however it might be compatible with other collapse mechanisms which attempt to give a reasonable solution to the measurement problem. In essence, the collapse hypothesis simply sustains that something intrinsic to the system, i.e., independent of observers, induces the collapse or reduction of the quantum mechanical state of the system. Various proposals of that sort have been considered [24, 25, 26, 27, 28], and might well be compatible with the self-induced collapse of the inflaton’s wave-function that we are considering. However, we are not following any previous proposed scheme as the intention at this point is to learn what characteristics are needed for it to work in the present context. The point is that, in the case at hand, the collapse hypothesis can be tested and exposed through strictly empirical analyses. The proposal is, at this stage of the analysis, a purely phenomenological scheme. It does not attempt to explain the process in terms of some specific new physical theory, but merely give a rather general parametrization of the quantum transition involved. We will refer to this phenomenological model as the _collapse scheme_. We will not further recapitulate the motivations and discussion of the original proposal and instead refer the reader to the above mentioned works. Previous works along these lines have focused on the times of collapse and the natural basis for the collapse [14], and the issue of fine-tuning of the inflaton potential in the collapse schemes [16]. However, so far the analysis has been based on the consideration of a single collapse of the inflaton’s wave-function for each mode. That limitation of scope has allowed the investigation to proceed without the post-collapse state being characterized beyond the specification of the expectation values of the field and the conjugate momentum in the corresponding modes. The motivation of this present paper is to extract more information about the collapse by considering the possibility that _multiple collapses_ occur in each mode, a consideration that requires a further specification of the post-collapse states; in particular, we are going to focus in models where the post-collapse states can be regarded as _coherent_ or _squeezed_ states. The article is organized as follows: In section 2 we briefly review the quantum mechanical treatment of the field’s fluctuations introducing the collapse hypothesis; we will emphasize how the self-induced collapse proposal is contrasted with the observations and, additionally, we will describe the three _collapse schemes_ that have been studied so far, namely: Independent, Newtonian and Wigner schemes. In section 3 we will generalize the collapse hypothesis of section 2 to the case of multiple collapses. In section 4, we will characterize the multiple post-collapse states and obtain new information about the parameters describing the post-collapse state. Finally in section 5 we will end with a discussion of the results obtained in the previous sections. Regarding notation we will use signature $(-+++)$ for the metric and Wald’s convention for the Riemann tensor. We will use units where $c=1$ but will keep the gravitational constant $G$ and $\hbar$ explicit throughout the paper. ## 2 The collapse model for the quantum fluctuations in the inflationary scenario In this section we will review the formalism used in analyzing the collapse process. The full formalism and motivation is presented in [10, 11, 12, 13]. We will use a semi-classical description of gravitation in interaction with quantum fields as reflected in the semi-classical Einstein’s equation $G_{ab}=8\pi G\langle\hat{T}_{ab}\rangle$, whereas the other fields are treated in the standard quantum field theory (in curved space-time) fashion. This is supposed to hold at all times except when a quantum gravity induced collapse of the wave function occurs. At that point, one would have to assume, that the excitation of the fundamental quantum gravitational degrees of freedom must be taken into account, with the corresponding breakdown of the semiclassical approximation (the possible breakdown of the semi-classical approximation is formally represented by the presence of a term $Q_{ab}$ in the left hand side of the semi-classical Einstein’s equation which is supposed to become nonzero only during the collapse of the quantum mechanical wave function of the matter fields, see [10] for the detailed discussion). The starting point is the action of a scalar field minimally coupled to gravity $S[\phi]=\int d^{4}x\sqrt{-g}\bigg{[}\frac{1}{16\pi G}R[g_{ab}]-\frac{1}{2}g^{ab}\nabla_{a}\phi\nabla_{b}\phi-V[\phi]\bigg{]}.$ (1) One then splits the corresponding fields into their homogeneous part and the perturbations. Thus the metric and the scalar fields are written as $g=g_{0}+\delta g$ and $\phi=\phi_{0}+\delta\varphi$. With the appropriate choice of gauge333Although the equations in this gauge are formally identical to the gauge-independent equations [29], the analysis done here requires the choosing of a specific gauge. One can not work with the so called ‘gauge invariant combinations’, because in the approach followed here, the metric and field fluctuations are treated on a different footing. The metric is considered a classical variable (taken to be describing, in an effective manner, the deeper fundamental degrees of freedom of the quantum gravity theory that one envisions, lies underneath), while the matter fields, specifically the inflaton field perturbations are given a standard quantum field (in curved space-time) treatment, with the two connected trough the semiclassical Einstein’s equations. The choice of gauge implies that the time coordinate is attached to some specific slicing of the perturbed space-time, and thus, our identification of the corresponding hypersurfaces (those of constant time) as the ones associated with the occurrence of collapses,–something deemed as an actual physical change–, turns what is normally a simple choice of gauge into a choice of the distinguished hypersurfaces, tied to the putative physical process behind the collapse. This naturally leads to tensions with the expected general covariance of a fundamental theory, a problem that afflicts all known collapse models, and which in the non-gravitational settings becomes the issue of compatibility with Lorentz or Poincare invariance of the proposals. We must acknowledge that this generic problem of collapse models is an open issue for the present approach. One would expect that its resolution would be tied to the uncovering of the actual physics behind what we treat here as the collapse of the wave function (which we view as a merely an effective description). As has been argued in related works, and in following ideas originally exposed by R. Penrose [20, 21, 22, 23], we hold that the physics that lies behind all this, ties the quantum treatment of gravitation with the foundational issues afflicting quantum theory in general, and in particular those with connection to the ‘measurement problem’. (we will work with the longitudinal gauge also referred to as the Newtonian gauge) and ignoring the vector and tensor part of the metric perturbations, the space-time metric can then be described by the line element $ds^{2}=a(\eta)^{2}[-(1+2\Psi(\eta,\textbf{x}))d\eta^{2}+(1-2\Psi(\eta,\textbf{x}))\delta_{ij}dx^{i}dx^{j}],$ (2) where $\Psi(\eta,\textbf{x})$ is referred to as the _Newtonian potential_. The inflationary regime is characterized by a scale factor $a(\eta)\approx-1/[H_{I}(1-\epsilon)\eta]$, with $H_{I}^{2}\approx 8\pi GV/3$ (which is Friedmann’s equation) and $\epsilon\equiv\frac{1}{2}(M_{P}^{2}/\hbar)(\partial_{\phi}V/V)^{2}$ the slow- roll parameter (which during inflation $\epsilon\ll 1$); $M_{P}$ the reduced Planck mass $M_{P}^{2}\equiv\hbar/(8\pi G)$. The normalization of the scale factor will be set so $a=1$ at the ‘present cosmological time’. The inflationary regime would end at $\eta=\eta_{r}$, a value which is negative and very small in absolute terms ($\eta_{r}\approx-10^{-22}$ Mpc). That is, the conformal time $\eta$ during the inflationary era is in the range $-\infty<\eta<\eta_{r}$, thus $\eta=0$ is a particular value of the conformal time that does not correspond to the inflationary period, in fact, it belongs to the radiation dominated epoch. The background scalar field $\phi_{0}$ will be considered in the slow-roll regime, i.e., $\phi_{0}^{\prime}=-(a^{3}/3a^{\prime})\partial_{\phi}V$, where the primes denotes $\\{\\}^{\prime}\equiv d/d\eta\\{\\}$. Combining the background equations with Einstein’s equations to first order in the perturbations we obtain $\nabla^{2}\Psi+\mu\Psi=4\pi G(u\delta\varphi+\phi_{0}^{\prime}\delta\varphi^{\prime}),$ (3) where $\mu\equiv\mathcal{H}^{2}-\mathcal{H}^{\prime}$; $u\equiv 3\mathcal{H}\phi_{0}^{\prime}+a^{2}\partial_{\phi}V[\phi]$ and $\mathcal{H}\equiv a^{\prime}(\eta)/a(\eta)$. If one uses the expressions for the scale factor during a de Sitter phase then $\mu=0$, while the slow-rolling approximation $\phi_{0}^{\prime}=-a^{2}\partial_{\phi}V/3\mathcal{H}$ corresponds to the condition $u=0$. Under those simplifying conditions the last equation becomes a Poisson-like equation $\nabla^{2}\Psi=4\pi G\phi_{0}^{\prime}\delta\varphi^{\prime}\equiv s\delta\varphi^{\prime},$ (4) with $s\equiv 4\pi G\phi_{0}^{\prime}$, which can be rewritten, by using the slow-roll parameter, the background equation for $\phi_{0}^{\prime}$ in the slow-roll regime and Friedmann’s equation, as $s\equiv a\hbar\sqrt{V\epsilon}/(\sqrt{6}M_{P}^{2})$. The next step involves the quantization of the field fluctuation. We emphasize that the background field $\phi_{0}$ is described in a classical444By _classical_ , in this context, we mean that the homogeneous background field $\phi_{0}(\eta)$ is taken as an approximated description of the quantum quantity $\langle\psi\|\hat{\phi}(x,\eta)\|\psi\rangle$, where the state $\|\psi\rangle$ is the vacuum state of $\hat{\delta\varphi}(x,\eta)$. fashion and it is only the fluctuation $\delta\varphi$ which is subjected to a quantum treatment. Actually, it is convenient to work with the auxiliary field $y=a\delta\varphi$. The equation of motion for this field is $y^{\prime\prime}-\bigg{(}\nabla^{2}+\frac{a^{\prime\prime}}{a}\bigg{)}y=0.$ (5) The conjugated canonical momentum of $y$ is $\pi=y^{\prime}-ya^{\prime}/a$. In order to avoid infrared problems we will consider a restriction of the system to a box of side L, with periodic boundary conditions. The field and its momentum can be decomposed in Fourier’s modes as $\hat{y}(\eta,\textbf{x})=\frac{1}{L^{3}}\sum_{\textbf{k}}e^{i\textbf{k}\cdot\textbf{x}}\hat{y}_{\textbf{k}}(\eta),\qquad\hat{\pi}(\eta,\textbf{x})=\frac{1}{L^{3}}\sum_{\textbf{k}}e^{i\textbf{k}\cdot\textbf{x}}\hat{\pi}_{\textbf{k}}(\eta),$ (6) with the wave vectors satisfying $k_{i}L=2\pi n_{i}$ for $i=1,2,3$. The field operator coefficients are further written as: $\hat{y}_{\textbf{k}}(\eta)\equiv y_{k}(\eta)\hat{a}_{\textbf{k}}+\overline{y}_{k}(\eta)\hat{a}_{-\textbf{k}}^{\dagger}$ and $\hat{\pi}_{\textbf{k}}(\eta)\equiv g_{k}(\eta)\hat{a}_{\textbf{k}}+\overline{g}_{k}(\eta)\hat{a}_{-\textbf{k}}^{\dagger}$. The functions $y_{k}(\eta)$ and $g_{k}(\eta)$ reflect the election of the vacuum state. In our case, as is customarily done in the field, we choose the so called Bunch-Davies vacuum [30], resulting from this choice $y_{k}(\eta)=\frac{1}{\sqrt{2k}}\bigg{(}1-\frac{i}{\eta k}\bigg{)}\exp(-ik\eta),\qquad g_{k}(\eta)=-i\sqrt{\frac{k}{2}}\exp(-ik\eta).$ (7) The vacuum state is defined by the condition $\hat{a}_{\textbf{k}}\|0\rangle=0$ for all k, and can be easily seen to be homogeneous and isotropic at all scales. The self collapse is assumed to operate in close analogy with a ‘measurement’ in the quantum-mechanical sense, but of course, without any external apparatus or observer that could be thought as performing the measurement. The self-induced collapse, is assumed to occur, independently, for each mode of the field. That is, one assumes that at a certain time $\eta_{k}^{c}$ (from now on we will refer to this particular time as the _time of collapse_) the state of each mode k of the field, which was initially the vacuum, changes spontaneously into another state. This self-collapse of the wave-function is inspired by Penrose’s ideas [20, 21, 22, 23], in which gravity plays a fundamental role on the collapse of the wave-function and it does not require outside observers who perform a measurement in order to induce the collapse. The collapse scheme as employed here, however, does not propose at this point a concrete physical mechanism behind it, although one envisions that a more profound theory, presumably derived from quantum gravity, will eventually account for it. These ideas and motivations are discussed in great detail in [10, 11, 12, 13]. In order to study the possibility of multiple collapses, we will see that more detailed specifications of the states after the collapse are needed in contrast with the works [10, 11, 12, 13]. Following [10] it is convenient to decompose the field $\hat{y}_{\textbf{k}}$ and its conjugated momentum $\hat{\pi}_{\textbf{k}}$ in their real and imaginary parts which are completely Hermitian $\hat{y}_{\textbf{k}}(\eta)=\hat{y}_{\textbf{k}}^{R}(\eta)+i\hat{y}_{\textbf{k}}^{I}(\eta)$ and $\hat{\pi}_{\textbf{k}}(\eta)=\hat{\pi}_{\textbf{k}}^{R}(\eta)+i\hat{\pi}_{\textbf{k}}^{I}(\eta)$ where $\hat{y}_{\textbf{k}}^{(R,I)}(\eta)=\frac{1}{\sqrt{2}}\bigg{(}y_{k}(\eta)\hat{a}_{\textbf{k}}^{(R,I)}+\overline{y}_{k}(\eta)\hat{a}_{\textbf{k}}^{{\dagger}(R,I)}\bigg{)},$ (8) $\hat{\pi}_{\textbf{k}}^{(R,I)}(\eta)=\frac{1}{\sqrt{2}}\bigg{(}g_{k}(\eta)\hat{a}_{\textbf{k}}^{(R,I)}+\overline{g}_{k}(\eta)\hat{a}_{\textbf{k}}^{{\dagger}(R,I)}\bigg{)},$ (9) where $\hat{a}_{\textbf{k}}^{R}\equiv\frac{1}{\sqrt{2}}(\hat{a}_{\textbf{k}}+\hat{a}_{-\textbf{k}}),\qquad\hat{a}_{\textbf{k}}^{I}\equiv\frac{-i}{\sqrt{2}}(\hat{a}_{\textbf{k}}-\hat{a}_{-\textbf{k}}).$ (10) The commutators of the real and imaginary annihilation and creation operators are $[\hat{a}_{\textbf{k}}^{R},\hat{a}_{\textbf{k}}^{R{\dagger}}]=\hbar L^{3}(\delta_{\textbf{k},\textbf{k}^{\prime}}+\delta_{\textbf{k},-\textbf{k}^{\prime}}),\qquad\qquad[\hat{a}_{\textbf{k}}^{I},\hat{a}_{\textbf{k}}^{I{\dagger}}]=\hbar L^{3}(\delta_{\textbf{k},\textbf{k}^{\prime}}-\delta_{\textbf{k},-\textbf{k}^{\prime}}).$ (11) A full characterization of the state of each mode of the field would require the specification all statistical moments. In previous works [10, 14, 16], the collapse has been characterized only in terms of the expectation values of field and of the momentum conjugate for the new quantum state. However, in this present work, as we are assuming the possibility of multiple collapses, we will need to focus on the first two statistical moments: the expectation value and the uncertainties (see section 3). For any state $\|{\Xi}\rangle$ of the field $\hat{y}$, we introduce the following quantities $d_{\textbf{k}}^{(R,I)}\equiv\langle{\hat{a}_{\textbf{k}}^{(R,I)}}\rangle_{\Xi},\qquad c_{\textbf{k}}^{(R,I)}\equiv\langle{(\hat{a}_{\textbf{k}}^{(R,I)})^{2}}\rangle_{\Xi},\qquad e_{\textbf{k}}^{(R,I)}\equiv\langle{\hat{a}_{\textbf{k}}^{(R,I){\dagger}}\hat{a}_{\textbf{k}}^{(R,I)}}\rangle_{\Xi}.$ (12) The expectation values of the field modes can be written as $\langle{\hat{y}_{\textbf{k}}^{(R,I)}(\eta)}\rangle_{\Xi}=\sqrt{2}\Re\left(y_{k}(\eta)d_{\textbf{k}}^{(R,I)}\right),\qquad\langle{\hat{\pi}_{\textbf{k}}^{(R,I)}(\eta)}\rangle_{\Xi}=\sqrt{2}\Re\left(g_{k}(\eta)d_{\textbf{k}}^{(R,I)}\right),$ (13) while their uncertainties are $\displaystyle(\Delta\hat{y}_{\textbf{k}}^{(R,I)}(\eta))^{2}_{\Xi}$ $\displaystyle=\Re\left(y_{k}^{2}(\eta)c_{\textbf{k}}^{(R,I)}\right)$ (14) $\displaystyle+\frac{1}{2}\|y_{k}(\eta)\|^{2}\left(\hbar L^{3}+2e_{\textbf{k}}^{(R,I)}\right)-2\left[\Re\left(y_{k}(\eta)d_{\textbf{k}}^{(R,I)}\right)\right]^{2},$ $\displaystyle(\Delta\hat{\pi}_{\textbf{k}}^{(R,I)}(\eta))^{2}_{\Xi}$ $\displaystyle=\Re\left(g_{k}^{2}(\eta)c_{\textbf{k}}^{(R,I)}\right)$ (15) $\displaystyle+\frac{1}{2}\|g_{k}(\eta)\|^{2}\left(\hbar L^{3}+2e_{\textbf{k}}^{(R,I)}\right)-2\left[\Re\left(g_{k}(\eta)d_{\textbf{k}}^{(R,I)}\right)\right]^{2},$ specifically for the vacuum state $\|0\rangle$ one has, as expected, $d_{\textbf{k}}^{(R,I)}=c_{\textbf{k}}^{(R,I)}=e_{\textbf{k}}^{(R,I)}=0$, and thus $\langle{\hat{y}_{\textbf{k}}^{(R,I)}(\eta)}\rangle_{0}=0$, $\langle{\hat{\pi}_{\textbf{k}}^{(R,I)}(\eta)}\rangle_{0}=0$, and their corresponding uncertainties $\left(\Delta\hat{y}_{\textbf{k}}^{(R,I)}(\eta)\right)^{2}_{0}=\frac{1}{2}\|y_{k}(\eta)\|^{2}\hbar L^{3},\qquad\left(\Delta\hat{\pi}_{\textbf{k}}^{(R,I)}(\eta)\right)_{0}^{2}=\frac{1}{2}\|g_{k}(\eta)\|^{2}\hbar L^{3}.$ (16) Once we specify the expectation value of the field’s modes $\hat{y}^{(R,I)}_{\textbf{k}}$ and $\hat{\pi}^{(R,I)}_{\textbf{k}}$ in the post collapse state $\|{\Xi}\rangle$ at the time of collapse $\eta^{c}_{k}$ ($\|{0}\rangle\to\|{\Xi}\rangle$) $\langle{\hat{y}^{(R,I)}_{\textbf{k}}(\eta_{k}^{c})}\rangle_{\Xi}\equiv\langle\Xi\|\hat{y}^{(R,I)}_{\textbf{k}}(\eta_{k}^{c})\|\Xi\rangle,\qquad\langle{\hat{\pi}^{(R,I)}_{\textbf{k}}(\eta_{k}^{c})}\rangle_{\Xi}\equiv\langle\Xi\|\hat{\pi}^{(R,I)}_{\textbf{k}}(\eta_{k}^{c})\|\Xi\rangle,$ (17) we can obtain the expectation values evolved at any time after the collapse, provided that there is no additional collapse. In fact, by comparing (17) with (13) we obtain $\langle\hat{\pi}_{\textbf{k}}^{(R,I)}(\eta)\rangle_{\Xi}=A(\eta,\eta^{c}_{k})\langle\hat{\pi}_{\textbf{k}}^{(R,I)}(\eta^{c}_{k})\rangle_{\Xi}+B(\eta,\eta^{c}_{k})\langle\hat{y}_{\textbf{k}}^{(R,I)}(\eta^{c}_{k})\rangle_{\Xi},$ (18a) $\langle\hat{y}_{\textbf{k}}^{(R,I)}(\eta)\rangle_{\Xi}=C(\eta,\eta^{c}_{k})\langle\hat{\pi}_{\textbf{k}}^{(R,I)}(\eta^{c}_{k})\rangle_{\Xi}+D(\eta,\eta^{c}_{k})\langle\hat{y}_{\textbf{k}}^{(R,I)}(\eta^{c}_{k})\rangle_{\Xi},$ (18b) where $A,B,C$ and $D$ are time dependent functions which describe the temporal evolution of the quantum system between $\eta^{c}_{k}$ to $\eta$. In particular, in the inflationary stage these functions are $A(\eta,\eta^{c}_{k})=\cos(k\eta-k\eta^{c}_{k})+\frac{\sin(k\eta-k\eta^{c}_{k})}{k\eta^{c}_{k}},$ (18sa) $B(\eta,\eta^{c}_{k})=-k\sin(k\eta-k\eta^{c}_{k}),$ (18sb) $C(\eta,\eta^{c}_{k})=\frac{\cos(k\eta-k\eta^{c}_{k})}{k}\left(\frac{1}{k\eta}-\frac{1}{k\eta^{c}_{k}}\right)+\frac{\sin(k\eta-k\eta^{c}_{k})}{k}\left(\frac{1}{k^{2}\eta\eta^{c}_{k}}+1\right),$ (18sc) $D(\eta,\eta^{c}_{k})=\cos(k\eta-k\eta^{c}_{k})-\frac{\sin(k\eta-k\eta^{c}_{k})}{k\eta}.$ (18sd) Equations (18a) and (18b) can be rewritten in matrix form $\Upsilon(\eta,\Xi)=\textbf{U}(\eta,\eta^{c}_{k})\Upsilon(\eta^{c}_{k},\Xi),$ (18st) where $\Upsilon(\eta,\Xi)\equiv\left(\begin{array}[]{c}\langle\pi_{\textbf{k}}^{(R,I)}(\eta)\rangle_{\Xi}\\\ \langle y_{\textbf{k}}^{(R,I)}(\eta)\rangle_{\Xi}\end{array}\right),$ (18su) $\textbf{U}(\eta,\eta^{c}_{k})\equiv\left(\begin{array}[]{lr}A(\eta,\eta^{c}_{k})&B(\eta,\eta^{c}_{k})\\\ C(\eta,\eta^{c}_{k})&D(\eta,\eta^{c}_{k})\end{array}\right),$ (18sv) $\Upsilon(\eta^{c}_{k},\Xi)\equiv\left(\begin{array}[]{c}\langle\pi_{\textbf{k}}^{(R,I)}(\eta^{c}_{k})\rangle_{\Xi}\\\ \langle y_{\textbf{k}}^{(R,I)}(\eta^{c}_{k})\rangle_{\Xi}\end{array}\right).$ (18sw) In this notation, it is clear that the matrix $\textbf{U}(\eta,\eta^{c}_{k})$ represents the standard unitary evolution (this refers to the standard quantum mechanical evolution of states or operators as it might be the case, and should not be taken to mean that the matrix $\textbf{U}(\eta,\eta^{c}_{k})$ is unitary. It is not, and there is no reason for it to be so), for the expectation value of the fields, from the time $\eta^{c}_{k}$ to the arbitrary time $\eta$. The evolution of the uncertainties $(\Delta\hat{y}_{\textbf{k}}^{(R,I)}(\eta))^{2}_{\Xi}$ and $(\Delta\hat{\pi}_{\textbf{k}}^{(R,I)}(\eta))^{2}_{\Xi}$ depends on the specific post-collapse state. In particular, the quantities $c_{\textbf{k}}^{(R,I)}$ and $e_{\textbf{k}}^{(R,I)}$ depend on the state after the collapse. That is, once we specify the post-collapse state (and thus the quantities $c_{\textbf{k}}^{(R,I)}$ and $e_{\textbf{k}}^{(R,I)}$ are fixed), we can use (14) and (15) to obtain the evolution of the uncertainties. ### 2.1 Connection to Observations In order to connect the predicted quantities with the observed ones, we start from (4) $\nabla^{2}\Psi(\eta,\textbf{x})=s\delta\varphi^{\prime}(\eta,\textbf{x}).$ For the mode $\Psi_{\textbf{k}}$, after a Fourier’s decomposition, we obtain $\Psi_{\textbf{k}}(\eta)=\frac{-s}{k^{2}}\delta\varphi_{\textbf{k}}^{\prime}(\eta).$ (18sx) After describing the parametrization of the collapse in the previous section, we proceed to evaluate the perturbed metric using the the semi-classical Einstein’s Field Equations: $G_{ab}=8\pi G\langle{\hat{T}_{ab}}\rangle$ which we described at the beginning of this section. To lowest order this set of equations reduces to $\Psi_{\textbf{k}}(\eta)=\frac{-s}{ak^{2}}\langle{\hat{\pi}_{\textbf{k}}(\eta)}\rangle,$ (18sy) where we used that $\langle{\hat{\delta\varphi}^{\prime}_{\textbf{k}}}\rangle_{\Xi}$ is connected to the expectation value of the momentum field by $\langle{\hat{\delta\varphi}^{\prime}_{\textbf{k}}}\rangle_{\Xi}=\langle{\hat{\pi}_{\textbf{k}}}\rangle_{\Xi}/a(\eta)$ on the state $\|{\Xi}\rangle$. Recalling that $s\equiv a\hbar\sqrt{V\epsilon}/\sqrt{6}M_{P}^{2}$, the expression for the Newtonian potential is $\Psi_{\textbf{k}}(\eta)=-\frac{\hbar}{k^{2}M_{P}^{2}}\sqrt{\frac{V\epsilon}{6}}\langle{\hat{\pi}_{\textbf{k}}(\eta)}\rangle.$ (18sz) We note that before the collapse occurs, the state of the field is the Bunch- Davies vacuum for which $\langle{\hat{\pi}_{\textbf{k}}(\eta)}\rangle=0$, consequently $\Psi_{\textbf{k}}(\eta)=0$, and the spacetime is homogeneous and isotropic (at that scale). However, after the collapse takes place, the new state will generically have $\langle{\hat{\pi}_{\textbf{k}}(\eta)}\rangle\not=0$ and the gravitational perturbations appear. That is, the onset of the inhomogeneity and anisotropy at each scale is associated with the first collapse of the corresponding mode. In order to obtain a theoretical prediction and contrast it with the observations, we strictly can not use the expression of $\Psi_{\textbf{k}}(\eta)$ as given in (18sz) because it was obtained using the slow-roll approximation which is only valid in the inflationary epoch, while the observations made today by our satellites depend on the Newtonian potential at the last scattering surface. That is, the observations rely on $\Psi(\eta_{D},\textbf{x}_{D})$, with $\eta_{D}$ the time of decoupling and $\textbf{x}_{D}=R_{D}(\sin\theta\sin\phi,\sin\theta\cos\phi,\cos\theta)$ where $R_{D}$ is the radius of the last scattering surface, $\theta,\phi$ are the standard spherical coordinates in the sky The conformal time of decoupling lies in the matter dominated epoch. Nevertheless, we will work with the expression for $\Psi_{\textbf{k}}(\eta)$ in the radiation dominated era, extending if one wants its range of validity which is from $\eta_{r}$ to $\eta_{eq}<\eta_{D}$ (where $\eta_{eq}$ is the conformal time of the radiation-matter equality epoch). The changes during the brief period from the start of ‘matter domination’ to ‘decoupling’ (where the scale factor changes only by a factor of 3, i.e., $a(\eta_{D})/a(\eta_{eq})\approx 3$), are naturally considered to be irrelevant for the issues concerning us here, and thus the approximated value for the quantities of observational interest obtained using $\Psi(\eta)$ in the radiation dominated regime should be a very good approximation for the exact value of these quantities. Therefore, our goal here is to obtain an estimate for $\Psi_{\textbf{k}}(\eta)$ during the radiation epoch. The analysis can be simplified by working with a quantity whose evolution is rather simple, the so called ‘intrinsic curvature perturbation’ [31, 32, 33] $\zeta$, which is defined as $\zeta\equiv\frac{2}{3(w+1)}\left(\mathcal{H}^{-1}\Psi^{\prime}+\Psi\right)+\Psi,$ (18saa) where $w\equiv P/\rho$. During the inflationary regime, $P=-\frac{1}{2}g^{ab}\partial_{a}\phi\partial_{b}\phi-V$ represents the ‘pressure’ of the scalar field and $\rho=-\frac{1}{2}g^{ab}\partial_{a}\phi\partial_{b}\phi+V$ the energy density. It is a known result [2, 34] that $\zeta$ is, for modes larger than the Hubble radius (commonly referred as modes ‘larger than the horizon’ i.e., modes with $k\ll\mathcal{H}$) and for ‘adiabatic perturbations’, roughly a ‘constant quantity’, irrespective of the cosmological regime and the nature of the dominant kind of matter. The constancy of this quantity is used to obtain a relation between the values of the Newtonian potential during the two relevant regimes: $\Psi^{inf}_{\textbf{k}}(\eta)$ and $\Psi^{rad}_{\textbf{k}}(\eta)$ $\zeta^{inf}=\zeta^{rad}\qquad\Rightarrow\qquad\Psi^{inf}_{\textbf{k}}\bigg{[}\frac{2}{3}\bigg{(}\frac{1}{w_{inf}+1}\bigg{)}+1\bigg{]}=\frac{3}{2}\Psi^{rad}_{\textbf{k}},$ (18sab) where, in obtaining the right hand side of (18sab) the use of the equation of state $P=\rho/3$ was made, and the left hand side was obtained using the equation of state $P=w_{inf}\rho$ where $w_{inf}+1=\phi_{0}^{\prime 2}/a^{2}\rho$. Finally, by relying on the assumption of validity of the slow- roll approximation during inflation, $\phi_{0}^{\prime 2}/a^{2}=\frac{2}{3}V\epsilon$, (18sab) becomes $\Psi^{rad}_{\textbf{k}}=\frac{2}{3}\frac{\Psi^{inf}_{\textbf{k}}}{\epsilon}.$ (18sac) Thus, substituting (18sz) in (18sac), the expression for the Newtonian potential, in the radiation dominated epoch, becomes $\Psi^{rad}_{\textbf{k}}(\eta)=\frac{-\hbar}{2M_{P}^{2}}\sqrt{\frac{8V}{27\epsilon}}\frac{\langle{\hat{\pi}_{\textbf{k}}(\eta)}\rangle}{k^{2}}.$ (18sad) The expression above is valid for modes with $k\ll\mathcal{H}$, which are actually the modes of interest from the observational point of view. That is, we need to consider that $k/\mathcal{H}\ll 1$ in $\langle{\hat{\pi}_{\textbf{k}}(\eta)}\rangle$. Furthermore, the result (18sad) shows that for a generic collapse scheme there is an amplification $1/\epsilon$ in the Newtonian potential, in accordance with the generic findings of the detailed study for the collapse scheme presented in [16]. In order to connect with the observations we note that the quantity that is observed is $\frac{\delta T}{T_{0}}(\theta,\phi)$, which is expressed in terms of its spherical harmonic decomposition $\sum_{lm}\alpha_{lm}Y_{lm}(\theta,\phi)$. The theoretical calculations make a prediction for the most likely value of the coefficients $\alpha_{lm}$ which are expressed in terms of the Newtonian potential on the 2-sphere corresponding to the intersection of our past light cone with the surface of last scattering $\alpha_{lm}=\int d^{2}\Omega\Psi(\eta_{D},\textbf{x}_{D})Y_{lm}(\theta,\phi),$ (18sae) After a Fourier decomposition of the Newtonian potential $\Psi(\eta_{D},\textbf{x}_{D})=\sum_{\textbf{k}}(\Psi_{\textbf{k}}(\eta_{D})/L^{3})e^{i\textbf{k}\cdot\textbf{x}_{D}}$, and using (18sad), we obtain $\alpha_{lm}=\int d^{2}\Omega\sum_{\textbf{k}}\frac{-\hbar}{2M_{P}^{2}k^{2}L^{3}}\sqrt{\frac{8V}{27\epsilon}}\langle{\hat{\pi}_{\textbf{k}}(\eta_{D})}\rangle Y^{\star}_{lm}(\theta,\phi)e^{i\textbf{k}\cdot\textbf{x}_{D}}.$ (18saf) Using standard spherical harmonic relations: $e^{i\textbf{k}\cdot\textbf{x}_{D}}=4\pi\sum_{lm}i^{l}j_{l}(kR_{D})Y_{lm}(\theta,\phi)Y^{\star}_{lm}(\hat{\textbf{k}})$, where $j_{l}$ are spherical Bessel functions, we get $\alpha_{lm}=4\pi i^{l}\sum_{\textbf{k}}\frac{-\hbar}{2M_{P}^{2}k^{2}L^{3}}\sqrt{\frac{8V}{27\epsilon}}\langle{\hat{\pi}_{\textbf{k}}(\eta_{D})}\rangle j_{l}(kR_{D})Y^{\star}_{lm}(\hat{\textbf{k}}).$ (18sag) The quantity $\alpha_{lm}$ is the sum of contributions from the collection of modes, each contribution being a complex number, leading to what is in effect a sort of ‘two-dimensional random walk’ whose total displacement corresponds to the observational quantity (this will be seen more clearly in the next section when we specify $\langle{\hat{\pi}_{\textbf{k}}(\eta^{c}_{k})}\rangle$) . It is clear that, as in the case of any random walk, such quantity can not be evaluated and the only thing that can be done is to evaluate the most likely value for such total displacement, with the expectation that the observed quantity will be close to that value. As is now standard in our treatments, we do this with the help of the imaginary ensemble of universes and the identification of the most likely value with the ensemble’s mean value $\|\alpha_{lm}\|^{2}_{M.L.}=\frac{(4\pi)^{2}}{L^{6}}\sum_{\textbf{k}\textbf{k}^{\prime}}\frac{2\hbar^{2}V}{27\epsilon M_{P}^{4}}\frac{1}{k^{2}k^{\prime 2}}\overline{\langle\hat{\pi}_{\textbf{k}}(\eta_{D})\rangle\langle\hat{\pi}^{\dagger}_{\textbf{k}^{\prime}}(\eta_{D})\rangle}j_{l}(kR_{D})j_{l}(k^{\prime}R_{D})Y^{\star}_{lm}(\hat{\textbf{k}})Y_{lm}(\hat{\textbf{k}}^{\prime}).$ (18sah) The rest of the present work focuses on obtaining the quantity $\overline{\langle\hat{\pi}_{\textbf{k}}(\eta)\rangle\langle\hat{\pi}^{\dagger}_{\textbf{k}^{\prime}}(\eta)\rangle}$ under specific conditions on the post-collapses states. An important observation follows directly from the point of view adopted to relate the metric effective description of gravity with the quantum aspect of the matter fields: The source of the fluctuations that lead to anisotropies and inhomogeneities lies in the quantum uncertainties for the scalar field, which collapses, due to some unknown quantum gravitational effect. Once collapsed, these density inhomogeneities and anisotropies feed into the gravitational degrees of freedom leading to nontrivial perturbations in the metric functions, in particular, the Newtonian potential. However, the metric itself is not a source of the quantum gravitational induced collapse. Therefore, as the scalar field does not act as a source for the gravitational tensor modes -at least not at the lowest order considered here-, the tensor modes can not be excited. Thus, as already discussed in [10, 11], the scheme naturally leads to the prediction555However, it is worthwhile pointing out that such conclusion is directly tied to our underlying approach that favours the semi-classical Einstein’s equations augmented with a collapse proposal as a way to deal with the gravity quantum interface faced in the current problem. It is of course conceivable, although seems harder to understand in a wider context (see the discussion in section 8 of [9]), that a collapse might be incorporated into a setting where both the gravitation and scalar filed perturbations are simultaneously treated at the quantum level. If the latter happened to be the correct approach, something that would be possible to ascertain when we have a fully satisfactory theory of quantum gravity, our conclusion about the tensor modes would be modified. of a zero -or at least a strongly suppressed- amplitude of gravitational waves to the CMB. ### 2.2 Quantum collapse schemes In order to proceed, we must specify the quantum collapse scheme which drives the inflaton field out of homogeneity and isotropy. In past works [10, 14] three different schemes were considered. Two of them, called Independent collapse and Newtonian collapse were presented in [10] and the last one, denominated Wigner’s collapse, was presented in [14]. In [14] these schemes are further studied but limiting the consideration to a single collapse. In the following, we will describe them briefly. #### 2.2.1 Independent collapse scheme In this scheme one assumes that the expectation values of the field’s mode $\hat{y}^{(R,I)}_{\textbf{k}}$, and their conjugate momentum $\hat{\pi}^{(R,I)}_{\textbf{k}}$, acquire ‘random’ independent values. The expectation value was considered as randomly selected $\langle{\hat{y}^{(R,I)}_{\textbf{k}}(\eta^{c}_{\textbf{k}})}\rangle_{\Xi}=x_{\textbf{k},I}^{(R,I)}\sqrt{\left(\Delta\hat{y}_{\textbf{k}}^{(R,I)}(\eta_{k}^{c})\right)^{2}_{0}},\qquad\langle{\hat{\pi}^{(R,I)}_{\textbf{k}}(\eta^{c}_{k})}\rangle_{\Xi}=x_{\textbf{k},II}^{(R,I)}\sqrt{\left(\Delta\hat{\pi}^{(R,I)}_{\textbf{k}}(\eta_{k}^{c})\right)^{2}_{0}}.$ (18sai) In this scheme the expectation value jumps to a random value $x_{\textbf{k}}^{(R,I)}$ multiplied by the uncertainty of the vacuum state of the field. The random variables $x_{\textbf{k},I}^{(R,I)}$, $x_{\textbf{k},II}^{(R,I)}$ are selected from a Gaussian distribution centered at zero, of spread one (normalized), and are statistically uncorrelated, that is the rationale of the name. This means that we are ignoring the natural correlation that exists in the conjugate fields in the pre-collapse state. #### 2.2.2 Newtonian collapse scheme This scheme is motivated by the observation that in the Poisson-like equation (18sz), only the expectation value of $\hat{\pi}^{(R,I)}$ appears. Thus, following Penrose’s ideas regarding the quantum uncertainties that the gravitational potential would be inheriting from the matter fields’ quantum uncertainties, as fundamental factors triggering the collapse, one is led to consider a scheme where ‘only $\hat{\pi}^{(R,I)}$ collapses’, leaving the expectation value of $\hat{y}^{(R,I)}$ unchanged $\langle{\hat{y}^{(R,I)}_{\textbf{k}}(\eta^{c}_{k})}\rangle_{\Xi}=0,\qquad\langle{\hat{\pi}^{(R,I)}_{\textbf{k}}(\eta^{c}_{k})}\rangle_{\Xi}=x_{\textbf{k},II}^{(R,I)}\sqrt{\left(\Delta\hat{\pi}^{(R,I)}_{\textbf{k}}(\eta_{k}^{c})\right)^{2}_{0}}.$ (18saj) As before, $x_{\textbf{k},II}^{(R,I)}$ represents a random Gaussian variable normalized and centered at zero. #### 2.2.3 Wigner’s collapse scheme The last collapse scheme considered in [14, 15] attempts to take into account the correlation between $\hat{y}^{(R,I)}$ and $\hat{\pi}^{(R,I)}$ existing in the pre-collapse state, and to characterize it in terms of the Wigner’s function. The Wigner’s function of the vacuum state of the inflaton is a bi- dimensional Gaussian function. This fact will be used to model the resulting collapse of the quantum field state. The assumption will be that, at a certain (conformal) time $\eta^{c}_{k}$, the part of the state characterizing the mode $k$ will collapse, leading to a new state in which the fields will have expectation values given by $\langle{\hat{y}^{(R,I)}_{\textbf{k}}(\eta_{k}^{c})}\rangle_{\Xi}=x^{(R,I)}_{\textbf{k}}\Lambda_{k}\cos\Theta_{k},\qquad\langle{\hat{\pi}^{(R,I)}_{\textbf{k}}(\eta_{k}^{c})}\rangle_{\Xi}=x^{(R,I)}_{\textbf{k}}\Lambda_{k}k\sin\Theta_{k},$ (18sak) where $x_{\textbf{k}}^{(R,I)}$ is a random variable, characterized by a Gaussian distribution centered at zero with a spread one. $\Lambda_{k}$ is given by the major semi-axis of the ellipse characterizing the bi-dimensional Gaussian function (the ellipse corresponds to the boundary of the region in ‘phase space’ where the Wigner function has a magnitude larger than 1/2 its maximum value), and $\Theta_{k}$ is the angle between that axis and the $\hat{y}_{\textbf{k}}^{(R,I)}$ axis. The quantities $\Lambda_{k}$ and $\Theta_{k}$ can be expressed in terms of $\eta^{c}_{k}$ [14] as $\Lambda_{k}=\frac{4\eta^{c}_{k}\sqrt{\hbar L^{3}k}}{\sqrt{1+5(k\eta^{c}_{k})^{2}-\sqrt{1+10(k\eta^{c}_{k})^{2}+9(k\eta^{c}_{k})^{4}}}},$ (18sal) $2\Theta_{k}=\arctan\left(\frac{4k\eta^{c}_{k}}{1-3(k\eta^{c}_{k})^{2}}\right).$ (18sam) ## 3 Multiple Quantum Collapses Once one hypothesizes that there is a new kind of physical process which affects the system under investigation, it seems logical to consider the possibility that it occurs more than once, and in circumstances different from those for which it was first proposed. The extensive study of that issue is well beyond the present manuscript and would require its merger with other studies of collapse models for more general circumstances. However, the cosmological situation is one where further analysis can be done with relative ease and where it is natural to assume that the effect must manifest in a rather unmodified version in all its occurrences. This work can be considered as the first exploration (see also [15]) of the effects of multiple collapses in the situation which lead to the first proposals suggesting that they play a fundamental role in cosmology. ### 3.1 Collapse Scheme Relations for Multiple Collapses We will analyze the provided schemes in the case of multiple quantum collapses by focusing on the expectation values of the relevant quantities at any time after exactly $n$ quantum collapses. First, we will generalize the notation in order to handle in a unified fashion the three different quantum collapse schemes (developed in [10, 14]). Let us assume that at time $\eta_{k}^{c_{1}}$ has occurred a single collapse taking the state $\|c_{0}\rangle$ to the state $\|{c_{1}}\rangle$666In this section, we have changed the notation slightly, the post-collapse state will be denoted $\|c_{n}\rangle$ instead of $\|\Xi\rangle$ as in the previous section. That is, the state $\|c_{o}\rangle$, represents the vacuum state; $\|c_{1}\rangle$ will denote the first collapse state and so on.. Then, a natural generalization of the expectation value of the field (and its conjugated momentum) in the post- collapse state $\|{c_{1}}\rangle$ will be assumed to be given by $\langle{\hat{y}^{(R,I)}_{\textbf{k}}(\eta_{k}^{c_{1}})}\rangle_{c_{1}}=x_{\textbf{k},I}^{(1)(R,I)}\sigma_{y}^{(R,I)}(\eta_{k}^{c_{1}},c_{0})+\langle{\hat{y}^{(R,I)}_{\textbf{k}}(\eta_{k}^{c_{1}})}\rangle_{c_{0}},$ (18san) $\langle{\hat{\pi}^{(R,I)}_{\textbf{k}}(\eta_{k}^{c_{1}})}\rangle_{c_{1}}=x_{\textbf{k},II}^{(1)(R,I)}\sigma_{\pi}^{(R,I)}(\eta_{k}^{c_{1}},c_{0})+\langle{\hat{\pi}^{(R,I)}_{\textbf{k}}(\eta_{k}^{c_{1}})}\rangle_{c_{0}},$ (18sao) where $x_{\textbf{k},I}^{(1)(R,I)}$ and $x_{\textbf{k},II}^{(1)(R,I)}$ stand for a random value characterizing the change in the expectation value of $\hat{y}^{(R,I)}_{\textbf{k}}$ and $\hat{\pi}^{(R,I)}_{\textbf{k}}$ respectively. The superscript (1) indicates that the random variables are associated with the first collapse while the quantities in the last term of the right hand side of (18san) and (18sao) represent the value of the corresponding operators if there had been no collapse. The correlation between these random variables depends on the particular collapse scheme. The functions $\sigma_{y}^{(R,I)}(\eta_{k}^{c_{1}},c_{0})$ and $\sigma_{\pi}^{(R,I)}(\eta_{k}^{c_{1}},c_{0})$ denote the uncertainties of the expectation values of the fields for the particular collapse scheme considered. The notation employed remind us the principal quantities that characterize the expectation values. That is, it depends on: the previous collapse state $\|{c_{0}}\rangle$ (which in the case of a single collapse is the vacuum state), the time of collapse $\eta_{k}^{c_{1}}$, and the random variables $x_{\textbf{k},I}^{(1)(R,I)},x_{\textbf{k},II}^{(1)(R,I)}$. Note that the left hand side (l.h.s.) of (18san) and (18sao) is in the post- collapse state $\|{c_{1}}\rangle$, while the right hand side (r.h.s.) is in the pre-collapse state $\|{c_{0}}\rangle$, i.e., the new state depends on the old state. Let us also note that the whole expressions (18san) and (18sao) are evaluated in $\eta_{k}^{c_{{1}}}$, the time at which the first collapse occurs. The second term in the r.h.s. is the expectation value of the mode in the state $\|c_{0}\rangle$ evolved up to $\eta_{k}^{c_{{1}}}$. This dynamical evolution is dictated by (18st). The generalization of these ideas allow us to write the collapse scheme for the $n$-th collapse $\|{c_{n-1}}\rangle\to\|{c_{n}}\rangle$ $\langle\hat{y}^{(R,I)}_{\textbf{k}}(\eta_{k}^{c_{{n}}})\rangle_{c_{n}}=x_{\textbf{k},I}^{(n)(R,I)}\sigma_{y}^{(R,I)}(\eta_{k}^{c_{n}},c_{n-1})+\langle\hat{y}^{(R,I)}_{\textbf{k}}(\eta_{k}^{c_{{n}}})\rangle_{c_{n-1}},$ (18sap) $\langle\hat{\pi}^{(R,I)}(\eta_{k}^{c_{{n}}})\rangle_{c_{n}}=x_{\textbf{k},II}^{(n)(R,I)}\sigma_{\pi}^{(R,I)}(\eta_{k}^{c_{n}},c_{n-1})+\langle\hat{\pi}^{(R,I)}_{\textbf{k}}(\eta_{k}^{c_{{n}}})\rangle_{c_{n-1}}.$ (18saq) The second term of the r.h.s. of (18sap) and (18saq) is the expectation value of the $(n-1)$-th collapse evaluated at the $n$-th collapse time $\eta_{k}^{c_{{n}}}$. If we employ the matrix notation introduced in the previous section, we can rewrite (18sap) and (18saq) as $\Upsilon(\eta_{k}^{c_{{n}}},c_{n})=\Delta(x_{\textbf{k},i}^{(n)(R,I)},\eta_{k}^{c_{{n}}},c_{n-1})+\Upsilon(\eta_{k}^{c_{{n}}},c_{n-1}),$ (18sar) where we introduced a new object $\Delta(x_{\textbf{k},i}^{(n)(R,I)},\eta_{k}^{c_{{n}}},c_{n-1})\equiv\left(\begin{array}[]{c}x_{\textbf{k},I}^{(n)(R,I)}\sigma_{y}^{(R,I)}(\eta_{k}^{c_{n}},c_{n-1})\\\ x_{\textbf{k},II}^{(n)(R,I)}\sigma_{\pi}^{(R,I)}(\eta_{k}^{c_{n}},c_{n-1})\end{array}\right),$ with $i=I,II$. ### 3.2 Evolution between collapses Equation (18st) characterizes the evolution of the state between two successive collapses, e.g., $n$ and $n-1$. In other words, this means that (18st) is valid from $\eta_{k}^{c_{{n-1}}}$ (their initial condition) to $\eta_{k}^{c_{{n}}}$. We can rewrite (18st), with the notation adopted in this section, in order to see this more clearly $\Upsilon(\eta,c_{n})=\textbf{U}(\eta,\eta_{k}^{c_{{n}}})\Upsilon(\eta_{k}^{c_{{n}}},c_{n}).$ (18sas) Thus, the evolution from $\eta_{k}^{c_{{n-1}}}$ to $\eta_{k}^{c_{{n}}}$, of the expectation values of the state $\|{c_{n-1}}\rangle$, is determined by (18sas) but using, as initial condition, the expectation value given by the collapse $n-2$, which evolved in a similar manner. This will lead us to a recursive relation for the dynamical equation of the field’s expectation value after $n$ collapses. Note that this description is just the orthodox quantum evolution following the standard rules of quantum mechanics: between ‘measurements’ the wave function evolves following Schrödinger’s equation, and at the times when the ‘measurements’ occur, the wave function is ‘collapsed’ or ‘reduced’. Then the wave function continues to evolve according to Schrödinger’s equation but now with the initial condition of the ‘post- measurement’ quantum state, etc. On account of the discussion above, we note that (18sas) depends on the $(n-1)^{th},(n-2)^{th},...,1^{st}$ collapse states. Therefore, we will obtain a new expression for (18sas) which will show this dependance explicitly. We start by substituting (18sar) in (18sas) obtaining $\Upsilon(\eta,c_{n})=\textbf{U}(\eta,\eta_{k}^{c_{{n}}})\Delta(x_{\textbf{k},i}^{(n)(R,I)},\eta_{k}^{c_{{n}}},c_{n-1})+\textbf{U}(\eta,\eta_{k}^{c_{{n}}})\Upsilon(\eta_{k}^{c_{{n}}},c_{n-1}).$ (18sat) The quantity $\Upsilon(\eta_{k}^{c_{{n}}},c_{n-1})$ contains information of the expectation value of the fields in the state $\|{c_{n-1}}\rangle$ at the time $\eta_{k}^{c_{{n}}}$, but (18sas) give us the value of $\Upsilon$ for any time $\eta$ and any state $\|{c_{n}}\rangle$. In other words, we can use (18sas) and the collapse ‘recipe’ (18sar) ( to obtain $\Upsilon(\eta_{k}^{c_{{n-1}}},c_{n-1})$) to calculate $\Upsilon(\eta_{k}^{c_{{n}}},c_{n-1})$. This calculation will result in a term $\Upsilon(\eta_{k}^{c_{{n-1}}},c_{n-2})$ which, again, can be computed from (18sas) and (18sar), therefore, (18sat) is a recursive relation which depends explicitly from the very first to the $(n-1)$-th post-collapse state. For example, if a single collapse occurs, we have $\Upsilon(\eta,c_{1})=\textbf{U}(\eta,\eta_{k}^{c_{{1}}})\Delta(x_{\textbf{k},i}^{(1)(R,I)},\eta_{k}^{c_{{1}}},c_{0}),$ (18sau) because $\|c_{0}\rangle$ is taken to be the vacuum and $\Upsilon(\eta_{k}^{c_{{1}}},c_{0})=0$. For two collapses one obtains $\Upsilon(\eta,c_{2})=\textbf{U}(\eta,\eta_{k}^{c_{{2}}})\Delta(x_{\textbf{k},i}^{(2)(R,I)},\eta_{k}^{c_{{2}}},c_{1})+\textbf{U}(\eta,\eta_{k}^{c_{{2}}})\textbf{U}(\eta_{k}^{c_{{2}}},\eta_{k}^{c_{{1}}})\Delta(x_{\textbf{k},i}^{(1)(R,I)},\eta_{k}^{c_{{1}}},c_{0}).$ (18sav) Thus, the general expression for $\Upsilon(\eta,c_{n})$ after $n$ collapses is $\displaystyle\Upsilon(\eta,c_{n})$ $\displaystyle=\textbf{U}(\eta,\eta_{k}^{c_{{n}}})\Delta(x_{\textbf{k},i}^{(n)(R,I)},\eta_{k}^{c_{{n}}},c_{n-1})$ (18saw) $\displaystyle+\textbf{U}(\eta,\eta_{k}^{c_{{n}}})\textbf{U}(\eta_{k}^{c_{{n}}},\eta_{k}^{c_{{n-1}}})\Delta(x_{\textbf{k},i}^{(n-1)(R,I)},\eta_{k}^{c_{{n-1}}},c_{n-2})$ $\displaystyle+\textbf{U}(\eta,\eta_{k}^{c_{{n}}})\textbf{U}(\eta_{k}^{c_{{n}}},\eta_{k}^{c_{{n-1}}})\textbf{U}(\eta_{k}^{c_{{n-1}}},\eta_{k}^{c_{{n-2}}})\Delta(x_{\textbf{k},i}^{(n-2)(R,I)},\eta_{k}^{c_{{n-2}}},c_{n-3})+\dots$ $\displaystyle+\textbf{U}(\eta,\eta_{k}^{c_{{n}}})\textbf{U}(\eta_{k}^{c_{{n}}},\eta_{k}^{c_{{n-1}}})\textbf{U}(\eta_{k}^{c_{{n-1}}},\eta_{k}^{c_{{n-2}}})\textbf{U}(\eta_{k}^{c_{{n-2}}},\eta_{k}^{c_{{n-3}}})\times\dots$ $\displaystyle\times\textbf{U}(\eta_{k}^{c_{{2}}},\eta_{k}^{c_{{1}}})\Delta(x_{\textbf{k},i}^{(1)(R,I)},\eta_{k}^{c_{{1}}},c_{0}).$ From (18sas) it is evident that the matrix $\textbf{U}(\eta_{k}^{c_{{n}}},\eta_{k}^{c_{{n-1}}})$ represents the unitary evolution for the expectation value of the fields in the state $\|{c_{n-1}}\rangle$ from $\eta_{k}^{c_{{n-1}}}$ to $\eta_{k}^{c_{{n}}}$. Because of the unitary evolution, we have $\textbf{U}(\eta,\eta_{k}^{c_{{n}}})\textbf{U}(\eta_{k}^{c_{{n}}},\eta_{k}^{c_{{n-1}}})\dots\textbf{U}(\eta_{k}^{c_{{2}}},\eta_{k}^{c_{{1}}})=\textbf{U}(\eta,\eta_{k}^{c_{{1}}})$. Using this property in (18saw), we finally obtain $\Upsilon(\eta,\eta_{k}^{c_{{n}}})=\sum_{m=1}^{n}\textbf{U}(\eta,\eta_{k}^{c_{{m}}})\Delta(x_{\textbf{k},i}^{(m)(R,I)},\eta_{k}^{c_{{m}}},c_{m-1}).$ (18sax) Equation (18sax) allows us to extract the evolution for the expectation value of of $\hat{\pi}^{(R,I)}_{\textbf{k}}(\eta)$ after $n$ collapses $\displaystyle\langle\hat{\pi}_{\textbf{k}}^{(R,I)}(\eta)\rangle_{c_{n}}$ $\displaystyle=\sum_{m=1}^{n}\Bigg{[}-k\sin(k\eta-k\eta_{k}^{c_{{m}}})x_{\textbf{k},I}^{(m)(R,I)}\sigma^{(R,I)}_{y}(\eta_{k}^{c_{{m}}},c_{m-1})$ (18say) $\displaystyle+\Bigg{(}\cos(k\eta-k\eta_{k}^{c_{{m}}})+\frac{\sin(k\eta-k\eta_{k}^{c_{{m}}})}{k\eta_{k}^{c_{{m}}}}\Bigg{)}x_{\textbf{k},II}^{(m)(R,I)}\sigma^{(R,I)}_{\pi}(\eta_{k}^{c_{{m}}},c_{m-1})\Bigg{]}.$ The result (18say) is the generalization of (18a) for multiple collapses during the inflationary epoch. We observe that the evolution of $\hat{\pi}^{(R,I)}_{\textbf{k}}$ resembles a superposition of many independent one-collapse evolutions of the expectation values of $\hat{\pi}^{(R,I)}_{\textbf{k}}$, any of which suffered a collapse at different times. As mentioned at the end of section 2.1, to connect the theoretical predictions with the observational quantities, we need to compute $\|\alpha_{lm}\|^{2}_{M.L.}$ as given in (18sah). That is, we need to obtain $\overline{\langle{\hat{\pi}_{\textbf{k}}(\eta)}\rangle_{c_{n}}\langle{\hat{\pi}^{{\dagger}}_{\textbf{k}^{\prime}}(\eta)}\rangle_{c_{n}}}$, which will be our next task. First, we note that, in the notation introduced in (18san) and (18sao), the expectation value of each collapse scheme was decomposed generically as $x_{\textbf{k},I}^{(1)(R,I)}\sigma_{y}^{(R,I)}(\eta_{k}^{c_{1}},c_{0})$ and $x_{\textbf{k},II}^{(1)(R,I)}\sigma_{\pi}^{(R,I)}(\eta_{k}^{c_{1}},c_{0})$, where the random variables are dimensionless, and $\sigma_{y},\sigma_{\pi}$ are the part of the collapse scheme carrying the units (e.g., in the independent collapse scheme $\sigma_{\pi}^{(R,I)}(\eta_{k}^{c_{1}},c_{0})=\sqrt{\hbar L^{3}/2}\|g_{k}(\eta)\|$ and $\sigma_{y}^{(R,I)}(\eta_{k}^{c_{1}},c_{0})=\sqrt{\hbar L^{3}/2}\|y_{k}(\eta)\|$). The mean value of the product of $x_{\textbf{k},i}^{(n)(R,I)}$ depends on the particular scheme considered. In the _Independent_ scheme, the product mean value is given by $\overline{x_{\textbf{k},i}^{(n)R}x_{\textbf{k}^{\prime},i}^{(m)R}}=(\delta_{\textbf{k},\textbf{k}^{\prime}}+\delta_{\textbf{k},-\textbf{k}^{\prime}})\delta_{m,n},\qquad\overline{x_{\textbf{k},i}^{(m)I}x_{\textbf{k}^{\prime},i}^{(n)I}}=(\delta_{\textbf{k},\textbf{k}^{\prime}}-\delta_{\textbf{k},-\textbf{k}^{\prime}})\delta_{m,n},$ (18saz) where $i=I,II$, and with all the other possible combinations equal zero. Meanwhile, in the _Newtonian_ scheme it is $\overline{x_{\textbf{k},II}^{(n)R}x_{\textbf{k}^{\prime},II}^{(m)R}}=(\delta_{\textbf{k},\textbf{k}^{\prime}}+\delta_{\textbf{k},-\textbf{k}^{\prime}})\delta_{m,n},\qquad\overline{x_{\textbf{k},II}^{(m)I}x_{\textbf{k}^{\prime},II}^{(n)I}}=(\delta_{\textbf{k},\textbf{k}^{\prime}}-\delta_{\textbf{k},-\textbf{k}^{\prime}})\delta_{m,n},$ (18sba) and in the _Wigner’s_ scheme we have $\overline{x_{\textbf{k}}^{(n)R}x_{\textbf{k}^{\prime}}^{(m)R}}=(\delta_{\textbf{k},\textbf{k}^{\prime}}+\delta_{\textbf{k},-\textbf{k}^{\prime}})\delta_{m,n},\qquad\overline{x_{\textbf{k}}^{(m)I}x_{\textbf{k}^{\prime}}^{(n)I}}=(\delta_{\textbf{k},\textbf{k}^{\prime}}-\delta_{\textbf{k},-\textbf{k}^{\prime}})\delta_{m,n}.$ (18sbb) The $\delta_{m,n}$ means that, in the three schemes, we are assuming independency among the random variables associated to different collapses respectively (e.g., the random variable $x_{\textbf{k},I}^{(1)I}$ is independent of $x_{\textbf{k},I}^{(2)I}$). After choosing a particular collapse scheme (with the corresponding characterization for the mean value of the random variables (18saz), (18sba), (18sbb)), and recalling that $\langle{\hat{\pi}_{\textbf{k}}(\eta)}\rangle_{c_{n}}=\langle{\hat{\pi}_{\textbf{k}}^{R}(\eta)}\rangle_{c_{n}}+i\langle{\hat{\pi}_{\textbf{k}}^{I}(\eta)}\rangle_{c_{n}}$ (using (18say)), one obtains the quantity $\overline{\langle{\hat{\pi}_{\textbf{k}}(\eta)}\rangle_{c_{n}}\langle{\hat{\pi}^{\dagger}_{\textbf{k}^{\prime}}(\eta)}\rangle_{c_{n}}}$ for each collapse scheme. The calculation will be simplified due to the fact that, as usual, the average over the random variables (in the three collapse schemes) will lead to a cancellation of the cross terms. Thus, after going to the continuum limit ($L\rightarrow\infty$), the expression for $\|\alpha_{lm}\|^{2}_{M.L.}$ (18sah), after $N$ collapses is given by $\|\alpha_{lm}\|_{M.L.}^{2}=\frac{4}{27\pi}\frac{V\hbar^{3}}{\epsilon M_{P}^{4}}\int\frac{dk}{k^{2}}\|j_{l}(kR_{D})\|^{2}\sum_{n=1}^{N}C^{(n)}_{l}(k,\eta_{D}),$ (18sbc) where $C^{(n)}_{l}(k,\eta_{D})$ depends on the collapse scheme considered. In the _independent_ scheme, this expression is $\displaystyle C^{(n)}_{l}(k,\eta_{D})$ $\displaystyle=\bigg{(}k\sin(k\eta_{D}-k\eta_{k}^{c_{{n}}})\bigg{)}^{2}\bigg{(}Y_{\textbf{k}}^{+}+(-1)^{l}Y_{\textbf{k}}^{-}\bigg{)}$ (18sbd) $\displaystyle+\bigg{(}\cos(k\eta_{D}-k\eta_{k}^{c_{{n}}})+\frac{\sin(k\eta_{D}-k\eta_{k}^{c_{{n}}})}{k\eta_{k}^{c_{{n}}}}\bigg{)}^{2}\bigg{(}\Pi_{\textbf{k}}^{+}+(-1)^{l}\Pi_{\textbf{k}}^{-}\bigg{)},$ meanwhile, in the case of the _Newtonian_ scheme $C^{(n)}_{l}(k,\eta_{D})=\bigg{(}\cos(k\eta_{D}-k\eta_{k}^{c_{{n}}})+\frac{\sin(k\eta_{D}-k\eta_{k}^{c_{{n}}})}{k\eta_{k}^{c_{{n}}}}\bigg{)}^{2}\bigg{(}\Pi_{\textbf{k}}^{+}+(-1)^{l}\Pi_{\textbf{k}}^{-}\bigg{)},$ (18sbe) the quantities $Y_{\textbf{k}}^{\pm}$ and $\Pi_{\textbf{k}}^{\pm}$ are defined as $Y_{\textbf{k}}^{\pm}\equiv(\Delta\hat{y}_{\textbf{k}}^{R}(\eta_{k}^{c_{n}}))^{2}_{c_{n-1}}\pm(\Delta\hat{y}_{\textbf{k}}^{I}(\eta_{k}^{c_{n}}))^{2}_{c_{n-1}},\quad\Pi_{\textbf{k}}^{\pm}\equiv(\Delta\hat{\pi}_{\textbf{k}}^{R}(\eta_{k}^{c_{n}}))^{2}_{c_{n-1}}\pm(\Delta\hat{\pi}_{\textbf{k}}^{I}(\eta_{k}^{c_{n}}))^{2}_{c_{n-1}}.$ (18sbf) Finally, in the _Wigner’s_ scheme, $C^{(n)}_{l}(k,\eta)$ is given by $\displaystyle C^{(n)}_{l}(k,\eta_{D})$ $\displaystyle=2k^{2}\Lambda_{k,n}^{2}\bigg{[}\sin(k\eta_{D}-k\eta_{k}^{c_{{n}}})\sin\Theta_{k,n}$ (18sbg) $\displaystyle+\bigg{(}\cos(k\eta_{D}-k\eta_{k}^{c_{{n}}})+\frac{\sin(k\eta_{D}-k\eta_{k}^{c_{{n}}})}{k\eta_{k}^{c_{{n}}}}\bigg{)}\cos\Theta_{k,n}\bigg{]}^{2},$ with $\Lambda_{k,n}$ and $\Theta_{k,n}$ defined as $\Lambda_{k,n}\equiv\frac{4\eta_{k}^{c_{{n}}}\sqrt{\hbar k}}{\sqrt{1+5(k\eta_{k}^{c_{{n}}})^{2}-\sqrt{1+10(k\eta_{k}^{c_{{n}}})^{2}+9(k\eta_{k}^{c_{{n}}})^{4}}}},$ (18sbha) $2\Theta_{k,n}\equiv\arctan\left(\frac{4k\eta_{k}^{c_{{n}}}}{1-3(k\eta_{k}^{c_{{n}}})^{2}}\right).$ (18sbhb) It is worthwhile to comment that uncertainties in (18sbc) are always evaluated at the $(n-1)$-th collapse state. Before discussing the physical implications of the general result (18sbc), let us start by analyzing the assumption of a single collapse in the _independent_ scheme, in this case (18sbc) reduces to $\|\alpha_{lm}\|_{M.L.}^{2}=\frac{4V\hbar^{3}}{54\pi\epsilon M_{P}^{4}}\int\frac{dk}{k}\|j_{l}(kR_{D})\|^{2}\left(1+2\frac{\sin^{2}(k\eta_{D}-k\eta_{k}^{c_{{1}}})}{(k\eta_{k}^{c_{{1}}})^{2}}+\frac{\sin 2(k\eta_{D}-k\eta_{k}^{c_{{1}}})}{k\eta_{k}^{c_{{1}}}}\right).$ (18sbhbi) Considering again a single collapse and working within the _Newtonian_ scheme, (18sbc) leads to $\displaystyle\|\alpha_{lm}\|_{M.L.}^{2}$ $\displaystyle=\frac{4V\hbar^{3}}{54\pi\epsilon M_{P}^{4}}\int\frac{dk}{k}\|j_{l}(kR_{D})\|^{2}$ (18sbhbj) $\displaystyle\times\left[1+\sin^{2}(k\eta_{D}-k\eta_{k}^{c_{{1}}})\left(\frac{1}{(k\eta_{k}^{c_{{1}}})^{2}}-1\right)+\frac{\sin 2(k\eta_{D}-k\eta_{k}^{c_{{1}}})}{k\eta_{k}^{c_{{1}}}}\right].$ Results (18sbhbi) and (18sbhbj) are consistent with the findings presented in [10] and [14]. The result obtained from (18sbc), for a single collapse in the _Wigner’s_ scheme, also corresponds with the one presented in [14]. We observe that, for the three schemes considered, in the case of a single collapse $N=1$, only the uncertainties of the vacuum state contribute to the integral in (18sbc). The point is that for a single collapse, (18sbc) does not contain any information characterizing the post-collapse state (the information that defines a particular post-collapse state is contained in the uncertainties evaluated in that precise state). That is, we do not need to specify the post-collapse state. However, if we assume multiple collapses, then the uncertainties of the post-collapse states will contribute to the integral in (18sbc), and since the uncertainties will depend on the pre- collapsed states, which are now different from the vacuum, then we will need to specify every pre-collapse state (which will be the subject of the next section). An important feature arises in the _independent_ and _Newtonian_ schemes, since for these cases, (18sbc) exhibits an explicit dependence of $l$ (there is also another dependence on $l$ in the term $\|j_{l}(kR_{D})\|^{2}$, however this dependence will not affect the compatibility of the theoretical predictions obtained in our approach with the ones from the standard treatment, since the latter, also involves this dependence on $l$ in the spherical Bessel function $j_{l}(kR_{D})$) in the terms $Y_{\textbf{k}}^{+}+(-1)^{l}Y_{\textbf{k}}^{-}$ and $\Pi_{\textbf{k}}^{+}+(-1)^{l}\Pi_{\textbf{k}}^{-}$. If $l$ is even, $Y_{\textbf{k}}^{+}+(-1)^{l}Y_{\textbf{k}}^{-}=2(\Delta\hat{y}_{\textbf{k}}^{R}(\eta_{k}^{c_{n}}))^{2}_{c_{n-1}}$ and $\Pi_{\textbf{k}}^{+}+(-1)^{l}\Pi_{\textbf{k}}^{-}=2(\Delta\hat{\pi}_{\textbf{k}}^{R}(\eta_{k}^{c_{n}}))^{2}_{c_{n-1}}$; if $l$ is odd, $Y_{\textbf{k}}^{+}+(-1)^{l}Y_{\textbf{k}}^{-}=2(\Delta\hat{y}_{\textbf{k}}^{I}(\eta_{k}^{cn}))^{2}_{c_{n-1}}$ and $\Pi_{\textbf{k}}^{+}+(-1)^{l}\Pi_{\textbf{k}}^{-}=2(\Delta\hat{\pi}_{\textbf{k}}^{I}(\eta_{k}^{c_{n}}))^{2}_{c_{n-1}}$. Thus, depending on the parity of $l$, the predicted quantity $\|\alpha_{lm}\|_{M.L.}^{2}$ will involve the uncertainty of the real or imaginary parts of $\hat{y}_{\textbf{k}}(\eta_{k}^{c_{n}})$ and $\hat{\pi}_{\textbf{k}}(\eta_{k}^{c_{n}})$ which is not entirely compatible with the standard prediction, namely a flat spectrum. In order to recover the standard theoretical prediction, the dependence of $l$ should be avoided, and the most natural option is that the uncertainties satisfy $(\Delta\hat{y}_{\textbf{k}}^{R}(\eta_{k}^{c_{n}}))^{2}_{c_{n-1}}=(\Delta\hat{y}_{\textbf{k}}^{I}(\eta_{k}^{c_{n}}))^{2}_{c_{n-1}},\qquad(\Delta\hat{\pi}_{\textbf{k}}^{R}(\eta_{k}^{c_{n}}))^{2}_{c_{n-1}}=(\Delta\hat{\pi}_{\textbf{k}}^{I}(\eta_{k}^{c_{n}}))^{2}_{c_{n-1}}.$ (18sbhbk) It is clear, of course, that this is not the most generic case777However, as we will show in the next section, if we assume that the post-collapse states are coherent states, this condition is fulfilled automatically., and needs not to be taken as a necessary condition for the compatibility of the theoretical predictions in our approach with the observations from the CMB, since we still need to consider the physics of the cosmological epochs after the end of the inflationary regime that leads to the so called acoustic oscillations. It is also interesting to note, that the condition on the uncertainties of the real and imaginary parts of $\hat{y}_{\textbf{k}}(\eta_{k}^{c_{n}})$ and $\hat{\pi}_{\textbf{k}}(\eta_{k}^{c_{n}})$, only applies to the _independent_ and _Newtonian_ schemes. In the _Wigner_ scheme we do not find a similar condition for the parameters $\Lambda_{k,n}$ and $\Theta_{k,n}$ that characterize the uncertainties in that case. Finally, we note that all the quantities involved in (18sbc) are positive. In other words, we have a sum of positive definite terms. Therefore, if we set $N\rightarrow\infty$ the sum will generically diverge, which implies that we can not set an infinite number of collapses because the predicted value for $\|\alpha_{lm}\|^{2}_{M.L.}$ will tend to infinity. Thus we must restrict consideration to the case with a finite number of collapses. It is important to note that the calculations that lead to result (18sbc) have not considered any particular post-collapse state. Of course (and we will do it in the next section), we can consider a particular post-collapse state and that information will enter in the uncertainties. However, the conclusion obtained from (18sbc) related to the finiteness of the collapses is valid for a generic post-collapse state in the three schemes considered. ## 4 Characterization of the post-collapse states The information that characterizes a particular post-collapse state will enter in the uncertainties of the field (and its momentum) through the parameters that characterize the post-collapse state. Thus, our first task will be to focus on obtaining the uncertainties for the coherent and squeezed states and afterwards we are going to use the results of the previous sections to obtain predicted values for the observational quantities. ### 4.1 Coherent states as post-collapse states A simple election for a post-collapse state is a coherent state. A coherent state is a specific state of the harmonic oscillator and its dynamic is very similar to the one of the classic harmonic oscillator. The coherent states $\|\xi\rangle$ are defined as the eigenstates of the annihilation operator $\hat{a}$, $\hat{a}\|\xi\rangle=\xi\|\xi\rangle,$ (18sbhbl) since $\hat{a}$ is not an Hermitian operator, $\xi$ is a complex number and can be represented in complex polar form $\xi=\|\xi\|e^{i\chi}$, where $\|\xi\|$ is the amplitude and $\chi$ is the phase. Equation (18sbhbl) physically implies that the coherent state $\|{\xi}\rangle$ is not affected by the detection and annihilation of one particle. In a coherent state the quantum uncertainties of $\hat{p}$ and $\hat{q}$ (the momentum and position of the quantum oscillator respectively) take the minimum value, i.e., $\Delta\hat{p}\Delta\hat{q}=\frac{1}{2}\hbar$. With the exception of the vacuum state $\|{0}\rangle$ (which is also a coherent state), every coherent state can be produced by the application of the _Displacement_ operator $\hat{D}(\xi)=\exp{(\xi\hat{a}^{\dagger}-\xi^{}\hat{a})}$ to the vacuum state $\|\xi\rangle=\hat{D}(\xi)\|0\rangle.$ Using the simple properties of the coherent states we can calculate the quantities $d^{(R,I)}_{\textbf{k}}$, $e^{(R,I)}_{\textbf{k}}$ and $c^{(R,I)}_{\textbf{k}}$ (12) when the post-collapse state of each mode of the field is a coherent state $\|{\xi_{\textbf{k}}}\rangle$, $d_{\textbf{k},}^{(R,I)}=\xi_{\textbf{k}}^{(R,I)},\qquad c_{\textbf{k},}^{(R,I)}=(\xi_{\textbf{k}}^{(R,I)})^{2},\qquad e_{\textbf{k},}^{(R,I)}=\|\xi_{\textbf{k}}^{(R,I)}\|^{2}.$ (18sbhbm) Expressions (18sbhbm), (14) and (15) allow us to obtain the evolution of the uncertainties of the field and its conjugate momentum for any coherent state. Making use of the same arguments that led (18st) to the generalization (18sas) in the case of multiple collapses, (14) and (15) can also be considered in conjunction with the assumption that every post-collapse state is a coherent state ($\|{\xi^{(n)}_{\textbf{k}}}\rangle=\|{c_{n}}\rangle$) $\displaystyle(\Delta\hat{y}_{\textbf{k}}^{(R,I)}(\eta))^{2}_{c_{n}}$ $\displaystyle=$ $\displaystyle\Re[y_{k}^{2}(\eta)(\xi^{(n)(R,I)}_{\textbf{k}})^{2}]+\frac{1}{2}\|y_{k}(\eta)\|^{2}(\hbar L^{3}+2\|\xi^{(n)(R,I)}_{\textbf{k}}\|^{2})-2\Re[y_{k}(\eta)\xi^{(n)(R,I)}_{\textbf{k}}]^{2}$ (18sbhbn) $\displaystyle=$ $\displaystyle\frac{1}{2}\|y_{k}(\eta)\|^{2}\hbar L^{3}=\frac{\hbar L^{3}}{4k}\left(1+\frac{1}{(k\eta)^{2}}\right),$ $\displaystyle(\Delta\hat{\pi}_{\textbf{k}}^{(R,I)}(\eta))^{2}_{c_{n}}$ $\displaystyle=$ $\displaystyle\Re[g_{k}^{2}(\eta)(\xi^{(n)(R,I)}_{\textbf{k}})^{2}]+\frac{1}{2}\|g_{k}(\eta)\|^{2}(\hbar L^{3}+2\|\xi^{(n)(R,I)}_{\textbf{k}}\|^{2})-2\Re[g_{k}(\eta)\xi^{(n)(R,I)}_{\textbf{k}}]^{2}$ (18sbhbo) $\displaystyle=$ $\displaystyle\frac{1}{2}\|g_{k}(\eta)\|^{2}\hbar L^{3}=\frac{k\hbar L^{3}}{4}.$ This last result shows that the uncertainties of the $n$-th coherent post- collapse state have the same form as those of the vacuum state. We also note that the uncertainty of the conjugate momentum is constant in the inflationary era. ### 4.2 Squeezed states as post-collapse states Squeezed states can be considered as a more general case of the coherent states. Qualitatively, a squeezed state is a state that has the minimal uncertainty, not in the standard position and momentum variables, but in a new pair of ‘rotated’ canonical variables (commonly referred as _quadrature_ variables [35]). Let us call them $\hat{Q}$ and $\hat{P}$. For a squeezed state one can have ‘more (or less)’ uncertainty in either $\hat{Q}$ or $\hat{P}$, as long as their product is equal to the minimum value allowed by Heisenberg’s principle. The parameters of the squeezed state control the angle of ‘rotation’ and the ‘squeezing’ of the uncertainties. The work with squeezed states is simplified by the introduction of the following operator $\hat{S}(\omega)\equiv\exp(\frac{1}{2}\omega^{\star}\hat{a}^{2}-\frac{1}{2}\omega\hat{a}^{{\dagger}2}),$ (18sbhbp) where the parameter $\omega$ is a complex number. In particular, $\omega$ can be written as $\omega=re^{i\theta}$. The operator $\hat{S}(\omega)$ is known as the _Squeeze_ Operator. Applying the Squeeze and Displacement operators to the vacuum state we obtain a squeezed state $\|{\xi\omega}\rangle\equiv\hat{D}(\xi)\hat{S}(\omega)\|0\rangle.$ (18sbhbq) We note that the _squeezed_ state $\|{\xi\omega}\rangle$ is completely defined by four parameters: $\|\xi\|,\chi,r,\theta$ Some well known properties of the operators $\hat{D}(\xi)$ and $\hat{S}(\omega)$ are 1. 1. $\hat{D}^{\dagger}(\xi)\hat{a}\hat{D}(\xi)=\hat{a}+{\xi}$. 2. 2. $\hat{D}^{\dagger}(\xi)\hat{a}^{\dagger}\hat{D}(\xi)=\hat{a}^{\dagger}+\overline{\xi}$. 3. 3. $\hat{S}^{\dagger}(\omega)\hat{a}\hat{S}(\omega)=\hat{a}\cosh{r}-\hat{a}^{{\dagger}}e^{-i\theta}\sinh{r}$. 4. 4. $\hat{S}^{\dagger}(\omega)\hat{a}^{\dagger}\hat{S}(\omega)=\hat{a}^{\dagger}\cosh{r}-\hat{a}e^{i\theta}\sinh{r}$. 5. 5. Both $\hat{D}$ and $\hat{S}$ are unitary operators. By regarding the post-collapse state of each mode as a _squeezed_ state and using the properties 1-5, one can obtain $d_{\textbf{k}}^{(R,I)}$, $c_{\textbf{k}}^{(R,I)}$ and $e_{\textbf{k}}^{(R,I)}$ from (12) $d_{\textbf{k}}^{(R,I)}=\xi_{\textbf{k}}^{(R,I)},$ (18sbhbr) $c_{\textbf{k}}^{(R,I)}=-\hbar L^{3}\cosh{r_{\textbf{k}}^{(R,I)}}\sinh{r_{\textbf{k}}^{(R,I)}}e^{-i\theta_{\textbf{k}}^{(R,I)}}+(\xi_{\textbf{k}}^{(R,I)})^{2},$ (18sbhbs) $e_{\textbf{k}}^{(R,I)}=\hbar L^{3}\sinh^{2}{r_{\textbf{k}}^{(R,I)}}+\|\xi_{\textbf{k}}^{(R,I)}\|^{2}.$ (18sbhbt) Equations (14) and (15) give us the evolution of the uncertainties, in terms of the quantities $d_{\textbf{k}}^{(R,I)}$, $c_{\textbf{k}}^{(R,I)}$ and $e_{\textbf{k}}^{(R,I)}$. As in the coherent state, (14) and (15) can be generalized straightforward to the case of multiple collapse. Thus, by considering the post-collapse states of each mode as _squeezed_ states, we can substitute (18sbhbr), (18sbhbs) and (18sbhbt) into (14) and (15), which for multiple collapses yields $\displaystyle(\Delta\hat{y}_{\textbf{k}}^{(R,I)}(\eta))^{2}_{c_{n}}$ $\displaystyle=\frac{\hbar L^{3}}{4k}\bigg{[}1+\frac{1}{(k\eta)^{2}}\bigg{]}\bigg{\\{}-\sinh(2r^{c_{n}(R,I)}_{\textbf{k}})$ (18sbhbu) $\displaystyle\times\cos\left[\theta^{c_{n}(R,I)}_{\textbf{k}}+2\arctan\left(\frac{1}{k\eta}\right)+2k\eta\right]+\cosh(2r^{c_{n}(R,I)}_{\textbf{k}})\bigg{\\}},$ $(\Delta\hat{\pi}_{\textbf{k}}^{(R,I)}(\eta))^{2}_{c_{n}}=\frac{\hbar L^{3}k}{4}\bigg{[}\sinh(2r^{c_{n}(R,I)}_{\textbf{k}})\cos(\theta^{c_{n}(R,I)}_{\textbf{k}}+2k\eta)+\cosh(2r^{c_{n}(R,I)}_{\textbf{k}})\bigg{]}.$ (18sbhbv) The squeezing parameters $r^{c_{n}(R,I)}_{\textbf{k}}$ and $\theta^{c_{n}(R,I)}_{\textbf{k}}$ refer to the squeeze parameters of the $n$-th post-collapse squeezed state of each mode. The situation at hand is totally different from the coherent case, in which the uncertainties are completely characterized by the vacuum state despite $n$ collapses have occurred. In the squeeze state case, it is evident from (18sbhbu) and (18sbhbv) that the dispersions $(\Delta\hat{y}_{\textbf{k}}^{(R,I)}(\eta))^{2}_{c_{n}},(\Delta\hat{\pi}_{\textbf{k}}^{(R,I)}(\eta))^{2}_{c_{n}}$ are determined by the squeeze parameters $r_{\textbf{k}}^{c_{n}(R,I)}$ and $\theta_{\textbf{k}}^{c_{n}(R,I)}$. This is a crucial difference with the coherent case in which the uncertainties are independent of the parameters characterizing the coherent state. ### 4.3 Connections with the observational quantities The uncertainties of the field are characterized by both, the particular post- collapse state and the collapse scheme. In the rest of this section we will focus on the _independent_ collapse scheme, however, similar conclusions as those obtained from these results can be derived when considering the other two collapse schemes that have been proposed so far. #### 4.3.1 Squeezed States as postcollapse states The connection with the observations will be made under the following assumptions: I) The wave-function of the field has collapsed $N$ times and the $N$ post-collapse states are squeezed states. II) The uncertainties $(\Delta\hat{y}_{\textbf{k}}^{R}(\eta_{k}^{c_{n}}))^{2}_{c_{n-1}}$ and $(\Delta\hat{y}_{\textbf{k}}^{I}(\eta_{k}^{c_{n}}))^{2}_{c_{n-1}}$ are equal (as well as the uncertainties $(\Delta\hat{\pi}_{\textbf{k}}^{R}(\eta_{k}^{c_{n}}))^{2}_{c_{n-1}}$ and $(\Delta\hat{\pi}_{\textbf{k}}^{I}(\eta_{k}^{c_{n}}))^{2}_{c_{n-1}}$), which is motivated by the discussion at the end of section 3. Under the assumption II), (18sbc) (recall that we are working under the _independent_ scheme) takes the simplified form $\displaystyle\|\alpha_{lm}\|_{M.L.}^{2}$ $\displaystyle=\frac{8}{27\pi}\frac{V\hbar^{3}}{\epsilon M_{P}^{4}}\int\frac{dk}{k^{2}}\|j_{l}(kR_{D})\|^{2}\sum_{n=1}^{N}\bigg{[}\bigg{(}k\sin(k\eta_{D}-k\eta_{k}^{c_{{n}}})\bigg{)}^{2}(\Delta\hat{y}_{\textbf{k}}(\eta_{k}^{c_{n}}))^{2}_{c_{n-1}}$ (18sbhbw) $\displaystyle+\bigg{(}\cos(k\eta_{D}-k\eta_{k}^{c_{{n}}})+\frac{\sin(k\eta_{D}-k\eta_{k}^{c_{{n}}})}{k\eta_{k}^{c_{{n}}}}\bigg{)}^{2}(\Delta\hat{\pi}_{\textbf{k}}(\eta_{k}^{c_{n}}))^{2}_{c_{n-1}}\bigg{]}.$ After a little algebra, the expression for $\|\alpha_{lm}\|_{M.L.}^{2}$ obtained by substituting (18sbhbu) and (18sbhbv) in (18sbhbw) becomes $\displaystyle\|\alpha_{lm}\|_{M.L.}^{2}$ $\displaystyle=\frac{2}{27\pi}\frac{V\hbar^{3}}{\epsilon M_{P}^{4}}\int\frac{dk}{k}\|j_{l}(kR_{D})\|^{2}\sum_{n=1}^{N}\Bigg{[}\bigg{(}1+\frac{\sin 2(k\eta_{D}-k\eta_{k}^{c_{{n}}})}{k\eta_{k}^{c_{{n}}}}\bigg{)}$ (18sbhbx) $\displaystyle\times\bigg{(}\cosh 2r^{c_{n-1}}_{\textbf{k}}+\sinh 2r^{c_{n-1}}_{\textbf{k}}\cos(\theta^{c_{n-1}}_{\textbf{k}}+2k\eta_{k}^{c_{{n}}})\bigg{)}+\frac{2\sin^{2}(k\eta_{D}-k\eta_{k}^{c_{{n}}})}{(k\eta_{k}^{c_{{n}}})^{2}}$ $\displaystyle\times\bigg{(}\cosh 2r^{c_{n-1}}_{\textbf{k}}+k\eta_{k}^{c_{{n}}}\sinh 2r^{c_{n-1}}_{\textbf{k}}\sin(\theta^{c_{n-1}}_{\textbf{k}}+2k\eta_{k}^{c_{{n}}})\bigg{)}\Bigg{]}.$ The above result lead us to conclude that, in order to obtain a reasonable power spectrum, that is, a nearly flat Harrison-Zel’dovich spectrum, there seems to be one simple case characterized by two particular conditions: First, for each one of the $n$ post-collapse states, $k\eta_{k}^{c_{{n}}}$ should be independent of $k$ but dependent of $n$. I.e., the time of collapse for the $n$ post-collapse states of the different modes should depend on the mode’s frequency according to $\eta^{c_{n}}_{k}=f_{n}/k$ (where $f_{n}$ is a real number that changes for each collapse). This condition is the generalization for $n$ collapses of the result presented in [10] where a single collapse was considered (a possible deviation of such ‘recipe’ for the time of collapse was studied in [14]). In other words, the result (18sbhbx) generalizes the condition $\eta^{c}_{k}\propto 1/k$ in the case of multiple collapses. Second, a nearly flat spectrum is recovered if the parameters characterizing the $n$ squeezed states are also independent of the mode’s frequency, that is, if $r^{c_{n}}_{\textbf{k}}=r^{c_{n}}$ and $\theta^{c_{n}}_{\textbf{k}}=\theta^{c_{n}}$ are independent of k but dependent of $n$. This does not mean that the uncertainties for each mode are all the same, because the uncertainties are also characterized by the time of collapse of each mode and its frequency, as can be seen in (18sbhbu) and (18sbhbv). #### 4.3.2 An upper bound for the number of collapse using coherent states as post-collapse states As already noted generically, the number of collapses in each mode must be finite, and we expect to provide a simple estimate in this subsection. We will continue the consideration of the _independent_ collapse scheme, but we will assume that all the $N$ post-collapse states are _coherent_ states (which, after all, are just a particular class of squeezed states with $r_{\textbf{k}}=0$). That is, we can use the uncertainties (18sbhbn) and (18sbhbo) to obtain a predicted value for $\|\alpha_{lm}\|_{M.L.}^{2}$. Note, however, that in the case of coherent states, assumption (II) of the subsection 4.3.1 is naturally obtained because the uncertainties of every coherent state are equal to the uncertainties of the vacuum state which automatically satisfy $(\Delta\hat{y}_{\textbf{k}}^{R}(\eta_{k}))^{2}_{c_{0}}=(\Delta\hat{y}_{\textbf{k}}^{I}(\eta_{k}))^{2}_{c_{0}}$ (as well as $(\Delta\hat{\pi}_{\textbf{k}}^{R}(\eta_{k}))^{2}_{c_{0}}=(\Delta\hat{\pi}_{\textbf{k}}^{I}(\eta_{k}))^{2}_{c_{0}}$). Substituting (18sbhbn) and (18sbhbo) in (18sbc) yields $\displaystyle\|\alpha_{lm}\|_{M.L.}^{2}=\frac{2}{27\pi}\frac{V\hbar^{3}}{\epsilon M_{P}^{4}}\int\frac{dk}{k}\|j_{l}(kR_{D})\|^{2}\sum_{n=1}^{N}\bigg{(}1+\frac{\sin 2(k\eta_{D}-k\eta_{k}^{c_{{n}}})}{k\eta_{k}^{c_{{n}}}}+\frac{2\sin^{2}(k\eta_{D}-k\eta_{k}^{c_{{n}}})}{(k\eta_{k}^{c_{{n}}})^{2}}\bigg{)}.$ From this last expression is a relatively simple task to obtain information regarding the maximum number of collapses allowed by observations. If we assume that $\|k\eta_{k}^{c_{{n}}}\|\gg k\eta_{D}$, that is, the time for the $1^{st},2^{nd},...,N^{th}$ collapse occurs at very early stage of the inflationary regime; (4.3.2) is approximated by $\|\alpha_{lm}\|_{M.L.}^{2}\approx\frac{2}{27\pi}\frac{V\hbar^{3}}{\epsilon M_{P}^{4}}\int\frac{dk}{k}\|j_{l}(kR_{D})\|^{2}N,$ (18sbhbz) using that $\int x^{-1}j^{2}_{l}(x)dx=\pi/l(l+1)$, the expression above reduces to $\|\alpha_{lm}\|_{M.L.}^{2}\approx\frac{2}{27}\frac{V\hbar^{3}}{\epsilon M_{P}^{4}}\frac{N}{l(l+1)}.$ (18sbhca) In general $\|\alpha_{lm}\|_{M.L.}^{2}$ is independent of $m$ and the quantity that is presented as the result of observations is $OB_{l}=l(l+1)C_{l}$, where $C_{l}=(2l+1)^{-1}\sum_{m}\|\alpha_{lm}^{obs}\|^{2}$. If we ignore the physics of the plasma that follows after the reheating era, $OB_{l}$ is essentially independent of $l$ corresponding to the amplitude of the metric perturbations (which is roughly $10^{-10}$). Thus, setting $OB_{l}\equiv A$, the maximum number of collapses $N_{max}$ allowed by the observations is $N_{max}\approx\frac{27\epsilon M_{P}^{4}A}{2V\hbar^{3}}.$ (18sbhcb) We believe that this constraint might be of great help in studying the viability of the actual proposals for the detail physical mechanism that lies behind the collapse we have been considering. ## 5 Discussion As first reviewed in [10], the inflationary account of the origin of cosmic structure posses a serious shortcoming, namely, the emergence of structure from an initial state that was homogeneous and isotropic. The proposal to address this existing issue was through the introduction of a modification of standard quantum theory corresponding to a dynamical reduction of the wave function. The present study represents a continuation of the investigation of such proposal. In this paper, we have examined the possibility that multiple collapses take place in each of the modes of the quantum field. This study required a much more detailed characterization of the post-collapse states. This, in turn, required the introduction of extra assumptions. We focused here in the possibility that the states are coherent or squeezed and under these assumptions we were able to further constrain, beyond the results of previous analyses, the features of the collapse hypothesis required for agreement with observations. These we will discuss in the following. The first result obtained in this manuscript is that in order to recover a flat spectrum, and assuming that multiple collapses occur, then the uncertainties of the real and imaginary parts of the fluctuation of the inflaton field, i.e., $\hat{y}_{\textbf{k}}$ and its conjugated momentum $\hat{\pi}_{\textbf{k}}$, must be equal. We can interpret this result as the most natural option for selecting simple candidates for post-collapse states since the uncertainty of each mode of the field and its conjugated momentum is characterized by specifying the post-collapse state. Therefore, given a particular state $\|\Xi\rangle$ for each mode k, one can calculate the uncertainties of the field and its conjugated momentum for that state. If the uncertainties for each mode satisfy the relation $(\Delta\hat{y}_{\textbf{k}}^{R})^{2}_{\Xi}=(\Delta\hat{y}_{\textbf{k}}^{I})^{2}_{\Xi}$ (as well as $(\Delta\hat{\pi}_{\textbf{k}}^{R})^{2}_{\Xi}=(\Delta\hat{\pi}_{\textbf{k}}^{I})^{2}_{\Xi})$ , then $\|\Xi\rangle$ can be regarded as a reasonable choice for a post- collapse state. In fact, in section 4.1, we found that, for coherent states, the relation between the uncertainties of the real and imaginary parts of $\hat{y}_{\textbf{k}}$ and $\hat{\pi}_{\textbf{k}}$ is satisfied automatically. Consequently, a coherent state is a natural candidate for a post-collapse state. The fact that a coherent state acts as a good candidate for a post-collapse state is consistent with the notion that a coherent state of the field is the closest quantum mechanical state to a classical description of the field, i.e., a state for which the semiclassical approximation of gravity given by $G_{ab}=8\pi G\langle\hat{T}_{ab}\rangle$ is valid in the sense of Ehrenfest’s theorem and thus qualifies for a reasonable candidate for a post-collapse state. Nevertheless, for a generic squeezed state $\|\Sigma\rangle$, $(\Delta\hat{y}_{\textbf{k}}^{R})^{2}_{\Sigma}\neq(\Delta\hat{y}_{\textbf{k}}^{I})^{2}_{\Sigma}$ and $(\Delta\hat{\pi}_{\textbf{k}}^{R})^{2}_{\Sigma}\neq(\Delta\hat{\pi}_{\textbf{k}}^{I})^{2}_{\Sigma}$, but this does not mean that post-collapse squeezed states are forbidden. That is, one can select a set of squeezed states, characterized by the squeezing parameters $r_{\textbf{k}}^{(R,I)}$ and $\theta_{\textbf{k}}^{(R,I)}$, such that $r_{\textbf{k}}^{R}=r_{\textbf{k}}^{I}$ and $\theta_{\textbf{k}}^{R}=\theta_{\textbf{k}}^{I}$ for which the relation in the uncertainties holds. Furthermore, in section 4.3.1 we argued that, given a collection of multiple post-collapse squeezed states characterized by $r_{\textbf{k}}^{c_{n}}$ and $\theta_{\textbf{k}}^{c_{n}}$, then, the simplest choice that allows the recovering of the standard flat spectrum, is that the squeezing parameters be independent of k. The point is that we again used the observations as a guide to uncover the particular characteristics of a squeezed state that could be regarded as a reasonable post-collapse state. We should note that, as discussed in [14], we can not expect such a strict pattern to be followed in an exact manner in a theory involving a collapse controlled by some fundamentally random events, and as such one can in principle investigate the effects of the expected deviations on the observational data. The investigation of the detail signature of those deviations, as well as the observational bounds on them (i.e. analogues of those considered in [14]), is part of our ongoing research program. Another important result from this work is that the number of collapses must be finite under generic conditions. However we could, in principle, select a set of post-collapse states and adjust the uncertainties of the field (and its conjugated momentum) and the times of collapse in a way that the predicted observational quantity (the sum in (18sbc)) would remain finite, even for an infinite number of collapses. Evidently, this would amount to a fine tune of the scheme which we do not see as an attractive choice. On the other hand, we should say that if the collapse of the state, which gives birth to the inhomogeneities observed in the CMB, is a process that keeps occurring indefinitely even after inflation ends, the Newtonian potentials would also be changing, thus affecting in a rather random way the propagation of photons from the last scattering surface to our satellites. These ideas might be considered as related, at least at the phenomenological level, to those explored in [36]. We did not investigate these issues here. In the present work we rather concentrated on the generic sort of conditions for the collapse during inflation and found, not only that the number of collapses should be finite, but obtained, – under the extra hypothesis on the form of post- collapse states– a rough estimate on the number of collapses in terms of the parameters of the inflaton potential. All of the previous discussion shows that, even though in principle we do not know precisely what is the nature of the physics behind what we call the collapse, we can, in fact, obtain some insights on the ‘rules’ that govern it, i.e., those determining the nature of post-collapse states and the number of collapses of each mode, by comparing the observations with our theoretical predictions. We are beginning to investigate the possible connection of our proposal with other more developed collapse mechanisms involving similar non unitary modifications of quantum theory. Henceforth, the path to follow in our future research is to explore the connections of our proposal with other collapse mechanisms compatible with the conclusions obtained in this and previous works. We believe that, in the case of the inflationary paradigm, we cannot content ourselves with the fact that calculations lead to results that match the observations but which can not be fully justified within the context of the interpretations provided by our current physical theories. We readily acknowledge that, although our proposal seems to offer a clearer picture of the emergence of the seeds of cosmic structure, it might be ultimately an incorrect proposal which might need to be replaced by something even more complex and distant from the established physical paradigms. What seems clear is that the standard account of the genesis of the cosmic structure, something intimately tied with the rise in the conditions that are a prerequisite for our own existence, is not fully satisfactory and that on the other hand, our present and future access to detailed empirical data makes the issue not only susceptible to scientific inquire, but from our point of view, one of the most promising fertile grounds where some fundamental questions can be explored. GL and DS acknowledge support from PAPPIT project IN 119808 and CONACyT project No 101712. DS was supported in part by sabbatical fellowships from CONACyT and DGAPA-UNAM and the hospitality of the IAFE. 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1012.2717	# Upper bounds on the photon mass Antonio Accioly accioly@cbpf.br Laboratório de Física Experimental (LAFEX), Centro Brasileiro de Pesquisas Físicas (CBPF), Rua Dr. Xavier Sigaud 150, Urca, 22290-180, Rio de Janeiro, RJ, Brazil Group of Field Theory from First Principles, São Paulo State University (UNESP), Rua Dr. Bento Teobaldo Ferraz 271, Bl. II-Barra Funda, 01140-070 São Paulo, SP, Brazil Instituto de Física Teórica (IFT), São Paulo State University (UNESP), Rua Dr. Bento Teobaldo Ferraz 271, Bl. II-Barra Funda, 01140-070 São Paulo, SP, Brazil José Helayël- Neto helayel@cbpf.br Laboratório de Física Experimental (LAFEX), Centro Brasileiro de Pesquisas Físicas (CBPF), Rua Dr. Xavier Sigaud 150, Urca, 22290-180, Rio de Janeiro, RJ, Brazil Group of Field Theory from First Principles, São Paulo State University (UNESP), Rua Dr. Bento Teobaldo Ferraz 271, Bl. II-Barra Funda, 01140-070 São Paulo, SP, Brazil Eslley Scatena scatena@ift.unesp.br Instituto de Física Teórica (IFT), São Paulo State University (UNESP), Rua Dr. Bento Teobaldo Ferraz 271, Bl. II-Barra Funda, 01140-070 São Paulo, SP, Brazil Group of Field Theory from First Principles, São Paulo State University (UNESP), Rua Dr. Bento Teobaldo Ferraz 271, Bl. II- Barra Funda, 01140-070 São Paulo, SP, Brazil ###### Abstract The effects of a nonzero photon rest mass can be incorporated into electromagnetism in a simple way using the Proca equations. In this vein, two interesting implications regarding the possible existence of a massive photon in nature, i.e., tiny alterations in the known values of both the anomalous magnetic moment of the electron and the gravitational deflection of electromagnetic radiation, are utilized to set upper limits on its mass. The bounds obtained are not as stringent as those recently found; nonetheless, they are comparable to other existing bounds and bring new elements to the issue of restricting the photon mass. ###### pacs: 13.40.Gp, 04.80.Cc ## I Introduction In general, systems of heavy vector bosons are non-renormalizable. There are, however, two important exceptions to this rule: (i) gauge theories with spontaneous symmetry breakdown, and (ii) Abelian theories with neutral vectorial bosons coupled to conserved currents 1 . The latter, i.e., the ‘conserved current models’, contain at least one massive boson, whose source is conserved. These systems can be constructed through the following general prescription 2 : * • Begin with a Lagrangian which is invariant under a nonsemisimple group of local gauge transformations (i.e., a group containing an invariant Abelian subgroup). * • Arrange for spontaneous symmetry breaking (if any) such that the vacuum expectation value of the scalar field is invariant under at least one invariant (single-parameter) Abelian subgroup (thus, at this stage the corresponding Abelian vector is massless and coupled to a conserved current). * • Add (in the $R$ gauge) an arbitrary mass term for the same Abelian vector. Massive electrodynamics (or, Proca electrodynamics), i.e, the electrodynamics that can be embedded into the standard $SU(2)\times U(1)$ model and in which the photon has a small mass, is the simplest system of this type, besides being the most straightforward extension of standard QED. Indeed, Proca’s electromagnetic field theory can be constructed in a unique way by adding a mass term to the Lagrangian for the electromagnetic field, namely, $\displaystyle{\cal L}=-\frac{1}{4}F_{\mu\nu}^{2}-J_{\mu}A^{\mu}+\frac{1}{2}m^{2}A_{\mu}^{2},$ (1) where $F_{\mu\nu}\;(=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})$ is the field strength, and $J_{\mu}$ is the (electric) current. The parameter $m$ can be interpreted as the photon rest mass. In this spirit, the characteristic scaling length $m^{-1}$ becomes the reduced Compton wavelength of the photon, which is the effective range of the electromagnetic interaction. Massive QED is not only simpler theoretically than the standard theory 3 , it also provides a fairly solid framework for analyzing (through the Proca equations) the far-reaching implications the existence of a massive photon would have for physics. Actually, some of these possible effects, such as variation of the speed of light, deviation in the behavior of static electromagnetic fields and longitudinal electromagnetic radiation, have been thoroughly studied by means of a number of different approaches over the past several decades 4 ; 5 ; 6 . It is worth mentioning that both the Aharonov-Bohm and the Aharonov-Casher effects are present in massive QED. The former was analyzed by Boulware and Deser 7 who showed that it reduces smoothly to the original result, while the latter was studied by Fuchs 8 . Nonetheless, the system of ‘Maxwell + photon mass + magnetic charge’ equations is not consistent 3 ; 4 . Interestingly enough, the possibility of a nonzero photon mass remains, as it was pointed out by Adelberger, Dvali, and Gruzinov 9 , one of the most important issues in physics, as it would shed a new light on some fundamental questions, such as charge conservation, charge quantization, the possibility of charged black holes and magnetic monopoles. We also remark that the popular view that gauge invariance implies a zero photon mass is not correct. In reality, a minimal dynamics obeying gauge invariance, i.e., the Maxwell action, does imply zero photon mass; nevertheless, by enlarging the dynamics, for instance, by adding another field interacting with the photon field, both gauge invariance and nonzero mass can be accommodated simultaneously 6 . The purpose of this paper is to set upper bounds on the photon mass supposing that it is described by Proca electrodynamics. To accomplish this goal we shall analyze two interesting but not yet explored consequences of the possible existence of a massive photon in nature: the very small alteration in the usual anomalous magnetic moment of the electron and the tiny change in the ordinary gravitational deflection of the electromagnetic radiation. These issues are analyzed in detail in Secs. II and III, respectively. To conclude, a discussion about the order of magnitude of the bounds estimated in the aforementioned sections, is presented in Sec. IV. In our conventions $\hbar=c=1$, and the signature is (+ - - -). ## II A QUANTUM BOUND As is well-known, QED predicts the anomalous magnetic moment of the electron correctly to ten decimal places. Therefore, it is perfectly reasonable that we use this astonishing result, one of the great triumphs of QED, to estimate a quantum bound on the photon mass. How can we do that? By computing the anomalous magnetic moment of the electron to order $\alpha$, where $\alpha$ is the fine structure constant, in the framework of massive QED and expanding afterward the result in powers of $(\frac{m}{\mu})^{2}$, where $m$ and $\mu$ are, respectively, the photon and the electron masses. The first term of this expansion must necessarily coincide with that calculated by Schwinger in 1948 10 , while the second one is the most important correction related to the parameter $m$ of massive QED. Now, taking into account that the latter must be less than $10^{-10}$ (the theoretical result predicted by QED for the anomalous magnetic moment of the electron 11 agrees in $1$ part in $10^{10}$ with the experimental one 12 ), we promptly find an upper bound for the photon mass. Let us then perform the computations. We begin by recalling that the anomalous magnetic moment of the electron stems from the vertex correction for the scattering of the electron by an external field, as it is shown in Fig. 1. Figure 1: Vertex correction for electron scattering by an external field. For an electron scattered by an external static magnetic field and in limit ${\bf q}\rightarrow{\bf 0}$, the gyromagnetic ratio is given by 13 $g=2[1+F_{2}(0)].$ The form factor of the electron, $F_{2}(0)$, corresponds to a shift in the $g-$factor, usually quoted in the form $F_{2}(0)\equiv\frac{g-2}{2}$, and yields the anomalous magnetic moment of the electron. On the other hand, from the quadratic part of Lagrangian (1) we immediately obtain the propagator for the massive QED, namely, $\displaystyle D_{\mu\nu}=-\frac{\eta_{\mu\nu}}{k^{2}-m^{2}}+\frac{k_{\mu}k_{\nu}}{m^{2}(k^{2}-m^{2})}.$ (2) By employing this expression in the calculation of the diagram in Fig. 1, it can be shown that $\displaystyle F_{2}(0)=\frac{\alpha}{\pi}\int_{0}^{\infty}d\alpha_{1}d\alpha_{2}d\alpha_{3}\delta(1-\Sigma\alpha_{i})\frac{\alpha_{1}(\alpha_{2}+\alpha_{3})}{(\alpha_{2}+\alpha_{3})^{2}+\lambda^{2}\alpha_{1}},$ where $\lambda^{2}\equiv(\frac{m}{\mu})^{2}$. We remark that the term $\frac{k_{\mu}k_{\nu}}{m^{2}(k^{2}-m^{2})}$ that appears in Eq. (2) was omitted ab initio from the calculations concerning $F_{2}(0)$ because the propagator for the massive photon always occurs coupled to conserved currents. In order to avoid unnecessary algebraic computations as far as the evaluation of $F_{2}(0)$ is concerned, we rewrite this expression as follows: $F_{2}(0)=X-Y,$ where $\displaystyle X\equiv\frac{\alpha}{\pi}\int_{0}^{\infty}d\alpha_{1}d\alpha_{2}d\alpha_{3}\delta(1-\Sigma\alpha_{i})\frac{\alpha_{1}}{\alpha_{2}+\alpha_{3}},$ (3) $\displaystyle Y\equiv$ $\displaystyle=$ $\displaystyle\frac{\alpha}{\pi}\int_{0}^{\infty}d\alpha_{1}d\alpha_{2}d\alpha_{3}\delta(1-\Sigma\alpha_{i})\left[\frac{\alpha_{1}}{\alpha_{2}+\alpha_{3}}\right.$ (4) $\displaystyle-\left.\frac{\alpha_{1}(\alpha_{2}+\alpha_{3})}{(\alpha_{2}+\alpha_{3})^{2}+\lambda^{2}\alpha_{1}}\right].$ Integrating the expression (3) first with respect to $\alpha_{3}$ and subsequently with respect to $\alpha_{2}$ gives $\displaystyle X$ $\displaystyle=$ $\displaystyle\frac{\alpha}{\pi}\int_{0}^{1}d\alpha_{1}\int_{0}^{1-\alpha_{1}}d\alpha_{2}\frac{\alpha_{1}}{1-\alpha_{1}}$ (5) $\displaystyle=$ $\displaystyle\frac{\alpha}{2\pi}.$ Similarly, the expression (4) yields $\displaystyle Y$ $\displaystyle=$ $\displaystyle\frac{\alpha}{\pi}\int_{0}^{1}d\alpha_{1}\frac{\lambda^{2}\alpha_{1}^{2}}{(1-\alpha_{1})^{2}+\lambda^{2}\alpha_{1}}$ (6) $\displaystyle=$ $\displaystyle\frac{\alpha}{\pi}\left[\lambda^{2}-(\lambda^{4}-2\lambda^{2})\ln\lambda\right.$ $\displaystyle+\left.\frac{\lambda^{5}-4\lambda^{3}+2\lambda}{\sqrt{4-\lambda^{2}}}\arctan\frac{\sqrt{4-\lambda^{2}}}{\lambda}\right].$ From (5) and (6) we get $\displaystyle F_{2}(0)$ $\displaystyle=$ $\displaystyle\frac{\alpha}{\pi}\left[\frac{1}{2}-\lambda^{2}(1+2\ln\lambda)+\lambda^{4}\ln\lambda\right.$ (7) $\displaystyle+\left.\frac{4\lambda^{3}-2\lambda-\lambda^{5}}{\sqrt{4-\lambda^{2}}}\arctan\frac{\sqrt{4-\lambda^{2}}}{\lambda}\right].$ Recalling that $\lambda\ll 1$, we arrive at the conclusion that $\displaystyle F_{2}(0)$ $\displaystyle\approx$ $\displaystyle\frac{\alpha}{2\pi}\left[1-\pi\left(\frac{m}{\mu}\right)-\left(\frac{3}{2}+4\ln\left(\frac{m}{\mu}\right)\right)\left(\frac{m}{\mu}\right)^{2}\right.$ (8) $\displaystyle+\left.{\cal\large O}\left(\left(\frac{m}{\mu}\right)^{3}\right)\right].$ As we have already commented, the first term in the above equation is equal to that calculated by Schwinger in 1948 (since then $F_{2}(0)$ has been calculated to order $\alpha^{8}$ for QED), while the second one is the most important correction concerning the parameter $m$ of massive QED. Since theory and experiment agree within errors to $\sim$ $1$ in $10^{10}$ for $F_{2}(0)$, we promptly obtain $\displaystyle\frac{\pi m}{\mu}<10^{-10},$ (9) implying $m<1.6\times 10^{-10}MeV.$ Recently, the measurement of the anomalous magnetic moment of the muon reached the fabulous relative precision of 0.5 ppm 14 ; 15 . Accordingly, it would be interesting to find another quantum bound on the photon mass using this phenomenon and make afterward a comparison with the bound estimated via the electron. Now, taking into account that for the muon 16 $F_{2}^{(\mathrm{exp})}(0)-F_{2}^{(\mathrm{SM})}(0)=(295\pm 88)\times 10^{-11},$ where $F_{2}^{(\mathrm{SM})}(0)$ denotes the prediction of the standard model 17 , we find $m<3.4\times 10^{-7}MeV$, three orders of magnitude higher than the bound derived from the anomalous magnetic moment of the electron. Consequently, we shall not consider this bound in our discussions. ## III A GRAVITATIONAL BOUND It is a generally acknowledged fact that the gravitational deflection of light by the sun can be measured more accurately at radio wavelengths with interferometry techniques than at visible wavelengths with available optical techniques 18 . Indeed, at present the Very Long Baseline Interferometry (VLBI) is the most accurate technique we have at our disposal for measuring radio-wave gravitational deflection 19 ; 20 ; 21 . The gravitational bending, in turn, is one of the most impressive predictions of general relativity. In addition, the recent measurements of the gravitational bending of radio waves using the VLBI have improved considerable on the previous results in the gravitational bending experiments near the solar limb 22 . Accordingly, we shall use these results to estimate an upper limit on the photon mass. To do that we need, in first place, the unpolarized differential cross section for the scattering of a massive photon (described by Proca’s electrodynamics) by an external weak gravitational field. On the other hand, it was recently shown that the unpolarized differential cross sections for the gravitational scattering of different quantum particles are spin dependent 23 (See Table I). Nonetheless, for small angles, the cross sections for the massive (massless) particles are one and the same, regardless of the spin. In fact, when the spin is ‘switched off’, i.e., for small angles ($\theta\ll 1$), it is fairly straightforward to see from Table I that for $m=0$, $\frac{d\sigma}{d\Omega}\approx\frac{16G^{2}M^{2}}{\theta^{4}}$, while for $m\neq 0$, $\frac{d\sigma}{d\Omega}\approx\frac{16G^{2}M^{2}}{\theta^{4}}\left(1+\frac{\lambda}{2}\right)^{2}.$ In short, for small angles the results of Table I are in perfect agreement with those predicted by Einstein’s geometrical theory. Consequently, the differential cross section we are searching for is independent of the spin of the massive particle and can be written as $\displaystyle\frac{d\sigma}{d\Omega}=\frac{16G^{2}M^{2}}{\theta^{4}}\left(1+\frac{\lambda}{2}\right)^{2}.$ (10) The above differential cross section can be related to a classical trajectory with impact parameter $b$ via the relation $\displaystyle bdb=-\frac{d\sigma}{d\Omega}\theta d\theta.$ (11) From (10) and (11), we arrive at the conclusion that $\displaystyle\theta=\frac{4MG}{b}\left(\frac{1-\frac{m^{2}}{2E^{2}}}{1-\frac{m^{2}}{E^{2}}}\right),$ (12) which in the ultrarelativistic limit, i.e., $E\gg m$, reduces to $\displaystyle\theta$ $\displaystyle=$ $\displaystyle\theta_{\mathrm{E}}\left(1+\frac{m^{2}}{2E^{2}}\right),$ (13) $\displaystyle=$ $\displaystyle\theta_{\mathrm{E}}\left(1+\frac{m^{2}}{8\pi^{2}\nu^{2}}\right),$ where $E$ and $\nu$ are, respectively, the energy and the frequency of the ingoing massive photon, and $\theta_{\mathrm{E}}\equiv\frac{4MG}{b}$. Table 1: Unpolarized differential cross sections for the scattering of different quantum particles by an external weak gravitational field generated by a static point particle of mass $M$. Here $m$ is the particle mass, $s$ the spin, $\theta$ the scattering angle, $G$ the Newtonian constant, and $\lambda\equiv\frac{m^{2}}{\mathbf{p}^{2}}=\frac{1-\mathbf{v}^{2}}{\mathbf{v}^{2}}$, with $\mathbf{v}$ and $\mathbf{p}$ being the velocity and three-momentum, in this order, of the incident particle. $m$ \| $s$ \| $\frac{d\sigma}{d\Omega}$ ---\|---\|--- 0 \| 0 \| $\Big{(}\frac{GM}{\sin^{2}{\frac{\theta}{2}}}\Big{)}^{2}$ $\neq 0$ \| 0 \| $\Big{(}\frac{GM}{\sin^{2}{\frac{\theta}{2}}}\Big{)}^{2}\Big{(}1+\frac{\lambda}{2}\Big{)}^{2}$ 0 \| $\frac{1}{2}$ \| $\Big{(}\frac{GM}{\sin^{2}{\frac{\theta}{2}}}\Big{)}^{2}\cos^{2}{\frac{\theta}{2}}$ $\neq 0$ \| $\frac{1}{2}$ \| $\Big{(}\frac{GM}{\sin^{2}{\frac{\theta}{2}}}\Big{)}^{2}\Big{[}\cos^{2}{\frac{\theta}{2}}+\frac{\lambda}{4}\Big{(}1+\lambda+3\cos^{2}{\frac{\theta}{2}}\Big{)}\Big{]}$ 0 \| 1 \| $\Big{(}\frac{GM}{\sin^{2}{\frac{\theta}{2}}}\Big{)}^{2}\cos^{4}{\frac{\theta}{2}}$ $\neq 0$ \| 1 \| $\Big{(}\frac{GM}{\sin^{2}{\frac{\theta}{2}}}\Big{)}^{2}\Big{[}\frac{1}{3}+\frac{2}{3}\cos^{4}{\frac{\theta}{2}}-\frac{\lambda}{3}\Big{(}1-\frac{3\lambda}{4}-4\cos^{2}{\frac{\theta}{2}}\Big{)}\Big{]}$ 0 \| 2 \| $\Big{(}\frac{GM}{\sin^{2}{\frac{\theta}{2}}}\Big{)}^{2}\Big{(}\sin^{8}{\frac{\theta}{2}}+\cos^{8}{\frac{\theta}{2}}\Big{)}$ The first term in the expression (13) coincides with that obtained by Einstein in 1916 by solving the equation of light propagation in the field of a static body 24 , whereas the second one is the most important correction due to the mass $m$ of the massive photon. On the other hand, the angle of gravitational bending measured by the experimental groups is expressed in general trough the relation 25 $\displaystyle\theta_{\mathrm{exp}}=\frac{1+\gamma}{2}\theta_{\mathrm{E}},$ (14) where $\gamma$ is the deflection parameter characterizing the contribution of space curvature to gravitational deflection. From Eqs. (13) and (14), we then get $\displaystyle\theta_{\mathrm{E}}\frac{m^{2}}{8\pi^{2}\nu^{2}}<\theta_{\mathrm{E}}\left(1-\frac{1+\gamma}{2}\right),$ (15) implying $\displaystyle m<2\pi\nu\sqrt{\left\|1-\gamma\right\|}.$ (16) Not long ago, Fomalont et al. 22 determined the deflection parameter $\gamma=0.9998\pm 0.0003$ (68% confidence level), using the VLBI at $43,23$ and $15GHz$ to measure the solar gravitational deflection of radio waves. Their results come mainly from $43GHz$ observations where the refraction effects of the solar corona were negligible beyond 3 degrees from the sun 26 . Using the result for the deflection parameter found by Fomalont et al. and assuming that the massive photon passing near the solar limb has a frequency $\nu=43GHz$ (which is perfectly justifiable in view of the argument previously provided), we conclude that $m<3.5\times 10^{-11}MeV$. We remark that Eq. (12) can also be deduced à la Einstein, namely, by finding an approximate solution to the geodesic equation of motion of a massive test particle in the Schwarzschild field. By adopting this approach, an expression for the angle of particle deflection by the sun was obtained to order $\left(\frac{GM}{b}\right)^{3}$ in Ref. 27. This kind of deduction, however, is a time-consuming work. On the other hand, Golowich, Gribosky, and Pal 28 , instead of taking the usual geometrical approach, considered the phenomenon of light bending as a quantum scattering problem. This treatment, which is not only instructive but also straightforward when the gravitational field is weak, allowed them to easily obtain an expression for the gravitational deflection of massive particles to order $\frac{GM}{b}$. An identical result was found by Mohany, Nieves, and Pal 29 using a method pioneered by Ohanian and Ruffini 30 . At this point, some comments are in order. * • According to general relativity, photons are not only deflected but also delayed by the curvature of space-time produced by any mass. And more, the bending and delay are proportional to $\gamma+1$. Consequently, time delay techniques can also be employed to set up bounds on the photon mass. It is interesting to note that a few years ago, Bertotti, Iess, and Tortora 31 reported a measurement of the frequency shift of radio photons to and from the Cassini spacecraft as they passed near the sun that led to a result for $\gamma$ which agrees with the predictions of standard general relativity with a sensitivity that approaches the level at which, theoretically, deviations are expected in some cosmological models 32 ; 33 . * • Equation (13) was derived on the assumption that the field responsible for the photon deflection is a static gravitational field. Nonetheless, as well-known, neither the sun nor the planets are at rest in the solar system. Actually, they are moving with respect to both the barycenter of the solar system and the observer. This motion will certainly bring about velocity-dependent corrections to the general-relativistic equation of the gravitational deflection of light. As a consequence, the aforementioned motion-induced correction to the gravitational deflection of light shall correlate with the correction to the photon’s mass exhibited in equation (13). This fact leads us to pose an important question: Currently, is modern technology sensitive enough to detect these tiny relativistic effects caused by the dependence of the gravitational field on time? Kopeikin 34 claims that ‘future gravitational light-ray deflection experiments 35 , radio ranging BepiColombo experiment 36 , laser ranging experiments ASTROD 37 and LATOR 38 will definitely reach the precision in measuring $\bar{\gamma}_{\mathrm{PPN}},\;\bar{\beta}_{\mathrm{PPN}}$ and $\bar{\delta}_{\mathrm{PPN}}$ that is comparable with the post-Newtonian corrections to the static time delay and to the deflection angle caused by the motion of the massive bodies in the solar system 39 .’ Here, deviation from general relativity are denoted with the comparative PPN parameter $\bar{\gamma}_{\mathrm{PPN}}\equiv\gamma_{\mathrm{PPN}}-1$, $\bar{\beta}_{\mathrm{PPN}}\equiv\beta_{\mathrm{PPN}}-1$ and $\bar{\delta}_{\mathrm{PPN}}\equiv\delta_{\mathrm{PPN}}-1$. On the other hand, one can show, using the equation for the post-post-Newtonian time delay, $\Delta t$, which was obtained by Kopeikin by coupling the PPN parameters with the velocity-dependent terms, that for gravitational experiments with light propagating in the field of the sun, $\Delta t\approx\left(1+\bar{\Gamma}\right)\ln\left(\frac{r_{1}+r_{2}+r_{12}}{r_{1}+r_{2}-r_{12}}\right),$ (17) with $\bar{\Gamma}\approx\bar{\gamma}_{\mathrm{PPN}}-2\beta_{\odot},$ (18) where $\beta_{\odot}$ ($=5.3\times 10^{-8}$) is the solar velocity (in natural units) with respect to the barycenter of the solar system, $r_{12}$ is the coordinate distance between the emission and observation points, $r_{1},\;r_{2}$ are radial distances to the emission and observation points, respectively. Now, noticing that the LATOR and ASTROD space missions are going to measure the $\bar{\gamma}_{\mathrm{PPN}}$ parameter with a precision approaching to $10^{-9}$ 37 ; 38 , we arrive at the conclusion that in the near-future, the explicit velocity-dependent correction to the static time delay in the solar gravitational field must apparently be taken into account. Let us then answer the question raised above. For the sake of simplicity we restrict our discussion to measurements of light bending by the sun obtained trough VLBI techniques. Currently the experimental groups have determined the parameter $\bar{\gamma}_{\mathrm{PPN}}$ using the VLBI with an accuracy of $10^{-4}$ 22 . Therefore, the alluded velocity-dependent correction is too small and can be neglected in the determination of $\bar{\gamma}_{\mathrm{PPN}}$. Actually, the detection of so small an effect is beyond current technology. * • Nowadays, as we have already pointed out, the VLBI is the most accurate technique we have at our disposal for measuring radio-wave gravitational deflection on a regular basis 19 ; 20 ; 21 . It was only superseded by the multiple frequency Doppler-tracking of Cassini spacecraft 31 . * • Measuring light deflection with optical techniques may turn out more advantageous for determining the parameter $\gamma$ in a foreseeable future 40 . ## IV Discussion We discuss now whether or not the bounds we have found could be improved. To begin with, we consider the quantum limit. A quick glance at Eq. (9) clearly shows that a better agreement between theory and experiment concerning the anomalous magnetic moment of the electron necessarily leads to an improvement on the quantum bound. Consequently, there is a great probability of obtaining a better quantum bound on the photon mass in the foreseeable future. We analyze in the sequel how a better limit on the photon mass might be obtained using Eq. (16). First, if the deflections measured using the VLBI could be made with greater accuracy the value of $\sqrt{\left\|1-\gamma\right\|}$ would be reduced giving, as a result, a better gravitational estimate. According to Fomalont $et\;al.$ 22 , a series of designed experiments with the VLBI could increase the accuracy of the future experiments by at least a factor of 4. Second, if deflection measurements can be obtained at lower frequencies, while maintaining the value of the deflection parameter $\gamma$, the gravitational bound will be improved in direct proportion to the frequency. This point, however, needs to be dealt with carefully. In fact, as we have already mentioned, up till now the best results obtained for the gravitational deflection via the VLBI are those that come mainly from $43GHz$ where the refraction effects of the solar corona are negligible beyond 3 degrees from the sun. Incidentally, the lowest frequency employed by the radio astronomers was $2GHz$. However, the measurements made at this frequency are less reliable because of the refraction effects of the solar corona. Actually, the radio astronomers use in their experiments a mixing of different frequencies but the most significant contributions come in general from $\sim 43GHz$. This possibility of increasing the gravitational limit is thence very limited. Certainly, the bounds we have found on the photon mass are higher than the recently recommended limit published by the Particle Data Group 12 . They are nevertheless comparable to other existing bounds (See Table II) and bring new elements to the issue of restricting the photon mass. Accordingly, they do have some merits. We discuss their main qualities in the following. Table 2: Some upper bounds on the photon mass obtained by measuring the dispersion in the speed of light in different ranges of the electromagnetic spectrum (in chronological order). Author \| Type of \| Limits on $m$ ---\|---\|--- (year) \| measurement \| ($MeV$) Froome \| Radio-wave \| $2.4\times 10^{-13}$ (1958)41 \| interferometer \| Warner et al. \| Observations on Crab \| $2.9\times 10^{-14}$ (1969)42 \| Nebula pulsar \| Bay et al. \| Pulsar emission \| $1.7\times 10^{-19}$ (1972)43 \| \| Brown et al. \| Short pulses \| $7.9\times 10^{-7}$ (1973)44 \| radiation \| Schaefer \| Gamma ray bursts \| $2.4\times 10^{-17}$ (1999)45 \| (GRB980703) \| \| Gamma ray bursts \| $3.4\times 10^{-12}$ \| (GRB930229) \| * • The theory adopted to describe the photon mass has the correct limit. * • The bounds are based on exact calculations performed in the framework of QED and general relativity, respectively; besides, the most accurate experimental data currently available have been taken as input. * • The conceptual approaches adopted to estimate the bounds are new. * • The methods used for placing the bounds are interesting in their own, although they do not lead to the most stringent limits. Indeed, the quantum bound is estimated using one of the most renowned predictions of QED — the anomalous magnetic moment of the electron, while the gravitational bound is obtained using the properties of gravity. Essentially, the point is that a massive photon is bent in a gravitational field by a different amount than a massless photon. Thus, observations of light bending by the sun allow one to place limits on the photon mass. * • The bounds are essentially a measurement of the agreement between theory and experiment. Since the two limits are of the same order, they may be used to give an idea of how much the theoretical prediction deviates from the experimental result. For the quantum and semiclassical bounds we have estimated this lower limit is $m^{-1}\sim 2cm$. Thus, the more the value of $m^{-1}$ increases, the more the concordance between theory and experiment increases. In other words, a null mass for the photon would imply a perfect agreement between theory and experiment * • Recently, Adelberger, Dvali, and Gruzinov 9 questioned the validity of some bounds on the photon mass available in the literature. They claim that if $m$ arises from a Higgs effect, these limits are invalid because the Proca vector potential of the galactic magnetic field may be neutralized by vortices giving a large-scale magnetic field that is effectively Maxwellian. However, these criticisms do not apply to our computations because they are based on the plausible assumption of large galactic vector potential; furthermore, in our case $m$ does not arise from a Higgs effect. Last but not least, we would like to draw the reader’s attention to the article by Barton and Dombey 46 in which it is demonstrated that the Casimir effect is not sensitive to a small photon mass. 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1012.2797	# Should Corpora be Big, Rich, or Dense? ###### Abstract In this paper, we ask what properties makes a large corpus more or less useful. We suggest that size, by itself, should not be the ultimate goal of building a corpus. Large-scale corpora are considered desirable because they offer statistical stability and rich variation. But this rich variation means more factors to control and evaluate, which can limit the advantages of size. We discuss the use of multi-channel data to complement large-scale speech corpora. Even though multi-channel data may limit the scale of a corpus (due to the complex and labor-intensive nature of data collection) they can offer information that allows us to tease apart various factors related to speech production. The reference for this paper is: Greg P. Kochanski, Chilin Shih, Ryan Shosted, “Should Corpora be Big, Rich, or Dense?”, presented at _New Tools and Methods for Very-Large-Scale Phonetics Research_ , University of Pennsylvania, January 28-31, 2011. http://www.ling.upenn.edu/phonetics/workshop/index.html To see a world in a grain of sand And a heaven in a wild flower, Hold infinity in the palm of your hand And eternity in an hour. William Blake - Auguries of Innocence Index Terms: corpora, experimental, linguistics, speech, articulation, large ## 1 Why use a Large Corpus? Not too long ago, the concept of a large linguistic corpus didn't exist; neither did the infrastructure necessary to build and maintain such a corpus. Recently, speech technology has opened up the possibility of conducting large experiments. Consider an enthusiastic human communicator who makes 200 hours of phone calls per month. Digitized at 16 bits, 16 kHz over a 90-year lifetime, this amounts to just 25 Terabytes, for a lifetime storage cost of $\approx$US$10,000.111Assuming November 2010 storage costs, no future price reductions, and a disk lifetime of 10 years. Given a suitable speech recognition system, that lifetime of data could be transcribed if a comparable amount of money were spent on computers and electricity. We are approaching the point where we can now investigate an entire language, rather than a small sample. Since the costs of such enormous corpora are suddenly within the realm of possibility, we ask how they should be designed. In the past, speech corpora have been small; increasing the size was intended to increase statistical power. If one is counting linguistic items222I.e. the frequency with which an word (or other linguistic items) occurs in a text. Or, more generally, the frequency of a particular word (phone, phrase, accent, …) combination in a particular context., 1000 examples ($N=1000$) are much more informative than one, because they allow you to estimate the frequency of the word precisely333The confidence intervals and statistical significance of frequency measurements can be modeled by Poisson statistics, where the fractional accuracy of a frequency measurement is $N^{-1/2}$, where $N$ is the number of occurrences of the items. So, $N=1000$ occurrences allows you to measure an item’s frequency within 3%., whereas a single example gives only the crudest possible idea of how common the word is. Similarly, a single measurement of an acoustic property means little, because from it we learn nothing about variability. Ten samples allow us to measure variability in one dimension; one hundred or a thousand samples allow us to come up with multidimensional correlations. In principle, more repetitions of a word will allow for a more precise measurement of the average properties of a sound, but the benefits of repetition taper off beyond $N=1000$. Currently, we don't know of two theories of speech variation that can be differentiated by measurements at this level of precision. It is possible that theories of speech variation will never be this precise because language is not part of the Newtonian “clockwork universe”, and some of the observed variation may be inherent to a stochastic communication system. In natural speech (or near approaches to it), the frequency distribution often follows Zipf’s law [1, 2]: There are a few items in a corpus with very high frequency, more items with lower frequencies, but most items have a very low frequency. One example is the distribution of words: 5% of an English text corpus is “the”, but most words are more like “haggard”, with frequencies near 0.0001%. Any particular word like “haggard” may not even appear in a corpus of less than a million words, even though such words (as a group) form much of the corpus. For applications that need a good representation of infrequent events, such as an automatic speech recognition system, it is crucial to train the systems using a very large corpus. This ensures correct recognition of infrequent words or unusual combinations of sounds in a variety of dialects. We can define a boundary between “small” and “large” corpora by asking whether the most common items occur often enough ($N>1000$) to allow for good measurements. In a small corpus, examples of all items are scarce; in a large corpus at least the most common items are sufficiently represented. The next natural step is a huge corpus, where most items have $N>1000$. Large corpora are appearing; huge corpora (except for phones) are still rare (Table 1). However, even the biggest current audio corpora, like the BNC [3] are just entering the “large” category if one wishes to study how one word affects another. Table 1: Large and Huge phonetic corpora. Research on: How big is a “large” corpus? …a “huge” corpus? Individual phones $>10^{3}$ words $>10^{5}$ words Triphones $>10^{5}$ words $>2\cdot 10^{6}$ words Triphones with prosody444 Assuming several prosodic factors, such as stress, focus, distance from speaker to listener, and noise level. $>10^{6}$ words $>4\cdot 10^{9}$ words Individual words $>3\cdot 10^{5}$ words $>10^{9}$ words Word bi-grams $>10^{7}$ words $>10^{15}$ words If one starts with a minimally large corpus, because of Zipf’s law there will be only a few items whose frequencies can be measured precisely. If we make the corpus bigger, this charmed circle of items with $N>1000$ will slowly expand. So, very large corpora help studies of rare items – and recall that most linguistic items are rare. As can be seen in Table 1, one would need to expand the corpus by factors of hundreds, thousands, or even millions to be able to study an entire language, instead of studying merely its most frequent items. ## 2 Natural Speech vs. Experiments The extreme amount of data needed for a huge corpus is a consequence of the rarity of many linguistic items (ie. Zipf’s law, interpreted broadly). But this is not a logical necessity, merely a description of the language that people produce in daily life. Techniques like sociolinguistic interviews (cf. [4]) and map tasks (cf. [5]) are useful to boost the frequency of a selected group of words while the speaker(s) still produce speech that is reasonably natural. These approaches are steps along a continuum towards a laboratory experiment, where the speech is under the experimenter’s control, and normally rare words and word combinations can be induced to occur as frequently as desired. So, for some purposes, laboratory experiments are far more efficient than a large corpus analysis. If a conclusion can be reached by examining a small fraction of the items in the whole language, and if these items can be easily induced, then an experiment may be appropriate. But, experiments have difficulties over and above the the possibility of phonetic differences between speech in a formal experiment and more natural situations (cf. [6]). An experiment (and the associated analysis) is often set up to decide between two possible hypotheses carefully chosen by the experimenter, based on the results of previous studies. When the null hypothesis is rejected, people may mistakenly assume the alternative is proven. This logic follows Sherlock Holmes’ famous dictum “When you have eliminated the impossible, whatever remains, however improbable, must be the truth.” [7]. While misleading, the dictum is not exactly wrong in the strict sense that the truth must be somewhere among whatever remains. However, Doyle (or Holmes?) was wrong to suggest that this was a useful way to solve difficult problems. It fails because when we apply it, our notion of “…whatever remains…” is limited by the human imagination, but the correct answer isn’t. The universe presents answers that people find hard to believe or imagine, so it is hard to design an experiment that anticipates them. In contrast, large speech corpora offer variations of language use and speech production that may be unexpected and hard to imagine. With large natural corpora, it is possible to break out of the limitations of one’s own imagination when one sees something unexpected. ## 3 Limits of Large Corpora In addition to their advantages, large corpora have disadvantages, too. Expanding a corpus often introduces extra factors into a statistical analysis. A small corpus might be very uniform: it might be acquired in a short time, in a restricted location, with a carefully defined dialect, in a uniform speaking style, under controlled recording conditions. Large corpora often allow some of these factors to vary, either for practical reasons, or intentionally, as a way to explore their effect. And, with each new factor, one should allocate some of the data towards understanding the effect of the factor. An (extreme) example can illustrate this point. Imagine a small corpus of English collected in Singapore, then double its size by adding American English. Singapore English is heavily influenced by its proximity to Chinese: it has different pronunciation, intonation, rhythm ([8], though see [9]) and word frequency. Any prosody research using the expanded corpus would probably be best done by partitioning the corpus into two halves, and analyzing each half separately. As a result, the expanded corpus will provide no better description of the prosody of Singapore English than the original.555Of course, the hypothetical enlarged corpus will allow dialect-to-dialect comparisons for whichever prosodic properties can be measured on the original corpus. However, we would only be able to measure and publish those comparisons if the corpus reliably separates speakers of the two dialects. Many do not, and fall back upon self-reporting and/or geographic information (e.g. the British National Corpus). This is an example where certain questions remain unanswerable, no matter how many dialects one adds to the corpus666Under some conditions, with a large and diverse corpus, the research questions can be broadened from (e.g.) “properties of a dialect” to “properties of the language” when more dialects are added. However, this should only be done in cases where it is reasonably clear that these average properties are relevant to real individuals who speak the language. For instance, “small” and “wee” are equivalent words in two British dialects, and British English as a whole might use “wee” 0.1% of the time (Google statistics for “wee child” vs. “small child”), but there may not be any actual individuals who use those two words interchangeably at the population average rate.. Sometimes, if there are confounds amongst the extra factors, they do not even yield interesting comparisons. For instance, one can imagine a corpus intended to sample the speech that the average British person would hear in the 1970s. It might be comprised of informal middle class speech in the local dialect and formal, RP speech from the BBC. Interpreting the difference between the two types of speech would be hindered because one would not know whether to attribute a difference to social class or to the formality of the presentation. Similar confounds between factors are common in speech data: the word pairs in a corpus are constrained by grammar, and the phone pairs in a word are limited to those present in the lexicon.777For instance, in a coarticulation experiment, one would like to be able to form all combinations of sounds to see how each sound affects all others. But most combinations are either unfamiliar to most speakers, or can only be formed across word boundaries. So, though size may have benefits, extra, uncontrolled factors often present in a larger corpus will erase some of the advantage: rich variation of a corpus is not necessarily an advantage unless the goal is to study variation. To an extent, one should think of a corpus in terms of the density of data per factor: the ratio between the size of a corpus and the number of combinations of relevant factors. If there is not enough data to support each factor, it will be impossible to find the best-fitting (possibly true), multi-factor explanation, no matter the size of the corpus. In other words, the design of the corpus can be more important than its size, especially as we move through the range of large, into huge corpora. ## 4 Multi-Channel Data Multi-channel data allow us to increase the data density of a corpus; such data can be used to complement controlled experiments and large, speech-only databases. Of course, having multiple data channels is nothing new to speech scientists, because any speech signal can be interpreted as a group of related signals, e.g. the power in various frequency bands may each be interpreted as separate signals.888As in a MFCC front end for a speech recognition system. By “multi-channel corpora” we mean corpora where the acoustics of speech are recorded along with other related signals. Data that can be recorded alongside speech acoustics include articulatory movement (Electromagnetic Articulography, ultrasound, fiberoscopy), linguopalatal contact (EPG), airflow and pressure, muscle activity (EMG), as well as facial and hand gestures.999 Part-of-speech annotation and other annotation might also count for something here, though such annotation carries relatively little information. In contrast to the large-scale speech corpus which are “horizontally rich” we view multi-channel data as “vertically” rich101010 Horizontally = large in terms of time; Vertically = large in terms of the number of measurements per time point. Data data are typically plotted on the y-axis against time.. Acoustic signals we record tell us something about the state of the oral articulators, but it is well-known that they render incomplete information. For instance, multiple articulatory configurations can generate virtually the same acoustic signal [10, 11]. This means, for instance, that one cannot deduce the state of the articulators from 100 milliseconds of a speech signal.111111Note that with longer speech signals, it is sometimes possible to use the idea that the motions of the articulators must be smooth and continuous to remove some ambiguities. See [12]. The ambiguity can become harder to resolve when one tries to deduce features of the language that are deeper than articulatory positions. For example, when an English speaker emphasizes a word, they may use a longer duration. But long durations are also associated with final lengthening and focus. So (absent other information), the case of a long syllable is ambiguous. Likewise, loudness can be associated with focus, emphasis, or low vowels, so observation of loudness alone cannot tell you the prosodic function. Fant put it neatly: “The translation from speech wave back to articulation is to some extent restricted by the existence of compensatory forms of articulation…A deeper insight into the potentialities of this aspect of the physiological interpretation of spectrograms must rely on extensive correlative work” [13, p. 209]. In some cases, the function of a gesture can be deduced by comparing several aspects of an acoustic signal. But humans experience richer communication in person than over the telephone, so there is good reason to believe that face, hand, and arm gestures are an important part of our communication. They may carry information of their own in addition to disambiguating the acoustics. To pick a trivial example, one cannot easily convey a shrug over the telephone. That information is either lost to the listener, or the speaker adapts to the communication channel and packages the information in some other form. Multi-channel data can be especially important when there are trade-off relationship between different factors. For example, while duration, loudness, and $f_{0}$ are recognized (across language) as important acoustic correlates of stress or emphasis, a speaker doesn’t need to use all factors at the same time to convey linguistic meaning. This might be implemented as a trade-off relationship where if a speaker lengthens the duration for emphasis, changes in loudness or $f_{0}$ would be unnecessary. Given such a trade-off, any one measurement (e.g. duration) would show large amounts of variation across emphasized syllables, but the correct combination of multiple properties would add up to some gestalt of emphasis with much less variability121212Strong trade-off relationships (to the extent that they exist) are important because they indicate that variability in certain combinations of acoustic parameters is linguistically unimportant. Absent knowledge of the trade-off, this variability would likely be interpreted as a difference in meaning or function.. Also, the articulatory-acoustic mapping is nonlinear (cf. [10, 14]) This means that (for instance) a 1mm closing gesture can be easily perceived in the confines of a narrow airway, but may be acoustically undetectable in an open airway. However, if one has formant information along with articulatory information, the formant information can provide a detailed view of the articulation near closure, and the articulatory measurements will constrain hypotheses about what may be going on when the airway is open.131313One might reasonably ask “why do the articulatory details matter when the airway is nearly open if it has no acoustic consequences?” First, your conversation partner may be watching you, so jaw opening may count as a facial gesture. Second, even for telephone speech, the width of opening is related to the velocity of the following closure, which may have audible consequences. Overall, adding data beyond audio measurements into a corpus can add substantial information that is not otherwise available. From the perspective of data density, this data brings along a minimum of extra factors because it is a simultaneous view of the exact same instance of a word. Contrast this with a horizontal expansion of a corpus: you can easily bring in new instances of the same word, but the new instances come without any reason to believe that they are equivalent to the instances you already have.141414I ndeed, if there is a relevant trade-off relationship that involves non-acoustic data, then one might well falsely conclude that two instances did not have equivalent meanings or functions. They are uttered in new conditions (typically we must introduce new factors to describe these conditions151515 Having metadata about the utterances will clearly help, but it should be noted that metadata derived from the audio is not strictly new, independent information. ), or simply uttered differently because of unexplained instance- to-instance variation. When you add a second instance of a word to a corpus, you cannot determine whether it is identical to the first word without spending some of the data’s explanatory power. In effect, one must introduce new factors that describe the differences between pairs of potentially identical words and new questions to answer161616Every pair of words comes with the implicit question “Are these words linguistically/functionally/phonologically equivalent or not?”. On the other hand, if you add multiple, simultaneous measures of a related signal, all for the same word, each measure corresponds to the exact same word you started out with. There is no question regarding the identity of the word, it is merely being viewed from a different angle.171717There will, typically, be some data spent to determine the relationship between acoustic and articulatory measurements. However, that is often more like an initial calibration, and one does not have a new increment of uncertainty with each new instance. ## 5 Conclusion It is generally agreed that multiple recordings of a given item will allow us to better understand variation, i.e. by revealing tendencies in the data from which we can make statistical inference. It follows that we should collect large numbers of items in order to make better predictions that generalize to the population. Corpus linguistics, as traditionally conceived, suggests that more observations of a phenomenon enable us to better understand the phenonmenon. While size generally helps, it is not always the case, and the design details can be very important. In some cases, a larger corpus raises more questions, and the increase in questions can cancel out the increase in size. Especially in cases where trade-offs are important or interpretation is ambiguous, multi-channel corpora with a relatively small number of items may have a comparable value to much larger acoustic-only corpora. ## 6 Acknowledgments We thank John Coleman for comments. Greg Kochanski appreciates support from the UK ESRC via RES-062-23-2566, RES-062-23-1172, and RES-062-23-1323. ## References * [3] Coleman, J., Liberman, M., Kochanksi, G., Burnard, L., and Yuan. J. ‘Mining a Year of Speech”, also submitted to this conference. * [7] Doyle, Sir Arthur. (1927). The Adventure of the Blanched Soldier, in The Casebook of Sherlock Holmes. * [10] Schroeder, M. R. (1967). “Determination of the vocal tract shape from measured formant frequencies”, J. Acoustic. Soc. Am. 41, pp. 1283–1294. * [13] Fant, Gunar. (1960). Acoustic Theory of Speech Production. Mouton & Co, The Hague, Netherlands. * [6] Kochanski, G., and Orphanidou, C. (2007) “Testing the Ecological Validity of Repetitive Speech” Proceedings of the International Congress of Phonetic Sciences (ICPhS XVI), Saarbr cken, Germany. http://www.icphs2007.de/conference/Papers/1632/1632.pdf * [9] Loukina, A., Kochanski, G., Shih, C., Keane, E., and Watson, I. (2009) “Rhythm measures with language-independent segmentation”, _Proceedings of the 10th Annual Conference of the International Speech Communication Association_ (Interspeech 2009). ISSN 1990-9772 Brighton, UK, 7–10 September 2009, pp 1531–1534. * [8] Ling, Low Ee, Grabe, Esther, and Nolan, Francis. (2000). “Quantitative characterizations of speech rhythm: Syllable-timing in Singapore English”. Language and Speech, 43 (4), pp. 377-401. * [4] Labov, William. (2006). The Social Stratification of English in New York City. Cambridge University Press. * [5] McAllister, J., Sotillo, C., Bard E.G., and Anderson, A.H. (1990). “Using the map task to investigate variability in speech,” Occasional paper, Department of Linguistics, University of Edinburgh. * [11] Mrayati, M., Carre, R., and Guerin, B. (1988) “Distinctive regions and modes: a new theory of speech production”, _Speech Communication_ 7(3), p. 257–286. * [12] Schroeter, Juergen and Sondhi, Mohan. (1994). Techniques for estimating vocal-track shapes from the speech signal. IEEE Transactions on Speech and Audio Processing, Vol 2, No 1, pp. 133-150. * [14] Stevens, Kenneth. (2000). Acoustic Phonetics. The MIT Press. * [1] Zipf, George K. (1935). The Psychobiology of Language. Houghton-Mifflin. * [2] Zipf, George K. (1949). Human Behavior and the Principle of Least Effort. Cambridge, MA: Addison-Wesley.	arxiv-papers	2010-12-13T16:44:46	2024-09-04T02:49:15.618638	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Greg P. Kochanski and Chilin Shih and Ryan Shosted", "submitter": "Greg P. Kochanski", "url": "https://arxiv.org/abs/1012.2797" }
1012.2908	Math. Model. Nat. Phenom. Vol. 6, No. 5, 2011, pp. 184-262 Quasichemical Models of Multicomponent Nonlinear Diffusion Alexander N. Gorbana111Corresponding author. E-mail: ag153@le.ac.uk, Hrachya P. Sargsyanb and Hafiz A. Wahaba a Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK b UNESCO Chair – Life Sciences International Postgraduate Educational Center (LSIPEC), Yerevan, Republic of Armenia Abstract. Diffusion preserves the positivity of concentrations, therefore, multicomponent diffusion should be nonlinear if there exist non-diagonal terms. The vast variety of nonlinear multicomponent diffusion equations should be ordered and special tools are needed to provide the systematic construction of the nonlinear diffusion equations for multicomponent mixtures with significant interaction between components. We develop an approach to nonlinear multicomponent diffusion based on the idea of the reaction mechanism borrowed from chemical kinetics. Chemical kinetics gave rise to very seminal tools for the modeling of processes. This is the stoichiometric algebra supplemented by the simple kinetic law. The results of this invention are now applied in many areas of science, from particle physics to sociology. In our work we extend the area of applications onto nonlinear multicomponent diffusion. We demonstrate, how the mechanism based approach to multicomponent diffusion can be included into the general thermodynamic framework, and prove the corresponding dissipation inequalities. To satisfy thermodynamic restrictions, the kinetic law of an elementary process cannot have an arbitrary form. For the general kinetic law (the generalized Mass Action Law), additional conditions are proved. The cell–jump formalism gives an intuitively clear representation of the elementary transport processes and, at the same time, produces kinetic finite elements, a tool for numerical simulation. Key words: diffusion, reaction mechanism, entropy production, detailed balance, complex balance, transport equation AMS subject classification: 35K57, 35Q82, 80A20, 80A30 ###### Contents 1. 1\. Introduction 1. 1.1. Linear Diffusion: from Graham and Fick to Einstein, Onsager and Teorell 1. 1.1.1. Fick’s Law 2. 1.1.2. Einstein’s Mobility 3. 1.1.3. Teorell Formula 4. 1.1.4. Onsager’s Linear Phenomenology 2. 1.2. Mechanisms of Nonlinear Diffusion 1. 1.2.1. Jumps on the Surface 2. 1.2.2. Diffusion in Solids as Reaction: from Frenkel to Eyring 3. 1.2.3. Ginzburg–Landau Free energy and Cahn–Hilliard equation 4. 1.2.4. Teorell Formula for Non-perfect Systems 3. 1.3. Main Ideas 1. 1.3.1. Mechanisms as Collections of Elementary Processes 2. 1.3.2. Discrete Kinetic Models and Lattice Automata 3. 1.3.3. Thermodynamics and Intermediate Complexes 2. 2\. Mass Action Law for Diffusion 1. 2.1. Mass Action Law 1. 2.1.1. Mass Action Law Kinetic Equations 2. 2.1.2. Existence and Uniqueness of Solutions 3. 2.1.3. Detailed Balance 4. 2.1.4. Complex Balance 2. 2.2. Mass Action Cell-Jump Formalism 1. 2.2.1. Stoichiometry of Diffusion Jumps 2. 2.2.2. MAL Equations for Diffusion 3. 2.2.3. Space Symmetry and Time Symmetry 4. 2.2.4. Arrested Diffusion and Boundary Equilibria 3. 2.3. Continuous Diffusion Equation 1. 2.3.1. MAL Diffusion Flux 2. 2.3.2. Examples 4. 2.4. Principle of Detailed Balance and Dissipation Inequality 1. 2.4.1. Detailed balance and Coupling of Direct and Reverse Processes 2. 2.4.2. The Dissipation Inequality and Detailed Balance 5. 2.5. Complex Balance in MAL Diffusion 1. 2.5.1. Complex Balance Conditions for MAL Diffusion 2. 2.5.2. The Dissipation Inequality and Complex Balance for MAL Diffusion 6. 2.6. Intermediate Summary 3. 3\. Generalized Mass Action Law for Diffusion 1. 3.1. Free Energy, Free Entropy, Chemical Potentials, Activities, and Generalized Mass Action Law 1. 3.1.1. Thermodynamic Potentials 2. 3.1.2. Markovian Microkinetics and Generalized Mass Action Law 2. 3.2. From Cell-Jump Models to Continuous Diffusion Equations for Generalized Mass Action Law 3. 3.3. Detailed Balance for the Generalized Mass Action Law and the Dissipation Inequality 1. 3.3.1. Isothermal conditions 2. 3.3.2. Generalization: Non-isothermal Processes 4. 3.4. Momentum and Center of Mass Conservation 1. 3.4.1. Mass Transfer and Heat Transfer 2. 3.4.2. Mechanisms of Transport and the General Forms of Macroscopic Equation 4. 4\. Conclusion ## 1\. Introduction ### 1.1. Linear Diffusion: from Graham and Fick to Einstein, Onsager and Teorell #### 1.1.1. Fick’s Law The first prominent equation of diffusion is Fick’s Law. According to this law, the diffusion flux $J$ is proportional to the antigradient of the concentration $c$: $J=-D\mathrm{grad}c\,.$ (1.1) The time derivative of the concentration is the negative of the divergence of the flux: $\frac{\partial c}{\partial t}=-\mathrm{div}J=D\Delta c\,,$ (1.2) where $\Delta$ is the Laplace operator. This statement is closely related to the Gauss–Ostrogradskii theorem $\iiint\limits_{V}\left(\mathrm{div}J\right)\,\mathrm{d}V=\iint\limits_{S}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\subset\\!\supset J\cdot n\,\mathrm{d}S\,.$ (1.3) The left side is an integral over the volume $V$, the right side is the surface integral over the boundary $S$ of the volume $V$, $S=\partial V$, and $n$ is the outward pointing unit normal field of $S$. The right-hand side represents the total flow across the boundary “out of the volume $V$”. The theorem was first discovered by J. L. Lagrange in 1762, then later independently rediscovered by C. F. Gauss in 1813, by G. Green in 1825 and in 1831 by M. V. Ostrogradsky. According to this theorem, the diffusion equation $\partial_{t}c=-\mathrm{div}J$ is just a conservation law: all changes of concentration are caused by the flux only. In his work on diffusion law, A. Fick used the conservation of matter and the analogy between diffusion and Fourier s law for heat conduction (1822), or Ohm s law for electricity (1827). Development of the fundamental law of diffusion was inspired by the Graham’s investigations on the diffusion of salts in water [55], in which he studied and compared the diffusibility of different salts. Before his study of diffusion in liquids, Graham studied diffusion in gases (1833). In 1863, J. C. Maxwell calculated the diffusion coefficients in gases from the Graham data. The results are amazing: “His coefficient of diffusion of CO2 in air is accurate at $\pm$5%. Isn’t it extraordinary?” [90]. Maxwell’s theory of diffusion was based on gas kinetics and mean fee path estimates. #### 1.1.2. Einstein’s Mobility In his theory of Brownian motion, A. Einstein [24] developed the microscopic theory of the diffusion coefficient for diluted particles in a liquid and compared two processes: the motion of particles in a liquid under a constant external force $K$, and diffusion. For a given $K$, each particle has the average velocity $\mathfrak{m}K$ where the coefficient $\mathfrak{m}$ characterizes mobility of particles. (We use $\mathfrak{m}$ for mobility and reserve $\mu$ for chemical potential.) For spherical particles in liquid, $\mathfrak{m}=\frac{1}{6\pi\eta r}\,,$ (1.4) where $\eta$ is the viscosity of the liquid, $r$ is the radius of particles, and ${6\pi\eta r}$ is the Stokes friction force. This approach results in a very useful relation for the diffusion coefficient: $D=\mathfrak{m}\frac{RT}{N_{A}}=\mathfrak{m}k_{B}T\,,$ (1.5) where $R$ is the gas constant, $N_{A}$ is the Avogadro constant, $k_{B}$ is the Boltzmann constant. The coefficient $\mathfrak{m}$ is called mobility or the Einstein mobility. Graham’s experimental research was extended to solids by W. C. Roberts-Austen [91]. He used Fick’s equation to determine the diffusion coefficient [79]. In 1922, S. Dushman and I. Langmuir [22] proposed to use the Arrhenius law for diffusion coefficient: $D=D_{0}\exp{(-Q/kT)}\,,$ (1.6) where $Q$ is a constant, which we now recognize as the activation Gibbs energy of diffusion $\Delta G$. More precisely, $\Delta G$ includes two terms: $\Delta G=\Delta H-T\Delta S$, where $\Delta H$ is the activation enthalpy and $\Delta S$ is the activation entropy. They checked this law by their own experiments with the diffusion of thorium through tungsten and found satisfactory agreement. Even better agreement was found with the published results of W. C. Roberts-Austen’s experiments. #### 1.1.3. Teorell Formula The mobility–based approach was further applied by T. Teorell [101]. In 1935, he studied the diffusion of ions through a membrane (Fig. 1). He considered a system of an ideally dilute solution of binary univalent strong electrolysis at the same temperature in water. The boundary is considered to be a membrane with strong a electrolyte in the presence of water. The solutions are assumed to be kept homogeneous on both sides of the membrane up to the boundary by some form of convection. The ionic mobilities within the membrane are assumed constant and may be different, and the membrane is not permeable for water. Heat effects, special membrane effects and chemical reactions are ignored. Figure 1: Teorell’s model [101]. A system of ideally dilute solutions in water of binary univalent strong electrolytes $DA$, $M^{\prime}B^{\prime}$, $M^{\prime\prime}B^{\prime\prime}$,… is considered. The boundary membrane is permeable for all ions (cations $D$, $M$,… and anions $A$, $B$,…) but the movement of water is prevented. The ionic concentrations outside are kept constant. Also $D_{i}^{+}$ is maintained constant. Accordingly, $DA$ is a steadily diffusing electrolyte. No other electric field is present besides that due to diffusion potential. No chemical reaction takes place. The steady state of a system of this nature with a steady diffusion is characterized by a constant ratio series: $\frac{M_{i}^{\prime}}{M_{o}^{\prime}}=\frac{M_{i}^{\prime\prime}}{M_{o}^{\prime\prime}}=\cdots=\frac{B_{i}^{\prime}}{B_{o}^{\prime}}=\frac{B_{i}^{\prime\prime}}{M_{o}^{\prime\prime}}=\cdots$. He formulated the essence of his approach in the formula: the flux is equal to mobility$\times$concentration$\times$force per gram ion. This is the so-called Teorell formula. The force consists of two parts: 1. 1. Diffusion force caused by concentration gradient: $-RT\frac{1}{c}\frac{\mathrm{d}c}{\mathrm{d}x}.$ 2. 2. Electrostatic force caused by electric potential gradient: $q\frac{\mathrm{d}\varphi}{\mathrm{d}x}.$ Here $R$ is the gas constant, $T$ is the absolute temperature, $c$ is the concentration, $q$ is the charge and $\varphi$ is the electric potential. In these notations, the Teorell formula for the flux $J$ is $J=\mathfrak{m}c\left(-\frac{RT}{c}\frac{\mathrm{d}c}{\mathrm{d}x}+q\frac{\mathrm{d}\varphi}{\mathrm{d}x}\right)$ (1.7) ($\mathfrak{m}$ denotes mobility; here we slightly modernize notations). It may be worthwhile to introduce the reference equilibrium concentrations vector $c^{}$ and write the diffusion force in the form $-\frac{RT}{c}\frac{\mathrm{d}c}{\mathrm{d}x}=-RT\frac{\mathrm{d}\ln(c/c^{})}{\mathrm{d}x}\,.$ (1.8) This expression allowed Teorell to find the concentration jump and the electric potential across the membrane caused by the joint action of diffusion and the electric field, when mobilities of various components are different. #### 1.1.4. Onsager’s Linear Phenomenology In 1931, L. Onsager [83, 84] included diffusion in the general context of linear non-equilibrium thermodynamics. For multi-component diffusion, $J_{i}=\sum_{j}L_{ij}X_{j}\,,$ (1.9) where $J_{i}$ is the flux of the $i$th component and $X_{j}$ is the $j$th thermodynamic force (for pure diffusion, this is the space antigradient of the $j$th chemical potential divided by $T$). After linearization near equilibrium, this approach gives for perfect systems (for which the chemical potential is $RT\ln(c/c^{})$): $\begin{split}&X_{j}=-\frac{1}{T}\mathrm{grad}\mu_{j}=-\frac{R}{c^{}_{j}}\mathrm{grad}c_{j};\;\\\ &J_{i}=-\sum_{j}L_{ij}\frac{R}{c^{}_{j}}\mathrm{grad}c_{j};\\\ &\frac{\partial c_{i}}{\partial t}=-\mathrm{div}J_{i}={R}\sum_{j}L_{ij}\frac{\Delta c_{j}}{c^{}_{j}}\,,\end{split}$ (1.10) where $c^{}_{j}$ are equilibrium constants ($c^{}$ is the point of linearization), deviations of $c_{j}$ from $c^{}_{j}$ are assumed to be small, $\Delta$ is the Laplace operator, and $L_{ij}=L_{ji}$ is the matrix of the coefficients. Its symmetry follows from microreversibility. The system (1.10) has one attractive property. Let us consider this system in a bounded domain $V$ with smooth boundary and with zero fluxes through its boundary: $(n,\mathrm{grad}c_{j})=0$ at any point of $\partial V$ at any time ($n$ is the vector of the outer normal). The positive quadratic functional $S_{2}=\frac{1}{2}\sum_{i}\int_{V}\frac{(c_{i}(x)-c^{}_{i})^{2}}{c_{i}^{}}\,\mathrm{d}x$ (1.11) is the second-order approximation to the relative entropy (or the so-called Kulback-Leubler divergence, see the review paper [47]) $S_{KL}=\sum_{i}\int_{V}c(x)\ln\left(\frac{c_{i}(x)}{c_{i}^{}}\right)\,\mathrm{d}x\,.$ (1.12) Let us calculate the time derivative of $S_{2}$ due to the system (1.10). Using the Gauss–Ostrogradskii formula (1.3) we get for the positive semidefinite matrix $L$: $\frac{\mathrm{d}S_{2}}{\mathrm{d}t}=-R\int_{V}\sum_{ij}{L_{ij}}\left(\frac{\nabla c_{i}}{c_{i}^{}},\frac{\nabla c_{j}}{c_{j}^{}}\right)\,\mathrm{d}x\leq 0\,,$ (1.13) where $\left(\frac{\nabla c_{i}}{c_{i}^{}},\frac{\nabla c_{j}}{c_{j}^{}}\right)$ is the inner product of the space vectors. Therefore, $\frac{\mathrm{d}S_{2}}{\mathrm{d}t}\leq 0$ if the symmetric coefficient matrix $L_{ij}$ is positive semidefinite (this means that for any vector $\xi$ the following inequality holds: $\sum_{ij}L_{ij}\xi_{i}\xi_{j}\geq 0$). The Onsager form of the diffusion equations is correct near the equilibrium but violates the obvious physical requirement: the diffusion flux of the $i$th component is zero if its concentration has zero value: the flux vanishes with the concentration. The Teorell formula satisfies this requirement. Fick’s law also satisfies this requirement in the following sense: if for nonnegative smooth $c(x)$ the concentration vanishes at some points, then at these points the flux vanishes too (because these points are minimizers of concentration and the gradient vanishes there). For isotropic non-perfect systems we have to use the generalized thermodynamic forces in Onsager’s form of the diffusion law: $X_{j}=-\left.\frac{\partial f}{\partial c_{j}}\right\|_{c=c^{}}\mathrm{grad}c_{j}\,,$ (1.14) where $Tf(c,T)$ is the free energy density. Let us denote $\Phi_{ij}=(\partial^{2}f/\partial c_{i}\partial c_{j})_{c=c^{}}$. In this notation, $\begin{split}&X_{j}=-\sum_{k}\Phi_{jk}\mathrm{grad}c_{k};\;J_{i}=\sum_{j}L_{ij}X_{j}=-\sum_{k}\left(\sum_{j}L_{ij}\Phi_{jk}\right)\mathrm{grad}c_{k};\\\ &\frac{\partial c_{i}}{\partial t}=-\mathrm{div}J_{i}=\sum_{k}\left(\sum_{j}L_{ij}\Phi_{jk}\right)\Delta c_{k}\,.\end{split}$ (1.15) The quadratic form $F_{2}=\frac{1}{2}\int\sum_{jk}\Phi_{jk}(c_{j}-c_{j}^{})(c_{k}-c_{k}^{})\,\mathrm{d}x$ is positive definite because $F$ is strictly convex. For positive definite $L$, $F_{2}$ decreases in time due to diffusion. Indeed, in a bounded domain $V$ with a smooth boundary and without fluxes through the boundary we get analogously to (1.13): $\frac{\mathrm{d}F_{2}}{\mathrm{d}t}=-\int_{V}\sum_{ij}\left(\sum_{k}\Phi_{ik}\nabla c_{k}\right){L_{ij}}\left(\sum_{l}\Phi_{jl}\nabla c_{l}\right)\,\mathrm{d}x\leq 0\,.$ (1.16) For non-isotropic diffusion (for example, in crystals), the coefficients $L$ have two pairs of indexes: $L_{i\alpha\,j\beta}$, where $i,j$ correspond to components and $\alpha,\beta$ correspond to the space coordinates. The forces and fluxes also have these two indexes and $J_{i\alpha}=\sum_{j\beta}L_{i\alpha\,j\beta}X_{j\beta}\,.$ In all cases, the diffusion equations in Onsager’s form do not describe the non-diagonal terms (the influence of gradients $c_{i}$ on fluxes of $c_{j}$ for $i\neq j$) properly near zeros of concentrations. These equations are applicable near a reference point $c^{}>0$ only. Non-diagonal diffusion must be non-linear. This simple remark is so important that we will explain it in detail. Let diffusion be non-diagonal and linear: $\partial_{t}c_{i}=\sum_{j}D_{ij}\Delta c_{j}\,.$ Assume that $D_{12}\neq 0$ and consider the state with $c_{2}=\ldots=c_{n}=0$. At this state, $\partial_{t}c_{2}=D_{12}\Delta c_{1}\,.$ If $D_{12}\Delta c_{1}(x)<0$ at some points then $c_{2}(x)$ becomes negative at these point in a short time. Therefore, linear non-diagonal diffusion does not preserve positivity of concentrations. ### 1.2. Mechanisms of Nonlinear Diffusion #### 1.2.1. Jumps on the Surface In 1980, A.N, Gorban, V.I. Bykov and G.S. Yablonskii [45] proposed a model for diffusion in monolayers of reagents on the surface of a catalyst, which is based on the jumps of the reagents on the nearest free places. This model was used for CO on Pt oxidation under low gas pressure. The system includes several reagents $A_{1},A_{2},\ldots A_{n}$ on the surface. Their surface concentrations are $c_{1},c_{2},\ldots c_{n}$. The surface is a lattice of the adsorbtion places. Each reagent molecule fills a place on the surface. Some of the places are free. We use $Z=A_{0}$ for a free place and the concentration is $z=c_{0}$. The sum of all $c_{i}$ (including free places) is constant: $\sum_{i=0}^{n}c_{i}=b=\mathrm{const}\,.$ The jump model gives for the diffusion flux of $A_{i}$ ($i=1,\ldots,n$): $J_{i}=-D_{i}[z\nabla c_{i}-c_{i}\nabla z]\,.$ (1.17) Therefore, the corresponding diffusion equation is: $\frac{\partial c_{i}}{\partial t}=-\mathrm{div}J_{i}=D_{i}[z\Delta c_{i}-c_{i}\Delta z]\,.$ (1.18) Due to the conservation law, $z=b-\sum_{i=1}^{n}c_{i}\,,$ and we have the system of $n$ diffusion equations: $\begin{split}&J_{i}=-D_{i}\left[\left(b-\sum_{i=1}^{n}c_{i}\right)\nabla c_{i}+c_{i}\nabla\left(\sum_{i=1}^{n}c_{i}\right)\right]\\\ &\frac{\partial c_{i}}{\partial t}=D_{i}\left[\left(b-\sum_{i=1}^{n}c_{i}\right)\Delta c_{i}+c_{i}\Delta\left(\sum_{i=1}^{n}c_{i}\right)\right]\,.\end{split}$ (1.19) It is straightforward to check that when $c_{i}\geq 0$ for all $x$ then $\partial_{t}c_{i}\geq 0$ for $c_{i}=0$. This is a necessary condition for preservation of positivity. If we assume that all particles can exchange their positions with their closest neighbors then a simple generalization of (1.17), (1.19) appears: $\begin{split}&J_{i}=-\sum_{j}D_{ij}[c_{j}\nabla c_{i}-c_{i}\nabla c_{j}]\,;\\\ &\frac{\partial c_{i}}{\partial t}=\sum_{j}D_{ij}[c_{j}\Delta c_{i}-c_{i}\Delta c_{j}]\,,\end{split}$ (1.20) where $D_{ij}=D_{ji}\geq 0$ is a symmetric matrix of coefficients which characterize the intensities of jumps. The entropic Lyapunov functional for (1.19), (1.20) has a simple traditional form of perfect relative entropy: for any reference vector of concentrations $c^{}$ ($c^{}_{i}\geq 0$) $S_{KL}=\int\sum_{i}c_{i}\ln\left(\frac{c_{i}}{c_{i}^{}}\right)\,\mathrm{d}x\,.$ (1.21) Remark: the free place entropy should be obligatorily included into $S_{KL}$. Simple algebra gives that in a bounded domain $V$ with a smooth boundary and without fluxes through boundaries $\frac{\mathrm{d}S_{KL}}{\mathrm{d}t}=-\sum_{ij}D_{ij}\int_{V}\left(\frac{c_{i}}{c_{j}}\nabla c_{j}-\frac{c_{j}}{c_{i}}\nabla c_{i}\right)^{2}\,\mathrm{d}x\leq 0\,.$ (1.22) This inequality provides the Lyapunov stability of diffusion. It is worth mentioning that the thermodynamic inequality (1.22) requires only the non-negativity of coefficients $D_{ij}$ and does not imply any requirements on the matrix $D$ as a whole (like positive definiteness). Another form of the thermodynamic inequality makes the formula for the entropy production more transparent: $\frac{\mathrm{d}S_{KL}}{\mathrm{d}t}=-\sum_{ij}D_{ij}\int_{V}\left(c_{i}\nabla\ln\left(\frac{c_{j}}{c_{j}^{}}\right)-c_{j}\nabla\ln\left(\frac{c_{i}}{c_{i}^{}}\right)\right)^{2}\,\mathrm{d}x\leq 0\,.$ (1.23) This system of models was further developed by A.N. Gorban and H.P. Sargsyan [53] and published in a book [46]. #### 1.2.2. Diffusion in Solids as Reaction: from Frenkel to Eyring The physical idea of the quasi-chemical representation of diffusion in solids belongs to Yakov Frenkel [36, 37]. He introduced both the vacancy and the interstitial mechanisms of diffusion and found some rate constants from experimental data. Thirty years later, F. C. Frank and D. Turnbull developed the Frenkel theory further [34]. They studied the diffusion of copper in germanium. This diffusivity is very rapid. They proposed that the copper could be dissolved in two states, interstitial and substitutional. For the interstitial state the solubility of copper is two orders of magnitude less and the diffusivity many orders of magnitude greater than in the substitutional state. The conversion of these states is effected by lattice vacancies. The quasi-chemical theory of diffusion and viscosity was developed also by H. Eyring with co-authors [66]. Eyring developed the theory of absolute reaction rates for chemical reactions in gases [28] and in condensed phase [116] and then applied these ideas to transport phenomena. In this theory, the transport process is represented by an ensemble of elementary events. Each elementary event is represented by the creation or disintegration of an activated complex. The rate of the elementary process is given by the concentration of activated complexes, multiplied by the rate at which they decompose. The main constructive hypothesis is that it is possible to calculate the concentration of activated complexes by equilibrium statistical thermodynamics: the complex concentration is in quasi-equilibrium with the stable components. Each complex has its “internal translational” degree of freedom. On the surface of potential energy this corresponds to the “reaction path”. Complexes move along this path. The velocity of this motion is assumed to be just a thermal velocity and is proportional to $\sqrt{T}$. The additional reaction path degree of freedom has its own kinetic energy and, therefore, increases the complex heat capacity. We have to take this into account in the calculation of the equilibrium constant. Collective models of diffusion were proposed too. One of the earliest collective model is the Z. Jeffries “ring mechanism” with 4 or more atoms. More on the history of solid-state diffusion is presented in the review [104] and in a modern textbook [78]. On the surface, there are various mechanisms for collective diffusion [88] as well. Elementary events for these mechanisms involve many atoms simultaneously. A dynamic description of nonlinear multicomponent diffusion requires a unified framework that should satisfy basic physical principles. #### 1.2.3. Ginzburg–Landau Free energy and Cahn–Hilliard equation The processes of phase separation has remained for a long time an important source of problems and ideas for the theory of nonlinear diffusion. The analogue for Fick’s equation is the Cahn–Hilliard equation [12]. The Cahn–Hilliard equation in its simplest form has the standard Onsager form, the flux is proportional to the force, the force is the gradient of the chemical potential: $J=-D\nabla\mu\,,\;\frac{\partial c}{\partial t}=-\mathrm{div}J=D\Delta\mu\,.$ (1.24) If we compare this equation to the Teorell formula then we immediately find the missed factor $c$ (concentration). We will return to the problem of the proper prefactor in the Cahn-Hilliard equations later. The main specificity of the Cahn-Hilliard equations is the form of the free energy and the chemical potential [12, 11] (the Ginzburg–Landau form): $f=f^{c}(c)+\gamma(\nabla c)^{2}\,,\;F=\int f\,\mathrm{d}x\,,\;\mu=\frac{\partial f^{c}(c)}{\partial c}-\gamma\Delta c\,.$ (1.25) The term $\gamma(\nabla c)^{2}$ in the free energy penalizes ovr sharp gradients and, in particular, models the interface energy. According to (1.24) and (1.25), the Cahn–Hilliard equation reads $J=-D\nabla\left(\frac{\partial f^{c}(c)}{\partial c}-\gamma\Delta c\right)\,,\;\frac{\partial c}{\partial t}=D\Delta\left(\frac{\partial f^{c}(c)}{\partial c}-\gamma\Delta c\right)\,.$ (1.26) If the chemical part of the free energy, $f^{c}(c)$ is not convex then phase separation is possible. If at the point $c$ this function is concave (spinodal) then without the term with $\Delta^{2}$ the constant solution $c(x)=c$ for the diffusion equation becomes unstable to any perturbation (negative diffusion coefficient). The term $-\gamma\Delta^{2}c$ in the right hand part of the Cahn–Hilliard equation regularizes solutions and the existence theorem was proved for the initial–boundary value problem given smooth initial data [26]. The proof relies essentially on the existence of a Lyapunov functional $F$ (1.25). The time derivative of $F$ in a domain $V$ with a smooth boundary and without external fluxes ($(J,n)=0$ on the boundary, where $n$ is the vector of outer normal to the boundary) is $\begin{split}\frac{\mathrm{d}F}{\mathrm{d}t}&=\int_{V}\mu\partial_{t}c\,\mathrm{d}x=\int_{V}\mu\mathrm{div}J\,\mathrm{d}x\\\ &=-\int_{V}(\nabla\mu,J)\,\mathrm{d}x=-D\int_{V}(\nabla\mu)^{2}\,\mathrm{d}x\leq 0\,.\end{split}$ (1.27) If we correct the Cahn–Hilliard equation by the “Teorell” factor $c$ then the dissipation inequality $\dot{F}\leq 0$ (1.27) persists: Due to the Teorell formula $\begin{split}&J=-\mathfrak{m}c\nabla\mu=-\mathfrak{m}c\nabla\left(\frac{\partial f^{c}(c)}{\partial c}-\gamma\Delta c\right)\,,\\\ &\frac{\partial c}{\partial t}=-\mathrm{div}J=\mathfrak{m}\,\mathrm{div}\left(c\,\mathrm{grad}\left(\frac{\partial f^{c}(c)}{\partial c}-\gamma\Delta c\right)\right)\,.\end{split}$ (1.28) Here, $\mathfrak{m}$ is the Einstein mobility. This equation should be called “The Cahn–Hilliard–Teorell” equation. In accordance with (1.28), $\begin{split}\frac{\mathrm{d}F}{\mathrm{d}t}&=\int_{V}\mu\partial_{t}c\,\mathrm{d}x=\int_{V}\mu\mathrm{div}J\,\mathrm{d}x\\\ &=-\int_{V}(\nabla\mu,J)\,\mathrm{d}x=-\mathfrak{m}\int_{V}c(\nabla\mu)^{2}\,\mathrm{d}x\leq 0\,.\end{split}$ (1.29) This dissipation inequality allows us to transfer all the results about solutions of the Cahn–Hilliard equation to the Cahn–Hilliard–Teorell equation. #### 1.2.4. Teorell Formula for Non-perfect Systems It seems very natural that the flux is proportional to the concentration of particles: the average velocity is proportional to the force and the total flux is the product of the average velocity and the amount of moving particles. This could be proved in the framework of non-equilibrium thermodynamics and the theory of absolute reaction rates when the concentration of moving particles is small. In perfect gases or in dilute solutions the chemical potential is $\mu=RT\ln c+\mu_{0}\,,$ where $\mu_{0}$ does not depend on $c$ (it is a function of $T$ and the state of the environment). In this case, we neglect the interaction between moving particles and use the Teorell formula (exactly as Einstein did in his theory of Brownian motion 30 years before Teorell [24]). When the concentration of moving particles $c$ is not small enough then the formula for perfect chemical potential is no longer valid and in front of $c$ in the flux a special activity coefficient $\alpha$ appears. Such coefficients were introduced for diffusion by Eyring at al in 1941 [66] and were used systematically for the theory of nonlinear diffusion by Gorban at al in the 1980–1986 [46]. Roughly speaking, the activity $a=\exp\left(\frac{\mu}{RT}\right)\,$ should substitute the concentration $c$ in the Teorell formula with the proper renormalization of the mobility coefficients: $J=\mathfrak{m}^{\prime}a(-\nabla\mu+(\mbox{external force per gram particle}))\,.$ The renormalized coefficient $\mathfrak{m}^{\prime}$ is defined by the condition: $(\mathfrak{m}^{\prime}a)/(\mathfrak{m}c)\to 1$ for $c\to 0$, $\nabla c\to 0$. This means that $\mathfrak{m}^{\prime}=\mathfrak{m}\exp\left(-\frac{\mu_{0}}{RT}\right)\,\mbox{ where }\mu_{0}=\lim_{c\to 0}(\mu-RT\ln c)\,,$ or we can write the Teorell formula for non-perfect systems using the usual Einstein mobility $\mathfrak{m}$ defined for small concentrations and this standard value of chemical potential, $\mu_{0}$: $J=\mathfrak{m}\exp\left(\frac{\mu-\mu_{0}}{RT}\right)(-\nabla\mu+(\mbox{external force per gram particle}))\,.$ (1.30) This formula is the main analogue of Fick’s law for monomolecular diffusion in non-perfect media. For the Cahn–Hilliard–Teorell equation (1.28), the Teorell formula for non- perfect systems (1.30) significantly changes the diffusion coefficient: the regularizing gradient term should appear in the activity coefficient. More details about activity coefficients in thermodynamics can be found, for example, in Chapter 9 of the classical book [21]. In the phase separation problem, the components are definitely non-perfect and the further correction of the Cahn–Hilliard–Teorell equation by the activity coefficients is necessary. The problem of the extension of the Cahn–Hilliard approach to multicomponent diffusion was discussed by various authors [76, 4]. Elastic forces and plasticity are also taken into account [4, 35]. Nevertheless, the problem of the proper equations of multicomponent nonlinear diffusion in highly non- homogeneous condensed phases is still open. From our point of view, there is no single “proper model” and the variety of possible models is very rich. In our work, we attempt to formulate the proper language for the description of the universe of these models similarly to chemical kinetics models. ### 1.3. Main Ideas #### 1.3.1. Mechanisms as Collections of Elementary Processes A complex process can be disassembled into several elementary processes. The dependence of the process rate (the flux) on the state (concentrations, chemical potentials and their gradients) is simple for elementary processes. The model of the whole process is assembled from these elementary “details”. This idea was developed in chemical kinetics. In 1862–1867, Guldberg and Waage proposed the mass action law for equilibrium. In 1879 they developed the mass action law for dynamics. This idea was developed further by many researchers and after several dozen years it was transformed into a technology for the representation of complex processes: A complex reaction is represented as an ensemble of elementary reactions. The reaction rate has a simple monomial dependence on concentration. Van’t Hoff [107] called the reactions that satisfy the mass action law “normal transformations” and found that “normal transformations take place very rarely”. Now we can say that most reactions are complex and the interaction of several elementary reactions causes non-trivial complex (“abnormal”) behavior. Van’t Hoff did not study complex reactions by disassembling them into several elementary reactions. Therefore, he was disappointed with the mass action law and finally wrote: “As a theoretical foundation I did not accept the concept of mass action, I had to abandon this concept in the course of my experiments…”. (“J’ai adopté pour la théorie, non la notion des masses actives, notion que j’ai dû abandonner dans le cours de mes expériences, …” [107], p. 7.) The set of elementary reactions which constitute a complex reactions is called the reaction mechanism. A mechanic analogue is obvious: the elementary reactions are the details of the mechanism that represents the complex reaction. This notion was, finally, introduced into chemistry in the 20th century due to the efforts of M. Bodenstein. M. Boudart described the “century of Bodenstein” in his paper [5]: “First came the data, then the rate equation, and finally the fitting of the data into the rate equation by means of a hypothesized mechanism with rate constants chosen for the best fit.” The great success of this approach was the theory of chain reactions. N.N. Semenov was awarded the Nobel prize for this theory [94]. The modern theory of complex chemical reactions is based on the idea of the detailed reaction mechanism and the simple kinetic law of elementary reactions [117]. Many authors proposed various particular mechanisms of nonlinear diffusion. One of our goals in this work is to repeat the way of chemical kinetics in application to multicomponent diffusion and to create a comprehensive theory of the mechanisms of diffusion. #### 1.3.2. Discrete Kinetic Models and Lattice Automata In the 1940s, S. Ulam and J. von Neumann proposed networks of interconnected finite-state automata for the modeling of complex systems. In the first period of study, research was focused on the abilities of these networks and, in particular, on the ability of self-reproduction [111]. The behavior of these cellular automata is so variable and surprising and its complexity is so high that Ulam proposed the idea of the computational experiment: we should regard our invention as a new sort of reality and study it by the experimental approach, as physics or chemistry does. Cellular automata were invented as an intellectual journey but soon were recognized as efficient tools for modeling [103]. Feynman’s attention to automata with local interactions as a tool for simulating physics, attracted much attention to this area: “Therefore my question is, can physics be simulated by a universal computer? I would like to have the elements of this computer locally interconnected, and therefore sort of think about cellular automata as an example (but I don’t want to force it). But I do want something involved with the locality of interaction. I would not like to think of a very enormous computer with arbitrary interconnections throughout the entire thing” [31]. The idea of modeling the natural world in terms of the behavior of sets of rules that can be embodied in simple automata with local interactions is now an important part of science. Sometimes it is called “the new science” to distinguish this approach from classical modeling by equations [115]. For the modeling of transport processes the lattice gas automata [114, 16] were invented and the lattice Boltzmann methods [98] became very popular: they are flexible and efficient. At the same time, the lattice Boltzmann methods are very simple for programming and parallelization. The essence of the lattice Boltzmann methods was formulated by S. Succi in the following maxim: “Nonlinearity is local, non-locality is linear” [100]. We should even strengthen this statement. Non-locality (a) is linear; (b) is exactly and explicitly solvable for all time steps; (c) space discretization is an exact operation. The lattice Boltzmann method is a discrete velocity method. The finite set of velocity vectors $\\{v_{i}\\}$ ($i=1,...m$) is selected, and a fluid is described by associating, with each velocity $v_{i}$, a single-particle distribution function $f_{i}=f_{i}(x,t)$ which is evolved by advection and interaction (collision) on a fixed computational lattice. The values $f_{i}$ are named _populations_. If we look at all lattice Boltzmann models, one finds that there are two steps: free flight for time $\delta t$ and a local collision operation. The free flight transformation for continuous space is $f_{i}(x,t+\delta t)=f_{i}(x-v_{i}\delta t,t).$ After the free flight step the collision step follows: $f_{i}(x)\mapsto F_{i}(\\{f_{j}(x)\\}),$ (1.31) or in the vector form $f(x)\mapsto F(f(x)).$ Here, the _collision operator_ $F$ is the set of functions $F_{i}(\\{f_{j}\\})$ ($i=1,...m$). Each function $F_{i}$ depends on all $f_{j}$ ($j=1,...m$): new values of the populations $f_{i}$ at a point $x$ are known functions of all previous population values at the same point. The lattice Boltzmann chain “free flight $\to$ collision $\to$ free flight $\to$ collision $\dotsb$” can be exactly restricted onto any space lattice which is invariant with respect to space shifts of the vectors $v_{i}\delta t$ ($i=1,\dotsc,m$). Indeed, free flight transforms the population values at sites of the lattice into the population values at sites of the same lattice. The collision operator (1.31) acts pointwise at each lattice site separately. Much effort has been applied to answer the questions: “how does the lattice Boltzmann chain approximate the transport equation for the moments $M$?”, and “how does one construct the lattice Boltzmann model for a given macroscopic transport phenomenon?” (a review is presented in the book [98]). The lattice Boltzmann models should describe the macroscopic dynamic, i.e., the dynamic of macroscopic variables. The macroscopic variables $M_{\ell}(x)$ are some linear functions of the population values at the same point: $M_{\ell}(x)=\sum_{i}m_{\ell i}f_{i}(x)$, or in the vector form, $M(x)=m(f(x))$. The macroscopic variables are invariants of collisions: $\sum_{i}m_{\ell i}f_{i}=\sum_{i}m_{\ell i}F_{i}(\\{f_{j}\\})\qquad\text{(or $m(f)=m(F(f))$).}$ The standard example of the macroscopic variables are hydrodynamic fields (density–velocity–energy density): $\\{n,u,E\\}(x):=\sum_{i}\\{1,v_{i},v_{i}^{2}/2\\}f_{i}(x)$. But this is not an obligatory choice. On the other hand, the athermal lattice Boltzmann models with a shortened list of macroscopic variables $\\{n,nu\\}$ are very popular. The quasiequilibrium is the positive fixed point of the collision operator for given macroscopic variables $M$. We assume that this point exists, is unique and depends smoothly on $M$. For the quasiequilibrium population vector for given $M$ we use the notation $f^{}_{M}$, or simply $f^{}$, if the correspondent value of $M$ is obvious. We use $\Pi^{}$ to denote the equilibration projection operation of a distribution $f$ into the corresponding quasiequilibrium state: $\Pi^{}(f)=f^{}_{m(f)}.$ Usually, collision operators are taken in the form: $F(f):=\Pi^{}(f)+A(\Pi^{}(f)-f)\,,$ (1.32) where $A$ is a linear operator, whose spectrum belongs to the interior of the unit circle. A special case of (1.32) is very popular, the lattice Bhatnagar–Gross–Krook (LBGK) model: $F(f):=f+\omega(\Pi^{}(f)-f)\,.$ (1.33) In this brief introduction of LBM we follow the paper [8]. The simplest LBGK realization of Fick’s law (in 1D) gives the following system. The discrete velocity set includes two elements only, $v$ and $-v$. The time step is $\tau$ the correspondent grid step is $h=v\tau$. The microscopic variables, the populations, are: $f^{-}$ for velocity $-v$ and $f^{+}$ for $v$. The macroscopic variable, the density, is $\rho=f^{-}+f^{+}$. The corresponding equilibrium is $\Pi^{}(f)=f^{}$: $f^{+}=f^{-}=\frac{f^{-}+f^{+}}{2}\,.$ For the non-negative populations, the equilibrium distribution is the maximizer of the entropy $S=-f^{-}\ln f^{-}+f^{+}\ln f^{+}$ under a given value of the macroscopic variable $\rho$. Let us take the LBGK collisions (1.33) with $\omega=1$, i.e. $F(f)^{\pm}=\Pi^{}(f)^{\pm}=\frac{f^{-}+f^{+}}{2}\,.$ (1.34) This particular case of the LBGK collision integral is an example of the so- called Ehrenfests’ coarse-graining. The idea of artificial partial equilibration steps was proposed by T. and P. Ehrenfest for the foundation of statistical physics [23] and further developed to a general formalism of nonequlibrium thermodynamics [48, 18, 49]. A review and comparative analysis of different approaches to coarse-graining was published in [44]. The LBGK chain for the collision integral (1.34) has a very simple form: $\begin{split}&f^{+}(nh,(m+1)\tau)=f^{-}(nh,(m+1)\tau)\\\ &=\frac{f^{+}((n-1)h,m\tau)+f^{-}((n+1)h,m\tau)}{2}\,.\end{split}$ (1.35) Therefore, for the density $\rho$ we get $\rho(nh,(m+1)\tau)=\frac{\rho((n-1)h,m\tau)+\rho((n+1)h,m\tau)}{2}\,.$ (1.36) This appears to be the one of the most common explicit finite difference methods for Fick’s diffusion equation. The diffusion coefficient is $D=h^{2}/(2\tau)=v^{2}\tau/2$ and depends explicitly on the lattice parameters. We can decouple $D$ and the lattice parameters if we use $\omega\in[1,2]$ in the LBGK collision integral (1.33). This lattice-gas scheme does not coincide with any of the finite difference schemes. Nevertheless, it also models diffusion and, to the first order in $\tau$, $D=v^{2}\tau\frac{2-\omega}{2\omega}$ [98, 44]. Now, the area of applications of the cellular and lattice Boltzmann automata is very wide and, in addition to classical fluid dynamics, includes many areas of chemistry [65], models of phase separation [92], dynamics of macromolecules and many other topics. We use cellular automata and lattice models of nonlinear multicomponent diffusion for two purposes: • As a tool for model creation (after that, this model could be translated into other languages, such as partial differential equations (PDE); * • As a tool for numerical simulation without the stage of PDE model. Elliott and Stuart [27] used the cell model of diffusion to study semilinear parabolic equations. They proved the existence of absorbing sets, bounded independently of the mesh size for discrete models. Discrete Lyapunov functions were constructed. We use the special quasichemical approach for the generation of the the cell models [46] that allowed us to construct the Lyapunov functions for semi-discrete systems and to prove stabilization of the solution in space and time under proper conditions. Figure 2: Cell Jump Model Let us consider our space divided into cells, a system represented as a chain of cells of homogeneous composition and elementary transfer processes between them. It is sufficient for our purposes to discuss two sells (Fig 2). Let us numerate these cells by the Roman numbers I and II and mark all the components and quantities related to them by the upper index I or II, correspondingly. The lists of the components for cells are different just by the upper index: $A_{1}^{\mathrm{I}},\ldots A_{n}^{\mathrm{I}}$, $A_{1}^{\mathrm{II}},\ldots A_{n}^{\mathrm{II}}$. The mechanism of diffusion is defined as a list of elementary transitions between cells described by their stoichiometric equation. Since diffusion is a sort of jumping reaction on the border, for these jumps the stoichiometric equation is written as, $\sum_{i}\alpha_{ri}^{\mathrm{I}}A_{i}^{\mathrm{I}}+\sum_{i}\alpha_{ri}^{\mathrm{II}}A_{i}^{\mathrm{II}}\rightarrow\sum_{i}\beta_{ri}^{\mathrm{I}}A_{i}^{\mathrm{I}}+\sum_{i}\beta_{ri}^{\mathrm{II}}A_{i}^{\mathrm{II}}\,,$ (1.37) where $r$ is the number of processes, $\alpha_{ri}^{\mathrm{I,II}},$ and $\beta_{ri}^{\mathrm{I,II}},$ are the stoichiometric coefficients which indicate the number of particles in cells involved in the process. The direction of changes in the elementary event (1.37) is defined by two stoichiometric vectors $\gamma_{ri}^{\mathrm{I}}=\beta_{ri}^{\mathrm{I}}-\alpha_{ri}^{\mathrm{I}}\,;\gamma_{ri}^{\mathrm{II}}=\beta_{ri}^{\mathrm{II}}-\alpha_{ri}^{\mathrm{II}}\,.$ Examples of elementary acts are presented in Fig. 3. (a) Simple diffusion: a particle from the cell I jumps into the cell II and inverse (b) Jumps to free places: a particle from the cell I jumps to the free place in cell II and inverse (c) Jumps with clustering: two particle attract the third one Figure 3: Elementary acts of diffusion, examples. Elementary events (1.37) should not include reactions. Therefore, for each $i$, the amount of $A_{i}$ in the system ($A_{i}^{\mathrm{I}}+A_{i}^{\mathrm{II}}$) should not change. This means exactly that for all $i,r$ $\gamma_{ri}^{\mathrm{I}}=-\gamma_{ri}^{\mathrm{II}}\,.$ Let us use the notation $\gamma_{ri}=\gamma_{ri}^{\mathrm{I}}=-\gamma_{ri}^{\mathrm{II}}\,.$ The composition of each cell is a vector $N^{\mathrm{I,II}}$. The components of this vector, $N_{i}^{\mathrm{I,II}}$ are the amounts of $A_{i}$ in the correspondent cell. We describe the dynamics of the compositions of two cells by the equations: $\frac{dN^{\mathrm{I}}}{dt}=-\frac{dN^{\mathrm{II}}}{dt}=S\sum_{r}\gamma_{r}w_{r},$ (1.38) where $S$ is the area of the boundary between two cells and $w_{r}$ is the rate of the process. For many cells the equations are the same, but with more pairs of cells interacting, and therefore there are more terms. The rates are intensive variables and should be defined as functions of concentrations or chemical potentials. The crucial question is: how to describe function $w_{r}(c^{\mathrm{I}},c^{\mathrm{II}})$, where $c^{\mathrm{I,II}}$ are concentrations components in cells. Figure 4: Cell Jump Model with first surroundings. The real physics of diffusion may be more complicated. For example, the intensity of jumps and the reaction rate $w_{r}(c^{\mathrm{I}},c^{\mathrm{II}})$ may depend not only on $(c^{\mathrm{I}},c^{\mathrm{II}})$ but on the surrounding. For example, direct simulation of the jumps on the surface [9] demonstrates that the influence of the surrounding is crucial for structures and critical effects on the surface. For each process (1.37) there is the space-inverted process that is defined just by changing I to II and vice versa. We mark the quantities for the space- inverted processes by ′. For example, $\gamma^{\prime}=-\gamma$. The detailed space-inversion symmetry requires that the rate functions for them should differ just by the transposition of the vectors of variables, $c^{\mathrm{I}},c^{\mathrm{II}}$: $w^{\prime}_{r}(c^{\mathrm{I}},c^{\mathrm{II}})=w_{r}(c^{\mathrm{II}},c^{\mathrm{I}})\,.$ (1.39) This requirement of detailed space symmetry allows us, in particular, to exclude various types of advection and transport driven by external force. Diffusion, by its definition, is driven by the gradients of the concentrations (or, in the thermodynamical approach, by the gradients of the chemical potentials). This is not the only way to formulate of pure diffusion equations without advection. Another possibility gives us, for example, the diffusion systems with complex balance (Section 2.5.). There are three ways to define the rate functions: from a phenomenological law (like the mass action law), from thermodynamics (like the generalized mass action law) or by direct stochastic simulation of particles jumps in cells (like in the Gillespie approach [40, 41]). In our research, we focus on the first two approaches. Therefore, we consider our lattice model as a semi-discrete model (discrete in space and continuous in time). For this semi-discrete model, the system of kinetic equations (1.38) describes diffusion. The continuous limit of these equations gives us the diffusion PDE. The discrete scheme by itself can serve as a computational model. A couple of simple examples can clarify our approach: * • Simple diffusion (Fig. 3(a)), $A_{i}^{\mathrm{I}}\to A_{i}^{\mathrm{II}}$ and $A_{i}^{\mathrm{II}}\to A_{i}^{\mathrm{I}}$ with the same rate constants. Particles jump into the neighbor cells. For perfect mixtures, $w_{r}=kc^{\mathrm{I}}_{i},\;w_{r}^{\prime}=kc^{\mathrm{II}}_{i}$ and in the continuous limit we get Fick’s law (1.2) as the first Taylor approximation. In this approximation, $D=kl$ where $l$ is the cell size. * • Jumps to free places (Fig. 3(b)), $A_{i}^{\mathrm{I}}+Z^{\mathrm{II}}\to A_{i}^{\mathrm{II}}+Z^{\mathrm{I}}$ and $A_{i}^{\mathrm{II}}+Z^{\mathrm{I}}\to A_{i}^{\mathrm{I}}+Z^{\mathrm{II}}$. According to the mass action law, $w_{r}(c^{\mathrm{I}},c^{\mathrm{II}})=kc_{i}^{\mathrm{I}}z^{\mathrm{II}}$, $w^{\prime}_{r}(c^{\mathrm{I}},c^{\mathrm{II}})=kc_{i}^{\mathrm{II}}z^{\mathrm{I}}$, where $z$ is the concentration of free places. In the first Taylor approximation, $J=-kl(z\nabla c_{i}-c_{i}\nabla z)$ and we get the model (1.17), (1.19). To get the continuous limit, we take $c^{\mathrm{I}}=c(x)$, $c^{\mathrm{II}}=c(x+l)$ and use the Taylor expansion: $c(x+l)=c(x)+l\partial_{x}c+o(l)$. If we the consider a sequence of cell representations of diffusion with various $l$ then, for the invariance of the first order, the scaling rule should be implemented: $D=kl$ does not change with the change of size, therefore, the rate constant $k$ depends on $l$: $k=D/l$. It is not always possible to keep to first order only. If this approach gives a negative diffusion coefficient then for regularity we have to keep the higher derivatives. For example, let us take the diffusion mechanism with attraction: $A_{i}^{\mathrm{I}}+2A_{i}^{\mathrm{II}}\to 3A_{i}^{\mathrm{II}}\,.$ (1.40) The space-inverted process in this case does not coincide with the inverse one. If we change the upper indexes (I to II and II to I) then we obtain $2A_{i}^{\mathrm{I}}+A_{i}^{\mathrm{II}}\to 3A_{i}^{\mathrm{I}}\,.$ (1.41) This mechanism means that 2 particles attract the third one. This mechanism is represented in Fig. 3(c). The reaction rates are: $w_{r}=k_{r}c_{i}^{\mathrm{I}}(c_{i}^{\mathrm{II}})^{2}\,,\;w_{r}^{\prime}=k_{r}(c_{i}^{\mathrm{I}})^{2}c_{i}^{\mathrm{II}}\,.$ The flux of $A_{i}$ from the first cell to the second one is $J=w_{r}-w_{r}^{\prime}=k_{r}c_{i}^{\mathrm{I}}c_{i}^{\mathrm{II}}(c_{i}^{\mathrm{II}}-c_{i}^{\mathrm{I}})\,.$ Therefore, to first order we have $J=klc^{2}\nabla c=\frac{1}{3}kl\nabla c^{3}\,;$ the sign is opposite to standard diffusion. This flux goes in the direction of gradients. The diffusion equation is $\frac{\partial c}{\partial t}=-kl\mathrm{div}(c^{2}\nabla c)=-kl\frac{1}{3}\Delta c^{3}\,.$ (1.42) Of course, if we take the mechanism ($n>1$) $A_{i}^{\mathrm{I}}+nA_{i}^{\mathrm{II}}\to(n+1)A_{i}^{\mathrm{I}}\,,\;A_{i}^{\mathrm{I}}+A_{i}^{\mathrm{II}}\to(n+1)A_{i}^{\mathrm{I}}\,,$ then we get the equation $\frac{\partial c}{\partial t}=-kl(n-1)\mathrm{div}(c^{n}\nabla c)=-kl\frac{n-1}{n+1}\Delta c^{n+1}\,.$ This diffusion process has two properties: first, it goes along gradients and all deviations from the uniform state will increase. Second, this diffusion is slow for small concentrations (the diffusion coefficient goes to 0 when $c$ approaches 0) and accelerates with the concentration growth. The equation $\partial_{t}c=-D\Delta c^{n}$ ($n>1$) admits a family of self- similar solutions with bounded support, which collapse in finite time. These solutions have the form $c(\tau)=\frac{A}{\rho^{q}}\phi\left(\frac{r}{\rho}\right)\,,$ where * • $\tau$ is the time till collapse; * • $q$ is the dimension of space (usually, $q=$1, 2 or 3); * • $\rho$ is the radius of the sphere, outside of which the solution is zero $\rho=B(D\tau)^{\frac{1}{q(n-1)+2}}\,;$ * • $\phi(\vartheta)=(1-\vartheta^{2})^{\frac{1}{n-1}}$ for $\vartheta<1$ and $\phi(\vartheta)=0$ if $\vartheta\geq 1$; * • The constants $A,B$ depend on $q$, $n$ and the total amount $N=\int c(x)\,\mathrm{d}x$. This is the so-called Barenblatt solution [1] for the equation of porous media $\partial_{\tau}c=+D\Delta c^{n}$. Such solutions were used in the analysis of an explosion which starts from a singularity for equations $\partial_{t}c=+D\Delta c^{n}$ (the classical review of self-similar solutions was published by Barenblatt and Zeldovich [2]). The cell model of diffusion with attraction (1.40) for a finite number of cells of a given size $l$ is a rather regular system of nonlinear ODE, but to first order of the Taylor expansion in $l$ the PDE (1.42) produces a singularity in an arbitrarily short time from smooth initial data. The second order Taylor approximation adds nothing because the even terms in $l$ cancel out if we take into account both the left and right neighbors of the cell. The third order Taylor expansion gives a regularized equation: $J=J=w_{r}-w_{r}^{\prime}=klc^{2}\frac{\partial}{\partial x}\left(c+\frac{l^{2}}{3}\frac{\partial^{2}c}{\partial x^{2}}\right)+o(l^{3})\,;$ $\frac{\partial c}{\partial t}=-kl\frac{\partial}{\partial x}c^{2}\frac{\partial}{\partial x}\left(c+\frac{l^{2}}{3}\frac{\partial^{2}c}{\partial x^{2}}\right)\,.$ This is an example of the Cahn–Hilliard type equation for spinodal decomposition with the regularizing term $-\mathrm{div}(c^{2}\mathrm{grad}\Delta c)$. In this equation, the cell size cannot be eliminated by scaling. The length $l$ is the “regularization length”. All inhomogeneities of size smaller than $l$ are smoothed by the biharmonic term. As we can see, the mass action law and the cell representation of the elementary acts of diffusion give the opportunity to model the Cahn–Hilliard type phase separation. Nevertheless, the approach based on the non-perfect thermodynamic potential (1.25) gives a better representation of the basic physics and does not require complicated elementary processes. Just the simplest Fick scheme, $A_{i}^{\mathrm{I}}\to A_{i}^{\mathrm{II}}\,,\;A_{i}^{\mathrm{II}}\to A_{i}^{\mathrm{I}}$ with the non-perfect Ginzburg–Landau free energy gives the Cahn–Hilliard equation (Sec. 3.2.). The diffusion mechanism with attraction (1.40) (Fig. 3(c)) differs from the elementary Fick mechanism (Fig. 3(c)) and from the mechanism of jumps to free places (Fig. 3(b)). The dynamical difference is obvious, the diffusion mechanism with attraction generates instabilities of the homogeneous state, clustering and singularities. On the other hand, Fick’s law and the mechanism of jumps to free places (Fig. 3(b)) allow a global Lyapunov functional and, in the systems without external fluxes, lead to homogeneous equilibrium. These mechanisms have also a very important structural difference. If we look at the direct and the space-inverted processes (Fig. 3) then we find that for the first two mechanisms, the space-inverted processes coincide with the inverse processes, which we get just by inversion of the arrow (or by the exchange $\alpha$ and $\beta$ coefficients in the stoichiometric equations (1.37)). For the elementary processes with attractions (Fig. 3(c)) the inverse processes are processes with repulsion: $3A_{i}^{\mathrm{II}}\to A_{i}^{\mathrm{I}}+2A_{i}^{\mathrm{II}}\,,\;3A_{i}^{\mathrm{I}}\to 2A_{i}^{\mathrm{I}}+A_{i}^{\mathrm{II}}\,.$ (1.43) The diffusion processes for which space-inverted elementary processes coincide with the inverse processes, have a fundamental property: The entropy production is positive for the corresponding mass action law diffusion equations. Let us consider a complex diffusion process in a bounded domain with smooth boundary and without external fluxes. We prove the following theorem in Section 2.2.3. Theorem 2. Let a complex diffusion process consist of elementary processes, which satisfy the following property: the space-inverted elementary process coincides with the inverse process. Then, for the mass action law equation of diffusion (2.25), the principle of detailed balance is valid, the global convex Lyapunov functional exists and the uniform distribution is asymptotically stable. This global Lyapunov functional may be selected in the form of the (minus) classical entropy, the sum of terms $c\ln c$ for all cells and components, or, for the continuous limit, $\sum_{i}\int c_{i}\ln c_{i}\,\mathrm{d}x\,.$ A particular case of the dissipation inequality for such processes is inequality (1.22) for the diffusion equations (1.20) that describe diffusion by exchange of positions. It is valid because the exchange mechanism satisfies this fundamental property: the space-inverted elementary processes coincide with the inverse processes. #### 1.3.3. Thermodynamics and Intermediate Complexes Thermodynamics is not always a good leader, but it is always a good judge. We cannot create nonlinear equations directly from thermodynamic principles, but we must always check whether our equations satisfy thermodynamics. They should satisfy the thermodynamic restrictions if we do not want to produce a perpetuum mobile in our theory. We also include some other fundamental restrictions like micro-reversibility in the thermodynamic requirements. It is not always simple to coordinate the lattice models with thermodynamics, nevertheless it is possible [64, 99]. There are two main approaches for the introduction of thermodynamics into kinetic models. First, we can start from general kinetic equations based on the representation of a complex process as an ensemble of elementary processes with a given simple kinetic law of elementary processes (for example, the mass action law). After that, we will find that the rate constants of the elementary processes are not independent. They must be coordinated to meet the thermodynamic requirements. Therefore, not all the possible kinetic systems are allowed thermodynamically. Another approach starts from the thermodynamic description of the system. We should find thermodynamic potentials which describe the system under given conditions. We have to know entropy, free energy (Helmholtz energy), or free enthalpy (Gibbs energy) for the proper set of independent variables [14]. After that, we define the rate of elementary process through the thermodynamic functions but with some arbitrariness: some constants remain free of thermodynamic restrictions. These constants are independent for different elementary processes. Which way is better? It is not a proper question: both are good for their purposes. The first approach (we start from kinetics and then add thermodynamics) is very flexible. In particular, it can be used when thermodynamic restrictions are not needed. For example, when we consider subsystems of open systems like the system of surface components in heterogeneous catalysis, then the constants of elementary processes include additional dependencies on some additional concentrations and are not the “proper” rate constants. Therefore, they do not satisfy the thermodynamic restrictions, and a subsystem may demonstrate non-thermodynamic behavior like non-decaying oscillations or bifurcations. The second approach is unavoidable for non-perfect systems. The kinetic law of elementary processes depends on the thermodynamic potential. For all perfect systems it is the same mass action law, but any deviation from the perfect thermodynamic function requires its own deviation of the kinetic law from the mass action law [68]. This deviation may be reformulated as the generalized mass action law with activities instead of concentrations but the activities are defined through thermodynamic potentials. In our work we follow both approaches: First, we formulate the mass action law for diffusion and study this with and without the thermodynamic restrictions. Secondly, we introduce the thermodynamic formalism for diffusion in non- perfect systems. The ideas for both approaches for diffusion were formulated in the early 1980s [53, 46]. The detailed analysis of the thermodynamic restrictions on chemical kinetics was performed by Gorban in 1982 [42]. Our work was influenced by the works of N.G. Van Kampen [105], M. Feinberg [29, 30] and Horn and Jackson [62]. In 1973, N.G. Van Kampen proposed a general formulation for the rates of irreversible processes as a combination of “unilateral transfer flows”. Each unilateral flow transfers energy and particles in one direction. Van Kampen decomposed the total in partial systems, each of which is in equilibrium and therefore possesses a well- defined temperature, entropy, and other thermodynamic quantities. Although the total system Y is not in equilibrium, it is still possible to attribute an entropy to it. Then Van Kampen studied the unilateral fluxes between subsystems. We decompose the Van Kampen unilateral processes further and represent them as a collection of essentially one-dimensional elementary processes with the simple kinetic mechanism, the mass action law or the generalized mass action law. We start from a similar representation of the total system and supplement it with the system of stoichiometric equations of elementary unilateral processes. To find the rate of the elementary processes we use an idea of intermediate complex (compound). This approach is borrowed from the theory of absolute reaction rates but we do not use the special idealization of the reaction pass and postulate the more general microscopic Markov kinetics instead. If the concentrations of compounds are small and the equilibrium between intermediates and other components is fast (both assumptions are important) then we approach the generalized mass action law, which is very similar to the Marselin–de Donder kinetics and the generalized mass action law studied by Feinberg, Horm and Jackson and other authors [29, 30, 62, 10]. This formalism is very convenient for implementation of the microreversibility consequences in the form of detailed balance conditions [75]. In addition, if there is no microreversibility then the thermodynamic behavior is also guaranteed by the special more general relations between kinetic constants, which follow from the Markov kinetics of intermediate complexes. First, the idea of such relations was proposed by Boltzmann as an answer to the Lorentz objections against Boltzmann’s proof of the $H$-theorem. Lorentz stated nonexistence of inverse collisions for polyatomic molecules. Boltzmann did not object to this argument but proposed the “cyclic balance” condition, that means balancing in cycles of transitions between states $S_{1}\to S_{2}\to\ldots\to S_{n}\to S_{1}$. Almost 100 years later, Cercignani and Lampis [15] demonstrated that the Lorenz arguments are wrong and this Boltzmann new relations are not needed for the polyatomic molecules under the microreversibility conditions. The detailed balance conditions should hold. Nevertheless, this Boltzmann’s idea is very seminal. It was studied further by Heitler [60] and Coester [19] and the results are sometimes cited as the “Heitler-Coestler theorem of semi-detailed balance”. In 1952 [97] proved these conditions for the Boltzmann equation. For the micro-description he used the $S$-matrix representation, which is in this case equivalent for the Markov microkinetics (see also [112]). Later, this sort of relation was rediscovered for chemical kinetics [30, 62]. The general proof for nonlinear nonequlibrium processes was presented recently [54]. In our analysis of these Boltzmann–…–Stueckelberg relations we follow the later. We extend the usual stoichiometric equations by additional reactions: an input linear combination of reagents forms a corresponding compound; this compound transforms into another compound that disintegrates into the corresponding output linear combination of reagents: $\sum_{i}\alpha_{\rho i}A_{i}\rightleftharpoons B_{\rho}^{-}\to B_{\rho}^{+}\rightleftharpoons\sum_{i}\beta_{\rho i}A_{i}\,.$ (1.44) Here $\rho$ is the elementary reaction number. Figure 5: A $2n$-tail scheme of an extended elementary process (1.44). It is useful to visualize the reaction scheme. In Fig. 5 we represent the $2n$-tail scheme of an elementary reaction sequence (1.44). This scheme was proposed in [54]. We assume that the amount of each compound $B_{\rho}$ is small enough to apply the perfect entropy formula, and that the equilibrium between each compound and the corresponding linear combinations of reagents is fast enough to apply the quasiequilibrium approximation [54] (the detailed analysis of this approximation was given in [52, 50, 51]). The main difference from the Eyring approach [66] is in the model of the hidden reaction of the “activated complexes”: * • Eyring used for each reaction one complex with a continuum of energetic states along the “reaction path”, whereas we use two compounds (or two states); * • Eyring modeled the reaction of the intermediate complex as a classical motion along the additional coordinate and even added this degree of freedom with classical kinetic energy of this motion to the free energy calculation, whereas we follow the Stueckelberg approach and model this reaction as a first order Markov kinetics, a Markov process with two states. The difference in the macroscopic consequences of these approaches seems to be not very large because the result of the Eyring approach is one relaxation time approximation for each reaction. From the dynamical point of view, this result coincides with the two-state Markov model. The main differences may arise in the hints which these approaches give to the microscopic calculation of the macroscopic quantities. In our work, we concentrate on the macroscopic dynamics. ## 2\. Mass Action Law for Diffusion ### 2.1. Mass Action Law #### 2.1.1. Mass Action Law Kinetic Equations This is an auxiliary subsection where we collect main definitions and results about the Mass Action Law (MAL). To construct a system of kinetic equations by MAL, one needs the following inputs: 1. 1. A list of components; 2. 2. A list of elementary reactions represented by their stoichiometric equations; 3. 3. A set of reaction rate constants. The list of components is just a set of symbols (the component names). We usually assume that this set is finite, $A_{1},A_{2},\ldots,A_{n}$. Elementary reactions are given by their stoichiometric equations, $\sum_{i}\alpha_{ri}A_{i}\to\sum_{i}\beta_{ri}A_{i}\,,$ (2.1) where $r$ is a reaction number, $\alpha_{ri}$ and $\beta_{ri}$ are nonnegative numbers, the stoichiometric coefficients. By default, they are assumed to be integer but, sometimes, there occurs a need in nonnegative real coefficients. For each elementary reaction (2.1), a stoichiometric vector is defined, $\gamma_{r}:\;\gamma_{ri}=\beta_{ri}-\alpha_{ri}\,.$ This is a “bookkeeping” vector, whose components are “gain minus loss” (or “income minus outcome”). We will also use the loss and gain vectors of elementary reactions: $\alpha_{r}$ (loss) with coordinates $\alpha_{ri}$ and $\beta_{r}$ (gain) with coordinates $\beta_{ri}$. Of course, $\gamma_{r}=\beta_{r}-\alpha_{r}\,.$ The stoichiometric matrix $\Gamma$ is the matrix with columns $\gamma_{i}$: $\Gamma_{ij}=\gamma_{ji}$, the first index in $\Gamma_{ij}$ corresponds to component and the second index corresponds to reaction. Reaction rate constants $k_{r}$ are non-negative numbers. They should be defined for all elementary reactions. For each component $A_{i}$, a real variable, concentration $c_{i}$ is defined. Vector of concentrations $c$ has coordinates $c_{i}$. The reaction rate for the elementary reaction (2.1) is the following function of $c$ $w_{r}=k_{r}\prod_{i=1}^{n}c_{i}^{\alpha_{ri}}\,.$ (2.2) The MAL kinetic equations are $\frac{\mathrm{d}c}{\mathrm{d}t}=\sum_{r}\gamma_{r}w_{r}\,.$ (2.3) From the physical point of view, these equations describe isochoric isothermal processes for perfect systems. For non-isochoric or non-isothermal processes it is necessary to introduce the volume (together with pressure) and the enthalpy (together with temperature) explicitly and describe their dynamics. For a given reaction mechanism, a linear stoichiometric conservation law is a linear functional $b(c)=\sum_{i}b_{i}c_{i}$ that annihilates all stoichiometric vectors: $b(\gamma_{r})=0\mbox{ for all }r\,.$ The stoichiometric conservation law is strictly positive, if all $b_{i}>0$. The assumption about existence of a positive stoichiometric conservation law plays an important role in the MAL kinetics. #### 2.1.2. Existence and Uniqueness of Solutions Let in the reaction mechanism all nonzero coordinates of the loss vectors be not less than 1: $\alpha_{ri}\geq 1\,\mbox{ or }\alpha_{ri}=0\,.$ This assumption is valid, for example, if all the stoichiometric coefficients are nonnegative integers. Let us assume also that there exists a strictly positive stoichiometric conservation law $b$. Then the following existence and uniqueness theorem for the MAL equation holds. ###### Theorem 1. For any nonnegative initial data $c(0)$ ($c_{i}(0)\geq 0$) there exists a unique solution of (2.3) $c(t)$ for all $t>0$. This solution is nonnegative ($c_{i}(t)\geq 0$) and satisfies the conservation law: $b(c(t))=b(c(0))$. This is a well known result (see, for example, [109]). The proof is quite simple. First, of all, let us consider a bounded neighborhood $U$ of the simplex $\Sigma_{0}$: $c_{i}\geq 0$, $b(c)=b(c(0))$. The right hand site of the MAL kinetic equations (2.3) has continuous first derivatives and these derivatives are bounded in $U$. Therefore, according to a standard existence and uniqueness theorem its solution exists on some time interval $t\in[0,T]$ and this $T$ is the same for a compact set of initial data $c(0)\in\Sigma_{0}\Subset U$. Secondly, let us mention that if $\gamma_{ri}<0$ then $\alpha_{ri}\geq 1$ and the reaction rate $w_{r}$ (2.2) includes the factor $c_{i}^{\alpha_{ri}}$. Therefore, $\sum_{\gamma_{ri}<0}\gamma_{ri}w_{r}=c_{i}g(c)\,,$ where $g(c)$ is continuous function. If $c_{i}\to 0$ then $\sum_{\gamma_{ri}<0}\gamma_{ri}w_{r}\to 0$. If $c_{i}=0$ then $\dot{c}_{i}=\sum_{r}\gamma_{ri}w_{r}\geq 0$. Therefore, the simplex $\Sigma_{0}$ is positively invariant with respect to equations (2.3): the existent solutions $c(t)$ do not leave this simplex for $t>0$ if $c(0)\in\Sigma_{0}$. Finally, this implies global existence of solutions in $\Sigma_{0}$. Both conditions of existence of a strictly positive stoichiometric conservation law $b$ and of coordinates of the loss vectors, $\alpha_{i}=0$ or $\alpha_{i}\geq 1$ are significant. Just for example, we can consider mechanisms that violate these conditions: $2A\to 3A$ and $\frac{1}{2}A\to A$. For the first mechanism, $\dot{c}=kc^{2}$ and there is no global existence, for the second system, $\dot{c}=k\sqrt{c}$ and there is no uniqueness of solution. #### 2.1.3. Detailed Balance The expected behavior of a system of physical or chemical kinetics is simple in the absence of external fluxes: everything goes to equilibrium. A Lyapunov function for this relaxation is the corresponding thermodynamic potential. The MAL kinetics (2.3) do not assume any thermodynamic properties “from scratch”. Moreover, this class of kinetic equations is so rich that it is dense in the class of all smooth semi-dynamical systems in $\Sigma_{0}$ for a given conservation law $b$ [46]. Additional assumptions are needed to guarantee the thermodynamic behavior. The most celebrated sufficient condition for the thermodynamic behavior of the MAL kinetics is the principle of detailed balance. This principle, as a realization of microreversibility was known for the Boltzmann equation [6] (since his proof of the $H$-theorem in 1872) long before Onsager’s reciprocal relations [83, 84]. A. Einstein used this principle for the linear kinetics of emission and absorption of radiation [25]. In 1901, R. Wegscheider published analysis of detailed balance for chemical kinetics [113]. To formulate the principle of detailed balance, it is convenient to join pairs of direct and inverse elementary reactions in (2.1) and write $\sum_{i}\alpha_{ri}A_{i}\rightleftharpoons\sum_{i}\beta_{ri}A_{i}\,.$ (2.4) If the inverse reaction does not exist in the original mechanism, we formally add it but assume that its rate constant is zero. We mark quantities for the direct and inverse reactions by the upper indexes + and - and write the MAL reaction rate: $\begin{split}&w_{r}=w_{r}^{+}-w_{r}^{-}\,,\\\ &w_{r}^{+}=k_{r}^{+}\prod_{i=1}^{n}c_{i}^{\alpha_{ri}}\,,\;w_{r}^{-}=k_{r}^{-}\prod_{i=1}^{n}c_{i}^{\beta_{ri}}\,.\end{split}$ (2.5) For the MAL kinetics the principle of detailed balance is: there exists a strictly positive point of detailed balance that is such vector of concentration $c^{}$ that $c^{}_{i}>0$ and $w_{r}^{+}(c^{})=w_{r}^{-}(c^{})\,(=w_{r}^{}>0)\,.$ (2.6) This means that at least at one positive point the direct elementary processes are equilibrated by the inverse elementary processes. Existence of one such point implies that all equilibria are also points of detailed balance (2.6) and, moreover, there exists a global Lyapunov function that has the form of relative entropy. This is, precisely, the analogue of the Boltzmann $H$-theorem for the MAL kinetics. For the formulation, use and proof of this theorem, it is convenient to rewrite the formulas for the direct and inverse reaction rates (2.5) using $c^{}$, $w_{r}^{}$ and the detailed balance relations (2.6): $\begin{split}&w_{r}=w_{r}^{+}-w_{r}^{-}\,,\\\ &w_{r}^{+}=w_{r}^{}\prod_{i=1}^{n}\left(\frac{c_{i}}{c_{i}^{}}\right)^{\alpha_{ri}}\,,\;w_{r}^{-}=w_{r}^{}\prod_{i=1}^{n}\left(\frac{c_{i}}{c_{i}^{}}\right)^{\beta_{ri}}\,.\end{split}$ (2.7) The Lyapunov function is: $G=\sum_{i}c_{i}\left(\ln\left(\frac{c_{i}}{c_{i}^{}}\right)-1\right)+\sum_{i}c_{i}^{}\,.$ (2.8) Here, the last constant term is added to satisfy $G(c^{})=0$. The partial derivatives of $G$ (the analogs of chemical potentials) are $\frac{\partial G}{\partial c_{i}}=\ln\left(\frac{c_{i}}{c_{i}^{}}\right)\,.$ (2.9) Therefore, we have one more form for the MAL kinetic law with detailed balance (2.7): $\begin{split}&w_{r}=w_{r}^{+}-w_{r}^{-}\,,\\\ &w_{r}^{+}=w_{r}^{}\exp\left(\sum_{i}\alpha_{ri}\frac{\partial G}{\partial c_{i}}\right)=w_{r}^{}\exp(\alpha_{r},\nabla_{c}G)\,,\\\ &w_{r}^{-}=w_{r}^{}\exp\left(\sum_{i}\beta_{ri}\frac{\partial G}{\partial c_{i}}\right)=w_{r}^{}\exp(\beta_{r},\nabla_{c}G)\,,\end{split}$ (2.10) It is worth mentioning that $\frac{w_{r}^{+}}{w_{r}^{-}}=\exp[-(\gamma_{r},\nabla_{c}G)]\,.$ For the time derivative of $G$ due to the MAL kinetics (2.3) with the detailed balance, simple algebra gives the dissipation inequality: $\frac{\mathrm{d}G}{\mathrm{d}t}=\sum_{r}w_{r}(\gamma_{r},\nabla_{c}G)=-\sum_{r}(w_{r}^{+}-w_{r}^{-})\ln\left(\frac{w_{r}^{+}}{w_{r}^{-}}\right)\leq 0\,.$ (2.11) The last inequality holds because $\ln x$ is a strictly monotone function and $\ln x-\ln y$ has the same sign as $x-y$ has. Obviously, $\dot{G}\|_{c}=0$ if and only if $c$ is a point of detailed balance. This equilibrium point may be different from the point $c^{}$, which was used for the definition. All the positive points of detailed balance for the MAL system (2.7) form a smooth manifold with dimension $n-\mathrm{rank}\\{\gamma_{1},\gamma_{2},\ldots\\}\,,$ where $n$ is the number of components and $\mathrm{rank}\\{\gamma_{1},\gamma_{2},\ldots\\}$ is the rank of the system of the stoichiometric vectors for the given reaction mechanism. If we fix values of all stoichiometric linear conservation laws then the strictly positive point of detailed balance is unique for the given values. Indeed, the dissipation inequality is valid for every pair of mutually inverse reactions and all the terms $w_{r}(\gamma_{r},\nabla_{c}G)$ in $\dot{G}$ (2.11) are non-positive. Therefore, at any strictly positive equilibrium point $c^{\mathrm{eq}}$, $(\gamma_{r},\nabla_{c}G)=0$ for all $r$. This means that $c^{\mathrm{eq}}$ is a critical point of $G$ in $c^{\mathrm{eq}}+\mathrm{span}\\{\gamma_{1},\gamma_{2},\ldots\\}$. The function $G$ is strictly convex at any positive point: its Hessian is positive definite, $\frac{\partial^{2}G}{\partial c_{i}\partial c_{j}}=\frac{1}{c_{i}}\delta_{ij}\,,$ where $\delta_{ij}$ is the Kronecker delta. Therefore, there may exist only one positive critical point of $G$ on a linear manifold. Due to the logarithmic singularity of the gradients at the boundary of $\mathbb{R}_{+}$ (where some of $c_{i}=0$), $G$ achieves its global minimum in $(c^{\mathrm{eq}}+\mathrm{span}\\{\gamma_{1},\gamma_{2},\ldots\\})\bigcap\mathbb{R}_{+}^{n}$ at a positive point. This point is a positive point of detailed balance. For any positive vector, $c^{0}$, the polyhedron $\mathcal{V}=(c^{0}+\mathrm{span}\\{\gamma_{1},\gamma_{2},\ldots\\})\bigcap\mathbb{R}_{+}^{n}\,,$ is positively invariant with respect to (2.3). For systems with detailed balance it includes one and only one positive point of detailed balance. This was first demonstrated by Zeldovich in 1938 (reprinted in 1996 [118]). In our analysis we mainly follow [109]. #### 2.1.4. Complex Balance In this subsection we consider the direct and inverse reactions separately as we did it before in (2.1), (2.2). Detailed balance is s sufficient but not necessary condition of the thermodynamic behavior. The simple example of thermodynamic behavior gives any monomolecular (linear) reaction mechanism, which consists of reactions $A_{i}\to A_{j}$. Let us use notation $k_{ji}$ for this reaction rate constant. The MAL equations for a monomolecular reaction mechanism are $\frac{\mathrm{d}c_{i}}{\mathrm{d}t}=\sum_{j,\,j\neq i}(k_{ij}c_{j}-k_{ji}c_{i})\,.$ (2.12) Let $c^{0}$ be a strictly positive steady state for these equations (not necessarily a point of detailed balance): $\sum_{j,\,j\neq i}k_{ij}c_{j}^{0}=c_{i}^{0}\sum_{j,\,j\neq i}k_{ji}\,.$ With this $c^{0}$ we can rewrite the second term in (2.12): $\sum_{j,\,j\neq i}k_{ji}=\sum_{j,\,j\neq i}k_{ij}\frac{c_{j}^{0}}{c_{i}^{0}}\,;$ $\sum_{j,\,j\neq i}k_{ji}c_{i}=\sum_{j,\,j\neq i}k_{ij}c_{j}^{0}\frac{c_{i}}{c_{i}^{0}}\,.$ Therefore, the kinetic equations (2.12) have the equivalent form for given $c^{0}$: $\frac{\mathrm{d}c_{i}}{\mathrm{d}t}=\sum_{j,\,j\neq i}k_{ij}c_{j}^{0}\left(\frac{c_{j}}{c^{0}_{j}}-\frac{c_{i}}{c^{0}_{i}}\right)\,.$ (2.13) Then we can define $G=\sum_{i}c_{i}\left(\ln\left(\frac{c_{i}}{c^{0}_{i}}\right)-1\right)\,.$ (2.14) After simple transformations, we find that due to (2.13) $\begin{split}\frac{\mathrm{d}G}{\mathrm{d}t}=\sum_{ij\,i\neq j}k_{ij}c_{j}^{0}&\left[\frac{c_{i}}{c_{i}^{0}}\left(\ln\left(\frac{c_{i}}{c_{i}^{0}}\right)-1\right)-\frac{c_{j}}{c_{j}^{0}}\left(\ln\left(\frac{c_{j}}{c_{j}^{0}}\right)-1\right)\right.\\\ &\left.+\ln\left(\frac{c_{i}}{c_{i}^{0}}\right)\left(\frac{c_{j}}{c_{j}^{0}}-\frac{c_{i}}{c_{i}^{0}}\right)\right]\leq 0\,.\end{split}$ (2.15) To prove this formula, it is worth mentioning that for any $n$ numbers $f_{i}$, $\sum_{ij\,i\neq j}k_{ij}c_{j}^{0}(f_{i}-f_{j})=0\,.$ This gives us the first two terms in the square brackets with $f_{i}=\frac{c_{i}}{c_{i}^{0}}\left(\ln\left(\frac{c_{i}}{c_{i}^{0}}\right)-1\right)\,.$ The last term, $\ln\left(\frac{c_{i}}{c_{i}^{0}}\right)\left(\frac{c_{j}}{c_{j}^{0}}-\frac{c_{i}}{c_{i}^{0}}\right)\,,$ appears in the straightforward computation of the time derivative of $G$ due to kinetic equations (2.13). The expressions in square brackets in (2.15) have the form $f(a)-f(b)+f^{\prime}(a)(b-a)$ for the convex function $f(x)=x(\ln x-1)$. This expression is always non- positive because of Jensen’s inequality. Linear MAL kinetics can obviously violate the principle of detailed balance. For example, an irreversible cycle $A_{1}\to A_{2}\to\ldots\to A_{n}\to A_{1}$ has always a positive steady state but never has a positive point of detailed balance if $n>2$. There is a nice generalization of the dissipation inequality (2.15) for the nonlinear MAL equations under some algebraic conditions on the kinetic constants. These conditions are strictly weaker than the principle of detailed balance. They were discovered for the Boltzmann equation by Stueckelberg [97] in 1952 and called later the “complex balancing condition” for the general MAL [62, 30]. To formulate this condition for the MAL kinetics, let us start from the function $G$ (2.14) and look for conditions that guarantee the inequality $\dot{G}\leq 0$. For a given $c^{0}$, we can rewrite the MAL reaction rate in the form $w_{r}=\varphi_{r}\exp(\alpha_{r},\nabla_{c}G)=\varphi_{r}\prod_{i}\left(\frac{c_{i}}{c_{i}^{0}}\right)^{\alpha_{ri}}\,,$ (2.16) where $\varphi_{r}=k_{r}\prod_{i}(c_{i}^{0})^{\alpha_{ri}}=w_{r}(c^{0})$. It is convenient to express $\dot{G}$ using an auxiliary function $\theta$ of an auxiliary variable $\lambda$: for any concentration vector, $\theta(\lambda)=\sum_{r}\varphi_{r}\exp[\lambda(\alpha_{r},\nabla_{c}G)+(1-\lambda)(\beta_{r},\nabla_{c}G)]\,.$ (2.17) This function is convenient because $\frac{\mathrm{d}\theta(\lambda)}{\mathrm{d}\lambda}=-\sum_{r}\varphi_{r}(\mu,(\beta_{r}-\alpha_{r}))\exp[\lambda(\alpha_{r},\nabla_{c}G)+(1-\lambda)(\beta_{r},\nabla_{c}G)]\,.$ (2.18) In this notation, $\dot{G}=-\theta^{\prime}(1)\,.$ The function $\theta(\lambda)$ is a sum of exponents. It is convex ($\theta^{\prime\prime}(\lambda)\geq 0$). Therefore, if $\theta(0)=\theta(1)$ then $\theta^{\prime}(1)\geq 0$. This condition, $\theta(0)=\theta(1)$ (for all positive $c$) is called the complex balancing condition and it is sufficient for the dissipation inequality: $\dot{G}=-\theta^{\prime}(1)\leq 0\,.$ The principle of detailed balance for the MAL equations has a form of existence of a special equilibrium point, a point of detailed balance. This existence implies important dynamical properties in the whole of $\mathbb{R}_{+}^{n}$ because of the very “rigid” monomial structure of MAL. The complex balancing condition also can be formulated as existence of a special “point of complex balance”. Let us reformulate it in this way. Some vectors $\alpha_{r}$, $\beta_{r}$ for a given reaction mechanism may coincide. Let us denote by $\\{y_{1},\ldots,y_{q}\\}$ the set of all different vectors $\alpha_{r}$, $\beta_{r}$. For each $y_{i}$ we define $R_{i}^{+}=\\{r\,\|\,\alpha_{r}=y_{i}\\}$, $R_{i}^{-}=\\{r\,\|\,\beta_{r}=y_{i}\\}$. In this notation, $\begin{split}\theta(1)=\sum_{i}\left(\sum_{r\in R_{i}^{+}}w_{r}(c^{0})\right)\exp(y_{i},\nabla_{c}G)\,,\\\ \theta(0)=\sum_{i}\left(\sum_{r\in R_{i}^{-}}w_{r}(c^{0})\right)\exp(y_{i},\nabla_{c}G)\,.\end{split}$ (2.19) For any finite set of (different) vectors $\\{y_{1},\ldots,y_{q}\\}$ the correspondent functions $\exp(y_{i},\nabla_{c}G)$ of $c\in\mathbb{R}_{+}^{n}$ are linearly independent because the Hessian of $G$ is strictly positive definite. Therefore, the condition $\theta(0)=\theta(1)$ (for all positive $c$) is equivalent to $\sum_{r\in R_{i}^{+}}w_{r}(c^{0})=\sum_{r\in R_{i}^{-}}w_{r}(c^{0})\,.$ (2.20) The point $c^{0}$ that satisfies (2.20) is called the point of complex balance and the complex balancing condition means that there exists such a strictly positive point of complex balance. In this case, the dissipation inequality, $\dot{G}\leq 0$, with $G$ defined by (2.14) holds for all positive points. Of course, a point of detailed balance is a point of complex balance as well. The reverse statement is not valid: for example, all the positive steady states of linear MAL kinetics (2.12) are the points of complex balance but they are not necessarily the points of detailed balance. The microscopic background for detailed balance is microreversibility, i.e. invariance of the microscopic classical or quantum equations with respect to the time inversion. The microscopic backgrounds for complex balance were formulated by Stueckelberg as unitarity of the $S$-matrix [97]. It is necessary to add that the validity of the scattering (or Markov) model of elementary reactions is also needed: see [54] and discussion in Section 3. ### 2.2. Mass Action Cell-Jump Formalism #### 2.2.1. Stoichiometry of Diffusion Jumps We represent the physical space as a network of compartments. Each compartment is modeled as a cubic cell with an edge size $l$. The stoichiometric equations of diffusion describe interaction of two neighboring cells. To distinguish the quantities related to these two cells we use the upper indexes I and II (Fig. 2). The general stoichiometric equation for an elementary event of diffusion is (1.37) $\sum_{i}\alpha_{ri}^{\mathrm{I}}A_{i}^{\mathrm{I}}+\sum_{i}\alpha_{ri}^{\mathrm{II}}A_{i}^{\mathrm{II}}\rightarrow\sum_{i}\beta_{ri}^{\mathrm{I}}A_{i}^{\mathrm{I}}+\sum_{i}\beta_{ri}^{\mathrm{II}}A_{i}^{\mathrm{II}}\,.$ (2.21) Coefficients $\alpha_{ri}^{\mathrm{I,II}}$, $\beta_{ri}^{\mathrm{I,II}}$ are nonnegative. Usually, we assume that they are integers but in some situations real numbers are needed. Elementary events (2.21) describe diffusion and do not include the transformation of components (reactions). Therefore, the total amounts of each component $A_{i}$ coincide in the left and the right hand sides of (2.21): $\alpha_{ri}^{\mathrm{I}}+\alpha_{ri}^{\mathrm{I}}=\beta_{ri}^{\mathrm{I}}+\beta_{ri}^{\mathrm{II}}\,.$ (2.22) Each elementary process (2.21) has two loss vectors, $\alpha_{r}^{\mathrm{I,II}}$ with coordinates $\alpha_{ri}^{\mathrm{I,II}}$ and two output vectors, $\beta_{r}^{\mathrm{I,II}}$ with coordinates $\beta_{ri}^{\mathrm{I,II}}$. Because of the conservation of particles of all types (2.22), the stoichiometric vectors of processes for the cells differ just by the sign of coordinates: $\gamma_{r}=\gamma_{r}^{\mathrm{I}}=\gamma_{r}^{\mathrm{II}}=\beta_{r}^{\mathrm{I}}-\alpha_{r}^{\mathrm{I}}\,.$ We define here a mechanism of diffusion as a system of stoichiometric equations for elementary events. The simple and basic examples are: * • Fick’s diffusion, $A_{i}^{\mathrm{I}}\to A_{i}^{\mathrm{II}}$, $A_{i}^{\mathrm{II}}\to A_{i}^{\mathrm{I}}$; * • The exchange of particles, $A_{i}^{\mathrm{I}}+A_{j}^{\mathrm{II}}\to A_{i}^{\mathrm{II}}+A_{j}^{\mathrm{I}}$; * • Clustering (diffusion with attraction), $A_{i}^{\mathrm{I}}+sA_{i}^{\mathrm{II}}\to(s+1)A_{i}^{\mathrm{II}}$, $sA_{i}^{\mathrm{I}}+A_{i}^{\mathrm{II}}\to(s+1)A_{i}^{\mathrm{I}}$, $s>1$; * • Diffusion with repulsion, $(s+1)A_{i}^{\mathrm{I}}\to sA_{i}^{\mathrm{I}}+A_{i}^{\mathrm{II}}$, $(s+1)A_{i}^{\mathrm{I}}\to A_{i}^{\mathrm{I}}+sA_{i}^{\mathrm{II}}$, ($s>0$). Formally, diffusion with repulsion is the time-inverted process of diffusion with attraction (the porous medium model) but for $s=1$ diffusion with attraction has no sense (the exactly uniform state cannot produce the nonuniform distribution). Therefore, the restrictions on $s$ are different. #### 2.2.2. MAL Equations for Diffusion Let us consider the system of stoichiometric equations (2.21) as a reaction mechanism for MAL (2.1). If we apply MAL then the rate of the elementary diffusion process is $w_{r}(c^{\mathrm{I}},c^{\mathrm{II}})=k_{r}\prod_{i}(c_{i}^{\mathrm{I}})^{\alpha^{\mathrm{I}}_{ri}}\prod_{i}(c_{i}^{\mathrm{II}})^{\alpha^{\mathrm{II}}_{ri}}\,.$ (2.23) For example, for Fick’s diffusion, we have two elementary processes, $A_{i}^{\mathrm{I}}\to A_{i}^{\mathrm{II}}$ and $A_{i}^{\mathrm{II}}\to A_{i}^{\mathrm{I}}$. The corresponding reaction rates are $k_{1}c_{i}^{\mathrm{I}}$ and $k_{2}c_{i}^{\mathrm{II}}$. The composition of each cell is vector $N^{\mathrm{I,II}}$. Components of this vector, $N_{i}^{\mathrm{I,II}}=V^{\mathrm{I,II}}c_{i}^{\mathrm{I,II}}$ are amounts of $A_{i}$ in the corresponding cell and $V^{\mathrm{I,II}}$ are volumes of cells. We describe the dynamics of the compositions of two cells by the equations: $\frac{dN^{\mathrm{I}}}{dt}=-\frac{dN^{\mathrm{II}}}{dt}=S^{\mathrm{I,II}}\sum_{r}\gamma_{r}w_{r}(c^{\mathrm{I}},c^{\mathrm{II}})\,,$ (2.24) where $S^{\mathrm{I,II}}$ is the area of the boundary between cells I and II. If there are many cells then $\frac{dN^{\mathrm{I}}}{dt}=\sum_{\mathrm{J}}S^{\mathrm{I,J}}\sum_{r}\gamma_{r}w_{r}(c^{\mathrm{I}},c^{\mathrm{J}})\,,$ (2.25) with summation through all interacting pairs (I,J). For example, for Fick’s diffusion, we have two elementary processes, $A_{i}^{\mathrm{I}}\to A_{i}^{\mathrm{II}}$ and $A_{i}^{\mathrm{II}}\to A_{i}^{\mathrm{I}}$. The corresponding reaction rates are $k_{1}c_{i}^{\mathrm{I}}$ and $k_{2}c_{i}^{\mathrm{II}}$. Equations (2.24) give $\frac{dN_{i}^{\mathrm{I}}}{dt}=-S^{\mathrm{I,II}}k_{1}c_{i}^{\mathrm{I}}+S^{\mathrm{I,II}}k_{2}c_{i}^{\mathrm{II}}\,.$ For several pairs, let us mention the symmetry in pairs: $k_{1}=k_{2}=k$ (we discuss this symmetry in more detail in the next subsection): $\frac{dN_{i}^{\mathrm{I}}}{dt}=\sum_{J}kS^{\mathrm{I,J}}(c_{i}^{\mathrm{J}}-c_{i}^{\mathrm{I}})\,.$ For example, on a straight line (two neighbors), this equation gives $\frac{dN_{i}^{\mathrm{I}}}{dt}=kS^{\mathrm{I,J}}(c_{i}^{\mathrm{I+1}}+c_{i}^{\mathrm{I-1}}-2c_{i}^{\mathrm{I}})\,.$ From this expression, the proper scaling of $k$ with the cell size is obvious: $\frac{dc_{i}^{\mathrm{I}}}{dt}=k\frac{S^{\mathrm{I,J}}l^{2}}{V}\frac{c_{i}^{\mathrm{I+1}}+c_{i}^{\mathrm{I-1}}-2c_{i}^{\mathrm{I}}}{l^{2}}\,,$ the fraction ${c_{i}^{\mathrm{I+1}}+c_{i}^{\mathrm{I-1}}-2c_{i}^{\mathrm{I}}}/{l^{2}}$ approximates the second derivative and, hence, ${kS^{\mathrm{I,J}}l^{2}}/{V}=\mathrm{const}$. For a cubic cell, $V=Sl$ and $kl=\mathrm{const}$. #### 2.2.3. Space Symmetry and Time Symmetry The system of elementary events should be symmetric with respect to space- inversion. For each elementary process (2.21) a space-inverted process is defined just by changing I to II and vice versa. We mark the quantities for the space-inverted processes by ′. For example, $\gamma^{\prime}=-\gamma\,.$ Space inversion is an involution: if we apply it two times then we return to the original process. The key condition is: the rate functions for the space-inverted processes should differ just by transposition by vectors of variables, $c^{\mathrm{I}},c^{\mathrm{II}}$ (1.39): $w^{\prime}_{r}(c^{\mathrm{I}},c^{\mathrm{II}})=w_{r}(c^{\mathrm{II}},c^{\mathrm{I}})\,.$ (2.26) For MAL this means that $k_{r}=k_{r}^{\prime}$. The requirement of space symmetry distinguishes diffusion from various types of advection and transport driven by external force. This condition is necessary for existence of diffusion equations when the cell size $l\to 0$ (see the next subsection). Inversion in time differs, in general, from the inversion in space. For example, for the elementary process $3A^{\mathrm{I}}\to 2A^{\mathrm{I}}+A^{\mathrm{II}}$ the space inversion gives $2A^{\mathrm{II}}\to A^{\mathrm{I}}+A^{\mathrm{II}}$ (we exchange the upper indexes, I$\to$II and II$\to$I), and the inversion of elementary events ($T$-transformation) gives $2A^{\mathrm{I}}+A^{\mathrm{II}}\to 3A^{\mathrm{I}}$ (here, we change direction of arrow). We have to stress that inversion in time assumes micro-reversion. At the macroscopic level it does not mean change $t$ to $-t$ in the kinetic equations but the transformation of the direct processes into reverse ones (inversion of collisions, for example). Time symmetry (microreversibility) means that the principle of detailed balance is valid. In this case, all the consequences of the principle of detailed balance are applicable (Section 2.1.3.), a global Lyapunov functional exists, and every positive equilibrium is a point of detailed balance. Microreversibility and symmetry of space are independent properties of the system. Nevertheless, for some elementary processes, the space-inverted process coincides with the reverse (the time-inverse) process. If a diffusion mechanism is constructed from such processes then the symmetry in space is equivalent to symmetry in time (to the principle of detailed balance). This specific class of diffusion mechanisms includes such mechanisms as Fick’s diffusion or the diffusion by exchange of particle positions. Indeed, for Fick’s diffusion, when we exchange the upper indexes in an elementary process $A^{\mathrm{I}}\to A^{\mathrm{II}}$ (inversion in space) then we get the reverse process as well, $A^{\mathrm{II}}\to A^{\mathrm{I}}$ (inversion of arrows). Analogously, for $A^{\mathrm{I}}+B^{\mathrm{II}}\to A^{\mathrm{II}}+B^{\mathrm{I}}$ inversion of space gives the reverse process as well: $A^{\mathrm{II}}+B^{\mathrm{I}}\to A^{\mathrm{I}}+B^{\mathrm{II}}$. This fundamental property is formulated in the following theorem. ###### Theorem 2. Let a complex diffusion process consist of elementary processes, which satisfy the following property: the space-inverted elementary process coincides with the inverse process. Then, for the mass action law equation of diffusion (2.25), the principle of detailed balance is valid, the global convex Lyapunov functional exists and the uniform distribution is asymptotically stable. Indeed, due to space symmetry, a uniform distribution is an equilibrium and each process is equilibrated at this state by its space-inverted process. At the same time, this distribution is a point of detailed balance. Therefore, the results about the principle of detailed balance are applicable. #### 2.2.4. Arrested Diffusion and Boundary Equilibria Existence of a uniform distribution, which is a point of detailed balance, existence of the global Lyapunov function and asymptotic stability of the uniform equilibrium distributions do not mean that there exist no nonuniform equilibria. The phenomenon of boundary equilibria is well known. For example, an autocatalytic reversible reaction $A+B\rightleftharpoons 2A$ has two equilibria for a given value of a stoichiometric conservation law, $b=c_{A}+c_{B}=$const. One equilibrium is strictly positive, with positive concentrations of $A$ and $B$. Another is a boundary equilibrium, $c_{A}=0$, $c_{B}=b$. The positive equilibrium is asymptotically stable, the boundary equilibrium is unstable but if the initial state is near the boundary ($c_{A}$ is close to zero) then slow relaxation occurs [43], and the motion may be arrested for a long time near this state. There are well known effects of arrested diffusion caused by changing of temperature. The solid solutions show the effects of diffusion, which has been arrested by chilling below a threshold temperature in a very short time [74]. Kinetic effects of arrested diffusion are also possible. For example, let us consider the diffusion mechanism by jumps to free places (Fig. 3(b)): any distribution of components $A_{i}$ for zero concentrations of free places $Z$ is stationary, and for a small concentration of free places diffusion is slow. For effects of arrested diffusion, the average concentration of free places should not be small. There may be, for example, a layer of particles with low mobility between a dense island of particles with high mobility and an island of free places. Formally, it is possible to construct many such situations. All of them may be characterized as follows: either a small change of concentrations or a small change of constants (or both) lead the system to a nonuniform equilibrium. At this nonuniform equilibrium some of the concentrations in some cells take zero values and, therefore, some of fluxes are also zero. Because of the appearance of these zero concentrations, such an equilibrium is called a boundary equilibrium. ### 2.3. Continuous Diffusion Equation #### 2.3.1. MAL Diffusion Flux Let us consider an elementary process together with its space-inverted process $\begin{split}\sum_{i}\alpha_{ri}^{\mathrm{I}}A_{i}^{\mathrm{I}}+\sum_{i}\alpha_{ri}^{\mathrm{II}}A_{i}^{\mathrm{II}}\rightarrow\sum_{i}\beta_{ri}^{\mathrm{I}}A_{i}^{\mathrm{I}}+\sum_{i}\beta_{ri}^{\mathrm{II}}A_{i}^{\mathrm{II}}\,,\\\ \sum_{i}\alpha_{ri}^{\mathrm{I}}A_{i}^{\mathrm{II}}+\sum_{i}\alpha_{ri}^{\mathrm{II}}A_{i}^{\mathrm{I}}\rightarrow\sum_{i}\beta_{ri}^{\mathrm{I}}A_{i}^{\mathrm{II}}+\sum_{i}\beta_{ri}^{\mathrm{II}}A_{i}^{\mathrm{I}}\,.\end{split}$ (2.27) The reaction rates are $\begin{split}&w_{r}(c^{\mathrm{I}},c^{\mathrm{II}})=k_{r}\prod_{i}(c_{i}^{\mathrm{I}})^{\alpha^{\mathrm{I}}_{ri}}\prod_{i}(c_{i}^{\mathrm{II}})^{\alpha^{\mathrm{II}}_{ri}}\,,\\\ &w_{r}^{\prime}(c^{\mathrm{I}},c^{\mathrm{II}})=w_{r}(c^{\mathrm{II}},c^{\mathrm{I}})=k_{r}\prod_{i}(c_{i}^{\mathrm{II}})^{\alpha^{\mathrm{I}}_{ri}}\prod_{i}(c_{i}^{\mathrm{I}})^{\alpha^{\mathrm{II}}_{ri}}\,,\end{split}$ (2.28) where we take $k^{\prime}_{r}=k_{r}$ due to the symmetry in space. To first order in $l$, the flux vector for $A_{i}$ in this process is $\begin{split}J_{ri}&=-\gamma_{ri}[w_{r}(c(x),c(x+l))-w_{r}(c(x+l),c(x))]\\\ &=-l\gamma_{ri}\sum_{j}\left(\left.\frac{\partial w_{r}(c^{\mathrm{I}},c^{\mathrm{II}})}{\partial c_{j}^{\mathrm{II}}}\right\|_{c^{\mathrm{I}}=c^{\mathrm{II}}=c(x)}-\left.\frac{\partial w_{r}(c^{\mathrm{I}},c^{\mathrm{II}})}{\partial c_{j}^{\mathrm{I}}}\right\|_{c^{\mathrm{I}}=c^{\mathrm{II}}=c(x)}\right)\nabla c_{j}(x)\\\ &=-l\gamma_{ri}w_{r}(c(x),c(x))\sum_{j}\frac{\alpha^{\mathrm{II}}_{rj}-\alpha^{\mathrm{I}}_{rj}}{c_{j}}\nabla c_{j}(x)\\\ &=-lk\gamma_{ri}\left(\prod_{q}c_{q}^{\alpha^{\mathrm{I}}_{rq}+\alpha^{\mathrm{II}}_{rq}}\right)\sum_{j}\frac{\alpha^{\mathrm{II}}_{rj}-\alpha^{\mathrm{I}}_{rj}}{c_{j}}\nabla c_{j}(x)\,.\end{split}$ (2.29) Here, $\gamma_{ri}=\beta_{ri}^{\mathrm{I}}-\alpha_{ri}^{\mathrm{I}}$ (input minus output in the first cell); the minus in front of the formula appears because the direction of flux from cell I to cell II (from $x$ to $(x+l)$) is positive. The factor $1/c_{j}$ never leads to a singularity in the flux because $c_{j}$ enters in the monomial $\prod_{q}c_{q}^{\alpha^{\mathrm{I}}_{rq}+\alpha^{\mathrm{II}}_{rq}}$ with the power $\alpha^{\mathrm{I}}_{rj}+\alpha^{\mathrm{II}}_{rj}$. This power is strictly positive if the coefficient $({\alpha^{\mathrm{II}}_{rj}-\alpha^{\mathrm{I}}_{rj}})$ is not zero. The proper scaling of $k$ for grid refinement or coarsening is $kl=d=const$ in order not to change the first order expression for flux (2.29). According to (2.29), the matrix of diffusion coefficients for the elementary process (2.27) (together with its space-inverted process) is $D_{r\,ij}(c)=d\left(\prod_{q}c_{q}^{\alpha^{\mathrm{I}}_{rq}+\alpha^{\mathrm{II}}_{rq}}\right)\frac{\gamma_{ri}(\alpha^{\mathrm{II}}_{rj}-\alpha^{\mathrm{I}}_{rj})}{c_{j}}\,,$ (2.30) where $d=\mathrm{const}(=kl)$. The corresponding diffusion equations have the divergent form: $\frac{\partial c}{\partial t}=\mathrm{div}(D(c)\nabla c)\,,$ (2.31) where $c$ is the vector of concentrations and $D$ is the matrix of diffusion coefficients (2.30). It might be useful to represent the flux (2.29) similarly to the Teorell formula (1.7). For this purpose, let us collect under $\nabla$ the terms which represent the chemical potential in perfect media: $\mu=RT\ln c+\mu_{0}$. We assume that $T$ and $\mu_{0}$ are constant in space. With these conditions, $J_{ri}=-\frac{lk}{RT}\gamma_{ri}\left(\prod_{q}c_{q}^{\alpha^{\mathrm{I}}_{rq}+\alpha^{\mathrm{II}}_{rq}}\right)\sum_{j}(\alpha^{\mathrm{II}}_{rj}-\alpha^{\mathrm{I}}_{rj})\nabla\mu_{j}(x)\,.$ (2.32) #### 2.3.2. Examples Let us illustrate application of formula (2.29) by several elementary examples. First of all, Fick’s law: $A^{\mathrm{I}}\to A^{\mathrm{II}}$ and $A^{\mathrm{II}}\to A^{\mathrm{I}}$. For this system, $\alpha^{\mathrm{I}}=1$, $\alpha^{\mathrm{II}}=0$, $\beta^{\mathrm{I}}=0$, $\beta^{\mathrm{II}}=1$ and $\gamma=-1$. Formula (2.29) gives $J=lkc\frac{-1}{c}\nabla c=-lk\nabla c\,.$ This is exactly the standard Fick law. The diffusion equation is $\partial_{t}c=d\Delta c$. Here and further in this subsection we use $d$ for $lk$. Exchange of positions: $A^{\mathrm{I}}+B^{\mathrm{II}}\to A^{\mathrm{II}}+B^{\mathrm{I}}$ together with the space-inverted process $A^{\mathrm{II}}+B^{\mathrm{I}}\to A^{\mathrm{I}}+B^{\mathrm{II}}$, which is the same as the reverse process. For this case, $\alpha_{A}^{\mathrm{I}}=1$, $\alpha_{B}^{\mathrm{I}}=0$, $\alpha_{A}^{\mathrm{II}}=0$, $\alpha_{B}^{\mathrm{II}}=1$, $\beta_{A}^{\mathrm{I}}=0$, $\beta_{B}^{\mathrm{I}}=1$, $\beta_{A}^{\mathrm{II}}=1$, $\beta_{B}^{\mathrm{II}}=0$, $\gamma_{A}=\beta_{A}^{\mathrm{I}}-\alpha_{A}^{\mathrm{I}}=-1$, and $\gamma_{B}=\beta_{B}^{\mathrm{I}}-\alpha_{B}^{\mathrm{I}}=1$. Due to (2.29), $\begin{split}&J_{A}=-d(-1)c_{A}c_{B}\left[\frac{-1}{c_{A}}\nabla c_{A}+\frac{1}{c_{B}}\nabla c_{B}\right]=-d(c_{B}\nabla c_{A}-c_{A}\nabla c_{B})\,,\\\ &J_{B}=-dc_{A}c_{B}\left[\frac{-1}{c_{A}}\nabla c_{A}+\frac{1}{c_{B}}\nabla c_{B}\right]=d(c_{B}\nabla c_{A}-c_{A}\nabla c_{B})\,.\end{split}$ (2.33) The diffusion equations are $\partial_{t}c_{A}=d(c_{B}\Delta c_{A}-c_{A}\Delta c_{B})\,,\;\partial_{t}c_{B}=d(c_{A}\Delta c_{B}-c_{B}\Delta c_{A})\,.$ Repulsion of components $A$ and $B$ ($A$ is mobile). The mechanism is $A^{\mathrm{I}}+B^{\mathrm{I}}\to A^{\mathrm{II}}+B^{\mathrm{I}}$ together with the space-inverted process $A^{\mathrm{II}}+B^{\mathrm{II}}\to A^{\mathrm{I}}+B^{\mathrm{II}}$, which does not coincide with the reverse process. For this mechanism, $\alpha_{A}^{\mathrm{I}}=1$, $\alpha_{B}^{\mathrm{I}}=1$, $\alpha_{A}^{\mathrm{II}}=0$, $\alpha_{B}^{\mathrm{II}}=0$, $\beta_{A}^{\mathrm{I}}=0$, $\beta_{B}^{\mathrm{I}}=1$, $\beta_{A}^{\mathrm{II}}=1$, $\beta_{B}^{\mathrm{II}}=0$, $\gamma_{A}=\beta_{A}^{\mathrm{I}}-\alpha_{A}^{\mathrm{I}}=-1$, and $\gamma_{B}=\beta_{B}^{\mathrm{I}}-\alpha_{B}^{\mathrm{I}}=0$. Formula (2.29) gives: $\begin{split}J_{A}&=-d(-1)c_{A}c_{B}\left[\frac{-1}{c_{A}}\nabla c_{A}+\frac{-1}{c_{B}}\nabla c_{B}\right]\\\ &=-d(c_{B}\nabla c_{A}+c_{A}\nabla c_{B})=-d\nabla(c_{A}c_{B})\,,\\\ J_{B}&=0\,.\end{split}$ (2.34) We can see that $B$ activates diffusion of $A$ (the term $c_{B}\nabla c_{A}$) and, at the same time, pushes $A$ in the area with lower concentration of $B$ (the term $c_{A}\nabla c_{B}$). The diffusion equation is $\partial_{t}c_{A}=d\Delta(c_{A}c_{B})\,.$ The no-flux steady states of these diffusion equations are given by the condition: $c_{A}c_{B}=\mathrm{const}$. If we assume that $B$ is also mobile by a similar mechanism then we get a system of equations (with two different diffusion coefficients): $J_{A}=-d_{A}\nabla(c_{A}c_{B})\,,\;J_{B}=-d_{B}\nabla(c_{A}c_{B})\,,$ (2.35) $\partial_{t}c_{A}=d_{A}\Delta(c_{A}c_{B})\,,\;\partial_{t}c_{B}=d_{B}\Delta(c_{A}c_{B})\,.$ (2.36) Let us change variables: $c_{+}=\frac{c_{A}}{d_{A}}+\frac{c_{B}}{d_{B}},\;c_{-}=\frac{c_{A}}{d_{A}}-\frac{c_{B}}{d_{B}}\,.$ In these variables, $c_{A}c_{B}=\frac{d_{A}d_{B}}{4}(c_{+}^{2}-c_{-}^{2})$, $\partial_{t}c_{-}=0$ and for $c_{+}$ we have the porous media equation: $\partial_{t}c_{+}=\frac{d_{A}d_{B}}{2}\Delta c_{+}^{2}\,.$ ### 2.4. Principle of Detailed Balance and Dissipation Inequality #### 2.4.1. Detailed balance and Coupling of Direct and Reverse Processes In this subsection, we formulate the principle of detailed balance for MAL diffusion. Physically, it follows from microreversibility. For every elementary process, $\sum_{i}\alpha_{ri}^{\mathrm{I}}A_{i}^{\mathrm{I}}+\sum_{i}\alpha_{ri}^{\mathrm{II}}A_{i}^{\mathrm{II}}\rightarrow\sum_{i}\beta_{ri}^{\mathrm{I}}A_{i}^{\mathrm{I}}+\sum_{i}\beta_{ri}^{\mathrm{II}}A_{i}^{\mathrm{II}}\,,$ (2.37) the reverse process is (just invert arrows): $\sum_{i}\beta_{ri}^{\mathrm{I}}A_{i}^{\mathrm{I}}+\sum_{i}\beta_{ri}^{\mathrm{II}}A_{i}^{\mathrm{II}}\to\sum_{i}\alpha_{ri}^{\mathrm{I}}A_{i}^{\mathrm{I}}+\sum_{i}\alpha_{ri}^{\mathrm{II}}A_{i}^{\mathrm{II}}\,.$ (2.38) Let us distinguish the quantities for the reverse and direct processes by the upper indices $\pm$. The simple algebraic relations hold: $\alpha_{ri}^{\mathrm{I,II}\mp}=\beta_{ri}^{\mathrm{I,II}\pm}\,\mbox{ and }\gamma_{ri}^{\mathrm{I,II}\mp}=-\gamma_{ri}^{\mathrm{I,II}\pm}\,.$ Therefore, $\alpha_{ri}^{\mathrm{I+}}+\alpha_{ri}^{\mathrm{II+}}=\beta_{ri}^{\mathrm{I-}}+\beta_{ri}^{\mathrm{II-}}=\alpha_{ri}^{\mathrm{I-}}+\alpha_{ri}^{\mathrm{II-}}=\beta_{ri}^{\mathrm{I+}}+\beta_{ri}^{\mathrm{II+}}\,.$ (2.39) The reaction rates are $\begin{split}&w_{r}^{+}(c^{\mathrm{I}},c^{\mathrm{II}})=k_{r}^{+}\prod_{i}(c_{i}^{\mathrm{I}})^{\alpha^{\mathrm{I}}_{ri}}\prod_{i}(c_{i}^{\mathrm{II}})^{\alpha^{\mathrm{II}}_{ri}}\,,\\\ &w_{r}^{-}(c^{\mathrm{I}},c^{\mathrm{II}})=k_{r}^{-}\prod_{i}(c_{i}^{\mathrm{I}})^{\beta^{\mathrm{I}}_{ri}}\prod_{i}(c_{i}^{\mathrm{II}})^{\beta^{\mathrm{II}}_{ri}}\,,\end{split}$ (2.40) Let there exist a uniform strictly positive point of detailed balance: such a strictly positive vector $c^{}$ that for all $r$ $w_{r}^{+}(c^{},c^{})=w_{r}^{-}(c^{},c^{})\,.$ This means that $k_{r}^{+}\prod_{i}(c_{i}^{})^{\alpha^{\mathrm{I}}_{ri}+\alpha^{\mathrm{II}}_{ri}}=k_{r}^{-}\prod_{i}(c^{}_{i})^{\beta^{\mathrm{I}}_{ri}+\beta^{\mathrm{II}}_{ri}}\,.$ Let us use the relations $\alpha^{\mathrm{I}}_{ri}+\alpha^{\mathrm{II}}_{ri}=\beta^{\mathrm{I}}_{ri}+\beta^{\mathrm{II}}_{ri}$. Therefore, the principle of detailed balance for MAL diffusion can be reformulated: for all $r$ $k_{r}^{+}=k_{r}^{-}\,.$ Let us join processes (2.37), (2.38) and write for them the diffusion flux analogously to (2.29): $\begin{split}J_{ri}&=-\gamma_{ri}[w_{r}^{+}(c(x),c(x+l))-w_{r}^{-}(c(x+l),c(x))]\\\ &=-l\gamma_{ri}\sum_{j}\left(\left.\frac{\partial w_{r}^{+}(c^{\mathrm{I}},c^{\mathrm{II}})}{\partial c_{j}^{\mathrm{II}}}\right\|_{c^{\mathrm{I}}=c^{\mathrm{II}}=c(x)}-\left.\frac{\partial w_{r}^{-}(c^{\mathrm{I}},c^{\mathrm{II}})}{\partial c_{j}^{\mathrm{I}}}\right\|_{c^{\mathrm{I}}=c^{\mathrm{II}}=c(x)}\right)\nabla c_{j}(x)\\\ &=-l\gamma_{ri}w_{r}(c(x),c(x))\sum_{j}\frac{\gamma_{rj}}{c_{j}}\nabla c_{j}(x)\\\ &=-lk\gamma_{ri}\left(\prod_{q}c_{q}^{\alpha^{\mathrm{I}}_{rq}+\alpha^{\mathrm{II}}_{rq}}\right)\sum_{j}\frac{\gamma_{rj}}{c_{j}}\nabla c_{j}(x)\,.\end{split}$ (2.41) According to this expression for the diffusion flux, (2.41), the matrix of diffusion coefficients for the elementary process (2.27) (together with its space-inverted process) is $D^{r}_{ij}(c)=d_{r}\left(\prod_{q}c_{q}^{\alpha^{\mathrm{I}}_{rq}+\alpha^{\mathrm{II}}_{rq}}\right)\frac{\gamma_{ri}\gamma_{rj}}{c_{j}}\,,$ (2.42) where $d=\mathrm{const}(=k_{r}l)$. This matrix is symmetric with respect to the inner product $\langle a,b\rangle_{c}=\sum_{i}\frac{a_{i}b_{i}}{c_{i}}\,.$ (2.43) This means that $\langle Da,b\rangle_{c}=\langle a,Db\rangle_{c}$ or in coordinates $\sum_{ij}\frac{D_{ij}a_{j}b_{i}}{c_{i}}=\sum_{ij}\frac{a_{i}D_{ij}b_{j}}{c_{i}}\,.$ Indeed, let $\widetilde{w}_{r}=d_{r}\left(\prod_{q}c_{q}^{\alpha^{\mathrm{I}}_{rq}+\alpha^{\mathrm{II}}_{rq}}\right)$. For two scalar products, $\langle Da,b\rangle_{c}$ and $\langle a,Db\rangle_{c}$ we get $\langle Da,b\rangle_{c}=\sum_{ij}\frac{D_{ij}a_{j}b_{i}}{c_{i}}=\widetilde{w}_{r}\sum_{ij}\frac{\gamma_{ri}\gamma_{rj}a_{j}b_{i}}{c_{j}c_{i}}\,$ and $\langle a,Db\rangle_{c}=\sum_{ij}\frac{a_{i}D_{ij}b_{j}}{c_{i}}=\widetilde{w}_{r}\sum_{ij}\frac{a_{i}\gamma_{ri}\gamma_{rj}b_{j}}{c_{i}c_{j}}\,,$ These two expressions differ only in the notation of dummy indexes, $i$ and $j$. Therefore, $D$ is symmetric with respect to the inner product (2.43). $D$ is also positive semi-definite. Indeed, for any vector $\xi$, $\langle\xi,D\xi\rangle=\sum_{i}\frac{\xi_{i}D^{r}_{ij}\xi_{j}}{c_{i}}=\widetilde{w}_{r}\langle\xi,\gamma)^{2}\geq 0\rangle\,.$ This expression may be zero at a positive state ($c_{i}>0$) if and only if the vector $\xi$ is orthogonal to the vector $\gamma_{r}$ in the inner product (2.43). If we summarize diffusion coefficients for all pairs of mutually inverse elementary processes then we get $D_{ij}=\sum_{r}D^{r}_{ij}(c)=\sum_{r}\widetilde{w}_{r}\frac{\gamma_{ri}\gamma_{rj}}{c_{j}}\,.$ (2.44) For this $D_{ij}$, $\sum\frac{\xi_{i}D_{ij}\xi_{j}}{c_{i}}\geq 0$ and is zero if and only if vector $\xi$ is orthogonal to all vectors $\gamma_{r}$ in the inner product (2.43). The corresponding diffusion equations have the divergent form: $\frac{\partial c}{\partial t}=\mathrm{div}(D(c)\nabla c)\,.$ (2.45) It might be useful to represent the flux (2.41) similarly to the Teorell formula (1.7). For this purpose, let us collect under $\nabla$ the terms which represent the chemical potential in perfect media: $\mu=RT\ln c+\mu_{0}$. We assume that $T$ and $\mu_{0}$ are constant in space. With these conditions, $J_{ri}=-\frac{lk}{RT}\gamma_{ri}\left(\prod_{q}c_{q}^{\alpha^{\mathrm{I}}_{rq}+\alpha^{\mathrm{II}}_{rq}}\right)\sum_{j}\gamma_{rj}\nabla\mu_{j}(x)\,.$ (2.46) The difference from the MAL Teorell formula (2.32) is obvious: for the systems with detailed balance (2.46) the matrix of the coefficients in the Teorell formula is symmetric for each elementary process together with its reverse process because it is a product of a number (in square brackets) and the symmetric matrix $\gamma_{ri}\gamma_{rj}$: $\left[\frac{lk}{RT}\left(\prod_{q}c_{q}^{\alpha^{\mathrm{I}}_{rq}+\alpha^{\mathrm{II}}_{rq}}\right)\right]\gamma_{ri}\gamma_{rj}\,.$ (2.47) For the general MAL diffusion (with detailed symmetry in space) the matrix these coefficients for the elementary process together with its space-inverted process is not symmetric in general: it is a scalar multiple of the matrix $\gamma_{ri}(\alpha^{\mathrm{II}}_{rj}-\alpha^{\mathrm{I}}_{rj})\,.$ Its symmetry may be guaranteed if the space inversion of the elementary process coincides with its reversion in time (i.e. $\alpha^{\mathrm{II}}_{r}$ coincides with $\beta^{\mathrm{I}}_{r}$). For MAL chemical kinetics, there is a sufficient algebraic condition for detailed balance that is independent of the microreversibility and follows just from the stoichiometric equations. Indeed, let us assume that all the reactions are reversible and all the stoichiometric vectors $\gamma_{r}$ in (2.7) are linearly independent. Then, at the equilibrium, from the condition $\dot{c}=\sum_{r}w_{r}\gamma_{r}=0$ we get $w_{r}=0$ for all $r$. That is the detailed balance condition. For MAL diffusion we have also a specific (“diffusion”) algebraic condition: if for all elementary processes the space-inverted reaction is the reverse reaction then the principle of detailed balance follows from the space symmetry condition. For such systems, the coupling “process–space inverted process” coincides with the coupling “process–reverse process” and equations (2.41) coincide with (2.29). For an elementary process (2.37), let the space-inverted process not coincide with the reverse process. With the condition of detailed balance, it is straightforward to check that for the pair of elementary processes (2.37), (2.38), the space-inverted process to (2.37) and the reverse process to this space-inverted one generate the same diffusion equations and the diffusion coefficient as the original pair does. These two couples together produce the flux that is twice as large as (2.41). #### 2.4.2. The Dissipation Inequality and Detailed Balance In this subsection, we construct the Lyapunov functional for the general MAL diffusion models and calculate its time derivative due to the diffusion equations. Let us select any strictly positive reference concentration vector $c^{}$ and take (2.8) $G=\sum_{i}c_{i}\left(\ln\left(\frac{c_{i}}{c_{i}^{}}\right)-1\right)+\sum_{i}c_{i}^{}\,.$ Let us consider this system in a bounded domain $V$ with smooth boundary and with zero fluxes through its boundary: $(n,J)=0$ at any point of $\partial V$ at any time ($n$ is the vector of the outer normal). Due to the definition of flux (2.29), it is sufficient that $(n,\mathrm{grad}c_{j})=0$ on $\partial V$ for all $j$ (but it is not necessary). The Lyapunov functional is $\mathbf{G}=\sum_{i}\int_{V}\left[c_{i}\left(\ln\left(\frac{c_{i}}{c_{i}^{}}\right)-1\right)+\sum_{i}c_{i}^{}\right]\,\mathrm{d}x\,.$ (2.48) Due to the boundary conditions and the Gauss–Ostrogradskii theorem $\frac{\mathrm{d}\mathbf{G}}{\mathrm{d}t}=-\sum_{i}\int_{V}\ln\left(\frac{c_{i}}{c_{i}^{}}\right)\mathrm{div}J_{i}\,\mathrm{d}x=\sum_{i}\int_{V}\left(\nabla_{x}\left(\ln\frac{c_{i}}{c_{i}^{}}\right),J_{i}\right)\,\mathrm{d}x\,.$ (2.49) Let us assume the principle of detailed balance and calculate $\dot{\mathbf{G}}$ (2.49) due to diffusion equation with the matrix of diffusion coefficients (2.44): $\begin{split}\frac{\mathrm{d}\mathbf{G}}{\mathrm{d}t}&=\sum_{i}\int_{V}\left(\nabla_{x}\left(\ln\frac{c_{i}}{c_{i}^{}}\right),J_{i}\right)\,\mathrm{d}x\,\\\ &=-\sum_{r}\int_{V}\widetilde{w}_{r}\sum_{ij}\left(\nabla_{x}(\ln{c_{i}}),\gamma_{ri}\gamma_{rj}\nabla_{x}(\ln{c_{j}})\right)\,\mathrm{d}x\,\\\ &=-\sum_{r}\int_{V}\widetilde{w}_{r}\left(\nabla_{x}\left(\sum_{i}\ln(\gamma_{ri}c_{i})\right),\nabla_{x}\left(\sum_{j}\ln(\gamma_{rj}c_{j})\right)\right)\,\mathrm{d}x\\\ &=-\sum_{r}\int_{V}\widetilde{w}_{r}(\nabla_{x}(\gamma_{r},\nabla_{c}G))^{2}\,\mathrm{d}x\leq 0\,.\end{split}$ (2.50) Here, $(\ ,\ )$ is the standard Euclidean inner product both in space and in the concentration space. As we can see from this dissipation inequality, the only positive equilibria for diffusion equations with detailed balance conditions satisfy the conditions $(\gamma_{r},\nabla_{c}G)=\mathrm{const}$ for all $r$. Another form of these conditions is $\prod_{i}c_{i}^{\gamma_{ri}}=\mathrm{const}\,.$ Inequality (2.50) demonstrates a significant difference between the two classes of diffusion mechanism. If the stoichiometric vectors $\gamma_{r}$ form a basis in the concentration space then all the equilibria are uniform because in this case the condition $(\gamma_{r},\nabla_{c}G)=\mathrm{const}$ (for all $r$) implies $\nabla_{c}G=\mathrm{const}$, that is, $\ln c_{i}=\mathrm{const}$ for all $i$, hence, $c_{i}=\mathrm{const}$ for all $i$. Let us call this mechanism the mechanisms with mixing. The first example is Fick’s mechanism: if the diffusion constant is not zero for all components then all the equilibria of the system are uniform. If $\mathrm{span}\\{\gamma_{r}\\}$ does not coincide with the concentration space then there exist the invariants of diffusion. They are given by the linear functionals that annihilate all the vectors $\gamma_{r}$. For example, in diffusion by jumps on the free places (Fig. 3(b)) the value of the sum $c_{Z}(x)+\sum_{i}c_{i}(x)$ is locally conserved. In the mechanism, $A^{\mathrm{I}}+B^{\mathrm{I}}\to A^{\mathrm{II}}+B^{\mathrm{I}}$, $A^{\mathrm{II}}+B^{\mathrm{II}}\to A^{\mathrm{I}}+B^{\mathrm{II}}$, concentration of $B$ is conserved. These locally conserved quantities together with the condition of positivity $c_{i}\geq 0$ define a convex body where the vector of concentrations may be situated at a given point. This body depends on the values of the conserved quantities, differs for different points, but does not change in time. ### 2.5. Complex Balance in MAL Diffusion #### 2.5.1. Complex Balance Conditions for MAL Diffusion The complex balance condition does not assume any space or time symmetry. The only microscopic assumption is the Markov fast microscopic kinetic with relatively small amount of active intermediate complexes [97, 54]. We discussed this condition for the MAL kinetics in Section 2.1.4., and now let us transform it into a condition for the MAL diffusion equation. First of all, we should abandon the symmetry conditions $k^{\prime}=k$ (space symmetry) and $k^{+}=k^{-}$ (microreversibility). Without these conditions, the zero- order terms in the expression for fluxes will not be annihilated by balance between the elementary processes with given pair of vectors $(\alpha_{r}^{\mathrm{I}},\alpha_{r}^{\mathrm{II}})$ and the processes with the same pair $(\beta_{r}^{\mathrm{I}},\beta_{r}^{\mathrm{II}})$ (2.20). Let the stoichiometric mechanism of the diffusion be given: $\sum_{i}\alpha_{ri}^{\mathrm{I}}A_{i}^{\mathrm{I}}+\sum_{i}\alpha_{ri}^{\mathrm{II}}A_{i}^{\mathrm{II}}\rightarrow\sum_{i}\beta_{ri}^{\mathrm{I}}A_{i}^{\mathrm{I}}+\sum_{i}\beta_{ri}^{\mathrm{II}}A_{i}^{\mathrm{II}}\,,$ (2.51) where all the elementary processes have different numbers $r$ and the space- inverted and reverse processes are represented separately. Let us consider all pairs of vectors, $(\alpha_{r}^{\mathrm{I}},\alpha_{r}^{\mathrm{II}})$ and $(\beta_{r}^{\mathrm{I}},\beta_{r}^{\mathrm{II}})$. Let us enumerate all the different pairs: $y_{1},y_{2},\ldots$, $y_{q}=(y_{q}^{\mathrm{I}},y_{q}^{\mathrm{II}})$. For each $y_{q}$ there are two sets of reactions, $R_{q}^{+}$, $R_{q}^{-}$: $R_{q}^{+}=\\{r\,\|\,(\alpha_{r}^{\mathrm{I}},\alpha_{r}^{\mathrm{II}})=y_{q}\\}\,,\;R_{q}^{-}=\\{r\,\|\,(\beta_{r}^{\mathrm{I}},\beta_{r}^{\mathrm{II}})=y_{q}\\}\,.$ The complex balance condition is: there exists a strictly positive vector $c^{}$ such that $\sum_{r\in R_{q}^{+}}w_{r}(c^{},c^{})=\sum_{r\in R_{q}^{-}}w_{r}(c^{},c^{})\,\mbox{ for all }q\,.$ (2.52) For MAL this means $\sum_{r\in R_{q}^{+}}k_{r}\prod_{i}(c^{}_{i})^{\alpha_{r}^{\mathrm{I}}+\alpha_{r}^{\mathrm{II}}}=\sum_{r\in R_{l}^{-}}k_{q}\prod_{i}(c^{}_{i})^{\alpha_{q}^{\mathrm{I}}+\alpha_{q}^{\mathrm{II}}}\,\mbox{ for all }l\,.$ (2.53) For $r\in R_{q}^{+}$, $\alpha_{r}^{\mathrm{I}}+\alpha_{r}^{\mathrm{II}}=y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}}\,,$ and for $r\in R_{l}^{-}$ $\beta_{r}^{\mathrm{I}}+\beta_{r}^{\mathrm{II}}=y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}}\,.$ Therefore, all the monomials coincide in the right and the left hand sides of (2.53) for the given $q$ and we can write the complex balance condition for diffusion in the following form: $\sum_{r\in R_{q}^{+}}k_{r}=\sum_{r\in R_{q}^{-}}k_{r}\,\mbox{ for all }q\,.$ (2.54) Let us calculate the flux $J=\sum_{r}\gamma_{r}w_{r}(c(x),c(x+l))$ at first order in $l$. We group terms in this sum and each group will corresponds to a pair $y_{q}$. For each elementary process with number $r$ there are two $q$, $q_{\alpha}(r)$ and $q_{\beta}(r)$: ${r\in R_{q_{\alpha}(r)}^{+}}$ and ${r\in R_{q_{\beta}(r)}^{-}}$. We split the term $\gamma_{r}w_{r}(c(x),c(x+l))$ in two terms: $\gamma_{r}w_{r}(c(x),c(x+l))=\beta_{r}w_{r}(c(x),c(x+l))-\alpha_{r}w_{r}(c(x),c(x+l))$ We associate the first of them with $y_{q_{\beta}(r)}$ and the second with $y_{q_{\alpha}(r)}$. The result is represented below: $\begin{split}J&=-\sum_{r}\gamma_{r}w_{r}(c(x),c(x+l))\\\ &=\sum_{q}\left(\sum_{r\in R_{q}^{+}}\alpha_{r}^{\mathrm{I}}w_{r}(c(x),c(x+l))-\sum_{r\in R_{q}^{-}}\beta_{r}^{\mathrm{I}}w_{r}(c(x),c(x+l))\right)\\\ &=\sum_{q}y_{q}^{\mathrm{I}}\left(\sum_{r\in R_{q}^{+}}w_{r}(c(x),c(x+l))-\sum_{r\in R_{q}^{-}}w_{r}(c(x),c(x+l))\right)\\\ &=\sum_{q}y_{q}^{\mathrm{I}}\prod_{i}c_{i}^{y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}}}\left(\sum_{r\in R_{q}^{+}}k_{r}-\sum_{r\in R_{q}^{-}}k_{r}\right)\\\ &\quad+l\sum_{q}y_{q}^{\mathrm{I}}\prod_{i}c_{i}^{y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}}}\left(\sum_{r\in R_{q}^{+}}k_{r}\sum_{j}\frac{\alpha^{\mathrm{II}}_{rj}}{c_{j}}\nabla c_{j}-\sum_{r\in R_{q}^{-}}k_{r}\sum_{j}\frac{\alpha^{\mathrm{II}}_{rj}}{c_{j}}\nabla c_{j}\right)+o(l)\,.\end{split}$ (2.55) The zero-order term is zero because of the complex balance condition (2.54). Let us take into account that both for ${r\in R_{q}^{+}}$ and ${r\in R_{q}^{-}}$ $\alpha^{\mathrm{I}}+\alpha^{\mathrm{II}}=y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}}\,.$ Therefore, if for two $y_{q},y_{s}$ $R_{q}^{+}\bigcap R_{s}^{-}\neq\emptyset$ then $y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}}=y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}}\,.$ For the set of all vectors $y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}}$ we use notation $Z$. For each $z\in Z$ $Y_{z}=\\{y_{q}\ \|\ y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}}=z\\}$ Let us group the terms with the same vectors $z=y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}}\in Z$ together and write the expression for the flux for the first order: $\begin{split}&J=l\sum_{z\in Z}\prod_{i}c_{i}^{z_{i}}J^{z}\,,\\\ &J^{z}=\sum_{y_{q}\in Y_{z}}y_{q}^{\mathrm{I}}\sum_{j}\left(\sum_{r\in R_{q}^{+}}k_{r}\frac{\alpha^{\mathrm{II}}_{rj}}{c_{j}}-\sum_{r\in R_{q}^{-}}k_{r}\frac{\alpha^{\mathrm{II}}_{rj}}{c_{j}}\right)\nabla c_{j}\,.\end{split}$ (2.56) Let us study the expression for $J^{z}$ for given $z$. First of all, $\sum_{y_{q}\in Y_{z}}\sum_{j}\left(\sum_{r\in R_{q}^{+}}k_{r}\frac{\alpha^{\mathrm{II}}_{rj}}{c_{j}}-\sum_{r\in R_{q}^{-}}k_{r}\frac{\alpha^{\mathrm{II}}_{rj}}{c_{j}}\right)\nabla c_{j}=0\,$ because each $k_{r}$ from this sum enters it twice, with the opposite signs, one time as an element of $R_{q}^{+}$ for some $y_{q}\in Y_{z}$ (with $+$) and the second time (with $-$) as an element of $R_{s}^{-}$ for another $y_{s}\in Y_{z}$ with the same sum $y_{s}^{\mathrm{I}}+y_{s}^{\mathrm{II}}=y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}}=z\,.$ We note that $y_{q}^{\mathrm{I}}=z-y_{q}^{\mathrm{II}}$ for $y_{q}\in Y_{z}$ in the sum (2.56). This allows us to rewrite the expression for $J^{z}$: $J^{z}=-\sum_{y_{q}\in Y_{z}}y_{q}^{\mathrm{II}}\sum_{j}\left(\sum_{r\in R_{q}^{+}}k_{r}\frac{\alpha^{\mathrm{II}}_{rj}}{c_{j}}-\sum_{r\in R_{q}^{-}}k_{r}\frac{\alpha^{\mathrm{II}}_{rj}}{c_{j}}\right)\nabla c_{j}\,.$ (2.57) #### 2.5.2. The Dissipation Inequality and Complex Balance for MAL Diffusion We would like to demonstrate some similarity of the expression for $J^{z}$ and some formulas from the theory of Markov chains. Vectors $y_{q}\in Y_{z}$ numerate the states. Elementary processes correspond to transitions between states. Each nonzero constant $k_{r}$ corresponds to two vectors $y_{q},y_{s}\in Y_{z}$: ${r\in R_{q}^{+}}$ and ${r\in R_{s}^{-}}$. We substitute the index $r$ by two indexes $q,s$ and use notation $k_{sq}$ (or even $k_{s\leftarrow q}$). If there is no nonzero constant for this pair $q,s$ then we take $k_{s\leftarrow q}=0$. In particular, $k_{qq}=0$. The complex balance condition (2.54) reads: $\sum_{q}k_{sq}=\sum_{q}k_{qs}\,.$ (2.58) This means that the constants $k_{sq}$ describe a continuous time Markov chain with the Master equation $\dot{\pi}_{q}=\sum_{s}(k_{qs}\pi_{s}-k_{sq}\pi_{q})\,$ (2.59) and equidistribution in equilibrium. Here $\pi_{q}$ is the probability to find the system in the state $q$ and $k_{qs}\pi_{s}$ is the probability flux from the state $s$ to the state $q$. We can use the steady state condition (2.58) and rewrite the Master equation (2.59): $\dot{\pi}_{q}=\sum_{s}k_{qs}(\pi_{s}-\pi_{q})\,.$ (2.60) From this form, it is easy to see that the functional $H=\frac{1}{2}\sum_{q}\pi_{q}^{2}$ monotonically decreases due to the system dynamics: $\frac{\mathrm{d}H}{\mathrm{d}t}=\sum_{sq}\pi_{q}k_{qs}(\pi_{s}-\pi_{q})\leq 0\,.$ (2.61) To prove (2.61), let us use the identity $\sum_{q}\dot{\pi}_{q}=\sum_{s}(k_{qs}\pi_{s}-k_{sq}\pi_{q})=\sum_{s}k_{qs}(\pi_{s}-\pi_{q})=0\,.$ (2.62) This condition holds for all values of numbers $\pi_{q}$ (this is obvious for the Master equation in the form (2.59)). In particular, $\sum_{s}k_{qs}\left(\frac{1}{2}\pi_{q}^{2}-\frac{1}{2}\pi_{s}^{2}\right)=0\,.$ Let us add this expression to the right hand side of (2.61): $\frac{\mathrm{d}H}{\mathrm{d}t}=\sum_{sq}k_{qs}\left(\frac{1}{2}\pi_{q}^{2}-\frac{1}{2}\pi_{s}^{2}+\pi_{q}(\pi_{s}-\pi_{q})\right)\leq 0$ (2.63) because $k_{qs}\geq 0$ and the expression in the parentheses is non-positive: $\frac{1}{2}\pi_{q}^{2}-\frac{1}{2}\pi_{s}^{2}+\pi_{q}(\pi_{s}-\pi_{q})=-\frac{1}{2}(\pi_{q}-\pi_{s})^{2}\leq 0\,.$ Therefore, we have proved the inequality for our set of coefficients $k_{qs}$: $\sum_{sq}\pi_{q}(k_{qs}\pi_{s}-k_{sq}\pi_{q})=\sum_{sq}\pi_{q}k_{qs}(\pi_{s}-\pi_{q})\leq 0\,$ (2.64) for any set of numbers $\pi_{q}$. This inequality means that $\mathrm{d}H/\mathrm{d}t\leq 0$ (2.61). It is zero if all $\pi_{q}$ coincide ($\pi_{s}=\pi_{q}$ for all $s$, $q$). For our purposes, it is important to know when the zero time derivative of $H$ ($\mathrm{d}H/\mathrm{d}t=0$) is equivalent to the equidistribution ($\pi_{s}=\pi_{q}$ for all $s$, $q$). They are equivalent if the Markov chain (2.59) is ergodic. The conditions of ergodicity are well known [95, 106]: the chain (2.59) is ergodic if for any two $s$, $q$ ($s\neq q$) there exists a oriented path from $s$ to $q$ in the graph of the network, that is such a sequence $r_{0},r_{1},\ldots,r_{g}\mbox{ that }s=r_{0},\,q=r_{g}\mbox{ and }k_{r_{j+1}r_{j}}>0\mbox{ fo all }j=0,\ldots,g\,.$ This means that the graph of of transitions of the Markov chain (2.59) is strongly connected. To apply this inequality to the proof of the dissipation inequality, we have to rewrite the expression for $J^{z}$ (2.57) using these notations, $k_{qs}$ instead of $k_{r}$: $J^{z}=\sum_{y_{q}\in Y_{z}}y_{q}^{\mathrm{II}}\sum_{j}\sum_{s}(k_{qs}{y_{sj}^{\mathrm{II}}}-k_{sq}{y_{qj}^{\mathrm{II}}})\nabla_{x}\ln c_{j}\,.$ (2.65) Now we are in the position to prove the dissipation inequality for MAL diffusion equation with complex balance. In a bounded domain $V$ with smooth boundary and without fluxes through boundary we have to estimate $\dot{\mathbf{G}}$ (2.48), (2.49). $\begin{split}\frac{\mathrm{d}\mathbf{G}}{\mathrm{d}t}&=-\int_{V}\sum_{j}\ln\left(\frac{c_{j}}{c_{j}^{}}\right)\mathrm{div}J_{j}\,\mathrm{d}x\\\ &=\int_{V}\sum_{j}(\nabla_{x}\ln c_{j},J_{j})\,\mathrm{d}x\\\ &=l\sum_{z\in Z}\int_{V}\prod_{i}c_{i}^{z_{i}}\sum_{j}(\nabla_{x}\ln c_{j},J_{j}^{z})\,\mathrm{d}x\,.\end{split}$ (2.66) Due to the representation of $J^{z}$ (2.65), $\sum_{j}(\nabla_{x}\ln c_{j},J_{j}^{z})=\sum_{q\,s}(\pi_{q},(k_{qs}\pi_{s}-k_{sq}\pi_{q}))\leq 0\,,$ (2.67) where $\pi_{q}$ is a space vector: $\pi_{q}=\sum_{i}y_{qi}^{\mathrm{II}}\nabla_{x}\ln c_{i}$ and for $y_{q}$, $y_{s}$ $y_{q,s}^{\mathrm{I}}+y_{q,s}^{\mathrm{II}}=z\,.$ This expression has exactly the form (2.64) for each space coordinate. Finally, $\sum_{j}(\nabla_{x}\ln c_{j},J_{j}^{z})\leq 0$ (2.68) and it is zero if all $\pi_{q}=\sum_{i}y_{qi}^{\mathrm{II}}\nabla_{x}\ln c_{i}$ coincide. The reverse statement, $\mbox{all }\pi_{q}=\sum_{i}y_{qi}^{\mathrm{II}}\nabla_{x}\ln c_{i}\mbox{ coincide if }\sum_{j}(\nabla_{x}\ln c_{j},J_{j}^{z})=0\,,$ is true if the auxiliary Markov chain is ergodic for given $z$ (i.e. the graph of transitions is strongly connected). Let us assume this ergodicity. For every $z$ we can define a linear subspace $E_{z}$ in the concentration space given by the system of equation $E_{z}=\\{e\ \|\ (y_{qi}^{\mathrm{II}},e)\mbox{ coincide for all }y_{q}\in Y_{z}\\}\,.$ If $\bigcap_{z\in Z}E_{z}=\\{0\\}$ then all the equilibria for this mechanism of diffusion are uniform. In particular, they are uniform if at least one $y_{qi}^{\mathrm{II}}=0$. ### 2.6. Intermediate Summary We presented the formal language of the stoichiometric mechanism for description of nonlinear diffusion (2.21), (2.27), (2.37), (2.38). The general construction of the diffusion equations under various conditions is given: • For systems with symmetry with respect to inversion in space ($k_{r}=k_{r}^{\prime}$) (2.29), (2.30); * • For systems with microreversibility ($k_{r}^{+}=k_{r}-$) (2.41), (2.42); * • For general systems with Markov microkinetics, which satisfy the complex balance conditions (2.54), (2.56). For systems with detailed balance (microreversibility) and with complex balance (Markov microkinetics) the explicit formula for the free energy–type Lyapunov functional $\mathbf{G}$ (2.48) is $\mathbf{G}=\sum_{i}\int_{V}\left[c_{i}\left(\ln\left(\frac{c_{i}}{c_{i}^{}}\right)-1\right)+\sum_{i}c_{i}^{}\right]\,\mathrm{d}x\,.$ We found that $\mathbf{\dot{G}}\leq 0$ for the systems with detailed or complex balance (2.49), (2.66). These inequalities guarantee the thermodynamic behavior of diffusion. The detailed symmetry with respect to the inversion in space is insufficient for such an inequality and the diffusion collapse for them is possible. The Mass Action Law by itself does not imply thermodynamics. In that sense, it is too flexible and needs additional requirements to respect the basic physics. This redundancy of MAL allows, at the same time, to use them in many other areas. The famous Lotka and Volterra models in Mathematical ecology [73, 110] are implementations of MAL for description of surviving and reproduction of animals, that is far from the initial area of MAL applications. Creation of mathematical genetics [33] and analysis of dynamical aspects of biological evolution [38] also use MAL in their backgrounds. Combination of MAL kinetics with nonlinear diffusion is very important for analysis of biological invasions and other phenomena in ecology [57, 61, 96, 89]. The MAL for diffusion can also generate equations that have no direct physical sense but may be used for modeling of some phenomena of non-physical nature. On the other hand, in this approach, we did not take into account some basic physical properties, namely, the momentum and the center of mass conservation. The diffusion transport should be coupled with the viscous transport or elastic deformation (or both) because two reasons: * • The mass average velocity of diffusion $u=\frac{\sum_{i}m_{i}J_{i}}{\sum_{i}m_{i}}\,,$ where $m_{i}$ is the molecular mass of the $A_{i}$ particles, is, in general, not zero; * • The change of the mixture composition implies the change of pressure and, hence, the viscous flux or the elastic deformation (or both). The careful analysis of these effects should give, for example a theory of the Kirkendall effect. In 1942, Kirkendall demonstrated experimentally that different atoms can migrate at different rates in an alloy, and this diffusion is accompanied by measurable local volume change and displacement of interfaces [67, 81]. The kinetic theory of this effect is still under development and there remain open problems and new ideas are needed [82]. We return to the problem of correct description of the general transport equation in next Section. The MAL formalism for diffusion is a flexible and effective tool for modeling. The semi-discrete MAL model may be used for numerical modeling directly as a sort of finite elements. Their coarse-graining and refinement should follow the main rule $kl=d$ where $l$ is the cell size, $k$ is a kinetic constant for the finite elements, $d$ is the invariant diffusion coefficient. For unstable processes, these provide a biharmonic regularization. These kinetic finite elements respect the basic physical properties like positivity of concentration, conservation laws and the second law of thermodynamics (under the relevant conditions of detailed or complex balance). ## 3\. Generalized Mass Action Law for Diffusion ### 3.1. Free Energy, Free Entropy, Chemical Potentials, Activities, and Generalized Mass Action Law #### 3.1.1. Thermodynamic Potentials In this Subsection, we present the thermodynamic approach to the generalized MAL. Exactly as in Section 2.1. we start from the chemical kinetic equations and then extend our approach to the transport processes. In the thermodynamic approach, the kinetic description of the multicomponent system requires the following inputs 1. 1. A list of components; 2. 2. A thermodynamic potential; 3. 3. A list of elementary reactions represented by their stoichiometric equations; 4. 4. A set of reaction rate constants. Exactly as it was in Section 2.1., the list of components is just a set of symbols (the component names). We usually assume that this set is finite, $A_{1},A_{2},\ldots,A_{n}$. The definitions of stoichiometric equation and the corresponding vectors $\alpha_{ri}$, $\beta_{ri}$ are also the same. There are many thermodynamic potentials and they form two series: energy and free energies and, on another hand, entropy and free entropies (the Massieu–Planck functions). Each of them has its own “natural variables” and if one of them is given in the natural variables then all other thermodynamic functions can be produced. The names and standard notations of variables are: * • Internal energy, $U$; * • Entropy, $S$; * • Enthalpy, $H$; * • Temperature, $T$; * • Volume, $V$; * • Pressure, $P$; * • Number of particles (or moles) composing the $i$th component $A_{i}$, $N_{i}$ ($N$ is vector with coordinates $N_{i}$, the vector of composition); * • Chemical potential of the $i$th component $A_{i}$, $\mu_{i}$. The first potential in the energetic series is the internal energy $U(S,V,N)$ in the natural coordinates $S$, $V$, $N$ and $\mathrm{d}U=T\mathrm{d}S-P\mathrm{d}V+\sum_{i}\mu_{i}\mathrm{d}N_{i}\,.$ The enthalpy, $H(S,P,N)=U+PV$ has the natural coordinates $S$, $P$, $N$ and $\mathrm{d}H=T\mathrm{d}S+V\mathrm{d}P+\sum_{i}\mu_{i}\mathrm{d}N_{i}\,.$ The free energy (the Helmholtz energy), $F(T,V,N)=U-TS$ and $\mathrm{d}F=-S\mathrm{d}T-P\mathrm{d}V+\sum_{i}\mu_{i}\mathrm{d}N_{i}\,.$ The free enthalpy (the Gibbs energy), $G(T,P,N)=H-TS$ and $\mathrm{d}G=-S\mathrm{d}T+V\mathrm{d}P+\sum_{i}\mu_{i}\mathrm{d}N_{i}\,.$ The grand potential, $\Omega(T,V,\mu)=U-TS-\sum_{i}\mu_{i}N_{i}$ and $\mathrm{d}\Omega=-S\mathrm{d}T-P\mathrm{d}V-\sum_{i}N_{i}\mathrm{d}\mu_{i}\,.$ The entropic series starts from the entropy $S(U,V,N)$ and $\mathrm{d}S=\frac{1}{T}\mathrm{d}U+\frac{P}{T}\mathrm{d}V-\sum_{i}\frac{\mu_{i}}{T}\mathrm{d}N_{i}\,.$ Therefore, the main set of the intensive variables for the entropic series is $\frac{1}{T}=\frac{\partial S(U,V,N)}{\partial U}\,,\frac{P}{T}=\frac{\partial S(U,V,N)}{\partial V}\,,-\frac{\mu_{i}}{T}=\frac{\partial S(U,V,N)}{\partial N_{i}}\,.$ The Massieu function, $\Phi(T^{-1},V,N)=S-T^{-1}U\ (=-F/T)$ and $\mathrm{d}\Phi=-U\mathrm{d}\left(\frac{1}{T}\right)+\frac{P}{T}\mathrm{d}V-\sum_{i}\frac{\mu_{i}}{T}\mathrm{d}N_{i}\,.$ The Planck function, $\Xi(T^{-1},T^{-1}P,N)=S-T^{-1}U-T^{-1}PV\ (=-G/T)$ and $\mathrm{d}\Phi=-U\mathrm{d}\left(\frac{1}{T}\right)-V\mathrm{d}\left(\frac{P}{T}\right)-\sum_{i}\frac{\mu_{i}}{T}\mathrm{d}N_{i}\,.$ All these functions are used for the definition of equilibrium. The main definition for an isolated system follows the R. Clausius two main laws formulated in 1865 [17]: 1. 1. The energy of the Universe is constant. 2. 2. The entropy of the Universe tends to a maximum. More precisely, the entropy of the isolated system tends to a maximum under given $U,V$ and values of other conservation laws. Let the conservation laws be given: $B_{j}=\sum_{i}b_{ji}N_{j}\,,$ then the equilibrium is the maximizer of the entropy under given values of $U$, $V$ and $B_{j}$: $S(U,V,N)\to\max\mbox{ subject to given }U,V\mbox{ and }\\{B_{j}\\}\,.$ (3.1) Gibbs [39] paid much attention to the dual formulation of this condition: $U(S,V,N)\to\min\mbox{ subject to given }S,V\mbox{ and }\\{B_{j}\\}\,.$ (3.2) Other definitions of the same equilibrium are available through the free energies and entropies. For the free energies this is condition of minimum. Analogously to (3.2) we get $\begin{split}&H(S,P,N)\to\min\mbox{ subject to given }S,P\mbox{ and }\\{B_{j}\\}\,,\\\ &F(T,V,N)\to\min\mbox{ subject to given }T,V\mbox{ and }\\{B_{j}\\}\,,\\\ &G(T,P,N)\to\min\mbox{ subject to given }T,P\mbox{ and }\\{B_{j}\\}\,.\end{split}$ (3.3) For the free entropies the equilibrium should be the maximizer: in addition to (3.1) $\begin{split}&\Phi\left(\frac{1}{T},V,N\right)\to\max\mbox{ subject to given }\frac{1}{T},V\mbox{ and }\\{B_{j}\\}\,,\\\ &\Xi\left(\frac{1}{T},\frac{P}{T},N\right)\to\max\mbox{ subject to given }\frac{1}{T},\frac{P}{T}\mbox{ and }\\{B_{j}\\}\,.\end{split}$ (3.4) In extensive thermodynamics, $V$, $U$, $S$, and $N$ are the extensive variables, that is they are directly proportional to the system volume if we just join several copies of the system with the proportional increase of the volume. Therefore, $U(S,V,N)$ is the homogeneous function of the first order, and the equation for $\mathrm{d}U$ can be easily integrated (this is the Euler theorem): $U=TS-PV+\sum_{i}\mu_{i}N_{i}\,.$ In this case, $H=U+PV=TS+\sum_{i}\mu_{i}N_{i}\,,$ $F=U-TS=-PV+\sum_{i}\mu_{i}N_{i}\,,$ $G=H-TS=\sum_{i}\mu_{i}N_{i}\,,$ $\Omega=U-TS-\sum_{i}\mu_{i}N_{i}=-PV\,,$ Free energies have a very important physical and technical sense. They measure the available work under given conditions. Free entropies coincide (up to some constant additions) with the entropies of the minimal isolated system, which includes the system under consideration. This statement was analyzed in detail in [42] but is still not very well known. Therefore, let us prove it. Physically, when we consider a system under isothermal condition, this means that the system is in contact with a large thermal bath. The state of the thermal bath is characterized by two variables, $U_{\mathrm{T}}$ and $V_{\mathrm{T}}$. The entropy of a thermal bath is $S_{\mathrm{T}}(U_{\mathrm{T}},V_{\mathrm{T}})$. The total entropy of the isolated system “a system + the thermal bath” is $\widetilde{S}(U,V,N,U_{\mathrm{T}},V_{\mathrm{T}})=S(U,V,N)+S_{\mathrm{T}}(U_{\mathrm{T}},V_{\mathrm{T}})\,.$ The equilibrium is the maximizer of the total entropy $\widetilde{S}$ for given total energy $\widetilde{U}=U+U_{\mathrm{T}}$, values of volumes $V$, $V_{\mathrm{T}}$ and linear conservation laws $\\{B_{j}\\}$. In particular, $\begin{split}&\frac{\partial[S(U,V,N)+S_{\mathrm{T}}(\widetilde{U}-U,V_{\mathrm{T}})]}{\partial U}=0\,,\\\ &\frac{\partial S(U,V,N)}{\partial U}=\left.\frac{\partial S_{\mathrm{T}}(U_{\mathrm{T}},V_{\mathrm{T}})}{\partial U}\right\|_{U_{\mathrm{T}}=\widetilde{U}-U}\,.\end{split}$ (3.5) This means that the temperatures of the thermal bath and the system are equal, $T=T_{\mathrm{T}}$. Let us take the conditional maximum function (under conditions $U+U_{\mathrm{T}}=\widetilde{U}$, $T=T_{\mathrm{T}}$: $\begin{split}S_{\Phi}(\widetilde{U},V,N,V_{\mathrm{T}})&=S(U,V,N)+S_{\mathrm{T}}(\widetilde{U}-U,V_{\mathrm{T}})\\\ &=S(U,V,N)+V_{\mathrm{T}}S_{\mathrm{T}}\left(\frac{\widetilde{U}-U}{V_{\mathrm{T}}},1\right)\\\ &=S(U,V,N)+V_{\mathrm{T}}S_{\mathrm{T}}\left(\frac{\widetilde{U}}{V_{\mathrm{T}}},1\right)-\frac{U}{T}\\\ &=\Phi+V_{\mathrm{T}}S_{\mathrm{T}}\left(\frac{\widetilde{U}}{V_{\mathrm{T}}},1\right)+O(V_{\mathrm{T}}^{-1})\end{split}$ (3.6) This function differs from the free entropy $\Phi$ by a constant $S_{\mathrm{T}}(\widetilde{U},V_{\mathrm{T}})$ and an infinitesimal $O(V_{\mathrm{T}}^{-1})$, which goes to zero when the bath increases. Therefore, the free entropy $\Psi$ (the Massieu function) is equal to the entropy of the minimal isolated system, which includes the system under consideration and the large thermal bath (up to a large constant and an infinitesimal additions). For the Planck function $\Xi$ we have to consider a system under a constant pressure in the contact with the same large thermal bath. The only difference is that instead of the internal energy of our system we have to take the enthalpy. The enthalpy is the energy of the system plus the device, which keeps the pressure constant (potential energy of a heavy piston of the given weight). So, the total energy is the energy of the minimal isolated system $\widetilde{U}=H+U_{\mathrm{T}}$ and everything else is the same as for $\Phi$: the free entropy is the entropy of the minimal isolated system, which includes the system under consideration, under condition of the partial equilibrium with auxiliary systems and up to a constant summand. For perfect systems, by definition, $\begin{split}&U=\sum_{i}N_{i}u_{i}(T)\,,\\\ &PV=RT\sum_{i}N_{i}\,,\end{split}$ (3.7) where $u_{i}(T)$ is the energy of one mole of $A_{i}$ at the temperature $T$. Under this assumption, the entropy $S$ is defined up to an arbitrary uniform function of first order $S_{0}(N)$: $S=RS_{0}(N)+\sum_{i}N_{i}\left[-R\ln c_{i}+\int_{T_{0}}^{T}\frac{1}{\tau}\frac{\mathrm{d}u_{i}(\tau)}{\mathrm{d}\tau}\,\mathrm{d}\tau\right]\,.$ (3.8) If we assume that $S_{0}(N)$ is a linear function, $S_{0}(N)=\sum_{i}\delta_{i}N_{i}$ then $S=\sum_{i}N_{i}\left[-R(\ln c_{i}-\delta_{i})+\int_{T_{0}}^{T}\frac{1}{\tau}\frac{\mathrm{d}u_{i}(\tau)}{\mathrm{d}\tau}\,\mathrm{d}\tau\right]\,,$ (3.9) where $c_{i}=N_{i}/V$ is the concentration, $T_{0}>0$ is a reference temperature (we assume that on the interval $[T_{0},T]$ the system is perfect (3.7)). It is necessary to stress that linearity of $S_{0}(N)$ does not follow from the assumption (3.7) and is an additional hypothesis. For a general perfect system (3.7) we can state that $S_{0}(N)$ is a uniform function of the first order only. Formulas (3.7), (3.9) allow us to express the free energy in the proper variables, $F(T,V,N)$: $\begin{split}F(T,V,N)&=U-TS\\\ &=\sum_{i}N_{i}u_{i}(T)+RT\sum_{i}N_{i}\left[\ln\left(\frac{N_{i}}{V}\right)-\delta_{i}-\int_{T_{0}}^{T}\frac{1}{R\tau}\frac{\mathrm{d}u_{i}(\tau)}{\mathrm{d}\tau}\,\mathrm{d}\tau\right]\,.\end{split}$ (3.10) From this formula for $F(T,V,N)$, all other thermodynamic functions can be expressed locally: $S=\frac{\partial F}{\partial T}\,,\;U=F-T\frac{\partial F}{\partial T}\,,\;P=-\frac{\partial F}{\partial V}\,,\;\mu_{i}=\frac{\partial F}{\partial N_{i}}\,,\ldots$ Generalizations of the free energy for non-perfect systems are often produced by transformations of (3.10). The first generalization describes a system of small admixtures to a general system. Let the “background” system have the extensive state variables $M$. Interaction of small admixtures is negligible and the formula $PV=RT\sum N_{i}$ describes the osmotic pressure of the admixtures.. Then we can write, analogously to (3.10): $\begin{split}F(T,V,N,M)=&F_{0}(T,V,M)+\sum_{i}N_{i}u_{i}\left(T,\frac{M}{V}\right)\\\ &+RT\sum_{i}N_{i}\left[\ln\left(\frac{N_{i}}{V}\right)-\delta_{i}\left(\frac{M}{V}\right)-\int_{T_{0}}^{T}\frac{1}{R\tau}\frac{\mathrm{d}u_{i}\left(\tau,\frac{M}{V}\right)}{\mathrm{d}\tau}\,\mathrm{d}\tau\right]\,.\end{split}$ (3.11) Here, the energies $u_{i}$ and parameters $\delta_{i}$ are functions of densities $M/V$. For each given value of these densities, this formula coincides with (3.10). The first model of non-perfect gases is the van der Waals gas. To write the free energy for this type of gas (or gas of admixtures) we have to take into account two effects: the excluded volume per mole of particles $A_{i}$, $v_{i}$, and the energy of attraction for particles $A_{i}$, $A_{j}$ with the energy density $\epsilon_{ij}=-a_{ij}c_{i}c_{j}\,$ (minus because this is attraction). For the free energy these effects give: $\begin{split}F(T,V,N)=&\sum_{i}N_{i}u_{i}(T)-V\sum_{ij}a_{ij}\frac{N_{i}}{V}\frac{N_{j}}{V}\\\ &+RT\sum_{i}N_{i}\left[\ln\left(\frac{N_{i}}{V-\sum_{i}v_{i}N_{i}}\right)-\delta_{i}-\int_{T_{0}}^{T}\frac{1}{R\tau}\frac{\mathrm{d}u_{i}(\tau)}{\mathrm{d}\tau}\,\mathrm{d}\tau\right]\,.\end{split}$ (3.12) For adsorbed particles on a surface, the model of an ideal adsorbed layer implies a lattice of places, and a multicomponent gas with components $A_{0}=Z,A_{1},\ldots,A_{n}$ where $Z$ ia a free place and $A_{i}$ are adsorbed particles on their places (each adsorbed particle occupies a place). There is a conservation law: $\sum_{i=0}^{n}c_{i}=\theta=\mathrm{const}\,,$ therefore, $c_{0}=\theta-\sum_{i=1}^{n}c_{i}$ and the free energy has the form of the energy of the Fermi-gas: $\begin{split}F(T,\sigma,N)=&F_{0}(T)\sigma+\sum_{i=1}^{n}N_{i}u_{i}(T)\\\ &+RT\sum_{i=1}^{n}N_{i}\left[\ln\left(\frac{N_{i}}{\sigma}\right)-\delta_{i}-\int_{T_{0}}^{T}\frac{1}{R\tau}\frac{\mathrm{d}u_{i}(\tau)}{\mathrm{d}\tau}\,\mathrm{d}\tau\right]\\\ &+\left(\sigma\theta-\sum_{i=1}^{n}N_{i}\right)\left(\ln\left(\theta-\sum_{i=1}^{n}\frac{N_{i}}{\sigma}\right)-\delta_{0}\right)\,,\end{split}$ (3.13) where $\sigma$ is the surface area, $F_{0}(T)\sigma$ is the free energy of empty surface. For the systems, distributed in space, the density of free energy may be expressed through the concentrations, for example, for the perfect system (3.10) $\begin{split}f(T,c)&=\frac{F(T,V,cV)}{V}\\\ &=\sum_{i}c_{i}u_{i}(T)+RT\sum_{i}c_{i}\left[\ln c_{i}-\delta_{i}-\int_{T_{0}}^{T}\frac{1}{R\tau}\frac{\mathrm{d}u_{i}(\tau)}{\mathrm{d}\tau}\,\mathrm{d}\tau\right]\,.\end{split}$ (3.14) Therefore, for non-uniform system $F=\int_{V}f(T(x),c(x))\,\mathrm{d}x\,.$ (3.15) This formula is applicable if the space gradients are not too sharp. If $f(T,c)$ is a convex function of $c$ then the minimizer of $F$ (3.15) under given $T$, $V$ and $N=\int_{V}c(x\,\mathrm{d}x$ (i.e. the equilibrium) is a uniform distribution and we should not expect spontaneous appearance of singularities on the way to the equilibrium. If $f(T,c)$ is not a convex function of $c$ then the minimizer of $F$ may be nonuniform, non-smooth and non-unique: phase transitions are possible. In that case, the simple integral of the density $f$ should be regularized by addition terms. Now, the standard approach gives the Ginzburg–Landau free energy: $\begin{split}&F=\int_{V}\psi(T(x),c(x),\nabla c(x))\,\mathrm{d}x\,,\\\ &\psi(T(x),c(x),\nabla c(x))=f(T(x),c(x))+\frac{1}{2}\sum_{i}\lambda_{i}(\nabla c_{i})^{2}\end{split}$ (3.16) More general dependencies of $\nabla c$ are also under consideration [58]. The chemical potentials $\mu_{i}$ for the free energy (3.16) are defined as variational derivatives of this functional, $\mu_{i}=\frac{\delta F}{\delta c_{i}}=\frac{\partial f(T,c)}{\partial c_{i}}-\lambda_{i}\Delta c_{i}\,.$ (3.17) Special analysis of the most general form of diffusion equations which provide the proper decrease of the Ginzburg–Landau free energy (3.16) was provided by Gurtin [58]. He introduced general nonlinear mobility matrices. The kinetic laws should satisfy the thermodynamic restrictions. That is, the dissipation should be positive. There are two physical forms of this law: (i) the available work should decrease and (ii) the entropy of the minimal isolated system, which includes the system under consideration, should increase. These two equivalent formulations correspond to two series of thermodynamic potentials: energetic or entropic series. There are several approaches to a general formalisms, which pretend to describe all systems that satisfy these monotonicity conditions [56, 87]. Such approaches form the special discipline, nonequilibrium thermodynamics [20, 59, 72] or beyond equilibrium thermodynamics [85]. Our goal is different: we construct a method for assembling of a complex transport process from a mechanism combined by simple elementary processes. This approach for physics should satisfy the basic thermodynamic requirements. #### 3.1.2. Markovian Microkinetics and Generalized Mass Action Law To satisfy the thermodynamic restrictions the kinetic law of the elementary reactions should have a special form, and the reaction rate constants for different elementary reactions should be harmonized. Let us start from a reaction mechanism given by the stoichiometric equations (2.1): $\sum_{i}\alpha_{ri}A_{i}\to\sum_{i}\beta_{ri}A_{i}\,,$ (3.18) where $r$ is a reaction number, $\alpha_{ri}$ and $\beta_{ri}$ are nonnegative numbers, the stoichiometric coefficients. In this Subsection, we return to some ideas of Michaelis and Menten [80] and to the Stueckelberg analysis of the Boltzmann equation [97] and represent the general kinetic law for elementary processes. Detailed analysis was provided recently in [54]. This law could be proved under the following assumptions: 1. 1. The elementary processes go through intermediate states (complexes or compounds) 1.44 (Fig. 5): $\sum_{i}\alpha_{ri}A_{i}\rightleftharpoons B_{r}^{-}\to B_{r}^{+}\rightleftharpoons\sum_{i}\beta_{ri}A_{i}\,,$ (3.19) where $\rho$ is the elementary process number; 2. 2. The amount of each compound $B_{\rho}$ is small enough to apply the perfect free energy formula (3.11) for them; 3. 3. The equilibrium between each compound and the corresponding linear combinations of reagents is fast enough to apply the quasiequilibrium approximation [54]; 4. 4. The transitions between compounds could be described by a continuous time Markov chain (the Master equation or the monomolecular kinetics). The first three items of these assumptions correspond exactly to the celebrated Michaelis–Menten work [80]. Later, Briggs and Haldane [7] abandoned the assumption of fast equilibria and produced the so-called Michaelis and Menten kinetics approximation. Original approach of Michaelis and Menten (fast equilibria with intermediate compounds + small amounts of compounds) was discovered again by Stueckelberg [97] almost 30 years later. It was applied not to the kinetics of catalytic reactions but to the collision in the gas kinetics, as an alternative background of the Boltzmann equation. Gorban and Shahzad [54] provided the detailed analysis of this approach to the general kinetic equation. Let the concentration of the intermediate compound $B_{\rho}$ (1.44) be $\varsigma_{\rho}$. The free energy (3.11) for the small admixture of compounds $B_{\rho}$ to the components $A_{i}$ may be represented in the form $F=Vf(c,T)+VRT\sum_{{\rho}=1}^{q}\varsigma_{\rho}\left(\ln\left(\frac{\varsigma_{\rho}}{\varsigma_{\rho}^{}(c,T)}\right)-1\right)\,,$ (3.20) where $\varsigma_{\rho}$ is the standard equilibrium concentrations of $B_{\rho}$. We assume that the standard equilibrium concentrations $\varsigma_{\rho}^{}(c,T)$ as well as the current concentrations $\varsigma_{\rho}$ are much smaller than the concentrations of $A_{i}$. To formulate the results of Michaelis–Menten–Stueckelberg–Gorban–Shahzad (MMSGS kinetics) we have to introduce the basic notions in more detail. First of all, some of the formal linear combinations $(\alpha_{r},A)=\sum_{i}\alpha_{ri}A_{i}\,,\;(\beta_{r},A)=\sum_{i}\beta_{ri}A_{i}\,$ may coincide. The same combination may be, simultaneously, the input combination of several reactions and the output combination of several other reactions. We assume that a fast intermediate compound $B_{\cdots}$ corresponds not to a reaction but to a formal complex of the form $(\alpha_{r},A)$ or $(\beta_{r},A)$ and this compound is the same for all reactions which include this complex. Let us call the formal linear combinations of the form $(\alpha_{r},A)$ or $(\beta_{r},A)$ the complexes and enumerate them: $\Theta_{1}$, $\Theta_{2}$, … , $\Theta_{q}$. For each complex $\Theta_{j}$, the corresponding vector of coefficients ($\alpha_{r}$ or $\beta_{r}$) is $y_{j}$: $\Theta_{j}=(y_{j},A)$. The reaction mechanism (3.18) may be represented in the form $\Theta_{j}\to\Theta_{s}$ for some pairs $(j,s)$. The additional component, the fast compound $B_{j}$, corresponds to each complex $\Theta_{j}$ and the reaction mechanism $\Theta_{j}\to\Theta_{s}$ (for some pairs $(j,s)$) (3.18) is extended to $\Theta_{j}\rightleftharpoons B_{j}\to B_{s}\rightleftharpoons\Theta_{s}\mbox{ for some pairs }(j,s)\,.$ (3.21) The fast equilibrium $\Theta_{j}\rightleftharpoons B_{j}$ gives $\vartheta_{j}=\sum_{i}y_{ji}\frac{\mu_{i}(c,T)}{RT}\,,$ (3.22) or $\varsigma_{j}=\varsigma^{}_{j}(c,T)\exp\left(\frac{\sum_{i}y_{ji}\mu_{i}(c,T)}{RT}\right)\,,$ (3.23) where $\mu_{i}=\frac{\partial f(c,T)}{\partial c_{i}}$ is the chemical potential of $A_{i}$ and $\vartheta_{j}=\ln\left(\frac{\varsigma_{j}}{\varsigma^{}_{j}}\right)$ ($RT\vartheta_{j}=\frac{1}{V}\frac{\partial F}{\partial\varsigma_{j}}$ is the chemical potential of $B_{j}$). For the systems with fixed volume, the stoichiometric conservation laws of the monomolecular system of reaction between compounds are sums of the concentrations of $B_{j}$ which belong to various connected components of the reaction graph. Under the hypothesis of weak reversibility there is no other conservation law. Let the graph of reactions $B_{j}\to B_{l}$ have $d$ connected components $C_{s}$ and let $V_{s}$ be the set of indexes of those $B_{j}$ which belong to $C_{s}$: $B_{j}\in C_{s}$ if and only if $j\in V_{s}$. For each $C_{s}$ there exists a stoichiometric conservation law $\beta_{s}=\sum_{j\in V_{s}}\varsigma_{j}\,.$ (3.24) For any set of positive values of $\beta_{s}$ ($s=1,\ldots,d$) and given $c,T$ there exists a unique conditional maximizer $\varsigma^{\mathrm{eq}}_{j}$ of the free energy (3.20): for the compound $B_{j}$ from the $s$th connected component ($j\in V_{s}$) this equilibrium concentration is $\varsigma^{\mathrm{eq}}_{j}=\beta_{s}\frac{\varsigma_{j}^{}(c,T)}{\sum_{l\in V_{s}}\varsigma_{j}^{}(c,T)}$ (3.25) Inversely, positive values of concentrations $\varsigma_{j}$ are the equilibrium concentrations (3.25) for some values of $\beta_{s}$ if and only if for any $s=1,\ldots,d$ $\vartheta_{j}=\vartheta_{l}\;\;{\mathrm{if}}\;\;j,l\in V_{s}\,$ (3.26) ($\vartheta_{j}=\ln(\varsigma_{j}/\varsigma_{j}^{})$). This system of equations together with the equilibrium conditions (3.23) constitute the equilibrium of the systems. All the equilibria form a linear subspace in the space with coordinates $\mu_{i}/RT$ ($i=1,\ldots,n$) and $\vartheta_{j}$ ($j=1,\ldots,q$). Our expression for the free energy does not assume anything special about the free energy of the mixture of $A_{i}$. For the compounds $B_{i}$, we assume that this is a very small addition to the mixture of $A_{i}$, neglect all quadratic terms in concentrations of $B_{i}$ and use the free energy of the perfect systems for this small admixture. The Master Equation for the concentration of $B_{j}$ gives: $\frac{\mathrm{d}\varsigma_{j}}{\mathrm{d}t}=\sum_{l,\,l\neq j}\left(\kappa_{jl}\varsigma_{l}-\kappa_{lj}\varsigma_{j}\right)\,.$ (3.27) This kinetic equation should respect thermodynamics. For the Master equation this means the equilibrium condition $\sum_{l,\,l\neq j}\kappa_{jl}\varsigma_{l}^{}=\sum_{l,\,l\neq j}\kappa_{lj}\varsigma_{j}^{}\,.$ (3.28) Under this condition, the Master Equation (3.27) has the equivalent form: $\frac{\mathrm{d}\varsigma_{j}}{\mathrm{d}t}=\sum_{l,\,l\neq j}\kappa_{jl}\varsigma_{l}^{}\left(\frac{\varsigma_{l}}{\varsigma_{l}^{}}-\frac{\varsigma_{j}}{\varsigma_{j}^{}}\right)\,.$ (3.29) In this form, it is obvious that $\varsigma_{j}^{}$ is equilibrium for the kinetic equation. All these expressions for concentrations of compounds and the Markov reaction rates result in the following kinetic law: 1. 1. The reaction rate of the reaction $\Theta_{j}\to\Theta_{s}$ is $w_{sj}=\phi_{sj}\exp\left(\frac{\sum_{i}y_{ji}\mu_{i}(c,T)}{RT}\right)\,,$ (3.30) where the quantity $\phi_{sj}\geq 0$ is a kinetic factor, in the Markov model it corresponds to $\kappa_{sj}\varsigma_{j}^{}$; 2. 2. The kinetic factors $\phi_{sj}$ satisfy the complex balance condition: $\sum_{s}\phi_{js}=\sum_{s}\phi_{sj}\mbox{ for all }j\,,$ (3.31) in the Markov model this identity corresponds to the equilibrium condition (3.28). This is the macroscopic MMSGS kinetics. For this kinetics, the free energy decreases in time. To demonstrate this dissipation inequality, let us formulate the macroscopic MMSGS kinetic equations in the original notations for the reaction mechanism (3.18). Formula (3.30) for the reaction rate gives $w_{r}=\phi_{r}\exp\left(\frac{\sum_{i}\alpha_{ri}\mu_{i}(c,T)}{RT}\right)\,,$ (3.32) The kinetic equation under the isochoric conditions (the constant volume) are $\frac{\mathrm{d}c}{\mathrm{d}t}=\sum_{r}\gamma_{r}w_{r}\,,$ (3.33) where the stoichiometric vector $\gamma_{r}=\beta_{r}-\alpha_{r}$. According to this equation, the dissipation rate is $\frac{\mathrm{d}F}{\mathrm{d}t}=V\sum_{r}(\mu,\gamma_{r})w_{r}=V\sum_{r}\phi_{r}(\mu,(\beta_{r}-\alpha_{r}))\exp\left(\frac{(\alpha_{r},\mu)}{RT}\right)\,,$ (3.34) where $\mu_{i}=\partial F/\partial N_{i}$ is the chemical potential. Let us introduce an auxiliary function like we did it for MAL (2.17): $\theta(\lambda)=\sum_{r}\phi_{r}\exp\left[\frac{\lambda(\alpha_{r},\mu)+(1-\lambda)(\beta_{r},\mu)}{RT}\right]\,.$ (3.35) This function is convenient because $\frac{\mathrm{d}\theta(\lambda)}{\mathrm{d}\lambda}=-\sum_{r}\phi_{r}(\mu,(\beta_{r}-\alpha_{r}))\exp\left[\frac{\lambda(\alpha_{r},\mu)+(1-\lambda)(\beta_{r},\mu)}{RT}\right]\,.$ (3.36) Therefore, for the dissipation rate we get $\frac{\mathrm{d}F}{\mathrm{d}t}=-V\theta^{\prime}(1)$ (3.37) The function $\theta(\lambda)$ is a sum of exponents. It is convex ($\theta^{\prime\prime}(\lambda)\geq 0$). Therefore, if $\theta(0)=\theta(1)$ then $\theta^{\prime}(1)\geq 0$. This means that the identity $\theta(0)\equiv\theta(1)$ (3.38) is a sufficient condition for the dissipation inequality $\frac{\mathrm{d}F}{\mathrm{d}t}\leq 0\,.$ Some of vectors $\alpha_{r}$, $\beta_{r}$ may coincide. Let there be $q$ different vectors among them. We denote them $y_{1},\ldots,y_{q}$. For each $y_{i}$ we define $R_{i}^{+}=\\{r\,\|\,\alpha_{r}=y_{i}\\}$, $R_{i}^{-}=\\{r\,\|\,\beta_{r}=y_{i}\\}$. The sufficient condition for the identity (3.38) is $\sum_{r\in R_{i}^{+}}\phi_{r}=\sum_{r\in R_{i}^{-}}\phi_{r}\mbox{ for all }i\,.$ (3.39) This condition is also necessary if we can vary $\phi_{r}$ and $\mu_{i}$ independently and the Jacobian $\|\partial\mu_{i}/\partial c_{j}\|$ has the full rank. This condition is the complex balance condition. Of course, the important particular case of the complex balance conditions is the detailed balance condition: $\phi_{r}^{+}=\phi_{r}^{-}\,,$ (3.40) where $\phi_{r}^{+}$, $\phi_{r}^{-}$ are the kinetic factors of the mutually reverse reactions: $\phi_{r}^{+}$ for $(\alpha_{r},A)\to(\beta_{r},A)$ and $\phi_{r}^{-}$ for $(\beta_{r},A)\to(\alpha_{r},A)$. ### 3.2. From Cell-Jump Models to Continuous Diffusion Equations for Generalized Mass Action Law Let us start again from the stoichiometric mechanism of the diffusion: $\sum_{i}\alpha_{ri}^{\mathrm{I}}A_{i}^{\mathrm{I}}+\sum_{i}\alpha_{ri}^{\mathrm{II}}A_{i}^{\mathrm{II}}\rightarrow\sum_{i}\beta_{ri}^{\mathrm{I}}A_{i}^{\mathrm{I}}+\sum_{i}\beta_{ri}^{\mathrm{II}}A_{i}^{\mathrm{II}}\,.$ (3.41) All the elementary processes have different numbers $r$ and the space-inverted and reverse processes are represented separately. Let us consider all pairs of vectors, $(\alpha_{r}^{\mathrm{I}},\alpha_{r}^{\mathrm{II}})$ and $(\beta_{r}^{\mathrm{I}},\beta_{r}^{\mathrm{II}})$ and numerate all the different pairs: $y_{1},y_{2},\ldots$, where $y_{q}=(y_{q}^{\mathrm{I}},y_{q}^{\mathrm{II}})$. For each $y_{q}$ there are two sets of reactions, $R_{q}^{+}$, $R_{q}^{-}$: $R_{q}^{+}=\\{r\,\|\,(\alpha_{r}^{\mathrm{I}},\alpha_{r}^{\mathrm{II}})=y_{q}\\}\,,\;R_{q}^{-}=\\{r\,\|\,(\beta_{r}^{\mathrm{I}},\beta_{r}^{\mathrm{II}})=y_{q}\\}\,.$ The reaction rate for the elementary process (3.41) is (3.32): $w_{r}=\phi_{r}\exp\left(\frac{(\alpha_{r}^{\mathrm{I}},\mu(c^{\mathrm{I}},T))+(\alpha_{r}^{\mathrm{II}},\mu(c^{\mathrm{II}},T))}{RT}\right)\,,$ (3.42) The kinetic factors are functions of $c^{\mathrm{I}}$, $c^{\mathrm{II}}$ and $T$. They should satisfy the identity of complex balance (3.39). The cell–jump model gives us $\begin{split}J&=-\sum_{r}\gamma_{r}w_{r}(c(x),c(x+l))\\\ &=\sum_{q}\left(\sum_{r\in R_{q}^{+}}\alpha_{r}^{\mathrm{I}}w_{r}(c(x),c(x+l))-\sum_{r\in R_{q}^{-}}\beta_{r}^{\mathrm{I}}w_{r}(c(x),c(x+l))\right)\\\ &=\sum_{q}y_{q}^{\mathrm{I}}\left(\sum_{r\in R_{q}^{+}}w_{r}(c(x),c(x+l))-\sum_{r\in R_{q}^{-}}w_{r}(c(x),c(x+l))\right)\end{split}$ (3.43) The zero order term in $l$ is $\sum_{q}y_{q}^{\mathrm{I}}\exp\left(\frac{(y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}},\mu)}{RT}\right)\left(\sum_{r\in R_{q}^{+}}\phi_{r}-\sum_{r\in R_{q}^{-}}\phi_{r}\right)=0\,.$ It vanishes because of the complex balance condition. The first order term gives $\begin{split}J=l\sum_{q}y_{q}^{\mathrm{I}}\exp\left(\frac{(y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}},\mu)}{RT}\right)&\left(\sum_{r\in R_{q}^{+}}\phi_{r}\left(\alpha^{\mathrm{II}}_{r},\nabla\left(\frac{\mu}{RT}\right)\right)\right.\\\ &-\left.\sum_{r\in R_{q}^{-}}\phi_{r}\sum_{j}\left(\alpha^{\mathrm{II}}_{r},\nabla\left(\frac{\mu}{RT}\right)\right)\right)\,.\end{split}$ (3.44) Each term in this sum consists of the positive scalar pre-factor $l\exp\left(\frac{(y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}},\mu)}{RT}\right)\,$ and the matrix $y_{qi}^{\mathrm{I}}\left(\sum_{r\in R_{q}^{+}}\phi_{r}\alpha^{\mathrm{II}}_{rj}-\sum_{r\in R_{q}^{-}}\phi_{r}\alpha^{\mathrm{II}}_{rj}\right)\,$ multiplied by $\nabla\left(\frac{\mu_{j}}{RT}\right)\,.$ This structure of the formula for the flux is very similar to (2.55). Let us follow the same logic as in Subsection 2.5. to find the more convenient form of the expression for the flux (3.44). First of all, let us group all terms with the same pre-factor. For the set of all vectors $y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}}$ we use notation $Z$. For each $z\in Z$ $Y_{z}=\\{y_{q}\ \|\ y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}}=z\\}\,.$ Analogously to (2.56) we get $\begin{split}&J=l\sum_{z\in Z}l\exp\left(\frac{(y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}},\mu)}{RT}\right)J^{z}\,,\\\ &J^{z}=\sum_{y_{q}\in Y_{z}}y_{q}^{\mathrm{I}}\sum_{j}\left(\sum_{r\in R_{q}^{+}}\phi_{r}\alpha^{\mathrm{II}}_{rj}-\sum_{r\in R_{q}^{-}}\phi_{r}\alpha^{\mathrm{II}}_{rj}\right)\nabla\left(\frac{\mu_{j}}{RT}\right)\,.\end{split}$ (3.45) Let us analyze $J^{z}$ for given $z$. First of all, $\sum_{y_{q}\in Y_{z}}\sum_{j}\left(\sum_{r\in R_{q}^{+}}\phi_{r}\alpha^{\mathrm{II}}_{rj}-\sum_{r\in R_{q}^{-}}\phi_{r}\alpha^{\mathrm{II}}_{rj}\right)\nabla\left(\frac{\mu_{j}}{RT}\right)=0\,$ because each $\phi_{r}$ from this sum enters it twice, with the opposite signs, one time as an element of the sum over $R_{q}^{+}$ for some $y_{q}\in Y_{z}$ (with $+$) and the second time (with $-$) as an element of the sum over $R_{s}^{-}$ for another $y_{s}\in Y_{z}$ with the same sum $y_{s}^{\mathrm{I}}+y_{s}^{\mathrm{II}}=y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}}=z\,.$ (Let us recall that $\alpha_{s}^{\mathrm{I}}+\alpha_{s}^{\mathrm{II}}=\beta_{s}^{\mathrm{I}}+\beta_{s}^{\mathrm{II}}$ for all $s$ and, hence, if $r\in R_{q}^{+}$ and $\alpha_{r}^{\mathrm{I}}+\alpha_{r}^{\mathrm{II}}=z$ then $\beta_{s}^{\mathrm{I}}+\beta_{s}^{\mathrm{II}}=z$ and $(\beta_{s}^{\mathrm{I}},\beta_{s}^{\mathrm{II}})\in Y_{z}$.) Let us mention that $y_{q}^{\mathrm{I}}=z-y_{q}^{\mathrm{II}}$. Therefore, $J^{z}=-\sum_{y_{q}\in Y_{z}}y_{q}^{\mathrm{II}}\sum_{j}\left(\sum_{r\in R_{q}^{+}}\phi_{r}\alpha^{\mathrm{II}}_{rj}-\sum_{r\in R_{q}^{-}}\phi_{r}\alpha^{\mathrm{II}}_{rj}\right)\nabla\left(\frac{\mu_{j}}{RT}\right)\,.$ (3.46) In this formula, all the stoichiometric vectors are for the same cell, for the second one. Let us demonstrate similarity of the expression (3.46) to some formulas from the theory of Markov chains. Let us follow Subsection 2.5. and numerate the states of the auxiliary Markov chain by vectors $y_{q}\in Y_{z}$. Elementary processes correspond to transitions between states. Each nonzero kinetic factor $\phi_{r}$ corresponds to two vectors $y_{q},y_{s}\in Y_{z}$: ${r\in R_{q}^{+}}$ and ${r\in R_{s}^{-}}$. Let us substitute the index $r$ by two indexes $q,s$ and use notation $\phi_{sq}$ for transitions $y_{q}\to y_{s}$ (i.e. for the case ${r\in R_{q}^{+}}$ and ${r\in R_{s}^{-}}$). The complex balance condition (3.39) reads: $\sum_{q}\phi_{sq}=\sum_{q}\phi_{qs}\,.$ (3.47) This is precisely the steady state condition for the Markov chain. Inequality (2.64) holds for the kinetic factors: $\sum_{sq}\pi_{q}(\phi_{qs}\pi_{s}-\phi_{sq}\pi_{q})=\sum_{sq}\pi_{q}\phi_{qs}(\pi_{s}-\pi_{q})\leq 0\,$ (3.48) for any set of numbers $\pi_{q}$. For the proof of the dissipation inequality it is convenient to rewrite $J^{z}$ in these notations: $J^{z}=\sum_{y_{q}\in Y_{z}}y_{q}^{\mathrm{II}}\sum_{j}\sum_{s}(\phi_{qs}{y_{sj}^{\mathrm{II}}}-\phi_{sq}{y_{qj}^{\mathrm{II}}})\nabla\left(\frac{\mu_{j}}{RT}\right)\,.$ (3.49) Free energy in a domain $V$ is $\mathbf{F}=\int_{V}f(c,T)\,\mathrm{d}x\,.$ Let us estimate $\dot{\mathbf{F}}$ in a bounded domain $V$ with smooth boundary and without fluxes through boundary for isothermal conditions. $\begin{split}\frac{\mathrm{d}\mathbf{F}}{\mathrm{d}t}&=RT\int_{V}\sum_{j}\left(\nabla_{x}\left(\frac{\mu_{j}}{RT}\right),J_{j}\right)\,\mathrm{d}x\\\ &=l\sum_{z\in Z}\int_{V}\exp\left(\frac{(y_{q}^{\mathrm{I}}+y_{q}^{\mathrm{II}},\mu)}{RT}\right)\sum_{j}\left(\nabla_{x}\left(\frac{\mu_{j}}{RT}\right),J_{j}^{z}\right)\,\mathrm{d}x\,.\end{split}$ (3.50) Due to the representation of $J^{z}$ (3.49), $\sum_{j}\left(\nabla_{x}\left(\frac{\mu_{j}}{RT}\right),J_{j}^{z}\right)=\sum_{q\,s}(\pi_{q},(\phi_{qs}\pi_{s}-\phi_{sq}\pi_{q}))\leq 0\,,$ (3.51) where $\pi_{q}$ is a space vector:$\pi_{q}=\sum_{i}y_{qi}^{\mathrm{II}}\nabla_{x}\left(\frac{\mu_{j}}{RT}\right)$ and for $y_{q}$, $y_{s}$ $y_{q,s}^{\mathrm{I}}+y_{q,s}^{\mathrm{II}}=z\,.$ This expression has exactly the form (3.48) for each space coordinate. Finally, $\sum_{j}\left(\nabla_{x}\left(\frac{\mu_{j}}{RT}\right),J_{j}^{z}\right)\leq 0$ (3.52) and it is zero if all $\pi_{q}=\sum_{i}y_{qi}^{\mathrm{II}}\nabla\ln c_{i}$ coincide. (The reverse statement is correct if the auxiliary Markov chain with the transition coefficients $\phi_{qs}$ is ergodic for given $z$.) Therefore, $\mathbf{\dot{F}}\leq 0\,$ for the generalized MAL (3.30) with the complex balance conditions (3.31). ### 3.3. Detailed Balance for the Generalized Mass Action Law and the Dissipation Inequality #### 3.3.1. Isothermal conditions Systems with detailed balance form an important subclass of the generalized MAL systems. Let us join each elementary process with its reverse process and represent the mechanism of diffusion by the pairs of mutually reverse processes: $\sum_{i}\alpha_{ri}^{\mathrm{I}}A_{i}^{\mathrm{I}}+\sum_{i}\alpha_{ri}^{\mathrm{II}}A_{i}^{\mathrm{II}}\rightleftharpoons\sum_{i}\beta_{ri}^{\mathrm{I}}A_{i}^{\mathrm{I}}+\sum_{i}\beta_{ri}^{\mathrm{II}}A_{i}^{\mathrm{II}}\,.$ (3.53) All the quantities for the direct process we mark by the upper index + and for the reverse process by the upper index -. The simple algebraic relations hold (see also (2.4.1.): $\alpha_{ri}^{\mathrm{I,II}\mp}=\beta_{ri}^{\mathrm{I,II}\pm}\,\mbox{ and }\gamma_{ri}^{\mathrm{I,II}\mp}=-\gamma_{ri}^{\mathrm{I,II}\pm}\,.$ (3.54) Therefore, $\alpha_{ri}^{\mathrm{I+}}+\alpha_{ri}^{\mathrm{II+}}=\beta_{ri}^{\mathrm{I-}}+\beta_{ri}^{\mathrm{II-}}=\alpha_{ri}^{\mathrm{I-}}+\alpha_{ri}^{\mathrm{II-}}=\beta_{ri}^{\mathrm{I+}}+\beta_{ri}^{\mathrm{II+}}\,.$ (3.55) Due to the principle of detailed balance, $\phi_{r}^{+}=\phi_{r}^{-}=\phi_{r}\,.$ The reaction rates are $\begin{split}&w_{r}^{+}(c^{\mathrm{I}},c^{\mathrm{II}})=\phi_{r}\exp\left(\sum_{i}{\alpha^{\mathrm{I}}_{ri}}\frac{\mu_{i}^{\mathrm{I}}}{RT}+\sum_{i}{\alpha^{\mathrm{II}}_{ri}}\frac{\mu_{i}^{\mathrm{II}}}{RT}\right)\,,\\\ &w_{r}^{+}(c^{\mathrm{I}},c^{\mathrm{II}})=\phi_{r}\exp\left(\sum_{i}{\beta^{\mathrm{I}}_{ri}}\frac{\mu_{i}^{\mathrm{I}}}{RT}+\sum_{i}{\beta^{\mathrm{II}}_{ri}}\frac{\mu_{i}^{\mathrm{II}}}{RT}\right)\,,\\\ \end{split}$ (3.56) The cell–jump model gives us (3.43) $\begin{split}J&=\sum_{r}J_{r}=-\sum_{r}\gamma_{r}(w_{r}^{+}(c^{\mathrm{I}},c^{\mathrm{II}})-w_{r}^{-}(c^{\mathrm{I}},c^{\mathrm{II}}))\\\ &=\sum_{r}\gamma_{r}\phi_{r}\left(\exp\left(\sum_{i}{\alpha^{\mathrm{I}}_{ri}}\frac{\mu_{i}^{\mathrm{I}}}{RT}+\sum_{i}{\alpha^{\mathrm{II}}_{ri}}\frac{\mu_{i}^{\mathrm{II}}}{RT}\right)\right.\\\ &\qquad\qquad\left.-\exp\left(\sum_{i}{\beta^{\mathrm{I}}_{ri}}\frac{\mu_{i}^{\mathrm{I}}}{RT}+\sum_{i}{\beta^{\mathrm{II}}_{ri}}\frac{\mu_{i}^{\mathrm{II}}}{RT}\right)\right)\,.\end{split}$ (3.57) In the first order approximation, we get analogously to (2.41): $\begin{split}&J_{ri}=-l\phi_{r}\exp\left(\sum_{j}(\alpha^{\mathrm{I}}_{rj}+\alpha^{\mathrm{II}}_{rj})\frac{\mu_{j}^{\mathrm{II}}}{RT}\right)\sum_{j}\gamma_{ri}\gamma_{rj}\nabla\left(\frac{\mu_{j}^{\mathrm{II}}}{RT}\right)\,,\\\ &J_{r}=-l\phi_{r}\exp\left(\frac{(\alpha^{\mathrm{I}}_{r}+\alpha^{\mathrm{II}}_{r},\mu)}{RT}\right)\gamma_{r}\left(\gamma_{r},\nabla\left(\frac{\mu}{RT}\right)\right)\,.\end{split}$ (3.58) This expression for $J_{r}$ has the form of the multicomponent Teorell formula with the symmetric matrix of coefficients. This symmetry for nonlinear diffusion gives us the generalization of the Onsager reciprocal symmetry [83, 84]. We represented the nonlinear multicomponent diffusion a sum of elementary processes. For each elementary process $J_{ri}=-d_{r}\sum_{j}\gamma_{ri}\gamma_{rj}\nabla\left(\frac{\mu_{j}^{\mathrm{II}}}{RT}\right)\,,$ (3.59) where the scalar coefficient $d_{r}=l\phi_{r}\exp\left(\sum_{j}(\alpha^{\mathrm{I}}_{rj}+\alpha^{\mathrm{II}}_{rj})\frac{\mu_{j}^{\mathrm{II}}}{RT}\right)>0$ is, from the thermodynamic point of view, an almost arbitrary positive quantity (because it includes the kinetic factor $\phi_{r}$). “Almost” here means that some conditions of zero values (and the orders of these zeros) at the boundary when some of $c_{i}=0$ are prescribed by the factor $\exp\left(\sum_{j}(\alpha^{\mathrm{I}}_{rj}+\alpha^{\mathrm{II}}_{rj})\frac{\mu_{j}^{\mathrm{II}}}{RT}\right)$ and the logarithmic singularity of $\mu_{i}$ when $c_{i}\to 0$. The internal symmetry of this formula makes the dissipation inequality obvious: in a bounded domain $V$ with smooth boundary and without fluxes through boundary for isothermal conditions $\begin{split}\frac{\mathrm{d}\mathbf{F}}{\mathrm{d}t}&=RT\int_{V}\sum_{j}\left(\nabla_{x}\left(\frac{\mu_{j}}{RT}\right),J_{j}\right)\,\mathrm{d}x\\\ &=-l\sum_{r}\phi_{r}\exp\left(\frac{(\alpha^{\mathrm{I}}_{r}+\alpha^{\mathrm{II}}_{r},\mu)}{RT}\right)\left(\gamma_{r},\nabla\left(\frac{\mu}{RT}\right)\right)^{2}\leq 0\,.\end{split}$ (3.60) #### 3.3.2. Generalization: Non-isothermal Processes Extension of the generalized MAL (3.30) on the non-isothermal processes is quite simple. Let us follow the paper [10] and include the “energetic component” $A_{U}$ in the list of components. Instead of the stoichiometric equations (2.1), (3.18) we get: $\sum_{i}\alpha_{ri}A_{i}+\alpha_{Q}A_{U}\to\sum_{i}\beta_{ri}A_{i}+\beta_{Q}A_{U}\,,$ (3.61) The corresponding macroscopic extensive variable for $A_{U}$ is the internal energy $U$ with the density (“concentration”) $u$. To consider energy as the additional extensive variable, we should take the main thermodynamic potential for the isolated system from the entropic series. This is the entropy $S$: $\mathrm{d}S=\frac{1}{T}\mathrm{d}U+\frac{P}{T}\mathrm{d}V-\sum_{i}\frac{\mu_{i}}{T}\mathrm{d}N_{i}\,.$ Let us extend the formulas for the generalized kinetics by additional component and take $-\frac{1}{RT}$ instead of $\frac{\mu_{i}}{RT}$. All the formulas including the dissipation inequalities remain the same. In isolated (isochoric) systems, $\dot{U}=0$ and $\dot{N}_{i}=\sum_{r}\gamma_{ri}w_{r}\,,$ where $w_{r}=\phi_{r}\exp\left[\left(\alpha_{r},\frac{\mu}{RT}\right)-\frac{\alpha_{Q}}{RT}\right]\,.$ For transport processes, conservation of energy gives the following relations: $\alpha_{Q}^{\mathrm{I}}+\alpha_{Q}^{\mathrm{II}}=\beta_{Q}^{\mathrm{I}}+\beta_{Q}^{\mathrm{II}}\,.$ The space gradient of $-\frac{1}{RT}$ enters the multicomponent Teorell formula as an additional force and the gradients of $\frac{\mu_{i}}{RT}$ also enter the formulas for the heat flux. In particular, the simplest mechanism of transport, $\alpha_{Q}A_{U}^{\mathrm{I}}\rightleftharpoons\alpha_{Q}A_{U}^{\mathrm{II}}\,,$ generates the Fourier law: $J_{Q}=-l\alpha_{Q}\phi\exp\left(-\frac{\alpha_{Q}}{RT}\right)\nabla\left(-\frac{1}{RT(x)}\right)=-\lambda(T)\nabla T\,.$ The thermodynamic consideration cannot produce the temperature dependence of the thermal conductivity $\lambda(T)>0$. From the thermodynamic point of view, $\phi$ here is an arbitrary positive quantity. The problem of temperature dependence of $\lambda$ and its relations with other constants like diffusivity is widely discussed from the kinetic point of view [77]. For computing thermal conductivity various methods were developed including direct simulation and the Green–Kubo approach [63]. These methods were compared in [93]. Thermodynamics may give the relations between different coefficients. For example, the principle of detailed balance produces the multicomponent Teorell formula with the symmetric matrix of coefficients (3.58). The nonlinear reciprocal relations (3.59) could be automatically extended to the heat flux: just use the heat flux $J_{Q}$ as additional flux and $-1/RT$ instead $\mu_{i}/RT$ for the component $A_{U}$. More relations we get for the specific mechanisms of transport. For example, for the activation mechanism of diffusion $A^{\mathrm{I}}+\alpha_{Q}A_{U}^{\mathrm{I}}\rightleftharpoons A^{\mathrm{II}}+\alpha_{Q}A_{U}^{\mathrm{II}}$ (3.62) the fluxes are: $\begin{split}&J_{A}=-l\phi\exp\left(\frac{\mu-\alpha_{Q}}{RT}\right)\left[\nabla\left(\frac{\mu}{RT}\right)+\alpha_{Q}\nabla\left(-\frac{1}{RT}\right)\right]\,,\\\ &J_{Q}=-l\phi\exp\left(\frac{\mu-\alpha_{Q}}{RT}\right)\left[\alpha_{Q}\nabla\left(\frac{\mu}{RT}\right)+\alpha_{Q}^{2}\nabla\left(-\frac{1}{RT}\right)\right]\,,\end{split}$ (3.63) or in the matrix form $\left(\begin{array}[]{l}J_{A}\\\ J_{Q}\end{array}\right)=-l\phi\exp\left(\frac{\mu-\alpha_{Q}}{RT}\right)\left(\begin{array}[]{cc}1&\alpha_{Q}\\\ \alpha_{Q}&\alpha_{Q}^{2}\end{array}\right)\left(\begin{array}[]{l}\nabla\left(\frac{\mu}{RT}\right)\\\ \nabla\left(-\frac{1}{RT}\right)\end{array}\right)\,.$ (3.64) For the mechanism (3.62), the heat flux $J_{Q}$ is proportional to the diffusion flux $J_{A}$ with the coefficient $\alpha_{Q}$, that is the activation heat. In this activation mechanism, the activation heat travels with the particle from the cell I to the cell II. We can, for example, assume different behavior of the activation heat: let $\beta_{Q}$ distribute symmetrically after the diffusion jump: $A^{\mathrm{I}}+\alpha_{Q}A_{U}^{\mathrm{I}}\rightleftharpoons A^{\mathrm{II}}+\frac{1}{2}\alpha_{Q}A_{U}^{\mathrm{I}}+\frac{1}{2}\alpha_{Q}A_{U}^{\mathrm{II}}\,.$ (3.65) For this mechanism, $\gamma_{U}=-\frac{1}{2}\alpha_{U}$ and $\left(\begin{array}[]{l}J_{A}\\\ J_{Q}\end{array}\right)=-l\phi\exp\left(\frac{\mu-\alpha_{Q}}{RT}\right)\left(\begin{array}[]{cc}1&\frac{1}{2}\alpha_{Q}\\\ \frac{1}{2}\alpha_{Q}&\frac{1}{4}\alpha_{Q}^{2}\end{array}\right)\left(\begin{array}[]{l}\nabla\left(\frac{\mu}{RT}\right)\\\ \nabla\left(-\frac{1}{RT}\right)\end{array}\right)\,.$ (3.66) The heat flux $J_{Q}$ should be supplemented by the heat transport together with particles, $\sum_{i}\mu_{i}J_{i}$, [70]. The total heat flux is $\mathbf{q}=J_{U}=J_{Q}+\sum_{i}\mu_{i}J_{i}\,.$ (3.67) To describe the energy balance properly we have to include the work of various forces. The proper framework for modeling of the energy transport gives continuum mechanics. In its simplest form, fluid mechanics, we present these equation in the next Section. ### 3.4. Momentum and Center of Mass Conservation #### 3.4.1. Mass Transfer and Heat Transfer In this subsection, we briefly discuss coupling of the diffusion and thermal conductivity with fluid dynamics. The heat and mass transfer should satisfy the general laws of mechanics and, in particular, does not violate the Newton laws. The diffusion and heat transfer equations do not present the complete theory and should be included into the context of continuum mechanics. First of all, let us introduce the mass average velocity. Let $m_{i}$ be the mass of mole (gram-molecule) for the component $A_{i}$. For each diffusion flux $J_{i}$ the associated flux of mass is $m_{i}J_{i}$. We introduced the fluxes $J_{i}$ with respect to a frame, which is connected to our cells. For continuum motion, this frame should also move and we have to introduce the velocity of the frame, $\mathbf{w}$. The flux of $A_{i}$ associated with $\mathbf{w}$ is $c_{i}\mathbf{w}$. The corresponding flux of mass is $m_{i}c_{i}\mathbf{w}$. The total flux of $A_{i}$ caused by diffusion and the frame motion is $\check{J}_{i}=J_{i}+c_{i}\mathbf{w}\,.$ The mass density is $\rho={\sum_{i}m_{i}c_{i}}\,;$ the momentum density is $\sum_{i}m_{i}\check{J}_{i}\,;$ the average mass velocity is $\mathbf{v}=\frac{\sum_{i}m_{i}\check{J}_{i}}{\sum_{i}m_{i}c_{i}}=\mathbf{w}+\frac{\sum_{i}m_{i}{J}_{i}}{\sum_{i}m_{i}c_{i}}\,.$ Both the frame velocity $\mathbf{w}$ and the average mass velocity $\mathbf{w}$ are unknown. They are connected by the simple identity $\mathbf{v}=\mathbf{w}+\frac{\sum_{i}m_{i}J_{i}}{\sum_{i}m_{i}c_{i}}\,,$ where the individual diffusion fluxes $J_{i}$ are given by the mechanism of diffusion. The standard definition of the diffusion fluxes includes fluxes in the local center of mass frame where the average mass velocity is zero. Therefore, let us introduce the “proper fluxes”: $\mathcal{J}_{i}=\check{J}_{i}-\mathbf{v}c_{i}=\check{J}_{i}-c_{i}\frac{\sum_{i}m_{i}\check{J}_{i}}{\sum_{i}m_{i}c_{i}}=J_{i}-c_{i}\frac{\sum_{i}m_{i}{J}_{i}}{\sum_{i}m_{i}c_{i}}\,.$ These fluxes do not depend on the frame velocity. They are not independent and are connected by the momentum conservation: the total momentum is zero, $\sum m_{i}\mathcal{J}_{i}=0\,.$ The heat flux in the local center of mass system is $\mathcal{J}_{U}=\left(J_{Q}-\frac{\sum_{i}m_{i}{J}_{i}}{\sum_{i}m_{i}c_{i}}u\right)+\sum_{i}\mu_{i}\mathcal{J}_{i}\,,$ where $J_{Q}$ is given by the transport processes mechanism. For the energy flux, the standard approach [70] gives $\mathbf{v}\left(\frac{\rho v^{2}}{2}+u+P\right)+\mathcal{J}_{U}\;+\mbox{viscosity terms}\,.$ The conservation laws give: $\begin{split}&\partial_{t}\rho+\mathrm{div}(\rho\mathbf{v})=0\,,\\\ &\partial_{t}(\rho\mathbf{v})+\mathrm{div}(\rho\mathbf{v}\otimes\mathbf{v})+\nabla P=\mathrm{div}\sigma\,,\\\ &\partial_{t}\left(\frac{\rho v^{2}}{2}+u\right)+\mathrm{div}\left[\mathbf{v}\left(\frac{\rho v^{2}}{2}+u+P\right)+\mathcal{J}_{U}\right]=\sigma:\nabla\mathbf{v}\,,\\\ &\partial_{t}c_{i}+\mathrm{div}(\mathbf{v}c_{i}+\mathcal{J}_{i})=0\,.\end{split}$ (3.68) Here, $\sigma$ is the viscous stress tensor and $\sigma:\nabla\mathbf{v}=\sum_{ij}\sigma_{ij}(\partial v_{i}/\partial x_{j})$. The pressure $P$ should be defined in accordance with thermodynamic properties of the mixture, for example, $P=-\frac{\partial F}{\partial V}=\sum_{i}c_{i}\frac{\partial f(c,T)}{\partial c_{i}}-f(c,T)\,,$ where $f(c,T)$ is the density of the free energy. The viscous stress tensor should be derived in (3.68) from the additional closure assumption. For the Newtonian liquid, $\sigma_{ij}=\mu\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{\partial v_{j}}{\partial x_{i}}\right)+\delta_{ij}\left(\zeta-\frac{2}{3}\mu\right)\mathrm{div}\mathbf{v}\,,$ where $\mu$ (here and only here) is the shear viscosity and $\zeta$ is the bulk viscosity. The individual equations in (3.68) are not independent. For example, the sum of the equations for conservation of $N_{i}$ with coefficients $m_{i}$ gives us the first equation, the conservation of mass. The elastic energy and the various viscoelastic terms may be added to this picture. This is necessary to do and it is in our future plans. #### 3.4.2. Mechanisms of Transport and the General Forms of Macroscopic Equation We developed a formalism of the mechanism of diffusion and heat conduction represented by the system of stoichiometric equations with the simple kinetic law $\exp(\alpha,\mu/RT)$ and the balance condition (complex balance for the general Markov microscopic kinetics and detailed balance for systems with microreversibility). This formalism produces equations which are particular cases of the general nonequilibrium thermodynamic equations [20, 85]. It is a very simple task to demonstrate that our transport equations are particular cases of the GENERIC formalism [56, 85]. Due to this two–generator formalism, evolution of any smooth function $A$ of the state variables $x$ is given by $\frac{\mathrm{d}A}{\mathrm{d}t}=\\{A,E\\}+[A,S]\,$ where $E$ and $S$ are the total energy and entropy, and $\\{\cdot,\cdot\\}$ and $[\cdot,\cdot]$ are Poisson and dissipative brackets, respectively. The formulas for fluxes produced in this Section have the form of dissipative brackets: $[A,S]=\frac{\delta A}{\delta x}M\frac{\delta S}{\delta x}\,,$ where $M$ is a symmetric positively semidefinite operator, “the friction matrix”. The general form of the “dissipative brackets with constraints” in application to multicomponent diffusion was produced very recently [86]. The flux of the $i$th component $J_{i}$ in that formalism was presented by formula (54) [86]: $J_{i}=-\sum_{j}\Lambda_{ij}^{c}\left[\nabla\left(\frac{\mu_{j}}{T}\right)+\Lambda^{\prime}_{j}\nabla\left(-\frac{1}{T}\right)\right]\,.$ Our formulas belong to this type and give particular expressions for coefficients $\Lambda$. In the paper [86] a precise comparison of this formula with the classical expressions from [20] was presented and the equivalence of these general forms was proven. Now, we can just refer to these results. In addition to the general form, our approach gives the possibility to build the model from the elementary processes. This construction also satisfies the “constrains” (conservation laws) of diffusion because these conservation laws are implemented in the algebra of the stoichiometric coefficients (2.22). ## 4\. Conclusion Chemical kinetics gave rise to the very seminal approach of the modeling of processes. This is, the stoichiometric algebra supplemented by the simple kinetic law. The results of this invention are now applied in may areas of science, from particle physics to sociology. In our work we extend the area of applications to nonlinear multicomponent diffusion. We demonstrated, how the mechanism based approach to multicomponent diffusion can be included within the general thermodynamic framework and proved the corresponding dissipation inequalities. To satisfy the thermodynamic restrictions, the kinetic law of an elementary process cannot have an arbitrary form. For the general kinetic law $\phi\exp(\alpha,\mu/RT)$ (the generalized Mass Action Law), additional conditions on the set of kinetic factors $\phi$ are needed. There are two main sets of these conditions. The historically first of them, the condition of detailed balance, follows from the special property of the underlying microscopic dynamics, microreversibility. It was used by Boltzmann for his equation and then by many researchers like Wegscheider [113], Einstein [25], and Onsager [83]. The second and more general property we now call “semi-detailed balance” or “complex balance” was proposed by Boltzmann in his discussion with Lorentz. The theoretic principles underneath these conditions were discovered by Stueckelberg in 1952 [97] in application to the Boltzmann equation. It received the name “complex balance” ten years later in the works of Horn and Jackson [62] and Feinberg [30]. 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In: Selected Works of Yakov Borisovich Zeldovich; Volume 1, Ostriker, J.P., Ed. Princeton University Press: Princeton, NJ, USA, 1996; pp. 144–148.	arxiv-papers	2010-12-14T00:58:51	2024-09-04T02:49:15.631289	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "A.N. Gorban, H.P. Sargsyan, H.A. Wahab", "submitter": "Alexander Gorban", "url": "https://arxiv.org/abs/1012.2908" }
1012.2931	Oscillator Variations of the Classical Theorem on Harmonic Polynomials 1112000 Mathematical Subject Classification. Primary 17B10, 17B15; Secondary 42B37. Cuiling Luoa and Xiaoping Xub a. College of Science, Hebei Polytechnic University, Tangshan, Hebei 063009, P. R. China. b. Corresponding author, Hua Loo-Keng Key Mathematical Laboratory, Institute of Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, 100190, P. R. China. ###### Abstract We study two-parameter oscillator variations of the classical theorem on harmonic polynomials, associated with noncanonical oscillator representations of $sl(n,\mathbb{F})$ and $o(n,\mathbb{F})$. We find the condition when the homogeneous solution spaces of the variated Laplace equation are irreducible modules of the concerned algebras and the homogeneous subspaces are direct sums of the images of these solution subspaces under the powers of the dual differential operator. This establishes a local $(sl(2,\mathbb{F}),sl(n,\mathbb{F}))$ and $(sl(2,\mathbb{F}),o(n,\mathbb{F}))$ Howe duality, respectively. In generic case, the obtained irreducible $o(n,\mathbb{F})$-modules are infinite-dimensional non-unitary modules without highest-weight vectors. As an application, we determine the structure of noncanonical oscillator representations of $sp(2n,\mathbb{C})$. When both parameters are equal to the maximal allowed value, we obtain an infinite family of explicit irreducible $({\cal G},{\cal K})$-modules for $o(n,\mathbb{F})$ and $sp(2n,\mathbb{C})$. Methodologically we have extensively used partial differential equations to solve representation problems. ## 1 Introduction Harmonic polynomials are important objects in analysis, differential geometry and physics. A fundamental theorem in classical harmonic analysis says that the spaces of homogeneous harmonic polynomials (solutions of Laplace equation) are irreducible modules of the corresponding orthogonal Lie group (algebra) and the whole polynomial algebra is a free module over the invariant polynomials generated by harmonic polynomials. Bases of these irreducible modules can be obtained easily (e.g., cf. [X]). The algebraic beauty of the above theorem is that Laplace equation characterizes the irreducible submodules of the polynomial algebra and the corresponding quadratic invariant gives a decomposition of the polynomial algebra into a direct sum of irreducible submodules. This actually forms an $(sl(2,\mathbb{F}),o(n,\mathbb{F}))$ Howe duality. Lie algebras (Lie groups) serve as the symmetries in quantum physics (e.g., cf. [FC, L, LF, G]). Their various representations provide distinct concrete practical physical models. Many important physical phenomena have been interpreted as the consequences of symmetry breaking (e.g., cf. [LF]). Harmonic oscillators are basic objects in quantum mechanics (e.g., cf. [FC, G]). Oscillator representations of finite-dimensional simple Lie algebras are the most fundamental ones in quantum physics. Their infinite-dimensional analogues are free field representations of affine Kac-Moody algebras. The aim of this work is to establish certain two-parameter oscillator variations of the classical theorem on harmonic polynomials, associated with noncanonical oscillator representations of special linear Lie algebras and orthogonal Lie algebras, which are obtained by swapping differential operators and multiplication operators in the canonical oscillator representations induced from the natural representations. The Howe duality does not hold on the whole polynomial algebras. But we find the condition when the homogeneous solution spaces of the variated Laplace equation are irreducible modules of the concerned algebras and the homogeneous subspaces are direct sums of the images of these solution subspaces under the powers of the dual differential operator. We may call this a local $(sl(2,\mathbb{F}),sl(n,\mathbb{F}))$ and $(sl(2,\mathbb{F}),o(n,\mathbb{F}))$ Howe duality, respectively. In particular, we obtain explicit infinite-dimensional non-unitary modules of orthogonal Lie algebras that are not of highest-weight type. As an application of our results on special linear Lie algebras, we prove that the homogeneous subspaces in noncanonical oscillator representations of symplectic Lie algebras are irreducible except some singular cases, in which the homogeneous subspaces are direct sums of exactly two explicitly given irreducible submodules. Explicit bases of all the above irreducible modules in generic case are obtained. Let ${\cal G}$ be a semisimple Lie algebra and let ${\cal K}$ be a maximal proper reductive Lie subalgebra of ${\cal G}$. An infinite-dimensional irreducible ${\cal G}$-module is said of $({\cal G},{\cal K})$-type if it is a direct sum of finite-dimensional irreducible ${\cal K}$-submodules. When both parameters are equal to the maximal allowed value, we obtain an infinite family of explicit irreducible $({\cal G},{\cal K})$-modules for orthogonal Lie algebras and symplectic Lie algebras. Since our representations are not unitary, the concerned modules are infinite-dimensional and we are dealing with pairs of dual invariant differential operators, traditional methods fail to solve our problems. In fact, we have extensively used the method of solving flag partial differential equations developed in [X] by the second author. Below we give a technical introduction. For convenience, we will use the notion $\overline{i,i+j}=\\{i,i+1,i+2,...,i+j\\}$ for integers $i$ and $j$ with $i\leq j$. Denote by $\mathbb{N}$ the additive semigroup of nonnegative integers. Let $E_{r,s}$ be the square matrix with 1 as its $(r,s)$-entry and 0 as the others. The compact orthogonal Lie algebra $o(n,\mathbb{R})=\sum_{1\leq r<s\leq n}\mathbb{R}(E_{r,s}-E_{s,r}),$ whose representation on the polynomial algebra ${\cal A}=\mathbb{R}[x_{1},...,x_{n}]$ is determined by $(E_{r,s}-E_{s,r})\|_{\cal A}=x_{r}\partial_{x_{s}}-x_{s}\partial_{x_{r}}$, which we call the canonical oscillator representation of $o(n,\mathbb{R})$ (e.g., cf. [FSS]). Denote by ${\cal A}_{k}$ the subspace of homogeneous polynomials in ${\cal A}$ with degree $k$. Recall that the Laplace operator $\Delta=\partial_{x_{1}}^{2}+\cdots+\partial_{x_{n}}^{2}$ and its corresponding invariant $\eta=x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}$. When $n\geq 3$, it is well known that the subspace of harmonic polynomials ${\cal H}_{k}=\\{f\in{\cal A}_{k}\mid\Delta(f)=0\\}$ $None$ forms an irreducible $o(n,\mathbb{R})$-module and ${\cal A}_{k}={\cal H}_{k}\oplus\eta{\cal A}_{k-2},$ which is equivalent to that ${\cal A}_{k}=\bigoplus_{i=1}^{\llbracket k/2\rrbracket}\eta^{i}{\cal H}_{k-2i}$ is a direct sum of irreducible submodules. Since the space $\mathbb{F}\Delta+\mathbb{F}[\Delta,\eta]+\mathbb{F}\eta$ forms an operator Lie algebra isomorphic to $sl(2,\mathbb{R})$, the above conclusion gives an $(sl(2,\mathbb{R}),o(n,\mathbb{R}))$ Howe duality. Below all the vector spaces are assumed over a field $\mathbb{F}$ with characteristic 0. Moreover, we always assume that $n\geq 2$ is an integer. Let us reconsider the canonical oscillator representation of $sl(n,\mathbb{F})$: $E_{i,j}\|_{\cal A}=x_{i}\partial_{j}\qquad\mbox{for}\;\;i,j\in\overline{1,n}.$ $None$ Fix $1\leq n_{1}<n$. Note $[\partial_{x_{r}},x_{r}]=1=[-x_{r},\partial_{x_{r}}].$ $None$ Changing operators $\partial_{x_{r}}\mapsto-x_{r}$ and $x_{r}\mapsto\partial_{x_{r}}$ in (1.2) for $r\in\overline{1,n_{1}}$, we obtain the following noncanonical oscillator representation of $sl(n,\mathbb{F})$ determined by: $E_{i,j}\|_{\cal A}=\left\\{\begin{array}[]{ll}-x_{j}\partial_{x_{i}}-\delta_{i,j}&\mbox{if}\;i,j\in\overline{1,n_{1}};\\\ \partial_{x_{i}}\partial_{x_{j}}&\mbox{if}\;i\in\overline{1,n_{1}},\;j\in\overline{n_{1}+1,n};\\\ -x_{i}x_{j}&\mbox{if}\;i\in\overline{n_{1}+1,n},\;j\in\overline{1,n_{1}};\\\ x_{i}\partial_{x_{j}}&\mbox{if}\;i,j\in\overline{n_{1}+1,n}.\end{array}\right.$ $None$ For any $k\in\mathbb{Z}$, we denote ${\cal A}_{\langle k\rangle}=\mbox{Span}\>\\{x^{\alpha}\mid\alpha\in\mathbb{N}\>^{n};\sum_{r=n_{1}+1}^{n}\alpha_{r}-\sum_{i=1}^{n_{1}}\alpha_{i}=k\\}.$ $None$ It was presented by Howe in his work [Ho] that for $m_{1},m_{2}\in\mathbb{N}$ with $m_{1}>0$, ${\cal A}_{\langle-m_{1}\rangle}$ is an irreducible highest- weight $sl(n,\mathbb{F})$-submodule with highest weight $m_{1}\lambda_{n_{1}-1}-(m_{1}+1)\lambda_{n_{1}}$ and ${\cal A}_{\langle m_{2}\rangle}$ is an irreducible highest-weight $sl(n,\mathbb{F})$-submodule with highest weight $-(m_{2}+1)\lambda_{n_{1}}+m_{2}(1-\delta_{n_{1},n-1})\lambda_{n_{1}+1}$. Denote ${\cal B}=\mathbb{F}[x_{1},...,x_{n},y_{1},...,y_{n}]$. Fix $n_{1},n_{2}\in\overline{1,n}$ with $n_{1}\leq n_{2}$. Changing operators $\partial_{x_{r}}\mapsto-x_{r},\;x_{r}\mapsto\partial_{x_{r}}$ for $r\in\overline{1,n_{1}}$ and $\partial_{y_{s}}\mapsto- y_{s},\;y_{s}\mapsto\partial_{y_{s}}$ for $s\in\overline{n_{2}+1,n}$, we get another noncanonical oscillator representation of $sl(n,\mathbb{F})$ on ${\cal B}$ determined by $E_{i,j}\|_{\cal B}=E_{i,j}^{x}-E_{j,i}^{y}\qquad\mbox{for}\;\;i,j\in\overline{1,n}$ $None$ with $E_{i,j}^{x}\|_{\cal B}=\left\\{\begin{array}[]{ll}-x_{j}\partial_{x_{i}}-\delta_{i,j}&\mbox{if}\;i,j\in\overline{1,n_{1}};\\\ \partial_{x_{i}}\partial_{x_{j}}&\mbox{if}\;i\in\overline{1,n_{1}},\;j\in\overline{n_{1}+1,n};\\\ -x_{i}x_{j}&\mbox{if}\;i\in\overline{n_{1}+1,n},\;j\in\overline{1,n_{1}};\\\ x_{i}\partial_{x_{j}}&\mbox{if}\;i,j\in\overline{n_{1}+1,n}\end{array}\right.$ $None$ and $E_{i,j}^{y}\|_{\cal B}=\left\\{\begin{array}[]{ll}y_{i}\partial_{y_{j}}&\mbox{if}\;i,j\in\overline{1,n_{2}};\\\ -y_{i}y_{j}&\mbox{if}\;i\in\overline{1,n_{2}},\;j\in\overline{n_{2}+1,n};\\\ \partial_{y_{i}}\partial_{y_{j}}&\mbox{if}\;i\in\overline{n_{2}+1,n},\;j\in\overline{1,n_{2}};\\\ -y_{j}\partial_{y_{i}}-\delta_{i,j}&\mbox{if}\;i,j\in\overline{n_{2}+1,n}.\end{array}\right.$ $None$ The related variated Laplace operator becomes ${\cal D}=-\sum_{i=1}^{n_{1}}x_{i}\partial_{y_{i}}+\sum_{r=n_{1}+1}^{n_{2}}\partial_{x_{r}}\partial_{y_{r}}-\sum_{s=n_{2}+1}^{n}y_{s}\partial_{x_{s}}$ $None$ and its dual $\eta=\sum_{i=1}^{n_{1}}y_{i}\partial_{x_{i}}+\sum_{r=n_{1}+1}^{n_{2}}x_{r}y_{r}+\sum_{s=n_{2}+1}^{n}x_{s}\partial_{y_{s}}.$ $None$ Set ${\cal B}_{\langle\ell_{1},\ell_{2}\rangle}=\mbox{Span}\\{x^{\alpha}y^{\beta}\mid\alpha,\beta\in\mathbb{N}\>^{n};\sum_{r=n_{1}+1}^{n}\alpha_{r}-\sum_{i=1}^{n_{1}}\alpha_{i}=\ell_{1};\sum_{i=1}^{n_{2}}\beta_{i}-\sum_{r=n_{2}+1}^{n}\beta_{r}=\ell_{2}\\}$ $None$ for $\ell_{1},\ell_{2}\in\mathbb{Z}$. Define ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}=\\{f\in{\cal B}_{\langle\ell_{1},\ell_{2}\rangle}\mid{\cal D}(f)=0\\}.$ $None$ The following is our first result: Theorem 1. For any $\ell_{1},\ell_{2}\in\mathbb{Z}$ such that $\ell_{1}+\ell_{2}\leq n_{1}-n_{2}+1-\delta_{n_{1},n_{2}}$, ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}$ is an irreducible highest-weight $sl(n,\mathbb{F})$-module and ${\cal B}_{\langle\ell_{1},\ell_{2}\rangle}=\bigoplus_{m=0}^{\infty}\eta^{m}({\cal H}_{\langle\ell_{1}-m,\ell_{2}-m\rangle})$ is a decomposition of irreducible submodules. In particular, ${\cal B}_{\langle\ell_{1},\ell_{2}\rangle}={\cal H}_{\langle\ell_{1},\ell_{2}\rangle}\oplus\eta({\cal B}_{\langle\ell_{1}-1,\ell_{2}-1\rangle})$. When $n_{1}+1<n_{2}<n$ and $\ell_{1}+\ell_{2}>n_{1}-n_{2}+1$, ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}$ is not irreducible and contains nonzero elements in $\eta({\cal B}_{\langle\ell_{1}-1,\ell_{2}-1\rangle})$. Although the space $\mathbb{F}{\cal D}+\mathbb{F}[{\cal D},\eta]+\mathbb{F}\eta$ forms an operator Lie algebra isomorphic to $sl(2,\mathbb{R})$, we do not have an $(sl(2,\mathbb{F}),sl(n,\mathbb{F}))$ Howe duality. We may call Theorem 1 an local $(sl(2,\mathbb{F}),sl(n,\mathbb{F}))$ Howe duality. Consider the split $o(2n,\mathbb{F})=\sum_{i,j=1}^{n}\mathbb{F}(E_{i,j}-E_{n+j,n+i})+\sum_{1\leq i<j\leq n}[\mathbb{F}(E_{i,n+j}-E_{j,n+i})+\mathbb{F}(E_{n+j,i}-E_{n+i,j})]$ $None$ and define a noncanonical oscillator representation of $o(2n,\mathbb{F})$ on ${\cal B}$ by $(E_{i,j}-E_{n+j,n+i})\|_{\cal B}=E_{i,j}^{x}\|_{\cal B}-E_{j,i}^{y}\|_{\cal B},$ $None$ $E_{i,n+j}\|_{\cal B}=\left\\{\begin{array}[]{ll}\partial_{x_{i}}\partial_{y_{j}}&\mbox{if}\;i\in\overline{1,n_{1}},\;j\in\overline{1,n_{2}},\\\ -y_{j}\partial_{x_{i}}&\mbox{if}\;i\in\overline{1,n_{1}},\;j\in\overline{n_{2}+1,n},\\\ x_{i}\partial_{y_{j}}&\mbox{if}\;i\in\overline{n_{1}+1,n},\;j\in\overline{1,n_{2}},\\\ -x_{i}y_{j}&\mbox{if}\;i\in\overline{n_{1}+1,n},\;j\in\overline{n_{2}+1,n}\end{array}\right.$ $None$ and $E_{n+i,j}\|_{\cal B}=\left\\{\begin{array}[]{ll}-x_{j}y_{i}&\mbox{if}\;j\in\overline{1,n_{1}},\;i\in\overline{1,n_{2}},\\\ -x_{j}\partial_{y_{i}}&\mbox{if}\;j\in\overline{1,n_{1}},\;i\in\overline{n_{2}+1,n},\\\ y_{i}\partial_{x_{j}}&\mbox{if}\;j\in\overline{n_{1}+1,n},\;i\in\overline{1,n_{2}},\\\ \partial_{x_{j}}\partial_{y_{i}}&\mbox{if}\;j\in\overline{n_{1}+1,n},\;i\in\overline{n_{2}+1,n}.\end{array}\right.$ $None$ Set ${\cal B}_{\langle k\rangle}=\bigoplus_{\ell_{1},\ell_{2}\in\mathbb{Z};\ell_{1}+\ell_{2}=k}{\cal B}_{\langle\ell_{1},\ell_{2}\rangle},\qquad{\cal H}_{\langle k\rangle}=\\{f\in{\cal B}_{\langle k\rangle}\mid{\cal D}(f)=0\\}.$ $None$ Below we always take ${\cal K}=\sum_{i,j=1}^{n}\mathbb{F}(E_{i,j}-E_{n+j,n+i})$. Our second results is: Theorem 2. For any $n_{1}-n_{2}+1-\delta_{n_{1},n_{2}}\geq k\in\mathbb{Z}$, ${\cal H}_{\langle k\rangle}$ is an irreducible $o(2n,\mathbb{F})$-submodule and ${\cal B}_{\langle k\rangle}=\bigoplus_{i=0}^{\infty}\eta^{i}({\cal H}_{\langle k-2i\rangle})$ is a decomposition of irreducible submodules. In particular, ${\cal B}_{\langle k\rangle}={\cal H}_{\langle k\rangle}\oplus\eta({\cal B}_{\langle k-2\rangle})$. The module ${\cal H}_{\langle k\rangle}$ under the assumption is of highest-weight type only if $n_{2}=n$. When $n_{1}=n_{2}=n$, all the irreducible modules ${\cal H}_{\langle k\rangle}$ with $0\geq k\in\mathbb{Z}$ are of $({\cal G},{\cal K})$-type. We may view Theorem 2 as an local $(sl(2,\mathbb{F}),o(2n,\mathbb{F}))$ Howe duality. Note the split $o(2n+1,\mathbb{F})=o(2n,\mathbb{F})\oplus\bigoplus_{i=1}^{n}[\mathbb{F}(E_{0,i}-E_{n+i,0})+\mathbb{F}(E_{0,n+i}-E_{i,0})].$ $None$ Let ${\cal B}^{\prime}=\mathbb{F}[x_{0},x_{1},...,x_{n},y_{1},...,y_{n}]$. We define a noncanonical oscillator representation of $o(2n+1,\mathbb{F})$ on ${\cal B}^{\prime}$ by the differential operators in (1.14)-(1.16) and $E_{0,i}\|_{{\cal B}^{\prime}}=\left\\{\begin{array}[]{ll}-x_{0}x_{i}&\mbox{if}\;i\in\overline{1,n_{1}},\\\ x_{0}\partial_{x_{i}}&\mbox{if}\;i\in\overline{n_{1}+1,n},\\\ x_{0}\partial_{y_{i}}&\mbox{if}\;i\in\overline{n+1,n+n_{2}},\\\ -x_{0}y_{i}&\mbox{if}\;i\in\overline{n+n_{2}+1,2n}\end{array}\right.$ $None$ and $E_{i,0}\|_{{\cal B}^{\prime}}=\left\\{\begin{array}[]{ll}\partial_{x_{0}}\partial_{x_{i}}&\mbox{if}\;i\in\overline{1,n_{1}},\\\ x_{i}\partial_{x_{0}}&\mbox{if}\;i\in\overline{n_{1}+1,n},\\\ y_{i}\partial_{x_{0}}&\mbox{if}\;i\in\overline{n+1,n+n_{2}},\\\ \partial_{x_{0}}\partial_{y_{i}}&\mbox{if}\;i\in\overline{n+n_{2}+1,2n}.\end{array}\right.$ $None$ Now the variated Laplace operator becomes ${\cal D}^{\prime}=\partial_{x_{0}}^{2}-2\sum_{i=1}^{n_{1}}x_{i}\partial_{y_{i}}+2\sum_{r=n_{1}+1}^{n_{2}}\partial_{x_{r}}\partial_{y_{r}}-2\sum_{s=n_{2}+1}^{n}y_{s}\partial_{x_{s}}$ $None$ and its dual operator $\eta^{\prime}=x_{0}^{2}+2\sum_{i=1}^{n_{1}}y_{i}\partial_{x_{i}}+2\sum_{r=n_{1}+1}^{n_{2}}x_{r}y_{r}+2\sum_{s=n_{2}+1}^{n}x_{s}\partial_{y_{s}}.$ $None$ Set ${\cal B}^{\prime}_{\langle k\rangle}=\sum_{i=0}^{\infty}{\cal B}_{\langle k-i\rangle}x_{0}^{i},\qquad{\cal H}^{\prime}_{\langle k\rangle}=\\{f\in{\cal B}^{\prime}_{\langle k\rangle}\mid{\cal D}^{\prime}(f)=0\\}.$ $None$ The following is our third result. Theorem 3. For any $n_{1}-n_{2}+1-\delta_{n_{1},n_{2}}\geq k\in\mathbb{Z}$, ${\cal H}^{\prime}_{\langle k\rangle}$ is an irreducible $o(2n+1,\mathbb{F})$-submodule and ${\cal B}^{\prime}_{\langle k\rangle}=\bigoplus_{i=0}^{\infty}(\eta^{\prime})^{i}({\cal H}^{\prime}_{\langle k-2i\rangle})$ is a decomposition of irreducible submodules. In particular, ${\cal B}^{\prime}_{\langle k\rangle}={\cal H}^{\prime}_{\langle k\rangle}\oplus\eta^{\prime}({\cal B}^{\prime}_{\langle k-2\rangle})$. The module ${\cal H}_{\langle k\rangle}^{\prime}$ under the assumption is of highest-weight type only if $n_{2}=n$. When $n_{1}=n_{2}=n$, all the irreducible modules ${\cal H}_{\langle k\rangle}^{\prime}$ with $0\geq k\in\mathbb{Z}$ are of $({\cal G},{\cal K})$-type. Again Theorem 2 can be viewed as an local $(sl(2,\mathbb{F}),o(2n+1,\mathbb{F}))$ Howe duality. Define a noncanonical oscillator representation of $sp(2n,\mathbb{F})$ on ${\cal B}$ by (1.14)-(1.16). Using some results in the proof of Theorem 1, we prove: Theorem 4. Let $k\in\mathbb{Z}$. If $n_{1}<n_{2}$ or $k\neq 0$, the subspace ${\cal B}_{\langle k\rangle}$ (cf. (1.17)) is an irreducible $sp(2n,\mathbb{F})$-submodule. When $n_{1}=n_{2}$, the subspace ${\cal B}_{\langle 0\rangle}$ is a direct sum of two irreducible $sp(2n,\mathbb{F})$-submodules. Moreover, each irreducible submodule is of highest-weight module only if $n_{2}=n$. When $n_{1}=n_{2}=n$, all the irreducible submodules are of $({\cal G},{\cal K})$-type. In addition, the explicit expressions for all the above irreducible modules are given. In the case of highest-weight type, the highest-weight vector and its weight of the corresponding irreducible modules are also presented. Since the representations with parameters $(n_{1},n_{2})$ are contragredient to those with parameters $(n-n_{2},n-n_{1})$, the case $n_{2}<n_{1}$ has virtually been handled. In Section 2, we present some preparatory works, in particular, the method of solving flag partial differential equations found in [X] by the second author. In Section 3, we prove Theorem 1 when $n_{1}<n_{2}$. Section 4 is devoted to the proof of Theorem 1 with $n_{1}=n_{2}$. In Sections 5, 6 and 7, we prove Theorems 2, 3 and 4, respectively. ## 2 Preparation It is very often that Lie group theorists characterize certain irreducible modules as kernels of a set of differential operators. But how to solve the corresponding systems of partial differential equations is in general unknown. It was realized by the second author that these equations are of “flag type” when the modules are of highest-weight type. A linear transformation (operator) $T$ on a vector space $V$ is called locally nilpotent if for any $v\in V$, there exists a positive integer $k$ such that $T^{k}(v)=0$. A partial differential equation of flag type is the linear differential equation of the form: $(d_{1}+f_{1}d_{2}+f_{2}d_{3}+\cdots+f_{n-1}d_{n})(u)=0,$ $None$ where $d_{1},d_{2},...,d_{n}$ are certain commuting locally nilpotent differential operators on the polynomial algebra $\mathbb{F}[x_{1},x_{2},...,x_{n}]$ and $f_{1},...,f_{n-1}$ are polynomials satisfying $d_{i}(f_{j})=0$ if $i>j.$ Many variable-coefficient (generalized) Laplace equations, wave equations, Klein-Gordon equations, Helmholtz equations are of this type. Solving such equations is also important in finding invariant solutions of nonlinear partial differential equations (e.g., cf. [I1, I2]). In representation theory, we are more interested in polynomial solutions of flag partial differential equations. The second author [X] found an effective way of solving for them. The following lemma is a slightly generalized form of Lemma 2.1 in [X]. Lemma 2.1 (Xu [X]). Let ${\cal B}$ be a commutative associative algebra and let ${\cal A}$ be a free ${\cal B}$-module generated by a filtrated subspace $V=\bigcup_{r=0}^{\infty}V_{r}$ (i.e., $V_{r}\subset V_{r+1}$). Let $T_{1}$ be a linear operator on ${\cal B}\oplus{\cal A}$ with a right inverse $T_{1}^{-}$ such that $T_{1}({\cal B},{\cal A}),\;T_{1}^{-}({\cal B},{\cal A})\subset({\cal B},{\cal A}),\;\;T_{1}(\eta_{1}\eta_{2})=T_{1}(\eta_{1})\eta_{2},\;\;T_{1}^{-}(\eta_{1}\eta_{2})=T_{1}^{-}(\eta_{1})\eta_{2}$ $None$ for $\eta_{1}\in{\cal B},\;\eta_{2}\in V$, and let $T_{2}$ be a linear operator on ${\cal A}$ such that $T_{2}(V_{r+1})\subset{\cal B}V_{r},\;\;T_{2}(f\zeta)=fT_{2}(\zeta)\qquad\mbox{for}\;\;r\in\mathbb{N},\;\;f\in{\cal B},\;\zeta\in{\cal A}.$ $None$ Then we have $\displaystyle\\{g\in{\cal A}\mid(T_{1}+T_{2})(g)=0\\}$ $\displaystyle=$ $\displaystyle\mbox{Span}\\{\sum_{i=0}^{\infty}(-T_{1}^{-}T_{2})^{i}(hg)\mid g\in V,\;h\in{\cal B};\;T_{1}(h)=0\\}.\hskip 110.96556pt(2.4)$ Set $\epsilon_{i}=(0,...,0,\stackrel{{\scriptstyle i}}{{1}},0,...,0)\in\mathbb{N}^{\>n}.$ $None$ For each $i\in\overline{1,n}$, we define the linear operator $\int_{(x_{i})}$ on ${\cal A}$ by: $\int_{(x_{i})}(x^{\alpha})=\frac{x^{\alpha+\epsilon_{i}}}{\alpha_{i}+1}\;\;\mbox{for}\;\;\alpha\in\mathbb{N}^{\>n}.$ $None$ Furthermore, we let $\int_{(x_{i})}^{(0)}=1,\qquad\int_{(x_{i})}^{(m)}=\stackrel{{\scriptstyle m}}{{\overbrace{\int_{(x_{i})}\cdots\int_{(x_{i})}}}}\qquad\mbox{for}\;\;0<m\in\mathbb{Z}$ $None$ and denote $\partial^{\alpha}=\partial_{x_{1}}^{\alpha_{1}}\partial_{x_{2}}^{\alpha_{2}}\cdots\partial_{x_{n}}^{\alpha_{n}},\;\;\int^{(\alpha)}=\int_{(x_{1})}^{(\alpha_{1})}\int_{(x_{2})}^{(\alpha_{2})}\cdots\int_{(x_{n})}^{(\alpha_{n})}\qquad\mbox{for}\;\;\alpha\in\mathbb{N}^{\>n}.$ $None$ Obviously, $\int^{(\alpha)}$ is a right inverse of $\partial^{\alpha}$ for $\alpha\in\mathbb{N}^{\>n}.$ We remark that $\int^{(\alpha)}\partial^{\alpha}\neq 1$ if $\alpha\neq 0$ due to $\partial^{\alpha}(1)=0$. In this paper, our $T_{1}$’s are of the type $\partial^{\alpha}$ and the right inverse $T_{1}^{-}=\int^{(\alpha)}$. Let $m_{1},m_{2},...,m_{n}$ be positive integers. Taking $T_{1}=\partial_{x_{1}}^{m_{1}},\;T_{2}=\partial_{x_{2}}^{m_{2}}+\cdots+\partial_{x_{n}}^{m_{n}}$ and $T_{1}^{-}=\int_{(x_{1})}^{(m_{1})}$, we find that the set $\displaystyle\\{\sum_{k_{2},...,k_{n}=0}^{\infty}(-1)^{k_{2}+\cdots+k_{n}}{k_{2}+\cdots+k_{k}\choose k_{2},...,k_{n}}\int_{(x_{1})}^{((k_{2}+\cdots+k_{n})m_{1})}(x_{1}^{\ell_{1}})$ $\displaystyle\times\partial_{x_{2}}^{k_{2}m_{2}}(x_{2}^{\ell_{2}})\cdots\partial_{x_{n}}^{k_{n}m_{n}}(x_{n}^{\ell_{n}})\mid\ell_{1}\in\overline{0,m_{1}-1},\;\ell_{2},...,\ell_{n}\in\mathbb{N}\\}\hskip 56.9055pt(2.9)$ forms a basis of the space of polynomial solutions for the equation $(\partial_{x_{1}}^{m_{1}}+\partial_{x_{2}}^{m_{2}}+\cdots+\partial_{x_{n}}^{m_{n}})(u)=0.$ $None$ When all $m_{i}=2$, we get a basis of the space of harmonic polynomials. Cao [C] used Lemma 2.1 to prove that the subspaces of homogeneous polynomial vector solutions of the $n$-dimensional Navier equations in elasticity are exactly direct sums of three explicitly given irreducible submodules when $n\neq 4$ and direct sums of four explicitly given irreducible submodules if $n=4$ of the corresponding orthogonal Lie group (algebra), and the whole polynomial vector space is also a free module over the invariant polynomials generated these solutions. The result can be viewed as a vector generalization of the classical theorem on harmonic polynomials. Moreover, Cao solved the initial value problem for the Navier equations based on the ideas in [X]. The idea of solving flag partial differential equation in [X] leads the second author to find a family of special functions functions ${\cal Y}_{r}(y_{1},...,y_{m})=\sum_{i_{1},...,i_{m}=0}^{\infty}{i_{1}+\cdots+i_{m}\choose i_{1},...,i_{m}}\frac{y_{1}^{i_{1}}y_{2}^{i_{2}}\cdots y_{m}^{i_{m}}}{(r+\sum_{s=1}^{m}si_{s})!},$ $None$ by which we can solve the initial value problem of the equation: $(\partial_{x_{1}}^{m}-\sum_{r=1}^{m}\partial_{x_{1}}^{m-i}f_{i}(\partial_{x_{2}},...,\partial_{x_{n}}))(u)=0,$ $None$ where $f_{i}(\partial_{x_{2}},...,\partial_{x_{n}})\in\mathbb{R}[\partial_{x_{2}},...,\partial_{x_{n}}].$ Let ${\cal A}=\mathbb{F}[x_{1},...,x_{n}]$ and let $gl(n,\mathbb{F})$ act on ${\cal A}$ by (1.4). With the notion in (1.5), ${\cal A}=\bigoplus_{k\in\mathbb{Z}}{\cal A}_{\langle k\rangle}$ is a $\mathbb{Z}$ graded algebra and each homogeneous subspace ${\cal A}_{\langle k\rangle}$ is infinite-dimensional. Set $\flat=\sum_{r=n_{1}+1}^{n}x_{r}\partial_{x_{r}}-\sum_{i=1}^{n_{1}}x_{i}\partial_{x_{i}}.$ $None$ Then ${\cal A}_{\langle k\rangle}=\\{f\in{\cal A}\mid\flat(f)=kf\\}.$ $None$ Moreover, we have $\flat E_{i,j}=E_{j,i}\flat\;\;\mbox{on}\;\;{\cal A}\qquad\mbox{for}\;i,j\in\overline{1,n}.$ $None$ Thus ${\cal A}_{\langle k\rangle}$ forms a ${\cal G}$-module for any subalgebra ${\cal G}$ of $gl(n,\mathbb{F})$. For $\alpha\in\mathbb{N}\>^{n}$, we denote $\alpha!=\prod_{i=1}^{n}\alpha_{i}!$ and define a symmetric bilinear form $(\cdot\|\cdot)$ on ${\cal A}$ by $(x^{\alpha}\|x^{\beta})=\delta_{\alpha,\beta}(-1)^{\sum_{i=1}^{n_{1}}\alpha_{i}}\alpha!\qquad\mbox{for}\;\;\alpha,\beta\in\mathbb{N}\>^{n}.$ $None$ Then we have: Lemma 2.2. For any $A\in gl(n,\mathbb{F})$ and $f,g\in\mathbb{\cal A}$, we have $(A(f)\|g)=(f\|A^{t}(g)),$ where $A^{t}$ denote the transpose of the matrix $A$. Proof. Let $\alpha,\beta\in\mathbb{N}\>^{n}$. For $i,j\in\overline{1,n_{1}}$, $(E_{i,j}(x^{\alpha})\|x^{\beta})=-\alpha_{i}(x^{\alpha+\epsilon_{j}-\epsilon_{i}}\|x^{\beta})-\delta_{i,j}(x^{\alpha}\|x^{\beta})$ $None$ and $(x^{\alpha}\|E_{j,i}(x^{\beta}))=-\beta_{j}(x^{\alpha}\|x^{\beta+\epsilon_{i}-\epsilon_{j}})-\delta_{i,j}(x^{\alpha}\|x^{\beta})$ $None$ by (1.4). Note $\displaystyle\hskip 28.45274pt\alpha_{i}(x^{\alpha+\epsilon_{j}-\epsilon_{i}}\|x^{\beta})$ $\displaystyle=$ $\displaystyle\delta_{\alpha+\epsilon_{j}-\epsilon_{i},\beta}(-1)^{\sum_{i=1}^{n_{1}}\alpha_{i}}(\alpha_{j}+1)\alpha!$ $\displaystyle=$ $\displaystyle\beta_{j}\delta_{\alpha,\beta+\epsilon_{i}-\epsilon_{j}}(-1)^{\sum_{i=1}^{n_{1}}\alpha_{i}}\alpha!=\beta_{j}(x^{\alpha}\|x^{\beta+\epsilon_{i}-\epsilon_{j}})\hskip 76.82234pt(2.19)$ by (2.16). Hence $(E_{i,j}(x^{\alpha})\|x^{\beta})=(x^{\alpha}\|E_{j,i}(x^{\beta})).$ $None$ If $i,j\in\overline{n_{1}+1,n}$, then (2.19) holds and so does (2.20). Consider $i\in\overline{1,n_{1}}$ and $j\in\overline{n_{1}+1,n}$. $(E_{i,j}(x^{\alpha})\|x^{\beta})=\alpha_{i}\alpha_{j}(x^{\alpha-\epsilon_{i}-\epsilon_{j}}\|x^{\beta})=-\delta_{\alpha-\epsilon_{i}-\epsilon_{j},\beta}(-1)^{\sum_{i=1}^{n_{1}}\alpha_{i}}\alpha!$ $None$ and $(x^{\alpha}\|E_{j,i}(x^{\beta}))=-(x^{\alpha}\|x^{\beta+\epsilon_{i}+\epsilon_{j}})=-\delta_{\alpha,\beta+\epsilon_{i}+\epsilon_{j}}(-1)^{\sum_{i=1}^{n_{1}}\alpha_{i}}\alpha!$ $None$ by (1.4) and (2.16). So (2.20) holds. Therefore, the lemma holds by the symmetry of the form.$\qquad\Box$ Let ${\cal G}$ be simple Lie subalgebra of $gl(n,\mathbb{F})$ such that $A^{t}\in{\cal G}$ if $A\in{\cal G}.$ Let $H$ be a Cartan subalgebra of ${\cal G}$ and assume that ${\cal A}$ forms a weighted ${\cal G}$-module with respect to $H$. Fix the positivity of roots and denote by ${\cal G}_{+}$ the sum of positive root subspaces. A singular vector is a weight vector annihilated by positive root vectors. From now on, we count the number of singular vectors up to a scalar multiple. Moreover, an element $g\in{\cal A}$ is called nilpotent with respect to ${\cal G}_{+}$ if there exist a positive integer $m$ such that $\xi_{1}\cdots\xi_{m}(g)=0\qquad\mbox{for any}\;\xi_{1},...,\xi_{m}\in{\cal G}_{+}.$ $None$ A subspace $V$ of ${\cal A}$ is called nilpotent with respect to ${\cal G}_{+}$ if all its elements are nilpotent with respect to ${\cal G}_{+}$. If the elements of ${\cal G}_{+}\|_{\cal A}$ are locally nilpotent and ${\cal G}_{+}({\cal A}_{i})\subset\sum_{r=0}^{i}{\cal A}_{r}$ for any $i\in\mathbb{N}$, then any element of ${\cal A}$ is nilpotent with respect to ${\cal G}_{+}$ by Engel’s Theorem. Lemma 2.3. If a submodule $N$ of ${\cal A}$ is nilpotent with respect to ${\cal G}_{+}$, $N$ contains only one singular vector $v$ and $(v\|v)\neq 0$, then $N$ is an irreducible summand of ${\cal A}$. Proof. Under the nilpotent assumption, any nonzero submodule of $N$ contains a singular vector. In particular, $N_{1}=U(\mathcal{G})(v)$ is an irreducible submodule by the uniqueness of singular vector. Set $\bar{N}_{1}^{\bot}=\\{u\in N\|(u\|w)=0\mid w\in N_{1}\\}.$ $None$ and ${\cal R}=\\{u\in N\|(u\|w)=0\mid w\in N\\}.$ $None$ Note that $\bar{N}_{1}^{\bot}$ and ${\cal R}$ are submodules of $N$ by Lemma 2.2. If ${\cal R}\neq 0$, it should contain a nonzero singular vector, which is impossible according to the assumption $(v\|v)\neq 0$. Therefore ${\cal R}=\\{0\\}$, and $N=N_{1}\bigoplus\bar{N}_{1}^{\bot}$. But $\bar{N}_{1}^{\bot}=0$ by the same argument, and so $N=N_{1}$. The fact ${\cal R}=\\{0\\}$ implies that ${\cal A}=N\oplus\\{f\in{\cal A}\mid(f\|g)=0\;\mbox{for}\;g\in N\\}$ $None$ is a direct sum of ${\cal G}$-submodules.$\qquad\Box$ Let ${\cal Q}=\mathbb{F}(x_{1},...,x_{n},y_{1},...,y_{n})$ be the space of rational functions in $x_{1},...,x_{n},y_{1},...,y_{n}$. Define a representation of $sl(n,\mathbb{F})$ on ${\cal Q}$ via $E_{i,j}\|_{\cal Q}=x_{i}\partial_{x_{j}}-y_{j}\partial_{y_{i}}\qquad\mbox{for}\;\;i,j\in\overline{1,n}.$ $None$ Set $\zeta=\sum_{i=1}^{n}x_{i}y_{i}.$ Then $\xi(\zeta)=0\qquad\mbox{for}\;\;\xi\in sl(n,\mathbb{F}).$ $None$ Take $H=\sum_{i=1}^{n-1}\mathbb{F}(E_{i,i}-E_{i+1,i+1})$ $None$ as a Cartan subalgebra of $sl(n,\mathbb{F})$ and the subspace spanned by positive root vectors: $sl(n,\mathbb{F})_{+}=\sum_{1\leq i<j\leq}\mathbb{F}E_{i,j}.$ $None$ The following lemma was proved in [X], which will be used in next section. Lemma 4. Any singular function in ${\cal Q}$ is a rational function in $x_{1},y_{n},\zeta$. ## 3 The $sl(n,\mathbb{F})$-Variation I: $n_{1}<n_{2}$ Fix $n_{1},n_{2}\in\overline{1,n}$ such that $n_{1}\leq n_{2}$. Recall that ${\cal Q}$ is the space of rational functions in $x_{1},...,x_{n},y_{1},...,y_{n}$. Define a representation of $sl(n,\mathbb{F})$ on ${\cal Q}$ determined by $E_{i,j}\|_{\cal Q}=E_{i,j}^{x}-E_{j,i}^{y}\qquad\mbox{for}\;\;i,j\in\overline{1,n}$ $None$ with $E_{i,j}^{x}\|_{\cal Q}=\left\\{\begin{array}[]{ll}-x_{j}\partial_{x_{i}}-\delta_{i,j}&\mbox{if}\;i,j\in\overline{1,n_{1}};\\\ \partial_{x_{i}}\partial_{x_{j}}&\mbox{if}\;i\in\overline{1,n_{1}},\;j\in\overline{n_{1}+1,n};\\\ -x_{i}x_{j}&\mbox{if}\;i\in\overline{n_{1}+1,n},\;j\in\overline{1,n_{1}};\\\ x_{i}\partial_{x_{j}}&\mbox{if}\;i,j\in\overline{n_{1}+1,n}\end{array}\right.$ $None$ and $E_{i,j}^{y}\|_{\cal Q}=\left\\{\begin{array}[]{ll}y_{i}\partial_{y_{j}}&\mbox{if}\;i,j\in\overline{1,n_{2}};\\\ -y_{i}y_{j}&\mbox{if}\;i\in\overline{1,n_{2}},\;j\in\overline{n_{2}+1,n};\\\ \partial_{y_{i}}\partial_{y_{j}}&\mbox{if}\;i\in\overline{n_{2}+1,n},\;j\in\overline{1,n_{2}};\\\ -y_{j}\partial_{y_{i}}-\delta_{i,j}&\mbox{if}\;i,j\in\overline{n_{2}+1,n}.\end{array}\right.$ $None$ Recall $\flat$ in (2.13) and define $\flat^{\prime}=\sum_{i=1}^{n_{2}}y_{i}\partial{y_{i}}-\sum_{r=n_{2}+1}^{n}y_{r}\partial{y_{r}}.$ $None$ Moreover, the deformed Laplace operator ${\cal D}$ in (1.9) and its dual $\eta$ in (1.10). Then $TE_{i,j}\|_{\cal Q}=E_{i,j}\|_{\cal Q}T\qquad\mbox{for}\;\;T=\flat,\flat^{\prime},{\cal D},\eta;\;i,j\in\overline{1,n}.$ $None$ In addition, $[\flat,{\cal D}]=[\flat^{\prime},{\cal D}]=-{\cal D},\qquad[\flat,\eta]=[\flat^{\prime},\eta]=\eta.$ $None$ By (3.1)-(3.3), we find $E_{i,r}\|_{\cal Q}=-x_{r}\partial_{x_{i}}-y_{r}\partial_{y_{i}}\qquad\mbox{for}\;\;1\leq i<r\leq n_{1},$ $None$ $E_{i,n_{1}+s}\|_{\cal Q}=\partial_{x_{i}}\partial_{x_{n_{1}+s}}-y_{n_{1}+s}\partial_{y_{i}}\qquad\mbox{for}\;\;i\in\overline{1,n_{1}},\;s\in\overline{1,n_{2}-n_{1}},$ $None$ $E_{r,s}\|_{\cal Q}=x_{r}\partial_{x_{s}}-y_{s}\partial_{y_{r}}\qquad\mbox{for}\;\;n_{1}<r<s\leq n_{2},$ $None$ $E_{n_{2},n_{2}+1}=x_{n_{2}}\partial_{x_{n_{2}+1}}-\partial_{y_{n_{2}}}\partial_{y_{n_{2}+1}},$ $None$ $E_{i,r}\|_{\cal Q}=x_{i}\partial_{x_{r}}+y_{i}\partial_{y_{r}}\qquad\mbox{for}\;\;n_{2}+1\leq i<r\leq n.$ $None$ The subalgebra $sl(n,\mathbb{F})_{+}$ in (2.30) is generated by the above $E_{i,j}$. Denote $\zeta_{1}=x_{n_{1}-1}y_{n_{1}}-x_{n_{1}}y_{n_{1}-1},\;\;\zeta=\sum_{r=n_{1}+1}^{n_{2}}x_{r}y_{r},\;\;\zeta_{2}=x_{n_{2}+1}y_{n_{2}+2}-x_{n_{2}+2}y_{n_{2}+1}.$ $None$ We will process according to three cases. Case 1. $n_{1}+1<n_{2}$ Assume $n_{1}+1<n_{2}<n$. Suppose that $f\in{\cal Q}$ is a singular vector. By Lemma 2.4, $f$ can be written as a rational function in $\\{x_{i},y_{r},\zeta_{1},\zeta,\zeta_{2}\mid n_{2}+2\neq i\in\overline{1,n_{1}+1}\bigcup\overline{n_{2}+1,n},\;n_{1}-1\neq r\in\overline{1,n_{1}}\bigcup\overline{n_{2},n}\\}.$ $None$ Note $E_{n_{1}-1,n_{1}}(f)=-x_{n_{1}}\partial_{x_{n_{1}-1}}(f)=0$ $None$ by (3.7) and $E_{n_{2}+1,n_{2}+2}(f)=y_{n_{2}+1}\partial_{y_{n_{2}+2}}(f)=0$ $None$ by (3.11). So $f$ is independent of $x_{n_{1}-1}$ and $y_{n_{2}+2}$. For $i\in\overline{1,n_{1}-2}$, we have $\displaystyle\qquad E_{i,n_{1}-1}(f)$ $\displaystyle=$ $\displaystyle- x_{n_{1}-1}\partial_{x_{i}}(f)-y_{n_{1}-1}\partial_{y_{i}}(f)$ $\displaystyle=$ $\displaystyle- x_{n_{1}-1}(\partial_{x_{i}}(f)+x_{n_{1}}^{-1}y_{n_{1}}\partial_{y_{i}}(f))+x_{n_{1}}^{-1}\zeta_{1}\partial_{y_{i}}(f)=0\hskip 71.13188pt(3.16)$ by (3.7). Since both $\partial_{x_{i}}(f)+x_{n_{1}}^{-1}y_{n_{1}}\partial_{y_{i}}(f)$ and $x_{n_{1}}^{-1}\zeta_{1}\partial_{y_{i}}(f)$ are independent of $x_{n_{1}-1}$, we have $\partial_{y_{i}}(f)=0$, which implies $\partial_{x_{i}}(f)$=0 by (3.16). Thus $f$ is independent of $\\{x_{i},y_{i}\mid i\in\overline{1,n_{1}-1}$. Similarly, we can prove that $f$ is independent of $\\{x_{i},y_{i}\mid i\in\overline{n_{2}+1,n}$. Therefore, $f$ only depends on $\\{x_{n_{1}},x_{n_{1}+1},x_{n_{2}+1},y_{n_{1}},y_{n_{2}},y_{n_{2}+1},\zeta_{1},\zeta,\zeta_{2}\\}.$ $None$ According to (3.8) and (3.12), $E_{n_{1},n_{1}+1}\|_{\cal Q}=\partial_{x_{n_{1}}}\partial_{x_{n_{1}+1}}-y_{n_{1}+1}\partial_{y_{n_{1}}}$ and $E_{n_{1},n_{1}+1}(f)=f_{x_{n_{1}}x_{n_{1}+1}}+y_{n_{1}+1}(f_{x_{n_{1}}\zeta}-y_{n_{1}-1}f_{\zeta_{1}\zeta}-f_{y_{n_{1}}}-x_{n_{1}-1}f_{\zeta_{1}})=0.$ $None$ Applying $E_{n_{1}+1,n_{2}}\|_{\cal Q}=x_{n_{1}+1}\partial_{x_{n_{2}}}-y_{n_{2}}\partial_{y_{n_{1}+1}}$ to the above equation, we get $-f_{x_{n_{1}}\zeta}+y_{n_{1}-1}f_{\zeta_{1}\zeta}+f_{y_{n_{1}}}+x_{n_{1}-1}f_{\zeta_{1}}=0$ $None$ by (3.9). According to (3.12), $x_{n_{1}-1}=y_{n_{1}}^{-1}\zeta_{1}+x_{n_{1}}y_{n_{1}}^{-1}y_{n_{1}-1}.$ $None$ Substituting it into (3.19), we get $y_{n_{1}-1}(f_{\zeta_{1}\zeta}+y_{n_{1}}^{-1}x_{n_{1}}f_{\zeta_{1}})+f_{y_{n_{1}}}+y_{n_{1}}^{-1}\zeta_{1}f_{\zeta_{1}}-f_{x_{n_{1}}\zeta}=0.$ $None$ Since $f$ is independent of $y_{n_{1}-1}$, we have $f_{\zeta_{1}\zeta}+y_{n_{1}}^{-1}x_{n_{1}}f_{\zeta_{1}}=0.$ $None$ Thus $f_{\zeta_{1}}=e^{-y_{n_{1}}^{-1}x_{n_{1}}\zeta}g$ $None$ for some function $g$ in the variables of (3.17) except $\zeta$, i.e., $g_{\zeta}=0$. But $f$ is a rational function in the variables of (3.17) and so is $f_{\zeta_{1}}$. Hence (3.23) forces $f_{\zeta_{1}}=0$, that is, $f$ is independent of $\zeta_{1}$. Similarly, we can prove that $f$ is independent of $\zeta_{2}$. Now $f$ only depends on $\\{x_{n_{1}},x_{n_{1}+1},x_{n_{2}+1},y_{n_{1}},y_{n_{2}},y_{n_{2}+1},\zeta\\}.$ $None$ Since $\zeta=\sum_{i=n_{1}+1}^{n_{2}}x_{i}y_{i}$, $f\in{\cal B}=\mathbb{F}[x_{1},...,x_{n},y_{1},...,y_{n}]$ if and only if $f$ is a polynomial in the variables (3.24). Now (3.18) and (3.19) are equivalent to $f_{x_{n_{1}}x_{n_{1}+1}}=0,\qquad f_{x_{n_{1}}\zeta}-f_{y_{n_{1}}}=0.$ $None$ Similarly, we can prove $f_{y_{n_{2}}y_{n_{2}+1}}=0,\qquad f_{y_{n_{2}+1}\zeta}-f_{x_{n_{2}+1}}=0.$ $None$ Set $\phi(m_{1},m_{2})=\sum_{i=0}^{\infty}\frac{y_{n_{1}}^{i}(\partial_{x_{n_{1}}}\partial_{\zeta})^{i}(x_{n_{1}}^{m_{1}}\zeta^{m_{2}})}{i!}\qquad\mbox{for}\;\;m_{1},m_{2}\in\mathbb{N}.$ $None$ By Lemma 2.1 with $T_{1}=\partial_{y_{n_{1}}},\;T_{1}^{-}=\int_{(y_{n_{1}})}$ (cf. (2.6)) and $T_{2}=-\partial_{x_{n_{1}}}\partial_{\zeta}$, the polynomial solution space of (3.25) is $[\mathbb{F}[x_{n_{1}+1},\zeta]+\sum_{m_{1}=1}^{\infty}\sum_{m_{2}=0}^{\infty}\mathbb{F}\phi(m_{1},m_{2})][\mathbb{F}[x_{n_{2}+1},y_{n_{2}},y_{n_{2}+1}]].$ $None$ Denote $\psi(m_{1},m_{2})=\sum_{i=0}^{\infty}\frac{x_{n_{2}+1}^{i}(\partial_{y_{n_{2}+1}}\partial_{\zeta})^{i}(y_{n_{2}+1}^{m_{1}}\zeta^{m_{2}})}{i!}\qquad\mbox{for}\;\;m_{1},m_{2}\in\mathbb{N},$ $None$ $\displaystyle\qquad\phi(m_{1},m_{2},m_{3})$ $\displaystyle=$ $\displaystyle\sum_{r=0}^{\infty}\frac{x_{n_{2}+1}^{r}(\partial_{y_{n_{2}+1}}\partial_{\zeta})^{r}(\phi(m_{1},m_{2})y_{n_{2}+1}^{m_{3}})}{r!}$ $\displaystyle=$ $\displaystyle\sum_{i,r=0}^{\infty}\frac{y_{n_{i}}^{i}x_{n_{2}+1}^{r}\partial_{x_{n_{1}}}^{i}\partial_{y_{n_{2}+1}}^{r}\partial_{\zeta}^{i+r}(x_{n_{1}}^{m_{1}}\zeta^{m_{2}}y_{n_{2}+1}^{m_{3}})}{i!r!}.\hskip 85.35826pt(3.30)$ Solving (3.26) by Lemma 2.1 with $T_{1}=\partial_{x_{n_{2}+1}}\;T_{1}^{-}=\int_{(x_{n_{2}+1})}$ (cf. (2.6)) and $T_{2}=-\partial_{y_{n_{2}+1}}\partial_{\zeta}$, we find the polynomial solution space of the system (3.25) and (3.26) is $\displaystyle\mathbb{F}[x_{n_{1}+1},y_{n_{2}},\zeta]+\sum_{m_{1},m_{3}=1}^{\infty}\sum_{m_{2}=0}^{\infty}\mathbb{F}\phi(m_{1},m_{2},m_{3})$ $\displaystyle+\sum_{m_{1}=1}^{\infty}\sum_{m_{2}=0}^{\infty}(\mathbb{F}[y_{n_{2}}]\phi(m_{1},m_{2})+\mathbb{F}[x_{n_{1}+1}]\psi(m_{1},m_{2})).\hskip 122.34692pt(3.31)$ According to (1.10), $x_{n_{1}+1}^{m_{1}}y_{n_{2}}^{m_{2}}\zeta^{m_{3}}=\eta^{m_{3}}(x_{n_{1}+1}^{m_{1}}y_{n_{2}}^{m_{2}}),$ $None$ $\eta^{m_{2}}(x_{n_{1}}^{m_{1}}y_{n_{2}}^{m_{3}})=(\zeta+y_{n_{1}}\partial_{x_{n_{1}}})^{m_{2}}(x_{n_{1}}^{m_{1}}y_{n_{2}}^{m_{3}})=\phi(m_{1},m_{2})y_{n_{2}}^{m_{3}},$ $None$ $\eta^{m_{2}}(y_{n_{2}+1}^{m_{1}}x_{n_{1}+1}^{m_{3}})=(\zeta+x_{n_{2}+1}\partial_{y_{n_{2}+1}})^{m_{2}}(y_{n_{2}+1}^{m_{1}}x_{n_{1}+1}^{m_{3}})=\psi(m_{1},m_{2})x_{n_{1}+1}^{m_{3}},$ $None$ $\eta^{m_{2}}(x_{n_{1}}^{m_{1}}y_{n_{2}+1}^{m_{3}})=(\zeta+y_{n_{1}}\partial_{x_{n_{1}}}+x_{n_{2}+1}\partial_{y_{n_{2}+1}})^{m_{2}}(x_{n_{1}}^{m_{1}}y_{n_{2}+1}^{m_{3}})=\phi(m_{1},m_{2},m_{3}).$ $None$ It can be verified that $\\{\eta^{m_{1}}(x_{i}^{m_{2}}y_{j}^{m_{3}})\mid m_{1},m_{2},m_{3}\in\mathbb{N};i=n_{1},n_{1}+1;j=n_{2},n_{2}+1\\}$ are singular vectors. By (3.31)-(3.35), the nonzero vectors in $\\{\mathbb{F}[\eta](x_{i}^{m_{1}}y_{j}^{m_{2}})\mid m_{1},m_{2}\in\mathbb{N};i=n_{1},n_{1}+1;j=n_{2},n_{2}+1\\}$ $None$ are all the singular vectors of $sl(n,\mathbb{F})$ in ${\cal B}=\mathbb{F}[x_{1},...,x_{n_{1}},y_{1},...,y_{n_{2}}]$. Similarly, when $n_{2}=n$ and $n_{1}\leq n-2$, the nonzero vectors in $\\{\mathbb{F}[\eta](x_{i}^{m_{1}}y_{n}^{m_{2}})\mid m_{1},m_{2}\in\mathbb{N};i=n_{1},n_{1}+1\\}$ $None$ are all the singular vectors of $sl(n,\mathbb{F})$ in ${\cal B}$. Denote ${\cal H}=\\{f\in{\cal B}\mid{\cal D}(f)=0\\}.$ $None$ By (3.5), ${\cal H}$ forms an $sl(n,\mathbb{F})$-submodule. Recall ${\cal B}_{\langle\ell_{1},\ell_{2}\rangle}$ defined in (1.11). Then ${\cal B}_{\langle\ell_{1},\ell_{2}\rangle}=\\{f\in{\cal B}\mid\flat(f)=\ell_{1}f;\flat^{\prime}(f)=\ell_{2}f\\}$ $None$ by (2.13) and (3.4). Moreover, ${\cal B}=\bigoplus_{\ell_{1},\ell_{2}\in\mathbb{Z}}{\cal B}_{\langle\ell_{1},\ell_{2}\rangle}$ becomes a $\mathbb{Z}^{2}$-graded algebra. According to (3.5), ${\cal B}_{\langle\ell_{1},\ell_{2}\rangle}$ forms an $sl(n,\mathbb{F})$-submodule, and so does ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}={\cal B}_{\langle\ell_{1},\ell_{2}\rangle}\bigcap{\cal H}.$ $None$ Next (1.9) and (1.10) imply $[{\cal D},\eta]=n_{2}-n_{1}+\flat+\flat^{\prime},\;\;{\cal D}(x_{i}^{m_{1}}y_{j}^{m_{2}})=0$ $None$ for $m_{1},m_{2}\in\mathbb{N},\;i=n_{1},n_{1}+1$ and $j=n_{2},n_{2}+1$. Thus $x_{n_{1}+1}^{m_{1}}y_{n_{2}}^{m_{2}}\in{\cal H}_{\langle m_{1},m_{2}\rangle},\qquad x_{n_{1}+1}^{m_{1}}y_{n_{2}+1}^{m_{2}}\in{\cal H}_{\langle m_{1},-m_{2}\rangle},$ $None$ $x_{n_{1}}^{m_{1}}y_{n_{2}}^{m_{2}}\in{\cal H}_{\langle- m_{1},m_{2}\rangle},\qquad x_{n_{1}}^{m_{1}}y_{n_{2}+1}^{m_{2}}\in{\cal H}_{\langle-m_{1},-m_{2}\rangle}.$ $None$ For any $g\in{\cal H}_{\langle\ell_{1},\ell_{2}\rangle}$and $0<m\in\mathbb{N}$, we have $\eta^{m}(g)\in{\cal B}_{\ell_{1}+m,\ell_{2}+m}$ and ${\cal D}(\eta^{m}(g))=m(n_{2}-n_{1}+\ell_{1}+\ell_{2}+m-1)\eta^{m-1}(g).$ $None$ Thus ${\cal D}(\eta^{m}(g))=0\;\;\mbox{if and only if}\;\;\ell_{1}+\ell_{2}\leq n_{1}-n_{2}\;\;\mbox{and}\;\;m=n_{1}-n_{2}-\ell_{1}-\ell_{2}+1.$ $None$ If so, $\eta^{m}(g)\in{\cal H}_{n_{1}-n_{2}-\ell_{2}+1,n_{1}-n_{2}-\ell_{1}+1}.$ $None$ Note $(n_{1}-n_{2}-\ell_{2}+1)+(n_{1}-n_{2}-\ell_{1}+1)=n_{1}-n_{2}+2+(n_{1}-n_{2}-\ell_{1}-\ell_{2})\geq n_{1}-n_{2}+2.$ $None$ Let $f_{\langle\ell_{1},\ell_{2}\rangle}\in{\cal H}_{\langle\ell_{1},\ell_{2}\rangle}$ be a singular vector in (3.42) and (3.43). Then the singular vectors in ${\cal H}$ are nonzero weight vectors in $\mbox{Span}\\{f_{\langle\ell_{1},\ell_{2}\rangle},\eta^{n_{1}-n_{2}+1-r_{1}-r_{2}}(f_{\langle r_{1},r_{2}\rangle})\mid\ell_{1},\ell_{2},r_{1},r_{2}\in\mathbb{Z};r_{1}+r_{2}\leq n_{1}-n_{2}\\}$ $None$ by (3.36), where $\eta^{n_{1}-n_{2}+1-r_{1}-r_{2}}(f_{\langle r_{1},r_{2}\rangle})\in{\cal H}_{\langle n_{1}-n_{2}+1-r_{2},n_{1}-n_{2}+1-r_{1}\rangle}.$ $None$ Thus when $n_{1}+1<n_{2}<n,$ we have ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}\;\mbox{has a unique singular vector if}\;\;\ell_{1}+\ell_{2}\leq n_{1}-n_{2}+1$ $None$ and ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}\;\mbox{has exactly two singular vectors if}\;\;\ell_{1}+\ell_{2}>n_{1}-n_{2}+1.$ $None$ In the case $n_{1}+1<n_{2}=n,$ ${\cal B}_{\langle\ell_{1},\ell_{2}\rangle}=0$ if $\ell_{2}<0$, and for $\ell_{1}\in\mathbb{Z}$, $\ell_{2}\in\mathbb{N}$, ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}\;\mbox{has a unique singular vector if}\;\;\ell_{1}\geq n_{1}-n+2\;\mbox{or}\;\ell_{1}+\ell_{2}\leq n_{1}-n+1.$ $None$ ${\cal H}_{\langle\ell_{1},\ell_{2}+1\rangle}\;\mbox{has exactly two singular vector if}\;\mbox{and}\;\;n_{1}-n+1-\ell_{2}\leq\ell_{1}\leq n_{1}-n+1.$ $None$ Observe that the symmetric bilinear form $(\cdot\|\cdot)$ on ${\cal B}$ is determined by $(x^{\alpha}y^{\beta}\|x^{\alpha_{1}}y^{\beta_{1}})=\delta_{\alpha,\alpha_{1}}\delta_{\beta,\beta_{1}}(-1)^{\sum_{i=1}^{n_{1}}\alpha_{i}+\sum_{r=n_{2}+1}^{n}\beta_{r}}\alpha!\beta!\qquad\mbox{for}\;\;\alpha,\beta,\alpha_{1},\beta_{1}\in\mathbb{N}\>^{n}.$ $None$ When $n_{1}+1<n_{2}<n$, Lemma 2.3 tells us that ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}$ for $\ell_{1},\ell_{2}\in\mathbb{Z}$ is an irreducible summand of ${\cal B}_{\ell_{1},\ell_{2}}$ if and only if $\ell_{1}+\ell_{2}\leq n_{1}-n_{2}+1$. It can be verified that $({\cal D}(x^{\alpha}y^{\beta})\|x^{\alpha_{1}}y^{\beta_{1}})=(x^{\alpha}y^{\beta}\|\eta(x^{\alpha_{1}}y^{\beta_{1}})).$ $None$ Recall that $f_{\langle\ell_{1},\ell_{2}\rangle}\in{\cal H}_{\langle\ell_{1},\ell_{2}\rangle}$ is a singular vector in (3.42) and (3.43). Thus $(f_{\langle\ell_{1},\ell_{2}\rangle}\|f_{\langle\ell_{1},\ell_{2}\rangle})\neq 0$ $None$ and $(f_{\langle\ell_{1},\ell_{2}\rangle}\|f_{\langle\ell_{1}^{\prime},\ell_{2}^{\prime}\rangle})=0\qquad\mbox{if}\;\;(\ell_{1},\ell_{2})\neq(\ell_{1}^{\prime},\ell_{2}^{\prime}).$ $None$ Recall $sl(n,\mathbb{F})_{+}$ in (2.30) and let $sl(n,\mathbb{F})_{-}=\sum_{1\leq i<j\leq n}\mathbb{F}E_{j,i}$ be the subalgebra spanned by the negative root vectors. Moreover, $(sl(n,\mathbb{F})_{-})^{t}=sl(n,\mathbb{F})_{+}.$ According to (3.7)-(3.11), ${\cal B}$ is nilpotent with respect to $sl(n,\mathbb{F})_{+}$. Thus all ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}$ with $\ell_{1}+\ell_{2}\leq n_{1}-n_{2}+1$ are irreducible $sl(n,\mathbb{F})$-submodules by Lemma 2.3 and (3.50), and so are $\eta^{m}({\cal H}_{\langle\ell_{1},\ell_{2}\rangle})$ for any $m\in\mathbb{N}$ by (3.5). We extend the transpose to an algebraic anti-isomorphism on $U(sl(n,\mathbb{F}))$ by $1^{t}=1$ and $(A_{1}A_{2}\cdots A_{r})^{t}=A_{r}^{t}\cdots A_{2}^{t}A_{1}^{t}\qquad\mbox{for}\;\;A_{i}\in sl(n,\mathbb{F}).$ $None$ By the irreducibility, ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}=U(sl(n,\mathbb{F})_{-})(f_{\langle\ell_{1},\ell_{2}\rangle})\qquad\mbox{if}\;\;\ell_{1}+\ell_{2}\leq n_{1}-n_{2}+1.$ $None$ Let $\ell_{1},\ell_{2},\ell_{1}^{\prime},\ell_{2}^{\prime}\in\mathbb{Z}$ such that $\ell_{1}+\ell_{2},\ell_{1}^{\prime}+\ell_{2}^{\prime}\leq n_{1}-n_{2}+1$ and $(\ell_{1},\ell_{2})\neq(\ell_{1}^{\prime},\ell_{2}^{\prime})$. Then $(w(f_{\langle\ell_{1},\ell_{2}\rangle})\|f_{\langle\ell_{1}^{\prime},\ell_{2}^{\prime}\rangle})=(f_{\langle\ell_{1},\ell_{2}\rangle}\|w^{t}(f_{\langle\ell_{1}^{\prime},\ell_{2}^{\prime}\rangle}))=0\qquad\mbox{for}\;\;w\in U(sl(n,\mathbb{F})_{-})sl(n,\mathbb{F})_{-}$ $None$ by Lemma 2.2. Since $f_{\langle\ell_{1},\ell_{2}\rangle}$ is a weight vector, we have $U(H)(f_{\langle\ell_{1},\ell_{2}\rangle})\subset\mathbb{F}f_{\langle\ell_{1},\ell_{2}\rangle}$ (cf. (2.29)). Thus for any $w_{1},w_{2}\in U(sl(n,\mathbb{F})_{-})$, $(w_{1}(f_{\langle\ell_{1},\ell_{2}\rangle})\|w_{1}(f_{\langle\ell_{1}^{\prime},\ell_{2}^{\prime}\rangle}))=(w_{2}^{t}w_{1}(f_{\langle\ell_{1},\ell_{2}\rangle})\|f_{\langle\ell_{1}^{\prime},\ell_{2}^{\prime}\rangle})=c(f_{\langle\ell_{1},\ell_{2}\rangle})\|f_{\langle\ell_{1}^{\prime},\ell_{2}^{\prime}\rangle})$ $None$ for some $c\in\mathbb{F}$ by (3.60). Hence (3.59) implies $({\cal H}_{\langle\ell_{1},\ell_{2}\rangle}\|{\cal H}_{\langle\ell_{1}^{\prime},\ell_{2}^{\prime}\rangle})=\\{0\\}.$ $None$ For $f\in{\cal H}_{\langle\ell_{1},\ell_{2}\rangle},\;g\in{\cal B}$ and $m,m_{1}\in\mathbb{N}$ such that $m\leq m_{1}$, we find $(\eta^{m}(f)\|\eta^{m_{1}}(g))=({\cal D}^{m_{1}}\eta^{m}(f)\|g)=\delta_{m_{1},m}m![\prod_{r=0}^{m-1}(\ell_{1}+\ell_{2}+n_{2}-n_{1}+r)](f\|g)$ $None$ by (3.44) and (3.55). In particular, the singular vectors $\eta^{n_{1}-n_{2}+1-r_{1}-r_{2}}(f_{\langle r_{1},r_{2}\rangle})$ for $r_{1},r_{2}\in\mathbb{Z}$ with $r_{1}+r_{2}\leq n_{1}-n_{2}$ are isotropic polynomials. Moreover, for $m,m_{1}\in\mathbb{N}$ and $\ell_{1},\ell_{2},\ell_{1}^{\prime},\ell_{2}^{\prime}\in\mathbb{Z}$ such that $\ell_{1}+\ell_{2},\ell_{1}^{\prime}+\ell_{2}^{\prime}\leq n_{1}-n_{2}+1$, $(\eta^{m}({\cal H}_{\langle\ell_{1},\ell_{2}\rangle})\|\eta^{m_{1}}({\cal H}_{\langle\ell_{1}^{\prime},\ell_{2}^{\prime}\rangle}))=\\{0\\}\qquad\mbox{if}\;\;(m,\ell_{1},\ell_{1})\neq(m_{1},\ell_{1}^{\prime},\ell_{1}^{\prime})$ $None$ by (3.62) and (3.63). On the other hand, $(\eta^{m}(f_{\langle\ell_{1},\ell_{2}\rangle})\|\eta^{m}(f_{\langle\ell_{1},\ell_{2}\rangle}))=m![\prod_{r=0}^{m-1}(\ell_{1}+\ell_{2}+n_{2}-n_{1}+r)](f_{\langle\ell_{1},\ell_{2}\rangle}\|f_{\langle\ell_{1},\ell_{2}\rangle})\neq 0$ $None$ by (3.63). Since the radical of $(\cdot\|\cdot)$ on $\eta^{m}({\cal H}_{\langle\ell_{1},\ell_{2}\rangle})$ is a proper submodule by Lemma 2.2, the irreducibility of $\eta^{m}({\cal H}_{\langle\ell_{1},\ell_{2}\rangle})$ implies that $(\cdot\|\cdot)\;\;\mbox{is nondegenerate rewtricted to}\;\;\eta^{m}({\cal H}_{\langle\ell_{1},\ell_{2}\rangle}).$ $None$ Fix $\ell_{1},\ell_{2}\in\mathbb{Z}$ with $\ell_{1}+\ell_{2}\leq n_{1}-n_{2}+1$. Set $\hat{\cal B}_{\langle\ell_{1},\ell_{2}\rangle}=\sum_{m=0}^{\infty}\eta^{m}({\cal H}_{\langle\ell_{1}-m,\ell_{2}-m\rangle}).$ $None$ By (3.64) and (3.66), the above sum is a direct sum and $(\cdot\|\cdot)$ is nondegenerate restricted to $\hat{\cal B}_{\langle\ell_{1},\ell_{2}\rangle}$. Hence ${\cal B}_{\langle\ell_{1},\ell_{2}\rangle}=\hat{\cal B}_{\langle\ell_{1},\ell_{2}\rangle}\oplus(\hat{\cal B}_{\langle\ell_{1},\ell_{2}\rangle}^{\perp}\bigcap{\cal B}_{\langle\ell_{1},\ell_{2}\rangle}).$ $None$ If $\hat{\cal B}_{\langle\ell_{1},\ell_{2}\rangle}^{\perp}\bigcap{\cal B}_{\langle\ell_{1},\ell_{2}\rangle}\neq\\{0\\}$, then it contains a singular vector, which must be of the form $\eta^{m_{1}}(f_{\langle\ell_{1}-m_{1},\ell_{2}-m_{1}\rangle})$ for some $m_{1}\in\mathbb{N}$ by (3.36). This contradicts (3.65). Thus $\hat{\cal B}_{\langle\ell_{1},\ell_{2}\rangle}^{\perp}\bigcap{\cal B}_{\langle\ell_{1},\ell_{2}\rangle}=\\{0\\}$, equivalently ${\cal B}_{\langle\ell_{1},\ell_{2}\rangle}=\bigoplus_{m=0}^{\infty}\eta^{m}({\cal H}_{\langle\ell_{1}-m,\ell_{2}-m\rangle})$ $None$ is completely reducible. Applying (3.69) to ${\cal B}_{\langle\ell_{1}-1,\ell_{2}-1\rangle}$, we have ${\cal B}_{\ell_{1},\ell_{2}}={\cal H}_{\langle\ell_{1},\ell_{2}\rangle}\oplus\eta({\cal B}_{\langle\ell_{1}-1,\ell_{2}-1\rangle})\qquad\mbox{if}\;\;\ell_{1}+\ell_{2}\leq n_{1}-n_{2}+1.$ $None$ Assume $n_{1}+1<n_{2}=n$. For $\ell_{1}\in\mathbb{Z}$ and $\ell_{2}\in\mathbb{N}$ such that $\ell_{1}\geq n_{1}-n+2\;\mbox{or}\;\ell_{1}+\ell_{2}\leq n_{1}-n+1$, all ${\cal H}_{\ell_{1},\ell_{2}}$ are irreducible submodules of ${\cal B}_{\ell_{1},\ell_{2}}$ by Lemma 2.3, (3.52) and (3.54). When $\ell_{1}+\ell_{2}\leq n_{1}-n_{2}+1$, (3.64), (3.66), (3.69) and (3.70) also hold by the same arguments as in the above. In summary, we have: Theorem 3.1. Suppose $n_{1}+1<n_{2}$. For $\ell_{1},\ell_{2}\in\mathbb{Z}$ with $\ell_{1}+\ell_{2}\leq n_{1}-n_{2}+1$ , ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}$ is an irreducible highest-weight $sl(n,\mathbb{F})$-module and ${\cal B}_{\langle\ell_{1},\ell_{2}\rangle}=\bigoplus_{m=0}^{\infty}\eta^{m}({\cal H}_{\langle\ell_{1}-m,\ell_{2}-m\rangle})$ $None$ is an orthogonal decomposition of irreducible submodules. In particular, ${\cal B}_{\langle\ell_{1},\ell_{2}\rangle}={\cal H}_{\langle\ell_{1},\ell_{2}\rangle}\oplus\eta({\cal B}_{\langle\ell_{1}-1,\ell_{2}-1\rangle})$. The symmetric bilinear form $(\cdot\|\cdot)$ restricted to $\eta^{m}({\cal H}_{\langle\ell_{1}-m,\ell_{2}-m\rangle})$ is nondengerate. If $n_{2}<n$, all ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}$ for $\ell_{1},\ell_{2}\in\mathbb{Z}$ with $\ell_{1}+\ell_{2}>n_{1}-n_{2}+1$ have exactly two singular vectors. Assume $n_{2}=n$. Then ${\cal B}_{\langle\ell,0\rangle}={\cal H}_{\langle\ell,0\rangle}$ with $\ell\in\mathbb{Z}$ are irreducible highest- weight $sl(n,\mathbb{F})$-modules. All ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}$ for $\ell_{1}\in\mathbb{Z}$ and $\ell_{2}\in\mathbb{N}$ such that $\ell_{1}\geq n_{1}-n+2$ are also irreducible highest-weight $sl(n,\mathbb{F})$-modules. Moreover, for $\ell_{2}\in 1+\mathbb{N}$ and $n_{1}-n_{2}+1+\ell_{2}\leq\ell_{1}\in\mathbb{Z}$, the orthogonal decompositions in (3.71) also holds. Furthermore, ${\cal H}_{\langle\ell_{1},\ell_{2}+1\rangle}$ for $\ell_{1}\in\mathbb{Z}$ and $\ell_{2}\in\mathbb{N}$ such that $n_{1}-n+1-\ell_{2}\leq\ell_{1}\leq n_{1}-n+1$ have exactly two singular vectors. Indeed, we have more detailed information. Suppose $n_{1}+1<n_{2}<n$. For $m_{1},m_{2}\in\mathbb{N}$ with $m_{1}+m_{2}\geq n_{2}-n_{1}-1$, ${\cal H}_{\langle-m_{1},-m_{2}\rangle}$ has a highest-weight vector $x_{n_{1}}^{m_{1}}y_{n_{2}+1}^{m_{2}}$ of weight $m_{1}\lambda_{n_{1}-1}-(m_{1}+1)\lambda_{n_{1}}-(m_{2}+1)\lambda_{n_{2}}+m_{2}(1-\delta_{n_{2},n-1})\lambda_{n_{2}+1}$. When $m_{1},m_{2}\in\mathbb{N}$ with $m_{2}-m_{1}\geq n_{2}-n_{1}-1$, ${\cal H}_{\langle m_{1},-m_{2}\rangle}$ has a highest-weight vector $x_{n_{1}+1}^{m_{1}}y_{n_{2}+1}^{m_{2}}$ of weight $-(m_{1}+1)\lambda_{n_{1}}+m_{1}\lambda_{n_{1}+1}-(m_{2}+1)\lambda_{n_{2}}+m_{2}(1-\delta_{n_{2},n-1})\lambda_{n_{2}+1}$. If $m_{1},m_{2}\in\mathbb{N}$ with $m_{1}-m_{2}\geq n_{2}-n_{1}-1$, ${\cal H}_{\langle-m_{1},m_{2}\rangle}$ is has a highest-weight vector $x_{n_{1}}^{m_{1}}y_{n_{2}}^{m_{2}}$ of weight $m_{1}\lambda_{n_{1}-1}-(m_{1}+1)\lambda_{n_{1}}+m_{2}\lambda_{n_{2}-1}-(m_{2}+1)\lambda_{n_{2}}$. Assume $n_{1}+1<n_{2}=n$. When $m_{1},m_{2}\in\mathbb{N}$, ${\cal H}_{\langle m_{1},m_{2}\rangle}$ has a highest-weight vector $x_{n_{1}+1}^{m_{1}}y_{n}^{m_{2}}$ of weight $-(m_{1}+1)\lambda_{n_{1}}+m_{1}\lambda_{n_{1}+1}+m_{2}\lambda_{n-1}$. If $m_{1},m_{2}\in\mathbb{N}$ with $m_{1}\leq n-n_{1}-2$ or $m_{2}-m_{1}\leq n_{1}-n+1$, ${\cal H}_{\langle-m_{1},m_{2}\rangle}$ has a highest-weight vector $x_{n_{1}}^{m_{1}}y_{n}^{m_{2}}$ of weight $m_{1}\lambda_{n_{1}-1}-(m_{1}+1)\lambda_{n_{1}}+m_{2}\lambda_{n-1}$. By Lemma 2.1 with $T_{1}=\partial_{x_{n_{1}+1}}\partial_{y_{n_{1}+1}},\;T_{1}^{-}=\int_{(x_{n_{1}+1})}\int_{(y_{n_{1}+1})}$ and $T_{2}={\cal D}-\partial_{x_{n_{1}}+1}\partial_{y_{n_{1}}+1}$, ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}$ has a basis $\displaystyle\big{\\{}\sum_{i=0}^{\infty}\frac{(x_{n_{1}+1}y_{n_{1}+1})^{i}({\cal D}-\partial_{x_{n_{1}}+1}\partial_{y_{n_{1}}+1})^{i}(x^{\alpha}y^{\beta})}{\prod_{r=1}^{i}(\alpha_{n_{1}+1}+r)(\beta_{n_{1}+1}+r)}\mid\alpha,\beta\in\mathbb{N}\>^{n};$ $\displaystyle\alpha_{n_{1}+1}\beta_{n_{1}+1}=0;\sum_{r=n_{1}+1}^{n}\alpha_{r}-\sum_{i=1}^{n_{1}}\alpha_{i}=\ell_{1};\sum_{i=1}^{n_{2}}\beta_{i}-\sum_{r=n_{2}+1}^{n}\beta_{r}=\ell_{2}\big{\\}}.\hskip 51.21504pt(3.72)$ Case 2. $n_{1}+1=n_{2}$ In this case, $\zeta=x_{n_{1}+1}y_{n_{1}+1}$. First we consider the subcase $n_{2}<n$. Suppose that $f\in{\cal Q}$ is a singular vector. According to the arguments in (3.13)-(3.17), $f$ is a rational function in $\\{x_{n_{1}},x_{n_{1}+1},x_{n_{1}+2},y_{n_{1}},y_{n_{1}+2},\zeta,\zeta_{1},\zeta_{2}\\}.$ $None$ Moreover, (3.18) holds. Substituting (3.20) and $y_{n_{1}+1}=x_{n_{1}+1}^{-1}\zeta$ into (3.18), we still get (3.22), which implies $f_{\zeta_{1}}=0$. Symmetrically, $f_{\zeta_{2}}=0$. Hence we can rewrite $f$ as a rational function in $\\{x_{n_{1}},x_{n_{1}+1},x_{n_{1}+2},y_{n_{1}},y_{n_{1}+1},y_{n_{1}+2}\\}.$ $None$ Now $f$ is a singular vector if and only if it is a weight vector satisfying the following system of differential equations $(\partial_{x_{n_{1}}}\partial_{x_{n_{1}+1}}-y_{n_{1}+1}\partial_{y_{n_{1}}})(f)=0,$ $None$ $(x_{n_{1}+1}\partial_{x_{n_{1}+2}}-\partial_{y_{n_{1}+1}}\partial_{y_{n_{1}+2}})(f)=0$ $None$ by (3.8) and (3.10) with $n_{2}=n_{1}+1$. Note $E_{n_{1},n_{1}+2}\|_{\cal Q}=[E_{n_{1},n_{1}+1}\|_{\cal Q},E_{n_{1}+1,n_{1}+2}\|_{\cal Q}]=\partial_{x_{n_{1}}}\partial_{x_{n_{1}+2}}-\partial_{y_{n_{1}}}\partial_{y_{n_{1}+2}}$ $None$ by (3.8) and (3.10) with $n_{2}=n_{1}+1$. So $(\partial_{x_{n_{1}}}\partial_{x_{n_{1}+2}}-\partial_{y_{n_{1}}}\partial_{y_{n_{1}+2}})(f)=0.$ $None$ For our purpose of representation, we only consider $f$ is a polynomial in $\\{x_{i},y_{i}\mid i=n_{1},n_{1}+1,n_{1}+2\\}$. Set $\displaystyle\qquad\phi_{m_{1},m_{2},m_{3}}$ $\displaystyle=$ $\displaystyle[\prod_{s=1}^{m_{2}}(m_{1}+s)]\sum_{i=0}^{\infty}\frac{x_{n_{1}}^{m_{1}+i}x_{n_{1}+2}^{i}(\partial_{y_{n_{1}}}\partial_{y_{n_{1}+2}})^{i}(y_{n_{1}}^{m_{2}}y_{n_{1}+2}^{m_{3}})}{i!\prod_{r=1}^{i}(m_{1}+r)}$ $\displaystyle=$ $\displaystyle(y_{n_{1}}\partial_{x_{n_{1}}}+x_{n_{1}+2}\partial_{y_{n_{1}+2}})^{m_{2}}(x_{n_{1}}^{m_{1}+m_{2}}y_{n_{1}+2}^{m_{3}})\hskip 122.34692pt(3.79)$ and $\displaystyle\qquad\psi_{m_{1},m_{2},m_{3}}$ $\displaystyle=$ $\displaystyle\frac{(m_{1}+m_{2})!\prod_{s=1}^{m_{1}}(m_{3}+s)}{m_{1}!}\sum_{i=0}^{\infty}\frac{x_{n_{1}}^{i}x_{n_{1}+2}^{m_{1}+i}(\partial_{y_{n_{1}}}\partial_{y_{n_{1}+2}})^{i}(y_{n_{1}}^{m_{2}}y_{n_{1}+2}^{m_{3}})}{i!\prod_{r=1}^{i}(m_{1}+r)}$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{m_{2}}{m_{2}\choose i}\frac{(m_{1}+m_{2})!x_{n_{1}}^{i}x_{n_{1}+2}^{m_{1}+i}y_{n_{1}}^{m_{2}-i}\partial_{y_{n_{1}+2}}^{m_{1}+i}(y_{n_{1}+2}^{m_{1}+m_{3}})}{(m_{1}+i)!}$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{m_{2}}{m_{2}\choose m_{2}-i}\frac{(m_{1}+m_{2})!x_{n_{1}}^{i}y_{n_{1}}^{m_{2}-i}(x_{n_{1}+2}\partial{y_{n_{1}+2}})^{m_{1}+i}(y_{n_{1}+2}^{m_{1}+m_{3}})}{(m_{1}+i)!}$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{m_{2}}\frac{(m_{1}+m_{2})!(y_{n_{1}}\partial_{x_{n_{1}}})^{m_{2}-i}(x_{n_{1}}^{m_{2}})(x_{n_{1}+2}\partial{y_{n_{1}+2}})^{m_{1}+i}(y_{n_{1}+2}^{m_{1}+m_{3}})}{(m_{2}-i)!(m_{1}+i)!}$ $\displaystyle=$ $\displaystyle\sum_{r=0}^{\infty}\frac{(m_{1}+m_{2})!(y_{n_{1}}\partial_{x_{n_{1}}})^{r}(x_{n_{1}}^{m_{2}})(x_{n_{1}+2}\partial{y_{n_{1}+2}})^{m_{1}+m_{2}-r}(y_{n_{1}+2}^{m_{1}+m_{3}})}{r!(m_{1}+m_{2}-r)!}$ $\displaystyle=$ $\displaystyle(y_{n_{1}}\partial_{x_{n_{1}}}+x_{n_{1}+2}\partial_{y_{n_{1}+2}})^{m_{1}+m_{2}}(x_{n_{1}}^{m_{2}}y_{n_{1}+2}^{m_{1}+m_{3}}).\hskip 108.12054pt(3.80)$ By Lemma 2.1 with $T_{1}=\partial_{x_{n_{1}}}\partial_{x_{n_{1}+2}},\;T_{1}^{-}=\int_{(x_{n_{1}})}\int_{(x_{n_{1}+2})}$ (cf. (2.6)) and $T_{2}=\partial_{y_{n_{1}}}\partial_{y_{n_{1}+2}}$, the polynomial solution space of (3.78) is $\mbox{Span}\\{\phi_{m_{1},m_{2},m_{3}}x_{n_{1}+1}^{m_{4}}y_{n_{1}+1}^{m_{5}},\psi_{m_{1}+1,m_{2},m_{3}}x_{n_{1}+1}^{m_{4}}y_{n_{1}+1}^{m_{5}}\mid m_{i}\in\mathbb{N}\\}.$ $None$ Note $\phi_{m_{1},m_{2},0}x_{n_{1}+1}^{m_{3}}y_{n_{1}+1}^{m_{4}}=[\prod_{r=1}^{m_{2}}(m_{1}+r)]x_{n_{1}}^{m_{1}}y_{n_{1}}^{m_{2}}x_{n_{1}+1}^{m_{3}}y_{n_{1}+1}^{m_{4}}\qquad\mbox{for}\;\;m_{i}\in\mathbb{N}$ $None$ and $x_{n_{1}+1}^{m_{1}}y_{n_{1}+1}^{m_{2}}\psi_{m_{3},0,m_{4}}=[\prod_{i=1}^{m_{3}}(m_{4}+i)]x_{n_{1}+1}^{m_{1}}y_{n_{1}+1}^{m_{2}}x_{n_{1}+2}^{m_{3}}y_{n_{1}+2}^{m_{4}}\qquad\mbox{for}\;\;m_{i}\in\mathbb{N}.$ $None$ In particular, all the polynomials in (3.83) are solutions of the equation (3.75). Now $\eta=\sum_{i=1}^{n_{1}}y_{i}\partial_{x_{i}}+x_{n_{1}+1}y_{n_{1}+1}+\sum_{s=n_{1}+2}^{n}x_{s}\partial_{y_{s}}.$ $None$ Write $\displaystyle\qquad h_{m_{1},m_{2},m_{3}}$ $\displaystyle=$ $\displaystyle\frac{(m_{1}+m_{2})!}{m_{1}!}\sum_{i=0}^{\infty}\frac{x_{n_{1}}^{i}x_{n_{1}+1}^{m_{1}+i}y_{n_{1}+1}^{m_{3}+i}\partial_{y_{n_{1}}}^{i}(y_{n_{1}}^{m_{2}})}{i!\prod_{r=1}^{i}(m_{1}+r)}$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{m_{2}}{m_{2}\choose i}\frac{(m_{1}+m_{2})!x_{n_{1}}^{i}x_{n_{1}+1}^{m_{1}+i}y_{n_{1}+1}^{m_{3}+i}y_{n_{1}}^{m_{2}-i}}{(m_{1}+i)!}$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{m_{2}}{m_{2}\choose m_{2}-i}\frac{(m_{1}+m_{2})!x_{n_{1}}^{i}x_{n_{1}+1}^{m_{1}+i}y_{n_{1}+1}^{m_{3}+i}y_{n_{1}}^{m_{2}-i}}{(m_{1}+i)!}$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{m_{2}}\frac{(m_{1}+m_{2})!(y_{n_{1}}\partial_{x_{n_{1}}})^{m_{2}-i}(x_{n_{1}}^{m_{2}})x_{n_{1}+1}^{m_{1}+i}y_{n_{1}+1}^{m_{3}+i}}{(m_{2}-i)!(m_{1}+i)!}$ $\displaystyle=$ $\displaystyle\eta^{m_{1}+m_{2}}(x_{n_{1}}^{m_{2}}y_{n_{1}+1}^{m_{3}-m_{1}})\hskip 221.93158pt(3.85)$ and calculate $\displaystyle\eta^{m_{2}}(x_{n_{1}}^{m_{1}+m_{2}})y_{n_{1}+1}^{m_{3}}=\eta^{m_{2}}(x_{n_{1}}^{m_{1}+m_{2}}y_{n_{1}+1}^{m_{3}})$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{m_{2}}{m_{2}\choose m_{2}-i}[\prod_{r=1}^{m_{2}-i}(m_{1}+i+r)]y_{n_{1}}^{m_{2}-i}x_{n_{1}}^{m_{1}+i}x_{n_{1}+1}^{i}y_{n_{1}+1}^{m_{3}+i}$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{m_{2}}{m_{2}\choose i}\frac{[\prod_{r=1}^{m_{2}}(m_{1}+r)]y_{n_{1}}^{m_{2}-i}x_{n_{1}}^{m_{1}+i}x_{n_{1}+1}^{i}y_{n_{1}+1}^{m_{3}+i}}{\prod_{s=1}^{i}(m_{1}+s)}$ $\displaystyle=$ $\displaystyle[\prod_{r=1}^{m_{2}}(m_{1}+r)]\sum_{i=0}^{m_{2}}\frac{x_{n_{1}}^{m_{1}+i}x_{n_{1}+1}^{i}y_{n_{1}+1}^{m_{3}+i}\partial_{y_{n_{1}}}^{i}(y_{n_{1}}^{m_{2}})}{i![\prod_{s=1}^{i}(m_{1}+s)]}.\hskip 147.95424pt(3.86)$ Lemma 2.1 with $T_{1}=\partial_{x_{n_{1}}}\partial_{x_{n_{1}+1}},\;T_{1}^{-}=\int_{(x_{n_{1}})}\int_{(x_{n_{1}+1})}$ (cf. (2.6)) and $T_{2}=-y_{n_{1}+1}\partial_{y_{n_{1}}}$ tells us that $\mbox{Span}\\{h_{m_{1},m_{2},m_{3}},\eta^{m_{2}}(x_{n_{1}}^{m_{1}+m_{2}})y_{n_{1}+1}^{m_{3}}\mid m_{i}\in\mathbb{N}\\}$ is the solution space of (3.75) in $\mbox{Span}\\{x_{n_{1}}^{m_{1}}y_{n_{1}}^{m_{2}}x_{n_{1}+1}^{m_{3}}y_{n_{1}+1}^{m_{4}}\mid m_{i}\in\mathbb{N}\\}.$ In particular, (3.85) and (3.86) can be viewed as algorithms of solving the equation (3.75). On the other hand, $\partial_{x_{n_{1}}}(\phi_{0,m_{2},m_{3}})=m_{2}\psi_{1,m_{2},m_{3}-1},$ $None$ $\partial_{x_{n_{1}}}(\phi_{m_{1},m_{2},m_{3}})=(m_{1}+m_{2})\phi_{m_{1}-1,m_{2},m_{3}}\qquad\mbox{if}\;m_{1}>0,$ $None$ $\partial_{x_{n_{1}}}(\psi_{m_{1},m_{2},m_{3}})=m_{2}\psi_{m_{1}+1,m_{2}-1,m_{3}-1},$ $None$ $\partial_{y_{n_{1}}}(\phi_{m_{1},m_{2},m_{3}})=m_{2}(m_{1}+m_{2})\phi_{m_{1},m_{2}-1,m_{3}},$ $None$ $\partial_{y_{n_{1}}}(\psi_{m_{1},m_{2},m_{3}})=m_{2}(m_{1}+m_{2})\psi_{m_{1},m_{2}-1,m_{3}}.$ $None$ Applying the algorithm (3.85) to (3.81), we get that $\hat{\phi}_{m_{1},m_{2},m_{3}}y_{n+1}^{m_{4}}$ and $\hat{\psi}_{m_{1},m_{2},m_{3}}y_{n+1}^{m_{4}}$ are the solutions of (3.75) by (3.87)-(3.91), where $\hat{\phi}_{m_{1},m_{2},m_{3}}=\sum_{i=0}^{\infty}{m_{2}\choose i}\phi_{m_{1}+i,m_{2}-i,m_{3}}(x_{n_{1}+1}y_{n_{1}+1})^{i}=\eta^{m_{2}}(x_{n_{1}}^{m_{1}+m_{2}}y_{n_{1}+2}^{m_{3}}),$ $None$ $\displaystyle\qquad\hat{\psi}_{m_{1},m_{2},m_{3}}$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{m_{1}}{m_{1}+m_{2}\choose i}\psi_{m_{1}-i,m_{2},m_{3}+i}(x_{n_{1}+1}y_{n_{1}+1})^{i}$ $\displaystyle+\sum_{r=1}^{m_{2}}{m_{1}+m_{2}\choose m_{1}+r}\phi_{m_{2}-r,r,m_{1}+m_{3}}(x_{n_{1}+1}y_{n_{1}+1})^{m_{1}+r}$ $\displaystyle=$ $\displaystyle\eta^{m_{1}+m_{2}}(x_{n_{1}}^{m_{2}}y_{n_{1}+2}^{m_{1}+m_{3}}).\hskip 219.08612pt(3.93)$ Using the algorithm (3.86), we find that the solution space of (3.75) in (3.81) is $\displaystyle\mbox{Span}\\{x_{n_{1}+1}^{m_{1}}y_{n_{1}+1}^{m_{2}}x^{m_{3}}_{n_{1}+2}y^{m_{4}}_{n_{1}+2},\hat{\phi}_{m_{1},m_{2},m_{3}}y_{n_{1}+1}^{m_{4}},h_{m_{1},m_{2},m_{3}},$ $\displaystyle\qquad\;\;\hat{\psi}_{m_{1}+1,m_{2}+1,m_{3}}y_{n_{1}+1}^{m_{4}}\mid m_{i}\in\mathbb{N}\\}.\hskip 187.78836pt(3.94)$ According to (3.84), (3.92) and (3.93), $\partial_{x_{n_{1}+2}}(\hat{\phi}_{m_{1},m_{2},m_{3}})=m_{2}m_{3}\hat{\phi}_{m_{1},m_{2}-1,m_{3}-1},$ $None$ $\partial_{y_{n_{1}+1}}(\hat{\phi}_{m_{1},m_{2},m_{3}})=m_{2}x_{n_{1}+1}\hat{\phi}_{m_{1},m_{2}-1,m_{3}},$ $None$ $\partial_{y_{n_{1}+2}}(\hat{\phi}_{m_{1},m_{2},m_{3}})=m_{3}\hat{\phi}_{m_{1},m_{2},m_{3}-1},$ $None$ $\partial_{x_{n_{1}+2}}(\hat{\psi}_{m_{1},m_{2},m_{3}})=(m_{1}+m_{2})(m_{1}+m_{3})\hat{\psi}_{m_{1}-1,m_{2},m_{3}},$ $None$ $\partial_{y_{n_{1}+1}}(\hat{\psi}_{m_{1},m_{2},m_{3}})=(m_{1}+m_{2})x_{n_{1}+1}\hat{\psi}_{m_{1}-1,m_{2},m_{3}+1},$ $None$ $\partial_{y_{n_{1}+2}}(\hat{\psi}_{m_{1},m_{2},m_{3}})=(m_{1}+m_{3})\hat{\psi}_{m_{1},m_{2},m_{3}-1}.$ $None$ Put $\displaystyle\qquad g_{m_{1},m_{2},m_{3}}$ $\displaystyle=$ $\displaystyle\frac{(m_{1}+m_{2})!}{m_{1}!}\sum_{i=0}^{\infty}\frac{y_{n_{1}+2}^{i}y_{n_{1}+1}^{m_{1}+i}x_{n_{1}+1}^{m_{3}+i}\partial_{x_{n_{1}+2}}^{i}(x_{n_{1}+2}^{m_{2}})}{i!\prod_{r=1}^{i}(m_{1}+r)}$ $\displaystyle=$ $\displaystyle\eta^{m_{1}+m_{2}}(y_{n_{1}+2}^{m_{2}}x_{n_{1}+1}^{m_{3}-m_{1}})\hskip 210.55022pt(3.101)$ and $\displaystyle\qquad g^{\prime}_{m_{1},m_{2},m_{3}}$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{\infty}\frac{[\prod_{s=1}^{m_{2}}(m_{1}+s)]y_{n_{1}+2}^{m_{1}+i}y_{n_{1}+1}^{i}x_{n_{1}+1}^{m_{3}+i}\partial_{x_{n_{1}+2}}^{i}(x_{n_{1}+2}^{m_{2}})}{i!\prod_{r=1}^{i}(m_{1}+r)}$ $\displaystyle=$ $\displaystyle\eta^{m_{2}}(y_{n_{1}+2}^{m_{1}+m_{2}}x_{n_{1}+1}^{m_{3}}).\hskip 221.93158pt(3.102)$ Symmetrically, $\mbox{ Span}\\{g_{m_{1},m_{2},m_{3}},g^{\prime}_{m_{1},m_{2},m_{3}}\mid m_{i}\in\mathbb{N}\\}$ is the solution space of (3.76) in Span$\\{x_{n_{1}+1}^{m_{1}}y_{n_{1}+1}^{m_{2}}x^{m_{3}}_{n_{1}+2}y^{m_{4}}_{n_{1}+2}\mid m_{i}\in\mathbb{N}\\}$ by Lemma 2.1 with $T_{1}=\partial_{y_{n_{1}+1}}\partial_{y_{n_{1}+2}},\;T_{1}^{-}=\int_{(y_{n_{1}+1})}\int_{(y_{n_{1}+2})}$ (cf. (2.6)) and $T_{2}=-x_{n_{1}+1}\partial_{x_{n_{1}+2}}$. Observe that $\\{\hat{\phi}_{m_{1},m_{2},m_{3}},\hat{\psi}_{m_{1},m_{2},m_{3}},\\\ h_{m_{1},m_{2},m_{3}}\mid m_{i}\in\mathbb{N}\\}$ are solutions of (3.76). Thus the solution space of (3.76) in (3.94) is $\displaystyle\mbox{Span}\\{g_{m_{1},m_{2},m_{3}},g^{\prime}_{m_{1},m_{2},m_{3}},h_{m_{1},m_{2},m_{3}},\hat{\phi}_{m_{1},m_{2},m_{3}},$ $\displaystyle\qquad\;\;\hat{\phi}_{m_{1},m_{2},0}y_{n_{1}+1}^{m_{3}},\hat{\psi}_{m_{1}+1,m_{2}+1,m_{3}}\mid m_{i}\in\mathbb{N}\\}\hskip 136.5733pt(3.103)$ by (3.97) and (3.100). Expressions (3.85), (3.92), (3.93), (3.101) and (3.102) imply that the solution space of the singular vectors in ${\cal B}$ is $\displaystyle\mbox{Span}\\{\eta^{m_{2}}(x_{i}^{m_{1}}y_{j}^{m_{3}}),x_{n_{1}+1}^{m_{1}}y_{n_{1}+1}^{m_{2}},\eta^{m_{1}+m_{2}}(x_{n_{1}}^{m_{2}}y_{n_{1}+1}^{m_{3}-m_{1}}),\eta^{m_{1}+m_{2}}(y_{n_{1}+2}^{m_{2}}x_{n_{1}+1}^{m_{3}-m_{1}})$ $\displaystyle\qquad\;\;\mid m_{r}\in\mathbb{N};(i,j)=(n_{1},n_{1}+1),(n_{1},n_{1}+2),(n_{1}+1,n_{1}+2)\\}.\hskip 59.75095pt(3.104)$ Remind that in this case, ${\cal D}=-\sum_{i=1}^{n_{1}}x_{i}\partial_{y_{i}}+\partial_{x_{n_{1}+1}}\partial_{y_{n_{1}+1}}-\sum_{s=n_{1}+2}^{n}y_{s}\partial_{x_{s}}.$ $None$ We have ${\cal D}[\eta^{m_{1}+m_{2}}(x_{n_{1}}^{m_{2}}y_{n_{1}+1}^{m_{3}-m_{1}})]=(m_{1}+m_{2})m_{3}\eta^{m_{1}+m_{2}-1}(x_{n_{1}}^{m_{2}}y_{n_{1}+1}^{m_{3}-m_{1}})$ $None$ by (3.44). Thus we find a singular $\eta^{m_{1}+m_{2}}(x_{n_{1}}^{m_{2}}y_{n_{1}+1}^{-m_{1}})\in{\cal H}_{\langle m_{1},m_{2}\rangle}$ $None$ of new type if $m_{1},m_{2}\geq 1$. Symmetrically, $\eta^{m_{1}+m_{2}}(y_{n_{1}+2}^{m_{2}}x_{n_{1}+1}^{-m_{1}})\in{\cal H}_{\langle m_{2},m_{1}\rangle}$ is a singular vector. Recall the singular vectors $f_{\langle-m_{1},-m_{2}\rangle}=x_{n_{1}}^{m_{1}}y_{n_{1}+2}^{m_{2}}\in{\cal H}_{\langle-m_{1},-m_{2}\rangle},\qquad f_{\langle- m_{1},m_{2}\rangle}=x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}}\in{\cal H}_{\langle- m_{1},m_{2}\rangle},$ $None$ $f_{\langle m_{1},-m_{2}\rangle}=x_{n_{1}+1}^{m_{1}}y_{n_{1}+2}^{m_{2}}\in{\cal H}_{\langle m_{1},-m_{2}\rangle}.$ $None$ Moreover, we have the singular vectors $\eta^{-\ell_{1}-\ell_{2}}(f_{\langle\ell_{1},\ell_{2}\rangle})\in{\cal H}_{\langle-\ell_{2},-\ell_{1}\rangle}\qquad\mbox{for}\;\;\ell_{1},\ell_{2}\in\mathbb{Z}\;\mbox{with}\;\ell_{1}+\ell_{2}\leq-1.$ $None$ Therefore, any singular vector in ${\cal H}$ (cf. (3.38)) is a nonzero weight vector in $\displaystyle\mbox{Span}\\{f_{\langle\ell_{1},\ell_{2}\rangle},\eta^{-\ell_{1}^{\prime}-\ell_{2}^{\prime}}(f_{\langle\ell_{1}^{\prime},\ell_{2}^{\prime}\rangle}),\eta^{m_{1}+m_{2}}(x_{n_{1}}^{m_{2}}y_{n_{1}+1}^{-m_{1}}),\eta^{m_{1}+m_{2}}(y_{n_{1}+2}^{m_{2}}x_{n_{1}+1}^{-m_{1}})$ $\displaystyle\qquad\;\;\mid\ell_{1},\ell_{2},\ell_{1}^{\prime},\ell_{2}^{\prime}\in\mathbb{Z},\;m_{1},m_{2}\in\mathbb{N}+1;\ell_{1}\leq 0\;\mbox{or}\;\ell_{2}\leq 0;\ell_{1}^{\prime}+\ell_{2}^{\prime}\leq-1\\}.\hskip 34.14322pt(3.111)$ Assume $n_{2}=n$. We similarly find that the solution space of the singular vectors in ${\cal B}$ is $\mbox{Span}\\{\eta^{m_{2}}(x_{n-1}^{m_{1}}y_{n}^{m_{3}}),x_{n}^{m_{1}}y_{n}^{m_{2}},\eta^{m_{1}+m_{2}}(x_{n-1}^{m_{2}}y_{n}^{m_{3}-m_{1}})\mid m_{i}\in\mathbb{N}\\}.$ $None$ In particular, any singular vector in ${\cal H}$ (cf. (3.38)) is a nonzero weight vector in $\displaystyle\mbox{Span}\\{x_{n-1}^{m_{1}}y_{n}^{m_{2}},x_{n}^{m_{1}},\eta^{m_{1}+1}(x_{n-1}^{m_{1}+m_{2}+1}y_{n}^{m_{2}}),$ $\displaystyle\qquad\;\;\eta^{m_{1}+m_{2}+2}(x_{n-1}^{m_{2}+1}y_{n}^{-m_{1}-1})\mid m_{1},m_{2}\in\mathbb{N}\\}.\hskip 147.95424pt(3.113)$ By the arguments of (3.55)-(3.70), we have: Theorem 3.2. Suppose $n_{1}+1=n_{2}$. For $\ell_{1},\ell_{2}\in\mathbb{Z}$ with $\ell_{1}+\ell_{2}\leq 0$ or $n_{2}=n$ and $0\leq\ell_{2}\leq\ell_{1}$, ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}$ is an irreducible highest-weight $sl(n,\mathbb{F})$-module and ${\cal B}_{\langle\ell_{1},\ell_{2}\rangle}=\bigoplus_{m=0}^{\infty}\eta^{m}({\cal H}_{\langle\ell_{1}-m,\ell_{2}-m\rangle})$ $None$ is an orthogonal decomposition of irreducible submodules. In particular, ${\cal B}_{\langle\ell_{1},\ell_{2}\rangle}={\cal H}_{\langle\ell_{1},\ell_{2}\rangle}\oplus\eta({\cal B}_{\langle\ell_{1}-1,\ell_{2}-1\rangle})$. The symmetric bilinear form $(\cdot\|\cdot)$ restricted to $\eta^{m}({\cal H}_{\langle\ell_{1}-m,\ell_{2}-m\rangle})$ is nondegenerate. Assume $n_{2}<n$. For $m_{1},m_{2}\in\mathbb{N}+1$, ${\cal H}_{\langle m_{1},m_{2}\rangle}$ has exactly three singular vectors. All the submodules ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}$ for $\ell_{1},\ell_{2}\in\mathbb{Z}$ such $\ell_{1}+\ell_{2}>0$ and $\ell_{1}\ell_{2}\leq 0$ have two singular vectors. Consider $n_{2}=n$. For $m_{1},m_{2}\in\mathbb{N}$ with $m_{1}<m_{2}$, ${\cal H}_{\langle m_{1},m_{2}\rangle}$ is also an irreducible highest-weight $sl(n,\mathbb{F})$-module. All submodules ${\cal H}_{\langle- m_{1},m_{1}+m_{2}+1\rangle}$ with $m_{1},m_{2}\in\mathbb{N}$ have have exactly two singular vectors. Indeed, we have more detailed information. Suppose $n_{2}<n$. For $m_{1},m_{2}\in\mathbb{N}$, ${\cal H}_{\langle-m_{1},-m_{2}\rangle}$ has a highest-weight vector $x_{n_{1}}^{m_{1}}y_{n_{1}+2}^{m_{2}}$ of weight $m_{1}\lambda_{n_{1}-1}-(m_{1}+1)\lambda_{n_{1}}-(m_{2}+1)\lambda_{n_{1}+1}+m_{2}(1-\delta_{n_{1},n-2})\lambda_{n_{1}+2}$. When $m_{1},m_{2}\in\mathbb{N}$ with $m_{2}-m_{1}\geq 0$, ${\cal H}_{\langle m_{1},-m_{2}\rangle}$ has a highest-weight vector $x_{n_{1}+1}^{m_{1}}y_{n_{1}+2}^{m_{2}}$ of weight $-(m_{1}+1)\lambda_{n_{1}}+(m_{1}-m_{2}-1)\lambda_{n_{1}+1}+m_{2}(1-\delta_{n_{1},n-2})\lambda_{n_{1}+2}$. If $m_{1},m_{2}\in\mathbb{N}$ with $m_{1}-m_{2}\geq 0$, ${\cal H}_{\langle- m_{1},m_{2}\rangle}$ is has a highest-weight vector $x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}}$ of weight $m_{1}\lambda_{n_{1}-1}+(m_{2}-m_{1}-1)\lambda_{n_{1}}-(m_{2}+1)\lambda_{n_{1}+1}$. Assume $n_{2}=n$. For $m_{1},m_{2}\in\mathbb{N}$ with $m_{2}\leq m_{1}$, ${\cal H}_{\langle-m_{1},m_{2}\rangle}$ has a highest-weight vector $x_{n-1}^{m_{1}}y_{n}^{m_{2}}$ of weight $m_{1}\lambda_{n-2}+(m_{2}-m_{1}-1)\lambda_{n-1}$. Moreover, ${\cal H}_{\langle m,0\rangle}$ has a highest-weight vector $x_{n-1}^{m}$ of weight $m\lambda_{n-2}-(m+1)\lambda_{n-1}$ for $m\in\mathbb{Z}$. For $m_{1},m_{2}\in\mathbb{N}+1$, ${\cal H}_{\langle m_{1},m_{2}\rangle}$ has a highest-weight vector $\eta^{m_{1}+m_{2}}(x_{n-1}^{m_{2}}y_{n}^{-m_{1}})$ of weight $m_{2}\lambda_{n-2}+(m_{1}-m_{2}-1)\lambda_{n-1}$. Again ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}$ has a basis of the form (3.72). ## 4 The $sl(n,\mathbb{F})$-Variation II: $n_{1}=n_{2}$ In this section, we continue the discussion from last section. Recall $n\geq 2$. Case 3. $n_{1}=n_{2}$. In this case, the variated Laplace operator ${\cal D}=-\sum_{i=1}^{n_{1}}x_{i}\partial_{y_{i}}-\sum_{s=n_{1}+1}^{n}y_{s}\partial_{x_{s}}$ $None$ and its dual $\eta=\sum_{i=1}^{n_{1}}y_{i}\partial_{x_{i}}+\sum_{s=n_{2}+1}^{n}x_{s}\partial_{y_{s}}.$ $None$ First we consider the subcase $1<n_{1}<n-1$. Suppose that $f\in{\cal Q}$ is a singular vector. According to the arguments in (3.13)-(3.17), $f$ is a rational function in $\\{x_{n_{1}},x_{n_{1}+1},y_{n_{1}},y_{n_{1}+1},\zeta_{1},\zeta_{2}\\}$ $None$ (cf. (3.12)). Note $E_{n_{1},n_{1}+1}\|_{\cal Q}=\partial_{x_{n_{1}}}\partial_{x_{n_{1}+1}}-\partial_{y_{n_{1}}}\partial_{y_{n_{1}+1}}$ $None$ by (3.1)-(3.3). Now $E_{n_{1},n_{1}+1}(f)=0$ implies $(\partial_{x_{n_{1}}}\partial_{x_{n_{1}+1}}-\partial_{y_{n_{1}}}\partial_{y_{n_{1}+1}})(f)=0,$ $None$ equivalently, $\displaystyle(x_{n_{1}-1}x_{n_{1}+2}-y_{n_{1}-1}y_{n_{1}+2})f_{\zeta_{1}\zeta_{2}}-y_{n_{1}-1}f_{\zeta_{1}x_{n_{1}+1}}-x_{n_{1}-1}f_{\zeta_{1}y_{n_{1}+1}}$ $\displaystyle+y_{n_{1}+2}f_{\zeta_{2}x_{n_{1}}}+x_{n_{1}+2}f_{\zeta_{2}y_{n_{1}}}+f_{x_{n_{1}}x_{n_{1}+1}}-f_{y_{n_{1}}y_{n_{1}+1}}=0.\hskip 113.81102pt(4.6)$ According to (3.12), $y_{n_{1}-1}=x_{n_{1}}^{-1}y_{n_{1}}x_{n_{1}-1}-x_{n_{1}}^{-1}\zeta_{1},\qquad y_{n_{1}+2}=x_{n_{1}+1}^{-1}\zeta_{2}+x_{n_{1}+1}^{-1}y_{n_{1}+1}x_{n_{1}+2}.$ $None$ Substituting (4.7) into (4.6), the coefficient of $x_{n_{1}-1}x_{n_{1}+2}$ implies $f_{\zeta_{1}\zeta_{2}}=0$. Thus $f=g+h\qquad\mbox{with}\;\;g_{\zeta_{2}}=h_{\zeta_{1}}=0.$ $None$ Now (4.6) becomes $\displaystyle x_{n_{1}}^{-1}\zeta_{1}g_{\zeta_{1}x_{n_{1}+1}}-(x_{n_{1}}^{-1}y_{n_{1}}g_{\zeta_{1}x_{n_{1}+1}}+g_{\zeta_{1}y_{n_{1}+1}})x_{n_{1}-1}+(x_{n_{1}+1}^{-1}y_{n_{1}+1}h_{\zeta_{2}x_{n_{1}}}+h_{\zeta_{2}y_{n_{1}}})x_{n_{1}+2}$ $\displaystyle+x_{n_{1}+1}^{-1}\zeta_{2}h_{\zeta_{2}x_{n_{1}}}+g_{x_{n_{1}}x_{n_{1}+1}}-g_{y_{n_{1}}y_{n_{1}+1}}+h_{x_{n_{1}}x_{n_{1}+1}}-h_{y_{n_{1}}y_{n_{1}+1}}=0,\hskip 76.82234pt(4.9)$ which implies $x_{n_{1}}^{-1}y_{n_{1}}g_{\zeta_{1}x_{n_{1}+1}}+g_{\zeta_{1}y_{n_{1}+1}}=0,\qquad x_{n_{1}+1}^{-1}y_{n_{1}+1}h_{\zeta_{2}x_{n_{1}}}+h_{\zeta_{2}y_{n_{1}}}=0.$ $None$ For the representation purpose, we assume that $g$ is polynomial in $\zeta_{1}$ with $g\|_{\zeta_{1}=0}=0$ and $h$ is polynomial in $\zeta_{2}$. Set $\zeta_{3}=x_{n_{1}}y_{n_{1}+1}-x_{n_{1}+1}y_{n_{1}}.$ $None$ By (4.10), $g\;\;\mbox{is a function in}\;\;x_{n_{1}},y_{n_{1}},\zeta_{1},\zeta_{3}.$ $None$ Moreover, (4.9) says $-x_{n_{1}}^{-1}y_{n_{1}}\zeta_{1}g_{\zeta_{1}\zeta_{3}}-y_{n_{1}}g_{x_{n_{1}}\zeta_{3}}-x_{n_{1}}g_{y_{n_{1}}\zeta_{3}}=0.$ $None$ Again we can assume that $g=\hat{g}+\tilde{g}$ is polynomial in $x_{n_{1}},y_{n_{1}},\zeta_{3}$ with $\hat{g}\|_{\zeta_{3}=0}=0$ and $\tilde{g}_{\zeta_{3}}=0$. Then (4.13) is equivalent to $y_{n_{1}}\zeta_{1}\hat{g}_{\zeta_{1}}+x_{n_{1}}y_{n_{1}}\hat{g}_{x_{n_{1}}}+x_{n_{1}}^{2}\hat{g}_{y_{n_{1}}}=0.$ $None$ This shows that $\hat{g}$ is a function in $\zeta_{1}/x_{n_{1}},x_{n_{1}}^{2}-y_{n_{1}}^{2},\zeta_{3}$. If $\hat{g}$ is a polynomial, then $\hat{g}=0$. So the polynomial solution of $g$ must be a polynomial in $x_{n_{1}},y_{n_{2}},\zeta_{1}$ with $g_{\zeta_{1}}\neq 0$. Similarly, if $h_{\zeta_{2}}\neq 0$ and $h\|_{\zeta_{2}=0}=0$, the polynomial solution of $h$ must be a polynomial in $x_{n+1},y_{n+1},\zeta_{2}$. Assume $h_{\zeta_{2}}=0$. Then $h_{x_{n_{1}}x_{n_{1}+1}}-h_{y_{n_{1}}y_{n_{1}+1}}=0$ $None$ by (4.9). By Lemma 2.1, (3.78)-(3.80) and (4.2), the polynomial solution of $h$ must be in $\mbox{Span}\\{\eta^{m_{3}}(x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}})\mid m_{1},m_{2},m_{3}\in\mathbb{N}\\}.$ $None$ Therefore, a singular vector in ${\cal B}$ must be a nonzero weight vector in $\mbox{Span}\\{x_{n_{1}}^{m_{1}}y_{n_{1}}^{m_{2}}\zeta_{1}^{m_{3}+1},x_{n_{1}+1}^{m_{1}}y_{n_{1}+1}^{m_{2}}\zeta_{2}^{m_{3}+1},\eta^{m_{3}}(x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}})\mid m_{i}\in\mathbb{N}\\}.$ $None$ Note $x_{n_{1}}^{m_{1}}y_{n_{1}}^{m_{2}}\zeta_{1}^{m_{3}+1}\in{\cal B}_{\langle- m_{1}-m_{3}-1,m_{2}+m_{3}+1\rangle},$ $None$ $x_{n_{1}+1}^{m_{1}}y_{n_{1}+1}^{m_{2}}\zeta_{2}^{m_{3}+1}\in{\cal B}_{\langle m_{1}+m_{3}+1,-m_{2}-m_{3}-1\rangle}.$ $None$ Moreover, ${\cal D}(x_{n_{1}}^{m_{1}}y_{n_{1}}^{m_{2}}\zeta_{1}^{m_{3}+1})=-m_{2}x_{n_{1}}^{m_{1}+1}y_{n_{1}}^{m_{2}-1}\zeta_{1}^{m_{3}+1}=0\Longleftrightarrow m_{2}=0$ $None$ and ${\cal D}(x_{n_{1}+1}^{m_{1}}y_{n_{1}+1}^{m_{2}}\zeta_{2}^{m_{3}+1})=-m_{1}x_{n_{1}+1}^{m_{1}-1}y_{n_{1}+1}^{m_{2}+1}\zeta_{2}^{m_{3}+1}=0\Longleftrightarrow m_{1}=0$ $None$ by (3.12) and (4.1). Furthermore, $x_{n_{1}}^{m_{1}}y_{n_{1}}^{m_{2}}\zeta_{1}^{m_{3}+1}=\frac{\eta^{m_{2}}(x_{n_{1}}^{m_{1}+m_{2}}\zeta_{1}^{m_{3}+1})}{\prod_{r=1}^{m_{2}}(m_{1}+r)},\;\;x_{n_{1}+}^{m_{1}}y_{n_{1}+1}^{m_{2}}\zeta_{2}^{m_{3}+1}=\frac{\eta^{m_{1}}(y_{n_{1}+1}^{m_{1}+m_{2}}\zeta_{2}^{m_{3}+1})}{\prod_{r=1}^{m_{1}}(m_{2}+r)}$ $None$ by (4.2). Indeed, $\eta^{m_{1}+1}(x_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}})=\eta^{m_{1}+1}(y_{n_{1}+1}^{m_{1}}\zeta_{2}^{m_{2}})=0\qquad\mbox{for}\;\;m_{1},m_{2}\in\mathbb{N}.$ $None$ Since $x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}}\in{\cal H}_{\langle- m_{1},-m_{2}\rangle}$, (3.45) says that $\eta^{m}(x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}})$ with $m>0$ is a singular vector only if $m=m_{1}+m_{2}+1$. But $\eta^{m_{1}+m_{2}+1}(x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}})=0$ by (4.2). Thus any singular vector in ${\cal H}$ (cf. (3.38)) is a nonzero weight vector in $\mbox{Span}\\{x_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}+1},y_{n_{1}+1}^{m_{1}}\zeta_{2}^{m_{2}+1},x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}}\mid m_{1},m_{2}\in\mathbb{N}\\}.$ $None$ Since ${\cal B}$ is nilpotent with respect to $sl(n,\mathbb{F})_{+}$ (cf. (2.30)), any nonzero submodule of ${\cal B}$ has a singular vector. The above fact implies ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}=\\{0\\}$ for $\ell_{1},\ell_{2}\in\mathbb{Z}$ such that $\ell_{1}+\ell_{2}>0.$ Observe that $\displaystyle(x_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}}\|x_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}})$ $\displaystyle=$ $\displaystyle(\sum_{i=0}^{m_{2}}{m_{2}\choose i}(-1)^{i}x_{n_{1}-1}^{m_{2}-i}x_{n_{1}}^{m_{1}+i}y_{n_{1}-1}^{i}y_{n_{1}}^{m_{2}-i}\|\sum_{i=0}^{m_{2}}{m_{2}\choose i}(-1)^{i}x_{n_{1}-1}^{m_{2}-i}x_{n_{1}}^{m_{1}+i}y_{n_{1}-1}^{i}y_{n_{1}}^{m_{2}-i})$ $\displaystyle=$ $\displaystyle(-1)^{m_{1}+m_{2}}m_{2}!\sum_{i=0}^{m_{2}}{m_{2}\choose i}(m_{1}+i)!(m_{2}-i))!\neq 0\hskip 147.95424pt(4.24)$ by (3.55). Similarly, $(y_{n_{1}+1}^{m_{1}}\zeta_{2}^{m_{2}}\|y_{n_{1}}^{m_{1}}\zeta_{2}^{m_{2}})\neq 0$. Next we assume $n_{1}=n_{2}=1$ and $n\geq 3$. By the arguments in the above, a singular vector in ${\cal B}$ must be a nonzero weight vector in $\mbox{Span}\\{x_{n_{1}+1}^{m_{1}}y_{n_{1}+1}^{m_{2}}\zeta_{2}^{m_{3}+1},\eta^{m_{3}}(x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}})\mid m_{i}\in\mathbb{N}\\}.$ $None$ Thus any singular vector in ${\cal H}$ (cf. (3.38)) is a nonzero weight vector in $\mbox{Span}\\{y_{n_{1}+1}^{m_{1}}\zeta_{2}^{m_{2}+1},x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}}\mid m_{1},m_{2}\in\mathbb{N}\\}.$ $None$ The above fact implies ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}=\\{0\\}$ for $\ell_{1},\ell_{2}\in\mathbb{Z}$ such that $\ell_{1}+\ell_{2}>0$ or $\ell_{2}>0.$ Consider the subcase $n_{1}=n_{2}=n-1$ and $n\geq 3$. A singular vector in ${\cal B}$ must be a nonzero weight vector in $\mbox{Span}\\{x_{n_{1}}^{m_{1}}y_{n_{1}}^{m_{2}}\zeta_{1}^{m_{3}+1},\eta^{m_{3}}(x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}})\mid m_{i}\in\mathbb{N}\\}.$ $None$ Thus any singular vector in ${\cal H}$ (cf. (3.38)) is a nonzero weight vector in $\mbox{Span}\\{x_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}+1},x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}}\mid m_{1},m_{2}\in\mathbb{N}\\}.$ $None$ The above fact implies ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}=\\{0\\}$ for $\ell_{1},\ell_{2}\in\mathbb{Z}$ such that $\ell_{1}+\ell_{2}>0$ or $\ell_{1}>0.$ Suppose $n_{1}=n_{2}=1$ and $n=2$. A singular vector in ${\cal B}$ must be a nonzero weight vector in $\mbox{Span}\\{\eta^{m_{3}}(x_{1}^{m_{1}}y_{2}^{m_{2}})\mid m_{i}\in\mathbb{N}\\}.$ $None$ Thus any singular vector in ${\cal H}$ (cf. (3.38)) is a nonzero weight vector in $\mbox{Span}\\{x_{1}^{m_{1}}y_{2}^{m_{2}}\mid m_{1},m_{2}\in\mathbb{N}\\}.$ $None$ The above fact implies ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}=\\{0\\}$ for $\ell_{1},\ell_{2}\in\mathbb{Z}$ such that $\ell_{1}>0$ or $\ell_{2}>0.$ Finally, we assume $n_{1}=n_{2}=n$. A singular vector in ${\cal B}$ must be a nonzero weight vector in $\mbox{Span}\\{x_{n_{1}}^{m_{1}}y_{n_{1}}^{m_{2}}\zeta_{1}^{m_{3}}\mid m_{i}\in\mathbb{N}\\}.$ $None$ Thus any singular vector in ${\cal H}$ (cf. (3.38)) is a nonzero weight vector in $\mbox{Span}\\{x_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}}\mid m_{1},m_{2}\in\mathbb{N}\\}.$ $None$ The above fact implies ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}=\\{0\\}$ for $\ell_{1},\ell_{2}\in\mathbb{Z}$ such that $\ell_{1}+\ell_{2}>0.$ Indeed, all ${\cal B}_{\langle-m_{1},m_{2}\rangle}$ with $m_{1},m_{2}\in\mathbb{N}$ are finite-dimensional and completely reducible by Weyl’s Theorem of complete reducibility. Moreover, their irreducible summands are completely determined by (4.31). By the arguments of (3.55)-(3.70), we obtain: Theorem 4.1. Suppose $n_{1}=n_{2}$. Let $\ell_{1},\ell_{2}\in\mathbb{Z}$ such that $\ell_{2}\geq 0$ when $n_{1}=n$. Assume $\ell_{1}+\ell_{2}\leq 0$ and: (a) $\ell_{2}\leq 0$ if $n_{1}=1$ and $n\geq 3$; (b) $\ell_{1}\leq 0$ if $n_{1}=n-1$ and $n\geq 3$; (c) $\ell_{1},\ell_{2}\leq 0$ when $n_{1}=1$ and $n=2$. Then ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}$ is an irreducible highest-weight $sl(n,\mathbb{F})$-module and ${\cal B}_{\langle\ell_{1},\ell_{2}\rangle}=\bigoplus_{m=0}^{\infty}\eta^{m}({\cal H}_{\langle\ell_{1}-m,\ell_{2}-m\rangle})$ $None$ is an orthogonal decomposition of irreducible submodules. The symmetric bilinear form restricted to $\eta^{m}({\cal H}_{\langle\ell_{1}-m,\ell_{2}-m\rangle})$. In particular, ${\cal B}_{\langle\ell_{1},\ell_{2}\rangle}={\cal H}_{\langle\ell_{1},\ell_{2}\rangle}\oplus\eta({\cal B}_{\langle\ell_{1}-1,\ell_{2}-1\rangle})$. If the conditions fails, ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}=\\{0\\}$. When $n_{1}=n_{2}=n$, all the above irreducible modules are of finite-dimensional. Suppose $n_{1}<n-1$. Let $m_{1},m_{2}\in\mathbb{N}$. The subspace ${\cal H}_{\langle-m_{1},-m_{2}\rangle}$ has a highest-weight vector $x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}}$ of weight $m_{1}(1-\delta_{1,n_{1}})\lambda_{n_{1}-1}-(m_{1}+m_{2}+2)\lambda_{n_{1}}+m_{2}\lambda_{n_{1}+1}$. If $n_{1}\geq 2$, the subspace ${\cal H}_{\langle- m_{1}-m_{2}-1,m_{2}+1\rangle}$ has a highest-weight vector $x_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}+1}$ of weight $(m_{2}+1)\lambda_{n_{1}-2}-m_{1}\lambda_{n_{1}-1}-(m_{1}+m_{2}+3)\lambda_{n_{1}}$. The subspace ${\cal H}_{\langle m_{1}+1,-m_{2}-m_{1}-1\rangle}$ has a highest- weight vector $y_{n_{1}+1}^{m_{2}}\zeta_{2}^{m_{1}+1}$ of weight $-(m_{1}+m_{2}+3)\lambda_{n_{1}}+m_{2}\lambda_{n_{1}+1}-(m_{1}+1)(1-\delta_{n_{1},n-2})\lambda_{n_{1}+2}$. Consider $n_{1}=n-1$. The subspace ${\cal H}_{\langle-m_{1},-m_{2}\rangle}$ has a highest-weight vector $x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}}$ of weight $m_{1}(1-\delta_{n,2})\lambda_{n-2}-(m_{1}+m_{2}+2)\lambda_{n-1}$. If $n\geq 3$, the subspace ${\cal H}_{\langle-m_{1}-m_{2}-1,m_{2}+1\rangle}$ has a highest-weight vector $x_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}+1}$ of weight $(m_{2}+1)(1-\delta_{n,3})\lambda_{n-3}-m_{1}\lambda_{n-2}-(m_{1}+m_{2}+3)\lambda_{n-1}$. Assume $n_{1}=n$. The subspace ${\cal H}_{\langle-m_{1}-m_{2},m_{2}\rangle}$ has a highest-weight vector $x_{n}^{m_{1}}\zeta_{1}^{m_{2}}$ of weight $m_{2}(1-\delta_{n,2})\lambda_{n-2}+m_{1}\lambda_{n-1}$. Now we want to find an explicit expression for ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}$ when it is irreducible. Set ${\cal G}^{\prime}=\sum_{i=1}^{n_{1}}\sum_{j=n_{1}+1}^{n}\mathbb{F}E_{j,i},$ $None$ $\hat{\cal G}=H+\sum_{r,s\in\overline{1,n_{1}}\;\mbox{or}\;r,s\in\overline{n_{1}+1,n};r\neq s}\mathbb{F}E_{r,s}+\sum_{i=1}^{n_{1}}\sum_{j=n_{1}+1}^{n}\mathbb{F}E_{i,j}.$ $None$ Then ${\cal G}^{\prime}$ and $\hat{\cal G}$ are Lie subalgebras of $sl(n,\mathbb{F})$ and $sl(n,\mathbb{F})={\cal G}^{\prime}\oplus\hat{\cal G}$. By PBW Theorem, $U(sl(n,\mathbb{F}))=U({\cal G}^{\prime})U(\hat{\cal G})$. According to (1.6)-(1.8), $E_{r,s}\|_{\cal B}=-x_{s}\partial_{x_{r}}-y_{s}\partial_{y_{r}},\qquad E_{p,q}\|_{\cal B}=x_{p}\partial_{x_{q}}+y_{p}\partial_{y_{q}},$ $None$ $E_{r,p}\|_{\cal B}=\partial_{x_{r}}\partial_{x_{p}}-\partial_{y_{r}}\partial_{y_{p}},\qquad E_{p,r}\|_{\cal B}=-x_{r}x_{p}+y_{r}y_{p}$ $None$ for $r,s\in\overline{1,n_{1}}$ and $p,q\in\overline{n_{1}+1,n}$. First we assume $n_{1}<n$. For $m_{1},m_{2}\in\mathbb{N}$, we have $\displaystyle\qquad{\cal H}_{\langle-m_{1},-m_{2}\rangle}$ $\displaystyle=$ $\displaystyle U(sl(n,\mathbb{F}))(x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}})=U({\cal G}^{\prime})U(\hat{\cal G})(x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}})$ $\displaystyle=$ $\displaystyle\mbox{Span}\\{[\prod_{r=1}^{n_{1}}x_{r}^{l_{r}}][\prod_{s=1}^{n-n_{1}}y_{n_{1}+s}^{k_{s}}][\prod_{r=1}^{n_{1}}\prod_{s=1}^{n-n_{1}}(x_{r}x_{n_{1}+s}-y_{r}y_{n_{1}+s})^{l_{r,s}}]$ $\displaystyle\qquad\;\;\mid l_{r},k_{s},l_{r,s}\in\mathbb{N};\sum_{r=1}^{n_{1}}l_{r}=m_{1};\sum_{s=1}^{n-n_{1}}k_{s}=m_{2}\\}\hskip 68.28644pt(4.38)$ by (4.36) and (4.37). Furthermore, we assume $n_{1}>1$. We let $\displaystyle{\cal H}_{\langle-m_{1}-m_{2},m_{2}\rangle}^{\prime}$ $\displaystyle=$ $\displaystyle\mbox{Span}\\{[\prod_{r=1}^{n_{1}}x_{r}^{l_{r}}][\prod_{1\leq p<q\leq n_{1}}(x_{p}y_{q}-x_{q}y_{p})^{k_{p,q}}][\prod_{r=1}^{n_{1}}\prod_{s=1}^{n-n_{1}}(x_{r}x_{n_{1}+s}-y_{r}y_{n_{1}+s})^{l_{r,s}}]$ $\displaystyle\qquad\;\;\mid l_{r},k_{p,q},l_{r,s}\in\mathbb{N};\sum_{r=1}^{n_{1}}l_{r}=m_{1};\sum_{1\leq p<q\leq n_{1}}k_{p,q}=m_{2}\\}.\hskip 113.81102pt(4.39)$ By (3.38), (3.40) and (4.1), we have ${\cal H}_{\langle- m_{1}-m_{2},m_{2}\rangle}^{\prime}\subset{\cal H}_{\langle- m_{1}-m_{2},m_{2}\rangle}$. Moreover, (4.37) and (4.38) yield ${\cal H}_{\langle- m_{1}-m_{2},m_{2}\rangle}=U(sl(n,\mathbb{F}))(x_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}})=U({\cal G}^{\prime})U(\hat{\cal G})(x_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}})\subset{\cal H}_{\langle-m_{1}-m_{2},m_{2}\rangle}^{\prime}.$ $None$ Thus ${\cal H}_{\langle-m_{1}-m_{2},m_{2}\rangle}^{\prime}={\cal H}_{\langle- m_{1}-m_{2},m_{2}\rangle}$. Symmetrically, if $n_{1}=n_{2}<n-1$, $\displaystyle{\cal H}_{\langle m_{2},-m_{1}-m_{2}\rangle}=\mbox{Span}\\{[\prod_{n_{1}+1\leq p<q\leq n}(x_{p}y_{q}-x_{q}y_{p})^{k_{p,q}}][\prod_{r=1}^{n_{1}}\prod_{s=n_{1}+1}^{n}(x_{r}x_{s}-y_{r}y_{s})^{l_{r,s}}]$ $\displaystyle\times[\prod_{r=1}^{n-n_{1}}y_{n_{1}+r}^{l_{r}}]\mid l_{r},k_{p,q},l_{r,s}\in\mathbb{N};\sum_{r=1}^{n_{1}}l_{r}=m_{1};\sum_{n_{1}+1\leq p<q\leq n}k_{p,q}=m_{2}\\}.\hskip 76.82234pt(4.41)$ When $n_{1}=n_{2}=n$, by the arguments between (4.39) and(4.40), $\displaystyle{\cal H}_{\langle-m_{1}-m_{2},m_{2}\rangle}$ $\displaystyle=$ $\displaystyle\mbox{Span}\\{[\prod_{r=1}^{n}x_{r}^{l_{r}}][\prod_{1\leq p<q\leq n}(x_{p}y_{q}-x_{q}y_{p})^{k_{p,q}}]$ $\displaystyle\qquad\;\;\mid l_{r},k_{p,q}\in\mathbb{N};\sum_{r=1}^{n}l_{r}=m_{1};\sum_{1\leq p<q\leq n}k_{p,q}=m_{2}\\},\hskip 71.13188pt(4.42)$ which is of finite-dimensional. ## 5 The $o(2n,\mathbb{F})$-Variation Recall that ${\cal B}=\mathbb{F}[x_{1},...,x_{n},y_{1},...,y_{n}]$ and the representation of $o(2n,\mathbb{F})$ on ${\cal B}$ defined by (1.14)-(1.16). It is easy to verify $T\xi=\xi T\;\;\mbox{on}\;\;{\cal B}\qquad\mbox{for}\;\;\xi\in o(2n,\mathbb{F});T=\flat,\flat^{\prime},{\cal D},\eta$ $None$ by (1.9), (1.10), (2.13) and (3.4). Recall the notions ${\cal B}_{\langle k\rangle}$ and ${\cal H}_{\langle k\rangle}$ defined in (1.17). The ${\cal B}=\bigoplus_{k\in\mathbb{Z}}{\cal B}_{\langle k\rangle}$ forms a $\mathbb{Z}$-graded algebra and ${\cal H}_{\langle k\rangle}=\bigoplus_{\ell_{1},\ell_{2}\in\mathbb{Z};\ell_{1}+\ell_{2}=k}{\cal H}_{\langle\ell_{1},\ell_{2}\rangle}.$ $None$ Moreover, ${\cal B}_{\langle k\rangle}$ and ${\cal H}_{\langle k\rangle}$ are $o(2n,\mathbb{F})$-submodules. Recall ${\cal K}=\sum_{i,j=1}^{n}\mathbb{F}(E_{i,j}-E_{n+j,n+i})$. Theorem 5.1. For any $n_{1}-n_{2}+1-\delta_{n_{1},n_{2}}\geq k\in\mathbb{Z}$, ${\cal H}_{\langle k\rangle}$ is an irreducible $o(2n,\mathbb{F})$-submodule and ${\cal B}_{\langle k\rangle}=\bigoplus_{i=0}^{\infty}\eta^{i}({\cal H}_{\langle k-2i\rangle})$ $None$ is an orthogonal decomposition of irreducible submodules. In particular, ${\cal B}_{\langle k\rangle}={\cal H}_{\langle k\rangle}\oplus\eta({\cal B}_{\langle k-2\rangle})$. Moreover, the bilinear form $(\cdot\|\cdot)$ restricted to $\eta^{i}({\cal H}_{\langle k-2i\rangle})$ is nondegenerate. Furthermore, ${\cal H}_{\langle k\rangle}$ has a basis $\displaystyle\big{\\{}\sum_{i=0}^{\infty}\frac{(x_{n_{1}+1}y_{n_{1}+1})^{i}({\cal D}-\partial_{x_{n_{1}}+1}\partial_{y_{n_{1}}+1})^{i}(x^{\alpha}y^{\beta})}{\prod_{r=1}^{i}(\alpha_{n_{1}+1}+r)(\beta_{n_{1}+1}+r)}\mid\alpha,\beta\in\mathbb{N}\>^{n};$ $\displaystyle\alpha_{n_{1}+1}\beta_{n_{1}+1}=0;-\sum_{i=1}^{n_{1}}\alpha_{i}+\sum_{r=n_{1}+1}^{n}\alpha_{r}+\sum_{i=1}^{n_{2}}\beta_{i}-\sum_{r=n_{2}+1}^{n}\beta_{r}=k\big{\\}}\hskip 71.13188pt(5.4)$ when $n_{1}<n_{2}$. The module ${\cal H}_{\langle k\rangle}$ under the assumption is of highest-weight type only if $n_{2}=n$, in which case $x_{n_{1}}^{-k}$ is a highest-weight vector with weight $-k\lambda_{n_{1}-1}+(k-1)\lambda_{n_{1}}+[(k-1)\delta_{n_{1},n-1}-2k\delta_{n_{1},n}]\lambda_{n}.$ When $n_{1}=n_{2}=n$, all the irreducible modules ${\cal H}_{\langle k\rangle}$ with $0\geq k\in\mathbb{Z}$ are of $({\cal G},{\cal K})$-type. Proof. Let $n_{1}-n_{2}+1\geq k\in\mathbb{Z}.$ Note $sl(n,\mathbb{F})\|_{\cal B}$ is a subalgebra of $o(2n,\mathbb{F})\|_{\cal B}$. Suppose $n_{1}+1<n_{2}<n$. By (5.2), Theorem 3.1 and the paragraph below, the $sl(n,\mathbb{F})$-singular vectors in ${\cal H}_{\langle k\rangle}$ are: for $m_{1},m_{2}\in\mathbb{N}$, $x_{n_{1}}^{m_{1}}y_{n_{2}+1}^{m_{2}}\qquad\mbox{with}\;-(m_{1}+m_{2})=k,$ $None$ $x_{n_{1}+1}^{m_{1}}y_{n_{2}+1}^{m_{2}}\qquad\mbox{with}\;m_{1}-m_{2}=k,$ $None$ $x_{n_{1}}^{m_{1}}y_{n_{2}}^{m_{2}}\qquad\mbox{with}\;-m_{1}+m_{2}=k.$ $None$ Note $(E_{n+n_{2}+1,n_{1}}-E_{n+n_{1},n_{2}+1})\|_{\cal B}=-x_{n_{1}}\partial_{y_{n_{2}+1}}-y_{n_{1}}\partial_{x_{n_{2}+1}}$ $None$ by (1.16). So $(E_{n+n_{2}+1,n_{1}}-E_{n+n_{1},n_{2}+1})^{m_{2}}(x_{n_{1}}^{m_{1}}y_{n_{2}+1}^{m_{2}})=(-1)^{m_{2}}m_{2}!x_{n_{1}}^{-k}$ $None$ for the vectors in (5.5). Moreover, $(E_{n+n_{2}+1,n_{1}+1}-E_{n+n_{1}+1,n_{2}+1})\|_{\cal B}=\partial_{x_{n_{1}+1}}\partial_{y_{n_{2}+1}}-y_{n_{1}+1}\partial_{x_{n_{2}+1}}$ $None$ again by (1.16), which implies $(E_{n+n_{2}+1,n_{1}+1}-E_{n+n_{1}+1,n_{2}+1})^{m_{2}}(x_{n_{1}+1}^{m_{1}}y_{n_{2}+1}^{m_{2}})=m_{1}![\prod_{r=0}^{m_{1}-1}(m_{2}-r)]y_{n_{2}+1}^{-k}$ $None$ for the vectors in (5.6). Furthermore, $(E_{n_{1},n+n_{2}}-E_{n_{2},n+n_{1}})\|_{\cal B}=\partial_{x_{n_{1}}}\partial_{y_{n_{2}}}-x_{n_{2}}\partial_{y_{n_{1}}}$ $None$ by (1.15), which implies $(E_{n_{1},n+n_{2}}-E_{n_{2},n+n_{1}})^{m_{2}}(x_{n_{1}}^{m_{1}}y_{n_{2}}^{m_{2}})=m_{2}![\prod_{r=0}^{m_{2}-1}(m_{1}-r)]x_{n_{1}}^{-k}$ $None$ for the vectors in (5.7). On the other hand, $(E_{n_{1},n+n_{2}+1}-E_{n_{2}+1,n+n_{1}})\|_{\cal B}=-y_{n_{2}+1}\partial_{x_{n_{1}}}-x_{n_{2}+1}\partial_{y_{n_{1}}}$ $None$ by (1.15), which implies $(E_{n_{1},n+n_{2}+1}-E_{n_{2}+1,n+n_{1}})^{m_{2}}(x_{n_{1}}^{-k})=(-1)^{m_{2}}[\prod_{r=0}^{m_{2}-1}(-k-r)]x_{n_{1}}^{m_{1}}y_{n_{2}+1}^{m_{2}}$ $None$ for the vectors in (5.5). Moreover, $(E_{n_{1}+1,n+n_{2}+1}-E_{n_{2}+1,n+n_{1}+1})\|_{\cal B}=-x_{n_{1}+1}y_{n_{2}+1}-x_{n_{2}+1}\partial_{y_{n_{1}+1}}$ $None$ by (1.15), which implies $(E_{n_{1}+1,n+n_{2}+1}-E_{n_{2}+1,n+n_{1}+1})^{m_{2}}(y_{n_{2}+1}^{-k})=(-1)^{m_{2}}x_{n_{1}}^{m_{1}}y_{n_{2}+1}^{m_{2}}$ $None$ for the vectors in (5.6). Furthermore, $(E_{n+n_{2},n_{1}}-E_{n+n_{1},n_{2}})\|_{\cal B}=-x_{n_{1}}y_{n_{2}}-y_{n_{1}}\partial_{x_{n_{2}}}$ $None$ by (1.16), which implies $(E_{n+n_{2},n_{1}}-E_{n+n_{1},n_{2}})^{m_{2}}(x_{n_{1}}^{-k})=(-1)^{m_{2}}x_{n_{1}}^{m_{1}}y_{n_{2}}^{m_{2}}$ $None$ for the vectors in (5.7). Thus for any two vectors in (5.5)-(5.7), there exists an element in the universal enveloping algebra $U(o(2n,\mathbb{F}))$ which carries one to another. On the other hand, the vectors in (5.5)-(5.7) have distinct weights (see the paragraph below Theorem 3.1). Thus any nonzero submodule of ${\cal H}_{\langle k\rangle}$ must contain one of the vectors in (5.5)-(5.7). Hence all the vectors in (5.5)-(5.7) are in the submodule by (5.8)-(5.19). Therefore, the submodule must be equal to ${\cal H}_{\langle k\rangle}$, that is, ${\cal H}_{\langle k\rangle}$ is irreducible. By (5.16) and (5.18), ${\cal H}_{\langle k\rangle}$ is not of highest-weight type. The equation (5.3) follows from Theorem 3.1 and (5.2). Assume $n_{1}+1=n_{2}<n$. By Theorem 3.2 and the paragraph below, the $sl(n,\mathbb{F})$-singular vectors in ${\cal H}_{\langle k\rangle}$ are those in (5.5)-(5.7). So the theorem holds. Suppose $n_{1}<n_{2}=n$. By Theorems 3.1, 3.2 and the paragraph below them, the $sl(n,\mathbb{F})$-singular vectors in ${\cal H}_{\langle k\rangle}$ are those in (5.7). Expressions (5.13) and (5.19) imply the conclusions in the theorem. Recall $\zeta_{1}=x_{n_{1}-1}y_{n_{1}}-x_{n_{1}}y_{n_{1}-1},\;\;\;\zeta_{2}=x_{n_{2}+1}y_{n_{2}+2}-x_{n_{2}+2}y_{n_{2}}.$ $None$ In the case $n_{1}=n_{2}<n-1$, Theorem 4.1 tell us that the $sl(n,\mathbb{F})$-singular vectors in ${\cal H}_{\langle k\rangle}$ are those in (5.5) and $x_{n_{1}}^{-k}\zeta_{1}^{m+1}\qquad\mbox{for}\;\;m\in\mathbb{N},$ $None$ $y_{n_{1}+1}^{-k}\zeta_{2}^{m+1}\qquad\mbox{for}\;\;m\in\mathbb{N}.$ $None$ Again all the singular vectors have distinct weights. If $N$ is a nonzero submodule of ${\cal H}_{\langle k\rangle}$, then $N$ must contain one of the above $sl(n,\mathbb{F})$-singular vectors. If $N$ contains a singular vector in (5.5), then $x_{n_{1}}^{-k}\in N$ by (5.9). Suppose $x_{n_{1}}^{-k}\zeta_{1}^{m+1}\in N$ for some $m\in\mathbb{N}$. Note $(E_{n_{1}-1,n+n_{1}}-E_{n_{1},n+n_{1}-1})\|_{\cal B}=\partial_{x_{n_{1}-1}}\partial_{y_{n_{1}}}-\partial_{x_{n_{1}}}\partial_{y_{n_{1}-1}}$ $None$ by (1.15). Thus $\displaystyle(E_{n_{1}-1,n+n_{1}}-E_{n_{1},n+n_{1}-1})^{m+1}(x_{n_{1}}^{-k}\zeta_{1}^{m+1})$ $\displaystyle=$ $\displaystyle\left[\sum_{r=0}^{m+1}(-1)^{r}{m+1\choose r}(\partial_{x_{n_{1}-1}}\partial_{y_{n_{1}}})^{m+1-r}(\partial_{x_{n_{1}}}\partial_{y_{n_{1}-1}})^{r}\right]$ $\displaystyle\left[\sum_{s=0}^{m+1}(-1)^{s}{m+1\choose s}(x_{n_{1}-1}y_{n_{1}})^{m+1-s}x_{n_{1}}^{-k+s}y_{n_{1}-1}^{s}\right]$ $\displaystyle=$ $\displaystyle\left(\sum_{r=0}^{m+1}{m+1\choose r}^{2}[(m+1-r!)]^{2}r![\prod_{i=1}^{r}(-k+i)]\right)x_{n_{1}}^{-k}$ $\displaystyle=$ $\displaystyle[(m+1)!]^{2}\left(\sum_{r=0}^{m+1}{-k+r\choose r}\right)x_{n_{1}}^{-k}\in N.\hskip 173.56198pt(5.23)$ So we have $x_{n_{1}}^{-k}\in N$ again. Symmetrically, it holds if $y_{n_{1}+1}^{-k}\zeta_{2}^{m+1}\in N$ for some $m\in\mathbb{N}$. Therefore, we always have $x_{n_{1}}^{-k}\in N$. According to (5.15), $N$ contains all the singular vectors in (5.5). Observe $(E_{n+n_{1}-1,n_{1}}-E_{n+n_{1},n_{1}-1})\|_{\cal B}=\zeta_{1},\;\;(E_{n_{1}+2,n+n_{1}+1}-E_{n_{1}+1,n+n_{1}+2})\|_{\cal B}=\zeta_{2}$ $None$ as multiplication operators on ${\cal B}$ by (1.15) and (1.16). Thus $(E_{n+n_{1}-1,n_{1}}-E_{n+n_{1},n_{1}-1})^{m+1}(x_{n_{1}}^{-k})=x_{n_{1}}^{-k}\zeta_{1}^{m+1},$ $None$ $(E_{n_{1}+2,n+n_{1}+1}-E_{n_{1}+1,n+n_{1}+2})^{m+1}(x_{n_{1}}^{-k})=x_{n_{1}}^{-k}\zeta_{2}^{m+1}\in N.$ $None$ Thus $N$ contains all the $sl(n,\mathbb{F})$-singular vectors in ${\cal H}_{\langle k\rangle}$, which implies that it contains all ${\cal H}_{\langle\ell_{1},\ell_{2}\rangle}\subset{\cal H}_{\langle k\rangle}$. So $N={\cal H}_{\langle k\rangle}$, that is, ${\cal H}_{\langle k\rangle}$ is an irreducible $o(2n,\mathbb{F})$-module, which is of $({\cal G},{\cal K})$-type if $n_{1}=n_{2}=n$ by (5.2). The basis (5.4) is obtained by (3.72) and (5.2). $\qquad\Box$ Finally, we want to find an expression for ${\cal H}_{\langle k\rangle}$ for $0\geq k\in\mathbb{Z}$ when $n_{1}=n_{2}$. First we assume $n_{1}=n_{2}=1$ and $n\geq 3$. According to (4.26), (4.38) and (4.41) $\displaystyle{\cal H}_{\langle-k\rangle}$ $\displaystyle=$ $\displaystyle\mbox{Span}\\{[\prod_{r=2}^{n}y_{r}^{\hat{l}_{r}}][\prod_{2\leq p<q\leq n}(x_{p}y_{q}-x_{q}y_{p})^{\hat{k}_{p,q}}][\prod_{s=2}^{n}(x_{1}x_{s}-y_{1}y_{s})^{\hat{l}_{s}}],x_{1}^{l}[\prod_{s=2}^{n}y_{s}^{k_{s}}]$ $\displaystyle\times[\prod_{s=2}^{n}(x_{1}x_{s}-y_{1}y_{s})^{l_{s}}]\mid l,k_{s},l_{s},\hat{l},\hat{k}_{p,q},\hat{l}_{s}\in\mathbb{N};l+\sum_{s=2}^{n}k_{s}=\sum_{r=2}^{n}\hat{l}_{r}=k\\}.\hskip 51.21504pt(5.27)$ Next we consider the subcase $1<n_{1}=n_{2}<n-1$. By (4.23), (4.38), (4.39) (note ${\cal H}_{\langle-m_{1}-m_{2},m_{2}\rangle}^{\prime}={\cal H}_{\langle- m_{1}-m_{2},m_{2}\rangle}$) and (4.41), we have $\displaystyle{\cal H}_{\langle-k\rangle}$ $\displaystyle=$ $\displaystyle\mbox{Span}\\{[\prod_{r=1}^{n_{1}}x_{r}^{l_{r}^{\prime}}][\prod_{1\leq p<q\leq n_{1}}(x_{p}y_{q}-x_{q}y_{p})^{k_{p,q}^{\prime}}][\prod_{r=1}^{n_{1}}\prod_{s=n_{1}+1}^{n}(x_{r}x_{s}-y_{r}y_{s})^{l_{r,s}^{\prime}}],$ $\displaystyle[\prod_{r=1}^{n-n_{1}}y_{n_{1}+r}^{\hat{l}_{r}}][\prod_{n_{1}+1\leq p<q\leq n}(x_{p}y_{q}-x_{q}y_{p})^{\hat{k}_{p,q}}][\prod_{r=1}^{n_{1}}\prod_{s=n_{1}+1}^{n}(x_{r}x_{s}-y_{r}y_{s})^{\hat{l}_{r,s}}],$ $\displaystyle[\prod_{r=1}^{n_{1}}x_{r}^{l_{r}}][\prod_{s=1}^{n-n_{1}}y_{n_{1}+s}^{k_{s}}][\prod_{r=1}^{n_{1}}\prod_{s=1}^{n-n_{1}}(x_{r}x_{n_{1}+s}-y_{r}y_{n_{1}+s})^{l_{r,s}}]\mid l_{r},k_{s},l_{r,s},l_{r}^{\prime},k_{p,q}^{\prime},$ $\displaystyle l_{r,s}^{\prime},\hat{l}_{r},\hat{k}_{p,q},\hat{l}_{r,s}\in\mathbb{N};\sum_{r=1}^{n_{1}}l_{r}+\sum_{s=1}^{n-n_{1}}k_{s}=\sum_{r=1}^{n_{1}}l_{r}^{\prime}=\sum_{r=1}^{n-n_{1}}\hat{l}_{r}=k\\}.\hskip 102.43008pt(5.28)$ Consider the subcase $n_{1}=n_{2}=n-1$ and $n\geq 3$. By (4.28), (4.38) and (4.39) (note ${\cal H}_{\langle-m_{1}-m_{2},m_{2}\rangle}^{\prime}={\cal H}_{\langle-m_{1}-m_{2},m_{2}\rangle}$), we obtain $\displaystyle{\cal H}_{\langle-k\rangle}$ $\displaystyle=$ $\displaystyle\mbox{Span}\\{[\prod_{r=1}^{n-1}x_{r}^{l_{r}^{\prime}}][\prod_{1\leq p<q\leq n-1}(x_{p}y_{q}-x_{q}y_{p})^{k_{p,q}^{\prime}}][\prod_{r=1}^{n-1}(x_{r}x_{n}-y_{r}y_{n})^{\bar{l}_{r}^{\prime}}],[\prod_{r=1}^{n-1}x_{r}^{l_{r}}]y_{n}^{\hat{k}}$ $\displaystyle\times[\prod_{r=1}^{n-1}(x_{r}x_{n}-y_{r}y_{n})^{\bar{l}_{r}}]\mid l_{r},\hat{k},\bar{l}_{r},l_{r}^{\prime},k_{p,q}^{\prime},\bar{l}_{r}^{\prime}\in\mathbb{N};\sum_{r=1}^{n-1}l_{r}+\hat{k}=\sum_{r=1}^{n-1}l_{r}^{\prime}=k\\}.\hskip 42.67912pt(5.29)$ Suppose $n_{1}=n_{2}=1$ and $n=2$. According to (4.30) and (4.38), ${\cal H}_{\langle-k\rangle}=\mbox{Span}\\{[x_{1}^{r}y_{2}^{s}(x_{1}x_{2}-y_{1}y_{2})^{l}\mid r,s,l\in\mathbb{N};r+s=k\\}.$ $None$ Finally we assume $n_{1}=n_{2}=n$. By (4.32) and (4.39) (note ${\cal H}_{\langle-m_{1}-m_{2},m_{2}\rangle}^{\prime}={\cal H}_{\langle- m_{1}-m_{2},m_{2}\rangle}$), ${\cal H}_{\langle-k\rangle}=\mbox{Span}\\{\prod_{r=1}^{n}x_{r}^{l_{r}}][\prod_{1\leq p<q\leq n}(x_{p}y_{q}-x_{q}y_{p})^{k_{p,q}}]\mid l_{r},k_{p,q}\in\mathbb{N};\sum_{r=1}^{n}l_{r}=k\\},$ $None$ whose $({\cal G},{\cal K})$-module structure is given by ${\cal H}_{\langle-k\rangle}=\bigoplus_{m=0}^{\infty}{\cal H}_{\langle-k-m,m\rangle}$ with ${\cal H}_{\langle-k-m,m\rangle}$ given in (4.42). ## 6 The $o(2n+1,\mathbb{F})$-Variation Recall $o(2n+1,\mathbb{F})=o(2n,\mathbb{F})\oplus\bigoplus_{i=1}^{n}[\mathbb{F}(E_{0,i}-E_{n+i,0})+\mathbb{F}(E_{0,n+i}-E_{i,0})]$ $None$ and ${\cal B}^{\prime}=\mathbb{F}[x_{0},x_{1},...,x_{n},y_{1},...,y_{n}]$. Fix $n_{1},n_{2}\in\overline{1,n}$ such that $n_{1}\leq n_{2}$. The representation of $o(2n+1,\mathbb{F})$ on ${\cal B}^{\prime}$ by the differential operators in (1.14)-(1.16), (1.19) and (1.20). Recall ${\cal B}^{\prime}_{\langle k\rangle}=\sum_{i=0}^{\infty}{\cal B}_{\langle k-i\rangle}x_{0}^{i}.$ Then all ${\cal B}^{\prime}_{\langle k\rangle}$ with $k\in\mathbb{Z}$ are $o(2n+1,\mathbb{F})$-submodules and ${\cal B}^{\prime}=\bigoplus_{k\in\mathbb{Z}}{\cal B}^{\prime}_{\langle k\rangle}$ forms a $\mathbb{Z}$-graded algebra. Moreover, the variated Laplace operator ${\cal D}^{\prime}=\partial_{x_{0}}^{2}+2{\cal D}$ by (1.21) and its dual $\eta^{\prime}=x_{0}^{2}+2\eta$ by (1.22). A straightforward verification shows ${\cal D}^{\prime}\xi=\xi{\cal D}^{\prime},\;\xi\eta^{\prime}=\eta^{\prime}\xi\;\mbox{on}\;{\cal B}^{\prime}\qquad\mbox{for}\;\;\xi\in o(2n+1,\mathbb{F}).$ $None$ As in the introduction, ${\cal H}^{\prime}_{\langle k\rangle}=\\{f\in{\cal B}^{\prime}{\langle k\rangle}\mid{\cal D}^{\prime}(f)=0\\}.$ According to (6.2), ${\cal H}^{\prime}_{\langle k\rangle}$ is an $o(2n+1,\mathbb{F})$-submodule. By Lemma 2.1 with $T_{1}=\partial_{x_{0}}^{2},\;T_{1}^{-}=\int_{(x_{0})}^{(2)}$ (cf. (2.6) and (2.7)) and $T_{2}=2{\cal D}$, we obtain ${\cal H}^{\prime}_{\langle k\rangle}=\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i}{\cal D}^{i}}{(2i)!}\right)({\cal B}_{\langle k\rangle})\oplus\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i+1}{\cal D}^{i}}{(2i+1)!}\right)({\cal B}_{\langle k-1\rangle}).$ $None$ Recall ${\cal K}=\sum_{i,j=1}^{n}\mathbb{F}(E_{i,j}-E_{n+j,n+i})$. Theorem 6.1. For any $n_{1}-n_{2}+1-\delta_{n_{1},n_{2}}\geq k\in\mathbb{Z}$, ${\cal H}^{\prime}_{\langle k\rangle}$ is an irreducible $o(2n+1,\mathbb{F})$-submodule and ${\cal B}^{\prime}_{\langle k\rangle}=\bigoplus_{i=0}^{\infty}(\eta^{\prime})^{i}({\cal H}^{\prime}_{\langle k-2i\rangle})$ $None$ is an orthogonal decomposition of irreducible submodules. In particular, ${\cal B}^{\prime}_{\langle k\rangle}={\cal H}^{\prime}_{\langle k\rangle}\oplus\eta^{\prime}({\cal B}^{\prime}_{\langle k-2\rangle})$. Moreover, the bilinear form $(\cdot\|\cdot)$ restricted to $(\eta^{\prime})^{i}({\cal H}_{\langle k-2i\rangle}^{\prime})$ is nondegenerate. Furthermore, ${\cal H}_{\langle k\rangle}$ has a basis $\displaystyle\big{\\{}\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i+\iota}{\cal D}^{i}(x^{\alpha}y^{\beta})}{(2i+\iota)!}\mid\alpha,\beta\in\mathbb{N}\>^{n};\iota=0,1;$ $\displaystyle-\sum_{i=1}^{n_{1}}\alpha_{i}+\sum_{r=n_{1}+1}^{n}\alpha_{r}+\sum_{i=1}^{n_{2}}\beta_{i}-\sum_{r=n_{2}+1}^{n}\beta_{r}=k-\iota\big{\\}}.\hskip 125.19194pt(6.5)$ The module ${\cal H}_{\langle k\rangle}^{\prime}$ under the assumption is of highest-weight type only if $n_{2}=n$, in which case $x_{n_{1}}^{-k}$ is a highest-weight vector with weight $-k\lambda_{n_{1}-1}+(k-1)\lambda_{n_{1}}+[(k-1)\delta_{n_{1},n-1}-2k\delta_{n_{1},n}]\lambda_{n}.$ When $n_{1}=n_{2}=n$, all the irreducible modules ${\cal H}_{\langle k\rangle}$ with $0\geq k\in\mathbb{Z}$ are of $({\cal G},{\cal K})$-type. Proof. Observe that $(x_{0}^{r}x^{\alpha}y^{\beta}\|x_{0}^{s}x^{\alpha_{1}}y^{\beta_{1}})=\delta_{r,s}\delta_{\alpha,\alpha_{1}}\delta_{\beta,\beta_{1}}(-1)^{\sum_{i=1}^{n_{1}}\alpha_{i}+\sum_{r=n_{2}+1}^{n}\beta_{r}}r!\alpha!\beta!$ $None$ for $r,s\in\mathbb{N}$ and $\alpha,\beta,\alpha_{1},\beta_{1}\in\mathbb{N}\>^{n}.$ By (1.21) and (1.22), $({\cal D}^{\prime}(f)\|g)=(f\|\eta^{\prime}(g))\qquad\mbox{for}\;\;f,g\in{\cal B}^{\prime}.$ $None$ Let $n_{1}-n_{2}+1\geq k\in\mathbb{Z}$. First by (5.3) and (6.3), ${\cal H}^{\prime}_{\langle k\rangle}=\bigoplus_{r=0}^{\infty}\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i}{\cal D}^{i}}{(2i)!}\right)(\eta^{r}({\cal H}_{\langle k-2r\rangle}))\oplus\bigoplus_{s=0}^{\infty}\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i+1}{\cal D}^{i}}{(2i+1)!}\right)(\eta^{s}({\cal H}_{\langle k-2s-1\rangle})).$ $None$ Let $N$ be a nonzero submodule of ${\cal H}^{\prime}_{\langle k\rangle}$. By comparing weights and the arguments in (5.5)-(5.13) and (5.21)-(5.23), we have $\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i}{\cal D}^{i}}{(2i)!}\right)(\eta^{m_{1}}(x_{n_{1}}^{-k+2m_{1}}))\in N$ $None$ for some $m_{1}\in\mathbb{N}$ or $\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i+1}{\cal D}^{i}}{(2i+1)!}\right)(\eta^{m_{2}}(x_{n_{1}}^{-k+2m_{2}+1}))\in N$ $None$ for some $m_{2}\in\mathbb{N}$. Note $(E_{n_{1},0}-E_{0,n+n_{1}})=\partial_{x_{0}}\partial_{x_{n_{1}}}-x_{0}\partial_{y_{n_{1}}}$ $None$ by (1.19) and (1.20). Recall ${\cal D}=-\sum_{i=1}^{n_{1}}x_{i}\partial_{y_{i}}+\sum_{r=n_{1}+1}^{n_{2}}\partial_{x_{r}}\partial_{y_{r}}-\sum_{s=n_{2}+1}^{n}y_{s}\partial_{x_{s}}$ $None$ and $\eta=\sum_{i=1}^{n_{1}}y_{i}\partial_{x_{i}}+\sum_{r=n_{1}+1}^{n_{2}}x_{r}y_{r}+\sum_{s=n_{2}+1}^{n}x_{s}\partial_{y_{s}}.$ $None$ Then (3.44) gives $\displaystyle(E_{n_{1},0}-E_{0,n+n_{1}})\left[\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i}{\cal D}^{i}}{(2i)!}\right)(\eta^{m_{1}}(x_{n_{1}}^{-k+2m_{1}}))\right]$ $\displaystyle=$ $\displaystyle\left(\sum_{i=0}^{\infty}\frac{(i+1)m_{1}(-k+2m_{1})(-2)^{i+1}x_{0}^{2i+1}{\cal D}^{i}}{(2i+1)!}\right)(\eta^{m_{1}-1}(x_{n_{1}}^{-k+2m_{1}-1}))$ $\displaystyle+\left(\sum_{i=0}^{\infty}\frac{(-k+2m_{1})(-2)^{i+1}x_{0}^{2i+1}{\cal D}^{i+1}}{(2i+1)!}\right)(\eta^{m_{1}}(x_{n_{1}}^{-k+2m_{1}-1}))$ $\displaystyle- m_{1}(-k+2m_{1})\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i+1}{\cal D}^{i}}{(2i)!}\right)(\eta^{m_{1}-1}(x_{n_{1}}^{-k+2m_{1}-1}))$ $\displaystyle=$ $\displaystyle m_{1}(-k+2m_{1})\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i+1}{\cal D}^{i}}{(2i+1)!}\right)(\eta^{m_{1}-1}(x_{n_{1}}^{-k+2m_{1}-1}))$ $\displaystyle+m_{1}(-k+2m_{1})(m_{1}-k+n_{1}-n_{2})\left(\sum_{i=0}^{\infty}\frac{(-2)^{i+1}x_{0}^{2i+1}{\cal D}^{i}}{(2i+1)!}\right)(\eta^{m_{1}-1}(x_{n_{1}}^{-k+2m_{1}-1}))$ $\displaystyle=$ $\displaystyle m_{1}(-k+2m_{1})(2m_{1}-2k+2n_{1}-2n_{2}+1)$ $\displaystyle\times\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i+1}{\cal D}^{i}}{(2i+1)!}\right)(\eta^{m_{1}-1}(x_{n_{1}}^{-k+2m_{1}-1})).\hskip 167.87108pt(6.14)$ Moreover, $\displaystyle(E_{n_{1},0}-E_{0,n+n_{1}})\left[\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i+1}{\cal D}^{i}}{(2i+1)!}\right)(\eta^{m_{2}}(x_{n_{1}}^{-k+2m_{2}+1}))\right]$ $\displaystyle=$ $\displaystyle\left(\sum_{i=0}^{\infty}\frac{im_{2}(-k+2m_{2}+1)(-2)^{i}x_{0}^{2i}{\cal D}^{i-1}}{(2i)!}\right)(\eta^{m_{2}-1}(x_{n_{1}}^{-k+2m_{2}}))$ $\displaystyle+\left(\sum_{i=0}^{\infty}\frac{(-k+2m_{2}+1)(-2)^{i}x_{0}^{2i}{\cal D}^{i}}{(2i)!}\right)(\eta^{m_{2}}(x_{n_{1}}^{-k+2m_{2}}))\hskip 199.16928pt$ $\displaystyle- m_{2}(-k+2m_{2}+1)\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i+2}{\cal D}^{i}}{(2i+1)!}\right)(\eta^{m_{2}-1}(x_{n_{1}}^{-k+2m_{2}}))$ $\displaystyle=$ $\displaystyle(-k+2m_{2}+1)\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i}{\cal D}^{i}}{(2i)!}\right)(\eta^{m_{2}}(x_{n_{1}}^{-k+2m_{2}})).\hskip 133.72786pt(6.15)$ Note $k\leq 0$ by our assumption. Using (6.9), (6.10), (6.14), (6.15) and induction, we obtain $x_{n_{1}}^{-k}\in N$. Observe $(E_{n+n_{1},0}-E_{0,n_{1}})\|_{{\cal B}^{\prime}}=x_{0}x_{n_{1}}+y_{n_{1}}\partial_{x_{0}}$ $None$ by (1.19) and (1.20). Then $(E_{n+n_{1},0}-E_{0,n_{1}})^{m}(x_{n_{1}}^{-k})=x_{0}^{m}x_{n_{1}}^{-k+m}+P_{m}\in N,$ $None$ where the degree of $P_{m}$ with respect to $x_{0}$ is less than $m$. For any $f\in{\cal H}_{\langle k-2m\rangle}$ and $g\in{\cal H}_{\langle k-2m-1\rangle}$ , (3.44) and (5.2) says that $\displaystyle\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i}{\cal D}^{i}}{(2i)!}\right)(\eta^{m}(f))$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{m}\frac{2^{i}x_{0}^{2i}\prod_{r=1}^{i}(m-r)(m-k+n_{1}-n_{2}+1+r)}{(2i)!}\eta^{m-i}(f)\hskip 85.35826pt(6.18)$ and $\displaystyle\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i+1}{\cal D}^{i}}{(2i)!}\right)(\eta^{m}(g))$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{m}\frac{2^{i}x_{0}^{2i+1}\prod_{r=1}^{i}(m-r)(m-k+n_{1}-n_{2}+2+r)}{(2i+1)!}\eta^{m-i}(g).\hskip 71.13188pt(6.19)$ This shows that if $x_{0}^{m}$ is the highest $x_{0}$-power of a nonzero element in ${\cal H}^{\prime}_{\langle k\rangle}$, then its coefficient must be in ${\cal H}_{\langle k-m\rangle}$ by (6.8). On the other hand, (6.17) implies that $\mbox{the coefficients of}\;x_{0}^{m}\;\mbox{in}\;U(o(2n,\mathbb{F}))[(E_{n+n_{1},0}-E_{0,n_{1}})^{m}(x_{n_{1}}^{-k})]={\cal H}_{\langle k-m\rangle},$ $None$ because it is an irreducible $o(2n,\mathbb{F})$-module by Theorem 5.1. By induction on $m$, we can prove ${\cal H}^{\prime}_{\langle k\rangle}\subset\sum_{m=0}^{\infty}U(o(2n,\mathbb{F}))[(E_{n+n_{1},0}-E_{0,n_{1}})^{m}(x_{n_{1}}^{-k})]\subset N.$ $None$ Thus $N={\cal H}^{\prime}_{\langle k\rangle}$. This shows that ${\cal H}^{\prime}_{\langle k\rangle}$ is irreducible. Since the bilinear form $(\cdot\|\cdot)$ restricted to ${\cal H}_{\langle k\rangle}\subset{\cal H}_{\langle k\rangle}^{\prime}$ is nondegenerate, the irreducibility of ${\cal H}^{\prime}_{\langle k\rangle}$ implies that the symmetric bilinear form $(\cdot\|\cdot)$ restricted to ${\cal H}_{\langle k\rangle}^{\prime}$ is nondegenerate. Next want to prove $\left({\cal H}_{\langle k\rangle}^{\prime}\|{\cal H}_{\langle k^{\prime}\rangle}^{\prime}\right)=\\{0\\}\qquad\mbox{for}\;\;n_{1}-n_{2}+1-\delta_{n_{1},n_{2}}\geq k,k^{\prime}\in\mathbb{Z}\;\mbox{such that}\;\;k\neq k^{\prime}.$ $None$ For any $f\in{\cal H}_{\langle k-2m\rangle}$ and $f^{\prime}\in{\cal H}_{\langle k^{\prime}-2m^{\prime}\rangle}$, (3.64). (5.2), (6.6) and (6.18) yield $\displaystyle\left(\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i}{\cal D}^{i}}{(2i)!}\right)(\eta^{2m}(f))\|\left(\sum_{r=0}^{\infty}\frac{(-2)^{r}x_{0}^{2r}{\cal D}^{r}}{(2r)!}\right)(\eta^{2m^{\prime}}(f^{\prime}))\right)$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{m}\frac{2^{2i}}{(2i)!}[\prod_{s=1}^{i}(m-s)(m-k+n_{1}-n_{2}+1+s)]$ $\displaystyle\times[\prod_{s^{\prime}=1}^{i}(m^{\prime}-s^{\prime})(m^{\prime}-k^{\prime}+n_{1}-n_{2}+1+s^{\prime})](\eta^{m-i}(f)\|\eta^{m^{\prime}-i}(f))$ $\displaystyle=$ $\displaystyle 0\qquad\mbox{if}\;\;(m,k-2m)\neq(m^{\prime},k^{\prime}-2m^{\prime}).\hskip 202.01474pt(6.23)$ Let $g\in{\cal H}_{\langle k-2m-1\rangle}$ and $g^{\prime}\in{\cal H}_{\langle k^{\prime}-2m^{\prime}-1\rangle}$. By (3.60). (5.2), (6.6) and (6.19), we have $\displaystyle\left(\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i+1}{\cal D}^{i}}{(2i+1)!}\right)(\eta^{2m+1}(g))\|\left(\sum_{r=0}^{\infty}\frac{(-2)^{r}x_{0}^{2r+1}{\cal D}^{r}}{(2r+1)!}\right)(\eta^{2m^{\prime}+1}(g^{\prime}))\right)$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{m}\frac{2^{2i}}{(2i+1)!}[\prod_{s=1}^{i}(m-s)(m-k+n_{1}-n_{2}+2+s)]$ $\displaystyle\times[\prod_{s^{\prime}=1}^{i}(m^{\prime}-s^{\prime})(m^{\prime}-k^{\prime}+n_{1}-n_{2}+2+s^{\prime})](\eta^{m-i}(g)\|\eta^{m^{\prime}-i}(g^{\prime}))$ $\displaystyle=$ $\displaystyle 0\qquad\mbox{if}\;\;(m,k-2m-1)\neq(m^{\prime},k^{\prime}-2m^{\prime}-1).\hskip 159.3356pt(6.24)$ Since $(x^{2i}_{0}\|x_{0}^{2i^{\prime}+1})=0$ for $i,i^{\prime}\in\mathbb{N}$, the elements of the form (6.18) are orthogonal to those of the form (6.19). Hence (6.22) holds by (6.8). For $g\in{\cal H}^{\prime}_{\langle k\rangle}$ and $m\in\mathbb{N}+1$, ${\cal D}^{\prime}[(\eta^{\prime})^{m}(g)]=2m[2(k+n_{2}-n_{1}+m-1)+1](\eta^{\prime})^{m-1}(g)$ $None$ by (3.44) and the facts ${\cal D}^{\prime}=\partial_{x_{0}}^{2}-2{\cal D}$ and its dual $\eta^{\prime}=x_{0}^{2}+2\eta$. This shows that $((\eta^{\prime})^{m}({\cal H}^{\prime}_{\langle k\rangle})\|(\eta^{\prime})^{m^{\prime}}({\cal H}^{\prime}_{\langle k^{\prime}\rangle}))=\\{0\\}\qquad\mbox{if}\;\;(m,k)\neq(m^{\prime},k^{\prime})$ $None$ for $n_{1}-n_{2}+1-\delta_{n_{1},n_{2}}\geq k,k^{\prime}\in\mathbb{Z}$ and $m,m^{\prime}\in\mathbb{N}$ by (6.7). Moreover, the symmetric bilinear form $(\cdot\|\cdot)$ restricted to $(\eta^{\prime})^{m}({\cal H}_{\langle k\rangle}^{\prime})$ is nondegenerate. Fix $n_{1}-n_{2}+1-\delta_{n_{1},n_{2}}\geq k\in\mathbb{Z}$. Denote $\hat{\cal B}^{\prime}_{\langle k\rangle}=\bigoplus_{i=0}^{\infty}(\eta^{\prime})^{i}({\cal H}^{\prime}_{\langle k-2i\rangle})$ $None$ Then the symmetric bilinear form $(\cdot\|\cdot)$ restricted to $\hat{\cal B}^{\prime}_{\langle k\rangle}$ is nondegenerate. Thus ${\cal B}^{\prime}_{\langle k\rangle}=\hat{\cal B}^{\prime}_{\langle k\rangle}\oplus(\hat{\cal B}^{\prime}_{\langle k\rangle})^{\perp}\bigcap{\cal B}^{\prime}_{\langle k\rangle}.$ $None$ According to Lemma 3.2, $(\hat{\cal B}^{\prime}_{\langle k\rangle})^{\perp}\bigcap{\cal B}^{\prime}_{\langle k\rangle}$ is an $o(2n+1,\mathbb{F})$-module. Assume $(\hat{\cal B}^{\prime}_{\langle k\rangle})^{\perp}\bigcap{\cal B}^{\prime}_{\langle k\rangle}\neq\\{0\\}$. By (5.2), (5.3), (5.8)-(5.13), (5.23) and (1.23), there exists a nonzero element in $(\hat{\cal B}^{\prime}_{\langle k\rangle})^{\perp}\bigcap{\cal B}^{\prime}_{\langle k\rangle}$ of the form: $f=\sum_{i=0}^{m}a_{i}x_{0}^{2i}(2\eta)^{m-i}(x_{n_{1}}^{-k+2m})$ $None$ or $g=\sum_{i=0}^{m}b_{i}x_{0}^{2i+1}(2\eta)^{m-i}(x_{n_{1}}^{-k+2m+1})$ $None$ for some $m\in\mathbb{N}+1$. Moreover, we assume that the exponent of $x_{n_{1}}$ is minimal. If (6.29) holds, then (6.11)) and (6.13) give $\displaystyle(E_{n_{1},0}-E_{0,n+n_{1}})(f)=(\partial_{x_{0}}\partial_{x_{n_{1}}}-x_{0}\partial_{y_{n_{1}}})(f)$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{m}2i(-k+2m)a_{i}x_{0}^{2i-1}(2\eta)^{m-i}(x_{n_{1}}^{-k+2m-1})$ $\displaystyle-\sum_{i=0}^{m-1}2(m-i)(-k+2m)a_{i}x_{0}^{2i+1}(2\eta)^{m-i-1}(x_{n_{1}}^{-k+2m-1})$ $\displaystyle=$ $\displaystyle 2(-k+2m)\sum_{i=0}^{m-1}[(i+1)a_{i+1}-(m-i)a_{i}]x_{0}^{2i+1}(2\eta)^{m-i-1}(x_{n_{1}}^{-k+2m-1})$ $\displaystyle=$ $\displaystyle 0\hskip 364.19536pt(6.31)$ by the minimality of the exponent of $x_{n_{1}}$, equivalently $(i+1)a_{i+1}=(m-i)a_{i}\qquad\mbox{for}\;\;i\in\overline{0,m-1}.$ $None$ Thus $a_{i}=a_{0}{m\choose i}\qquad\mbox{for}\;\;i\in\overline{0,m}.$ $None$ So $f=\sum_{i=0}^{m}a_{0}{m\choose i}x_{0}^{2i}(2\eta)^{m-i}(x_{n_{1}}^{-k+2m})=a_{0}(\eta^{\prime})^{m}(x_{n_{1}}^{-k+2m})\in\hat{\cal B}^{\prime}_{\langle k\rangle},$ $None$ which contradicts (6.28). Suppose that (6.30) holds. Note $x_{0}x_{n_{1}}^{-k+2m+1}\in{\cal H}^{\prime}_{\langle k-2m\rangle}$ by (1.23). Expressions (6.11) and (6.13) deduce $\displaystyle(E_{n_{1},0}-E_{0,n+n_{1}})(g)=(\partial_{x_{0}}\partial_{x_{n_{1}}}-x_{0}\partial_{y_{n_{1}}})(g)$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{m}(2i+1)(-k+2m+1)b_{i}x_{0}^{2i}(2\eta)^{m-i}(x_{n_{1}}^{-k+2m})$ $\displaystyle-\sum_{i=0}^{m-1}2(m-i)(-k+2m+1)b_{i}x_{0}^{2i+2}(2\eta)^{m-i-1}(x_{n_{1}}^{-k+2m})$ $\displaystyle=$ $\displaystyle(-k+2m+1)\big{\\{}\sum_{i=0}^{m-1}[(2i+3)b_{i+1}-2(m-i)b_{i}]x_{0}^{2i+2}(2\eta)^{m-i-1}(x_{n_{1}}^{-k+2m})$ $\displaystyle+b_{0}(2\eta)^{m}(x_{n_{1}}^{-k+2m})\big{\\}}=0\hskip 247.53888pt(6.35)$ by the minimality of the exponent of $x_{n_{1}}$, equivalently $b_{0}=0,\;\;(2i+3)b_{i+1}=2(m-i)b_{i}\qquad\mbox{for}\;\;i\in\overline{0,m-1}.$ $None$ Thus $b_{i}=0$ for $i\in\overline{0,m},$ that is, $g=0$. This contradicts our choice of nonzero element. Hence $(\hat{\cal B}^{\prime}_{\langle k\rangle})^{\perp}\bigcap{\cal B}^{\prime}_{\langle k\rangle}=\\{0\\}$. Then (6.28) gives (6.4). Furthermore, (6.5) is obtained by Lemma 3.1 with $T_{1}=\partial_{x_{0}}^{2},\;T_{1}^{-}=\int_{(x_{0})}^{(2)}$ (cf. (2.6) and (2.7)) and $T_{2}=2{\cal D}$. When $n_{1}=n_{2}$, an expression of ${\cal H}_{\langle k\rangle}^{\prime}$ can be obtained via (5.3), (5.27)-(5.31), (6.8), (6.18) and (6.19). In particular, when $n_{1}=n_{2}=n$, the $({\cal G},{\cal K})$-module structure is given by $\displaystyle\qquad{\cal H}^{\prime}_{\langle-k\rangle}$ $\displaystyle=$ $\displaystyle\bigoplus_{m,r=0}^{\infty}\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i}{\cal D}^{i}}{(2i)!}\right)(\eta^{r}({\cal H}_{\langle-k-2r-m,m\rangle}))$ $\displaystyle\oplus\bigoplus_{l,s=0}^{\infty}\left(\sum_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i+1}{\cal D}^{i}}{(2i+1)!}\right)(\eta^{s}({\cal H}_{\langle-k-2s-1-l,l\rangle})),\hskip 102.43008pt(6.37)$ where ${\cal H}_{\langle-m_{1}-m_{2},m_{2}\rangle}$ given in (4.42). $\qquad\Box$ ## 7 Noncanonical Representations of $sp(2n,\mathbb{F})$ In this section, we use the results in Sections 3 and 4 to study noncanonical polynomial representation of $sp(2n,\mathbb{F})$. Recall the symplectic Lie algebra $\displaystyle\hskip 28.45274ptsp(2n,\mathbb{F})$ $\displaystyle=$ $\displaystyle\sum_{i,j=1}^{n}\mathbb{F}(E_{i,j}-E_{n+j,n+i})+\sum_{i=1}^{n}(\mathbb{F}E_{i,n+i}+\mathbb{F}E_{n+i,i})$ $\displaystyle+\sum_{1\leq i<j\leq n}[\mathbb{F}(E_{i,n+j}+E_{n+j,i})+\mathbb{F}(E_{n+i,j}+E_{n+j,i})].\hskip 68.28644pt(7.1)$ Again we take the Cartan subalgebra $H=\sum_{i=1}^{n}\mathbb{F}(E_{i,i}-E_{n+i,n+i})$ and the subspace spanned by positive root vectors $sp(2n,\mathbb{F})_{+}=\sum_{1\leq i<j\leq n}[\mathbb{F}(E_{i,j}-E_{n+j,n+i})+\mathbb{F}(E_{i,n+j}+E_{n+j,i})]+\sum_{i=1}^{n}\mathbb{F}E_{i,n+i}.$ $None$ Fix $1\leq n_{1}\leq n_{2}\leq n$. The noncanonical oscillator representation of $sp(2n,\mathbb{F})$ on ${\cal B}=\mathbb{F}[x_{1},...,x_{n},y_{1},...,y_{n}]$ is defined via (1.14)-(1.16). Recall ${\cal K}=\sum_{i,j=1}^{n}\mathbb{F}(E_{i,j}-E_{n+j,n+i})$. Theorem 7.1. Let $k\in\mathbb{Z}$. If $n_{1}<n_{2}$ or $k\neq 0$, the subspace ${\cal B}_{\langle k\rangle}$ (cf. (1.17)) is an irreducible $sp(2n,\mathbb{F})$-module. Moreover, it is a highest-weight module only if $n_{2}=n$, in which case for $m\in\mathbb{N}$, $x_{n_{1}}^{-m}$ is a highest- weight vector of ${\cal B}_{\langle-m\rangle}$ with weight $-m\lambda_{n_{1}-1}+(m-1)\lambda_{n_{1}}$, $x_{n_{1}+1}^{m+1}$ is a highest- weight vector of ${\cal B}_{\langle m+1\rangle}$ with weight $-(m+2)\lambda_{n_{1}}+(m+1)\lambda_{n_{1}+1}+(m+1)\delta_{n_{1},n-1}\lambda_{n}$ if $n_{1}<n$ and $y_{n}^{m+1}$ is a highest-weight vector of ${\cal B}_{\langle m+1\rangle}$ with weight $(m+1)\lambda_{n-1}-2(m+1)\lambda_{n}$ when $n_{1}=n$. When $n_{1}=n_{2}$, the subspace ${\cal B}_{\langle 0\rangle}$ is a direct sum of two irreducible $sp(2n,\mathbb{F})$-submodules. If $n_{1}=n_{2}=n$, they are highest-weight modules with a highest-weight vector $1$ of weight $-2\lambda_{n}$ and with a highest-weight vector $x_{n-1}y_{n}-x_{n}y_{n-1}$ of weight $\delta_{n,2}\lambda_{n-2}-4\lambda_{n}$, respectively. If $n_{1}=n_{2}=n$, all the irreducible modules are of $({\cal G},{\cal K})$-type. Proof. Recall that we embed $sl(n,\mathbb{F})$ into $sp(2n,\mathbb{F})$ via $E_{i,j}\mapsto E_{i,j}-E_{n+j,n_{i}}$. Moreover, ${\cal B}$ is nilpotent with respect to $sl(n,\mathbb{F})_{+}$ (cf. (2.30)) and $\eta=\sum_{i=1}^{n_{1}}y_{i}\partial_{x_{i}}+\sum_{r=n_{1}+1}^{n_{2}}x_{r}y_{r}+\sum_{s=n_{2}+1}^{n}x_{s}\partial_{y_{s}}.$ $None$ Note $(E_{i,n+j}+E_{j,n+i})\|_{\cal B}=\partial_{x_{i}}\partial_{y_{j}}+\partial_{x_{j}}\partial_{y_{i}},\;\;(E_{i,n+r}+E_{r,n+i})\|_{\cal B}=\partial_{x_{i}}\partial_{y_{r}}+x_{r}\partial_{y_{i}},$ $None$ $(E_{r,n+s}+E_{s,n+r})\|_{\cal B}=x_{r}\partial_{y_{s}}+x_{s}\partial_{y_{r}}$ $None$ for $i,j\in\overline{1,n_{1}}$ and $r,s\in\overline{n_{1}+1,n_{2}}$ by (1.15). Moreover, $(E_{i,j}-E_{n+j,n+i})\|_{\cal B}=-x_{j}\partial_{x_{i}}-y_{j}\partial_{y_{i}}-\delta_{i,j}$ $None$ and $(E_{i,r}-E_{n+r,n+i})\|_{\cal B}=\partial_{x_{i}}\partial_{x_{r}}-y_{r}\partial_{y_{i}}$ $None$ for $i,j\in\overline{1,n_{1}}$ and $r\in\overline{n_{1}+1,n_{2}}$ by (1.7), (1.8) and (1.14). We will process our arguments in two steps. Step 1. $n_{2}=n$. Under the assumption, ${\cal B}$ is nilpotent with respect to $sp(2n,\mathbb{F})_{+}$ by (7.4)-(7.7). First we assume $n_{1}+1<n$. According to (3.37), the nonzero weight vectors in $\mbox{Span}\\{\eta^{m_{3}}(x_{i}^{m_{1}}y_{n}^{m_{2}})\mid m_{r}\in\mathbb{N};i=n_{1},n_{1}+1\\}$ $None$ are all the singular vectors of $sl(n,\mathbb{F})$ in ${\cal B}$. The singular vectors of $sp(2n,\mathbb{F})$ in ${\cal B}$ must be among them. Moreover, the subalgebra $sp(2n,\mathbb{F})_{+}$ is generated by $sl(n,\mathbb{F})_{+}$ and $E_{n,2n}$. According to (7.5), $E_{n,2n}\|_{\cal B}=x_{n}\partial_{y_{n}}$. Hence $E_{n,2n}(\eta^{m_{3}}(x_{i}^{m_{1}}y_{n}^{m_{2}}))=x_{n}[m_{3}x_{n}\eta^{m_{3}-1}(x_{i}^{m_{1}}y_{n}^{m_{2}})+m_{2}\eta^{m_{3}}(x_{i}^{m_{1}}y_{n}^{m_{2}-1})]$ $None$ for $i=n_{1},n_{1}+1$ by (7.3). Considering weights, we conclude that the vectors $\\{x_{n_{1}}^{m},x_{n_{1}+1}^{m+1}\mid m\in\mathbb{N}$ are all the singular vectors of $sp(2n,\mathbb{F})$ in ${\cal B}$. Furthermore, $x_{n_{1}}^{m}\in{\cal B}_{\langle-m\rangle}\;\;\mbox{and}\;\;x_{n_{1}+1}^{m+1}\in{\cal B}_{\langle m+1\rangle}\qquad\mbox{for}\;\;m\in\mathbb{N}.$ $None$ Thus each ${\cal B}_{\langle k\rangle}$ has a unique non-isotropic singular vector for $k\in\mathbb{Z}$. By Lemma 3.3, all ${\cal B}_{\langle k\rangle}$ with $k\in\mathbb{Z}$ are irreducible highest-weight $sp(2n,\mathbb{F})$-submodules. Consider the case $n_{1}+1=n$. According to (3.112), the nonzero weight vectors in $\mbox{Span}\\{\eta^{m_{2}}(x_{n-1}^{m_{1}}y_{n}^{m_{3}}),x_{n}^{m_{1}}y_{n}^{m_{2}},\eta^{m_{1}+m_{2}}(x_{n-1}^{m_{2}}y_{n}^{m_{3}-m_{1}})\mid m_{i}\in\mathbb{N}\\}$ $None$ are all the singular vectors of $sl(n,\mathbb{F})$ in ${\cal B}$. Recall $E_{n,2n}\|_{\cal B}=x_{n}\partial_{y_{n}}$. We have $E_{n,2n}(x_{n}^{m_{1}}y_{n}^{m_{2}})=m_{2}x_{n}^{m_{1}+1}y_{n}^{m_{2}-1}.$ $None$ By (7.11) and considering weights, we again conclude that the vectors $\\{x_{n-1}^{m},x_{n}^{m+1}\mid m\in\mathbb{N}$ are all the singular vectors of $sp(2n,\mathbb{F})$ in ${\cal B}$. Again all ${\cal B}_{\langle k\rangle}$ with $k\in\mathbb{Z}$ are irreducible highest-weight $sp(2n,\mathbb{F})$-submodules. Suppose $n_{1}=n$. By (7.4), we have $E_{n,2n}=\partial_{x_{n}}\partial_{y_{n}}$ in this case. According to (4.31), the nonzero weight vectors in $\mbox{Span}\\{x_{n}^{m_{1}}y_{n}^{m_{2}}\zeta_{1}^{m_{3}}\mid m_{i}\in\mathbb{N}\\}$ $None$ are all the singular vectors of $sl(n,\mathbb{F})$ in ${\cal B}$, where $\zeta_{1}=x_{n-1}y_{n}-x_{n}y_{n-1}$ in this case. $\displaystyle E_{n,2n}(x_{n}^{m_{1}}y_{n}^{m_{2}}\zeta_{1}^{m_{3}})$ $\displaystyle=$ $\displaystyle m_{1}m_{2}x_{n}^{m_{1}-1}y_{n}^{m_{2}-1}\zeta_{1}^{m_{3}}+m_{1}m_{3}x_{n-1}x_{n}^{m_{1}-1}y_{n}^{m_{2}}\zeta_{1}^{m_{3}-1}$ $\displaystyle- m_{2}m_{3}y_{n-1}x_{n}^{m_{1}}y_{n}^{m_{2}-1}\zeta_{1}^{m_{3}-1}-m_{3}(m_{3}-1)x_{n-1}y_{n-1}x_{n}^{m_{1}}y_{n}^{m_{2}}\zeta_{1}^{m_{3}-2}$ $\displaystyle=$ $\displaystyle m_{1}(m_{2}+m_{3})x_{n}^{m_{1}-1}y_{n}^{m_{2}-1}\zeta_{1}^{m_{3}}+m_{3}(m_{1}-m_{2}-m_{3}+1)y_{n-1}x_{n}^{m_{1}}y_{n}^{m_{2}-1}\zeta_{1}^{m_{3}-1}$ $\displaystyle- m_{3}(m_{3}-1)y_{n-1}^{2}x_{n}^{m_{1}+1}y_{n}^{m_{2}-1}\zeta_{1}^{m_{3}-2}.\hskip 213.39566pt(7.14)$ Considering weights, we again conclude that the vectors $\\{x_{n}^{m},y_{n}^{m+1},\zeta_{1}\mid m\in\mathbb{N}\\}$ are all the singular vectors of $sp(2n,\mathbb{F})$ in ${\cal B}$. Moreover, $x_{n}^{m}\in{\cal B}_{\langle-m\rangle},\;\;\zeta_{1}\in{\cal B}_{\langle 0\rangle}\;\;\mbox{and}\;\;y_{n}^{m+1}\in{\cal B}_{\langle m+1\rangle}\qquad\mbox{for}\;\;m\in\mathbb{N}.$ $None$ Thus each ${\cal B}_{\langle k\rangle}$ with $k\neq 0$ has a unique non- isotropic singular vector for $k\in\mathbb{Z}$. By Lemma 3.3, all ${\cal B}_{\langle k\rangle}$ with $0\neq k\in\mathbb{Z}$ are irreducible highest- weight $sp(2n,\mathbb{F})$-submodules. Set ${\cal B}_{\langle 0,1\rangle}=\mbox{Span}\\{[\prod_{1\leq r\leq s\leq n}(x_{r}y_{s}+x_{s}y_{r})^{l_{r,s}}]\mid l_{r,s}\in\mathbb{N}\\}$ $None$ and ${\cal B}_{\langle 0,2\rangle}=\mbox{Span}\\{[\prod_{1\leq r\leq s\leq n}(x_{r}y_{s}+x_{s}y_{r})^{l_{r,s}}](x_{p}y_{q}-x_{q}y_{p})\mid l_{r,s}\in\mathbb{N};1\leq p<q\leq n\\}.$ $None$ Let ${\cal G}^{\prime}=\sum_{1\leq r\leq s\leq n}\mathbb{F}(E_{n+s,r}+E_{n+r,s})$ $None$ and $\hat{\cal G}=\sum_{i,j=1}^{n}\mathbb{F}(E_{i,j}-E_{n+j,n+i})+\sum_{1\leq r\leq s\leq n}\mathbb{F}(E_{r,n+s}+E_{s,n+r}).$ $None$ Then ${\cal G}^{\prime}$ and $\hat{\cal G}$ are Lie subalgebras of $sp(2n,\mathbb{F})$ and $sp(2n,\mathbb{F})={\cal G}^{\prime}\oplus\hat{\cal G}.$ By PBW Theorem $U(sp(2n,\mathbb{F}))=U({\cal G}^{\prime})U(\hat{\cal G}).$ $None$ Note $(E_{n+s,r}+E_{n+r,s})_{\cal B}=-(x_{r}y_{s}+x_{s}y_{r})\qquad\mbox{for}\;\;r,s\in\overline{1,n}$ $None$ by (1.16). According to (7.4), (7.6) and (7.23), ${\cal B}_{\langle 0,1\rangle}=U({\cal G}^{\prime})(1)=U(sp(2n,\mathbb{F}))(1)$ $None$ and ${\cal B}_{\langle 0,2\rangle}=\sum_{1\leq p<q\leq n}U({\cal G}^{\prime})(x_{p}y_{q}-x_{q}y_{p})=U(sp(2n,\mathbb{F}))(\zeta_{1})$ $None$ are $sp(2n,\mathbb{F})$-submodules. It is obvious, $1\not\in{\cal B}_{\langle 0,2\rangle}$. On the other hand, $({\cal B}_{\langle 0,1\rangle}\|x_{n-1}y_{n}-x_{n}y_{n-1})=\\{0\\}$. Hence $x_{n-1}y_{n}-x_{n}y_{n-1}\not\in{\cal B}_{\langle 0,1\rangle}$. Thus ${\cal B}_{\langle 0,1\rangle}$ and ${\cal B}_{\langle 0,0\rangle}$ have a unique non-isotropic singular vector. By Lemma 3.3, they are irreducible. Since $1$ and $x_{n-1}y_{n}-x_{n}y_{n-1}$ are the only singular vectors in ${\cal B}_{\langle 0\rangle}$ which is nilpotent with respect to $sp(2n,\mathbb{F})_{+}$, Lemma 2.3 yields ${\cal B}_{\langle 0\rangle}={\cal B}_{\langle 0,1\rangle}\oplus{\cal B}_{\langle 0,2\rangle}$ $None$ by the similar arguments as those from (3.67) to (3.69). Step 2. $n_{2}<n$. We set $\displaystyle\hskip 28.45274pt{\cal G}_{1}$ $\displaystyle=$ $\displaystyle\sum_{i,j=1}^{n_{2}}\mathbb{F}(E_{i,j}-E_{n+j,n+i})+\sum_{i=1}^{n_{2}}(\mathbb{F}E_{i,n+i}+\mathbb{F}E_{n+i,i})$ $\displaystyle+\sum_{1\leq i<j\leq n_{2}}[\mathbb{F}(E_{i,n+j}+E_{n+j,i})+\mathbb{F}(E_{n+i,j}+E_{n+j,i})]\hskip 102.43008pt(7.27)$ and $\displaystyle\hskip 28.45274pt{\cal G}_{2}$ $\displaystyle=$ $\displaystyle\sum_{i,j=n_{1}+1}^{n}\mathbb{F}(E_{i,j}-E_{n+j,n+i})+\sum_{i=n_{1}+1}^{n}(\mathbb{F}E_{i,n+i}+\mathbb{F}E_{n+i,i})$ $\displaystyle+\sum_{n_{1}+1\leq i<j\leq n}[\mathbb{F}(E_{i,n+j}+E_{n+j,i})+\mathbb{F}(E_{n+i,j}+E_{n+j,i})].\hskip 85.35826pt(7.28)$ Then ${\cal G}_{1}=sp(2n_{2},\mathbb{F})$ and ${\cal G}_{2}\cong sp(2(n-n_{1}),\mathbb{F})$ are Lie subalgebras of $sp(2n,\mathbb{F})$. Denote ${\cal M}^{1}=\mathbb{F}[x_{1},...,x_{n_{2}},y_{1},...,y_{n_{2}}],\qquad{\cal M}^{2}=\mathbb{F}[x_{n_{1}+1},...,x_{n},y_{n_{1}+1},...,y_{n}].$ $None$ Observe that ${\cal M}^{1}$ is exactly the ${\cal G}_{1}$-module as ${\cal B}$ in Step 1 with $n\rightarrow n_{2}$ and ${\cal M}^{2}$ is exactly the ${\cal G}_{1}$-module as ${\cal B}$ in Step 1 with $n_{1}=n_{2}$ and $n\rightarrow n-n_{1}$. Moreover, we set ${\cal M}^{3}=\mathbb{F}[x_{1},...,x_{n_{1}},y_{1},...,y_{n_{1}}],\qquad{\cal M}^{4}=\mathbb{F}[x_{n_{1}+1},...,x_{n_{2}},y_{n_{1}+1},...,y_{n_{2}}].$ $None$ Let ${\cal M}^{i}_{\langle k\rangle}={\cal M}^{i}\bigcap{\cal B}_{\langle k\rangle}\qquad\mbox{for}\;\;i\in\overline{1,4},\;k\in\mathbb{Z}.$ $None$ Then ${\cal M}^{1}_{\langle k\rangle}=\bigoplus_{r\in\mathbb{Z}}{\cal M}^{3}_{\langle r\rangle}{\cal M}^{4}_{\langle k-r\rangle}\qquad\mbox{for}\;\;k\in\mathbb{Z}.$ $None$ Next we prove the theorem case by case. Case 1. $n_{1}+1<n_{2}$ According to (3.36), the nonzero weight vectors in $\mbox{Span}\\{\eta^{m_{3}}(x_{i}^{m_{1}}y_{j}^{m_{2}})\mid m_{r}\in\mathbb{N};i=n_{1},n_{1}+1;j=n_{2},n_{2}+1\\}$ $None$ are all the singular vectors of $sl(n,\mathbb{F})$ in ${\cal B}$. Fix $k\in\mathbb{N}$. Then the singular vectors of $sl(n,\mathbb{F})$ in ${\cal B}_{\langle-k\rangle}$ are $\displaystyle\\{\eta^{m_{3}}(x_{n_{1}}^{k+m_{2}+2m_{3}}y_{n_{2}}^{m_{2}}),\eta^{m_{3}}(x_{n_{1}+1}^{m_{1}}y_{n_{2}+1}^{k+m_{1}+2m_{3}}),$ $\displaystyle\eta^{m_{3}}(x_{n_{1}}^{m_{4}}y_{n_{2}+1}^{m_{5}})\mid m_{i}\in\mathbb{N};m_{4}+m_{5}-2m_{3}=k\\}.\hskip 119.50148pt(7.34)$ Let $M$ be a nonzero $sp(2n,\mathbb{F})$-submodule of ${\cal B}_{\langle-k\rangle}$. Then $M$ contains a singular of $sl(n,\mathbb{F})$. Suppose some $\eta^{m_{3}}(x_{n_{1}}^{k+m_{2}+2m_{3}}y_{n_{2}}^{m_{2}})\in M$. We have $E_{n_{1},n+n_{1}}\|_{\cal B}=\partial_{x_{n_{1}}}\partial_{y_{n_{1}}}$ and $E_{n_{1},n+n_{1}}^{m_{3}}[\eta^{m_{3}}(x_{n_{1}}^{k+m_{2}+2m_{3}}y_{n_{2}}^{m_{2}})]=m_{3}![\prod_{r=1}^{2m_{3}}(k+m_{2}+r)]x_{n_{1}}^{k+m_{2}}y_{n_{2}}^{m_{2}}\in M$ $None$ by (7.3) and (7.4). Moreover, $(E_{n_{1},n+n_{2}}+E_{n_{2},n+n_{1}})\|_{\cal B}=\partial_{x_{n_{1}}}\partial_{y_{n_{2}}}+x_{n_{2}}\partial_{y_{n_{1}}}$ and $(E_{n_{1},n+n_{2}}+E_{n_{2},n+n_{1}})^{m_{2}}(x_{n_{1}}^{k+m_{2}}y_{n_{2}}^{m_{2}})=m_{2}![\prod_{r=1}^{m_{2}}(k+r)]x_{n_{1}}^{k}\in M$ $None$ by (7.4). Thus $x_{n_{1}}^{k}\in M.$ $None$ Assume some $\eta^{m_{3}}(x_{n_{1}+1}^{m_{1}}y_{n_{2}+1}^{k+m_{1}+2m_{3}})\in M$. According to (1.16), $(E_{n+i,j}+E_{n+j,i})\|_{\cal B}=\partial_{x_{i}}\partial_{y_{j}}+\partial_{x_{j}}\partial_{y_{i}}\qquad\mbox{for}\;\;i\in\overline{n_{2}+1,n}.$ $None$ So $E_{n+n_{2}+1,n_{2}+1}^{m_{3}}[\eta^{m_{3}}(x_{n_{1}+1}^{m_{1}}y_{n_{2}+1}^{k+m_{1}+2m_{3}})]=m_{3}![\prod_{r=1}^{2m_{3}}(k+m_{1}+r)]x_{n_{1}+1}^{m_{1}}y_{n_{2}+1}^{k+m_{1}}\in M.$ $None$ Moreover, $(E_{n+n_{2}+1,n_{1}+1}+E_{n+n_{1}+1,n_{2}+1})\|_{\cal B}=\partial_{x_{n_{1}+1}}\partial_{y_{n_{2}+1}}+y_{n_{1}+1}\partial_{x_{n_{2}+1}}$ $None$ by (1.16). Hence $(E_{n+n_{2}+1,n_{1}+1}+E_{n+n_{1}+1,n_{2}+1})^{m_{1}}(x_{n_{1}+1}^{m_{1}}y_{n_{2}+1}^{k+m_{1}})=m_{1}![\prod_{r=1}^{m_{1}}(k+r)]y_{n_{2}+1}^{k}\in M.$ $None$ Furthermore, $(E_{n+n_{2}+1,n_{1}}+E_{n+n_{1},n_{2}+1})\|_{\cal B}=-x_{n_{1}}\partial_{y_{n_{2}+1}}+y_{n_{1}}\partial_{x_{n_{2}+1}}$ $None$ by (1.16). Thus $(E_{n+n_{2}+1,n_{1}}+E_{n+n_{1},n_{2}+1})^{k}(y_{n_{2}+1}^{k})=(-1)^{k}k!x_{n_{1}}^{k}\in M.$ $None$ Thus (7.37) holds again. Consider $\eta^{m_{3}}(x_{n_{1}}^{m_{4}}y_{n_{2}+1}^{m_{5}})$ for some $m_{3},m_{3},m_{4}\in\mathbb{N}$ such that $m_{4}+m_{5}-2m_{3}=k$. Note that $E_{n_{1}+1,n+n_{1}+1}\|_{\cal B}=x_{n_{1}+1}\partial_{y_{n_{1}+1}}$ by (7.5) and $E_{n_{1}+1,n+n_{1}+1}^{m_{3}}[\eta^{m_{3}}(x_{n_{1}}^{m_{4}}y_{n_{2}+1}^{m_{5}})]=m_{3}!x_{n_{1}+1}^{2m_{3}}x_{n_{1}}^{m_{4}}y_{n_{2}+1}^{m_{5}}\in M.$ $None$ There exists $r_{1},r_{2}\in\mathbb{N}$ such that $r_{1}+r_{2}=2m_{3}$ and $r_{1}\leq m_{4},\;r_{2}\leq m_{5}$. Moreover, $(E_{n_{1},n_{1}+1}-E_{n+n_{1}+1,n+n_{1}})\|_{\cal B}=\partial_{x_{n_{1}}}\partial_{x_{n_{1}+1}}-y_{n_{1}+1}\partial_{y_{n_{1}}}$ $None$ by (1.7), (1.8) and (1.14). Moreover, (7.40) and (7.45) yield $\displaystyle(E_{n_{1},n_{1}+1}-E_{n+n_{1}+1,n+n_{1}})^{r_{1}}(E_{n+n_{2}+1,n_{1}+1}+E_{n+n_{1}+1,n_{2}+1})^{r_{2}}(x_{n_{1}+1}^{2m_{3}}x_{n_{1}}^{m_{4}}y_{n_{2}+1}^{m_{5}})$ $\displaystyle=$ $\displaystyle(2m_{3})![\prod_{s_{1}=0}^{r_{1}-1}(m_{4}-s_{1})][\prod_{s_{2}=0}^{r_{2}-1}(m_{5}-s_{2})]x_{n_{1}}^{m_{4}-r_{1}}y_{n_{2}+1}^{m_{5}-r_{2}}\in M.\hskip 91.04872pt(7.46)$ Furthermore, (7.42) yields $\displaystyle(E_{n+n_{2}+1,n_{1}}+E_{n+n_{1},n_{2}+1})^{m_{5}-r_{2}}(x_{n_{1}}^{m_{4}-r_{1}}y_{n_{2}+1}^{m_{5}-r_{2}})$ $\displaystyle=$ $\displaystyle(-1)^{m_{5}-r_{2}}(m_{5}-r_{2})!x_{n_{1}}^{k}\in M.\hskip 187.78836pt(7.47)$ Thus we always have $x_{n_{1}}^{k}\in M$. Note that ${\cal M}^{1}_{\langle-k\rangle}\ni x_{n_{1}}^{k}$ is an irreducible ${\cal G}_{1}$-module (cf. (7.27) and (7.29)) by Step 1. So ${\cal M}^{1}_{\langle-k\rangle}\subset M.$ $None$ Let $r\in\mathbb{Z}$. According to (7.32), ${\cal M}^{3}_{\langle r\rangle}{\cal M}^{4}_{\langle-k-r\rangle}\subset{\cal M}^{1}_{\langle-k\rangle}\subset M.$ $None$ Moreover, ${\cal M}^{2}_{\langle-k-r\rangle}\supset{\cal M}^{4}_{\langle- k-r\rangle}$ is an irreducible ${\cal G}_{2}$-module (cf. (7.28) and (7.29)) by Step 1. Thus ${\cal M}^{3}_{\langle r\rangle}{\cal M}^{2}_{\langle-k-r\rangle}=U({\cal G}_{2})({\cal M}^{3}_{\langle r\rangle}{\cal M}^{4}_{\langle- k-r\rangle})\subset M.$ $None$ Then ${\cal B}_{\langle-k\rangle}=\bigoplus_{r\in\mathbb{Z}}{\cal M}^{3}_{\langle r\rangle}{\cal M}^{2}_{\langle-k-r\rangle}\subset M$ $None$ by (7.29) and (7.30). Therefore, $M={\cal B}_{\langle-k\rangle}$, that is, ${\cal B}_{\langle-k\rangle}$ is an irreducible $sp(2n,\mathbb{F})$-submodule. Fix $0<k\in\mathbb{N}$. Then the singular vectors of $sl(n,\mathbb{F})$ in ${\cal B}_{\langle k\rangle}$ are $\displaystyle\\{\eta^{m_{2}}(x_{n_{1}+1}^{k+m_{1}-2m_{2}}y_{n_{2}+1}^{m_{1}}),\eta^{m_{2}}(x_{n_{1}}^{m_{1}}y_{n_{2}}^{k+m_{1}-2m_{2}}),\eta^{m_{3}}(x_{n_{1}+1}^{m_{4}}y_{n_{2}}^{m_{5}})$ $\displaystyle\mid m_{i}\in\mathbb{N};2m_{2}\leq k+m_{1};m_{4}+m_{5}+2m_{3}=k\\}\hskip 145.10922pt(7.52)$ by (7.33). Let $M$ be a nonzero $sp(2n,\mathbb{F})$-submodule of ${\cal B}_{\langle k\rangle}$. Then $M$ contains a singular of $sl(n,\mathbb{F})$. Suppose some $\eta^{m_{2}}(x_{n_{1}+1}^{k+m_{1}-2m_{2}}y_{n_{2}+1}^{m_{1}})\in M$ with $2m_{2}\leq k+m_{1}$. We have $E_{n_{1}+1,n+n_{1}+1}\|_{\cal B}=x_{n_{1}+1}\partial_{y_{n_{1}+1}}$ and $E_{n_{1}+1,n+n_{1}+1}^{m_{2}}[\eta^{m_{2}}(x_{n_{1}+1}^{k+m_{1}-2m_{2}}y_{n_{2}}^{m_{1}})]=m_{2}!x_{n_{1}+1}^{k+m_{1}}y_{n_{2}+1}^{m_{1}}\in M$ $None$ by (7.3) and (7.5). Moreover, (7.40) gives $(E_{n+n_{2}+1,n_{1}+1}+E_{n+n_{1}+1,n_{2}+1})^{m_{1}}(x_{n_{1}+1}^{k+m_{1}}y_{n_{2}+1}^{m_{1}})=m_{1}![\prod_{r=1}^{m_{1}}(k+r)]x_{n_{1}+1}^{k}\in M.$ $None$ Thus $x_{n_{1}+1}^{k}\in M.$ $None$ Assume some $\eta^{m_{2}}(x_{n_{1}}^{m_{1}}y_{n_{2}}^{k+m_{1}-2m_{2}})\in M$ with $2m_{2}\leq k+m_{1}$. Observe $E_{n+n_{2},n_{2}}=y_{n_{2}}\partial_{x_{n_{2}}}$ by (1.16). So $E_{n+n_{2},n_{2}}^{m_{2}}[\eta^{m_{2}}(x_{n_{1}}^{m_{1}}y_{n_{2}}^{k+m_{1}-2m_{2}})]=m_{2}!x_{n_{1}}^{m_{1}}y_{n_{2}}^{k+m_{1}}\in M.$ $None$ Moreover, (7.4) gives that $(E_{n_{1},n+n_{2}}+E_{n_{2},n+n_{1}})\|_{\cal B}=\partial_{x_{n_{1}}}\partial_{y_{n_{2}}}+x_{n_{2}}\partial_{y_{n_{1}}}$ and $(E_{n_{1},n+n_{2}}+E_{n_{2},n+n_{1}})^{m_{1}}(x_{n_{1}}^{m_{1}}y_{n_{2}}^{k+m_{1}})=m_{1}![\prod_{r=1}^{m_{1}}(k+r)]y_{n_{2}}^{k}\in M.$ $None$ Furthermore, (7.5) yields that $(E_{n_{1}+1,n+n_{2}}+E_{n_{2},n+n_{1}+1})\|_{\cal B}=x_{n_{1}+1}\partial_{y_{n_{2}}}+x_{n_{2}}\partial_{y_{n_{1}+1}}$ and $(E_{n_{1}+1,n+n_{2}}+E_{n_{2},n+n_{1}+1})^{k}(y_{n_{2}}^{k})=k!x_{n_{1}+1}^{k}\in M.$ $None$ Thus (7.55) holds again. Consider $\eta^{m_{3}}(x_{n_{1}+1}^{m_{4}}y_{n_{2}}^{m_{5}})$ for some $m_{3},m_{3},m_{4}\in\mathbb{N}$ such that $m_{4}+m_{5}+2m_{3}=k$. Note $E_{n_{1}+1,n+n_{1}+1}=x_{n_{1}+1}\partial_{y_{n_{1}+1}}$ by (7.5). So $E_{n_{1}+1,n+n_{1}+1}^{m_{3}}[\eta^{m_{3}}(x_{n_{1}+1}^{m_{4}}y_{n_{2}}^{m_{5}})]=m_{3}!x_{n_{1}+1}^{m_{4}+2m_{3}}y_{n_{2}}^{m_{5}}\in M.$ $None$ According to (7.5), $(E_{n_{1}+1,n+n_{2}}+E_{n_{2},n+n_{1}+1})^{m_{5}}(x_{n_{1}+1}^{m_{4}+2m_{3}}y_{n_{2}}^{m_{5}})=m_{5}!x_{n_{1}+1}^{k}\in M.$ $None$ Therefore, we always have $x_{n_{1}+1}^{k}\in M$. Observe that ${\cal M}^{2}_{\langle k\rangle}\ni x_{n_{1}+1}^{k}$ is an irreducible ${\cal G}_{2}$-module (cf. (7.28) and (7.29)) by Step 1. So ${\cal M}^{2}_{\langle k\rangle}\subset M.$ $None$ Let $r\in\mathbb{Z}$. Denote ${\cal M}^{5}=\mathbb{F}[x_{n_{2}+1},...,x_{n},y_{n_{2}+1},...,y_{n}],\qquad{\cal M}^{5}_{\langle k\rangle}={\cal M}^{5}\bigcap{\cal B}_{\langle k\rangle},\;\;k\in\mathbb{Z}.$ $None$ Then ${\cal M}^{2}_{\langle k\rangle}=\bigoplus_{r\in\mathbb{Z}}{\cal M}^{4}_{\langle r\rangle}{\cal M}^{5}_{\langle k-r\rangle}$ $None$ (cf. (7.30)). Fix $r\in\mathbb{Z}$. ${\cal M}^{4}_{\langle r\rangle}{\cal M}^{5}_{\langle k-r\rangle}\subset{\cal M}^{2}_{\langle k\rangle}\subset M.$ $None$ Moreover, ${\cal M}^{1}_{\langle r\rangle}\supset{\cal M}^{4}_{\langle r\rangle}$ is an irreducible ${\cal G}_{1}$-module (cf. (7.27) and (7.29)) by Step 1. Thus ${\cal M}^{1}_{\langle r\rangle}{\cal M}^{5}_{\langle k-r\rangle}=U({\cal G}_{1})({\cal M}^{4}_{\langle r\rangle}{\cal M}^{5}_{\langle k-r\rangle})\subset M.$ $None$ Furthermore, ${\cal B}_{\langle k\rangle}=\bigoplus_{r\in\mathbb{Z}}{\cal M}^{1}_{\langle r\rangle}{\cal M}^{5}_{\langle k-r\rangle}\subset M$ $None$ by (7.27) and (7.65). Therefore, $M={\cal B}_{\langle k\rangle}$, that is, ${\cal B}_{\langle k\rangle}$ is an irreducible $sp(2n,\mathbb{F})$-submodule. Case 2. $n_{2}=n_{1}+1$. According to (3.104), the nonzero weight vectors in $\displaystyle\mbox{Span}\\{\eta^{m_{2}}(x_{i}^{m_{1}}y_{j}^{m_{3}}),x_{n_{1}+1}^{m_{1}}y_{n_{1}+1}^{m_{2}},\eta^{m_{1}+m_{2}}(x_{n_{1}}^{m_{2}}y_{n_{1}+1}^{m_{3}-m_{1}}),\eta^{m_{1}+m_{2}}(y_{n_{1}+2}^{m_{2}}x_{n_{1}+1}^{m_{3}-m_{1}})$ $\displaystyle\qquad\;\;\mid m_{r}\in\mathbb{N};(i,j)=(n_{1},n_{1}+1),(n_{1},n_{1}+2),(n_{1}+1,n_{1}+2)\\}.\hskip 71.13188pt(7.67)$ are all the singular vectors of $sl(n,\mathbb{F})$ in ${\cal B}$. Fix $k\in\mathbb{N}$. Then the singular vectors of $sl(n,\mathbb{F})$ in ${\cal B}_{\langle-k\rangle}$ are those in (7.34). According to the arguments in Case 1, ${\cal B}_{\langle-k\rangle}$ is an irreducible $sp(2n,\mathbb{F})$-submodule. Let $0<k\in\mathbb{N}$. Then the singular vectors of $sl(n,\mathbb{F})$ in ${\cal B}_{\langle k\rangle}$ are $\displaystyle\\{\eta^{m_{2}}(x_{n_{1}+1}^{k+m_{1}-2m_{2}}y_{n_{1}+2}^{m_{1}}),\eta^{m_{2}}(x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{k+m_{1}-2m_{2}}),\eta^{m_{5}+m_{6}}(x_{n_{1}}^{m_{6}}y_{n_{1}+1}^{m_{7}-m_{5}}),\eta^{m_{5}+m_{6}}(y_{n_{1}+2}^{m_{6}}x_{n_{1}+1}^{m_{7}-m_{5}}),\hskip 17.07182pt$ $\displaystyle\\!\\!\\!\\!\\!x_{n_{1}+1}^{m_{3}}y_{n_{1}+1}^{m_{4}}\mid m_{i}\in\mathbb{N};2m_{2}\leq k+m_{1};m_{3}+m_{4}=k=m_{5}+m_{5}+m_{7}\\}\hskip 59.75095pt(7.68)$ by (7.67). Let $M$ be a nonzero $sp(2n,\mathbb{F})$-submodule of ${\cal B}_{\langle k\rangle}$. As an $sl(n,\mathbb{F})$-module, $M$ contains a singular of $sl(n,\mathbb{F})$. If $x_{n_{1}+1}^{m_{3}}y_{n_{1}+1}^{m_{4}}\in M$ with $m_{3}+m_{4}=k$, then $E_{n_{1}+1,n+n_{1}+1}\|_{\cal B}=x_{n_{1}+1}\partial_{y_{n_{1}+1}}$ and $E_{n_{1}+1,n+n_{1}+1}^{m_{4}}(x_{n_{1}+1}^{m_{3}}y_{n_{1}+1}^{m_{4}})=m_{4}!x_{n_{1}+1}^{k}\in M\Longrightarrow x_{n_{1}+1}^{k}\in M$ $None$ by (7.5). Suppose some $\eta^{m_{5}+m_{6}}(x_{n_{1}}^{m_{6}}y_{n_{1}+1}^{m_{7}-m_{5}})\in M$ with $m_{5}+m_{5}+m_{7}=k$. According to (1.16), $E_{n+n_{1}+1,n_{1}+1}=y_{n_{1}+1}\partial_{x_{n_{1}+1}}$. So $E_{n+n_{1}+1,n_{1}+1}^{m_{5}+m_{6}}[\eta^{m_{5}+m_{6}}(x_{n_{1}}^{m_{6}}y_{n_{1}+1}^{m_{7}-m_{5}})]=(m_{5}+m_{6})!x_{n_{1}}^{m_{6}}y_{n_{1}+1}^{k+m_{6}}\in M.$ $None$ Moreover, (7.4) yields that $(E_{n_{1},n+n_{1}+1}+E_{n_{1}+1,n+n_{1}})\|_{\cal B}=\partial_{x_{n_{1}}}\partial_{y_{n_{1}+1}}+x_{n_{1}+1}\partial_{y_{n_{1}}}$ and $(E_{n_{1},n+n_{1}+1}+E_{n_{1}+1,n+n_{1}})^{m_{6}}(x_{n_{1}}^{m_{6}}y_{n_{1}+1}^{k+m_{6}})=m_{6}![\prod_{r=1}^{m_{6}}(k+r)]y_{n_{1}+1}^{k}\in M.$ $None$ Assume some $\eta^{m_{5}+m_{6}}(y_{n_{1}+2}^{m_{6}}x_{n_{1}+1}^{m_{7}-m_{5}})\in M$ with $m_{5}+m_{5}+m_{7}=k$. By (7.3) and (7.5), $E_{n_{1}+1,n+n_{1}+1}^{m_{5}+m_{6}}[\eta^{m_{5}+m_{6}}(y_{n_{1}+2}^{m_{6}}x_{n_{1}+1}^{m_{7}-m_{5}})]=(m_{5}+m_{6})!y_{n_{1}+2}^{m_{6}}x_{n_{1}+1}^{k+m_{6}}\in M.$ $None$ Observe $(E_{n+n_{1}+2,n_{1}+1}+E_{n+n_{1}+1,n_{1}+2})\|_{\cal B}=\partial_{x_{n_{1}+1}}\partial_{y_{n_{1}+2}}+y_{n_{1}+1}\partial_{x_{n_{1}+2}}$ $None$ by (1.16). Hence $(E_{n+n_{1}+2,n_{1}+1}+E_{n+n_{1}+1,n_{1}+2})^{m_{6}}(y_{n_{1}+2}^{m_{6}}x_{n_{1}+1}^{k+m_{6}})=m_{6}![\prod_{r=1}^{m_{6}}(k+r)]x_{n_{1}+1}^{k}\in M.$ $None$ Expressions (7.53)-(7.60), (7.69), (7.71) and (7.74) show that we always have $x_{n_{1}+1}^{k}\in M$. Furthermore, (7.61)-(7.66) imply that ${\cal B}_{\langle k\rangle}$ is an irreducible $sp(2n,\mathbb{F})$-module. Case 3. $n_{1}=n_{2}$. In this case, $\eta=\sum_{i=1}^{n_{1}}y_{i}\partial_{x_{i}}+\sum_{s=n_{2}+1}^{n}x_{s}\partial_{y_{s}}.$ $None$ First we consider the subcase $1<n_{1}<n-1$. Expression (4.17) says that the nonzero weight vectors in $\mbox{Span}\\{x_{n_{1}}^{m_{1}}y_{n_{1}}^{m_{2}}\zeta_{1}^{m_{3}+1},x_{n_{1}+1}^{m_{1}}y_{n_{1}+1}^{m_{2}}\zeta_{2}^{m_{3}+1},\eta^{m_{3}}(x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}})\mid m_{i}\in\mathbb{N}\\}$ $None$ are all the singular vectors of $sl(n,\mathbb{F})$ in ${\cal B}$, where $\zeta_{1}=x_{n_{1}-1}y_{n_{1}}-x_{n_{1}}y_{n_{1}-1},\;\;\zeta_{2}=x_{n_{1}+1}y_{n_{1}+2}-x_{n_{1}+2}y_{n_{1}+1}.$ $None$ Fix $k\in\mathbb{N}+1$. Then the singular vectors of $sl(n,\mathbb{F})$ in ${\cal B}_{\langle-k\rangle}$ are $\displaystyle\\{x_{n_{1}}^{k+m_{1}}y_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}+1},x_{n_{1}+1}^{m_{1}}y_{n_{1}+1}^{k+m_{1}}\zeta_{2}^{m_{2}+1},\eta^{m_{3}}(x_{n_{1}}^{m_{4}}y_{n_{1}+1}^{m_{5}})$ $\displaystyle\mid m_{i}\in\mathbb{N};m_{4}+m_{5}-2m_{3}=k\\}.\hskip 219.08612pt(7.78)$ Let $M$ be a nonzero $sp(2n,\mathbb{F})$-submodule of ${\cal B}_{\langle-k\rangle}$. As an $sl(n,\mathbb{F})$-module, $M$ contains a singular vector of $sl(n,\mathbb{F})$. Suppose some $x_{n_{1}}^{k+m_{1}}y_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}+1}\in M$. Note $E_{n_{1},n+n_{1}}\|_{\cal B}=\partial_{x_{n_{1}}}\partial_{y_{n_{1}}}$ by (7.4), and so $\displaystyle E_{n_{1},n+n_{1}}(x_{n_{1}}^{k+m_{1}}y_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}})$ $\displaystyle=$ $\displaystyle(k+m_{1})m_{1}x_{n_{1}}^{k+m_{1}-1}y_{n_{1}}^{m_{1}-1}\zeta_{1}^{m_{2}}-m_{2}(m_{2}-1)x_{n_{1}}^{k+m_{1}}y_{n_{1}}^{m_{1}}x_{n_{1}-1}y_{n_{1}-1}\zeta_{1}^{m_{2}-2}$ $\displaystyle+(k+m_{1})m_{2}x_{n_{1}}^{k+m_{1}-1}y_{n_{1}}^{m_{1}}x_{n_{1}-1}\zeta_{1}^{m_{2}-1}-m_{1}m_{2}x_{n_{1}}^{k+m_{1}}y_{n_{1}}^{m_{1}-1}y_{n_{1}-1}\zeta_{1}^{m_{2}-1}.\hskip 48.36958pt(7.79)$ Moreover, $(E_{n_{1}-1,n_{1}}-E_{n+n_{1},n+n_{1}-1})\|_{\cal B}=-(x_{n_{1}}\partial_{x_{n_{1}-1}}+y_{n_{1}}\partial_{y_{n_{1}-1}})$ $None$ by (1.7), (1.8) and (1.14). Thus $\displaystyle(E_{n_{1}-1,n_{1}}-E_{n+n_{1},n+n_{1}-1})^{2}E_{n_{1},n+n_{1}}(x_{n_{1}}^{k+m_{1}}y_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}})$ $\displaystyle=$ $\displaystyle-2m_{2}(m_{2}-1)x_{n_{1}}^{k+m_{1}+1}y_{n_{1}}^{m_{1}+1}\zeta_{1}^{m_{2}-2}\in M.\hskip 145.10922pt(7.81)$ Hence $x_{n_{1}}^{k+m_{1}+1}y_{n_{1}}^{m_{1}+1}\zeta_{1}^{m_{2}-2}\in M\qquad\mbox{if}\;\;m_{2}>1.$ $None$ Furthermore, $(E_{n_{1}-1,n_{1}}-E_{n+n_{1},n+n_{1}-1})E_{n_{1},n+n_{1}}(x_{n_{1}}^{k+m_{1}}y_{n_{1}}^{m_{1}}\zeta_{1})=-kx_{n_{1}}^{k+m_{1}}y_{n_{1}}^{m_{1}}\in M.$ $None$ So we always have $x_{n_{1}}^{k+m}y_{n_{1}}^{m}\in M$ for some $m\in\mathbb{N}$ by induction on $m_{2}$. Observe $E_{n_{1},n+n_{1}}(x_{n_{1}}^{k+m}y_{n_{1}}^{m})=\partial_{x_{n_{1}}}\partial_{y_{n_{1}}}(x_{n_{1}}^{k+m}y_{n_{1}}^{m})=m![\prod_{r=1}^{m}(k+r)]x_{n_{1}}^{k}$ $None$ by (7.4). Thus $x_{n_{1}}^{k}\in M.$ $None$ Symmetrically, if some $x_{n_{1}+1}^{m_{1}}y_{n_{1}+1}^{k+m_{1}}\zeta_{2}^{m_{2}+1}\in M$, we have $y_{n_{1}+1}^{k}\in M$. But $(E_{n+n_{1}+1,n_{1}}+E_{n+n_{1},n_{1}+1})\|_{\cal B}=-x_{n_{1}}\partial_{y_{n_{1}+1}}+y_{n_{1}}\partial_{x_{n_{1}+1}}$ $None$ by (1.16), which gives $(E_{n+n_{1}+1,n_{1}}+E_{n+n_{1},n_{1}+1})^{k}(y_{n_{1}+1}^{k})=(-1)^{k}k!x_{n_{1}}^{k}\in M.$ $None$ Thus (7.85) holds again. Assume that some $\eta^{m_{3}}(x_{n_{1}}^{m_{4}}y_{n_{1}+1}^{m_{5}})\in M$ with $m_{4}+m_{5}-2m_{3}=k$. Note there exists $r_{1},r_{2}\in\mathbb{N}$ such that $r_{1}+r_{2}=m_{3}$ and $2r_{1}\leq m_{4},\;2r_{2}\leq m_{5}$. Moreover, $E_{n_{1},n+n_{1}}\|_{\cal B}=\partial_{x_{n_{1}}}\partial_{y_{n_{1}}}$ by (7.4) and $E_{n+n_{1}+1,n_{1}+1}\|_{\cal B}=\partial_{x_{n_{1}+1}}\partial_{y_{n_{1}+1}}$ by (1.16). Thus $\displaystyle E_{n_{1},n+n_{1}}^{r_{1}}E_{n+n_{1}+1,n_{1}+1}^{r_{2}}[\eta^{m_{3}}(x_{n_{1}}^{m_{4}}y_{n_{1}+1}^{m_{5}})]$ $\displaystyle=$ $\displaystyle m_{3}![\prod_{s_{1}=0}^{2r_{1}-1}(m_{4}-s_{1})][\prod_{s_{2}=0}^{2r_{2}-1}(m_{5}-s_{2})]x_{n_{1}}^{m_{4}-2r_{1}}y_{n_{1}+1}^{m_{5}-2r_{2}}\in M.\hskip 91.04872pt(7.88)$ Furthermore, (1.16) gives $(E_{n+n_{1}+1,n_{1}}+E_{n+n_{1},n_{1}+1})\|_{\cal B}=-x_{n_{1}}\partial_{y_{n_{1}+1}}+y_{n_{1}}\partial_{x_{n_{1}+1}}$ , and so $\displaystyle(E_{n+n_{1}+1,n_{1}}+E_{n+n_{1},n_{1}+1})^{m_{5}-2r_{2}}(x_{n_{1}}^{m_{4}-2r_{1}}y_{n_{1}+1}^{m_{5}-2r_{2}})$ $\displaystyle=$ $\displaystyle(-1)^{m_{5}-2r_{2}}(m_{5}-2r_{2})!x_{n_{1}}^{k}\in M.\hskip 176.407pt(7.89)$ Thus we always have $x_{n_{1}}^{k}\in M$. Now $(E_{n_{1},n+n_{1}+1}+E_{n_{1}+1,n+n_{1}})\|_{\cal B}=-y_{n_{1}+1}\partial_{x_{n_{1}}}+x_{n_{1}+1}\partial_{y_{n_{1}}}$ $None$ by (7.4). For any $r\in\mathbb{N}+1$, $\frac{(-1)^{r}}{\prod_{s=0}^{r-1}(k-r)}(E_{n_{1},n+n_{1}+1}+E_{n_{1}+1,n+n_{1}})^{r}(x_{n_{1}}^{k})=x_{n_{1}}^{k-r}y_{n_{1}+1}^{r}\in M.$ $None$ If $k\geq 2$ and $r\in\overline{1,k-1}$, then ${\cal M}^{1}_{\langle-k+r\rangle}{\cal M}^{2}_{\langle-r\rangle}=U({\cal G}_{1})U({\cal G}_{2})(x_{n_{1}}^{k-r}y_{n_{1}+1}^{r})\subset M$ $None$ because ${\cal M}^{1}_{\langle-k+r\rangle}$ is an irreducible ${\cal G}_{1}$-module and ${\cal M}^{2}_{\langle-r\rangle}$ is an irreducible ${\cal G}_{2}$-module by Step 1. Moreover, ${\cal M}^{1}_{\langle-k\rangle}=U({\cal G}_{1})(x_{n_{1}}^{k}),\;{\cal M}^{2}_{\langle-k\rangle}=U({\cal G}_{2})(y_{n_{1}+1}^{k})\subset M.$ $None$ Furthermore, ${\cal M}^{1}_{\langle-k\rangle}{\cal M}^{2}_{\langle 0\rangle}=U({\cal G}_{1})U({\cal G}_{2})(x_{n_{1}}^{k})\subset M\;\;\mbox{if}\;\;n_{1}=n-1$ $None$ and ${\cal M}^{1}_{\langle 0\rangle}{\cal M}^{2}_{\langle-k\rangle}=U({\cal G}_{1})U({\cal G}_{2})(y_{n_{1}+1}^{k})\subset M\;\;\mbox{if}\;\;n_{1}=1.$ $None$ Note $(E_{r,i}-E_{n+i,n+r})\|_{\cal B}=y_{i}y_{r}-x_{i}x_{r}\qquad\mbox{for}\;\;i\in\overline{1,n_{1}},\;r\in\overline{n_{1}+1,n}$ $None$ by (1.7), (1.8) and (1.14). In particular, if $k>1$ or $n_{1}=1$, we have $(E_{n_{1}+1,n_{1}}-E_{n+n_{1},n+n_{1}+1})(x_{n_{1}}^{k})=y_{n_{1}}x_{n_{1}}^{k}y_{n_{1}+1}-x_{n_{1}}^{k+1}x_{n_{1}+1}\in M.$ $None$ Since $y_{n_{1}}x_{n_{1}}^{k}y_{n_{1}+1}\in{\cal M}^{1}_{\langle-k+1\rangle}{\cal M}^{2}_{\langle-1\rangle}\subset M,$ $None$ we get $x_{n_{1}}^{k+1}x_{n_{1}+1}\in M.$ $None$ Suppose $k=1$ and $n_{1}>1$. By (7.93), $\zeta_{1}x_{n_{1}}=(x_{n_{1}-1}y_{n_{1}}-x_{n_{1}}y_{n_{1}-1})x_{n_{1}}\in M.$ $None$ Observe $(E_{n_{1}+1,n+n_{1}-1}+E_{n_{1}-1,n+n_{1}+1})\|_{\cal B}=x_{n_{1}+1}\partial_{y_{n_{1}-1}}-y_{n_{1}+1}\partial_{x_{n_{1}-1}}$ $None$ by (1.15). So $-(E_{n_{1}+1,n+n_{1}-1}+E_{n_{1}-1,n+n_{1}+1})(\zeta_{1}x_{n_{1}})=x_{n_{1}}^{2}x_{n_{1}+1}-x_{n_{1}}y_{n_{1}}y_{n_{1}+1}\in M.$ $None$ On the other hand, (1.16) gives $(E_{n+i,j}+E_{n+j,i})\|_{\cal B}=-(x_{i}y_{j}+x_{j}y_{i})\qquad\mbox{for}\;\;i,j\in\overline{1,n_{1}},$ $None$ which implies $-E_{n+n_{1},n_{1}}(y_{n_{1}+1})=x_{n_{1}}y_{n_{1}}y_{n_{1}+1}\in M.$ $None$ By (7.102), we have $x_{n_{1}}^{2}x_{n_{1}+1}\in M.$ So (7.99) always holds. By Step 1, ${\cal M}^{1}_{\langle-k-1\rangle}{\cal M}^{2}_{\langle 1\rangle}=U({\cal G}_{1})U({\cal G}_{2})(x_{n_{1}}^{k+1}x_{n_{1}+1})\subset M.$ $None$ Suppose ${\cal M}^{1}_{\langle-k-i\rangle}{\cal M}^{2}_{\langle i\rangle}\subset M$ $None$ for $1\leq i\leq m$. Then $\displaystyle(E_{n_{1}+1,n_{1}}-E_{n+n_{1},n+n_{1}+1})(x_{n_{1}}^{k+m}x_{n_{1}+1}^{m})$ $\displaystyle=$ $\displaystyle y_{n_{1}}x_{n_{1}}^{k+m}x_{n_{1}+1}^{m}y_{n_{1}+1}-x_{n_{1}}^{k+m+1}x_{n_{1}+1}^{m+1}\in M\hskip 142.26378pt(7.107)$ by (7.96). If $m>1$, we have $y_{n_{1}}x_{n_{1}}^{k+m}x_{n_{1}+1}^{m}y_{n_{1}+1}\in{\cal M}^{1}_{\langle-k-(m-1)\rangle}{\cal M}^{2}_{\langle m-1\rangle}\subset M.$ $None$ Note $(E_{r,n+s}+E_{s,n+r})\|_{\cal B}=-(x_{r}y_{s}+x_{s}y_{r})\qquad\mbox{for}\;\;r,s\in\overline{n_{1}+1,n}$ $None$ by (1.15). If $m=1$, we have $y_{n_{1}}x_{n_{1}}^{k+1}x_{n_{1}+1}y_{n_{1}+1}=-E_{n_{1}+1,n+n_{1}+1}(y_{n_{1}}x_{n_{1}}^{k+1})\subset E_{n_{1}+1,n+n_{1}+1}({\cal M}^{1}_{\langle-k\rangle})\subset M.$ $None$ Then (7.107), (7.108) and (7.110) give $x_{n_{1}}^{k+m+1}x_{n_{1}+1}^{m+1}\in M.$ $None$ Furthermore, ${\cal M}^{1}_{\langle-k-m-1\rangle}{\cal M}^{2}_{\langle m+1\rangle}=U({\cal G}_{1})U({\cal G}_{2})(x_{n_{1}}^{k+m+1}x_{n_{1}+m+1})\subset M.$ $None$ Thus (7.106) holds for any $i\in\mathbb{N}+1$. Symmetrically, we have ${\cal M}^{1}_{\langle i\rangle}{\cal M}^{2}_{\langle-k-i\rangle}\subset M\qquad\mbox{for}\;\;i\in\mathbb{N}+1.$ $None$ Suppose $n_{1}<n-1$. Then $x_{n_{1}}^{k+1}x_{n_{1}+1}\zeta_{2}\in M$ by (7.105). Moreover, $(k+1)y_{n_{1}}x_{n_{1}}^{k}y_{n_{1}+1}\zeta_{2}=-(k+1)E_{n+n_{1},n_{1}}(x_{n_{1}}^{k-1}y_{n_{1}+1}\zeta_{2})\in M$ $None$ by (7.92) and (7.103). According (1.7), (1.8) and (1.14), $(E_{n_{1},n_{1}+1}-E_{n+n_{1}+1,n+n_{1}})\|_{\cal B}=\partial_{x_{n_{1}}}\partial_{x_{n_{1}+1}}-\partial_{y_{n_{1}}}\partial_{y_{n_{1}+1}}.$ $None$ Thus $\displaystyle(E_{n_{1},n_{1}+1}-E_{n+n_{1}+1,n+n_{1}})[(x_{n_{1}}^{k+1}x_{n_{1}+1}-(k+1)y_{n_{1}}x_{n_{1}}^{k}y_{n_{1}+1})\zeta_{2}]$ $\displaystyle=$ $\displaystyle 3(k+1)x_{n_{1}}^{k}\zeta_{2}\in M\hskip 273.14662pt(7.116)$ by (7.77). Hence ${\cal M}^{1}_{\langle-k\rangle}{\cal M}^{2}_{\langle 0\rangle}=U({\cal G}_{1})U({\cal G}_{2})(x_{n_{1}}^{k})+U({\cal G}_{1})U({\cal G}_{2})(x_{n_{1}}^{k}\zeta_{2})\subset M$ $None$ by (7.26) and(7.85). Symmetrically, ${\cal M}^{1}_{\langle 0\rangle}{\cal M}^{2}_{\langle-k\rangle}\subset M.$ $None$ By (7.92)-(7.95), (7.106), (7.112), (7.113), (7.117) and (7.118), ${\cal M}^{1}_{\langle-k-r\rangle}{\cal M}^{2}_{\langle r\rangle}\subset M\qquad\mbox{for}\;\;r\in\mathbb{Z}.$ $None$ Therefore, ${\cal B}_{\langle-k\rangle}=\bigoplus_{r\in\mathbb{Z}}{\cal M}^{1}_{\langle- k-r\rangle}{\cal M}^{2}_{\langle r\rangle}\subset M.$ $None$ We get $M={\cal B}_{\langle-k\rangle}$, that is, ${\cal B}_{\langle-k\rangle}$ is an irreducible $sp(2n,\mathbb{F})$-module. We can similarly prove that ${\cal B}_{\langle k\rangle}$ is an irreducible $sp(2n,\mathbb{F})$-module. Finally, we study ${\cal B}_{\langle 0\rangle}$. We first consider the generic case $1<n_{1}<n-1$. Set $\displaystyle\qquad{\cal B}_{\langle 0,1\rangle}$ $\displaystyle=$ $\displaystyle\mbox{Span}\\{[\prod_{1\leq r\leq s\leq n_{1}\;\mbox{or}\;n_{1}+1\leq r\leq s\leq n}(x_{r}y_{s}+x_{s}y_{r})^{l_{r,s}}]$ $\displaystyle\qquad\;\;\times[\prod_{p=1}^{n_{1}}\prod_{q=n_{1}+1}^{n}(x_{p}x_{q}-y_{p}y_{q})^{k_{p,q}}]\mid l_{r,s},k_{p,q}\in\mathbb{N}\\}\hskip 85.35826pt(7.121)$ and ${\cal B}_{\langle 0,2\rangle}=\sum_{1\leq r<s\leq n_{1}\;\mbox{or}\;n_{1}+1\leq r<s\leq n}{\cal B}_{\langle 0,1\rangle}(x_{r}y_{s}-x_{s}y_{r})+\sum_{p=1}^{n_{1}}\sum_{q=n_{1}+1}^{n}{\cal B}_{\langle 0,1\rangle}(x_{p}x_{q}+y_{p}y_{q}).$ $None$ We want to prove that ${\cal B}_{\langle 0,1\rangle}$ and ${\cal B}_{\langle 0,2\rangle}$ forms $sp(2n,\mathbb{F})$-submodules. Let $\displaystyle\qquad{\cal G}^{\prime}$ $\displaystyle=$ $\displaystyle\sum_{1\leq r\leq s\leq n_{1}}\mathbb{F}(E_{n+s,r}+E_{n+r,s})+\sum_{n_{1}+1\leq p\leq q\leq n}\mathbb{F}(E_{p,n+q}+E_{q,n+p})$ $\displaystyle+\sum_{r=1}^{n_{1}}\sum_{p=n_{1}+1}^{n}\mathbb{F}(E_{p,r}-E_{n+r,n+p})\hskip 199.16928pt(7.123)$ and $\displaystyle\hat{\cal G}$ $\displaystyle=$ $\displaystyle\sum_{i,j=1}^{n_{1}}\mathbb{F}(E_{i,j}-E_{n+j,n+i})+\sum_{r,s=n_{1}+1}^{n}\mathbb{F}(E_{r,s}-E_{n+s,n+r})+\sum_{1\leq r\leq s\leq n_{1}}\mathbb{F}(E_{r,n+s}+E_{s,n+r})$ $\displaystyle+\sum_{n_{1}+1\leq p\leq q\leq n}\mathbb{F}(E_{n+q,p}+E_{n+p,q})+\sum_{r=1}^{n_{1}}\sum_{p=n_{1}+1}^{n}[\mathbb{F}(E_{r,p}-E_{n+p,n+r})$ $\displaystyle+\mathbb{F}(E_{r,n+p}+E_{p,n+r})+\mathbb{F}(E_{n+r,p}-E_{n+p,r})].\hskip 165.02606pt(7.124)$ Then ${\cal G}^{\prime}$ and $\hat{\cal G}$ are Lie subalgebras of $sp(2n,\mathbb{F})$ and $sp(2n,\mathbb{F})={\cal G}^{\prime}\oplus\hat{\cal G}.$ By PBW Theorem $U(sp(2n,\mathbb{F}))=U({\cal G}^{\prime})U(\hat{\cal G}).$ By (7.96), (7.103) and (7.109), $U({\cal G}^{\prime})\|_{\cal B}={\cal B}_{\langle 0,1\rangle}\;\;\mbox{as multiplication operators on}\;\;{\cal B}.$ $None$ Moreover, $(E_{r,s}-E_{n+s,n+r})\|_{\cal B}=x_{r}\partial_{x_{s}}+y_{r}\partial_{y_{s}}+\delta_{r,s},$ $None$ $(E_{n+r,s}+E_{n+s,r})\|_{\cal B}=\partial_{x_{r}}\partial{y_{s}}+\partial_{x_{s}}\partial{y_{r}},$ $None$ $(E_{n+r,i}+E_{n+i,r})\|_{\cal B}=-x_{i}\partial{y_{r}}+y_{i}\partial{x_{r}},$ $None$ $(E_{i,n+r}+E_{r,n+i})\|_{\cal B}=-y_{r}\partial{x_{i}}+x_{r}\partial{y_{i}},$ $None$ $(E_{i,r}-E_{n+r,n+i})\|_{\cal B}=\partial_{x_{i}}\partial{x_{r}}-\partial_{y_{i}}\partial{y_{r}}$ $None$ for $i\in\overline{1,n_{1}}$ and $r,s\in\overline{n_{1}+1,n}$. According to (7.4), (7.6), (7.124) and (7.126)-(7.130), $U(\hat{\cal G})(1)=\mathbb{F}.$ Thus ${\cal B}_{\langle 0,1\rangle}=U({\cal G}^{\prime})(1)=U(sp(2n,\mathbb{F}))(1)$ $None$ forms an $sp(2n,\mathbb{F})$-submodule. Let $W=\sum_{1\leq r<s\leq n_{1}\;\mbox{or}\;n_{1}+1\leq r<s\leq n}\mathbb{F}(x_{r}y_{s}-x_{s}y_{r})+\sum_{p=1}^{n_{1}}\sum_{q=n_{1}+1}^{n}\mathbb{F}(x_{p}x_{q}+y_{p}y_{q}).$ $None$ By (7.4), (7.6) and (7.126)-(7.130), we can verify that $W$ forms an irreducible $\hat{\cal G}$-submodule. Hence ${\cal B}_{\langle 0,2\rangle}=U({\cal G}^{\prime})(W)=U(sp(2n,\mathbb{F}))(W)$ $None$ forms an $sp(2n,\mathbb{F})$-submodule. Moreover, ${\cal B}_{\langle 0,1\rangle}\bigcap W=\\{0\\}.$ $None$ Next we want to prove that ${\cal B}_{\langle 0,1\rangle}$ and ${\cal B}_{\langle 0,2\rangle}$ are irreducible $sp(2n,\mathbb{F})$-submodules. According to (7.78), the singular vectors of $sl(n,\mathbb{F})$ in ${\cal B}_{\langle 0\rangle}$ are $\displaystyle\\{x_{n_{1}}^{m_{1}}y_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}+1},x_{n_{1}+1}^{m_{1}}y_{n_{1}+1}^{m_{1}}\zeta_{2}^{m_{2}+1},\eta^{m_{3}}(x_{n_{1}}^{m_{4}}y_{n_{1}+1}^{m_{5}})$ $\displaystyle\mid m_{i}\in\mathbb{N};m_{4}+m_{5}=2m_{3}\\}.\hskip 233.3125pt(7.135)$ Let $M$ be a nonzero submodule of ${\cal B}_{\langle 0,1\rangle}$. Then $M$ contains a singular vector of $sl(n,\mathbb{F})$. Suppose some $x_{n_{1}}^{m_{1}}y_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}}\in M$. By (7.79)-(7.82), we can assume $m_{2}=0,1$. If $m_{2}=0$, (7.84) yields $1\in M$. Then $M={\cal B}_{\langle 0,1\rangle}$ by (7.131). Suppose $m_{2}=1$. We have $E_{n_{1},n+n_{1}}\|_{\cal B}=\partial_{x_{n_{1}}}\partial_{y_{n_{1}}}$ by (7.4), and $E_{n_{1},n+n_{1}}[x_{n_{1}}^{m_{1}}y_{n_{1}}^{m_{1}}\zeta_{1}]=m_{1}(m_{1}+1)x_{n_{1}}^{m_{1}-1}y_{n_{1}}^{m_{1}-1}\zeta_{1}$ $None$ by (7.79). By induction on $m_{1}$, we have $\zeta_{1}\in M\subset{\cal B}_{\langle 0,1\rangle}$, which contradicts (7.134). Similarly, if some $x_{n_{1}+1}^{m_{1}}y_{n_{1}+1}^{m_{1}}\zeta_{2}^{m_{2}+1}\in M$, we have $M={\cal B}_{\langle 0,1\rangle}$. Assume some $\eta^{m_{3}}(x_{n_{1}}^{m_{4}}y_{n_{1}+1}^{m_{5}})\in M$ with $m_{4}+m_{5}=2m_{3}$. Note $m_{4}$ and $m_{5}$ are both even or odd. If $m_{4}=2r_{1}$ and $m_{5}=2r_{2}$ are even, then (7.88) gives $1\in M$, equivalently $M={\cal B}_{\langle 0,1\rangle}$. Suppose that $m_{4}=2r_{1}+1$ and $m_{5}=2r_{2}+1$ are odd. Expression (7.75) yields $\eta(x_{n_{1}}y_{n_{1}+1})=x_{n_{1}}x_{n_{1}+1}+y_{n_{1}}y_{n_{1}+1}\in M\subset{\cal B}_{\langle 0,1\rangle},$ $None$ which contradicts (7.134) again. Thus we always have $M={\cal B}_{\langle 0,1\rangle}$, that is, ${\cal B}_{\langle 0,1\rangle}$ is irreducible. Similarly, we can prove that ${\cal B}_{\langle 0,2\rangle}$ is irreducible. If $n_{1}=1$ and $n=2$, we let ${\cal B}_{\langle 0,1\rangle}=\mbox{Span}\\{[\prod_{i=1}^{n}(x_{i}y_{i})^{m_{i}}](x_{1}x_{2}-y_{1}y_{2})^{m_{3}}]\mid m_{i}\in\mathbb{N}\\}$ $None$ and ${\cal B}_{\langle 0,2\rangle}={\cal B}_{\langle 0,1\rangle}(x_{1}x_{2}+y_{1}y_{2}).$ When $n_{1}=1$ and $n>2$, we set ${\cal B}_{\langle 0,1\rangle}=\mbox{Span}\\{[(x_{1}y_{1})^{l}\prod_{2\leq r\leq s\leq n}(x_{r}y_{s}+x_{s}y_{r})^{l_{r,s}}][\prod_{q=2}^{n}(x_{1}x_{q}-y_{1}y_{q})^{k_{q}}]\mid l,l_{r,s},k_{q}\in\mathbb{N}\\}$ $None$ and ${\cal B}_{\langle 0,2\rangle}=\sum_{2\leq r<s\leq n}{\cal B}_{\langle 0,1\rangle}(x_{r}y_{s}-x_{s}y_{r})+\sum_{q=2}^{n}{\cal B}_{\langle 0,1\rangle}(x_{1}x_{q}+y_{1}y_{q}).$ $None$ In the case $1<n_{1}=n-1$, we put $\displaystyle\qquad{\cal B}_{\langle 0,1\rangle}$ $\displaystyle=$ $\displaystyle\mbox{Span}\\{(x_{n}y_{n})^{l}[\prod_{1\leq r\leq s\leq n-1}(x_{r}y_{s}+x_{s}y_{r})^{l_{r,s}}]$ $\displaystyle\qquad\;\;\times[\prod_{p=1}^{n-1}(x_{p}x_{n}-y_{p}y_{n})^{k_{p}}]\mid l,l_{r,s},k_{p}\in\mathbb{N}\\}\hskip 119.50148pt(7.141)$ and ${\cal B}_{\langle 0,2\rangle}=\sum_{1\leq r<s\leq n_{1}}{\cal B}_{\langle 0,1\rangle}(x_{r}y_{s}-x_{s}y_{r})+\sum_{p=1}^{n_{1}}{\cal B}_{\langle 0,1\rangle}(x_{p}x_{n}+y_{p}y_{n}).$ $None$ The above corresponding partial arguments show that ${\cal B}_{\langle 0,1\rangle}$ and ${\cal B}_{\langle 0,2\rangle}$ are irreducible in the corresponding case. Now $1$ is a non-isotropic element in ${\cal B}_{\langle 0,1\rangle}$ and $x_{n_{1}}x_{n_{1}+1}+y_{n_{1}}y_{n_{1}+1}$ a non-isotropic element in ${\cal B}_{\langle 0,2\rangle}$ by (3.54). By Lemma 2.3, the symmetric bilinear form $(\cdot\|\cdot)$ restricted to them are nondegenrate. Since $(1\|{\cal B}_{\langle 0,2\rangle})=\\{0\\}$, ${\cal B}_{\langle 0,1\rangle}$ is orthogonal to ${\cal B}_{\langle 0,2\rangle}$. Thus the symmetric bilinear form $(\cdot\|\cdot)$ restricted ${\cal B}_{\langle 0,1\rangle}+{\cal B}_{\langle 0,2\rangle}$ is nondegenerate. Then ${\cal B}_{\langle 0\rangle}=({\cal B}_{\langle 0,1\rangle}+{\cal B}_{\langle 0,2\rangle})\oplus({\cal B}_{\langle 0,1\rangle}+{\cal B}_{\langle 0,2\rangle})^{\perp}\bigcap{\cal B}_{\langle 0\rangle}.$ $None$ If $({\cal B}_{\langle 0,1\rangle}+{\cal B}_{\langle 0,2\rangle})^{\perp}\bigcap{\cal B}_{\langle 0\rangle}\neq\\{0\\}$, then it contains a singular vector of $sl(n,\mathbb{F})$. Our above arguments in proving the irreducibility of ${\cal B}_{\langle 0,1\rangle}$ show that it contains either ${\cal B}_{\langle 0,1\rangle}$ or ${\cal B}_{\langle 0,2\rangle}$, which is absurd. Therefore, ${\cal B}_{\langle 0\rangle}={\cal B}_{\langle 0,1\rangle}\oplus{\cal B}_{\langle 0,2\rangle}$ is an orthogonal decomposition of irreducible $sp(2n,\mathbb{F})$-submodules. Suppose $n_{1}=n_{2}=n$. For $k\in\mathbb{N}+1$, (4.21) and (4.31) imply that ${\cal B}_{\langle k\rangle}=\bigoplus_{m=0}^{\infty}\bigoplus_{r=\llbracket(k+1)/2\rrbracket}^{\infty}\eta^{r}({\cal H}_{\langle k-2r-m,m\rangle})$ $None$ and ${\cal B}_{\langle-k\rangle}=\bigoplus_{m,r=0}^{\infty}\eta^{r}({\cal H}_{\langle-k-2r-m,m\rangle})$ $None$ are $({\cal G},{\cal K})$-structures, where ${\cal H}_{\langle- m_{1}-m_{2},m_{2}\rangle}$ is given in (4.42). Moreover, ${\cal B}_{\langle 0,1\rangle}=\bigoplus_{m,r=0}^{\infty}\eta^{r}({\cal H}_{\langle-2r-2m,2m\rangle})$ $None$ and ${\cal B}_{\langle 0,2\rangle}=\bigoplus_{m,r=0}^{\infty}\eta^{r}({\cal H}_{\langle-2r-2m-1,2m+1\rangle})$ $None$ are $({\cal G},{\cal K})$-structures by the arguments in (7.79)-(7.82), (7.84) and (7.136) (cf. (7.24), (7.25)). $\qquad\Box$ References [C] B. Cao, Solutions of Navier Equations and Their Representation Structure, Advances in Applied Mathematics, 43 (2009), 331-374. [DES] M. Davidson, T. Enright, and R. Stanke, Differential Operators and Highest Weight Representations, Memoirs of American Mathematical Society 94, no. 455, 1991. [FC] F. M. Fernández and E. A. Castro, Algebraic Methods in Quantum Chemistry and Physics, CRC Press, Inc., 1996. [FSS] L.Frappat, A.Sciarrino and P.sorba, Dictionary on Lie Algebras and Superalgebras, Academic Press,2000 [G] H. Georgi, Lie Algebras in Particle Physics, Second Edition, Perseus Books Group, 1999. [Ho] R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv), 1-182, Israel Math. Conf. Proc., 8, Bar-Ilan Univ., Ramat Gan, 1995. [Hu] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag New York Inc., 1972. [K] V. G. Kac, Infinite Dimensional Lie Algebras, Third edition, Cambridge University Press, 1990. [I1] N. H. Ibragimov, Transformation groups applied to mathematical physics, Nauka, 1983. [I2] N. H. Ibragimov, Lie Group Analysis of Differential Equations, Volume 2, CRC Handbook, CRC Press, 1995. [L] F. S. Levin, An Introduction to Quantum Theory, Cambridge University Press, 2002. [LF] W. Ludwig and C. Falter, Symmetries in Physics, Second Edition, Springer- Verlag, Berlin/Heidelberg, 1996. [X] X.Xu, Flag partial differential equations and representations of Lie algebras, Acta Appl Math 102 (2008), 249–280.	arxiv-papers	2010-12-14T04:53:58	2024-09-04T02:49:15.658788	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Cuiling Luo and Xiaoping Xu", "submitter": "Xiaoping Xu", "url": "https://arxiv.org/abs/1012.2931" }
1012.2939	# Maximum Mass of the Hot Neutron Star with the Quark Core T. Yazdizadeh1 and G. H. Bordbar2,3 1Islamic Azad University, Bafgh Branch, Bafgh, Iran 2Department of Physics, Shiraz University, Shiraz 71454, Iran 3Research Institute for Astronomy and Astrophysics of Maragha, P.O. Box 55134-441, Maragha, Iran ###### Abstract We have considered a hot neutron star with a quark core, a mixed phase of quark-hadron matter, and a hadronic matter crust and have determined the equation of state of the hadronic phase and the quark phase, we have then found the equation of state of the mixed phase under the Gibbs conditions. Finally, we have computed the structure of hot neutron star with the quark core and compared our results with those of the neutron star without the quark core. For the quark matter calculations, we have used the MIT bag model in which the total energy of the system is considered as the kinetic energy of the particles plus a bag constant. For the hadronic matter calculations, we have used the lowest order constrained variational (LOCV) formalism. Our calculations show that the results for the maximum gravitational mass of the hot neutron star with the quark core are substantially different from those of without the quark core. ###### pacs: 21.65.-f, 26.60.-c, 64.70.-p ## I Introduction A hot neutron star is born following the gravitational collapse of the core of a massive star just after the supernova explosion. The interior temperature of a neutron star at its birth is of order $20-50\ MeV$ burrows . Therefore, the high temperature of these stages cannot be neglected with respect to the Fermi temperature throughout the calculation of its structure. This shows that the equation of state of the hot dense matter is very important for investigating the structure of a newborn neutron star. Depending on the total number of nucleons, a newborn neutron star evolves either to a black hole or to a stable neutron star strobel . Hence, calculation of the maximum mass of hot neutron star is of special interest in astrophysics. As we go from surface to the center of a neutron star, at sufficiently high densities, the matter is expected to undergo a transition from hadronic matter, where the quarks are confined inside the hadrons, to a state of deconfined quarks. Finally, there are up, down and strange quarks in the quark matter. Other quarks have high masses and do not appear in this state. Glendenning has shown that a proper construction of the hadron-quark phase transition inside the neutron stars implies the coexistence of nucleonic matter and quark matter over a finite range of the pressure. Therefore, a mixed hadron-quark phase exists in the neutron star and its energy is lower than those of the quark matter and nucleonic matter glen1 . These show that we can consider a neutron star as composed of a hadronic matter layer, a mixed phase of quarks and hadrons and, in core, a quark matter. Recent Chandra observations also imply that the objects RX J185635-3754 and 3C58 could be neutron stars with the quark core prakash . Burgio et al. have investigated the structure of neutron stars with the quark core at zero burgio1 and finite temperature burgio2 with the Brueckner- Bethe-Goldstone formalism to determine the equation of state of the hadronic matter, they have used . We have calculated the structure properties of the cold neutron star by considering a quark phase at its core b1 and compared the results with our previous calculations for the neutron star without the quark core bh . In these works, we have employed the lowest order constrained variational (LOCV) method for the hadronic matter calculations. In the present paper, we intend to extend these calculations for the hot neutron star with the quark core. ## II Equation of State As it was mentioned in the previous section, we consider a neutron star composed of a hadronic matter (hadron phase), a mixed phase of quarks and hadrons, and a quark core (quark phase). Therefore, we should calculate the equation of state of these phases separately as follows. ### II.1 Hadron Phase For this phase of the neutron star matter, we consider the total energy per nucleon as the sum of contributions from the leptons and nucleons, $E=E_{lep}+E_{nucl}\cdot$ (1) The contribution from the energy of leptons (electrons and muons) is $E_{lep}=E_{e}+E_{\mu},$ (2) where $E_{e}$ and $E_{\mu}$ are the energies of electrons and muons, respectively, $E_{i}=\frac{m_{i}^{4}c^{5}}{\pi^{2}n\hbar^{3}}\int_{0}^{\infty}\frac{\sqrt{1+x^{2}}}{1+Exp\\{\beta[m_{i}c^{2}\sqrt{1+x^{2}}-\mu_{i}]\\}}x^{2}dx.$ (3) Here $\mu_{i}$ and $m_{i}$ are the chemical potential and mass of particle $i$, $\beta=\frac{1}{k_{B}T}$ ($T$ is the temperature), $n$ is the total number density of nucleons ($n=n_{p}+n_{n}$), $c$ is speed of light and $x$ is as follows, $x=\frac{\hbar k}{m_{i}c}.$ (4) In our calculations, the equation of state of hot nucleonic matter is determined using the lowest order constrained variational (LOCV) method as follows b2 ; b3 ; b4 ; b5 ; b6 ; b7 ; b8 ; b9 . We adopt a trail wave function as $\psi=F\phi,$ (5) where $\phi$ is the Slater determinant of the single-particle wave function and $F$ is the correlation function which is taken to be $F={\cal S}\prod_{i>j}f(ij).$ (6) ${\cal S}$ is a symmetrizing operator. For the energy of nucleonic matter, we consider up to the two-body term in the cluster expansion, $E_{nucl}=E_{1}+E_{2}.$ (7) The one body term $E_{1}$ for the hot asymmetrical nucleonic matter that consists of $Z$ protons and $N$ neutrons is simply the fermi gas kinetic energy, $E_{1}=\sum_{i=1,2}{\cal E}_{i}$ (8) Label 1 and 2 are used instead of proton and neutron, respectively, and ${\cal E}_{i}$ is ${\cal E}_{i}=\sum_{k}\frac{\hbar^{2}k^{2}}{2m_{i}}f_{i}(k,T,n_{i}),$ (9) where $f(k,T,n_{i})$ is the Fermi-Dirac distribution function fetter , $f(k,T,n_{i})=\frac{1}{e^{\beta[\epsilon_{i}(k,T,n_{i})-\mu_{i}(T,n_{i})]}+1}.$ (10) In the above equation, $n_{i}$ are the number densitis and $\epsilon_{i}$ are the single particle energies associated with the protons and neutrons, $\epsilon_{i}(k,T,n_{i})=\frac{\hbar^{2}k^{2}}{2m_{i}^{\ast}(T,n_{i})},$ (11) where $m_{i}^{\ast}$ are the effective masses. The two-body energy, $E_{2}$, is $E_{2}=\frac{1}{2A}\sum_{ij}<ij\|\nu(12)\|ij-ji>,$ (12) where $\nu(12)=-\frac{\hbar^{2}}{2m}[f(12),[\nabla^{2}_{12},f(12)]]+f(12)V(12)f(12).$ (13) $f(12)$ and $V(12)$ are the two-body correlation and inter-nucleonic potential. We note that the conditions of charge neutrality and beta stability impose the following constraints on the number densities and chemical potentials, $\displaystyle n_{p}=n_{e}+n_{\mu}$ (14) $\displaystyle\mu_{n}-\mu_{p}=\mu_{e}=\mu_{\mu}.$ (15) The procedure to calculate the nucleonic matter has been fully discussed in the Refs. b2 ; b3 . ### II.2 Quark Phase We use the MIT bag model for the quark matter calculations. In this model, the energy density is the kinetic energy of quarks plus a bag constant (${\cal B}$) which is interpreted as the difference between the energy densities of non interacting quarks and interacting ones farhi1 , ${\cal E}_{tot}={\cal E}_{u}+{\cal E}_{d}+{\cal E}_{s}+{\cal B},$ (16) where ${\cal E}_{i}$ is the kinetic energy per volume of particle $i$, ${\cal E}_{i}=\frac{g}{2\pi^{2}}\int_{0}^{\infty}{(m_{i}^{2}c^{4}+\hbar^{2}k^{2}c^{2})^{1/2}}{f(k,T,n_{i})}{k^{2}dk}.$ (17) In above equation, $g$ is the degeneracy number of the system and $n_{i}$ is the number density of particle $i$, $n_{i}=\frac{g}{2\pi^{2}}\int_{0}^{\infty}{f(k,T,n_{i})}{k^{2}dk}.$ (18) For the quark phase, the Fermi-Dirac distribution function, $f(k,T,n_{i})$, is given by $f(k,T,n_{i})=\frac{1}{Exp\\{\beta((m_{i}^{2}c^{4}+\hbar^{2}k^{2}c^{2})^{1/2}-\mu_{i})\\}+1}.$ (19) We assume that the up and down quarks are massless, the strange quark has a mass equal to $150\ MeV$ and ${\cal B}=90\ MeVfm^{-3}$. Now, by applying the beta stability and charge neutrality conditions, we get the following relations for the chemical potentials and number densities, $\mu_{d}=\mu_{u}+\mu_{l},$ (20) $\mu_{s}=\mu_{u}+\mu_{l},$ (21) $\Rightarrow\mu_{d}=\mu_{s},$ (22) $\frac{2}{3}n_{u}-\frac{1}{3}n_{d}-\frac{1}{3}n_{s}-n_{l}=0,$ (23) $n_{B}=\frac{1}{3}(n_{u}+n_{d}+n_{s}),$ (24) where $n_{l}$ and $\mu_{l}$ are the leptonic number density and chemical potential, and $n_{B}$ is the baryonic number density. The pressure of the system is calculated from free energy using the following equation, $P=\sum_{i}{n_{i}\frac{\partial{\cal F}_{i}}{\partial n_{i}}-{\cal F}_{i}},$ (25) where the Helmholtz free energy per volume (${\cal F}$) is given by ${\cal F}={\cal E}_{tot}-T{\cal S}_{tot}.$ (26) The entropy of quark matter (${\cal S}_{tot}$) can be written as follows, ${\cal S}_{tot}={\cal S}_{u}+{\cal S}_{d}+{\cal S}_{s},$ (27) where ${\cal S}_{i}$ is the entropy of particle $i$, $\displaystyle{\cal S}_{i}(n_{i},T)$ $\displaystyle=$ $\displaystyle-\frac{3}{\pi^{2}}k_{B}\int_{0}^{\infty}[f(k,T,n_{i})\ln(f(k,T,n_{i}))$ (28) $\displaystyle+(1-f(k,T,n_{i}))\ln(1-f(k,T,n_{i}))]k^{2}dk.$ ### II.3 Mixed phase For the mixed phase, where the fraction of space occupied by quark matter smoothly increases from zero to unity, we have a mixture of hadrons, quarks and electrons. In the mixed phase, according the Gibss equilibrium condition, the temperatures, pressures and chemical potentials of the hadron phase (H) and quark phase (Q) are equal glen1 . Here, for each temperature we let the pressure to be an independent variable. The Gibss conditions implies that $\mu^{Q}_{n}=\mu^{H}_{n},$ (29) $\mu^{Q}_{p}=\mu^{H}_{p},$ (30) where $\mu^{H}_{n}$ and $\mu^{Q}_{n}$ ($\mu^{H}_{p}$ and $\mu^{Q}_{p}$) are the neutron (proton) chemical potentials in the hadron phase and the quark phase, respectively, $\mu_{n}=\frac{\partial{\cal E}}{\partial{n_{n}}},$ (31) $\mu_{p}=\frac{\partial{\cal E}}{\partial{n_{p}}}.$ (32) In above equations, ${\cal E}$ is the energy density of the system, ${\cal E}=n(E+mc^{2}).$ (33) To obtain $\mu^{H}_{p}$ and $\mu^{H}_{n}$ for the hadronic matter in mixed phase, we use the semiempirical mass formula kutschera ; langris ; wiringa , $E=T(n,x)+V_{0}(n)+(1-2x)^{2}V_{2}(n),$ (34) where $x=\frac{n_{p}}{n}$ is the proton fraction. $T(n,x)$ is kinetic energy contribution and the functions $V_{0}$ and $V_{2}$ represent the interaction energy contributions which are determined from the energies of the symmetric nuclear matter $(x=\frac{1}{2})$ and pure neutron matter $(x=0)$. We calculate $V_{0}$ and $V_{2}$ using our results for the LOCV calculation of nucleonic matter with the $UV_{14}+TNI$ nuclear potential which is discussed in section II.1. Now, we can obtain the chemical potentials of neutrons and protons from Eqs. (31)-(34) as follows, $\displaystyle\mu^{H}_{p}$ $\displaystyle=$ $\displaystyle T(n,x)+n\frac{\partial{T(n,x)}}{\partial{n}}+\frac{\partial{T(n,x)}}{\partial{x}}+V_{0}(n)+n{V_{0}}^{\prime}(n)$ (35) $\displaystyle+(-3+8x-4x^{2})V_{2}(n)+(1-2x)^{2}n{V_{2}}^{\prime}(n)+mc^{2},$ $\displaystyle\mu^{H}_{n}$ $\displaystyle=$ $\displaystyle T(n,x)+n\frac{\partial{T(n,x)}}{\partial{n}}-\frac{\partial{T(n,x)}}{\partial{x}}+V_{0}(n)+n{V_{0}}^{\prime}(n)$ (36) $\displaystyle+(1-4x^{2})V_{2}(n)+(1-2x)^{2}n{V_{2}}^{\prime}(n)+mc^{2}.$ For the quark matter in mixed phase, we have $\mu_{p}^{Q}=2\mu_{u}+\mu_{d},$ (37) $\mu_{n}^{Q}=\mu_{u}+2\mu_{d}.$ (38) At a certain pressure, we calculate $\mu_{u}$ for different $\mu_{d}$ under the condition that the densities yield this certain pressure. By calculating $\mu_{u}$ and $\mu_{d}$, we obtain $\mu^{Q}_{p}$ and $\mu^{Q}_{n}$. Now, we plot $\mu_{p}$ versus $\mu_{n}$ for both hadron and quark phases, the cross point of the two curves satisfies the Gibss conditions. In the mixed phase, as the chemical potentials determine the densities, the volume fraction occupied by quark matter, $\chi$, can be obtained by the requirement of global charge neutrality, $\chi(\frac{2}{3}n_{u}-\frac{1}{3}n_{d}-\frac{1}{3}n_{s})+(1-\chi)n_{p}-n_{e}=0.$ (39) Finally, we can calculate the baryonic density of the mixed phase (M), $n_{B}=\chi n_{Q}+(1-\chi)n_{H},$ (40) and then the total energy density of mixed phase is found, ${\cal E}_{M}=\chi{\cal E}_{Q}+(1-\chi){\cal E}_{H}.$ (41) ### II.4 Results We have shown our results for the energy densities of hadron phase, quark phase and mixed phase in Figs. 1 and 2 at two different temperatures. Figs. 1 and 2 show that at low densities the energy density of the hadronic matter is lower than those of other phases. However, as the density increases, at first the energy of mixed phase and finally the energy of quark phase is lower than those of other phases. We also see that there is a mixed phase for a range of densities. Below (beyond) this range, we have the pure hadron (quark) phase. By comparing Figs. 1 and 2, we see that for a given value of the density, the energies of all phases increases by increasing the temperature. Using the above calculated energy density, we can determine the equation of state and finally the structure of the hot neutron star with the quark core which is discussed in the next section. ## III Structure of the Hot Neutron Star with the Quark Core The structure of neutron star is determined by numerically integrating the Tolman-Oppenheimer-Volkoff equation (TOV) shapiro ; glen2 ; weber ; alder , $\frac{dP}{dr}=-\frac{G[{\cal E}(r)+\frac{P(r)}{c^{2}}][m(r)+\frac{4\pi r^{3}P(r)}{c^{2}}]}{r^{2}[1-\frac{2Gm(r)}{rc^{2}}]},$ (42) $\frac{dm}{dr}=4\pi r^{2}{\cal E}(r),$ (43) where $P$ is the pressure and ${\cal E}$ is the total energy density. For a given equation of state in the form $P({\cal E})$, the TOV equation yields the mass and radius of star as a function of the central mass density. In our calculations for the structure of hot neutron star with the quark core, we use the following equations of state: (i) Below the density of $0.05\ fm^{3}$, we use the equation of state calculated by Baym baym . (ii) From the density of $0.05\ fm^{3}$ up to the density where the mixed phase starts, we use the equation of state of pure hadron phase calculated in section II.1. (iii) In the range of densities in which there is the mixed phase, we use the equation of state calculated in section II.3. (iv) Beyond the density of end point of the mixed phase, we use the equation of state of pure quark phase calculated in section II.2. All calculations are done for ${\cal B}=90\ MeVfm^{-3}$ at two different temperatures $T=10$ and $20\ MeV$. Our results are as follows. The gravitational mass as a function of the central mass density for the hot neutron star with the quark core at two different temperatures has been presented in Figs. 3 and 4. It is seen that for both relevant temperatures, the gravitational mass increases by increasing the central mass density and finally reaches a limiting value (maximum mass). In Figs. 3 and 4, our results for the case of neutron star without the quark core have been also given for comparison. We see that by including the quark core for the neutron star, our results for the gravitational mass are substantially affected. For the neutron star with the quark core, our results for the gravitational mass at three different temperatures ($T=0$, $10$ and $20\ MeV$) have been compared in Fig. 5. It is seen that the gravitational mass increases by increasing the temperature. Figs. 6 and 7 show the gravitational mass versus the radius for both cases of the neutron star with and without the quark core at two different temperatures. At each temperature, it is seen that there is a reasonable difference between the mass-radius relations of these two cases of the neutron star. However, for both cases, we see that the radius decreases as the mass increases. By comparing Figs. 6 and 7, we can see that the decreasing rate of the radius versus the mass is substantially different for different temperatures. Our results for the maximum gravitational mass of the hot neutron star with the quark core and the corresponding values of radius and central mass density have been given in Tables 1 and 2 for two different temperatures. Our results for the case of hot neutron star without quark core have been also presented for comparison. For different temperatures, it is seen that the inclusion of the quark core considerably reduces the maximum mass of the hot neutron star. This is due to the fact that by including the quark core for the neutron star, the equation of state becomes softer than that without the quark core. However, we do not see any substantial changes for the radius and central mass density of these two cases of the hot neutron star. ## IV Summary and Conclusion For the hot neutron star, from the surface toward the center, we have considered a pure hadronic matter layer, a mixed phase of quarks and hadrons in a range of densities which are determined by employing the Gibbs conditions, and a pure quark matter in the core, to calculate its equation of state at finite temperature. For calculating the equation of state of the hot hadronic matter, we have applied the lowest order constrained variational (LOCV) method at finite temperature. The equation of state of the hot quark matter has been computed using the MIT bag model with the bag constant ${\cal B}=90\ MeVfm^{-3}$. Using this equation of state, we have solved the TOV equation by numerical method to determine the structure properties of the hot neutron star with the quark core at $T=10$ and $20\ MeV$. Then, we have compared the results of these calculations with those for the hot neutron star without the quark core. It is found that our results for the maximum gravitational mass of the neutron star with a quark core are less than those of the neutron star without the quark core. ###### Acknowledgements. This paper is derived from research project entitled: Structure of neutron stars with a quark core at finite temperature. Financial support from Islamic Azad University, Bafgh Branch, Research council is gratefully acknowledged. G. H. Bordbar wishes to thank the Shiraz University Research council. G. H. Bordbar also wishes to thank the Research Institute for Astronomy and Astrophysics of Maragha for its financial support. ## References * (1) A. Burrows and J. M. Lattimer, Astrophys. J. 307(1986) 178. * (2) K. Strobel and M. K. Weigel, Astron. Astrophys. 367 (2001) 582\. * (3) N. K. Glendenning, Phys. Rev. D 46 (1992) 1274. * (4) M. Prakash, J. M. Lattimer, A.W. Steiner and D. Page, Nuch. Phys. A 715 (2003) 835. * (5) G. F. Burgio, M. Baldo, P. K. Sahu, A. B. Santra and H. -J. Schulze, Phys. Lett. B 526 (2002) 19. * (6) G. F. Burgio, M. Baldo, O. E. Nicotra and H. -J. Schulze, Astrophys. Space Sci. 308 (2007) 387. * (7) G. H. Bordbar, M. Bigdeli and T. Yazdizadeh, Int. J. Mod. Phys. A 21 (2006) 5991. * (8) G. H. Bordbar and M. Hayati, Int. J. Mod. Phys. A 21 (2006) 1555. * (9) G. H. Bordbar and M. Modarres, J. Phys. G: Nucl. Phys. 23 (1997) 1631. * (10) G. H. Bordbar and M. Modarres, Phys. Rev. C 57 (1998) 714. * (11) M. Modarres and G. H. Bordbar, Phys. Rev. C 58 (1998) 2781. * (12) G. H. Bordbar and M. Bigdeli, Phys. Rev. C 75 (2007) 045804. * (13) G. H. Bordbar and M. Bigdeli, Phys. Rev. C 76 (2007) 035803. * (14) G. H. Bordbar and M. Bigdeli, Phys. Rev. C 77 (2008) 015805. * (15) G. H. Bordbar and M. Bigdeli, Phys. Rev. C 78 (2008) 054315. * (16) M. Bigdeli, G. H. Bordbar and Z. Rezaei, Phys. Rev. C 80 (2009) 034310. * (17) E. Farhi and R. L. Jaffe, Phys. Rev. D 30 (1984) 2379. * (18) A. L. Fetter and J. D. Walecka, _Quantum Theory of Many-Body System_ , (McGraw-Hill, New York, 1971). * (19) M. Kutschera and J. Niemiec, Phys. Rev. C 62 (2000) 025802\. * (20) I. Lagaris and V. R. Pandharipande, Nuch. Phys. A 369 (1981) 470. * (21) R. B. Wiringa, V. Fiks and A. Fabrocin, Phys. Rev. C 38 (1988) 1010. * (22) S. L. Shapiro and S. A. Teukolsky, _Black Holes, White Dwarfs and Neutron Stars_ (New York, 1983). * (23) N. K. Glendenning, _Compact Stars: Nuclear Physics, Particle Physics, and General Relativity_ (Springer, New York, 2000). * (24) F. Weber, _Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics_ (Institute of Physics, Bristol, 1999). * (25) R. Adler, M. Bazin and M. Schiffer, _Introduction to General Relativity_ (McGraw-Hill, New York, 1965) * (26) G. Baym, C.Pethick and P. Sutherland, Astrophys. J. 170 ( 1971) 299. Table 1: Maximum gravitational mass $(M_{\max})$, and the corresponding radius ($R$) and central mass density $(\varepsilon_{c})$ of the hot neutron star without (NS) and with (NS+Q ) the quark core at $T=10\ MeV$. \| $M_{\max}(M_{\odot})$ \| $R(Km)$ \| $\varepsilon_{c}(10^{14}gr/cm^{3})$ ---\|---\|---\|--- NS \| 2.07 \| 10.22 \| 26.94 NS+Q \| 1.76 \| 10.45 \| 27.38 Table 2: As Table 1 but at $T=20\ MeV$. \| $M_{\max}(M_{\odot})$ \| R(Km) \| $\varepsilon_{c}(10^{14}gr/cm^{3})$ ---\|---\|---\|--- NS \| 2.09 \| 10.64 \| 27.01 NS+Q \| 1.78 \| 11 \| 27.37 Figure 1: Energy density versus the baryonic density at $T=10\ MeV$ for the hadron phase (solid curve), quark phase (dotted curve) and mixed phase (dashed curve). Figure 2: As Fig. 1 but at $T=20\ MeV$. Figure 3: Gravitational mass versus the central mass density for the neutron star with (dotted curve) and neutron star without (solid curve) the quark core at $T=10\ MeV$. Figure 4: As Fig. 3 but at $T=20\ MeV$. Figure 5: Gravitational mass versus the central mass density for the neutron star with the quark core at $T=0$ (dotted-dashed curve), $10$ (solid curve) and $20\ MeV$ (dotted curve). Figure 6: Mass- radius relation for the neutron star with (dotted curve) and without (solid curve) the quark core at $T=10\ MeV$. Figure 7: As Fig. 6 but at $T=20\ MeV$.	arxiv-papers	2010-12-14T05:38:34	2024-09-04T02:49:15.677357	{ "license": "Public Domain", "authors": "T. Yazdizadeh and G. H. Bordbar", "submitter": "Gholam Hossein Bordbar", "url": "https://arxiv.org/abs/1012.2939" }
1012.3160	# A fundamental equation for Supermassive Black Holes ANTONIO FEOLI Department of Engineering, University of Sannio Corso Garibaldi n. 107, Palazzo Bosco Lucarelli 82100 - Benevento, Italy feoli@unisannio.it LUIGI MANCINI Max Planck Institute for Astronomy Königstuhl 17 69117 - Heidelberg, Germany mancini@mpia-hd.mpg.de ###### Abstract We developed a theoretical model able to give a common origin to the correlations between the mass $M_{\bullet}$ of supermassive black holes and the mass, velocity dispersion, kinetic energy and momentum parameter of the corresponding host galaxies. Our model is essentially based on the transformation of the angular momentum of the interstellar material, which falls into the black hole, into the angular momentum of the radiation emitted in this process. In this framework, we predict the existence of a relation of the form $M_{\bullet}\propto R_{\mathrm{e}}\sigma^{3}$, which is confirmed by the experimental data and can be the starting point to understand the other popular scaling laws too. ###### keywords: Black hole physics; galaxies: general. ## 1 Introduction At present, thanks to the improved angular resolution of modern telescopes, the experimental evidence indicates that a variety of nearby galaxies host a supermassive black hole (SMBH; $M_{\bullet}>10^{6}M_{\odot}$) at their center[1][2]. The subsequent discovery of a large number of scaling laws, in which the mass of SMBHs correlates with the properties of the host galaxies (bulges)[3][11], demonstrates a link between the process of accretion of SMBHs and the formation and evolution of their galaxy. Even if many analytical and numerical models have been proposed at the same time in order to explain the observed scaling relationships (see for example Refs. churazov02–lusso10), the physical origin of these correlations, as well as the answer to the question “what is the most fundamental one?”, are still unclear and under debate[18][19]. Here we propose a simple theoretical model able to give a common origin for the correlations between the mass $M_{\bullet}$ of SMBHs and the mass, velocity dispersion, kinetic energy and momentum parameter of the corresponding bulges. Starting from a principle of conservation of the angular momentum and using, as a first approximation, the Bondi–Hoyle–Lyttleton (BHL) theory of accretion[20][22], we found a fundamental equation of the form $M_{\bullet}\propto R_{\mathrm{e}}\sigma^{3}$, where $R_{\mathrm{e}}$ and $\sigma$ are the bulge effective radius and effective stellar velocity dispersion, respectively. From the projections of this fundamental plane, using suitable correlations, we easily derive the other popular scaling laws. Despite the drastic hypotheses and hence the simplicity of the model, our results show an excellent agreement between its predictions and the experimental data, indicating that we have found a basic law for galaxies and their SMBHs. ## 2 The model Marconi and Hunt in 2003 were the first to note that $M_{\bullet}$ is significantly correlated both with $\sigma$, and with $R_{\mathrm{e}}$. The conclusion of their study was that a combination of $\sigma$ and $R_{\mathrm{e}}$ is necessary to drive the correlations between $M_{\bullet}$ and other bulge properties[6]. This topic was then deeply investigated through simulations of major galaxy mergers, which defined a fundamental plane, analogous to the fundamental plane of elliptical galaxies, of the form $M_{\bullet}\propto R_{\mathrm{e}}^{1/2}\sigma^{3}$ or $M_{\bullet}\propto M^{1/2}_{\star}\sigma^{2}$, and $M_{\bullet}\propto(M_{\star}\sigma^{2})^{0.7}$, where $M_{\star}$ is the bulge stellar mass[23][24]. These scaling laws are very similar to what was really found observationally[25][28]. Following this path we build a theoretical model able to explain all the famous relations linking the mass of SMBHs with the properties of their bulges. Our model is mainly based on two hypotheses: the conservation of angular momentum and a suitable velocity field of the gas. Then we need to make an approximation to estimate the accretion radius of the black hole, which can be found either in a rough way or recurring to the BHL theory of accretion. Let us consider a black hole of mass $M_{\bullet}$ emitting radiation at rate $L_{\varepsilon}$, and accreting from a distribution $\rho$ of gas, whose inward velocity is $V_{\mathrm{in}}$. First, we assume that the angular momentum is conserved in such a way that a part ($M_{\mathrm{acc}}$) of the total mass of the gas contained in the galactic bulge will be captured by the black hole, converting its angular momentum into the angular momentum of the perpendicularly emitted radiation (with an effective mass $M_{\mathrm{rad}}$). Of course, there are other mechanisms of conversion, transport or dissipation of angular momentum (viscosity, change in the spin of central black hole, etc.) that, in the past, have contributed with a different weight and importance to the accretion of black holes and probably are still acting now, but our drastic hypothesis focuses the attention only on the above described process. More complicated and detailed models should involve effects due to the other mechanisms that in our approximation are neglected. An order of magnitude estimate of the angular momentum for the galactic bulge[29][30] leads to the conservation equation: $M_{\mathrm{acc}}\,R_{\mathrm{e}}\,V_{\mathrm{rot}}=c\,M_{\mathrm{rad}}\,R_{\mathrm{A}},$ (1) where $R_{\mathrm{e}}$ is the effective radius of the bulge, $V_{\mathrm{rot}}$ is the mass-weighted mean rotational velocity of the gas, $c$ is the speed of light and $R_{\mathrm{A}}$ the accretion radius of the Black hole. Then, we suppose that the velocity of the gas in the galactic bulge is related to the effective stellar velocity dispersion $\sigma$ in such a way that $V_{\mathrm{in}}=\sigma$ (see Refs. soker10 and soker09) and $V_{\mathrm{rot}}=A\,\sigma$, where $A$ is a constant[32]. In the ideal case of the isothermal sphere, $A$ is equal to $\sqrt{2}$. Now we must estimate the accretion radius and as a first approximation we can use the Bondi–Hoyle–Lyttleton theory[20, 21]. In their model the rate at which the gas is accreted onto the black hole is $\dot{M}_{\mathrm{acc}}=\pi R_{\mathrm{B}}^{2}V_{\mathrm{in}}\rho,$ (2) where $R_{\mathrm{B}}=2GM_{\bullet}/V_{\mathrm{in}}^{2}$ (3) is the BHL radius and $G$ is the gravitational constant. Of course the BHL theory is a realistic model in the case of radial accretion with low transverse velocity but it can be used as a good approximation even in presence of angular momentum (see the Appendix A). Anyway, also without considering the BHL theory, the expression of $R_{\mathrm{B}}$ in Eq. (3), with $V_{\mathrm{in}}\simeq\sigma$, roughly corresponds to the well known gravitational radius of influence of a black hole, which we can adopt as an estimate of $R_{\mathrm{A}}$. Hence, we consider $R_{\mathrm{A}}\simeq R_{\mathrm{B}}$ (see the Appendix A), and substitute Eq. (3) in Eq. (1). Finally, recalling our hypothesis on the velocity field, we obtain the fundamental equation for supermassive black holes we were seeking: $M_{\bullet}=\frac{A\,R_{\mathrm{e}}\sigma^{3}}{2\,\varepsilon\,G\,c}\simeq 4.4\times 10^{7}\left(\frac{A}{\sqrt{2}}\right)\left(\frac{0.1}{\varepsilon}\right)\left(\frac{R_{\mathrm{e}}}{\mathrm{kpc}}\right)\left(\frac{\sigma}{200\,\mathrm{km/s}}\right)^{3},$ (4) where $\varepsilon$ is the mass to energy conversion efficiency, which is set by the amount of rest mass energy of matter accreted onto the black hole that is extracted and radiated outward ($\varepsilon=M_{\mathrm{rad}}/M_{\mathrm{acc}}$). The accreting black hole liberates energy at a rate $L_{\varepsilon}=\varepsilon\dot{M}_{\mathrm{acc}}c^{2}=\ell L_{\mathrm{Edd}}.$ (5) It is commonly assumed that the accretion of matter onto a black hole releases energy at $10\%$ efficiency so that a fixed value of $\varepsilon=0.1$ is usually adopted[33], even if the range $0.001\leq\varepsilon\leq 0.1$ has been recently investigated[17]. This radiated luminosity is related to the Eddington limit, $L_{\mathrm{Edd}}$. In particular, for $\ell=1$ the central black hole is radiating at its Eddington limit: $L_{\mathrm{Edd}}=\frac{4\pi GMc\,m_{\mathrm{p}}}{\sigma_{\mathrm{T}}}=\frac{4\pi GMc}{\kappa_{\mathrm{Edd}}}.$ (6) Here $m_{\mathrm{p}}$ is the proton mass, $\sigma_{\mathrm{T}}$ the Thomson scattering cross–section of the electron, and $\kappa_{\mathrm{Edd}}=\sigma_{\mathrm{T}}/m_{\mathrm{p}}$ the opacity of the fully ionized hydrogen. Substituting Eq. (2) and Eq. (6) in Eq. (5), and recalling our two hypotheses, we can determine also the gas density at the effective radius in a form: $\rho=\frac{2\,\ell}{\kappa_{\mathrm{Edd}}\,A\,R_{\mathrm{e}}},$ (7) similar to that found in Ref. begelman05. While in other models the gas density is given among the hypotheses, in our approach it is a consequence of the theory. We have tested the effectiveness of our model on a sample[34] of 58 nearby galaxies ($z\sim 0$). By using these experimental data and the corresponding errors, we report in Table 1 the best–fitting values for the slope $m$ and the normalization $b$ for the linear relations used in this work. These values have been calculated by the routine FITEXY[35] for the relation $y=b+mx$, by minimizing the $\chi^{2}$. The estimates of the $\chi_{\mathrm{red}}^{2}=\chi^{2}/(58-2)$ and the Pearson linear correlation $r$ for each relation are also shown. Errors have been calculated by using the formula reported in Appendix A of Ref. feoli09. The results of the fits for the $M_{\bullet}-M_{\mathrm{G}}$ and $M_{\bullet}-M_{\mathrm{G}}\sigma^{2}$ relations are in agreement with the ones obtained in a previous paper using three different samples of data[28] and with the results obtained by other authors while the $M_{\bullet}-\sigma$ relation requires a more careful discussion (see Appendix B). Black hole–bulge correlations and fitting parameters for the considered galaxy sample. Relation $b\pm\Delta b$ $m\pm\Delta m$ $\chi_{\mathrm{red}}^{2}$ $\epsilon_{0}$ $r$ $M_{\bullet}-R_{\mathrm{e}}\sigma_{200}^{3}$ $8.11\pm 0.03$ $1.00\pm 0.02$ $5.00$ $0.39$ $0.87$ $M_{\bullet}-M_{\mathrm{G}}\sigma^{2}/c^{2}$ $4.56\pm 0.10$ $0.87\pm 0.02$ $6.03$ $0.40$ $0.87$ $M_{\bullet}-\sigma_{200}$ $8.21\pm 0.02$ $5.83\pm 0.15$ $6.22$ $0.40$ $0.87$ $M_{\bullet}-M_{\mathrm{G}}\sigma/c$ $0.73\pm 0.19$ $1.01\pm 0.03$ $6.61$ $0.42$ $0.87$ $M_{\bullet}-M_{\mathrm{G}}$ $8.74\pm 0.03$ $1.21\pm 0.03$ $7.71$ $0.45$ $0.85$ $R_{\mathrm{e}}-\sigma_{200}$ $0.12\pm 0.01$ $2.72\pm 0.09$ $10.31$ $0.32$ $0.75$ $R_{\mathrm{e}}-M_{\mathrm{G}}\sigma^{2}/c^{2}$ $-1.77\pm 0.06$ $0.45\pm 0.01$ $3.26$ $0.16$ $0.92$ The best–fitting line for the new relationship $M_{\bullet}-R_{\mathrm{e}}\sigma_{200}^{3}$ in a log–log plane is: $\log_{10}M_{\bullet}=(8.11\pm 0.03)+(1.00\pm 0.02)\log_{10}(R_{\mathrm{e}}\sigma_{200}^{3}),$ (8) where $\sigma_{200}$ is the velocity dispersion in $200$ km sec-1 units, while $R_{\mathrm{e}}$ is expressed in kpc. The linear relation (8) has a slope equal to the unity, which is exactly the value predicted by our model in the Eq. (4). We remark also that this relation has the best $\chi^{2}$ and $r$ among the relations in the upper part of Table 1 involving the black hole mass . We also report the intrinsic scatter $\epsilon_{0}$, finding that the relation $M_{\bullet}-R_{\mathrm{e}}\sigma^{3}$ has $\epsilon_{0}=0.39$, whereas the other ones have values of $\epsilon_{0}\geq 0.40$. Furthermore, from the normalization we can calculate the effective efficiency coefficient, which turns out to be equal to $\hat{\varepsilon}=2\varepsilon/A=0.048\pm 0.003$, a value close to $0.05$ estimated in Ref. churazov02, and to $0.06$ used in the case of Schwarzschild’s metric[33]. Of course if the coefficient $\hat{\varepsilon}$ depends on $R_{e}$ or $\sigma$ the relation (8) will still exist but its slope will be different from the unity. In Figure 1, we report the $M_{\bullet}-R_{\mathrm{e}}\sigma_{200}^{3}$ diagram in a log -log plot (we associated a particular symbol to each galaxy according to its morphology[34]). The best-fitting line is also shown. We also tested Eq. (4) by using an old sample of 37 objects[6], obtaining a slope equal to $0.90\pm 0.04$, which is in good agreement with the slope of Eq. (8). Using a subset of 27 galaxies of the same old sample, Hopkins et al.[25] obtained a fundamental plane $M_{\bullet}\propto R_{\mathrm{e}}^{0.43}\sigma^{3.00}$ and Graham[26] $M_{\bullet}\propto R_{\mathrm{e}}^{0.28}\sigma^{3.65}$. The difference with our result is due to a different fitting method and to the fact that these authors have considered three free parameters. With a different sample also Aller and Richstone[10] studied the same three parameters fit obtaining $M_{\bullet}\propto R_{\mathrm{e}}^{0.28}\sigma^{3.16}$. The fact that we predict a relation of the kind $M_{\bullet}\propto R_{\mathrm{e}}\sigma^{3}$ and not the Hopkins result[23] $M_{\bullet}\propto R_{\mathrm{e}}^{0.5}\sigma^{3}$ does not depend on an oversimplification of our model. To obtain the latter relation one must start from different hypotheses and only increasing the number of experimental data it will be possible to distinguish the better approach. file=f1.eps,width=12.cm Figure 1: The $M_{\bullet}-R_{\mathrm{e}}\sigma_{200}^{3}$ relation for the galaxies of the considered data set, where $\sigma_{200}$ is the bulge velocity dispersion in units of 200 km s-1. The symbols represent elliptical galaxies (red ellipses), lenticular galaxies (green circles), barred lenticular galaxies (dark green circles), spiral galaxy (blue spirals), barred spiral galaxies (dark blue barred spirals), and dwarf elliptical galaxies (orange ellipses). The black line is the line of best fit for the sample of galaxies considered. ## 3 The origin of the other scaling relations The dynamical masses of bulges, $M_{\mathrm{G}}$, can be estimated by $M_{\mathrm{G}}=\frac{kR_{\mathrm{e}}\sigma^{2}}{G},$ (9) where $k$ is a model dependent dimensionless constant. We stress that the isothermal model (i.e. $\sigma$ is a constant throughout the galaxy) is not a hypothesis of our framework. We make use of this approximation in order to test our model. As in Refs. marconi03 and hu09, we use $k=3$ to compute the “isothermal masses” of the galaxy sample considered[34] (in a more detailed model $k$ is dependent on the Sersic index). Thanks to the fundamental Eq. (4), coupled with Eq. (9), in the following we derive all the most famous scaling relationships between the black hole mass and the parameters of the host galaxy. ### 3.1 The $M_{\bullet}-M_{\mathrm{G}}\,\sigma/c$ relation Replacing Eq. (9) in Eq. (4), we get: $M_{\bullet}=\frac{1}{\hat{\varepsilon}k}\left(\frac{M_{\mathrm{G}}\sigma}{c}\right),$ (10) in optimum agreement with the corresponding relation in Table 1, where the value of the slope $1.01$ is close to unity as we expected (the quantity $M_{\mathrm{G}}\sigma/c$ is also known as the momentum parameter[11]). If we impose the exponent of the momentum to be equal to 1.00, then by refitting the data we obtain a normalization $b=0.82\pm 0.02$, from which we derive $\hat{\varepsilon}=0.05$. ### 3.2 The $M_{\bullet}-\sigma$ relation Substituting the experimental relation between $R_{\mathrm{e}}$ and $\sigma$ (see Table 1) in Eq. (4), we obtain: $M_{\bullet}=\frac{10^{0.12}}{\hat{\varepsilon}c\,G}\left(\sigma_{200}\right)^{5.72}=10^{8.23}\left(\sigma_{200}\right)^{5.72},$ (11) which is in agreement, inside the errors, with the corresponding law in Table 1: $M_{\bullet}=10^{8.21\pm 0.02}(\sigma_{200})^{5.83\pm 0.15}.$ (12) ### 3.3 The $M_{\bullet}-M_{\mathrm{G}}\sigma^{2}/c^{2}$ relation Using Eq. (9), Eq. (4) can be written in terms of the kinetic energy of random motions: $M_{\bullet}=\frac{1}{\hat{\varepsilon}}\left(\frac{c^{2}R_{\mathrm{e}}}{Gk^{3}}\right)^{1/4}\left(\frac{M_{\mathrm{G}}\sigma^{2}}{c^{2}}\right)^{3/4}.$ (13) Replacing $R_{\mathrm{e}}$ in Eq. (13) with the value $R_{\mathrm{e}}=10^{-1.77}(M_{\mathrm{G}}\sigma^{2}/c^{2})^{0.45}$, taken from the experimental relation in Table 1, we obtain: $M_{\bullet}=10^{4.60}(M_{\mathrm{G}}\sigma^{2}/c^{2})^{0.86},$ (14) where we have used the value of $0.048$ for $\hat{\varepsilon}$. We point out the good match between the relation in Eq. (14) and the corresponding one reported in Table 1: $M_{\bullet}=10^{4.56\pm 0.10}(M_{\mathrm{G}}\sigma^{2}/c^{2})^{0.87\pm 0.02}.$ (15) As shown in previous papers[27, 28], this relation has the best $\chi^{2}$ compared to $M_{\bullet}-\sigma$ and $M_{\bullet}-M_{\mathrm{G}}$ laws. ### 3.4 The $M_{\bullet}-M_{\mathrm{G}}$ relation Starting from the Eq. (10) and expressing $\sigma$ in terms of $M_{\bullet}$ (see Table 1), we obtain: $M_{\bullet}=10^{-4.52}M_{\mathrm{G}}^{1.21},$ (16) in excellent agreement with the relation: $M_{\bullet}=10^{-4.57\pm 0.34}M_{\mathrm{G}}^{1.21\pm 0.03}.$ (17) ## 4 Summary To sum up, we have shown that the model proposed in §2 works very well since, given a consistent set of data, it perfectly predicts the slope of a new relation (Eq. 8). Moreover, the most investigated scaling relationships can be easily obtained as projections of the plane identified by the fundamental Eq. (4). Unfortunately, we cannot infer this correlation at redshift $z>>0$, because we are limited by the small number of observable hosts. Other mechanisms of accretion, different from the one presented in this work, may have acted in the past, and hence other fundamental equations may have ruled the first Gigayears of the life of SMBHs. Future detection of new SMBHs, especially at higher redshift, and measurements of their masses will enable us to confirm the universality of our law or if it holds just for a restricted period of the cosmic time. At this stage it is early to predict the consequences of our equation in the context of the models about the co–evolution of galaxies and black holes. First of all we must check the validity of the approach using an enlarged sample of data. Recently Sani et al. collected a new interesting (but not larger) galaxy sample[36] and we intend to present a complete analysis of their data in a forthcoming paper. A preliminary result is that a tight relation $M_{\bullet}-R_{\mathrm{\mathrm{e}}}\sigma^{3}$ exists, but its slope oscillates from 0.78 to 1.2 depending on the number of pseudobulges111Essentially, a pseudobulge is a bulge that shows photometric and kinematic evidence for disk–like dynamics considered in the fit. This aspect requires a deeper investigation before drawing any conclusions about the evolution of supermassive black holes and galaxies. ## Acknowledgments We are grateful to Gaetano Scarpetta, and Sidney van den Bergh for their very useful suggestions and to the anonymous referee whose comments have contributed to improve our paper. L.M. acknowledges support for this work by research funds of the University of Sannio, University of Salerno, and the International Institute for Advanced Scientific Studies. A.F. acknowledges support for this work by research funds of the University of Sannio. ## Appendix A We underline that our model is not based on the BHL theory assumption. We can simply estimate the value of the accretion radius recurring to the concept of radius of influence of a black hole. Actually in our paper the BHL theory is equivalently used as another approach to have an estimate of the accretion radius even if it is well known that in presence of angular momentum the BHL approach is only a fair approximation. In their original paper, Hoyle and Lyttleton suppose that an element of volume of the gas cloud has an initial angular momentum, but it “loses this momentum through its constituent particles suffering collisions” at a radius $R_{\mathrm{B}}$[20]. That is, the transverse velocities of the particles, which reach the accretion line from opposite directions, annihilate reciprocally, whereas the radial component, if it is not bigger than the escape velocity (and this occurs at a distance $d\leq R_{\mathrm{B}}$ from the hole), makes possible that the particles were captured by the black hole. Then they show that the collisions occur with sufficient frequency to be effective in reducing the angular momentum. Anyway, it is possible to take into account the angular momentum, but the final result does not drastically change. In fact, in a generalization of BHL approach, the author in Ref. horedt00 considers the orbital velocity as the source of a lateral pressure on the accretion column. Having included this pressure term in the dynamical equations, he obtained an accretion radius $R_{\mathrm{A}}$ of the same order of magnitude of the BHL radius: $0.3\,R_{\mathrm{B}}<R_{\mathrm{A}}<1.75\,R_{\mathrm{B}}$. Both BHL and Horedt analyzed a case of pure accretion of the central object without considering the energy eventually radiated outward. Our framework is also different compared with the spherical Bondi accretion[37], where a sound speed of the gas is considered, and a numerical correction factor $\alpha_{\mathrm{B}}\approx 100$ (see Ref. springel05), which depends on the mass profile and gas equation of state, is inserted in Eq. (2). Some authors, by analyzing the role of angular momentum, refer to that special case of spherical Bondi accretion, finding that it fails systematically to reproduce their numerical simulations[38]. On the other hand, other authors, by using Chandra X-ray observations of several nearby elliptical galaxies, observed a tight correlation between the Bondi accretion rates (calculated from the observed gas temperature and density profiles as well as from the estimated black hole masses) and the power emerging from these systems in relativistic jets[39]. They concluded that the Bondi formulae provide a reasonable description of the accretion process in these systems, despite the likely presence of angular momentum in the accreting gas. We can study the effect of making the BHL theory a real assumption of our model. Let us analyze a different case (based on different hypotheses) from the one considered in section 2, assuming that the angular momentum is very small so the BHL theory is not only a fair approximation, but it works very well. Actually we can suppose that the photon is emitted by the black hole with a velocity $\vec{c}$ in the same direction of the arriving gas particle on the accretion line. It means that $\vec{c}$ forms an angle $\alpha$ with this line and with the radial component of gas velocity $V_{\mathrm{in}}$ and hence $A=V_{\mathrm{rot}}/V_{\mathrm{in}}=\tan(\alpha)$. In this case the conservation of angular momentum can be written $M_{\mathrm{acc}}\,R_{\mathrm{e}}\,V_{\mathrm{rot}}=M_{\mathrm{acc}}\,R_{\mathrm{e}}\,V_{\mathrm{in}}\tan(\alpha)=c\,M_{\mathrm{rad}}\,R_{\mathrm{A}}\sin(\alpha).$ (18) If we suppose that the accretion occurs mainly for particles with low angular momentum ($A<<1$), we can simplify the above equation because for $\alpha\simeq 0$ we have $\tan(\alpha)\simeq\sin(\alpha)$. In this particular case, our fundamental Eq. (4) becomes $M_{\bullet}=\frac{R_{\mathrm{e}}\sigma^{3}}{2\,\varepsilon\,G\,c}$ (19) and the effect to consider the BHL theory as a real hypothesis of the model reduces to the disappearing of the factor $A$ from the fundamental Eq. (4), and all the calculations of the first part of the paper are still valid considering $\hat{\varepsilon}=2\varepsilon$. We reported again the data of the Hu sample[34] in Figure 2 together with the lines representing the values of the efficiency coefficient $\varepsilon$ predicted by Eq. (19). So, given the SMBH mass and the effective radius and dispersion velocity of the host galaxy, the equation (19) allows a quick estimate of the efficiency of a black hole. In this way the resulting diagram (Fig. 2) can be used to classify the black holes in terms of their efficiency provided that the host galaxies satisfy the hypotheses of the model. For example, in our sample all the galaxies have a predicted $\varepsilon<0.25$, but the prediction could be not valid, as expected, for some pseudobulges for which the approximation of the model $A<<1$ does not work. file=f2.eps,width=12.cm Figure 2: The data of the considered sample in the $M_{\bullet}-R_{\mathrm{e}}\sigma_{200}^{3}$ plane. The symbols are the same of Figure 1. Lines of constant values of the efficiency coefficient $\varepsilon$ are shown, see Eq. (19). All the galaxies have $\varepsilon<0.25$. ## Appendix B The slopes of the $M_{\bullet}-M_{\mathrm{G}}$ and $M_{\bullet}-M_{\mathrm{G}}\sigma^{2}$ relations remain very stable changing the sample of data or the fitting method. In Ref 28 we have used one fitting method and three different samples and the resulting slopes were (1.18, 1.22, 1.27) for the $M_{\bullet}-M_{\mathrm{G}}$ relation and (0.83, 0.86, 0.91) for the $M_{\bullet}-M_{\mathrm{G}}\sigma^{2}$ relation. Viceversa, in Ref. 27 we have used only one sample and three different fitting procedures and the slopes are still stable with values (1.15, 0.98, 1.07) and (0.80, 0.74, 0.78) respectively. These values are in agreement with the results of other authors. For instance, Hopkins et al.[25] find 1.05 for $M_{\bullet}-M_{\mathrm{G}}$ and 0.71 for $M_{\bullet}-M_{\mathrm{G}}\sigma^{2}$, whereas Soker et al.[11] find 1.07 and 0.74 respectively. The $M_{\bullet}-\sigma$ relation has a different behavior since its slope strongly depends on the fitting method (5.06, 4.46, 4.25) in Ref. feoli09, and on different samples (5.26, 4.99, 5.83) in Ref. feoli10. In this case Hopkins et al.[25] find 3.96, Soker[11] 4.18, Ferrarese and Ford[40] 4.86 and 5.1 and Hu[41] a slope that changes from 4.01 to 5.62 depending on the considered subsample of data. The oscillation of the slope between 3.9 and 5.9 could be due in part to the fitting method (FITEXY finds the first minimum of $\chi^{2}$ and gives a slope; if one introduces the intrinsic scatter and refits until the reduced $\chi^{2}=1$, a smaller slope is generally obtained[42]) and in part to the presence of pseudobulges in the sample. In their last paper Sani et. al.[36] find the same value (4.00) for the slope using three different methods, but selecting only the classical bulges. 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Bender et al., ApJ 574 (2002) 740	arxiv-papers	2010-12-14T20:57:24	2024-09-04T02:49:15.686672	{ "license": "Public Domain", "authors": "Antonio Feoli and Luigi Mancini", "submitter": "Luigi Mancini", "url": "https://arxiv.org/abs/1012.3160" }
1012.3221	# Existence of Doubly-Weighted Pseudo Almost Periodic Solutions to Some Classes of Nonautonomous Differential Equations Toka Diagana Department of Mathematics, Howard University, 2441 6th Street N.W., Washington, D.C. 20059, USA tdiagana@howard.edu ###### Abstract. The main objective of this paper is twofold. We first show that if the doubly- weighted Bohr spectrum of an almost periodic function exists, then it is either empty or coincides with the Bohr spectrum of that function. Next, we investigate the problem which consists of the existence of doubly-weighted pseudo-almost periodic solutions to some nonautonomous abstract differential equations. ###### Key words and phrases: weight, weighted pseudo-almost periodic, doubly-weighted Bohr spectrum, almost periodic, doubly-weighted pseudo-almost periodic. ###### 2000 Mathematics Subject Classification: primary 35B15; secondary 34D09; 58D25; 42A75; 37L05 ## 1\. Introduction Motivated by the functional structure of the so-called weighted Morrey spaces [16], in Diagana [10], a new concept called doubly-weighted pseudo-almost periodicity, which generalizes in a natural fashion the notion of weighted pseudo-almost periodicity is introduced and studied. Among other things, in [10], properties of these new functions have been studied including the stability of the convolution operator, the translation-invariance, the existence of a doubly-weighted mean for almost periodic functions under some reasonable assumptions, the uniqueness of the decomposition involving these new functions as well as some results on the composition of these new functions have been studied. The main objective of this paper is twofold. We first show if the doubly- weighted Bohr spectrum of an almost periodic function exists, then it is either empty or coincides with the Bohr spectrum of that function. Next, we investigate the problem which consists of the existence of doubly-weighted pseudo-almost periodic solutions to the nonautonomous abstract differential equations (1.1) $\displaystyle u^{\prime}(t)=A(t)u(t)+g(t,u(t)),\;\;t\in\mathbb{R},$ where $A(t)$ for $t\in\mathbb{R}$ is a family of closed linear operators on $D(A(t))$ satisfying the well-known Acquistapace and Terreni conditions, and $g:\mathbb{R}\times\mathbb{X}\mapsto\mathbb{X}$ is doubly weighted pseudo- almost periodic in $t\in\mathbb{R}$ uniformly in the second variable. It is well-known that in that case, there exists an evolution family ${\mathcal{U}}=\\{U(t,s)\\}_{t\geq s}$ associated with the family of linear operators $A(t)$. Assuming that the evolution family ${\mathcal{U}}=\\{U(t,s)\\}_{t\geq s}$ is exponentially dichotomic and under some additional assumptions it will be shown that Eq. (1.1) has a unique doubly-weighted pseudo-almost periodic solution. The existence of weighted pseudo-almost periodic, weighted pseudo-almost automorphic, and pseudo-almost periodic solutions to differential equations constitutes one of the most attractive topics in qualitative theory of differential equations due to possible applications. Some contributions on weighted pseudo-almost periodic functions, their extensions, and their applications on differential equations have recently been made, among them are for instance [1], [5], [7], [8], [12], [13], [14], [15], [18], [20], [21], [28], and [29] and the references therein. However, the problem which consists of the existence of doubly-weighted pseudo-almost periodic(mild) solutions to evolution equations in the form Eq. (1.1) is quite new and untreated and thus constitutes one of the main motivations of the present paper. The paper is organized as follows: Section 2 is devoted to preliminaries results related to the existence of an evolution family, intermediate spaces, properties of weights, and basic definitions and results on the concept of doubly-weighted pseudo-almost periodic functions. Section 3 is devoted to the existence of a doubly-weighted Bohr spectral theory for almost periodic functions while Section 4 is devoted to the existence of doubly-weighted pseudo-almost periodic solutions to Eq. (1.1). ## 2\. Preliminaries Let $(\mathbb{X},\\|\cdot\\|)$ be a Banach space. If $C$ is a linear operator on $\mathbb{X}$, then $D(C)$, $\rho(C)$, and $\sigma(C)$ stand respectively for the domain, resolvent, and spectrum of $C$. Similarly, one sets $R(\lambda,C):=(\lambda I-C)^{-1}$ for all $\lambda\in\rho(C)$ where $I$ is the identity operator for $\mathbb{X}$. Furthermore, we set $Q=I-P$ for a projection $P$. We denote the Banach algebra of bounded linear operators on $\mathbb{X}$ equipped with its natural norm by $B(\mathbb{X})$. If $\mathbb{Y}$ is another Banach space, we then let $BC(\mathbb{R},\mathbb{X})$ (respectively, $BC(\mathbb{R}\times\mathbb{Y},\mathbb{X})$) denote the collection of all $\mathbb{X}$-valued bounded continuous functions and equip it with the sup norm (respectively, the space of jointly bounded continuous functions $F:\mathbb{R}\times\mathbb{Y}\mapsto\mathbb{X}$). The space $BC(\mathbb{R},\mathbb{X})$ equipped with the sup norm is a Banach space. Furthermore, $C(\mathbb{R},\mathbb{Y})$ (respectively, $C(\mathbb{R}\times\mathbb{Y},\mathbb{X})$) denotes the class of continuous functions from $\mathbb{R}$ into $\mathbb{Y}$ (respectively, the class of jointly continuous functions $F:\mathbb{R}\times\mathbb{Y}\mapsto\mathbb{X}$). ### 2.1. Evolution Families The setting of this Subsection follows that of Baroun et al. [3] and Diagana [14]. Fix once and for all a Banach space $(\mathbb{X},\\|\cdot\\|)$. ###### Definition 2.1. A family of closed linear operators $A(t)$ for $t\in\mathbb{R}$ on $\mathbb{X}$ with domain $D(A(t))$ (possibly not densely defined) satisfy the so-called Acquistapace and Terreni conditions, if there exist constants $\omega\in\mathbb{R}$, $\theta\in(\frac{\pi}{2},\pi)$, $L>0$ and $\mu,\nu\in(0,1]$ with $\mu+\nu>1$ such that (2.1) $\Sigma_{\theta}\cup\\{0\\}\subset\rho(A(t)-\omega)\ni\lambda,\;\qquad\\|R(\lambda,A(t)-\omega)\\|\leq\frac{K}{1+\|\lambda\|}\ \ \ \mbox{for all}\ t\in\mathbb{R},$ and (2.2) $\\|(A(t)-\omega)R(\lambda,A(t)-\omega)\,[R(\omega,A(t))-R(\omega,A(s))]\\|\leq L\,\frac{\|t-s\|^{\mu}}{\|\lambda\|^{\nu}}$ for $t,s\in\mathbb{R}$, $\displaystyle\lambda\in\Sigma_{\theta}:=\\{\lambda\in\mathbb{C}\setminus\\{0\\}:\|\arg\lambda\|\leq\theta\\}$. For a given family of linear operators $A(t)$, the existence of an evolution family associated with it is not always guaranteed. However, if $A(t)$ satisfies Acquistapace-Terreni, then there exists a unique evolution family ${\mathcal{U}}=\\{U(t,s):t,s\in\mathbb{R}\ \ \mbox{such that}\ \ t\geq s\\}$ on $\mathbb{X}$ associated with $A(t)$ such that $U(t,s)\mathbb{X}\subseteq D(A(t))$ for all $t,s\in\mathbb{R}$ with $t\geq s$, and 1. (a) $U(t,s)U(s,r)=U(t,r)$ for $t,s\in\mathbb{R}$ such that $t\geq s\geq s$; 2. (b) $U(t,t)=I$ for $t\in\mathbb{R}$ where $I$ is the identity operator of $\mathbb{X}$; 3. (c) $(t,s)\mapsto U(t,s)\in B(\mathbb{X})$ is continuous for $t>s$; 4. (d) $U(\cdot,s)\in C^{1}((s,\infty),B(\mathbb{X}))$, $\displaystyle\frac{\partial U}{\partial t}(t,s)=A(t)U(t,s)$ and $\displaystyle\left\\|A(t)^{k}U(t,s)\right\\|$ $\displaystyle\leq K\,(t-s)^{-k}$ for $0<t-s\leq 1$ and $k=0,1$. ###### Definition 2.2. An evolution family ${\mathcal{U}}=\\{U(t,s):t,s\in\mathbb{R}\ \ \mbox{such that}\ \ t\geq s\\}$ is said to have an exponential dichotomy (or is hyperbolic) if there are projections $P(t)$ ($t\in\mathbb{R}$) that are uniformly bounded and strongly continuous in $t$ and constants $\delta>0$ and $N\geq 1$ such that 1. (e) $U(t,s)P(s)=P(t)U(t,s)$; 2. (f) the restriction $U_{Q}(t,s):Q(s)\mathbb{X}\to Q(t)\mathbb{X}$ of $U(t,s)$ is invertible (we then set $\widetilde{U}_{Q}(s,t):=U_{Q}(t,s)^{-1}$); and 3. (g) $\left\\|U(t,s)P(s)\right\\|\leq Ne^{-\delta(t-s)}$ and $\left\\|\widetilde{U}_{Q}(s,t)Q(t)\right\\|\leq Ne^{-\delta(t-s)}$ for $t\geq s$ and $t,s\in\mathbb{R}$. This setting requires some estimates related to ${\mathcal{U}}=\\{U(t,s)\\}_{t\geq s}$. For that, we introduce the interpolation spaces for $A(t)$. Let $A$ be a sectorial operator on $\mathbb{X}$ (in Definition 2.1, replace $A(t)$ with $A$) and let $\alpha\in(0,1)$. Define the real interpolation space $\displaystyle\mathbb{X}^{A}_{\alpha}:=\Big{\\{}x\in\mathbb{X}:\\|x\\|^{A}_{\alpha}:=\sup\nolimits_{r>0}\\|r^{\alpha}(A-\omega)R(r,A-\omega)x\\|<\infty\Big{\\}},$ which, by the way, is a Banach space when endowed with the norm $\\|\cdot\\|^{A}_{\alpha}$. For convenience we further write $\mathbb{X}_{0}^{A}:=\mathbb{X},\ \\|x\\|_{0}^{A}:=\\|x\\|,\ \mathbb{X}_{1}^{A}:=D(A)$ and $\\|x\\|^{A}_{1}:=\\|(\omega-A)x\\|$. Moreover, let $\hat{\mathbb{X}}^{A}:=\overline{D(A)}$ of $\mathbb{X}$. ###### Definition 2.3. Given a family of linear operators $A(t)$ for $t\in\mathbb{R}$ satisfying the Acquistapace-Terreni conditions, we set $\mathbb{X}^{t}_{\alpha}:=\mathbb{X}_{\alpha}^{A(t)}$ and $\hat{\mathbb{X}}^{t}:=\hat{\mathbb{X}}^{A(t)}$ for $0\leq\alpha\leq 1$ and $t\in\mathbb{R}$, with the corresponding norms. ###### Proposition 2.4. [3] For $x\in\mathbb{X}$, $0\leq\alpha\leq 1$ and $t>s,$ the following hold: 1. (i) There is a constant $c(\alpha),$ such that (2.3) $\\|U(t,s)P(s)x\\|_{\alpha}^{t}\leq c(\alpha)e^{-\frac{\delta}{2}(t-s)}(t-s)^{-\alpha}\\|x\\|.$ 2. (ii) There is a constant $m(\alpha),$ such that (2.4) $\\|\widetilde{U}_{Q}(s,t)Q(t)x\\|_{\alpha}^{s}\leq m(\alpha)e^{-\delta(t-s)}\\|x\\|,\qquad t\leq s.$ ### 2.2. Properties of Weights This subsection is similar to the one given in Diagana [10] except that most of all the proofs will be omitted. Let $\mathbb{U}$ denote the collection of functions (weights) $\rho:\mathbb{R}\mapsto(0,\infty)$, which are locally integrable over $\mathbb{R}$ such that $\rho>0$ almost everywhere. In the rest of the paper, if $\mu\in\mathbb{U}$, $T>0$, and $a\in\mathbb{R}$, we then set $Q_{T}:=[-T,T]$, $Q_{T}+a:=[-T+a,T+a]$, and $\mu(Q_{T}):=\int_{Q_{T}}\mu(x)dx.$ Here as in the particular case when $\mu(x)=1$ for each $x\in\mathbb{R}$, we are exclusively interested in the weights $\mu$ for which, $\displaystyle\lim_{T\to\infty}\mu(Q_{T})=\infty.$ Consequently, we define the space of weights $\mathbb{U}_{\infty}$ by $\mathbb{U}_{\infty}:=\Bigg{\\{}\mu\in\mathbb{U}:\ \inf_{x\in\mathbb{R}}\mu(x)=\mu_{0}>0\ \ \mbox{and}\ \ \lim_{T\to\infty}\mu(Q_{T})=\infty\Bigg{\\}}.$ In addition to the above, we define the set of weights $\mathbb{U}_{B}$ by $\displaystyle\mathbb{U}_{B}:=\Bigg{\\{}\mu\in\mathbb{U}_{\infty}:\ \sup_{x\in\mathbb{R}}\mu(x)=\mu_{1}<\infty\Bigg{\\}}.$ We also need the following set of weights, which makes the spaces of weighted pseudo-almost periodic functions translation-invariant, $\displaystyle\mathbb{U}_{\infty}^{\rm Inv}:=\Bigg{\\{}\mu\in\mathbb{U}_{\infty}:\ \lim_{x\to\infty}\frac{\mu(x+\tau)}{\mu(x)}<\infty\ \ \mbox{and}\ \ \lim_{T\to\infty}\frac{\mu(Q_{T+\tau})}{\mu(Q_{T})}<\infty\ \mbox{for all}\ \tau\in\mathbb{R}\Bigg{\\}}.$ Let $\mathbb{U}_{\infty}^{c}$ denote the collection of all continuous functions (weights) $\mu:\mathbb{R}\mapsto(0,\infty)$ such that $\mu>0$ almost everywhere. Define $\displaystyle\mathbb{U}_{\infty}^{s}:=\Bigg{\\{}\mu\in\mathbb{U}_{\infty}^{c}\cap\mathbb{U}_{\infty}:\ \lim_{x\to\infty}\frac{\mu(x+\tau)}{\mu(x)}<\infty\ \ \mbox{for all}\ \tau\in\mathbb{R}\Bigg{\\}}.$ ###### Lemma 2.5. [10, Diagana] The inclusion $\mathbb{U}_{\infty}^{s}\subset\mathbb{U}_{\infty}^{\rm Inv}$ holds. ###### Definition 2.6. Let $\mu,\nu\in\mathbb{U}_{\infty}$. One says that $\mu$ is equivalent to $\nu$ and denote it $\mu\prec\nu$, if $\displaystyle\frac{\mu}{\nu}\in\mathbb{U}_{B}.$ Let $\mu,\nu,\gamma\in\mathbb{U}_{\infty}.$ It is clear that $\mu\prec\mu$ (reflexivity); if $\mu\prec\nu$, then $\nu\prec\mu$ (symmetry); and if $\mu\prec\nu$ and $\nu\prec\gamma$, then $\mu\prec\gamma$ (transitivity). Therefore, $\prec$ is a binary equivalence relation on $\mathbb{U}_{\infty}$. We have ###### Proposition 2.7. Let $\mu,\nu\in\mathbb{U}_{\infty}^{\rm Inv}$. If $\mu\prec\nu$, then $\sigma=\mu+\nu\in\mathbb{U}_{\infty}^{\rm Inv}$. ###### Proposition 2.8. Let $\mu,\nu\in\mathbb{U}_{\infty}^{s}$. Then their product $\pi=\mu\nu\in\mathbb{U}_{\infty}^{s}$. Moreover, if $\mu\prec\nu$, then $\sigma:=\mu+\nu\in\mathbb{U}_{\infty}^{s}$. The next theorem describes all the nonconstant polynomials belonging to the set of weights $\mathbb{U}_{\infty}$. ###### Theorem 2.9. [10, Diagana] If $\mu\in\mathbb{U}_{\infty}$ is a nonconstant polynomial of degree $N$, then $N$ is necessarily even ($N=2n^{\prime}$ for some nonnegative integer $n^{\prime}$). More precisely, $\mu$ can be written in the following form: $\mu(x)=a\prod_{k=0}^{n}(x^{2}+a_{k}x+b_{k})^{m_{k}}$ where $a>0$ is a constant, $a_{k}$ and $b_{k}$ are some real numbers satisfying $a_{k}^{2}-4b_{k}<0$, and $m_{k}$ are nonnegative integers for $k=0,...,n$. Furthermore, the weight $\mu$ given above belongs to $\mathbb{U}_{\infty}^{s}$. ### 2.3. Doubly-Weighted Pseudo-Almost Periodic Functions ###### Definition 2.10. A function $f\in C(\mathbb{R},\mathbb{X})$ is called (Bohr) almost periodic if for each $\varepsilon>0$ there exists $l(\varepsilon)>0$ such that every interval of length $l(\varepsilon)$ contains a number $\tau$ with the property that $\\|f(t+\tau)-f(t)\\|<\varepsilon\ \ \mbox{for each}\ \ t\in\mathbb{R}.$ The collection of all almost periodic functions will be denoted $AP(\mathbb{X})$. ###### Definition 2.11. A function $F\in C(\mathbb{R}\times\mathbb{Y},\mathbb{X})$ is called (Bohr) almost periodic in $t\in\mathbb{R}$ uniformly in $y\in\mathbb{Y}$ if for each $\varepsilon>0$ and any compact $K\subset\mathbb{Y}$ there exists $l(\varepsilon)$ such that every interval of length $l(\varepsilon)$ contains a number $\tau$ with the property that $\\|F(t+\tau,y)-F(t,y)\\|<\varepsilon\ \ \mbox{for each}\ \ t\in\mathbb{R},\ \ y\in K.$ The collection of those functions is denoted by $AP(\mathbb{Y},\mathbb{X})$. If $\mu,\nu\in\mathbb{U}_{\infty}$, we then define $PAP_{0}(\mathbb{X},\mu,\nu):=\Bigg{\\{}f\in BC(\mathbb{R},\mathbb{X}):\ \ \lim_{T\to\infty}\displaystyle{\frac{1}{\mu(Q_{T})}}\int_{Q_{T}}\left\\|f(\sigma)\right\\|\,\nu(\sigma)\,d\sigma=0\Bigg{\\}}.$ Similarly, we define $PAP_{0}(\mathbb{Y},\mathbb{X},\mu,\nu)$ as the collection of jointly continuous functions $F:\mathbb{R}\times\mathbb{Y}\mapsto\mathbb{X}$ such that $F(\cdot,y)$ is bounded for each $y\in\mathbb{Y}$ and $\lim_{T\to\infty}\displaystyle{\frac{1}{\mu(Q_{T})}}\left\\{\int_{Q_{T}}\\|F(s,y)\\|\,\nu(s)\,ds\right\\}=0$ uniformly in $y\in\mathbb{Y}$. ###### Definition 2.12. Let $\mu,\nu\in\mathbb{U}_{\infty}$. A function $f\in C(\mathbb{R},\mathbb{X})$ is called doubly-weighted pseudo-almost periodic if it can be expressed as $f=g+\phi,$ where $g\in AP(\mathbb{X})$ and $\phi\in PAP_{0}(\mathbb{X},\mu,\nu)$. The collection of such functions will be denoted by $PAP({\mathbb{X}},\mu,\nu)$. ###### Definition 2.13. Let $\mu,\nu\in\mathbb{U}_{\infty}$. A function $F\in C(\mathbb{R}\times\mathbb{Y},\mathbb{X})$ is called doubly-weighted pseudo- almost periodic if it can be expressed as $F=G+\Phi,$ where $G\in AP(\mathbb{Y},\mathbb{X})$ and $\Phi\in PAP_{0}(\mathbb{Y},\mathbb{X},\mu,\nu)$. The collection of such functions will be denoted by $PAP(\mathbb{Y},{\mathbb{X}},\mu,\nu)$. ###### Proposition 2.14. [10, Diagana] Let $\mu\in\mathbb{U}_{\infty}$ and let $\nu\in\mathbb{U}_{\infty}^{\rm Inv}$ such that (2.5) $\displaystyle\displaystyle\sup_{T>0}\Bigg{[}\displaystyle{\frac{\nu(Q_{T})}{\mu(Q_{T})}}\Bigg{]}<\infty.$ Let $f\in PAP_{0}(\mathbb{R},\mu,\nu)$ and let $g\in L^{1}(\mathbb{R})$. Suppose (2.6) $\displaystyle\displaystyle\lim_{T\to\infty}\Bigg{[}\displaystyle{\frac{\mu(Q_{T+\|\tau\|})}{\mu(Q_{T})}}\Bigg{]}<\infty\ \ \mbox{for all}\ \ \tau\in\mathbb{R}.$ Then $f\ast g$, the convolution of $f$ and $g$ on $\mathbb{R}$, belongs to $PAP_{0}(\mathbb{R},\mu,\nu)$. ###### Proof. It is clear that if $f\in PAP_{0}(\mathbb{R},\mu,\nu)$ and $g\in L^{1}(\mathbb{R})$, then their convolution $f\ast g\in BC(\mathbb{R},\mathbb{R})$. Now setting $J(T,\mu,\nu):=\displaystyle{\frac{1}{\mu(Q_{T})}}\int_{Q_{T}}\int_{-\infty}^{+\infty}\|f(t-s)\|\,\|g(s)\|\nu(t)\,dsdt$ it follows that $\displaystyle\displaystyle{\frac{1}{\mu(Q_{T})}}\int_{Q_{T}}\|(f\ast g)(t)\|\nu(t)dt$ $\displaystyle\leq$ $\displaystyle J(T,\mu,\nu)$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{+\infty}\|g(s)\|\left(\displaystyle{\frac{1}{\mu(Q_{T})}}\int_{Q_{T}}\|f(t-s)\|\nu(t)dt\right)ds$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{+\infty}\|g(s)\|\phi_{T}(s)ds,$ where $\displaystyle\displaystyle\phi_{T}(s)$ $\displaystyle=$ $\displaystyle\frac{1}{\mu(Q_{T})}\int_{Q_{T}}\|f(t-s)\|\nu(t)dt$ $\displaystyle=$ $\displaystyle\frac{\mu(Q_{T+\|s\|})}{\mu(Q_{T})}\,.\,\frac{1}{\mu(Q_{T+\|s\|})}\int_{Q_{T}}\|f(t-s)\|\nu(t)dt$ $\displaystyle\leq$ $\displaystyle\frac{\mu(Q_{T+\|s\|})}{\mu(Q_{T})}\,.\,\frac{1}{\mu(Q_{T+\|s\|})}\int_{Q_{T+\|s\|}}\|f(t)\|\nu(t+s)dt.$ Using the fact that $\nu\in\mathbb{U}_{\infty}^{\rm Inv}$ and Eq. (2.6), one can easily see that $\phi_{T}(s)\mapsto 0$ as $T\mapsto\infty$ for all $s\in\mathbb{R}$. Next, since $\phi_{T}$ is bounded, i.e., $\|\phi_{T}(s)\|\leq\\|f\\|_{\infty}\,.\,\displaystyle\sup_{T>0}\displaystyle{\frac{\nu(Q_{T})}{\mu(Q_{T})}}<\infty$ and $g\in L^{1}(\mathbb{R})$, using the Lebesgue Dominated Convergence Theorem it follows that $\lim_{T\to\infty}\left\\{\int_{-\infty}^{+\infty}\|g(s)\|\phi_{T}(s)ds\right\\}=0,$ and hence $f\ast g\in PAP_{0}(\mathbb{R},\mu,\nu)$. ∎ ###### Corollary 2.15. Let $\mu\in\mathbb{U}_{\infty}$ and let $\nu\in\mathbb{U}_{\infty}^{\rm Inv}$ such that Eqs. (2.5) – (2.6) hold. If $f\in PAP(\mathbb{R},\mu,\nu)$ and $g\in L^{1}(\mathbb{R})$, then $f\ast g$ belongs to $PAP(\mathbb{R},\mu,\nu)$. ###### Theorem 2.16. [10, Diagana] If $\mu,\nu\in\mathbb{U}_{\infty}$ such that the space $PAP_{0}(\mathbb{X},\mu,\nu)$ is translation-invariant and if (2.7) $\displaystyle\displaystyle\inf_{T>0}\Bigg{[}\displaystyle{\frac{\nu(Q_{T})}{\mu(Q_{T})}}\Bigg{]}=\delta_{0}>0,$ then the decomposition of doubly-weighted pseudo-almost periodic functions is unique. ###### Theorem 2.17. [10, Diagana] Let $\mu,\nu\in\mathbb{U}_{\infty}$ and let $f\in PAP(\mathbb{Y},\mathbb{X},\mu,\nu)$ satisfying the Lipschitz condition $\\|f(t,u)-f(t,v)\\|\leq L\,.\,\\|u-v\\|_{\mathbb{Y}}\ \ \mbox{for all}\ \ u,v\in\mathbb{Y},\ t\in\mathbb{R}.$ If $h\in PAP(\mathbb{Y},\mu,\nu)$, then $f(\cdot,h(\cdot))\in PAP(\mathbb{X},\mu,\nu)$. ## 3\. Existence of a Doubly-Weighted Mean for Almost Periodic Functions Let $\mu,\nu\in\mathbb{U}_{\infty}$. If $f:\mathbb{R}\mapsto\mathbb{X}$ is a bounded continuous function, we define its doubly-weighted mean, if the limit exists, by ${\mathcal{M}}(f,\mu,\nu):=\lim_{T\to\infty}\frac{1}{\mu(Q_{T})}\int_{Q_{T}}f(t)\nu(t)dt.$ It is well-known that if $f\in AP(\mathbb{X})$, then its mean defined by ${\mathcal{M}}(f):=\lim_{T\to\infty}\frac{1}{2T}\int_{Q_{T}}f(t)dt$ exists [6]. Consequently, for every $\lambda\in\mathbb{R}$, the following limit $a(f,\lambda):=\lim_{T\to\infty}\frac{1}{2T}\int_{Q_{T}}f(t)e^{-i\lambda t}dt$ exists and is called the Bohr transform of $f$. It is well-known that $a(f,\lambda)$ is nonzero at most at countably many points [6]. The set defined by $\sigma_{b}(f):=\Big{\\{}\lambda\in\mathbb{R}:a(f,\lambda)\not=0\Big{\\}}$ is called the Bohr spectrum of $f$ [19]. ###### Theorem 3.1. (Approximation Theorem) [17, 19] Let $f\in AP(\mathbb{X})$. Then for every $\varepsilon>0$ there exists a trigonometric polynomial $P_{\varepsilon}(t)=\sum_{k=1}^{n}a_{k}e^{i\lambda_{k}t}$ where $a_{k}\in\mathbb{X}$ and $\lambda_{k}\in\sigma_{b}(f)$ such that $\\|f(t)-P_{\varepsilon}(t)\\|<\varepsilon$ for all $t\in\mathbb{R}$. In Liang et al. [18], the original question which consists of the existence of a weighted mean for almost periodic functions was raised. In particular, Liang et al. have shown through an example that there exist weights for which a weighted mean for almost periodic functions may not exist. In this section we investigate the broader question, which consists of the existence of a doubly- weighted mean for almost periodic functions. Namely, we give some sufficient conditions, which do guarantee the existence of a doubly-weighted mean for almost periodic functions. Moreover, under those conditions, it will be shown that the doubly-weighted mean and the classical (Bohr) mean are proportional (Theorem 3.2). Further, it will be shown that if the doubly-weighted Bohr spectrum of an almost periodic function exists, then it is either empty or coincides with the Bohr spectrum of that function. We have ###### Theorem 3.2. Let $\mu,\nu\in\mathbb{U}_{\infty}$ and suppose that $\displaystyle\lim_{T\to\infty}\frac{\nu(Q_{T})}{\mu(Q_{T})}=\theta_{\mu\nu}$. If $f:\mathbb{R}\mapsto\mathbb{X}$ is an almost periodic function such that (3.1) $\displaystyle\lim_{T\to\infty}\Bigg{\|}\frac{1}{\mu(Q_{T})}\int_{Q_{T}}e^{i\lambda t}\nu(t)dt\Bigg{\|}=0$ for all $0\not=\lambda\in\sigma_{b}(f)$, then the doubly-weighted mean of $f$, ${\mathcal{M}}(f,\mu,\nu)=\lim_{T\to\infty}\frac{1}{\mu(Q_{T})}\int_{Q_{T}}f(t)\nu(t)dt$ exists. Furthermore, ${\mathcal{M}}(f,\mu,\nu)=\theta_{\mu\nu}{\mathcal{M}}(f)$. ###### Proof. If $f$ is a trigonometric polynomial, say, $\displaystyle f(t)=\sum_{k=0}^{n}a_{k}e^{i\lambda_{k}t}$ where $a_{k}\in\mathbb{X}-\\{0\\}$ and $\lambda_{k}\in\mathbb{R}$ for $k=1,2,...,n$, then $\sigma_{b}(f)=\\{\lambda_{k}:\ k=1,2,...,n\\}$. Moreover, $\displaystyle\frac{1}{\mu(Q_{T})}\int_{Q_{T}}f(t)\nu(t)dt$ $\displaystyle=$ $\displaystyle a_{0}\frac{\nu(Q_{T})}{\mu(Q_{T})}+\frac{1}{\mu(Q_{T})}\int_{Q_{T}}\Big{[}\sum_{k=1}^{n}a_{k}e^{i\lambda_{k}t}\Big{]}\nu(t)dt$ $\displaystyle=$ $\displaystyle a_{0}\frac{\nu(Q_{T})}{\mu(Q_{T})}+\sum_{k=1}^{n}a_{k}\Big{[}\frac{1}{\mu(Q_{T})}\int_{Q_{T}}e^{i\lambda_{k}t}\nu(t)dt\Big{]}$ and hence $\displaystyle\left\\|\frac{1}{\mu(Q_{T})}\int_{Q_{T}}f(t)\nu(t)dt- a_{0}\frac{\nu(Q_{T})}{\mu(Q_{T})}\right\\|$ $\displaystyle\leq$ $\displaystyle\sum_{k=1}^{n}\left\\|a_{k}\right\\|\Big{\|}\frac{1}{\mu(Q_{T})}\int_{Q_{T}}e^{i\lambda_{k}t}\nu(t)dt\Big{\|}$ which by Eq. (3.1) yields $\left\\|\frac{1}{\mu(Q_{T})}\int_{Q_{T}}f(t)\nu(t)dt- a_{0}\theta_{\mu\nu}\right\\|\to 0\ \ \mbox{as}\ \ T\to\infty$ and therefore ${\mathcal{M}}(f,\mu,\nu)=a_{0}\theta_{\mu\nu}=\theta_{\mu\nu}M(f)$. If in the finite sequence of $\lambda_{k}$ there exist $\lambda_{n_{k}}=0$ for $k=1,2,...l$ with $a_{m}\in\mathbb{X}-\\{0\\}$ for all $m\not=n_{k}$ ($k=1,2,...,l$), it can be easily shown that $\displaystyle{\mathcal{M}}(f,\mu,\nu)=\theta_{\mu\nu}\sum_{k=1}^{l}a_{n_{k}}=\theta_{\mu\nu}M(f).$ Now if $f:\mathbb{R}\mapsto\mathbb{X}$ is an arbitrary almost periodic function, then for every $\varepsilon>0$ there exists a trigonometric polynomial (Theorem 3.1) $P_{\varepsilon}$ defined by $P_{\varepsilon}(t)=\sum_{k=1}^{n}a_{k}e^{i\lambda_{k}t}$ where $a_{k}\in\mathbb{X}$ and $\lambda_{k}\in\sigma_{b}(f)$ such that (3.2) $\displaystyle\left\\|f(t)-P_{\varepsilon}(t)\right\\|<\varepsilon$ for all $t\in\mathbb{R}$. Proceeding as in Bohr [6] it follows that there exists $T_{0}$ such that for all $T_{1},T_{2}>T_{0}$, $\displaystyle\Big{\\|}\frac{1}{\mu(Q_{T_{1}})}\int_{Q_{T_{1}}}P_{\varepsilon}(t)\nu(t)dt-\frac{1}{\mu(Q_{T_{2}})}\int_{Q_{T_{2}}}P_{\varepsilon}(t)\nu(t)dt\Big{\\|}=\theta_{\mu\nu}\Big{\\|}M(P_{\varepsilon})-M(P_{\varepsilon})\Big{\\|}=0<\varepsilon.$ In view of the above it follows that for all $T_{1},T_{2}>T_{0}$, $\displaystyle\noindent\Big{\\|}\frac{1}{\mu(Q_{T_{1}})}\int_{Q_{T_{1}}}f(t)\nu(t)dt-\frac{1}{\mu(Q_{T_{2}})}\int_{Q_{T_{2}}}f(t)\nu(t)dt\Big{\\|}$ $\displaystyle\leq$ $\displaystyle\frac{1}{\mu(Q_{T_{1}})}\int_{Q_{T_{1}}}\\|f(t)-P_{\varepsilon}(t)\\|\nu(t)dt$ $\displaystyle+\Big{\\|}\frac{1}{\mu(Q_{T_{1}})}\int_{Q_{T_{1}}}P_{\varepsilon}(t)\nu(t)dt-\frac{1}{\mu(Q_{T_{2}})}\int_{Q_{T_{2}}}P_{\varepsilon}(t)\nu(t)dt\Big{\\|}$ $\displaystyle+\frac{1}{\mu(Q_{T_{2}})}\int_{Q_{T_{2}}}\\|f(t)-P_{\varepsilon}(t)\\|\nu(t)dt<3\varepsilon.$ ∎ ###### Example 3.3. Fix a natural number $N>1$. Let $\mu(t)=e^{\|t\|}$ and $\nu(t)=(1+\|t\|)^{N}$ for all $t\in\mathbb{R}$, which yields $\theta_{\mu\nu}=0$. If $\varphi:\mathbb{R}\mapsto\mathbb{X}$ is a (nonconstant) almost periodic function, then according to the previous theorem, its doubly-weighted mean ${\mathcal{M}}(\varphi,\mu,\nu)$ exists. Moreover, $\lim_{T\to\infty}\frac{1}{2(e^{T}-1)}\int_{Q_{T}}f(t)(1+\|t\|)^{N}dt=0.\lim_{T\to\infty}\frac{1}{2T}\int_{Q_{T}}f(t)dt=0.$ Consider the set of weights $\mathbb{U}_{\infty}^{0}$ defined by $\mathbb{U}_{\infty}^{0}=\Bigg{\\{}\mu\in\mathbb{U}_{\infty}:D_{\tau}:=\lim_{\|t\|\to\infty}\frac{\mu(Q_{t+\tau})}{\mu(Q_{t})}<\infty\ \ \mbox{for all}\ \ \tau\in\mathbb{R}\Bigg{\\}}.$ Setting, $\displaystyle C_{\tau}=\lim_{\|t\|\to\infty}\frac{\mu(Q_{t}+\tau)}{\mu(Q_{t})}$, one can easily see that $C_{\tau}\leq D_{\tau}<\infty$ for all $\tau\in\mathbb{R}$. ###### Corollary 3.4. Fix $\mu,\nu\in\mathbb{U}_{\infty}^{0}$ and suppose that $\displaystyle\lim_{T\to\infty}\frac{\nu(Q_{T})}{\mu(Q_{T})}=\theta_{\mu\nu}$. If $f:\mathbb{R}\mapsto\mathbb{X}$ is an almost periodic function such that Eq. (3.1) holds, then (3.3) $\displaystyle{\mathcal{M}}(f_{a},\mu,\nu_{a})=C_{-a}\theta_{\mu\nu}{\mathcal{M}}(f)=C_{-a}{\mathcal{M}}(f,\mu,\nu)$ uniformly in $a\in\mathbb{R}$, where ${\mathcal{M}}(f_{b},\mu,\nu_{b})=\lim_{T\to\infty}\frac{1}{\mu(Q_{T})}\int_{Q_{T}}f_{b}(t)\nu_{b}(t)dt=\lim_{T\to\infty}\frac{1}{\mu(Q_{T})}\int_{Q_{T}}f(t+b)\nu(t+b)dt$ for each $b\in\mathbb{R}$. ###### Proof. Clearly, the existence of ${\mathcal{M}}(f,\mu,\nu)$ is guaranteed by Theorem 3.2. Without lost of generality, suppose $a>0$. Now since $f\in AP(\mathbb{X})$ it follows that $f_{a}:t\mapsto f(t+a)$ belongs to $AP(\mathbb{X})$. Moreover, the weight $\nu_{a}$ defined by $\nu_{a}(t)=\nu(t+a)$ for all $t\in\mathbb{R}$ belongs to $\mathbb{U}_{\infty}^{0}$. Now $\Big{\|}\int_{Q_{T}}e^{i\lambda t}\nu_{a}(t)dt\Big{\|}=\Big{\|}\int_{Q_{T}-a}e^{i\lambda(t-a)}\nu(t)dt\Big{\|}=\Big{\|}\int_{Q_{T}-a}e^{i\lambda t}\nu(t)dt\Big{\|}\leq\Big{\|}\int_{Q_{T+a}}e^{i\lambda t}\nu(t)dt\Big{\|}$ and hence $\displaystyle\lim_{T\to\infty}\Big{\|}\frac{1}{\mu(Q_{T})}\int_{Q_{T}}e^{i\lambda t}\nu_{a}(t)dt\Big{\|}$ $\displaystyle=$ $\displaystyle\lim_{T\to\infty}\Big{\|}\frac{1}{\mu(Q_{T})}\int_{Q_{T}-a}e^{i\lambda t}\nu(t)dt\Big{\|}$ $\displaystyle\leq$ $\displaystyle\lim_{T\to\infty}\Big{\|}\frac{1}{\mu(Q_{T})}\int_{Q_{T+a}}e^{i\lambda t}\nu(t)dt\Big{\|}$ $\displaystyle=$ $\displaystyle\lim_{T\to\infty}\Big{\|}\frac{\mu(Q_{T+a})}{\mu(Q_{T})}\frac{1}{\mu(Q_{T+a})}\int_{Q_{T+a}}e^{i\lambda t}\nu(t)dt\Big{\|}$ $\displaystyle=$ $\displaystyle D_{a}\lim_{T\to\infty}\Big{\|}\frac{1}{\mu(Q_{T+a})}\int_{Q_{T+a}}e^{i\lambda t}\nu(t)dt\Big{\|}$ $\displaystyle=$ $\displaystyle 0.$ Now $\displaystyle\lim_{T\to\infty}\frac{\nu_{a}(Q_{T})}{\mu(Q_{T})}=C_{-a}\theta_{\mu\nu}.$ Using Theorem 3.2 it follows that for every $\varphi\in AP(\mathbb{X})$, ${\mathcal{M}}(\varphi_{a},\mu,\nu_{a})=\lim_{T\to\infty}\frac{1}{\mu(Q_{T})}\int_{Q_{T}}\varphi_{a}(t)\nu_{a}(t)dt$ exists. Furthermore, ${\mathcal{M}}(\varphi_{a},\mu,\nu_{a})=C_{-a}\theta_{\mu\nu}{\mathcal{M}}(\varphi_{a})$ for all $a\in\mathbb{R}$. In particular, ${\mathcal{M}}(f_{a},\mu,\nu_{a})=C_{-a}\theta_{\mu\nu}{\mathcal{M}}(f_{a})$ uniformly in $a\in\mathbb{R}$. Now from Bohr [6], ${\mathcal{M}}(f_{a})={\mathcal{M}}(f)$ uniformly in $a\in\mathbb{R}$, which completes the proof. ∎ ###### Definition 3.5. Fix $\mu,\nu\in\mathbb{U}_{\infty}$ and suppose that $\displaystyle\lim_{T\to\infty}\frac{\nu(Q_{T})}{\mu(Q_{T})}=\theta_{\mu\nu}$. If $f:\mathbb{R}\mapsto\mathbb{X}$ is an almost periodic function such that Eq. (3.1) holds, we then define its doubly-weighted Bohr transform as $\widehat{a}_{\mu\nu}(f)(\lambda):=\lim_{T\to\infty}\frac{1}{\mu(Q_{T})}\int_{Q_{T}}f(t)e^{-i\lambda t}\nu(t)dt\ \ \mbox{for all}\ \ \lambda\in\mathbb{R}.$ Now since $t\mapsto g_{\lambda}(t):=f(t)e^{-i\lambda t}\in AP(\mathbb{X})$ it follows that $\widehat{a}_{\mu\nu}(f)(\lambda)=\theta_{\mu\nu}{\mathcal{M}}(f(\cdot)e^{-i\lambda\cdot})=\theta_{\mu\nu}a(f,\lambda).$ That is, under Eq. (3.1), $\widehat{a}_{\mu\nu}(f)(\lambda):=\lim_{T\to\infty}\frac{1}{\mu(Q_{T})}\int_{Q_{T}}f(t)e^{-i\lambda t}\nu(t)dt=\theta_{\mu\nu}\lim_{T\to\infty}\frac{1}{2T}\int_{Q_{T}}f(t)e^{-i\omega t}dt=\theta_{\mu\nu}a(f,\lambda)$ for all $\lambda\in\mathbb{R}.$ In summary, there are two possibilities for the doubly-weighted Bohr spectrum of an almost periodic function. Indeed, 1) If $\displaystyle\lim_{T\to\infty}\frac{\nu(Q_{T})}{\mu(Q_{T})}=\theta_{\mu\nu}=0$, then $\widehat{a}_{\mu\nu}(f)(\lambda)=\theta_{\mu\nu}a(f,\lambda)=0$ for all $\lambda\in\mathbb{R}$. In that event, the doubly-weighted Bohr spectrum of $f$ is $\sigma_{b}^{\mu\nu}(f):=\Big{\\{}\lambda\in\mathbb{R}:\widehat{a}_{\mu\nu}(f)(\lambda)\not=0\Big{\\}}=\emptyset.$ 2) If $\displaystyle\lim_{T\to\infty}\frac{\nu(Q_{T})}{\mu(Q_{T})}=\theta_{\mu\nu}\not=0$, then $\widehat{a}_{\mu\nu}(f)(\lambda)=\theta_{\mu\nu}a(f,\lambda)$ exists for all $\lambda\in\mathbb{R}$ and is nonzero at most at countably many points. In that event, the doubly-weighted Bohr spectrum of $f$ is $\sigma_{b}^{\mu\nu}(f):=\Big{\\{}\lambda\in\mathbb{R}:\widehat{a}_{\mu\nu}(f)(\lambda)\not=0\Big{\\}}=\Big{\\{}\lambda\in\mathbb{R}:a(f,\lambda)\not=0\Big{\\}},$ that is, $\sigma_{b}^{\mu\nu}(f)=\sigma_{b}(f).$ In particular, $\sigma_{b}^{\mu\mu}(f)=\sigma_{b}(f).$ ## 4\. Existence of Doubly-Weighted Pseudo-Almost Periodic Solutions to Some Differential Equations In this Section, we fix two weights $\mu,\nu\in\mathbb{U}_{\infty}$ such that $PAP(\mathbb{X},\mu,\nu)$ is translation-invariant and Eq. (2.7) holds. Under these assumptions, it can be easily shown that $PAP(\mathbb{X},\mu,\nu)$ is a Banach space when equipped with the sup norm. In what follows, we denote by $\Gamma_{1}$ and $\Gamma_{2}$, the nonlinear integral operators defined by $(\Gamma_{1}u)(t):=\int_{-\infty}^{t}U(t,s)P(s)g(s,u(s))ds,\ \mbox{and}$ and $(\Gamma_{2}u)(t):=\int_{t}^{\infty}U_{Q}(t,s)Q(s)g(s,u(s))ds.$ To study the existence of doubly-weighted pseudo-almost periodic solutions to Eq. (1.1) we will assume that the following assumptions hold: 1. (H.1) The family of closed linear operators $A(t)$ for $t\in\mathbb{R}$ on $\mathbb{X}$ with domain $D(A(t))$ (possibly not densely defined) satisfy Acquistapace and Terreni conditions, that is, there exist constants $\omega\in\mathbb{R}$, $\theta\in\Big{(}\frac{\pi}{2},\pi\Big{)}$, $L>0$ and $\mu,\nu\in(0,1]$ with $\mu+\nu>1$ such that $\Sigma_{\theta}\cup\Big{\\{}0\Big{\\}}\subset\rho\Big{(}A(t)-\omega\Big{)}\ni\lambda,\;\qquad\\\ \\|R(\lambda,A(t)-\omega)\\|\leq\frac{K}{1+\|\lambda\|}\ \ \ \mbox{for all}\ t\in\mathbb{R},$ and $\\|(A(t)-\omega)R(\lambda,A(t)-\omega)\,[R(\omega,A(t))-R(\omega,A(s))]\\|\leq L\,\frac{\|t-s\|^{\mu}}{\|\lambda\|^{\nu}}$ for $t,s\in\mathbb{R}$, $\displaystyle\lambda\in\Sigma_{\theta}:=\\{\lambda\in\mathbb{C}\setminus\\{0\\}:\|\arg\lambda\|\leq\theta\\}$. 2. (H.2) The evolution family ${\mathcal{U}}=\\{U(t,s)\\}_{t\geq s}$ generated by $A(\cdot)$ has an exponential dichotomy with constants $N,\delta>0$ and dichotomy projections $P(t)$ for $t\in\mathbb{R}$. 3. (H.3) There exists $0\leq\alpha<1$ such that $\mathbb{X}_{\alpha}^{t}=\mathbb{X}_{\alpha}$ for all $t\in\mathbb{R},$ with uniform equivalent norms. 4. (H.4) $R(\omega,A(\cdot))\in AP(B(\mathbb{X}_{\alpha}))$. 5. (H.5) The function $g:\mathbb{R}\times\mathbb{X}\mapsto\mathbb{X}$ belongs to $PAP(\mathbb{X},\mathbb{X},\mu,\nu)$. Moreover, the functions $g$ are uniformly Lipschitz with respect to the second argument in the following sense: there exists $K>0$ such that $\\|g(t,u)-g(t,v)\\|\leq K\\|u-v\\|$ for all $u,v\in\mathbb{X}$ and $t\in\mathbb{R}$. If $0<\alpha<1$, then the nonnegative constant $k$ will denote the bounds of the embedding $\mathbb{X}_{\alpha}\hookrightarrow\mathbb{X}$, that is, $\\|x\\|\leq k\\|x\\|_{\alpha}$ for all $x\in\mathbb{X}_{\alpha}$. To study the existence and uniqueness of doubly-weighted pseudo-almost periodic solutions to Eq. (1.1) we first introduce the notion of mild solution. ###### Definition 4.1. A continuous function $u:\mathbb{R}\mapsto\mathbb{X}_{\alpha}$ is said to be a mild solution to Eq. (1.1) if $\displaystyle u(t)=U(t,s)u(s)+\int_{s}^{t}U(t,s)P(s)g(s,u(s))ds-\int_{t}^{s}U(t,s)Q(s)g(s,u(s))ds$ for $t\geq s$ and for all $t,s\in\mathbb{R}$. Under previous assumptions (H.1)-(H.5), it can be easily shown Eq. (1.1) has a unique mild solution given by $\displaystyle u(t)=\int_{-\infty}^{t}U(t,s)P(s)g(s,u(s))ds-\int_{t}^{\infty}U_{Q}(t,s)Q(s)g(s,u(s))ds$ for each $t\in\mathbb{R}$. ###### Lemma 4.2. Under assumptions (H.1)—(H.5), the integral operators $\Gamma_{1}$ and $\Gamma_{2}$ defined above map $PAP(\mathbb{X}_{\alpha},\mu,\nu)$ into itself. ###### Proof. Let $u\in PAP(\mathbb{X}_{\alpha},\mu,\nu)$. Setting $h(t)=g(t,u(t))$ and using the theorem of composition of doubly-weighted pseudo-almost periodic functions (Theorem 2.17) it follows that $h\in PAP(\mathbb{X},\mu,\nu)$. Now write $h=\phi+\zeta$ where $\phi\in AP(\mathbb{X})$ and $\zeta\in PAP_{0}(\mathbb{X},\mu,\nu)$. The nonlinear integral operator $\Gamma_{1}u$ can be rewritten as $\displaystyle(\Gamma_{1}u)(t)$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{t}U(t,s)P(s)\phi(s)ds+\int_{-\infty}^{t}U(t,s)P(s)\zeta(s)ds.$ Set $\displaystyle\Phi(t)=\int_{-\infty}^{t}U(t,s)P(s)\phi(s)ds$ and $\displaystyle\Psi(t)=\int_{-\infty}^{t}U(t,s)P(s)\zeta(s)ds$ for each $t\in\mathbb{R}$. The next step consists of showing that $\Phi\in AP(\mathbb{X}_{\alpha})$ and $\Psi\in PAP_{0}(\mathbb{X}_{\alpha},\mu,\nu)$. Obviously, $\Phi\in AP(\mathbb{X}_{\alpha})$. Indeed, since $\phi\in AP(\mathbb{X})$, for every $\varepsilon>0$ there exists $l(\varepsilon)>0$ such that for every interval of length $l(\varepsilon)$ contains a $\tau$ with the property $\\|\phi(t+\tau)-\phi(t)\\|<\varepsilon C\ \ \mbox{for each}\ t\in\mathbb{R},$ where $\displaystyle C=\frac{\delta^{1-\alpha}}{c(\alpha)2^{1-\alpha}\Gamma(1-\alpha)}$ with $\Gamma$ being the classical $\Gamma$ function. Now $\displaystyle\Phi(t+\tau)-\Phi(t)$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{t+\tau}U(t+\tau,s)P(s)\phi(s)ds-\int_{-\infty}^{t}U(t,s)P(s)\phi(s)ds$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{t}U(t+\tau,s+\tau)P(s+\tau)\phi(s+\tau)ds-\int_{-\infty}^{t}U(t,s)P(s)\phi(s)ds$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{t}U(t+\tau,s+\tau)P(s+\tau)\phi(s+\tau)ds$ $\displaystyle-$ $\displaystyle\int_{-\infty}^{t}U(t+\tau,s+\tau)P(s+\tau)\phi(s)ds$ $\displaystyle+$ $\displaystyle\int_{-\infty}^{t}U(t+\tau,s+\tau)P(s+\tau)\phi(s)ds-\int_{-\infty}^{t}U(t,s)P(s)\phi(s)ds$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{t}U(t+\tau,s+\tau)P(s+\tau)\Big{(}\phi(s+\tau)-\phi(s)\Big{)}ds$ $\displaystyle+$ $\displaystyle\int_{-\infty}^{t}\Big{(}U(t+\tau,s+\tau)P(s+\tau)-U(t,s)P(s)\Big{)}\phi(s)ds.$ Using [4, 22] it follows that $\left\\|\int_{-\infty}^{t}\Big{[}U(t+\tau,s+\tau)P(s+\tau)-U(t,s)P(s)\Big{]}\phi(s)ds\right\\|_{\alpha}\leq\frac{2\\|\phi\\|_{\infty}}{\delta}\varepsilon.$ Similarly, using (2.3), it follows that $\left\\|\int_{-\infty}^{t}U(t+\tau,s+\tau)P(s+\tau)(\phi(s+\tau)-\phi(s))ds\right\\|_{\alpha}\leq\varepsilon.$ Therefore, $\\|\Phi(t+\tau)-\Phi(t)\\|_{\alpha}<\Big{(}1+\frac{2\\|\phi\\|_{\infty}}{\delta}\Big{)}\varepsilon\ \ \mbox{for each}\ t\in\mathbb{R},$ and hence, $\Phi\in AP(\mathbb{X}_{\alpha})$. To complete the proof for $\Gamma_{1}$, we have to show that $\Psi\in PAP_{0}(\mathbb{X}_{\alpha},\mu,\nu)$. First, note that $s\mapsto\Psi(s)$ is a bounded continuous function. It remains to show that $\lim_{T\to\infty}\displaystyle{\frac{1}{\mu(Q_{T})}}\ \int_{Q_{T}}\\|\Psi(t)\\|_{\alpha}\nu(t)dt=0.$ Again using Eq. (2.3) it follows that $\displaystyle\displaystyle\lim_{T\to\infty}\displaystyle{\frac{1}{\mu(Q_{T})}}\ \int_{Q_{T}}\\|\Psi(t)\\|_{\alpha}\nu(t)dt$ $\displaystyle\leq$ $\displaystyle\lim_{T\to\infty}\displaystyle{\frac{c(\alpha)}{\mu(Q_{T})}}\ \int_{Q_{T}}\int_{0}^{+\infty}s^{-\alpha}e^{-\frac{\delta}{2}s}\\|\zeta(t-s)\\|\nu(t)dsdt$ $\displaystyle\leq$ $\displaystyle\lim_{T\to\infty}\displaystyle c(\alpha)\int_{0}^{+\infty}s^{-\alpha}e^{-\frac{\delta}{2}s}\frac{1}{\mu(Q_{T})}\int_{Q_{T}}\\|\zeta(t-s)\\|\nu(t)dtds.$ Set $\displaystyle\Gamma_{s}(T)=\frac{1}{\mu(Q_{T})}\int_{Q_{T}}\\|\zeta(t-s)\\|\nu(t)dt.$ Since $PAP_{0}(\mathbb{X},\mu,\nu)$ is assumed to be translation invariant and that Eq. (2.7) holds, it follows that $t\mapsto\zeta(t-s)$ belongs to $PAP_{0}(\mathbb{X},\mu,\nu)$ for each $s\in\mathbb{R}$, and hence $\lim_{T\mapsto\infty}\frac{1}{\mu(Q_{T})}\int_{Q_{T}}\\|\zeta(t-s)\\|\nu(t)dt=0$ for each $s\in\mathbb{R}$. One completes the proof by using the well-known Lebesgue Dominated Convergence Theorem and the fact $\Gamma_{s}(T)\mapsto 0$ as $T\to\infty$ for each $s\in\mathbb{R}$. The proof for $\Gamma_{2}u(\cdot)$ is similar to that of $\Gamma_{1}u(\cdot)$. However one makes use of Eq. (2.4) rather than Eq. (2.3). ∎ ###### Theorem 4.3. Under assumptions (H.1)—(H.5), then Eq. (1.1) has a unique doubly-weighted pseudo-almost periodic mild solution whenever $K$ is small enough. ###### Proof. Consider the nonlinear operator ${\mathbb{M}}$ defined on $PAP(\mathbb{X}_{\alpha},\mu,\nu)$ by $\displaystyle{\mathbb{M}}u(t)=\int_{-\infty}^{t}U(t,s)P(s)g(s,u(s))ds-\int_{t}^{\infty}U_{Q}(t,s)Q(s)g(s,u(s))ds$ for each $t\in\mathbb{R}$. In view of Lemma 4.2, it follows that ${\mathbb{M}}$ maps $PAP(\mathbb{X}_{\alpha},\mu,\nu)$ into itself. To complete the proof one has to show that ${\mathbb{M}}$ has a unique fixed-point. If $v,w\in PAP(\mathbb{X}_{\alpha},\mu,\nu)$, then $\displaystyle\\|\Gamma_{1}(v)(t)-\Gamma_{1}(w)(t)\\|_{\alpha}$ $\displaystyle\leq$ $\displaystyle\int_{-\infty}^{t}\\|U(t,s)P(s)\left[g(s,v(s))-g(s,w(s))\right]\\|_{\alpha}ds$ $\displaystyle\leq$ $\displaystyle\int_{-\infty}^{t}c(\alpha)(t-s)^{-\alpha}e^{-\frac{\delta}{2}(t-s)}\\|g(s,v(s))-g(s,w(s))\\|ds$ $\displaystyle\leq$ $\displaystyle Kc(\alpha)\int_{-\infty}^{t}(t-s)^{-\alpha}e^{-\frac{\delta}{2}(t-s)}\\|v(s)-w(s)\\|ds$ $\displaystyle\leq$ $\displaystyle kKc(\alpha)\int_{-\infty}^{t}(t-s)^{-\alpha}e^{-\frac{\delta}{2}(t-s)}\\|v(s)-w(s)\\|_{\alpha}ds$ $\displaystyle\leq$ $\displaystyle kKc(\alpha)2^{1-\alpha}\,\Gamma(1-\alpha)\delta^{\alpha-1}\\|v-w\\|_{\alpha,\infty},$ and $\displaystyle\\|\Gamma_{2}(v)(t)-\Gamma_{2}(w)(t)\\|_{\alpha}$ $\displaystyle\leq$ $\displaystyle\int_{t}^{\infty}\\|U_{Q}(t,s)Q(s)\left[g(s,v(s))-g(s,w(s))\right]\\|_{\alpha}ds$ $\displaystyle\leq$ $\displaystyle\int_{t}^{\infty}m(\alpha)e^{\delta(t-s)}\\|g(s,v(s))-g(s,w(s))\\|ds$ $\displaystyle\leq$ $\displaystyle\int_{t}^{\infty}m(\alpha)Ke^{\delta(t-s)}\\|v(s)-w(s)\\|ds$ $\displaystyle\leq$ $\displaystyle km(\alpha)K\int_{t}^{\infty}e^{\delta(t-s)}\\|v(s)-w(s)\\|_{\alpha}ds$ $\displaystyle\leq$ $\displaystyle Kkm(\alpha)\\|v-w\\|_{\alpha,\infty}\int_{t}^{+\infty}e^{\delta(t-s)}ds$ $\displaystyle=$ $\displaystyle Kkm(\alpha)\delta^{-1}\\|v-w\\|_{\alpha,\infty},$ where $\displaystyle\left\\|u\right\\|_{\alpha,\infty}:=\sup_{t\in\mathbb{R}}\left\\|u(t)\right\\|_{\alpha}.$ Combining previous approximations it follows that $\\|\mathbb{M}v-\mathbb{M}w\\|_{\infty,\alpha}\leq KC(\alpha,\delta)\,.\,\\|v-w\\|_{\alpha,\infty},$ where $C(\alpha,\delta)=km(\alpha)\delta^{-1}+kc(\alpha)2^{1-\alpha}\,\Gamma(1-\alpha)\delta^{\alpha-1}>0$ is a constant, and hence if the Lipschitz $K$ is small enough, then Eq. (1.1) has a unique solution, which obviously is its only doubly-weighted pseudo- almost periodic mild solution. ∎ ## References * [1] R. P. Agarwal, B. de Andrade, and C. Cuevas, Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations. Nonlinear Anal. (RWA) 11 (2010), no. 5, 3532–3554. * [2] B. Amir and L. Maniar, Composition of pseudo-almost periodic functions and Cauchy problems with operator of nondense domain. Ann. Math. Blaise Pascal 6 (1999), no. 1, 1–11. * [3] M. Baroun, S. Boulite, T. Diagana, and L. Maniar, Almost periodic solutions to some semilinear non-autonomous thermoelastic plate equations. J. Math. Anal. Appl. 349(2009), no. 1, 74–84. * [4] M. Baroun, S. Boulite, G. M. N’Guérékata, and L. Maniar, Almost automorphy of semilinear parabolic evolution equations. Electron. J. Differential Equations. 2008 (2008), no. 60, 1–9. * [5] J. Blot, G. M. Mophou, G. M. N’Guérékata, and D. 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Schnaubelt, Almost periodicity of inhomogeneous parabolic evolution equations, Lecture Notes in Pure and Appl. Math. 234 (2003), Dekker, New York, 299- 318. * [23] T-J. Xiao, J. Liang, and J. Zhang, Pseudo-almost automorphic solutions to semilinear differential equations in Banach spaces. Semigroup Forum 76 (2008), no. 3, 518–524. * [24] T. J. Xiao, X-X. Zhu, and J. Liang, Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications. Nonlinear Anal. 70 (2009), no. 11, 4079–4085. * [25] C. Y. Zhang, Pseudo-almost periodic solutions of some differential equations. J. Math. Anal. Appl 181 (1994), no. 1, 62–76. * [26] C. Y. Zhang, Pseudo-almost periodic solutions of some differential equations. II. J. Math. Anal. Appl 192 (1995), no. 2, 543–561. * [27] C. Y. Zhang, Integration of vector-valued pseudo-almost periodic functions, Proc. Amer. Math. Soc 121 (1994), no. 1, pp. 167–174. * [28] L. Zhang and Y. 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1012.3490	# CHARM 2010: Experiment Summary and Future Charm Facilities JEFFREY A. APPEL Fermilab,PO Box 500 Batavia, IL 60510, USA appel@fnal.gov (Day Month Year; Day Month Year) ###### Abstract The CHARM 2010 meeting had over 30 presentations of experimental results, plus additional future facilities talks just before this summary talk. Since there is not enough time even to summarize all that has been shown from experiments and to recognize all the memorable plots and results, this summary will give a few personal observations, an overview at a fairly high level of abstraction. ###### keywords: Charm; conference summary; experiments; facilities. PACS numbers: 13.85.Ni, 13.88.+e, 14.20.Lq, 14.20.Lb, 14.65.Dw ## 1 Introduction This CHARM 2010 at IHEP in Beijing is only the 4th International Workshop on Charm Physics! The previous meetings were held in 2006 here at IHEP (Beijing), in 2007 at Cornell University (Ithaca, NY); and in 2009 in Leimen, Germany. Over 30 presentations of experimental results were presented here before the future facilities talks. Clearly, there is not enough time even to summarize all that we have seen from experiments, to recognize all the memorable plots and results - tempting as it is to reproduce the many clean signals and data vs theory figures, the quantum correlations plots, and the $D$-mixing plots before and after the latest CLEO-c data is added. So, this review will give only my personal observations, exposing my prejudices and my areas of ignorance, no doubt. This overview will be at a fairly high level of abstraction - not re-showing individual plots or results. I ask the forgiveness of those who will have been slighted in this way - meaning all the presenters. ## 2 Renewed Interest in Charm Now Rereading the text on the CHARM 2010 first bulletin, you miss what I think is the most profound hot topic in charm physics today, the observation of charm mixing in the neutral $D$ meson system, the possibility of this being due to new physics, and of the resulting need to look for $CP$ violation there. Not since the discovery of charm and its immediate impact (belief in quarks for real!) has charm had so much interest. This current interest is focused on the size of $D^{0}$ mixing. Given its size, mixing in the $D^{0}$ system could be due to new physics. If so, observation of $CP$ violation in this mixing would be a demonstration that the observed large mixing is due to physics beyond the standard model. Before, standard-model mixing in charm was thought to be too small to be interesting. However, this small background from standard-model effects was coming to be seen as a plus before large mixing was observed (relative to $B$ mesons where the standard-model mixing is now seen as an annoying background to any signal for new physics). More fundamentally, charm is the only up-type quark with mixing possible, a unique sensitivity to beyond-standard-model physics. Models with minium flavor violation have been made popular due to the good agreement between measurements with kaon and bottom mesons and standard-model predictions (ignoring some less than or about 3 $\sigma$ ”tensions”). ## 3 Experiment Presentations and Their Lessons We have seen truly impressive numbers of events in plots. Note that we often need the pressure of such copious data to force us to think creatively about underlying physics, to change our prejudices. I remember well how increased data forced E791 collaborators in Rio de Janeiro to propose S-wave resonances (sigma and kappa) to explain the otherwise unfittable Dalitz-plot decay distributions - though in hindsight, earlier data sets had shown evidence of the same need, just not as dramatically. Similarly, data forced FOCUS to see the interference with the S-wave under the $K^{}$ in $D$ semileptonic decays. At the same time, we should not forget the lesson cited by Will Johns who remembered how FOCUS ”learned more about the realities of the higher- statistics environment”.[1] In my experience these realities have included how to take, manage, and analyze the added data - as well as solving physics problems about which the new data may cry out. Is the disagreement at high $q^{2}$ between the LQCD form factors and data trying to tell us something important? Many results presented at CHARM 2010 are the first such observation or first such measurement - even now in this arguably mature field! Think of: New hadronic, radiative, and semileptonic decay modes New excited states of charm mesons Wide resonances, visible above background with enough data Form factors for Cabibbo-suppressed $D$ decays $A_{CP}$ measurements in new decay modes; so far, all consistent with no $CP$-violating asymmetry A surprising number of new results, even among the most interesting new results, have systematic errors which are significantly smaller than the statistical errors - even from the full CLEO-c, BaBar, or Belle data sets. So, the case for new facilities is very strong on that basis. There is room and utility for much more data. More data (and more analyses of existing data) are also needed to help reinforce or remove states from the growing list that need to be explained, and to see additional decay modes of states already indicated - even those multiply confirmed. We have all been uncomfortable with the idea that QCD would only choose to make states of quark-antiquark pairs and three quarks of different color. Yet, we seem to be forced against our will to accept other states that we have every reason to believe must exist. On the other hand, it is unlikely that every newly-observed state will survive an onslaught of new data. Possible states near thresholds need to be tested against other explanations: e.g., the possibility of fluctuations in threshold-enhancement- shaped backgrounds and/or fluctuations of backgrounds otherwise incompletely modeled as phase-space shaped. With the LHC really just starting its turn-on, we are getting a whiff of what may lie ahead from ATLAS, CMS, LHCb, and ALICE. Nevertheless, it has been useful for charm data that the LHC turn-on has been slower than some optimists have expected. This may be our only chance to see the low-$p_{t}$ production region at 7 $TeV$. We will have to see how charm-physics goals fit into LHC ”full-luminosity” trigger menus. In spite of all the exciting things we have seen at CHARM 2010, there are important things we have not seen, things that are sorely missed. For example, we have not seen any detailed analyses of the systematic errors in measurements with an extrapolation into the next generation of experiments. Just how far will we be able to push mixing and $CP$-violation measurements before we hit a wall of systematic uncertainty? Of course, we will need to experience the additional real data to be certain about this. However, knowing the likely-most-productive modes and avenues to pursue first is always useful. Will techniques have to change to stay competitive? Will we be able to use 10 times more data? 100 times more? ## 4 An Alert Many of the results we have seen have been the result of a tour de force - ”an army of researchers working for a couple of years” (David Asner).[1] There is concern about the future of doing charm physics, even with new facilities replacing or upgrading old ones. Let me emphasize, however, that the additional data should allow new analyses to be done, new questions to be asked. The ”golden times ahead” proclaimed by Ulf Meissner[1] for BEPCII and FAIR - and I would add others - will not be automatic. Bring the new data on! Force us to think harder. ## 5 If $CP$ Violation is Observed in Charm Decays, … If observed, $CP$ violation in charm mixing will be a ”game-changer” (forcing paradigm change). Motivation for charm physics will increase beyond the often cited justification of helping to understand or certify $B$-physics applications (”to the rescue” per Jernej Kamenik[1]). ## 6 Comments on Charm Production Is the color-octet model on its way out as a major source of charm production? It was proposed as an explanation of the historic theory underestimate of the observed production of charmonium and open charm. The model has detailed predictions for polarization of charmonium, predictions which have not been born out by data. To be viable, it also should have had a universality that has not been seen in charm production at HERA and the Tevatron. I have always been uncomfortable with the appearance of an easy acceptance of any suggested correction to theory that increased cross-section predictions. Which enhancements of the simplest calculations will survive? What will provide universality in matrix elements, and correctly describe onium polarization as a function of $p_{t}$? Are next-to-leading-order (NLO) and relativistic corrections enough to explain earlier cross-section discrepancies? To flip the size and sign of charmonium polarization? Joan Soto stated that ”Important discrepancies with experiment have been resolved”;[1] for example, the factor of two in the NLO prediction with respect to the leading-order. However, will the next-to-next-to-leading-order (NNLO) contribuion really be negligible on this scale? Theory errors, even before estimating NNLO, etc. remain too large to have confidence yet. Of course, we also want to see resolution of the experimental situation in polarization measurements within CDF (current and earlier) and with DZero. ## 7 Spectroscopy: Hidden Charm and Other Spectroscopy There is apparent progress since the last CHARM symposium in terms of the observation of various states, both for added decay modes and new states. However, is there any real progress in understanding? Questions in spectroscopy are multiplying still, though some patterns may be appearing. At the same time, charm is providing input to help understand light-meson spectroscopy. A personal favorite of mine is the use of charm decay as a source of information on low mass (e.g., scalar) mesons. Also, charm decays provide clean laboratories for the spectroscopy of excited kaon states. Many of the new states still require confirmation or more precise mass and width measurements. As more data become available from LHCb and from a future Super-$B$ factory, analyses similar to the ones presented here can further elucidate light-meson spectroscopy. ## 8 Fermilab as a Charm Facility Just prior to this review, motivations and plans were presented for future facilities for charm physics experiments. I will not repeat or summarize these reports now. However, I should probably comment on the situation at Fermilab since it is not otherwise reported. For now, the only new Fermilab data on charm physics continues to come from CDF and DZero at the Tevatron Collider. The current data taking, Run II, is scheduled to end in September, 2011. However, there is a proposal to extend Run II for three more years, through September of 2014. The US Particle Physics Program Prioritization Panel, P5, just considered this proposal and is to give its recommendation to the High Energy Physics Advisory Panel (HEPAP) on October 26. This is just one more of the hurdles which will have to be surmounted along a possible path to approval. HEPAP will make its comments in transmitting the report to the Department of Energy, and funding may appear in the President’s budget for the next fiscal year, which will be public in February, 2011. Fermilab has asked for approval of a plan which requires additional funding for a Run II extension to happen - so as not to jepordize other programs currently funded nor to unduly delay the approved program at the High Intensity Frontier. Stay tuned. There are two other options for future charm physics experiments at Fermilab being discussed: Proposal Number 986 - ”Medium-Energy Antiproton Physics with The Antiproton Annihilation Spectrometer (TApAS)”[2] A new fixed target experiment using the high-energy Tevatron beam[3] The first is a serious proposal, submitted to Fermilab and scheduled for review by the Fermilab Physics Advisory Committee (PAC) at its meeting, November 4-6. The second is only an attempt to keep alive the possibility of a future Tevatron experiment. Dan Kaplan is spokesperson for the former, serious proposal; Alan Schwartz and I have led the discussion of the latter. Both options require use of facilities scheduled for decommissioning and/or reuse for other programs at Fermilab. Again, stay tuned. ## 9 Final Comments Finally, we owe great thanks to our hosts for an exceptionally well-organized and enjoyable meeting. We also owe great thanks to all the presenters and their collaborators for their efforts. Charm remains a fascinating and vibrant area of research, one with the potential to teach us new things and be hotly pursued in many places. ## References [1] Quoted from his oral presentation at CHARM 2010. * [2] Proposal Number 986 - ”Medium-Energy Antiproton Physics with The Antiproton Annihilation Spectrometer (TApAS).” See link from the Fermilab PAC web page: http://www.fnal.gov/directorate/program_planning/phys_adv_com/PACdates.html * [3] T. Adams, et al., ”Renaissance of the 1-TeV Fixed-Target Program.” Int. J. Mod. Phys. A 25, 777 (2010). e-Print: arXiv:0905.3004 [hep-ex].	arxiv-papers	2010-12-15T22:55:23	2024-09-04T02:49:15.709887	{ "license": "Public Domain", "authors": "Jeffrey A. Appel", "submitter": "Jeffrey A. Appel", "url": "https://arxiv.org/abs/1012.3490" }
1012.3541	On the polygonal diameter of the interior, resp. exterior, of a simple closed polygon in the plane _Yaakov S. Kupitz 111Partially supported by the Landau Center at the Mathematics Institute of the Hebrew University of Jerusalem $($supported by Minerva Foundation, Germany$)$, and by Deutsche Forschungsgemeinschaft.$\,{}^{}$, Horst Martini ∗∗, Micha A. Perles ∗_ ∗ Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, ISRAEL, kupitz@math.huji.ac.il; perles@math.huji.ac.il ∗∗ Faculty of Mathematics, University of Technology, 09107 Chemnitz, GERMANY, martini@mathematik.tu-chemnitz.de ###### Abstract We give a tight upper bound on the polygonal diameter of the interior, resp. exterior, of a simple $n$-gon, $n\geq 3$, in the plane as a function of $n$, and describe an $n$-gon $(n\geq 3)$ for which both upper bounds (for the interior and the exterior) are attained _simultaneously_. Keywords: Jordan-Brouwer theorem, Jordan exterior (interior), Jordan’s curve theorem, polygonal diameter, raindrop proof, simple closed polygon MSC(2000): 51M05, 52B70, 57M50, 57N05 ## 1 Introduction The following is well known ###### Theorem 1.1. (The Jordan theorem) Let $f:[0,1]\to{\mathbb{R}}^{2}$ be a simple closed curve in the plane ($f$ is continous, $f(0)=f(1)$ and $f(u)\not=f(v)$ for $0<u<v\leq 1$). Define $P=_{\rm def}$ image$f=\\{f(u):0\leq u\leq 1\\}$, the image of $f$. Then ${\mathbb{R}}^{2}\setminus P=U_{0}\cup U_{1}$, where $U_{0},U_{1}$ are connected open, non-empty mutually disjoint sets, $U_{0}$ is bounded (interior), $U_{1}$ is unbounded (exterior), and $P={\rm bd}(U_{0})={\rm bd}(U_{1})$. The proof of this theorem is not easy; see [3], [8], [11], [9, p. 37 ff.], [1, vol. I, pp. 39-64], [7, pp. 285 ff.], and the survey [5]. When the curve $P$ is polygonal, however, i.e., when $f$ is piecewise affine, the theorem becomes elementary: ###### Theorem 1.2. (The piecewise affine Jordan theorem) Let $p_{0},p_{1},\dots,p_{n-1},p_{n}=p_{0},n\geq 3$, be ($n$ distinct) points in ${\mathbb{R}}^{2}$. Assume that the polygon $P=_{\rm def}\bigcup\limits^{n}_{i=1}[p_{i-1},p_{i}]$ is simple, i.e., the segments $[p_{i-1},p_{i}]$ do not intersect except for common endpoints: $\\{p_{i}\\}=[p_{i-1},p_{i}]\cap[p_{i},p_{i+1}]$ for $1\leq i\leq n-1,\\{p_{0}\\}=[p_{0},p_{1}]\cap[p_{n-1},p_{0}]$. Then ${\mathbb{R}}^{2}\setminus P=U_{0}\cup U_{1}$ with the same properties of $U_{0},U_{1}$ listed above (Theorem 1.1). ###### Definition 1.1. A polygon $P$ satisfying the conditions of Theorem 1.2 is a _simple closed $n$-gon_. The bounded [resp. unbounded] domain $U_{0}$ [resp. $U_{1}$] is the _interior_ [resp. _exterior_], denoted by int$P$ [resp. ext$P$], of $P$. A particularly simple proof of Theorem 1.2 is known as the “raindrop proof”, see [4, pp. 267-269], [6, pp. 281-285], [2, pp. 27-29], or [9, pp. 16-18]. We reproduce this proof in a somewhat more complete and formal form than usually given in the literature for later reference to some of its parts. So we first prove Theorem 1.2 (in Paragraphs 2 and 3 below). Then, squeezing this proof, a _tight_ upper bound on the polygonal diameter of int$P$ [resp. ext$P$] (see Definition 3.2 below) is given as a function of $n$, and an $n$-gon $(n\geq 3)$ for which both upper bounds are attained _simultaneously_ is described (see Theorem 4.1 below). The $d$-dimensional analogue $(d\geq 2)$ of this problem was discussed in [10, Theorem 3.2]. There we gave upper bounds on the polygonal diameter of int${\mathcal{C}}$, resp. ext${\mathcal{C}}$, for a polyhedral $(d-1)$-pseudomanifold ${\mathcal{C}}$ in ${\mathbb{R}}^{d}$ as a function of the number $n$ of its facets and $d$. The bounds given there are shown to be _almost_ tight (see [10, Section 4]), whereas the bounds given here (for $d=2$) are tight. Another novelty of the present paper is that there is an $n$-gon $P$ in ${\mathbb{R}}^{2}$ for which _both_ upper bounds (on the polygonal diameter of int$P$ and ext$P$) are attained (simultanously), as said above, whereas for $d\geq 3$ the examples given in [10, Section 4] (namely one for int${\mathcal{C}}$ and another one for ext${\mathcal{C}}$) are _different_ from each other. For the sake of the proof of Theorem 1.2, we split it into two statements: Let $P$ be a simple closed polygon in ${\mathbb{R}}^{2}$. (E) (separation): ${\mathbb{R}}^{2}\setminus P$ is the disjoint union of two open sets, int$P$ and ext$P$. The boundary of each one of these sets is $P$; int$P$ is bounded and ext$P$ is unbounded. (F) (connectivity): The sets int$P$ and ext$P$ are [polygonally] connected. We shall prove (E) (Paragraph 2) by constructing a continuous function $f:{\mathbb{R}}^{2}\setminus P\to\\{0,1\\}$ which attains both values $0$ and $1$ in every neighborhood of every point $x\in P$, and defining ext$P=f^{-1}(0)$, int$P=f^{-1}(1)$. Statement (F) (polygonal connectivity of int$P$ and of ext$P$) follows from Theorem 3.1 below. ## 2 A “raindrop” proof of (E) The construction of $f$ will be performed in three steps: Preliminary step: Choosing a “generic” direction. Choose an orthogonal basis $(u,v)$ for ${\mathbb{R}}^{2}$ so that no two vertices of $P$ have the same $x$-coordinate. Intuitively: the polygon $P$ is drawn as a paper; rotate the paper so that no two vertices lie one above the other. Formally: let $L_{1},\dots,L_{t}$ be all lines spanned by subsets of $\\{p_{1},\dots,p_{n}\\}$. For $i=1,\dots,t$ let $L^{0}_{i}=_{\rm def}L_{i}-L_{i}$ be the linear ($1$-dimensional) subspace parallel to $L_{i}$. Choose a unit vector $v\in{\mathbb{R}}^{2}\setminus\bigcup\limits^{t}_{i=1}L^{0}_{i}$ (“$v$” for “vertical”). The vector $v$ is our direction “up”, and $-v$ is pointing “down”. By our choice of $v$, a line $L$, spanned by the vertices of $P$, will meet a line parallel to $v$ in at most one point. For a point $p\in{\mathbb{R}}^{2}\setminus P$ denote by $R(p)$ the closed vertical “pointing down” half-line $R(p)=_{\rm def}\\{p-\lambda v:0\leq\lambda<\infty\\}$. $R(p)$ is the path of a “raindrop” emanating from $p$. We divide ${\mathbb{R}}^{2}\setminus P$ into two disjoint sets $\begin{array}[]{lll}S_{0}&=_{\rm def}&\\{p\in{\mathbb{R}}^{2}\setminus P:R(p)\,\mbox{ does not meet any vertex of }\,P\\}\,,\\\ S_{1}&=_{\rm def}&\\{p\in{\mathbb{R}}^{2}\setminus P:R(p)\,\mbox{ meets exactly one vertex of }\,P\\}\,.\end{array}$ (By our choice of $v$, we have ${\mathbb{R}}^{2}\setminus P=S_{0}\cup S_{1}$.) We shall define $f$ on $S_{0}$ (= Step I), then extend it (continuously) to $S_{1}$ (= Step II). The following notation will be used: For a set $A\subset{\mathbb{R}}^{2}$, $A^{+}=_{\rm def}\\{a+\lambda v:a\in A,\lambda\geq 0\\}$. Thus $A^{+}$ is the set of points that lie “above” $A$. If $A$ is closed, then $A^{+}$ is closed. Note that (for all $p\in{\mathbb{R}}^{2}$ and $A\subset{\mathbb{R}}^{2}$): $R(p)\,\mbox{ meets }\,A\,\mbox{ iff }\,p\in A^{+}\,.$ (1) Step I: Define $f$ on $S_{0}$. For $p\in S_{o}$ denote by $r(p)$ the number of edges of $P$ met by $R(p)$, and define $f(p)=_{\rm def}{\rm par}(r(p))=_{\rm def}\frac{1}{2}(1-(-1)^{r(p)})$, the parity of $r(p)$ ($f(p)=0$ if $r(p)$ is even, $1$ if $r(p)$ is odd). Fig. 1: the function $r(p)$ Fig. 2: the parity function $f(p)={\rm par}(r(p))$ Next we show that $S_{0}$ is a dense open subset of ${\mathbb{R}}^{2}$, and that $f:S_{0}\to\\{0,1\\}$ is a continuous, hence locally constant function. Using vert$P$ for the set of vertices of $P$, we have in view of (1) $S_{0}={\mathbb{R}}^{2}\setminus(P\cup(\mbox{vert}P)^{+})\,.$ (2) The set $({\rm vert}P)^{+}$ is closed, same as $P$. Thus $S_{0}$ is an open subset of ${\mathbb{R}}^{2}$. Moreover, the set $P\cup({\rm vert}P)^{+}$ can be covered by a finite number of lines in ${\mathbb{R}}^{2}$. It follows that $S_{0}$ is dense in $R^{2}$. Continuity of $f$: Assume $x\in S_{0}$. Let $\varepsilon$ be the (positive) distance from $x$ to $P\cup({\rm vert}P)^{+}(={\mathbb{R}}^{2}\setminus S_{0})$. If $x^{\prime}\in{\mathbb{R}}^{2},\\|x-x^{\prime}\\|<\varepsilon$, then the segment $[x,x^{\prime}]$ does not meet $P\cup({\rm vert}P)^{+}$. Let $e=[p_{i-1},p_{i}]\,(1\leq i\leq n)$ be any edge of $P$. The set $e^{+}$ is a closed, convex, unbounded and full-dimensional polyhedral subset of ${\mathbb{R}}^{2}$, whose boundary consists of the lower edge $e$ and the side edges $p^{+}_{i-1},p^{+}_{i}$. Thus bd$e^{+}\subset P\cup({\rm vert}P)^{+}$, and therefore the segment $[x,x^{\prime}]$ does not meet the boundary of $e^{+}$. It follows that $x^{\prime}\in e^{+}$ iff $x\in e^{+}$, i.e., $R(x)$ meets $e$ iff $R(x^{\prime})$ meets $e$. This is true for all edges $e$ of $P$. Therefore $r(x)=r(x^{\prime})$, hence $f(x)=f(x^{\prime})$. This shows that the function $f:S_{0}\to\\{0,1\\}$ is locally constant, hence continuous (in $S_{0})$. Step II: Extend $f$ continuously from $S_{0}$ to $S_{0}\cup S_{1}={\mathbb{R}}^{2}\setminus P$. Suppose $p\in S_{1}$. Let $p_{i}$ be the unique vertex of $P$ that meets $R(p)$, i.e., $p\in p^{+}_{i}$. Note that $p\not=p_{i}$, i.e., $p\in\mbox{ relint }p^{+}_{i}$. Let $e_{1}=[p_{i-1},p_{i}],e_{2}=[p_{i},p_{i+1}]$ be the two edges of $P$ incident with $p_{i}$. Define $L=p+{\mathbb{R}}v$. $L$ is the vertical line through $p$. Denote by $L^{-},L^{+}$ the two closed half-planes of ${\mathbb{R}}^{2}$ bounded by $L$. None of the edges $e_{1},e_{2}$ is included in $L$, and they may be either in the same half-plane $L^{-}$ or $L^{+}$, or in different half-planes. Choose the notation so that either $(\alpha)$ $e_{1}\subset L^{-},e_{2}\subset L^{+}$ (Fig. 3) or $(\beta)$ $e_{1}\cup e_{2}\subset L^{+}$ (Fig. 4). Fig. 3: case $\alpha$ Fig. 4: case $\beta$ A glance on Figures 3 and 4 shows that for a point $x$ in the vicinity of $p$, but not lying on $L$, the parity of $r(x)$ is the same in either side of $L$. Hence we can extend the definition of $f$ to $p$ by defining $f(p)$ to be this parity. To make this into a formal argument consider the closed set $\triangle=_{\rm def}P\cup({\rm vert}P\setminus\\{p_{i}\\})^{+}$. This set includes the boundary of $e^{+}$, for every edge $e$ of $P$, except for $e^{+}_{1}$ and $e^{+}_{2}$. It also includes the boundaries of $e^{+}_{1}$ and $e^{+}_{2}$, except for $p^{+}_{i}\setminus\\{p_{i}\\}$, and it does not contain the point $p$. Put $\varepsilon=_{\rm def}\mbox{ dist}(p,\triangle)>0$, and define $U=_{\rm def}\\{x\in{\mathbb{R}}^{2}:\\|x-p\\|<\varepsilon\\}={\rm int}B^{2}(p,\varepsilon)$. Note that if $x\in U$, then the closed interval $[p,x]$ misses $\triangle$. Now make the following observations. 1. (I) If $e$ is any edge of $P$, other than $e_{1}$ and $e_{2}$, then the interval $[p,x]$ does not meet the boundary of $e^{+}$, and therefore $p$ and $x$ are either both in $e^{+}$, or both not in $e^{+}$. 2. (II) If, say, $e_{1}\subset L^{-}$ and $x\in{\rm int}L^{-}$ then, moving along the interval $[p,x]$ from $p$ to $x$, we start at a point $p\in p^{+}_{i}\subset{\rm bd}e^{+}_{1}$, move into int$e^{+}_{1}$, and do not hit the boundary of $e^{+}_{1}$ again. Therefore $x\in{\rm int}e^{+}_{1}$. The same holds with $L^{-}$ replaced by $L^{+}$, and/or $e_{1}$ replaced by $e_{2}$. It follows that in case $(\alpha)$: if $x\in U\setminus L$, then $x$ belongs to exactly one of the sets $e^{+}_{1},e^{+}_{2}$. And it follows that in case $(\beta)$: if $x\in U\cap{\rm int}L^{-}$, then $x$ belongs to none of the sets $e^{+}_{1},e^{+}_{2}$; if $x\in U\cap L^{+}$, then $x$ belongs to both of them. 3. (III) If $p_{j}\in{\rm vert}P\setminus\\{p_{i}\\}$, then $p^{+}_{j}\subset\triangle$, and therefore $x\notin p^{+}_{j}$, who-ever $x\in U$. 4. (IV) If $x\in U\setminus L$, then clearly $x\notin p^{+}_{i}$. If $x\in U\cap L$, then the interval $[p,x]$ lies on $L$, contains a point $p\in p^{+}_{i}\setminus\\{p_{i}\\}$ and does not meet $p_{i}$; therefore $x\in p^{+}_{i}\setminus\\{p_{i}\\}$ (= relint$p^{+}_{i}$). From these observations we infer: 1. (A) $U\setminus L\subset S_{0}$ and $f$ is constant on $U\setminus L$. 2. (B) $U\cap L\subset S_{1}$. Now define $f(p)$ to be the constant value that $f$ takes on $U\setminus L$. Clearly, if we apply the same procedure to any point $p^{\prime}\in U\cap L$, we will end up with a value $f(p^{\prime})$ equal to the value $f(p)$ just defined. (Note that any $\varepsilon^{\prime}$-neighborhood of $p^{\prime}\,(\varepsilon^{\prime}>0)$ contains points of $U\setminus L$.) Thus we have extended $f$ to a locally constant, hence continuous function $f:{\mathbb{R}}^{2}\setminus P\to\\{0,1\\}$. To complete the proof of statement (E), we define, as indicated after (F) above, the sets ext$P=_{\rm def}f^{-1}(0)$ and int$P=_{\rm def}f^{-1}(1)$. These are clearly two disjoint open sets in ${\mathbb{R}}^{2}$, whose union is dom$f={\mathbb{R}}^{2}\setminus P$. Note that ${\mathbb{R}}^{2}\setminus{\rm conv}P\subset{\rm ext}P$ and, therefore, int$P\subset{\rm conv}P$. Thus ext$P$ is unbounded and int$P$ is bounded. We still have to show that every point of $P$ is a boundary point of both int$P$ and ext$P$ (and therefore int$P\not=\emptyset,{\rm ext}P\not=\emptyset$). Since the boundaries of int$P$ and of ext$P$ are closed sets, it suffices to show that the common boundary points of int$P$ and ext$P$ are dense in $P$. For any vertex $p_{i}\,(1\leq i\leq n)$ the intersection of the vertical line $p_{i}+{\mathbb{R}}v$ with an edge $e$ of $P$ is at most a singleton. Thus $e\setminus\cup\\{p_{i}+{\mathbb{R}}v:1\leq i\leq n\\}$ is dense in $e$, and $P\setminus\cup\\{p_{i}+{\mathbb{R}}v:1\leq i\leq n\\}$ is dense in $P$. If $x\in P\setminus\cup\\{p_{i}+{\mathbb{R}}v:1\leq i\leq n\\}$, then $x$ belongs to the relative interior of some edge $e$ of $P$. If $\varepsilon>0$ is sufficiently small, then the points $x+\varepsilon v,x-\varepsilon v$ are both in $S_{0}$, the half-line $R(x+\varepsilon v)$ meets $e$, in addition to all edges met by $R(x-\varepsilon v)$. Thus $r(x+\varepsilon v)=1+r(x-\varepsilon v)$, and $f(x+\varepsilon v)\not=f(x-\varepsilon v)$, i.e., $\\{f(x-\varepsilon v),f(x+\varepsilon v)\\}=\\{0,1\\}$. Thus $x$ is a common boundary point of int$P$ and ext$P$. This finishes the proof of (E). ## 3 Proof of (F) Put $I_{i}=_{\rm def}[p_{i-1},p_{i}],1\leq i\leq n$, the edges of $P$, and for $i=1,2,\dots,n$ let $u_{i}$ be a unit vector perpendicular to aff$I_{i}$. Choose the orientation of $u_{i}$ in such a way that for each point $b\in{\rm relint}I_{i}$ and for all sufficiently small positive value of $\varepsilon,b+\varepsilon u_{i}\in{\rm ext}P$ and $b-\varepsilon u_{i}\in{\rm int}P$. Define $u_{i,i+1}=_{\rm def}u_{i}+u_{i+1},\,1\leq i\leq n$ (the indices are taken modulo $n$, i.e., $p_{n}=p_{0},u_{n+1}=u_{1},u_{n,n+1}=u_{n,1}=u_{n}+u_{1}$). ###### Lemma 3.1. If $\varepsilon$ is a sufficiently small positive number, then $p_{i}+\varepsilon u_{i,i+1}\in{\rm ext}P$, and $p_{i}-\varepsilon u_{i,i+1}\in{\rm int}P$ for $1\leq i\leq n$. Proof: The edges $I_{i},I_{i+1}$ lie in two rays (half-lines) $L_{i},L_{i+1}$ bounded by $p_{i}$, say $L_{i}=p_{i}+{\mathbb{R}}^{+}v_{i},L_{i+1}=p_{i}+{\mathbb{R}}^{+}v_{i+1}$, where $v_{i},v_{i+1}$ are suitable unit vectors orthogonal to $u_{i}$, $u_{i+1}$, respectively. 000000(a) (b) (c) Fig. 5 If $\varepsilon$ is a sufficiently small positive number $(0<\varepsilon<{\rm dist}(p_{i},P\setminus({\rm relint}(I_{i}\cup I_{i+1}))$, then $B^{2}(p_{i},\varepsilon)\setminus P=B^{2}(p_{i},\varepsilon)\setminus(L_{i}\cup L_{i+1})$. The union $L_{i}\cup L_{i+1}$ divides $B^{2}(p_{i},\varepsilon)$ into two open sectors, $B^{2}(p_{i},\varepsilon)\cap{\rm int}P$ and $B^{2}(p_{i},\varepsilon)\cap{\rm ext}P$. If $L_{i},L_{i+1}$ are collinear $(v_{i+1}=-v_{i})$, then each one of these two sectors is an open half disc. In this case $u_{i}=u_{i+1}$ (Fig. 5(a)), $u_{i,i+1}=2u_{i}=2u_{i+1}$, and the lemma holds trivially. If $u_{i},u_{i+1}$ are not collinear, then one of the sectors is larger than a half disc, and the other is smaller. In both cases we have $\langle u_{i},v_{i+1}\rangle=\langle u_{i+1},v_{i}\rangle=\sin\alpha\,,$ (3) where $\alpha$ is the central angle of the sector $B^{2}(p_{i},\varepsilon)\cap{\rm ext}P$ at $p_{i}\,(0\leq\alpha\leq 360^{o})$. If $\langle u_{i},v_{i+1}\rangle<0$, then $B^{2}(p_{i},\varepsilon)\cap{\rm ext}P$ is the larger sector (Fig. 5(b)), and if $\langle u_{i},v_{i+1}\rangle>0$, then $B^{2}(p_{i},\varepsilon)\cap{\rm int}P$ is the larger sector (Fig. 5(c)). Summing up the equalities $\begin{array}[]{lll}u_{i}&=&\langle u_{i},u_{i+1}\rangle u_{i+1}+\langle u_{i},v_{i+1}\rangle v_{i+1}\,,\\\ u_{i+1}&=&\langle u_{i+1},u_{i}\rangle u_{i}+\langle u_{i+1},v_{i}\rangle v_{i}\end{array}$ and using (3), we find $(1-\langle u_{i},u_{i+1}\rangle)\,(u_{i}+u_{i+1})=\sin\alpha\,(v_{i}+v_{i+1})$. If $u_{i}\not=u_{i+1}$, then $1-\langle u_{i},u_{i+1}\rangle>0$, and $u_{i,i+1}=u_{i}+u_{i+1}=\frac{\sin\alpha}{1-\langle u_{i},u_{i+1}\rangle}\cdot(v_{i}+v_{i+1})\,.$ Thus $u_{i,i+1}$ is a positive [resp., negative] multiple of $v_{i}+v_{i+1}$ when $\sin\alpha>0$ [resp., $\sin\alpha<0$]. In both cases, $u_{i,i+1}$ points towards ext$P$, and $-u_{i,i+1}$ towards int$P$. ###### Lemma 3.2. (“Push away from $P$”) 1. (a) Fix $i,\,1\leq i\leq n$, suppose $b\in{\rm relint}I_{i}$ and $u$ is a vector satisfying $\langle u,u_{i}\rangle>0$. Define $I^{0}=_{\rm def}[b,p_{i}],I^{\varepsilon}=_{\rm def}[b+\varepsilon u,p_{i}+\varepsilon u_{i,i+1}]$ ($u_{i},u_{i+1}$ and $u_{i,i+1}=u_{i}+u_{i+1}$ denote the same vectors as in the previous lemma). If $\varepsilon$ is a sufficiently small positive number, then $I^{\varepsilon}\subset{\rm ext}P$ and $I^{-\varepsilon}\subset{\rm int}P$. (The required smallness of $\varepsilon$ may depend on the choice of the point $b$ and of the vector $u$.) 2. (b) Fix $i,1\leq i\leq n$, and define $J^{0}=_{\rm def}[p_{i},p_{i+1}]=I_{i+1},J^{\varepsilon}=_{\rm def}[p_{i}+\varepsilon u_{i,i+1},p_{i+1}+\varepsilon u_{i+1,i+2}]$. If $\varepsilon$ is a sufficiently small positive number, then $J^{\varepsilon}\in{\rm ext}P$ and $J^{-\varepsilon}\in{\rm int}P$. Proof: 1. (a) First note that $I^{0}$ does not meet any edge of $P$ except $I_{i}$ and $I_{i+1}$. The same holds for $I^{\varepsilon}$, provided $\|\varepsilon\|<\min\left(\frac{1}{2},\frac{1}{\\|u\\|}\right)\cdot{\rm dist}\left(I^{0},P\setminus({\rm relint}(I_{i}\cup I_{i+1}))\right)\,.$ By Lemma 3.1, $p_{i}+\varepsilon u_{i,i+1}\in{\rm ext}P$ and $p_{i}-\varepsilon u_{i,i+1}\in{\rm int}P$, provided $\varepsilon$ is positive and sufficiently small. To complete the proof, it suffices to show that $I^{\varepsilon}\cap I_{i}=\emptyset$ and $I^{\varepsilon}\cap I_{i+1}=\emptyset$ (for sufficiently small $\|\varepsilon\|,\,\varepsilon\not=0$). As for $I_{i}:\langle u_{i},u\rangle>0$ (given) and $\langle u_{i},u_{i,i+1}\rangle=1+\langle u_{i},u_{i+1}\rangle>0$. Therefore, for any $\varepsilon\not=0$ both endpoints of $I^{\varepsilon}$ lie (strictly) on the same side of the line aff$I_{i}$, hence $I_{i}\cap I^{\varepsilon}=\emptyset$. As for $I_{i+1}$: If $I_{i+1}$ and $I_{i}$ lie on the same line $(u_{i}=u_{i+1})$, then the previous argument shows that $I_{i+1}\cap I^{\varepsilon}=\emptyset$ for all $\varepsilon\not=0$ as well. If $u_{i}\not=u_{i+1}$, consider first the case $\langle u_{i},v_{i+1}\rangle<0$. (Fig. 5(b)). For $\varepsilon>0,I^{\varepsilon}$ lies in the open half-plane $\\{x\in{\mathbb{R}}^{2}:\langle u_{i},x\rangle>\langle u_{i},p_{i}\rangle\\}$, whereas $I_{i+1}$ lies in the closed half-plane $\\{x\in{\mathbb{R}}^{2}:\langle u_{i},x\rangle\leq\langle u_{i},p_{i}\rangle\\}$. Therefore $I^{\varepsilon}\cap I_{i+1}=\emptyset$. For $\varepsilon<0$, $\langle u_{i+1},p_{i}+\varepsilon u_{i,i+1}\rangle=\langle u_{i+1},p_{i}\rangle+\varepsilon(1+\langle u_{i},u_{i+1}\rangle)<\langle u_{i+1},p_{i}\rangle\,.$ On the other hand, $\langle u_{i+1},b\rangle<\langle u_{i+1},p_{i}\rangle$ (for any point $b\in{\rm relint}I_{i}$, since $\langle u_{i+1},v_{i}\rangle<0$), and therefore $\langle u_{i+1},b+\varepsilon u\rangle<\langle u_{i+1},p_{i}\rangle$ for sufficiently small $\|\varepsilon\|,\varepsilon\not=0$. Thus both endpoints of $I^{\varepsilon}$ lie on the same open side of the line aff$I_{i+1}$, hence $I^{\varepsilon}\cap I_{i+1}=\emptyset$. In the case $\langle u_{i},v_{i+1}\rangle>0$ (Fig. 5(c) above), just repeat the previous argument with the roles of $\varepsilon>0$ and $\varepsilon<0$ interchanged. 2. (b) The proof is similar to that of (a). First, note that $J^{0}$ does not meet any edge of $P$ except $I_{i},I_{i+1}$ and $I_{i+2}$. The same holds for $J^{\varepsilon}$, provided $\|\varepsilon\|<\min\left(\frac{1}{2},\frac{1}{\\|u\\|}\right)\cdot{\rm dist}\left(J^{0},P\setminus{\rm relint}(I_{i}\cup I_{i+1}\cup I_{i+2})\right)\,.$ By Lemma 3.1, $p_{i}+\varepsilon u_{i,i+1},p_{i+1}+\varepsilon u_{i+1,i+2}\in{\rm ext}P$ and $p_{i}-\varepsilon u_{i,i+1},p_{i+1}-\varepsilon u_{i+1,i+2}\in{\rm int}P$, provided $\varepsilon$ is positive and sufficiently small. To complete the proof, it suffices to show that $J^{\varepsilon}\cap I_{i}=\emptyset,J^{\varepsilon}\cap I_{i+1}=\emptyset$ and $J^{\varepsilon}\cap I_{i+2}=\emptyset$ (for sufficiently small $\|\varepsilon\|,\varepsilon\not=0$). As for $I_{i+1}\\!:\langle u_{i+1},u_{i,i+1}\rangle=1+\langle u_{i+1},u_{i}\rangle>0$ and $\langle u_{i+1},u_{i+1,i+2}\rangle=1+\langle u_{i+1},u_{i+2}\rangle>0$. Therefore, for any $\varepsilon>0$, both endpoints of $J^{\varepsilon}$ lie on the same open side of the line aff$I_{i+1}$, hence $I_{i+1}\cap J^{\varepsilon}=\emptyset$. As for $I_{i}$: If $I_{i+1}$ and $I_{i}$ lie in the same line $(u_{i}=u_{i+1})$, then the previous argument shows that $I_{i}\cap J^{\varepsilon}=\emptyset$ for all $\varepsilon\not=0$ as well. If $u_{i}\not=u_{i+1}$, consider first the case $\langle u_{i},v_{i+1}\rangle<0$ (Fig. 5(b)). For $\varepsilon>0,J^{\varepsilon}$ lies in the open half-plane $\\{x\in{\mathbb{R}}^{2}:\langle u_{i+1},x\rangle>\langle u_{i+1},p_{i}\rangle\\}$, whereas $I_{i}$ lies in the closed half-plane $\\{x\in{\mathbb{R}}^{2}:\langle u_{i+1},x\rangle\leq\langle u_{i+1},p_{i}\rangle\\}$. Therefore, $J^{\varepsilon}\cap I_{i}=\emptyset$. For $\varepsilon<0$, we have $\langle u_{i},p_{i}+\varepsilon u_{i,i+1}\rangle=\langle u_{i},p_{i}\rangle+\varepsilon(1+\langle u_{i},u_{i+1}\rangle)<\langle u_{i},p_{i}\rangle$. On the other hand, $\langle u_{i},p_{i+1}\rangle<\langle u_{i},p_{i}\rangle$ (since $\langle u_{i},v_{i+1}\rangle<0$), and therefore $\langle u_{i},p_{i+1}+\varepsilon u_{i+1,i+2}\rangle<\langle u_{i},p_{i}\rangle$ for sufficiently small $\|\varepsilon\|$. Thus both endpoints of $J^{\varepsilon}$ lie on the same open side of the line aff$I_{i}$, hence $J^{\varepsilon}\cap I_{i}=\emptyset$. In the case $\langle u_{i},v_{i+1}\rangle>0$ (Fig. 5(c)), just repeat the previous argument with the roles of $\varepsilon>0$ and $\varepsilon<0$ interchanged. As for $I_{i+2}$: Since the roles of $I_{i}$ and $I_{i+2}$ are interchangeable, the statement proved above for $I_{i}$ applies to $I_{i+2}$ as well. ###### Definition 3.1. Let $p$ be a point in ${\mathbb{R}}^{2}\setminus P$ (= ${\rm ext}P\cup{\rm int}P$), and $I$ be an edge of $P$. We say that $p$ _sees_ $I$ if, for some point $a\in{\rm relint}\ I,[p,a]\cap P=\\{a\\}$. ###### Lemma 3.3. Assume $p\in{\mathbb{R}}^{2}\setminus P$. Then $p$ sees at least one edge of $P$. Proof: Assume, w.l.o.g., that $p\in{\rm ext}P$. Let $q$ be a point in int$P$. Let $U$ be a neighborhood of $q$ that lies entirely in int$P$. Choose a point $q^{\prime}\in U$ such that the line aff$(p,q^{\prime})$ does not meet any vertex of $P$. (This condition can be met by avoiding a finite number of lines through $p$.) Then the line segment $[p,q^{\prime}]$ must meet $P$. Let $a$ be the first point of $P$ on $[p,q^{\prime}]$ (starting from $p$). Then $a$ is a relative interior point of some edge $I$ of $P_{i}$, and $[p,a]\cap P=\\{a\\}$. ###### Definition 3.2. (poldiam($\cdot$)): For a set $S\subset{\mathbb{R}}^{2}$ and points $a,b\in S$, denote by $\pi_{S}(a,b)$ the smallest number of edges of a polygonal path that connects $a$ to $b$ within $S$ ($\pi_{S}(a,b)=_{\rm def}\infty$ if no such polygonal path exists). If $S$ is polygonally connected, then $\pi_{S}(\cdot,\cdot)$ is an integer valued metric on $S$. The _polygonal diameter_ of $S$ is defined as poldiam$(S)=_{\rm def}$ ${\rm sup}\\{\pi_{S}(a,b):a,b\in S\\}$. To prove (F) in Section 1 above, it suffices to show that poldiam(int$P$)$<\infty$ and poldiam(ext$P$)$<\infty$. The following theorem does it. ###### Theorem 3.1. (straightforward upper bound on poldiam(int$P$) and poldiam(ext$P$)) If $P$ is a simple closed $n$-gon $(n\geq 3)$ in ${\mathbb{R}}^{2}$, then we have that poldiam(int$P$) and poldiam(ext$P$) are both $\leq\lfloor\frac{n}{2}\rceil+3$. Proof: Assume that $a,b$ are two points in the same component (int$P$ or ext$P$) of ${\mathbb{R}}^{2}\setminus P$. By Lemma 3.2, $a\,[b]$ sees at least one edge $I^{\prime}\,[I^{\prime\prime}]$ of $P$ via ${\mathbb{R}}^{2}\setminus P$ (possibly $I^{\prime}=I^{\prime\prime}$). The set $P\setminus({\rm relint}(I^{\prime}\cup I^{\prime\prime}))$ consists of at most two simple polygonal paths $P^{\prime},P^{\prime\prime}$, the shorter one of which, say $P^{\prime}$, concatenated by $I^{\prime},I^{\prime\prime}$ in both of its endpoints is of the form $\langle J_{0},J_{1},\dots,J_{m},J_{m+1}\rangle$, where $m\leq\lfloor\frac{n-2}{2}\rfloor=\lfloor\frac{n}{2}\rfloor-1,J_{0},J_{1}\dots,J_{m+1}$ are edges of $P\,(\\{J_{0},J_{m+1}\\}=\\{I^{\prime},I^{\prime\prime}\\})$, $J_{i-1}$ and $J_{i}$ share a vertex $q_{i}$ for $i=1,\,2,\dots,m+1$, $a$ sees via ${\mathbb{R}}^{2}\setminus P$ a point $a^{\prime}\in{\rm relint}J_{0}$, and $b$ sees via ${\mathbb{R}}^{2}\setminus P$ a point $b^{\prime}\in{\rm relint}J_{m+1}$. Thus $\langle a,a^{\prime},q_{1},q_{2},\dots,q_{m},q_{m+1},b^{\prime},b\rangle$ is a polygonal path of $m+4\leq\lfloor\frac{n}{2}\rfloor-1+4=\lfloor\frac{n}{2}\rfloor+3$ edges that connects $a$ to $b$ and runs along $P$ except for $[a,a^{\prime}]$ and $[b^{\prime},b]$. By Lemma 3.2, this path can be pushed away from $P$ into ${\mathbb{R}}^{2}\setminus P$, thus producing a polygonal path of $m+4\leq\lfloor\frac{n}{2}\rfloor+3$ edges that connects $a$ to $b$ via ${\mathbb{R}}^{2}\setminus P$. ## 4 Tight upper bounds on poldiam(int$P$) and on poldiam(ext$P$) Theorem 3.1 gives a upper bound on poldiam(int$P$) [poldiam(ext$P$)] which is somewhat “naive”, but sufficient to prove (F) in Section 1 above. Here we “squeeze” the proof of Theorem 3.1 to obtain a tight result. ###### Theorem 4.1. (Main Theorem) Let $P$ be a simple closed $n$-gon in ${\mathbb{R}}^{2},n\geq 3$. Then 1. (a) the polygonal diameter of int$P$ is $\leq\lfloor\frac{n}{2}\rfloor$, and the polygonal diameter of ext$P$ is $\leq\lceil\frac{n}{2}\rceil$; 2. (b) for every $n\geq 3$, there is an $n$-gon $P_{n}$ for which _both_ bounds are attained. Proof of Theorem 4.1(a): First note that if $P$ is a convex polygon, then poldiam(int$P)=1\leq\lfloor\frac{n}{2}\rfloor$, and it can be easily checked that poldiam(ext$P)=2\leq\lceil\frac{n}{2}\rceil$. (If we consider the closures, however, we find that poldiam(cl int$P)=1$, whereas poldiam(cl ext$P)=3$ if $P$ has parallel edges, and equals $2$ otherwise.) This settles the case $n=3$ ($P_{3}$ is just a triangle). If $n=4$ and $P$ is not convex, then ext$P$ is the union of three convex sets (two open half-planes and a wedge), each two having a point in common, and therefore poldiam (ext$P)=2=\lceil\frac{n}{2}\rceil$. This settles the case $n=4$ for ext$P$. In view of the proof of Theorem 3.1 and the foregoing discussion, we can establish the bounds on poldiam(int$P$) and poldiam(ext$P$) as claimed in Theorem 4.1(a) by showing the following: ###### Theorem 4.2. Let $P$ be a closed simple $n$-gon in ${\mathbb{R}}^{2}$. 1. (i) If $n\geq 4$ and $a,b\in{\rm int}P$, then there are two vertices $a^{\prime},b^{\prime}$ of $P$ such that a sees $a^{\prime}$ via int$P$, $b$ sees $b^{\prime}$ via int$P$, and $a^{\prime},b^{\prime}$ are at most $\lfloor\frac{n}{2}\rfloor-2$ edges apart on $P$. (Recall that “$a$ sees $a^{\prime}$ via int$P$” means just: $]a,a^{\prime}[\subset{\rm int}P$.) 2. (ii) If $n\geq 5$ and $a,b\in{\rm ext}P$, then there are two vertices $a^{\prime},b^{\prime}$ of $P$ such that a sees $a^{\prime}$ via ext$P$, $b$ sees $b^{\prime}$ via ext$P$, and $a^{\prime},b^{\prime}$ are at most $\lceil\frac{n}{2}\rceil-2$ edges apart on $P$, _or:_ $\pi_{{\rm extP}}(a,b)\leq 3\left(\leq\lceil\frac{n}{2}\rceil\right.$ for $\left.n\geq 5\right)$. ###### Remark 4.1. The condition $n\geq 5$ in the first part of Theorem 4.2 (ii) cannot be relaxed to $n\geq 4$: Let $P_{4}=\langle p_{0},p_{1},p_{2},p_{3}\rangle$ be a convex quadrilateral, and let $a,b\in{\rm ext}P_{4}$, $a$ close to $[p_{0},p_{1}]$ and $b$ close to $[p_{2},p_{3}]$. Then $a$ and $b$ do not see a common vertex of $P_{4}$. ###### Lemma 4.1. Let $P$ be a simple closed polygon in ${\mathbb{R}}^{2}$. Let $\lceil b^{\prime},p\rceil$ be an edge of $P$, $a,b$ two points such that $a\in{\mathbb{R}}^{2}\setminus P$, $b\in]b^{\prime},p]$ $($=$[b^{\prime},p]\setminus\\{b^{\prime}\\})$ and a sees $b$ $($via ${\mathbb{R}}^{2}\setminus P$$)$. Then a sees $($via ${\mathbb{R}}^{2}\setminus P$$)$ a vertex of $P$ included in $[a,b^{\prime},b]\setminus[a,b]$. Proof: If a sees $b^{\prime}$ then we are done. Otherwise the polygon $P\setminus]b^{\prime},p[$ meets the set $[a,b,b^{\prime}]\setminus[b^{\prime},b]$. For $0\leq\lambda\leq 1$, define $b(\lambda)=_{\rm def}(1-\lambda)b+\lambda b^{\prime}$, and let $\lambda_{0}$ be the smallest value of $\lambda$, $0\leq\lambda\leq 1$, such that $[a,b(\lambda)]\cap(P\setminus]b^{\prime},p[)\not=\emptyset$ $(0<\lambda_{0}\leq 1;\lambda_{0}=1$ is possible). Let $c^{\prime}$ be the point of $[a,b(\lambda_{0})]\cap P$ nearest to $a$. Then $c^{\prime}$ is a vertex of $P$, $c^{\prime}\in[a,b,b^{\prime}]\setminus[a,b]$ and $a$ sees $c^{\prime}$. ###### Corollary 4.1. Let $P$ be a simple closed $n$-gon, $n\geq 3$, in ${\mathbb{R}}^{2}$. Every point $a\in{\mathbb{R}}^{2}\setminus P$ sees via ${\mathbb{R}}^{2}\setminus P$ at least two vertices of $P$. Proof: Let $R$ be a ray emanating from $a$ that meets $P$. By a slight rotation of $R$ around $a$ we may assume that $R$ does not meet any vertex of $P$, but still $R\cap P\not=\emptyset$. Let $b$ be the first point of $R$ that belongs to $P$ (starting from $a$). By assumption $b\in[b^{\prime},b^{\prime\prime}[$ for some edge $[b^{\prime},b^{\prime\prime}]$ of $P$. By Lemma 4.1, $a$ sees via ${\mathbb{R}}^{2}\setminus P$ a vertex $c^{\prime}$ $[c^{\prime\prime}]$ of $P$ included in $[a,b,b^{\prime}]\setminus[a,b]$ [included in $[a,b,b^{\prime\prime}]\setminus[a,b]$], and clearly $c^{\prime}\not=c^{\prime\prime}$. ###### Lemma 4.2. Let $P$ be a simple closed $n$-gon, $n\geq 4$, in ${\mathbb{R}}^{2}$, and let $a\in{\mathbb{R}}^{2}\setminus P$. If every ray emanating from a meets $P$, then a sees via ${\mathbb{R}}^{2}\setminus P$ two _non-adjacent_ vertices of $P$. ###### Remark 4.2. The condition that every ray emanating from $a$ meets $P$ is met by every point $a\in{\rm int}P$. Proof: By Corollary 4.1, $a$ sees a vertex $c$ of $P$ via ${\mathbb{R}}^{2}\setminus P$. Consider the ray $R=_{\rm def}\\{a+\lambda(a-c):\lambda\geq 0\\}$ that emanates from $a$ in a direction _opposite_ to $c$. By our assumption, $R$ meets $P$. Let $b$ be the first point of $R$ that belongs to $P$. If $b$ is a vertex of $P$, then a sees the two vertices $b,c$ via ${\mathbb{R}}^{2}\setminus P$. These vertices are _not adjacent_ , since $[c,b]\cap P=\\{c,b\\}$. Otherwise, if $b$ is not a vertex of $P$, then $b$ is a relative interior point of an edge $[b^{\prime},b^{\prime\prime}]$ of $P$ $(R\cap]b^{\prime},b^{\prime\prime}[=\\{b\\}$). By Lemma 4.1, $a$ sees via ${\mathbb{R}}^{2}\setminus P$ a vertex $c^{\prime}$ $[c^{\prime\prime}]$ of $P$ included in $[a,b,b^{\prime}]\setminus[a,b]$ [included in $[a,b,b^{\prime\prime}]\setminus[a,b]$]. Clearly, $c^{\prime}\not=c^{\prime\prime}$ and $c^{\prime},c^{\prime\prime}$ are non- adjacent in $P$ unless $c^{\prime}=b^{\prime}$ and $c^{\prime\prime}=b^{\prime\prime}$. In this case $a$ sees via ${\mathbb{R}}^{2}\setminus P$ both couples of vertices $\\{c,b^{\prime}\\}$ and $\\{c,b^{\prime\prime}\\}$. At least one of these couples is _non- adjacent_ in $P$, otherwise $P$ would be a triangle, contrary to the assumption that $n\geq 4$. Proof of Theorem 4.2: 1. (i) Suppose $P$ is a simple closed $n$-gon, $n\geq 4$, in ${\mathbb{R}}^{2}$. Define $S=_{\rm def}{\rm int}P$, and assume $a,b\in S$. If $n=4,5$, then cl$S$ (=$P\cup{\rm int}P$) is starshaped with respect to a vertex of $P$. (If $n=5$, then $S$ can be triangulated by two interior diagonals with a common vertex.) In this case $a$ and $b$ see via $S$ a common vertex $a^{\prime}$ of $P$. Define $b^{\prime}=_{\rm def}a^{\prime}$; we find that $a^{\prime},b^{\prime}$ are at zero edges apart on $P$. But $0\leq 0=\lfloor\frac{n}{2}\rfloor-2$ for $n=4,5$. Assume, therefore, that $n\geq 6$, and that $a$ and $b$ do not see a common vertex of $P$ via $S$. By Lemma 4.2, $a$ sees via $S$ two non-adjacent vertices $a^{\prime},a^{\prime\prime}$ of $P$. These vertices divide $P$ into two paths $P_{1},P_{2}$, each having $\leq n-2$ edges. Applying Lemma 4.2 again, we find that $b$ sees via $S$ two non-adjacent vertices $b^{\prime},b^{\prime\prime}$ of $P$ and $\\{a^{\prime},a^{\prime\prime}\\}\cap\\{b^{\prime},b^{\prime\prime}\\}=\emptyset$. If both $b^{\prime}$ and $b^{\prime\prime}$ are interior vertices of the same path, say $P_{1}$, then they divide $P_{1}$ into three parts. The middle part has at least two edges, and the two extreme parts together have at most $n-4$ edges. The shorter extreme part, with endpoints (say) $a^{\prime},b^{\prime}$, has at most $\lfloor\frac{n-4}{2}\rfloor=\lfloor\frac{n}{2}\rfloor-2$ edges. If, however, $b^{\prime}$ is an interior vertex of $P_{1}$ and $b^{\prime\prime}$ is an interior vertex of $P_{2}$, then they divide $P_{1}$ and $P_{2}$ into four polygonal paths, each one of which having one endpoint $b^{\prime}$ or $b^{\prime\prime}$. The shortest of these paths has at most $\lfloor\frac{n}{4}\rfloor$ edges. But $\lfloor\frac{n}{4}\rfloor\leq\lfloor\frac{n}{2}\rfloor-2$ for $n\geq 6$. 2. (ii) Assume $n\geq 5$, define $T={\rm ext}P$, and let $a,b\in T$. Then either 1. (A1) every ray emanating from $a$ meets $P$, _or_ 2. (A2) some ray emanating from $a$ misses $P$. Similarly, either 1. (B1) every ray emanating from $b$ meets $P$, _or_ 2. (B2) some ray emanating from $b$ misses $P$. If (A1) and (B1) hold, then both $a$ and $b$ see via $T$ two non-adjacent vertices of $P$ (Lemma 4.2). If $n\geq 6$, this implies that $a[b]$ sees a vertex $a^{\prime}$ $[b^{\prime}]$ of $P$ such that $a^{\prime},b^{\prime}$ are at most $\lfloor\frac{n-4}{2}\rfloor=\lfloor\frac{n}{2}\rfloor-2\leq\lceil\frac{n}{2}\rceil-2$ or $\lfloor\frac{n}{4}\rfloor\leq\lfloor\frac{n}{2}\rfloor-2\leq\lceil\frac{n}{2}\rceil-2$ edges apart on $P$, as in the proof of part (i) above. If $n=5$, then $a$ sees via $T$ a vertex $a^{\prime}$ of $P$, and $b$ sees via $T$ a vertex $b^{\prime}$ of $P$, where $a^{\prime}$ and $b^{\prime}$ are either equal or adjacent, i.e., $a^{\prime},b^{\prime}$ are at most one edge apart on $P$. But for $n=5$ one has $1\leq\lceil\frac{n}{2}\rceil-2$. If (A2) and (B2) hold, then, due to the compactness of $P$, we can find rays $R_{a}=\\{a+\lambda u:\lambda\geq 0\\}$ and $R_{b}=\\{b+\lambda v:\lambda\geq 0\\}$ that miss $P$, where the direction vectors $u$ and $v$ are _linearly independent_. When $\lambda$ is sufficiently large, the segment $[a+\lambda u,b+\lambda u]$ misses $P$. Therefore $\pi_{T}(a,b)\leq 3\left(\leq\lceil\frac{n}{2}\rceil\right.$ for $\left.n\geq 5\right)$ if $R_{a}\cap R_{b}=\emptyset$, and $\pi_{T}(a,b)=2<3\left(\leq\lceil\frac{n}{2}\rceil\mbox{ for }n\geq 5\right)$ if $R_{a}\cap R_{b}\not=\emptyset$. If (A1) and (B2) hold, then $a$ sees via $T$ two non-adjacent vertices $a^{\prime},a^{\prime\prime}$ of $P$, which divide $P$ into two paths $P_{1},P_{2}$ (with disjoint relative interiors) each one of which having $\leq n-2$ edges. The point $b$, however, sees two distinct vertices $b^{\prime},b^{\prime\prime}$ of $P$, which may be adjacent (Corollary 4.1). If $\\{a^{\prime},a^{\prime\prime}\\}\cap\\{b^{\prime},b^{\prime\prime}\\}\not=\emptyset$, then again $\pi_{T}(a,b)\leq 2<3\left(\leq\lceil\frac{n}{2}\rceil\right.$ for $\left.n\geq 5\right)$. If $\\{a^{\prime},a^{\prime\prime}\\}\cap\\{b^{\prime},b^{\prime\prime}\\}=\emptyset$, then $b^{\prime}$ and $b^{\prime\prime}$ are interior vertices of $P_{1}$ or $P_{2}$, or both. If $b^{\prime}$ and $b^{\prime\prime}$ belong to different paths, then (as in the proof of part (i) above) they divide $P_{1}$ and $P_{2}$ into four polygonal paths, each having one endpoint $b^{\prime}$ or $b^{\prime\prime}$. The shortest one of these paths has at most $\lfloor\frac{n}{4}\rfloor$ edges. But $\lfloor\frac{n}{4}\rfloor\leq\lceil\frac{n}{2}\rceil-2$ for $n\geq 5$. If both $b^{\prime}$ and $b^{\prime\prime}$ are interior vertices of the same path, say $P_{1}$, then (as in the proof of part (i) above) they divide $P_{1}$ into three parts. The two extreme parts together have at most $n-2-1=n-3$ edges. The shortest extreme part with endpoints (say) $a^{\prime},b^{\prime}$ has at most $\lfloor\frac{n-3}{2}\rfloor$ edges. But $\lfloor\frac{n-3}{2}\rfloor=\lfloor\frac{n-1}{2}\rfloor-1=\lceil\frac{n}{2}\rceil-2$ for all $n\in{\mathbb{N}}$. The same applies when (A2) and (B1) hold. This finishes the proof of Theorem 4.2. By this also the proof of Theorem 4.1(a) is finished. Proof of Theorem 4.1(b): We split our examples into two cases, namely even $n$ and odd $n$, $n\geq 3$. ###### Example 4.1. $n=2m$ (even), $m\geq 2$. Figure 6 shows the example for the case $m=3\,(n=6)$. Fig. 6: $m=3\,(n=6)$ Here we have $\pi_{{\rm int}P}(a,b)=m\,(=3)=\lfloor\frac{n}{2}\rfloor$ and $\pi_{{\rm ext}P}(c,d)=m\,(=3)=\lceil\frac{n}{2}\rceil$. One can extend the figure inward beyond vertex $\\#4$. ###### Example 4.2. $n=2m+1$ (odd), $m\geq 1$. Figure 7 shows the example for the case $m=3$ $(n=7)$ Fig. 7: $m=3\,(n=7)$ We have $\pi_{{\rm int}P}(a,b)=m\,(=3)=\lfloor\frac{n}{2}\rfloor$ and $\pi_{{\rm ext}P}(c,d)=m+1\,(=4)=\lceil\frac{n}{2}\rceil$. Again, one can extend the figure inward beyond vertex $\\#4$. ## References [1] Aleksandrov, P. S.: _Combinatorial Topology_ (in three volumes), translated from the Russian, 1947. Kombinatornaya Topologiya, by Harace Komm, Graylock Press, Rochester (1956). Reproduced by Dover Publications, New York (1998). * [2] Benson, R. V.: _Euclidean Geometry and Convexity_ , McGraw-Hill, New York (1966). * [3] Bertoglio, N., Chuaqui, R.: An elementary geometric nonstandard proof of the Jordan curve theorem, _Geom. Dedicata_ 51 (1994), 15-27. * [4] Courant, R., Robbins, H.: _What is Mathematics_? 4th ed., Oxford University Press, London (2003). * [5] Dostál, M., Tindell, R.: The Jordan curve theorem revisited, _Jahresber. Deutsch. Math.-Ver._ 80 (1978), 111-128. * [6] Hille, E.: _Analytic Function Theory_ , Vol. I, Ginn and Company, Boston (1959), (Chelsea, 1973). * [7] Kuratowski, K.: _Introduction to Set Theory and Topology_ , 2nd ed., Pergamon Press and Polish Scientific Publications, Warsaw (1972). * [8] Lawson, T.: _Topology: A Geometric Approach_ , Oxford Graduate Texts in Mathematics, Vol. 9, Oxford University Press, London (2003). * [9] Moise, E. E.: _Geometric Topology in Dimension 2 and 3_ , Springer, New York (1977). * [10] Perles, M. A., Martini, H., Kupitz, Y. S.: A Jordan-Brouwer separation theorem for polyhedral pseudomanifolds, _Discrete Comput. Geom._ 42 (2009), 277-304. * [11] Thomassen, C.: The Jordan-Schönflies theorem and the classification of surfaces, _Amer. Math. Monthly_ 99 (1992), 116-130.	arxiv-papers	2010-12-16T09:34:36	2024-09-04T02:49:15.716476	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Yaakov S. Kupitz, Horst Martini, Micha A. Perles", "submitter": "Margarita Spirova", "url": "https://arxiv.org/abs/1012.3541" }
1012.3589	11institutetext: Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK # Topics in statistical data analysis for high-energy physics G. Cowan ###### Abstract These lectures concern two topics that are becoming increasingly important in the analysis of High Energy Physics (HEP) data: Bayesian statistics and multivariate methods. In the Bayesian approach we extend the interpretation of probability to cover not only the frequency of repeatable outcomes but also to include a degree of belief. In this way we are able to associate probability with a hypothesis and thus to answer directly questions that cannot be addressed easily with traditional frequentist methods. In multivariate analysis we try to exploit as much information as possible from the characteristics that we measure for each event to distinguish between event types. In particular we will look at a method that has gained popularity in HEP in recent years: the boosted decision tree (BDT). ## 0.1 Introduction When a high-energy physics experiment enters the phase of data collection and analysis, the daily tasks of its postgraduate students are often centred not around the particle physics theories one is trying to test but rather on statistical methods. These methods are the tools needed to compare data with theory and quantify the extent to which one stands in agreement with the other. Of course one must understand the physical basis of the models being tested and so the theoretical emphasis in postgraduate education is no doubt well founded. But with the increasing cost of HEP experiments it has become important to exploit as much of the information as possible in the hard-won data, and to quantify as accurately as possible the inferences one draws when confronting the data with model predictions. Despite efforts to make the lectures self contained, some familiarity with basic ideas of statistical data analysis is assumed. Introductions to the subject can be found, for example, in the reviews of the Particle Data Group [1] or in the texts [2, 3, 4, 5, 6]. In these two lectures we will discuss two topics that are becoming increasingly important: Bayesian statistics and multivariate methods. In Section 0.2 we will review briefly the concept of probability and see how this is used differently in the frequentist and Bayesian approaches. Then in Section 0.2.2 we will discuss a simple example, the fitting of a straight line to a set of measurements, in both the frequentist and Bayesian approaches and compare different aspects of the two. This will include in Section 0.2.2 a brief description of Markov Chain Monte Carlo (MCMC), one of the most important tools in Bayesian computation. We generalize the treatment in Section 0.2.3 to include systematic errors. In Section 0.3 we take up the general problem of how to distinguish between two classes of events, say, signal and background, on the basis of a set of characteristics measured for each event. We first describe how to quantify the performance of a classification method in the framework of a statistical test. Although the Neyman–Pearson lemma indicates that this problem has an optimal solution using the likelihood ratio, this usually cannot be used in practice and one is forced to seek other methods. In Section 0.3.1 we look at a specific example of such a method, the boosted decision tree. Using this example we describe several issues common to many classification methods, such as overtraining. Finally, some conclusions are mentioned in Section 0.4. ## 0.2 Bayesian statistical methods for high-energy physics In this section we look at the basic ideas of Bayesian statistics and explore how these can be applied in particle physics. We will contrast these with the corresponding notions in frequentist statistics, and to make the treatment largely self contained, the main ideas of the frequentist approach will be summarized as well. ### 0.2.1 The role of probability in data analysis We begin by defining probability with the axioms written down by Kolmogorov [7] using the language of set theory. Consider a set $S$ containing subsets $A,B,\ldots$. We define the probability $P$ as a real-valued function with the following properties: 1. 1. For every subset $A$ in $S$, $P(A)\geq 0$; 2. 2. For disjoint subsets (i.e., $A\cap B=\emptyset$), $P(A\cup B)=P(A)+P(B)$; 3. 3. $P(S)=1$. In addition, we define the conditional probability $P(A\|B)$ (read $P$ of $A$ given $B$) as $P(A\|B)=\frac{P(A\cap B)}{P(B)}\;.$ (1) From this definition and using the fact that $A\cap B$ and $B\cap A$ are the same, we obtain Bayes’ theorem, $P(A\|B)=\frac{P(B\|A)P(A)}{P(B)}\;.$ (2) From the three axioms of probability and the definition of conditional probability, we can derive the law of total probability, $P(B)=\sum_{i}P(B\|A_{i})P(A_{i})\;,$ (3) for any subset $B$ and for disjoint $A_{i}$ with $\cup_{i}A_{i}=S$. This can be combined with Bayes’ theorem (2) to give $P(A\|B)=\frac{P(B\|A)P(A)}{\sum_{i}P(B\|A_{i})P(A_{i})}\;,$ (4) where the subset $A$ could, for example, be one of the $A_{i}$. The most commonly used interpretation of the subsets of the sample space are outcomes of a repeatable experiment. The probability $P(A)$ is assigned a value equal to the limiting frequency of occurrence of $A$. This interpretation forms the basis of frequentist statistics. The subsets of the sample space can also be interpreted as hypotheses, i.e., statements that are either true or false, such as “The mass of the $W$ boson lies between 80.3 and 80.5 GeV.” In the frequency interpretation, such statements are either always or never true, i.e., the corresponding probabilities would be 0 or 1. Using subjective probability, however, $P(A)$ is interpreted as the degree of belief that the hypothesis $A$ is true. Subjective probability is used in Bayesian (as opposed to frequentist) statistics. Bayes’ theorem can be written $P(\hbox{theory}\|\hbox{data})\propto P(\hbox{data}\|\hbox{theory})P(\hbox{theory})\;,$ (5) where ‘theory’ represents some hypothesis and ‘data’ is the outcome of the experiment. Here $P(\hbox{theory})$ is the prior probability for the theory, which reflects the experimenter’s degree of belief before carrying out the measurement, and $P(\hbox{data}\|\hbox{theory})$ is the probability to have gotten the data actually obtained, given the theory, which is also called the likelihood. Bayesian statistics provides no fundamental rule for obtaining the prior probability; this is necessarily subjective and may depend on previous measurements, theoretical prejudices, Once this has been specified, however, Eq. (5) tells how the probability for the theory must be modified in the light of the new data to give the posterior probability, $P(\hbox{theory}\|\hbox{data})$. As Eq. (5) is stated as a proportionality, the probability must be normalized by summing (or integrating) over all possible hypotheses. The difficult and subjective nature of encoding personal knowledge into priors has led to what is called objective Bayesian statistics, where prior probabilities are based not on an actual degree of belief but rather derived from formal rules. These give, for example, priors which are invariant under a transformation of parameters or which result in a maximum gain in information for a given set of measurements. For an extensive review see, for example, Ref. [8]. ### 0.2.2 An example: fitting a straight line In Section 0.2.2 we look at the example of a simple fit in both the frequentist and Bayesian frameworks. Suppose we have independent data values $y_{i}$, $i=1,...,n$, that are each made at a given value $x_{i}$ of a control variable $x$. Suppose we model the $y_{i}$ as following a Gaussian distribution with given standard deviations $\sigma_{i}$ and mean values $\mu_{i}$ given by a function that we evaluate at the corresponding $x_{i}$, $\mu(x;\theta_{0},\theta_{1})=\theta_{0}+\theta_{1}x\;.$ (6) We would like to determine values of the parameters $\theta_{0}$ and $\theta_{1}$ such that the model best describes the data. The ingredients of the analysis are illustrated in Fig. 1(a). (a)(b) Figure 1: (a) Illustration of fitting a straight line to data (see text). (b) The $\chi^{2}$ as a function of the parameter $\theta_{0}$, illustrating the method to determine the estimator $\hat{\theta}_{0}$ and its standard deviation $\sigma_{\hat{\theta}_{0}}$. Now suppose the real goal of the analysis is only to estimate the parameter $\theta_{0}$. The slope parameter $\theta_{1}$ must also be included in the model to obtain a good description of the data, but we are not interested in its value as such. We refer to $\theta_{0}$ as the parameter of interest, and $\theta_{1}$ as a nuisance parameter. In the following sections we treat this problem using both the frequentist and Bayesian approaches. #### The frequentist approach Our model states that the measurements are Gaussian distributed, i.e., the probability density function (pdf) for the $i$th measurement $y_{i}$ is $f(y_{i};\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})=\frac{1}{\sqrt{2\pi}\sigma_{i}}e^{-(y_{i}-\mu(x_{i};\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}))^{2}/2\sigma_{i}^{2}}\;,$ (7) where $\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}=(\theta_{0},\theta_{1})$. The likelihood function is the joint pdf for all of the $y_{i}$, evaluated with the $y_{i}$ obtained and regarded as a function of the parameters. Since we are assuming that the measurements are independent, the likelihood function is in this case given by the product $L(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})=\prod_{i=1}^{n}f(y_{i};\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})=\prod_{i=1}^{n}\frac{1}{\sqrt{2\pi}\sigma_{i}}e^{-(y_{i}-\mu(x_{i};\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}))^{2}/2\sigma_{i}^{2}}\;.$ (8) In the frequentist approach we construct estimators $\hat{\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}}$ for the parameters $\textstyle\bf\theta$, usually by finding the values that maximize the likelihood function. (We will write estimators for parameters with hats.) In this case one can see from (8) that this is equivalent to minimizing the quantity $\chi^{2}(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})=\sum_{i=1}^{n}\frac{(y_{i}-\mu(x_{i};\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}))^{2}}{\sigma_{i}^{2}}=-2\ln L(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})+C\;,$ (9) where $C$ represents terms that do not depend on the parameters. Thus for the case of independent Gaussian measurements, the maximum likelihood (ML) estimators for the parameters coincide with those of the method of least squares (LS). Suppose first that the slope parameter $\theta_{1}$ is known exactly, and so it is not adjusted to maximize the likelihood (or minimize the $\chi^{2}$) but rather held fixed. The quantity $\chi^{2}$ versus the single adjustable parameter $\theta_{0}$ would be as shown in Fig. 1(b), where the minimum indicates the value of the estimator $\hat{\theta}_{0}$. Methods for obtaining the standard deviations of estimators — the statistical errors of our measured values — are described in many references such as [1, 2, 3, 4, 5, 6]. Here in the case of a single fitted parameter the rule boils down to moving the parameter away from the estimate until $\chi^{2}$ increases by one unit (i.e., $\ln L$ decreases from its maximum by $1/2$) as indicated in the figure. It may be, however, that we do not know the value of the slope parameter $\theta_{1}$, and so even though we do not care about its value in the final result, we are required to treat it as an adjustable parameter in the fit. Minimizing $\chi^{2}(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})$ results in the estimators $\hat{\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}}=(\hat{\theta}_{0},\hat{\theta}_{1})$, as indicated schematically in Fig. 2(a). Now the recipe to obtain the statistical errors, however, is not simply a matter of moving the parameter away from its estimated value until the $\chi^{2}$ goes up by one unit. Here the standard deviations must be found from the tangent lines (or in higher- dimensional problems, the tangent hyperplanes) to the contour defined by $\chi^{2}(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})=\chi^{2}_{\rm min}+1$, as shown in the figure. (a)(b) Figure 2: Contour of $\chi^{2}(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})=\chi^{2}_{\rm min}+1$ centred about the estimates $(\hat{\theta}_{0},\hat{\theta}_{1})$ (a) with no prior measurement of $\theta_{1}$ and (b) when a prior measurement of $\theta_{1}$ is included. The tilt of the contour in Fig. 2(a) reflects the correlation between the estimators $\hat{\theta}_{0}$ and $\hat{\theta}_{1}$. A useful estimate for the inverse of the matrix of covariances $V_{ij}=\mbox{cov}[V_{i},V_{j}]$ can be found from the second derivative of the log-likelihood evaluated at its maximum, $\widehat{V}^{-1}_{ij}=-\left.\frac{\partial^{2}\ln L}{\partial\theta_{i}\partial\theta_{j}}\right\|_{{\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}}=\hat{\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}}}\;.$ (10) More information on how to extract the full covariance matrix from the contour can be found, for example, in Refs. [1, 2, 3, 4, 5, 6]. The point to note here is that the correlation between the estimators for the parameter of interest and the nuisance parameter has the result of inflating the standard deviations of both. That is, if $\theta_{1}$ were known exactly, then the distance one would have to move $\theta_{0}$ away from its estimated value to make the $\chi^{2}$ increase by one unit would be less, as one can see from the figure. So although we can improve the ability of a model to describe the data by including additional nuisance parameters, this comes at the price of increasing the statistical errors. This is an important theme which we will encounter often in data analysis. Now consider the case where we have a prior measurement of $\theta_{1}$. For example, we could have a measurement $t_{1}$ which we model as following a Gaussian distribution centred about $\theta_{1}$ and having a given standard deviation $\sigma_{t_{1}}$. If this measurement is independent of the other $y_{i}$ values, then the full likelihood function is obtained simply by multiplying the original one by a Gaussian, and so when we find the new $\chi^{2}$ from $-2\ln L$ there is an additional term, namely, $\chi^{2}(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})=\sum_{i=1}^{n}\frac{(y_{i}-\mu(x_{i};\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}))^{2}}{\sigma_{i}^{2}}+\frac{(\theta_{1}-t_{1})^{2}}{\sigma_{t_{1}}^{2}}\;.$ (11) As shown in Fig. 2(b), the new (solid) contour of $\chi^{2}=\chi^{2}_{\rm min}+1$ is compressed relative to the old (dashed) one in the $\theta_{1}$ direction, and this compression has the effect of decreasing the error in $\theta_{0}$ as well. The lesson is: by better constraining nuisance parameters, one improves the statistical accuracy of the parameters of interest. #### The Bayesian approach To treat the example above in the Bayesian framework, we write Bayes’ theorem (2) as $p(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}})=\frac{L(\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})\pi(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})}{\int L(\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})\pi(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})\,d\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}}\;.$ (12) Here $\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}=(\theta_{0},\theta_{1})$ symbolizes the hypothesis whose probability we want to determine. The likelihood $L(\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})$ is the probability to obtain the data $\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}}=(y_{1},\ldots,y_{n})$ given the hypothesis, and the prior probability $\pi(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}})$ represents our degree of belief about the parameters before seeing the outcome of the experiment. The posterior probability $p(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})$ encapsulates all of our knowledge about $\textstyle\bf\theta$ when the data $\textstyle\bf y$ is combined with our prior beliefs. The denominator in (12) serves to normalize the posterior pdf to unit area. The likelihood $L(\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})$ is the same as the $L(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})$ that we used in the frequentist approach above. The slightly different notation here simply emphasizes its role as the conditional probability for the data given the parameter. To proceed we need to write down a prior probability density $\pi(\theta_{0},\theta_{1})$. This phase of a Bayesian analysis, sometimes called the elicitation of expert opinion, is in many ways the most problematic, as there are no universally accepted rules to follow. Here we will explore some of the important issues that come up. In general, prior knowledge about one parameter might affect knowledge about the other, and if so this must be built into $\pi(\theta_{0},\theta_{1})$. Often, however, one may regard the prior knowledge about the parameters as independent, in which case the density factorizes as $\pi(\theta_{0},\theta_{1})=\pi_{0}(\theta_{0})\pi_{1}(\theta_{1})\;.$ (13) For purposes of the present example we will assume that this holds. For the parameter of interest $\theta_{0}$, it may be that we have essentially no prior information, so the density $\pi_{0}(\theta_{0})$ should be very broad. Often one takes the limiting case of a broad distribution simply to be a constant, i.e., $\pi_{0}(\theta_{0})=\mbox{const.}\;.$ (14) Now one apparent problem with Eq. (14) is that it is not normalizable to unit area, and so does not appear to be a valid probability density. It is said to be an improper prior. The prior always appears in Bayes’ theorem multiplied by the likelihood, however, and as long as this falls off quickly enough as a function of the parameters, then the resulting posterior probability density can be normalized to unit area. A further problem with uniform priors is that if the prior pdf is flat in $\textstyle\bf\theta$, then it is not flat for a nonlinear function of $\textstyle\bf\theta$, and so a different parametrization of the problem would lead in general to a non-equivalent posterior pdf. For the special case of a constant prior, one can see from Bayes’ theorem (12) that the posterior is proportional to the likelihood, and therefore the mode (peak position) of the posterior is equal to the ML estimator. The posterior mode, however, will change in general upon a transformation of parameter. A summary statistic other than the mode may be used as the Bayesian estimator, such as the median, which is invariant under a monotonic parameter transformation. But this will not in general coincide with the ML estimator. For the prior $\pi_{1}(\theta_{1})$, let us assume that our prior knowledge about this parameter includes the earlier measurement $t_{1}$, which we modelled as a Gaussian distributed variable centred about $\theta_{1}$ with standard deviation $\sigma_{t_{1}}$. If we had taken, even prior to that measurement, a constant prior for $\theta_{1}$, then the “intermediate-state” prior that we have before looking at the $y_{i}$ is simply this flat prior times the Gaussian likelihood, i.e., a Gaussian prior in $\theta_{1}$: $\pi_{1}(\theta_{1})=\frac{1}{\sqrt{2\pi}\sigma_{t_{1}}}e^{-(\theta_{1}-t_{1})^{2}/2\sigma_{t_{1}}^{2}}\;.$ (15) Putting all of these ingredients into Bayes’ theorem gives $p(\theta_{0},\theta_{1}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}})\propto\prod_{i=1}^{n}\frac{1}{\sqrt{2\pi}\sigma_{i}}e^{-(y_{i}-\mu(x_{i};\theta_{0},\theta_{1}))^{2}/2\sigma_{i}^{2}}\,\pi_{0}\,\frac{1}{\sqrt{2\pi}\sigma_{t_{1}}}e^{-(\theta_{1}-t_{1})^{2}/2\sigma_{t_{1}}^{2}}\;,$ (16) where $\pi_{0}$ represents the constant prior in $\theta_{0}$ and the equation has been written as a proportionality with the understanding that the final posterior pdf should be normalized to unit area. What Bayes’ theorem gives us is the full joint pdf $p(\theta_{0},\theta_{1}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}})$ for both the parameter of interest $\theta_{0}$ as well as the nuisance parameter $\theta_{1}$. To find the pdf for the parameter of interest only, we simply integrate (marginalize) the joint pdf, i.e., $p(\theta_{0}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}})=\int p(\theta_{0},\theta_{1}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}})\,d\theta_{1}\;.$ (17) In this example, it turns out that we can do the integral in closed form. We find a Gaussian posterior, $p(\theta_{0}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}})=\frac{1}{\sqrt{2\pi}\sigma_{\theta_{0}}}e^{-(\theta_{0}-\hat{\theta}_{0})^{2}/2\sigma_{\theta_{0}}^{2}}\;,$ (18) where $\hat{\theta}_{0}$ is in fact the same as the ML (or LS) estimator found above with the frequentist approach, and $\sigma_{\theta_{0}}$ is the same as the standard deviation of that estimator $\sigma_{\hat{\theta}_{0}}$. So we find something that looks just like the frequentist answer, although here the interpretation of the result is different. The posterior pdf $p(\theta_{0}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}})$ gives our degree of belief about the location of the parameter in the light of the data. We will see below how the Bayesian approach can, however, lead to results that differ both in interpretation as well as in numerical value from what would be obtained in a frequentist calculation. First, however, we need to pause for a short digression on Bayesian computation. #### Bayesian computation and MCMC In most real Bayesian calculations, the marginalization integrals cannot be carried out in closed form, and if the number of nuisance parameters is too large then they can also be difficult to compute with standard Monte Carlo methods. However, Markov Chain Monte Carlo (MCMC) has become the most important tool for computing integrals of this type and has revolutionized Bayesian computation. In-depth treatments of MCMC can be found, for example, in the texts by Robert and Casella [9], Liu [10], and the review by Neal [11]. The basic idea behind using MCMC to marginalize the joint pdf $p(\theta_{0},\theta_{1}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}})$ is to sample points $\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}=(\theta_{0},\theta_{0})$ according to the posterior pdf but then only to look at the distribution of the component of interest, $\theta_{0}$. A simple and widely applicable MCMC method is the Metropolis-Hastings algorithm, which allows one to generate multidimensional points $\textstyle\bf\theta$ distributed according to a target pdf that is proportional to a given function $p(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})$, which here will represent our posterior pdf. It is not necessary to have $p(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})$ normalized to unit area, which is useful in Bayesian statistics, as posterior probability densities are often determined only up to an unknown normalization constant, as is the case in our example. To generate points that follow $p(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})$, one first needs a proposal pdf $q(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}};\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{0})$, which can be (almost) any pdf from which independent random values $\textstyle\bf\theta$ can be generated, and which contains as a parameter another point in the same space $\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{0}$. For example, a multivariate Gaussian centred about $\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{0}$ can be used. Beginning at an arbitrary starting point $\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{0}$, the Hastings algorithm iterates the following steps: 1. 1. Generate a value $\textstyle\bf\theta$ using the proposal density $q(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}};\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{0})$; 2. 2. Form the Hastings test ratio, $\alpha=\min\left[1,\frac{p(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})q(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{0};\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})}{p(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{0})q(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}};\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{0})}\right]$; 3. 3. Generate a value $u$ uniformly distributed in $[0,1]$; 4. 4. If $u\leq\alpha$, take $\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{1}=\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}$. Otherwise, repeat the old point, i.e., $\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{1}=\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{0}$. If one takes the proposal density to be symmetric in $\textstyle\bf\theta$ and $\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{0}$, then this is the Metropolis–Hastings algorithm, and the test ratio becomes $\alpha=\min[1,p(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})/p(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{0})]$. That is, if the proposed $\textstyle\bf\theta$ is at a value of probability higher than $\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}_{0}$, the step is taken. If the proposed step is rejected, hop in place. Methods for assessing and optimizing the performance of the algorithm are discussed, for example, in Refs. [9, 10, 11]. One can, for example, examine the autocorrelation as a function of the lag $k$, i.e., the correlation of a sampled point with one $k$ steps removed. This should decrease as quickly as possible for increasing $k$. Generally one chooses the proposal density so as to optimize some quality measure such as the autocorrelation. For certain problems it has been shown that one achieves optimal performance when the acceptance fraction, that is, the fraction of points with $u\leq\alpha$, is around 40%. This can be adjusted by varying the width of the proposal density. For example, one can use for the proposal pdf a multivariate Gaussian with the same covariance matrix as that of the target pdf, but scaled by a constant. For our example above, MCMC was used to generate points according to the posterior pdf $p(\theta_{0},\theta_{1})$ by using a Gaussian proposal density. The result is shown in Fig. 3. (a)(b)(c) Figure 3: MCMC marginalization of the posterior pdf $p(\theta_{0},\theta_{1}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}})$: (a) scatter-plot of points in $(\theta_{0},\theta_{1})$ plane and the marginal distribution of (b) the parameter of interest $\theta_{0}$ and (c) the nuisance parameter $\theta_{1}$. From the $(\theta_{0},\theta_{1})$ points in the scatter plot in Fig. 3(a) we simply look at the distribution of the parameter of interest, $\theta_{0}$ [Fig. 3(b)]. The standard deviation of this distribution is what we would report as the statistical error in our measurement of $\theta_{0}$. The distribution of the nuisance parameter $\theta_{1}$ from Fig. 3(c) is not directly needed, although it may be of interest in some other context where that parameter is deemed interesting. In fact one can go beyond simply summarizing the width of the distributions with the a statistic such as the standard deviation. The full form of the posterior distribution of $\theta_{0}$ contains useful information about where the parameter’s true value is likely to be. In this example the distributions will in fact turn out to be Gaussian, but in a more complex analysis there could be non-Gaussian tails and this information can be relevant in drawing conclusions from the result. #### Sensitivity analysis The posterior distribution of $\theta_{0}$ obtained above encapsulates all of the analyst’s knowledge about the parameter in the light of the data, given that the prior beliefs were reflected by the density $\pi(\theta_{0},\theta_{1})$. A different analyst with different prior beliefs would in general obtain a different posterior pdf. We would like the result of a Bayesian analysis to be of value to the broader scientific community, not only to those that share the prior beliefs of the analyst. And therefore it is important in a Bayesian analysis to show by how much the posterior probabilities would change upon some reasonable variation in the prior. This is sometimes called the sensitivity analysis and is an important part of any Bayesian calculation. In the example above, we can imagine a situation where there was no prior measurement $t_{1}$ of the parameter $\theta_{1}$, but rather a theorist had told us that, based on considerations of symmetry, consistency, aesthetics, etc., $\theta_{1}$ was “almost certainly” positive, and had a magnitude “probably less than 0.1 or so”. When pressed to be precise, the theorist sketches a curve roughly resembling an exponential with a mean of 0.1. So we can express this prior as $\pi_{1}(\theta_{1})=\frac{1}{\tau}e^{-\theta_{1}/\tau}\quad(\theta_{1}\geq 0)\;,$ (19) with $\tau\approx 0.1$. We can substitute this prior into Bayes’ theorem (16) to obtain the joint pdf for $\theta_{0}$ and $\theta_{1}$, and then marginalize to find the pdf for $\theta_{0}$. Doing this numerically with MCMC results in the posterior distributions shown in Fig. 4(a). (a)(b) Figure 4: Posterior probability densities for the parameter $\theta_{0}$ obtained using (a) an exponential prior for $\theta_{0}$ of different widths and (b) several different functional forms for the prior. Now the theorist who proposed this prior for $\theta_{1}$ may feel reluctant to be pinned down, and so it is important to recall (and to reassure the theorist about) the “if-then” nature of a Bayesian analysis. One does not have to be absolutely certain about the prior in Eq. (19). Rather, Bayes’ theorem simply says that if one were to have these prior beliefs, then we obtain certain posterior beliefs in the light of the data. One simple way to vary the prior here is to try different values of the mean $\tau$, as shown in Fig. 4(a). We see here the same basic feature as shown already in the frequentist analysis, namely, that when one increases the precision about the nuisance parameter, $\theta_{1}$, then the knowledge about the parameter of interest, $\theta_{0}$, is improved. Alternatively (or in addition) we may try different functional forms for the prior, as shown in Fig. 4(b). In this case using a uniform distribution for $\pi_{1}(\theta_{1})$ with $0\leq\theta_{1}\leq 0.5$ or Gaussian with $\sigma=0.1$ truncated for $\theta_{1}<0$ both give results similar to the exponential with a mean of $0.1$. So one concludes that the result is relatively insensitive to the detailed nature of the tails of $\pi_{1}(\theta_{1})$. ### 0.2.3 A fit with systematic errors We can now generalize the example of Section 0.2.2 to explore some further aspects of a Bayesian analysis. Let us suppose that we are given a set of $n$ measurements as above, but now in addition to the statistical errors we also are given systematic errors. That is, we are given $y_{i}\pm\sigma_{i}^{\rm stat}\pm\sigma_{i}^{\rm sys}$ for $i=1,\ldots,n$ where the measurements as before are each carried out for a specified value of a control variable $x$. More generally, instead of having $y_{i}\pm\sigma_{i}^{\rm stat}\pm\sigma_{i}^{\rm sys}$ it may be that the set of measurements comes with an $n\times n$ covariance matrix $V^{\rm stat}$ corresponding to the statistical errors and another matrix $V^{\rm sys}$ for the systematic ones. Here the square roots of the diagonal elements give the errors for each measurement, and the off-diagonal elements provide information on how they are correlated. As before we assume some functional form $\mu(x;\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})$ for the expectation values of the $y_{i}$. This could be the linear model of Eq. (6) or something more general, but in any case it depends on a vector of unknown parameters $\textstyle\bf\theta$. In this example, however, we will allow that the model is not perfect, but rather could have a systematic bias. That is, we write that the true expectation value of the $i$th measurement can be written $E[y_{i}]=\mu(x_{i};\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})+b_{i}\;,$ (20) where $b_{i}$ represents the bias. The $b_{i}$ can be viewed as the systematic errors of the model, present even when the parameters $\textstyle\bf\theta$ are adjusted to give the best description of the data. We do not know the values of the $b_{i}$. If we did, we would account for them in the model and they would no longer be biases. We do not in fact know that their values are nonzero, but we are allowing for the possibility that they could be. The reported systematic errors are intended as a quantitative measure of how large we expect the biases to be. As before, the goal is to make inferences about the parameters $\textstyle\bf\theta$; some of these may be of direct interest and others may be nuisance parameters. In Section 0.2.3 we will try to do this using the frequentist approach, and in Section 0.2.3 we will use the Bayesian method. #### A frequentist fit with systematic errors If we adopt the frequentist approach, we need to write down a likelihood function such as Eq. (8), but here we know in advance that the model $\mu(x;\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})$ is not expected to be fully accurate. Furthermore it is not clear how to insert the systematic errors. Often, perhaps without a clear justification, one simply adds the statistical and systematic errors in quadrature, or in the case where one has the covariance matrices $V^{\rm stat}$ and $V^{\rm sys}$, they are summed to give a sort of ‘full’ covariance matrix: $V_{ij}=V_{ij}^{\rm stat}+V_{ij}^{\rm sys}\;.$ (21) One might then use this in a multivariate Gaussian likelihood function, or equivalently it could be used to construct the $\chi^{2}$, $\chi^{2}(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})=(\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}}-\mathchoice{\mbox{\boldmath$\displaystyle\bf\mu$}}{\mbox{\boldmath$\textstyle\bf\mu$}}{\mbox{\boldmath$\scriptstyle\bf\mu$}}{\mbox{\boldmath$\scriptscriptstyle\bf\mu$}}(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}))^{T}V^{-1}(\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}}-\mathchoice{\mbox{\boldmath$\displaystyle\bf\mu$}}{\mbox{\boldmath$\textstyle\bf\mu$}}{\mbox{\boldmath$\scriptstyle\bf\mu$}}{\mbox{\boldmath$\scriptscriptstyle\bf\mu$}}(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}))\;,$ (22) which is then minimized to find the LS estimators for $\textstyle\bf\theta$. In Eq. (22) the vector $\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}}=(y_{1},\ldots,y_{n})$ should be understood as a column vector, $\mathchoice{\mbox{\boldmath$\displaystyle\bf\mu$}}{\mbox{\boldmath$\textstyle\bf\mu$}}{\mbox{\boldmath$\scriptstyle\bf\mu$}}{\mbox{\boldmath$\scriptscriptstyle\bf\mu$}}(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})=(\mu(x_{1};\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}),\ldots,\mu(x_{n};\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}))$ is the corresponding vector of model values, and the superscript $T$ represents the transpose (row) vector. Minimizing this $\chi^{2}$ gives the generalized LS estimators $\hat{\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}}$, and the usual procedures can be applied to find their covariances, which now in some sense include the systematics. But in what sense is there any formal justification for adding the covariance matrices in Eq. (21)? Next we will treat this problem in the Bayesian framework and see that there is indeed some reason behind this recipe, but with limitations, and further we will see how to get around these limitations. #### The equivalent Bayesian fit In the corresponding Bayesian analysis, one treats the statistical errors as given by $V^{\rm stat}$ as reflecting the distribution of the data $\textstyle\bf y$ in the likelihood. The systematic errors, through $V^{\rm sys}$, reflect the width of the prior probabilities for the bias parameters $b_{i}$. That is, we take $\displaystyle L(\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}},\mathchoice{\mbox{\boldmath$\displaystyle\bf b$}}{\mbox{\boldmath$\textstyle\bf b$}}{\mbox{\boldmath$\scriptstyle\bf b$}}{\mbox{\boldmath$\scriptscriptstyle\bf b$}})$ $\displaystyle\propto$ $\displaystyle\exp\left[-\matrix{}{1}{2}(\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}}-\mathchoice{\mbox{\boldmath$\displaystyle\bf\mu$}}{\mbox{\boldmath$\textstyle\bf\mu$}}{\mbox{\boldmath$\scriptstyle\bf\mu$}}{\mbox{\boldmath$\scriptscriptstyle\bf\mu$}}(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})-\mathchoice{\mbox{\boldmath$\displaystyle\bf b$}}{\mbox{\boldmath$\textstyle\bf b$}}{\mbox{\boldmath$\scriptstyle\bf b$}}{\mbox{\boldmath$\scriptscriptstyle\bf b$}})^{T}V_{\rm stat}^{-1}(\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}}-\mathchoice{\mbox{\boldmath$\displaystyle\bf\mu$}}{\mbox{\boldmath$\textstyle\bf\mu$}}{\mbox{\boldmath$\scriptstyle\bf\mu$}}{\mbox{\boldmath$\scriptscriptstyle\bf\mu$}}(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})-\mathchoice{\mbox{\boldmath$\displaystyle\bf b$}}{\mbox{\boldmath$\textstyle\bf b$}}{\mbox{\boldmath$\scriptstyle\bf b$}}{\mbox{\boldmath$\scriptscriptstyle\bf b$}})\right]\;,$ (23) $\displaystyle\pi_{b}(\mathchoice{\mbox{\boldmath$\displaystyle\bf b$}}{\mbox{\boldmath$\textstyle\bf b$}}{\mbox{\boldmath$\scriptstyle\bf b$}}{\mbox{\boldmath$\scriptscriptstyle\bf b$}})$ $\displaystyle\propto$ $\displaystyle\exp\left[-\matrix{}{1}{2}\mathchoice{\mbox{\boldmath$\displaystyle\bf b$}}{\mbox{\boldmath$\textstyle\bf b$}}{\mbox{\boldmath$\scriptstyle\bf b$}}{\mbox{\boldmath$\scriptscriptstyle\bf b$}}^{T}V_{\rm sys}^{-1}\mathchoice{\mbox{\boldmath$\displaystyle\bf b$}}{\mbox{\boldmath$\textstyle\bf b$}}{\mbox{\boldmath$\scriptstyle\bf b$}}{\mbox{\boldmath$\scriptscriptstyle\bf b$}}\right]\;,\quad\quad\pi_{\theta}(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})=\mbox{const.}\;,$ (24) $\displaystyle p(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}},\mathchoice{\mbox{\boldmath$\displaystyle\bf b$}}{\mbox{\boldmath$\textstyle\bf b$}}{\mbox{\boldmath$\scriptstyle\bf b$}}{\mbox{\boldmath$\scriptscriptstyle\bf b$}}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}})$ $\displaystyle\propto$ $\displaystyle L(\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}},\mathchoice{\mbox{\boldmath$\displaystyle\bf b$}}{\mbox{\boldmath$\textstyle\bf b$}}{\mbox{\boldmath$\scriptstyle\bf b$}}{\mbox{\boldmath$\scriptscriptstyle\bf b$}})\pi_{\theta}(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}})\pi_{b}(\mathchoice{\mbox{\boldmath$\displaystyle\bf b$}}{\mbox{\boldmath$\textstyle\bf b$}}{\mbox{\boldmath$\scriptstyle\bf b$}}{\mbox{\boldmath$\scriptscriptstyle\bf b$}})\;,$ (25) where in (25), Bayes’ theorem is used to obtain the joint probability for the parameters of interest, $\textstyle\bf\theta$, and also the biases $\textstyle\bf b$. To obtain the probability for $\textstyle\bf\theta$ we integrate (marginalize) over $\textstyle\bf b$, $p(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}})=\int p(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}},\mathchoice{\mbox{\boldmath$\displaystyle\bf b$}}{\mbox{\boldmath$\textstyle\bf b$}}{\mbox{\boldmath$\scriptstyle\bf b$}}{\mbox{\boldmath$\scriptscriptstyle\bf b$}}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}})\,d\mathchoice{\mbox{\boldmath$\displaystyle\bf b$}}{\mbox{\boldmath$\textstyle\bf b$}}{\mbox{\boldmath$\scriptstyle\bf b$}}{\mbox{\boldmath$\scriptscriptstyle\bf b$}}\;.$ (26) One finds that the mode of $p(\mathchoice{\mbox{\boldmath$\displaystyle\bf\theta$}}{\mbox{\boldmath$\textstyle\bf\theta$}}{\mbox{\boldmath$\scriptstyle\bf\theta$}}{\mbox{\boldmath$\scriptscriptstyle\bf\theta$}}\|\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}})$ is at the same position as the least-squares estimates, and its covariance will be the same as obtained from the frequentist analysis where the full covariance matrix was given by the sum $V=V^{\rm stat}+V^{\rm sys}$. So this can be taken in effect as the formal justification for the addition in quadrature of statistical and systematic errors in a least-squares fit. #### The error on the error If one stays with the prior probabilities used above, the Bayesian and least- squares approaches deliver essentially the same result. An advantage of the Bayesian framework, however, is that it allows one to refine the assessment of the systematic uncertainties as expressed through the prior probabilities. For example, the least-squares fit including systematic errors is equivalent to the assumption of a Gaussian prior for the biases. A more realistic prior would take into account the experimenter’s own uncertainty in assigning the systematic error, i.e., the ‘error on the error’. Suppose, for example, that the $i$th measurement is characterized by a reported systematic uncertainty $\sigma^{\rm sys}_{i}$ and an unreported factor $s_{i}$, such that the prior for the bias $b_{i}$ is $\pi_{b}(b_{i})=\int\frac{1}{\sqrt{2\pi}s_{i}\sigma^{\rm sys}_{i}}\exp\left[-\frac{1}{2}\frac{b_{i}^{2}}{(s_{i}\sigma^{\rm sys}_{i})^{2}}\right]\pi_{s}(s_{i})\,ds_{i}\;.$ (27) Here the ‘error on the error’ is encapsulated in the prior for the factor $s$, $\pi_{s}(s)$. For this we can take whatever function is deemed appropriate. For some types of systematic error it could be close to the ideal case of a delta function centred about unity. Many reported systematics are, however, at best rough guesses, and one could easily imagine a function $\pi_{s}(s)$ with a mean of unity but a standard deviation of, say, $0.5$ or more. Here we show examples using a Gamma distribution for $\pi_{s}(s)$, which results in substantially longer tails for the prior $\pi_{b}(b)$ than those of the Gaussian. This can be seen in Fig. 5, which shows $\ln\pi_{b}(b)$ for different values of the standard deviation of $\pi_{s}(s)$, $\sigma_{s}$. Related studies using an inverse Gamma distribution can be found in Refs. [12, 13], which have the advantage that the posterior pdf can be written down in closed form. Figure 5: The log of the prior pdf for a bias parameter $b$ for different values of the standard deviation of $\pi_{s}(s)$. Using a prior for the biases with tails longer than those of a Gaussian results in a reduced sensitivity to outliers, which arise when an experimenter overlooks an important source of systematic uncertainty in the estimated error of a measurement. As a simple test of this, consider the sample data shown in Fig. 6(a). Suppose these represent four independent measurements of the same quantity, here a parameter called $\mu$, and the goal is to combine the measurements to provide a single estimate of $\mu$. That is, we are effectively fitting a horizontal line to the set of measured $y$ values, where the control variable $x$ is just a label for the measurements. In this example, suppose that each measurement $y_{i}$, $i=1,\ldots 4$, is modelled as Gaussian distributed about $\mu$, having a standard deviation $\sigma_{\rm stat}=0.1$, and furthermore each measurement has a systematic uncertainty $\sigma_{\rm sys}=0.1$, which here is taken to refer to the standard deviation of the Gaussian component of the prior $\pi_{b}(b_{i})$. This is then folded together with $\pi_{s}(s_{i})$ to get the full prior for $b_{i}$ using Eq. (27), and the joint prior for the vector of bias parameters is simply the product of the corresponding terms, as the systematic errors here are treated as being independent. These ingredients are then assembled according to the recipe of Eqs. (23)–(26) to produce the posterior pdf for $\mu$, $p(\mu\|\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}})$. Results of the exercise are shown in Fig. 6. In Fig. 6(a), the four measurements $y_{i}$ are reasonably consistent with each other. Figure 6(b) shows the corresponding posterior $p(\mu\|\mathchoice{\mbox{\boldmath$\displaystyle\bf y$}}{\mbox{\boldmath$\textstyle\bf y$}}{\mbox{\boldmath$\scriptstyle\bf y$}}{\mbox{\boldmath$\scriptscriptstyle\bf y$}})$ for two values of $\sigma_{s}$, which reflect differing degrees of non-Gaussian tails in the prior for the bias parameters, $\pi_{b}(b_{i})$. For $\sigma_{s}=0$, the prior for the bias is exactly Gaussian, whereas for $\sigma_{s}=0.5$, the non- Gaussian tails are considerably longer, as can be seen from the corresponding curves in Fig. 5. The posterior pdfs for both cases are almost identical, as can be see in Fig. 6(a). Determining the mean and standard deviation of the posterior for each gives $\hat{\mu}=1.000\pm 0.71$ for the case of $\sigma_{s}=0$, and $\hat{\mu}=1.000\pm 0.72$ for $\sigma_{s}=0.5$. So assuming a 50% “error on the error” here one only inflates the error of the averaged result by a small amount. (a)(b)(c)(d) Figure 6: (a) Data values which are relatively consistent and (b) a data set with an outlier; the horizontal lines indicate the posterior mean for two different values of the parameter $\sigma_{s}$. (c) and (d) show the posterior distributions corresponding to (a) and (b), respectively. (The dashed and solid curves in (a) and (c) overlap.) Now consider the case where one of the measured values is substantially different from the other three, as shown in Fig. 6(c). Here using the same priors for the bias parameters results in the posteriors shown in Fig. 6(d). The posterior means and standard deviations are $\hat{\mu}=1.125\pm 0.71$ for the case of $\sigma_{s}=0$, and $\hat{\mu}=1.093\pm 0.089$ for $\sigma_{s}=0.5$. When we assume a purely Gaussian prior for the bias ($\sigma_{s}=0.0$), the presence of the outlier has in fact no effect on the width of the posterior. This is rather counter-intuitive and results from our assumption of a Gaussian likelihood for the data and a Gaussian prior for the bias parameters. The posterior mean is however pulled substantially higher than the three other measurements, which are clustered around $1.0$. If the priors $\pi_{b}(b_{i})$ have longer tails, as occurs when we take $\sigma_{s}=0.5$, then the posterior is broader, and furthermore it is pulled less far by the outlier, as can be seen in Fig. 6(d). The fact is that the width of the posterior distribution, which effectively tells us the uncertainty on the parameter of interest $\mu$, becomes coupled to the internal consistency of the data. In contrast, in the (frequentist) least-squares method, or in the Bayesian approach using a Gaussian prior for the bias parameters, the final uncertainty on the parameter of interest is unaffected by the presence of outliers. And in many cases of practical interest, it would be in fact appropriate to conclude that the presence of outliers should indeed increase one’s uncertainty about the final parameter estimates. The example shown here can be generalized to cover a wide variety of model uncertainties by including prior probabilities for an enlarged set of model parameters. ### 0.2.4 Summary on Bayesian methods In these lectures we have seen how Bayesian methods can be used in parameter estimation, and this has also given us the opportunity to discuss some aspects of Bayesian computation, including the important tool of Markov Chain Monte Carlo. Although Bayesian and frequentist methods may often deliver results that agree numerically, there is an important difference in their interpretation. Furthermore, Bayesian methods allow one to incorporate prior information that may be based not on other measurements but rather on theoretical arguments or purely subjective considerations. And as these considerations may not find universal agreement, it is important to investigate how the results of a Bayesian analysis would change for a reasonable variation of the prior probabilities. It is important to keep in mind that in the Bayesian approach, all information about the parameters is encapsulated in the posterior probabilities. So if the analyst also wants to set upper limits or determine intervals that cover the parameter with a specified probability, then this is a straightforward matter of finding the parameter limits such that the integrated posterior pdf has the desired probability content. A discussion of Bayesian methods to the important problem of setting upper limits on a Poisson parameter is covered in Ref. [1] and references therein; we will not have time in these lectures to go into that question here. We will also unfortunately not have time to explore Bayesian model selection. This allows one to quantify the degree to which the the data prefer one model over the other using a quantity called the Bayes factor. These have not yet been widely used in particle physics but should be kept in mind as providing important complementary information to the corresponding outputs of frequentist hypothesis testing such as $p$-values. A brief description of Bayes factors can be found in Ref. [1] and a more in-depth treatment is given in Ref. [14]. ## 0.3 Topics in multivariate analysis In the second part of these lectures we will take a look at the important topic of multivariate analysis. In-depth information on this topic can be found in the textbooks [15, 16, 17, 18]. In a particle physics context, multivariate methods are often used when selecting events of a certain type using some potentially large number of measurable characteristics for each event. The basic framework we will use to examine these methods is that of a frequentist hypothesis test. The fundamental unit of data in a particle physics experiment is the ‘event’, which in most cases corresponds to a single particle collision. In some cases it could be instead a decay, and the picture does not change much if we look, say, at individual particles or tracks. But to be concrete let us suppose that we want to search for events from proton–proton collisions at the LHC that correspond to some interesting ‘signal’ process, such as supersymmetry. When running at full intensity, the LHC should produce close to a billion events per second. After a quick sifting, the data from around 200 per second are recorded for further study, resulting in more than a billion events per year. But only a tiny fraction of these are of potential interest. If one of the speculative theories such as supersymmetry turns out to be realized in Nature, then this will result in a subset of events having characteristic features, and the SUSY events will simply be mixed in randomly with a much larger number of Standard Model events. The relevant distinguishing features depend on what new physics Nature chooses to reveal, but one might see, for example, high $p_{\rm T}$ jets, leptons, missing energy. Unfortunately, background processes (e.g., Standard Model events) can often mimic these features and one will not be able to say with certainty that a given event shows a clear evidence for something new such as supersymmetry. For example, even Standard Model events can contain neutrinos which also escape undetected. The typical amount and pattern of missing energy in these events differs on average, however, from what a SUSY event would give, and so a statistical analysis can be applied to test whether something besides Standard Model events is present. In a typical analysis there is a class of event we are interested in finding (signal), and these, if they exist at all, are mixed in with the rest of the events (background). The data for each event is some collection of numbers $\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}=(x_{1},\ldots,x_{n})$ representing particle energies, momenta, etc. We will refer to these as the input variables of the problem. And the probabilities are joint densities for $\textstyle\bf x$ given the signal (s) or background (b) hypotheses: $f(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}\|\mbox{s})$ and $f(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}\|\mbox{b})$. To illustrate the general problem, consider the scatterplots shown in Fig. 7. These show the distribution of two variables, $x_{1}$ and $x_{2}$, which represent two out of a potentially large number of quantities measured for each event. The blue circles could represent the sought after signal events, and the red triangles the background. In each of the three figures there is a decision boundary representing a possible way of classifying the events. (a)(b)(c) Figure 7: Scatter plots of two variables corresponding to two hypotheses: signal and background. Event selection could be based, e.g., on (a) cuts, (b) a linear boundary, (c) a nonlinear boundary. Figure 7(a) represents what is commonly called the ‘cut-based’ approach. One selects signal events by requiring $x_{1}<c_{1}$ and $x_{2}<c_{2}$ for some suitably chosen cut values $c_{1}$ and $c_{2}$. If $x_{1}$ and $x_{2}$ represent quantities for which one has some intuitive understanding, then this can help guide one’s choice of the cut values. Another possible decision boundary is made with a diagonal cut as shown in Fig. 7(b). One can show that for certain problems a linear boundary has optimal properties, but in the example here, because of the curved nature of the distributions, neither the cut-based nor the linear solution is as good as the nonlinear boundary shown in Fig. 7(c). The decision boundary is a surface in the $n$-dimensional space of input variables, which can be represented by an equation of the form $y(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}})=y_{\rm cut}$, where $y_{\rm cut}$ is some constant. We accept events as corresponding to the signal hypothesis if they are on one side of the boundary, e.g., $y(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}})\leq y_{\rm cut}$ could represent the acceptance region and $y(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}})>y_{\rm cut}$ could be the rejection region. Equivalently we can use the function $y(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}})$ as a scalar test statistic. Once its functional form is specified, we can determine the pdfs of $y(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}})$ under both the signal and background hypotheses, $p(y\|\mbox{s})$ and $p(y\|\mbox{b})$. The decision boundary is now effectively a single cut on the scalar variable $y$, as illustrated in Fig. 8. Figure 8: Distributions of the scalar test statistic $y(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}})$ under the signal and background hypotheses. To quantify how good the event selection is, we can define the efficiency with which one selects events of a given type as the probability that an event will fall in the acceptance region. That is, the signal and background efficiencies are $\displaystyle\varepsilon_{\rm s}$ $\displaystyle=$ $\displaystyle P(\mbox{accept event}\|\mbox{s})=\int_{\rm A}f(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}\|\mbox{s})\,d\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}=\int_{-\infty}^{y_{\rm cut}}p(y\|\mbox{s})\,dy\;,$ (28) $\displaystyle\varepsilon_{\rm b}$ $\displaystyle=$ $\displaystyle P(\mbox{accept event}\|\mbox{b})=\int_{\rm A}f(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}\|\mbox{b})\,d\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}=\int_{-\infty}^{y_{\rm cut}}p(y\|\mbox{b})\,dy\;,$ (29) where the region of integration A represents the acceptance region. Dividing the space of input variables into two regions where one accepts or rejects the signal hypothesis is essentially the language of a frequentist statistical test. If we regard background as the ‘null hypothesis’, then the background efficiency is the same as what in a statistical context would be called the significance level of the test, or the rate of ‘type-I error’. Viewing the signal process as the alternative, the signal efficiency is then what a statistician would call the power of the test; it is the probability to reject the background hypothesis if in fact the signal hypothesis is true. Equivalently, this is one minus the rate of ‘type-II error’. The use of a statistical test to distinguish between two classes of events (signal and background), comes up in different ways. Sometimes both event classes are known to exist, and the goal is to select one class (signal) for further study. For example, proton–proton collisions leading to the production of top quarks are a well-established process. By selecting these events one can carry out precise measurements of the top quark’s properties such as its mass. In other cases, the signal process could represent an extension to the Standard Model, say, supersymmetry, whose existence is not yet established, and the goal of the analysis is to see if one can do this. Rejecting the Standard Model with a sufficiently high significance level amounts to discovering something new, and of course one hopes that the newly revealed phenomena will provide important insights into how Nature behaves. What the physicist would like to have is a test with maximal power with respect to a broad class of alternative hypotheses. For two specific signal and background hypotheses, it turns out that there is a well defined optimal solution to our problem. The Neyman–Pearson lemma states that one obtains the maximum power relative for the signal hypothesis for a given significance level (background efficiency) by defining the acceptance region such that, for $\textstyle\bf x$ inside the region, the likelihood ratio, i.e., the ratio of pdfs for signal and background, $\lambda(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}})=\frac{f(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}\|\mbox{s})}{f(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}\|\mbox{b})}\;,$ (30) is greater than or equal to a given constant, and it is less than this constant everywhere outside the acceptance region. This is equivalent to the statement that the ratio (30) represents the test statistic with which one obtains the highest signal efficiency for a given background efficiency, or equivalently, for a given signal purity. In principle the signal and background theories should allow us to work out the required functions $f(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}\|\mbox{s})$ and $f(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}\|\mbox{b})$, but in practice the calculations are too difficult and we do not have explicit formulae for these. What we have instead of $f(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}\|\mbox{s})$ and $f(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}\|\mbox{b})$ are complicated Monte Carlo programs, that is, we can sample $\textstyle\bf x$ to produce simulated signal and background events. Because of the multivariate nature of the data, where $\textstyle\bf x$ may contain at least several or perhaps even hundreds of components, it is a nontrivial problem to construct a test with a power approaching that of the likelihood ratio. In the usual case where the likelihood ratio (30) cannot be used explicitly, there exists a variety of other multivariate classifiers that effectively separate different types of events. Methods often used in HEP include neural networks or Fisher discriminants. Recently, further classification methods from machine learning have been applied in HEP analyses; these include probability density estimation (PDE) techniques, kernel-based PDE (KDE or Parzen window), support vector machines, and decision trees. Techniques such as ‘boosting’ and ‘bagging’ can be applied to combine a number of classifiers into a stronger one with greater stability with respect to fluctuations in the training data. Descriptions of these methods can be found, for example, in the textbooks [15, 16, 17, 18] and in Proceedings of the PHYSTAT conference series [19]. Software for HEP includes the TMVA [20] and StatPatternRecognition [21] packages, although support for the latter has unfortunately been discontinued. As we will not have the time to examine all of the methods mentioned above, in the following section we look at a specific example of a classifier to illustrate some of the main ideas of a multivariate analysis: the boosted decision tree (BDT). ### 0.3.1 Boosted decision trees Boosted decision trees exploit relatively recent developments in machine learning and have gained significant popularity in HEP. First in Section 0.3.1 we describe the basic idea of a decision tree, and then in Section 0.3.1 we will say how the the technique of ‘boosting’ can be used to improve its performance. #### Decision trees A decision tree is defined by a collection of successive cuts on the set of input variables. To determine the appropriate cuts, one begins with a sample of $N$ training events which are known to be either signal or background, e.g., from Monte Carlo. The set of $n$ input variables measured for each event constitutes a vector $\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}=(x_{1},\ldots x_{n})$. Thus we have $N$ instances of $\textstyle\bf x$, $\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}_{1},\ldots\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}_{N}$, as well as the corresponding $N$ true class labels $y_{1},\ldots,y_{N}$. It is convenient to assign numerical values to the labels so that, e.g., $y=1$ corresponds to signal and $y=-1$ for background. In addition we will assume that each event can be assigned a weight, $w_{i}$, with $i=1,\ldots,N$. For any subset of the events and for a set of weights, the signal fraction (purity) is taken to be $p=\frac{\sum_{i\in\mbox{s}}w_{i}}{\sum_{i\in\mbox{s}}w_{i}+\sum_{i\in\mbox{b}}w_{i}}\;,$ (31) where s and b refer to the signal and background event types. The weights are not strictly speaking necessary for a decision tree, but will be used in connection with boosting in Section 0.3.1. For a decision tree without boosting we can simply take all the weights to be equal. To quantify the degree of separation achieved by a classifier for a selected subset of the events one can use, for example, the Gini coefficient [22], which historically has been used as a measure of dispersion in economics and is defined as $G=p(1-p)\;.$ (32) The Gini coefficient is zero if the selected sample is either pure signal or background. Another measure is simply the misclassification rate, $\varepsilon=1-\mbox{max}(p,1-p)\;.$ (33) The idea behind a decision tree is illustrated in Fig. 9, from an analysis by the MiniBooNE neutrino oscillation experiment at Fermilab [23]. Figure 9: Illustration of a decision tree used by the MiniBooNE experiment [23] (see text). One starts with the entire sample of training events in the root node, shown in the figure with 52 signal and 48 background events. Out of all of the possible input variables in the vector $\textstyle\bf x$, one finds the component that provides the best separation between signal and background by use of a single cut. This requires a definition of what constitutes ‘best separation’, and there are a number of reasonable choices. For example, for a cut that splits a set of events $a$ into two subsets $b$ and $c$, one can define the degree of separation through the weighted change in the Gini coefficients, $\Delta=W_{a}G_{a}-W_{b}G_{b}-W_{c}G_{c}\;.$ (34) where $W_{a}=\sum_{i\in a}w_{i}\;,$ (35) and similarly for $W_{b}$ and $W_{c}$. Alternatively one may use a quantity similar to (34) but with the misclassification rate (33), for example, instead of the Gini coefficient. More possibilities can be found in Ref. [20]. For whatever chosen measure of degree of separation, $\Delta$, one finds the cut on the variable amongst the components of $\textstyle\bf x$ that maximizes it. In the example of the MiniBooNE experiment shown in Fig. 9, this happened to be a cut on the number of PMT hits with a value of 100. This splits the training sample into the two daughter nodes shown in the figure, one of which is enhanced in signal and the other in background events. The algorithm requires a stopping rule based, for example, on the number of events in a node or the misclassification rate. If, for example, the number of events or the misclassification rate in a given node falls below a certain threshold, then this is defined as a terminal node or ‘leaf’. It is classified as a signal or background leaf based on its predominant event type. In Fig. 9, for example, the node after the cut on PMT hits with 4 signal and 37 background events is classified as a terminal background node. For nodes that have not yet reached the stopping criterion, one iterates the procedure and finds, as before, the variable that provides the best separation with a single cut. In Fig. 9 this is an energy cut of $0.2\,\mbox{GeV}$. The steps are continued until all nodes reach the stopping criterion. The resulting set of cuts effectively divides the $\textstyle\bf x$ space into two regions: signal and background. To provide a numerical output for the classifier we can define $f(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}})=\left\\{\\!\\!\begin{array}[]{ll}1&\quad\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}\mbox{ in signal region},\\\\[5.69046pt] -1&\quad\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}\mbox{ in background region}.\end{array}\right.$ (36) Equation (36) defines a decision tree classifier. In this form, these tend to be very sensitive to statistical fluctuations in the training data. One can easily see why this is, for example, if two of the components of $\textstyle\bf x$ have similar discriminating power between signal and background. For a given training sample, one variable may be found to give the best degree of separation and is chosen to make the cut, and this affects the entire further structure of the tree. In a different statistically independent sample of training events, the other variable may be found to be better, and the resulting tree could look very different. Boosting is a technique that can decrease the sensitivity of a classifier to such fluctuations, and we describe this in the following section. #### Boosting Boosting is a general method of creating a set of classifiers which can be combined to give a new classifier that is more stable and has a smaller misclassification rate than any individual one. It is often applied to decision trees, precisely because they suffer from sensitivity to statistical fluctuations in the training sample, but the technique can be applied to any classifier. Let us suppose as above that we have a sample of $N$ training events, i.e., $N$ instances of the data vector, $\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}_{1},\ldots,\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}_{N}$, and $N$ true class labels $y_{1},\ldots,y_{N}$, with $y=1$ for signal and $y=-1$ for background. Also as above assume we have $N$ weights $w_{1}^{(1)},\ldots,w_{N}^{(1)}$, where the superscript $(1)$ refers to the fact that this is the first training set. We initially set the weights equal and normalized such that $\sum_{i=1}^{N}w_{i}^{(1)}=1\;.$ (37) The idea behind boosting is to create from the initial sample, a series of further training samples which differ from the initial one in that the weights will be changed according to a specific rule. A number of boosting algorithms have been developed, and these differ primarily in the rule used to update the weights. We will describe the AdaBoost algorithm of Freund and Schapire [24], as it was one of the first such algorithms and its properties have been well studied. One begins with the initial training sample and from it derives a classifier. We have in mind here a decision tree, but it could be any type of classifier for where the training employs the event weights. The resulting function $f_{1}(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}})$ will have a certain misclassification rate $\varepsilon_{1}$. In general for the $k$th classifier (i.e., based on the $k$th training sample), we can write the error rate as $\varepsilon_{k}=\sum_{i=1}^{N}w_{i}^{(k)}I(y_{i}f_{k}(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}_{i})\leq 0)\;,$ (38) where $I(X)=1$ if the Boolean expression $X$ is true, and is zero otherwise. We then assign a score to the classifier based on its error rate. For the AdaBoost algorithm this is $\alpha_{k}=\ln\frac{1-\varepsilon_{k}}{\varepsilon_{k}}\;,$ (39) which is positive as long as the error rate is lower than 50%, i.e,. the classifier does better than random guessing. Having carried out these steps for the initial training sample, we define the second training sample by updating the weights. More generally, the weights for step $k+1$ are found from those for step $k$ by $w_{i}^{(k+1)}=w_{i}^{(k)}\,\frac{e^{-\alpha_{k}f_{k}(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}_{i})y_{i}/2}}{Z_{k}}\;,$ (40) where the factor $Z_{k}$ is chosen so that the sum of the updated weights is equal to unity. Note that if an event is incorrectly classified, then the true class label $y_{i}$ and the value $f_{k}(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}_{i})$ have opposite signs, and thus the new weights are greater than the old ones. Correctly classified events have their weights decreased. This means that the updated training set will pay more attention in the next iteration to those events that were not correctly classified, the idea being that it should try harder to get it right the next time around. After $K$ iterations of this procedure one has classifiers $f_{1}(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}}),\ldots,f_{K}(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}})$, each with a certain error rate and score based on Eqs. (38) and (39). In the case of decision trees, the set of new trees is called a forest. From these one defines an averaged classifier as $y(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}})=\sum_{k=1}^{K}\alpha_{k}f_{k}(\mathchoice{\mbox{\boldmath$\displaystyle\bf x$}}{\mbox{\boldmath$\textstyle\bf x$}}{\mbox{\boldmath$\scriptstyle\bf x$}}{\mbox{\boldmath$\scriptscriptstyle\bf x$}})\;.$ (41) Equation (41) defines a boosted decision tree (or more generally, a boosted version of whatever classifier was used). One of the important questions to be addressed is how many boosting iterations to use. One can show that for a sufficiently large number of iterations, a boosted decision tree will eventually classify all of the events in the training sample correctly. Similar behaviour is found with any classification method where one can control to an arbitrary degree the flexibility of the decision boundary. The user can arrange it so that the boundary twists and turns so as to get all of the events on the right side. In the case of a neural network, for example, one can increase the number of hidden layers, or the number of nodes in the hidden layers; for a support vector machine, one can adjust the width of the kernel function and the regularization parameter to increase the flexibility of the boundary. An example is shown in Fig. 10(a), where an extremely flexible classifier has managed to enclose all of the signal events and exclude all of the background. (a)(b) Figure 10: Scatter plot of events of two types and the decision boundary determined by a particularly flexible classifier. Plot (a) shows the events used to train the classifier, and (b) shows an independent sample of test data. Of course if we were now to take the decision boundary shown in Fig. 10(a) and apply it to a statistically independent data sample, there is no reason to believe that the contortions that led to such good performance on the training sample will still work. This can be seen in Fig. 10(b), which shows the same boundary with a new data sample. In this case the classifier is said to be overtrained. Its error rate calculated from the same set of events used to train the classifier underestimates the rate on a statistically independent sample. To deal with overtraining, one estimates the misclassification rate not only with the training data sample but also with a statistically independent test sample. We can then plot these rates as a function of the parameters that regulate the flexibility of the decision boundary, e.g., the number of boosting iterations used to form the BDT. For a small number of iterations, one will find in general that the error rates for both samples drop. The error rate based on the training sample will continue to drop, eventually reaching zero. But at some point the error rate from the test sample will cease to decrease and in general will increase. One chooses the architecture of the classifier (number of boosting iterations, number of nodes or layers in a neural network, etc.) to minimize the error rate on the test sample. As the test sample is used to choose between a number of competing architectures based on the minimum observed error rate, this in fact gives a biased estimate of the true error rate. In principle one should use a third validation sample to obtain an unbiased estimate of the error rate. In many cases the bias is small and this last step is omitted, but one should be aware of its potential existence. In some applications, the training data is relatively inexpensive; one simply generates more events with Monte Carlo. But often event generation can take a prohibitively long time and one may be reluctant to use only a fraction of the events for training and the other half for testing. In such cases, procedures such as cross validation (see, e.g., Refs. [15, 16]) can be used where the available events are partitioned in a number of different ways into training and test samples and the results averaged. Boosted decision trees have become increasingly popular in particle physics in recent years. One of their advantages is that they are relatively insensitive to the number of input variables used in the data vector $\textstyle\bf x$. Components that provide little or no separation between signal and background are rarely chosen as for the cut that provides separation, i.e., to split the tree, and thus they are effectively ignored. Decision trees have no difficulty in dealing with different types of data; these can be real, integer, or they can simply be labels for which there is no natural ordering (categorical data). Furthermore, boosted decision trees are surprisingly insensitive to overtraining. That is, although the error rate on the test sample will not decrease to zero as one increases the number of boosting iterations (as is the case for the training sample), it tends not to increase. Further discussion of this point can be found in Ref. [25]. ### 0.3.2 Summary on multivariate methods The boosted decision tree is an example of a relatively modern development in Machine Learning that has attracted substantial attention in HEP. Support Vector Machines (SVMs) represent another such development and will no doubt also find further application in particle physics; further discussion on SVMs can be found in Refs. [15, 16] and references therein. Linear classifiers and neural networks will no doubt continue to play an important role, as will probability density estimation methods used to approximate the likelihood ratio. Multivariate methods have the advantage of exploiting as much information as possible out of all of the quantities measured for each event. In an environment of competition between experiments, this can be a natural motivation to use them. Some caution should be exercised, however, before placing too much faith in the performance of a complicated classifier, to say nothing of a combination of complicated classifiers. These may have decision boundaries that indeed exploit nonlinear features of the training data, often based on Monte Carlo. But if these features have never been verified experimentally, then they may or may not be present in the real data. There is thus the risk of, say, underestimating the rate of background events present in a region where one looks for signal, which could lead to a spurious discovery. Simpler classifiers are not immune to such dangers either, but in such cases the problems may be easier to control and mitigate. One should therefore keep in mind the following quote, often heard in the multivariate analysis community: > Keep it simple. As simple as possible. Not any simpler. > > — A. Einstein To this we can add the more modern variant, > If you believe in something you don’t understand, you suffer, … > > —Stevie Wonder Having made the requisite warnings, however, it seems clear that multivariate methods will play an important role in the discoveries we hope to make at the LHC. One can easily imagine, for example, that 5-sigma evidence for New Physics from a highly performant, and complicated, classifier would be regarded by the community with some scepticism. But if this is backed up by, say, 4-sigma significance from a simpler, more transparent analysis, then the conclusion would be more easily accepted, and the team that pursues both approaches may well win the race. ## 0.4 Summary and conclusions In these lectures we have looked at two topics in statistics, Bayesian methods and multivariate analysis, which will play an important role in particle physics in the coming years. Bayesian methods provide important tools for analysing systematic uncertainties, where prior information may be available that does not necessarily stem solely from other measurements, but rather from theoretical arguments or other indirect means. The Bayesian framework allows one to investigate how the posterior probabilities change upon variation of the prior probabilities. Through this type of sensitivity analysis, a Bayesian result becomes valuable to the broader scientific community. As experiments become more expensive and the competition more intense, one will always be looking for ways to exploit as much information as possible from the data. Multivariate methods provide a means to achieve this, and advanced tools such as boosted decision trees have in recent years become widely used. And while their use will no doubt increase as the LHC experiments mature, one should keep in mind that a simple analysis also has its advantages. As one studies the advanced multivariate techniques, however, their properties become more apparent and the community will surely find ways of using them so as to maximize the benefits without excessive risk. ## Acknowledgements I wish to convey my thanks to the students and organizers of the 2009 European School of High-Energy Physics in Bautzen for a highly stimulating environment. The friendly atmosphere and lively discussions created a truly enjoyable and productive school. ## References * [1] C. Amsler et al. 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Stork, Pattern Classification, 2nd ed. (Wiley, New York, 2001). * [18] A. Webb, Statistical Pattern Recognition, 2nd ed. (Wiley, Chichester, 2002). * [19] Links to the Proceedings of the PHYSTAT conference series (Durham 2002, Stanford 2003, Oxford 2005, and Geneva 2007) can be found at phystat.org. * [20] A. Höcker et al., TMVA Users Guide, physics/0703039 (2007); software available from tmva.sf.net. * [21] I. Narsky, StatPatternRecognition: A C++ Package for Statistical Analysis of High Energy Physics Data, physics/0507143 (2005); software available from sourceforge.net/projects/statpatrec. * [22] C.W. Gini, Variabilità e Mutabilità, Studi Economicogiuridici Università di Cagliari, III, 2a, Bologna, (1912) 1–156. * [23] B. Roe et al., Boosted decision trees as an alternative to artificial neural networks for particle Identification, Nucl. Instrum. Methods Phys. Res. A543 (2005) 577–584; H.J. Yang, B. Roe and J. Zhu, Studies of boosted decision trees for MiniBooNE particle Identification, Nucl. Instrum. Methods Phys. Res. A555 (2005) 370–385. * [24] Y. Freund and R. E. Schapire, A decision-theoretic generalization of on-line learning and an application to boosting, J. Comput. Syst. Sci. 55 (1997) 119–139. * [25] Y. Freund and R. E. Schapire, A short introduction to boosting, J. Jpn. Soc. Artif. Intell. 14 (1999) 771–780.	arxiv-papers	2010-12-16T13:00:11	2024-09-04T02:49:15.725868	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "G. Cowan (Royal Holloway)", "submitter": "Scientific Information Service Cern", "url": "https://arxiv.org/abs/1012.3589" }
1012.3678	# Components of the Extragalactic Gamma Ray Background Floyd W. Stecker11affiliation: Astrophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771 and Tonia M. Venters22affiliation: NASA Postdoctoral Program Fellow, Goddard Space Flight Center, Greenbelt, MD 20771 ###### Abstract We present new theoretical estimates of the relative contributions of unresolved blazars and star-forming galaxies to the extragalactic $\gamma$-ray background (EGB) and discuss constraints on the contributions from alternative mechanisms such as dark matter annihilation and truly diffuse $\gamma$-ray production. We find that the Fermi source count data do not rule out a scenario in which the EGB is dominated by emission from unresolved blazars, though unresolved star-forming galaxies may also contribute significantly to the background, within order-of-magnitude uncertainties. In addition, we find that the spectrum of the unresolved star-forming galaxy contribution cannot explain the EGB spectrum found by EGRET at energies between $50$ and $200$ MeV, whereas the spectrum of unresolved FSRQs, when accounting for the energy- dependent effects of source confusion, could be consistent with the combined spectrum of the low-energy EGRET EGB measurements and the Fermi-LAT EGB measurements. ###### Subject headings: Gamma rays: diffuse background – Gamma rays: galaxies – Galaxies: general – Galaxies: active ## 1\. Introduction Studies of the extragalactic $\gamma$-ray background (EGB) can provide insight into high energy processes in the universe and, as such, has been the subject of much debate, particularly concerning the roles of extragalactic astrophysical sources and new physics. Recent data from the Large Area Telescope (LAT)111Hereafter we shall refer to the Fermi-LAT instrument simply as Fermi. on board the Fermi Gamma Ray Space Telescope allow for a reassessment of the possible astrophysical origins of the EGB, which could improve our understanding of $\gamma$-ray production in these objects and provide more robust constraints on the more exotic scenarios. However, in order to determine the strength and spectrum of this isotropic background one needs to proceed from the raw photon count data by determining to the best extent possible the detector sensitivity, the intrinsic events produced by the larger charged particle flux impinging on the detector, and the much larger $\gamma$-ray foreground within our Galaxy resulting from cosmic-ray interactions with photons and gas nuclei. Such analyses have been made for both the Energetic $\gamma$-ray Experiment Telescope (EGRET) aboard the Compton Gamma Ray Observatory (Sreekumar et al., 1998; Strong et al., 2004a) and Fermi (Abdo et al., 2010j). Various extragalactic $\gamma$-ray production scenarios have been explored theoretically as candidate components that could contribute significantly to the observed background. Among those considered are the unresolved astronomical sources, such as active galactic nuclei (AGN) (Padovani et al., 1993; Stecker et al., 1993; Salamon & Stecker, 1994; Chiang et al., 1995; Stecker & Salamon, 1996; Kazanas & Perlman, 1997; Chiang & Mukherjee, 1998; Mukherjee & Chiang, 1999; Mücke & Pohl, 2000; Giommi et al., 2006; Narumoto & Totani, 2006; Dermer, 2007; Pavlidou & Venters, 2008; Inoue & Totani, 2009; Venters et al., 2009; Venters, 2010; Abazajian et al., 2010), star-forming galaxies (Pavlidou & Fields, 2002; Fields et al., 2010; Makiya et al., 2011), and starburst galaxies (Thompson et al., 2007; Stecker, 2007; Makiya et al., 2011). The large majority of associated extragalactic sources thus far detected by both EGRET and Fermi are blazars (Hartmann et al., 1999; Abdo et al., 2010f), i.e., those AGN for which the jet is closely aligned with the observer’s line-of-sight (Blandford & Königl, 1979), including $\gamma$-ray loud flat spectrum radio quasars (FSRQs) and BL Lacertae-type objects. It is expected that since blazars comprise the largest class of identified extragalactic $\gamma$-ray sources, unresolved blazars should contribute significantly to the EGB. Additionally, just as our Galaxy produces $\gamma$-rays, it is expected that $\gamma$-rays are produced in other galaxies, and as such, unresolved galaxies might also contribute to the EGB with the most significant contribution originating from the population of actively star-forming galaxies (Stecker, 1975; Pavlidou & Fields, 2001, 2002; Fields et al., 2010; Makiya et al., 2011). Interesting truly diffuse mechanisms that could contribute to the EGB involve cosmic ray interactions with intergalactic gas and the cosmic background radiation (Fazio et al., 1966; Stecker, 1973; Dar, 2007; Keshet et al., 2003) and electromagnetic cascades produced by interactions of very high and ultrahigh energy particles with the extragalactic background light (Kalashev et al., 2009; Berezinsky et al., 2011; Ahlers et al., 2010; Venters, 2010), as well as more exotic scenarios such as dark matter annihilation (Silk & Srednicki, 1984; Stecker et al., 1985; Rudaz & Stecker, 1988; Stecker & Tylka, 1989a, b; Rudaz & Stecker, 1991; Ullio et al., 2002) and decay (Olive & Silk, 1985; Stecker, 1986; Ibarra & Tran, 2008)222For reviews on dark matter annihilation, see Jungman et al. (1996) and Bertone & Silk (2005).. In this paper, we estimate the contributions to the EGB from unresolved extragalactic $\gamma$-ray sources of various types and compare them with the EGB obtained from analysis of Fermi data. In doing so, we also take into consideration the effects of both the completeness of the Fermi flux limited blazar survey and the important effect of source confusion owing to the energy dependent angular resolution of the Fermi-LAT detector. We will then briefly discuss the implications of possible truly diffuse emission mechanisms to the EGB. ## 2\. EGRET and Fermi Resolved Sources and the $\gamma$-ray Log N - log S relation In Figure 1 we plot the number of blazars observed per square degree versus blazar flux integrated above $100$ MeV for both EGRET (Reimer & Thompson, 2001) and Fermi (Abdo et al., 2010f). In the case of Fermi, the source spectra were extrapolated from a power-law fit above a fiducial energy of $1$ GeV (Abdo et al., 2010k). The offset between the resolved source count data obtained by the two detectors is a result of differences in their sensitivity calibrations(Thompson et al., 1993; Abdo et al., 2010f). The fluxes of the Fermi sources are reported to be systematically brighter than those reported by EGRET. We note that the 50% completeness in the Fermi survey is estimated to be reached at an effective flux limit of $\sim 2\times 10^{-8}$ cm-2 s-1 (Abdo et al., 2010k). The flux limit varies with galactic latitude and source hardness. Nevertheless, the effective flux limit indicated by both the turnover in Figure 1 and the Monte Carlo simulation in Abdo et al. (2010k) is $\sim 2\times 10^{-8}$ cm-2 s-1. By assuming that the Fermi survey is 100% complete at $\sim 7\times 10^{-8}$ cm-2 s-1 (Abdo et al., 2010k) and comparing the data for EGRET and the Fermi shown in Figure 1, we estimate that the EGRET survey is 50% complete at $\sim 8\times 10^{-8}$ cm-2 s-1. Figure 1.— Source count distributions for Fermi (black triangles; Abdo et al., 2010f) and EGRET (red circles; Reimer & Thompson, 2001) blazars. One may note that the flux level attained for 50% completeness for the Fermi extragalactic blazar survey is only about four times fainter than that for the EGRET survey even though the Fermi-LAT is sensitive to sources $\sim 30$ times fainter than that of EGRET. It is important to note that there are several factors that determine the efficiency of a $\gamma$-ray telescope for detecting extragalactic sources, among which are: (1) the flux of the source, (2) the spectral index of the source, (3) the intrinsic detector background from cosmic-ray induced events, (4) the foreground from the Milky Way, and (5) the diffuse extragalactic background. The Fermi-LAT was designed to reach its optimal effective area for $\gamma$-rays with energies near and above $1$ GeV, whereas EGRET was designed to reach its optimal effective area for $\gamma$-rays with energies near and above $100$ MeV. As such, Fermi is more sensitive to sources with hard spectral indices, particularly for the faintest sources observable by Fermi. Thus, it is difficult to make a direct comparison between EGRET and Fermi. However, we note that the positions of the turnovers in the data presented in Figure 1 provide a good indication of the observational situation. ## 3\. Determination of the EGB from Point Sources In general, the total contribution of a given population of sources to the EGB is found by convolving the source spectrum, $F_{\rm ph}(E_{0},z,L_{\gamma})$, measured at energy, $E_{0}$, of one source of $\gamma$-ray luminosity, $L_{\gamma}$, at a given redshift, $z$, with the comoving number density of sources at that redshift per luminosity interval, $n(L_{\gamma},z)=d^{2}N/dL_{\gamma}dV_{\rm com}$. We then integrate over the comoving volume, $V_{\rm com}$, and $\gamma$-ray luminosity: $I_{E}(E_{0})=\int_{0}^{z_{\rm max}}\int_{L_{\gamma,{\rm min}}}^{L_{\gamma,{\rm max}}}F_{\rm ph}(E_{0},z,L_{\gamma})n(L_{\gamma},z)\frac{d^{2}V_{\rm com}}{dzd\Omega}dL_{\gamma}dz\,,$ (1) where $I_{E}(E_{0})$ is the EGB intensity given in units of photons ${\rm cm^{-2}\,\,s^{-1}\,\,sr^{-1}\,\,GeV^{-1}}$, $L_{\gamma,{\rm max}}$ depends on the redshift, source spectrum, and the detector sensitivity, and we have differentiated the comoving volume with respect to redshift and solid angle, $\Omega$. However, as the different types of likely contributing $\gamma$-ray sources have distinct spectral characteristics and redshift and luminosity distributions, we must consider each case separately. Note that in Equation 1 we have neglected the effect of $\gamma$-ray absorption due to pair-production interactions with the extragalactic UV photons. The inclusion of $\gamma$-ray absorption would result in a steepening in the collective point source spectrum at the high-energy end (Salamon & Stecker, 1998; Venters et al., 2009), though a possible contribution from electromagnetic cascade photons might mitigate the steepening at higher energies (Venters, 2010). However, recent Fermi constraints on the $\gamma$-ray opacity imply that the UV background is likely to be fairly low (Abdo et al., 2010e), and as such, the absorption and the resulting cascades will only have a small effect on the EGB. ### 3.1. The Contribution to the EGB from Unresolved Blazars In determining the blazar contribution to the EGB, we follow the procedure outlined in Venters et al. (2009). We approximate the blazar $\gamma$-ray spectrum as a power law in energy defined by the _photon_ spectral index, $\Gamma$ ($F_{\rm ph}\propto E^{-\Gamma}$). The spectral indices of the population of blazars form a distribution with some spread (the spectral index distribution, SID; Stecker & Salamon, 1996; Venters & Pavlidou, 2007); hence, the number density of blazars is defined as $n(L_{\gamma},z,\Gamma)=\rho_{\gamma}(L_{\gamma},z)p_{L}(\Gamma)=\frac{d^{3}N}{dL_{\gamma}dV_{\rm com}d\Gamma}\,,$ (2) where $\rho_{\gamma}(L_{\gamma},z)=d^{2}N/dL_{\gamma}dV_{\rm com}$ is the blazar $\gamma$-ray luminosity function (GLF) giving the comoving number density of blazars per luminosity interval and $p_{L}(\Gamma)=dN/d\Gamma$ is the normalized blazar SID accounting for spectral bias (see Section 4). $L_{\gamma}$ is the $\gamma$-ray luminosity at the fiducial energy, $E_{f}$ (taken to be $100$ MeV), defined as $E_{f}^{2}$ times the differential photon luminosity, $L_{\rm ph}=d^{2}N_{\gamma}/dtdE$ measured at $E_{f}$. $L_{\gamma}$ is related to the integral flux greater than $E_{f}$, $F(>E_{f})$, by $L_{\gamma}=4\pi D^{2}(\Gamma-1)(1+z)^{\Gamma}E_{f}F(>E_{f})\,,$ (3) where $D$ is the distance measure for the Friedman-Robertson-Walker cosmology333We take $H_{0}=70\mbox{ km s}^{-1}\mbox{ Mpc}^{-1}$, $\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$, and $\Omega_{r}\ll 1$.: $D=\frac{c}{H_{0}}\int_{0}^{z}\left[\Omega_{\Lambda}+\Omega_{m}(1+z^{\prime})^{3}\right]^{-1/2}\,dz^{\prime}\,.$ (4) A given blazar of $\gamma$-ray luminosity, $L_{\gamma}$, at a redshift, $z$, with a $\gamma$-ray photon spectral index, $\Gamma$, has a measured photon flux of $F_{\rm ph}(E_{0},z,L_{\gamma},\Gamma)=\frac{L_{\gamma}}{4\pi E_{f}^{2}[d_{L}(z)]^{2}}(1+z)^{2-\Gamma}\left(\frac{E_{0}}{E_{f}}\right)^{-\Gamma}\,,$ (5) where $d_{L}(z)=D(1+z)$ is the luminosity distance. Thus, the total contribution to the EGB at a given energy, $E_{0}$, from unresolved blazars (the collective unresolved blazar intensity) is determined by integrating the contribution from each individual blazar fainter than the detector sensitivity, $I^{\rm bl}_{E}(E_{0})=\int_{-\infty}^{\infty}\int_{0}^{z_{\rm max}}\int_{L_{\gamma,{\rm min}}}^{L_{\gamma,{\rm max}}}F_{\rm ph}(E_{0},z,L_{\gamma},\Gamma)\rho_{\gamma}p_{L}(\Gamma)\frac{d^{2}V_{\rm com}}{dzd\Omega}\,dL_{\gamma}\,dz\,d\Gamma\,,$ (6) where $z_{\rm max}=5.0$ and $L_{\gamma,{\rm max}}$ is determined from Equation 3 taking $F(>E_{f})=F_{\rm min}$, with $F_{\rm min}$ being the minimum flux capable of being resolved by Fermi. We have also integrated Equation 1 over the blazar spectral index. ### 3.2. The Contribution to the EGB from Unresolved Star-forming Galaxies By applying the same procedure that we use to determine the background from unresolved blazars (see Section 3.1), we calculate the contribution from unresolved star-forming galaxies by determining the $\gamma$-ray photon flux as a function of energy for one galaxy and then convolving with and integrating over the appropriate cosmological distributions. Is is expected that, as in the Milky Way, the $\gamma$-ray emission for a star-forming galaxy comes mainly from the decay of $\pi^{0}$ mesons produced by cosmic-ray interactions with interstellar gas (Stecker, 1970). The resulting $\gamma$-ray production spectrum has been calculated by many authors (Stecker, 1970, 1973, 1979; Cavallo & Gould, 1971; Stephens & Badhwar, 1981; Dermer, 1986; Mori, 1997; Strong et al., 2004b, 2007, 2010; Kelner et al., 2006; Kamae et al., 2006; Mori, 2009). For our calculation we adopt the $\pi^{0}$ emissivity given by Stecker (1979) renormalized upwards by 25$\%$ to be consistent with the local emissivity measured by Fermi Abdo et al. (2009b). We note that there is also emission arising from electron bremsstrahlung, but the contribution is likely to be small, particularly above $100$ MeV (Abdo et al., 2009b). We also neglect the emission from Compton interactions that may contribute significantly to galaxy spectra above 10 GeV, particularly for starburst galaxies (Stecker, 1977; Hunter et al., 1997; Abdo et al., 2009c; Strong et al., 2010). We also note that the cosmic ray spectrum should, in fact, vary from galaxy to galaxy since it depends on energy dependent leakage and each galaxy has a different morphology and magnetic field configuration. This uncertainty can affect the predicted slope of the EGB spectrum at high energies, but it does not affect the absolute value of the predicted background at $\sim 200$ MeV. The $\gamma$-ray photon luminosity is related to the $\gamma$-ray production spectrum per hydrogen atom, $q_{\rm H}(E)$, by $L_{\rm ph}(E)=\left<q_{\rm H}(E)\right>N_{\rm H}\,,$ (7) where $\left<q_{\rm H}(E)\right>$ is found by averaging $q_{\rm H}(E)$, the differential $\gamma$-ray production spectrum per hydrogen atom, over the galaxy, and $N_{\rm H}$ is the number of hydrogen atoms in the galaxy in the form of both atomic (HI) and molecular (${\rm H_{2}}$) hydrogen. Thus, the $\gamma$-ray photon flux for a galaxy at redshift $z$ as observed at energy $E_{0}$ is $F_{\rm ph}(E_{0},z)=\frac{1}{4\pi D^{2}}\left<q_{\rm H}[E_{0}(1+z)]\right>N_{\rm H}(z)\,.$ (8) The rate of production of $\gamma$-rays from $\pi^{0}$ decay is proportional to the flux of cosmic rays, which we assume to be proportional to the supernova rate. The supernova rate is expected to be proportional to the rate of formation of higher mass stars, which is, in turn, proportional to the overall star formation rate assuming a universal initial mass function. Assuming then that the rate of production of $\gamma$-rays from $\pi^{0}$ decay is proportional to the star formation rate (SFR) for a galaxy, we can relate the $\left<q_{\rm H}\right>$ of the galaxy to that of the Milky Way, $\left<q^{\rm MW}_{\rm H}\right>$: $\frac{\left<q_{\rm H}\right>}{\left<q^{\rm MW}_{\rm H}\right>}=\frac{\Psi(z)}{\Psi(z=0)}\,,$ (9) where $\Psi$ is the SFR of the galaxy, and we take $\left<q^{\rm MW}_{\rm H}\right>$ to be some fraction, $f_{q}$, of the locally measured444That is, since $q^{\rm MW}_{\rm H}$ is calculated in the literature assuming the cosmic ray flux as measured in the solar neighborhood. $q^{\rm MW}_{\rm H}$. We determine $f_{q}$ by integrating the radial profile of the flux of cosmic rays weighted by $r^{2}$. Using the radial profiles calculated by Stecker & Jones (1977), we obtain $f_{q}\sim 0.825$. Thus, the galaxy spectrum becomes $F_{\rm ph}(E_{0},z)=\frac{1}{4\pi D^{2}}f_{q}q^{\rm MW}_{\rm H}[E_{0}(1+z)]\frac{\Psi(z)}{\Psi(0)}N_{\rm H}(z)\,.$ (10) The calculation of the amount of gas in a galaxy is subject to a considerable degree of uncertainty, especially at high redshifts (see Section 4.2). As such, rather than focusing on one particular model, we calculate the star- forming galaxy contribution for three different models arising from different sets of assumptions. In so doing, we seek to explore various possibilities and highlight the uncertainty. #### 3.2.1 Galaxy Contribution Determined from the Schecter Function and An Evolving Gas Fraction One method for determining the number of hydrogen atoms in a galaxy is to assume that the mass of gas in the galaxy is some fraction of its stellar mass: $N_{\rm H}(z)=\frac{f_{\rm gas}(z)}{1-f_{\rm gas}(z)}\frac{M_{\ast}}{m_{\rm H}}\,,$ (11) where $f_{\rm gas}(z)=M_{\rm gas}/(M_{\rm gas}+M_{\ast})\propto(1+z)^{0.9}$ is the gas fraction given in Papovich et al. (2010), and we have neglected the possible contribution of helium to the gas mass of a galaxy. Substituting Equation (11) into Equation (10) and approximating $\Psi(z)/\Psi(0)\sim\dot{\rho}_{\rm SFR}(z)/\dot{\rho}_{\rm SFR}(0)$, we get: $F_{\rm ph}(E_{0},z,M_{\ast})=\frac{f_{q}q^{\rm MW}_{\rm H}[E_{0}(1+z)]}{4\pi m_{\rm H}D^{2}}\frac{\dot{\rho}_{\rm SFR}(z)}{\dot{\rho}_{\rm SFR}(0)}\frac{f_{\rm gas}(z)}{1-f_{\rm gas}(z)}M_{\ast}\,,$ (12) where $\dot{\rho}_{\rm SFR}(z)$ is the cosmic SFR (CSFR) given by $\log(\dot{\rho}_{\rm SFR}(z))=-2.06+3.39\log(1+z)$ for $z<1.3$ and $\dot{\rho}_{\rm SFR}(z)\sim\mbox{const.}$ for $1.3\leq z\leq 4.0$ (Ly et al., 2011). To get the total contribution to the EGB, we convolve with the comoving number density of star-forming galaxies per stellar mass interval as a function of redshift and integrate: $I^{\rm gal}_{E}(E_{0})=\int_{0}^{z_{\rm max}}\\!\\!\\!\\!\\!\\!\\!\\!dz\,\frac{d^{2}V_{\rm com}}{dzd\Omega}\\!\\!\int_{M^{\prime}_{\rm min}}^{M^{\prime}_{\rm max}}\\!\\!\\!\\!\\!\\!\\!\\!\\!dM^{\prime}F_{\rm ph}(E_{0},z,M^{\prime})\Phi(z,M^{\prime})\,,$ (13) where $\Phi(z,M^{\prime})=d^{2}N/dM^{\prime}dV_{\rm com}$ is the Schecter function for stellar mass with parameters as determined in Elsner et al. (2008), $M^{\prime}$ is given by $M_{\ast}=10^{M^{\prime}}M_{\odot}$, and we take $M^{\prime}_{\rm min}=8.0$ and $M^{\prime}_{\rm max}=12.0$. #### 3.2.2 Galaxy Contribution Determined from IR Luminosity Functions Alternatively, we can determine the $\gamma$-ray spectrum of a galaxy by assuming that the $\gamma$-ray luminosity of the galaxy is proportional to some power of its SFR (Fields et al., 2010; Makiya et al., 2011; Abdo et al., 2010g): $L_{\rm ph}\propto\Psi^{\alpha}$. Since $L_{\rm ph}(E)=\left<q_{\rm H}(E)\right>N_{\rm H}$ and $\left<q_{\rm H}(E)\right>\propto\Psi$ (as demonstrated in Section 3.2.1), $N_{\rm H}=\left(\frac{A\Psi_{\rm MW}}{\int\left<q^{\rm MW}_{\rm H}(E)\right>dE}\right)\Psi^{\alpha-1}\,,$ (14) where $A$ and $\alpha$ are the best-fit parameters of the above power law determined from Fermi observations of star-forming galaxies in the Local Group and their SFRs (see Section 4.2.2). Assuming the Chabrier (2003) initial mass function, the SFR of a galaxy is related to its total infrared luminosity, $L_{\rm IR}$, $L_{\rm IR}=1.1\times 10^{10}L_{\odot}\left(\frac{\Psi}{M_{\odot}{\rm\,\,yr^{-1}}}\right)$ (15) (Hopkins et al., 2010). The $\gamma$-ray photon flux for a galaxy at redshift $z$ is given by $F_{\rm ph}(E_{0},z,L_{\rm IR})=\frac{1}{4\pi D^{2}}\left(\frac{A}{\int q^{\rm MW}_{\rm H}dE}\right)\left(\frac{L_{\rm IR}}{1.1\times 10^{10}L_{\odot}}\right)^{\alpha}q^{\rm MW}_{\rm H}[E_{0}(1+z)].$ (16) Then, the total contribution to the EGB is found by convolving the galaxy photon flux with an infrared luminosity function, $\Phi(z,L_{\rm IR})=d^{2}N/dL_{\rm IR}dV_{\rm com}$ and integrating over infrared luminosity and redshift: $I^{\rm gal}_{E}(E_{0})=\int_{0}^{z_{\rm max}}\\!\\!\\!\\!\\!\\!\\!\\!dz\,\frac{d^{2}V_{\rm com}}{dzd\Omega}\\!\\!\int_{L_{\rm IR,min}}^{L_{\rm IR,max}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!dL_{\rm IR}F_{\rm ph}(E_{0},z,L_{\rm IR})\Phi(z,L_{\rm IR})\,,$ (17) where we take $L_{\rm IR,min}=10^{10}L_{\odot}$ and $L_{\rm IR,max}=10^{15}L_{\odot}$. #### 3.2.3 Galaxy Contribution Determined from the Cosmic Star-formation Rate and the Star-formation Efficiency Another alternative is to relate the cosmic density of hydrogen in star- forming galaxies to the cosmic star formation rate. Given that stars are formed in giant molecular clouds (GMCs), it is reasonable to assume $\dot{\rho}_{\rm SFR}\sim\xi({\rm H_{2}})\rho_{\rm H_{2}}\,,$ (18) where $\dot{\rho}_{\rm SFR}$ is the cosmic star formation rate density (see Section 3.2.1), $\rho_{\rm H_{2}}$ is the cosmic molecular hydrogen density in star-forming galaxies, and $\xi({\rm H_{2}})$ is the star formation “efficiency” (SFE) of molecular hydrogen (Bigiel et al., 2008; Gnedin et al., 2009; Bauermeister et al., 2010). Leroy et al. (2008) measure the SFE to be $\sim(5.25\pm 2.5)\times 10^{-10}{\rm yr^{-1}}$ and to be roughly constant over a wide range of conditions555More precisely, $\xi({\rm H_{2}})\sim\epsilon/\tau_{\rm ff}$, where $\epsilon$ is the percentage of gas involved in forming stars and $\tau_{\rm ff}\propto\rho^{-1/2}$ is the _local_ free-fall timescale of the gas. However, using this relation requires knowledge of the local density of the gas and a better understanding of the formation of GMCs than presently exists (see Section 4.2). We also note that the measurements obtained by Leroy et al. (2008) were taken from a sample of low-redshift galaxies. The SFE could actually evolve with redshift (Bauermeister et al., 2010).. We can relate the density of atomic hydrogen to the density of molecular hydrogen through the average mass ratio of atomic and molecular hydrogen ($\mathcal{R}=\left<M_{\rm HI}/M_{\rm H_{2}}\right>$) in star-forming galaxies, $\rho_{\rm HI}\sim\mathcal{R}\rho_{\rm H_{2}}$. The average mass ratio of atomic and molecular hydrogen can be found by integrating radial profiles of the gas surface densities of star-forming galaxies found in Leroy et al. (2008), resulting in $\mathcal{R}\sim 0.9$. Note that in so doing, we only integrate the profiles out to the optical radius since recent surveys indicate that star formation is extremely inefficient beyond this radius (Bigiel et al., 2010)666We should note that even though star formation is extremely inefficient beyond the isophotal radius, there is still gas beyond this radius. However based on the radial profiles determined in Stecker & Jones (1977), we do not expect cosmic rays to propagate much beyond the isophotal radius; hence, we expect gamma-ray production beyond this radius to be low.. Thus, with appropriate modifications to Equation 10, we find the $\gamma$-ray flux from a particular redshift $F_{\rm ph}(E_{0},z)=\frac{f_{q}(1+\mathcal{R})}{4\pi m_{\rm H}\xi({\rm H_{2}})D^{2}}q^{\rm MW}_{\rm H}[E_{0}(1+z)]\frac{\dot{\rho}^{2}_{\rm SFR}(z)}{\dot{\rho}_{\rm SFR}(0)}\frac{dV_{\rm com}}{dz}dz\,.$ (19) Differentiating with respect to solid angle $\Omega$ and integrating over redshift results in an equation for the total contribution to the EGB: $I^{\rm gal}_{E}(E_{0})=\frac{f_{q}(1+\mathcal{R})}{4\pi m_{\rm H}\xi({\rm H_{2}})\dot{\rho}_{\rm SFR}(0)}\int_{0}^{z_{\rm max}}\\!\\!\\!\\!\frac{1}{D^{2}}q^{\rm MW}_{\rm H}[E_{0}(1+z)]\dot{\rho}^{2}_{\rm SFR}(z)\frac{d^{2}V_{\rm com}}{d\Omega dz}dz\,.$ (20) ## 4\. Observational Inputs and Considerations ### 4.1. $\gamma$-ray Blazars Of the nearly $1500$ resolved point sources observed by Fermi in the first year, $573$ are associated with blazars (First Fermi catalog (1FGL); Abdo et al. 2010f). Thus, blazars comprise the largest class of astrophysical objects associated with $\gamma$-ray sources. Naturally, unresolved blazars have long been suspected of providing, at least, a substantial contribution to the EGB, though the exact amount remains in debate and depends on various assumptions as to constructing GLFs and redshift distributions (Padovani et al., 1993; Stecker et al., 1993; Salamon & Stecker, 1994; Chiang et al., 1995; Stecker & Salamon, 1996; Kazanas & Perlman, 1997; Chiang & Mukherjee, 1998; Sreekumar et al., 1998; Mukherjee & Chiang, 1999; Mücke & Pohl, 2000; Giommi et al., 2006; Narumoto & Totani, 2006; Dermer, 2007; Kneiske & Mannheim, 2008; Pavlidou & Venters, 2008; Inoue & Totani, 2009; Venters et al., 2009; Abdo et al., 2010k; Venters, 2010). A detailed discussion of all of the assumptions that go into these calculations is beyond the scope of this paper (though, for a detailed discussion of the Chiang & Mukherjee 1998 calculation, see Stecker & Salamon 2001), and it is likely that many of these models will be updated in light of Fermi data. However, we note that as discussed in Abdo et al. (2010k), the source counts predicted by Dermer (2007) and Mücke & Pohl (2000) fall short of the Fermi observations of resolved sources above $5\times 10^{-8}\,{\rm ph}\,{\rm cm}^{-2}\,{\rm s}^{-1}$. In calculating the blazar contribution to the EGB, we assume functional forms for the blazar GLF and SID and fit them to 1FGL data, accounting for errors in measurement of blazar spectral indices and the spectral bias inherent in a flux-limited catalog. #### 4.1.1 Source Counts for Faint Unresolved Blazars: Theory Meets Observations Determining the GLF from observations relies on the ability to associate $\gamma$-ray blazars with lower energy counterparts for which redshifts can be measured (for discussion, see Venters et al., 2009; Venters, 2010). However, making the necessary association can be complicated by the angular resolution of the Fermi-LAT, which is limited by electron scattering in the LAT detector and is much poorer than that of more traditional telescopes (see Figure 3). The resulting wide point-spread function results in significant source confusion at energies below $\sim$ 1 GeV even for fluxes well above the Fermi- LAT sensitivity. In principle, one could construct source counts from fluxes integrated above an energy for which source confusion is less of a hinderance, but doing so would limit the already suppressed blazar number statistics. Thus, rather than construct a luminosity function solely from $\gamma$-ray blazars with redshifts, we employ the Stecker & Salamon (1996)777See also Narumoto & Totani (2006). approach of determining the luminosity function from wavebands with larger samples and smaller positional error circles. As in Stecker & Salamon (1996), we take the functional form of the FSRQ luminosity function from radio observations (Dunlop & Peacock, 1990), but corrected for the present cosmological parameters. The $\gamma$-ray luminosity of a blazar is then determined from its radio luminosity. The average correlation between the radio and $\gamma$-ray luminosities of blazars is determined by fitting the bright end of modeled source counts888In doing so, we also include the blazar SID (see Section 4.1.3). to that of the observed $\gamma$-ray source counts ($\chi^{2}_{\rm reduced}\sim 0.4$). In so doing, we find that $\gamma$-ray luminosity integrated from $100$ MeV to $100$ GeV is $\sim 10^{3.2}$ times $\nu L_{\nu}$ in radio, in agreement with the results obtained using recent Fermi observations999One might be concerned that the effect of the new radio-$\gamma$ correlation would be to increase the blazar background with respect to the Stecker & Salamon (1996) model. However, we note that Stecker & Salamon (1996) distinguished between “quiescent” and “flaring” blazars and used separate radio-$\gamma$ correlations (and spectral properties) for each subpopulation. In this paper, we make no such distinction, so a comparison between our results and those of the Stecker & Salamon (1996) model is not straightforward. Using the Stecker & Salamon (1996) radio-$\gamma$ correlation for quiescent blazars from Stecker & Salamon (1996) would yield a smaller blazar background, but it would also under- predict the bright end of the $\gamma$-ray source counts. To compensate, one would have to add a flaring component (as per Stecker & Salamon (1996)) to fit the data, which would also contribute to the background. In effect, such a procedure is equivalent to our method of fitting the radio-$\gamma$ correlation to the data. (Giroletti et al., 2010; Abdo et al., 2010j; Ghirlanda et al., 2010; Mahony et al., 2010). We should also note that we include sources out to $z\sim 5$, but as demonstrated in Venters et al. (2009), the emission is dominated by sources with $z\lesssim 2$ (consistent with expectations from the redshift distribution predicted by the GLF). As such, we do not expect this choice to significantly impact the results. The resulting modeled source count distribution for FSRQs (solid) is presented in Figure 2 along with the distributions of the brighter FSRQs resolved by Fermi (light data points) and all blazars (dark data points). We also show the forms of the unresolved source count distributions obtained from a Monte Carlo modeled Fermi-LAT sensitivity calculation performed by Abdo et al. (2010k) as dashed lines. All of the models separate from the data at fainter fluxes as the source counts fall off very rapidly due to the survey incompleteness and source confusion. Thus, the determination of the _true_ faint-end shape directly from the source counts can be severely hindered. In an effort to mitigate the effect of survey completeness, Abdo et al. (2010k) modeled the Fermi-LAT efficiency from a Monte Carlo simulation and then divided the differential source counts by this efficiency. Thus, the calculated source counts for faint sources strongly depend on such modeled efficiencies. Indeed at fluxes near the Fermi-LAT sensitivity limit, the efficiency is extremely small and model dependent; hence, in flux bins with low number statistics, the source counts are multiplied by a very large and uncertain number. The result of such a procedure is that even though one source is not statistically different from two sources, whether one source is seen or two could result in different modeled counts for faint sources. It has also been argued that unassociated $\gamma$-ray sources are likely to be dominated by known classes of $\gamma$-ray sources, especially blazars (Mirabal et al., 2010) and will have a contribution to the EGB even though they will not have been included in the source count distributions. It is also important to note that in determining the blazar contribution to the EGB from measured source count distributions, source confusion could not be taken into account since its effect depends on the source density (see Section 4.1.2), which is exactly the unknown quantity that the observer seeks to determine. As such, it is likely that the blazar contribution to the EGB will be underestimated in analyses based solely on the measured source count distributions. Since in this paper we use a theoretically determined source count distribution, our model gives a source count density from which one can determine the effect of source confusion (see Section 4.1.2). Such differences between our analysis and that given in Abdo et al. (2010k) result in different calculations of the unresolved blazar contribution to the EGB: Abdo et al. (2010k) conclude that blazars can only account for less than 25% of the EGB101010In Section 8 of Abdo et al. (2010k) a value of 40% of the EGB is obtained if one extrapolates to zero flux., while our analysis indicates that blazars could possibly account for the bulk of the EGB. The $F_{100}$ fluxes included in the source count distributions presented in Figure 1 are not actually _measured_ $F_{100}$ fluxes since the 1FGL catalog does not include $F_{100}$ fluxes. In order to determine the $F_{100}$ fluxes, we make use of Equation 1 of Abdo et al. (2010k) to _extrapolate_ $F_{100}$ fluxes from measured differential fluxes determined at the pivot energies. Abdo et al. (2010f) define the pivot energy as that energy for which the differential flux is minimal. We note that for 1FGL blazars, the average of the pivot energies is $\sim 1$ GeV. Thus, the source count distributions might be more representative of source counts for sources brighter than the Fermi sensitivity above $1$ GeV rather than $100$ MeV. In effect (and also as the result of spectral bias; see Section 4.1.3), the source counts could underestimate the number of blazars with $F_{100}$ fluxes above the Fermi sensitivity, mostly impacting the faint end of the source count distributions. As such, analyses on unresolved blazars based solely on source counts likely underestimate the blazar contribution to the EGB. This effect could provide an explanation for the fact that Fermi observes roughly as many BL Lacs as FSRQs even though BL Lacs are intrinsically fainter than FSRQs. Since the $\gamma$-ray spectra of BL Lacs are harder than those of FSRQs they are more easily observed by Fermi than FSRQs. Figure 2.— Bright end of the source count distributions for all blazars (including BL Lacs; black data points) and FSRQs (light green data points). Our model fit to the data is shown by the solid (purple) line. The dashed lines are the faint-end slopes determined in Abdo et al. (2010k) by including a modeled Monte Carlo Fermi-LAT efficiency. #### 4.1.2 The Fermi-LAT and EGRET Angular Resolutions and Source Confusion As mentioned previously, the large angular resolutions of pair-production $\gamma$-ray detectors such as EGRET and the Fermi-LAT111111http://www- glast.slac.stanford.edu/software/IS/glast_ lat_performance.htm . result in significant source confusion, particularly for faint sources and for energies below $\sim 1$ GeV. Since the angular resolution for the Fermi-LAT is similar to that of EGRET at $\sim 100-200$ MeV, the capability of Fermi to resolve faint sources is similar to that of EGRET at these energies. Hence, if the EGB does indeed consist of unresolved sources, then the EGB measurements of the detectors should be similar at $\sim 100-200$ MeV, whereas at $\sim 1$ GeV, the improved angular resolution of Fermi with respect to that of EGRET should result in a lower measurement of the EGB by Fermi than that of EGRET owing to the enhanced ability of Fermi to resolve point sources (Stecker & Salamon, 1999). The probability, $\cal{P}$, of finding a nearest neighboring source with $S\geq S_{\rm lim}$ within the minimum angular separation, $\theta_{\rm min}$, for a source density, $N$, is given by ${\cal{P}}(\leq\theta_{\rm min}(E))=1-\exp(-\pi N\theta^{2}_{\rm min}(E)).$ (21) For our source confusion criterion, we take the acceptable probability limit, ${\cal{P}}_{\rm min}\sim 0.1$, and $\theta_{\rm min}(E)\sim\theta_{67\%}(E)$ approximately given by $\theta_{67\%}(E)=5.12^{\circ}\times\left(\frac{E}{100\,{\rm MeV}}\right)^{-0.8}$ (22) (Atwood et al., 2009) (see Figure 3). Then, the source density criterion (SDC) is found by inverting Equation 21: $N_{\rm SDC}=-\frac{\ln(1-{\cal{P}}_{\rm min})}{\pi\theta^{2}_{\rm min}(E)}\,.$ (23) The limiting source flux, $S_{\rm SDC}$, is then determined from the modeled source counts. If one were to think of the Fermi-LAT as an ordinary telescope with $\theta_{\rm min}=\theta_{67\%}$ where $\theta_{67\%}$ is the half angle for a beam the contains $67\%$ of the photons. The source density criterion would correspond to $\sim 1/10$ sources per beam. For sources with $F_{100}\lesssim 1\times 10^{-7}\,\mbox{ph cm}^{-2}\mbox{ s}^{-1}$, the probability for finding another $\gamma$-ray source of similar or greater flux within the error circle is quite high; hence, many sources that, in principle, should be resolvable are, in fact, _unresolved_. At fluxes close to the sensitivity limit, this probability is so high that faint sources are _indistinguishable_ and will contribute to the measured EGB of the detector in question (for the Fermi-LAT resolution at $100$ MeV, the source criterion corresponds to $N_{\rm SDC}\sim 1.3\times 10^{-3}\,{\rm sources}/{\rm deg}^{2}$; for reference, at $F\sim 2\times 10^{-9}\,{\rm ph}\,{\rm cm}^{-2}\,{\rm s}^{-1}$, our model predicts a much larger source density of $\sim 0.3\,{\rm sources}/{\rm deg}^{2}$). Thus, the effect of source confusion compounded with the separate effect of detector sensitivity is to flatten the faint end of an observationally derived source count distribution. We should emphasize that this definition of source confusion is not the same as that employed by Abdo et al. (2010k). Source confusion as discussed in this section refers to the probability that for a given source density, a source has a nearest neighbor within the angular resolution of the detector with a flux greater than or equal to the flux limit. In Abdo et al. (2010k), the term is applied to a _detected_ source associated with a given _real_ source for which the measured flux of the detected source is greater than the flux of the real source (plus three standard deviations) by a given amount (the analysis was performed on simulated data to determine the impact on the actual data). In effect, the former definition identifies the limit at which the source density is sufficiently large so that individual sources _cannot_ be resolved, and only fluctuations are observed. The latter definition applies to the probability that a given detected source is in fact the superposition of several sources. However, the Abdo et al. (2010k) criterion is based on the assumption that a source can be _resolved_. As such, the source would have to be significantly brighter than the background, including the sources within the error circle of the detector. In the case of many sources with similar fluxes, none of the sources would be resolved. In the case of one bright source and several fainter sources, the flux of the detected source would be dominated by the flux of the brightest source and likely would escape the Abdo et al. (2010k). Thus, under this criterion, it is not surprising that they conclude that there is very little source confusion. A common treatment of source confusion in measured source counts is a fluctuation analysis, but as yet, such an analysis has not be performed on Fermi source counts. Figure 3.— The angular resolution as a function of energy for the Fermi-LAT (solid black line; Atwood et al., 2009) and EGRET (dashed red line; Thompson et al., 1993). #### 4.1.3 Blazar Spectra The spectral indices of the population of blazars form a distribution with a given finite spread (Stecker & Salamon, 1996; Venters & Pavlidou, 2007), resulting in curvature in the spectrum of the collective intensity of unresolved blazars due to the increasing relative importance at high energies of blazars with harder spectral indices (Stecker & Salamon, 1996; Pavlidou & Venters, 2008). Thus, the determination of the blazar SID is crucial in determining the correct spectrum of the unresolved blazar contribution. In determining the blazar SID from survey data, one must carefully account for uncertainties in measurement of the spectral indices. In so doing, we follow the likelihood method of Venters & Pavlidou (2007), fitting Fermi 1FGL FSRQs to a gaussian SID. We determined the maximum-likelihood gaussian SID parameters (mean, $\Gamma_{0}$, and spread, $\sigma_{0}$) to be $\Gamma_{0}=2.45$ and $\sigma_{0}=0.15$. We must also account for the effect of spectral bias of the 1FGL catalog. Even in a flux-limited catalog, low-luminosity, high-redshift blazars that would be most likely to appear are those with spectral indices that are harder than most of the population. The 1FGL catalog is not actually flux limited (see Section 2), rather it consists of sources above a given threshold in test statistic121212For 1FGL, this threshold is $25$, corresponding to a statistical significance of $\sim 4\sigma$., which depends on source spectra and the background. However, as demonstrated in Venters & Pavlidou (2010), applying the likelihood analysis to the sample of FSRQs with fluxes $\gtrsim 7\times 10^{-8}\,{\rm ph\,cm^{-2}\,s^{-1}}$ and galactic latitudes $\gtrsim 10^{\circ}$ (as per Abdo et al., 2010k) does not appreciably change the SID parameters ($\Gamma_{0}=2.45$, $\sigma_{0}=0.16$)131313See also Abdo et al. (2010k).. Thus, even though the 1FGL catalog is not flux limited, we can assume that the sample of 1FGL FSRQs is approximately flux limited. We can then follow the method of Venters et al. (2009) in correcting for the sample bias inherent in a flux-limited catalog. In doing so, we apply a correction factor, $\hat{M}(\alpha)$, to the SID (for derivation, see Venters et al., 2009): $p_{L}(\alpha)=\frac{\hat{p}(\alpha)}{\hat{M}(\alpha)}\,,$ (24) where $\hat{p}(\alpha)$ is the SID corrected for measurement uncertainty in the spectral indices, and $\hat{M}(\alpha)\propto\int_{F_{\gamma,\rm min}}^{\infty}dF_{\gamma}\frac{1}{F_{\gamma}}\int_{z=0}^{\infty}dz\hat{\rho}_{\gamma}(\alpha,z,F_{\gamma})\frac{dV_{\rm com}}{dz}(z)\,.$ (25) #### 4.1.4 Summary of Differences with the Stecker & Salamon (1996) Blazar Model To summarize, our approach in calculating the blazar contribution to the EGB is similar to that of Stecker & Salamon (1996) with some notable differences: 1. 1. The model has been updated to make use of the current cosmological parameters. The change in cosmology has little impact on the results. 2. 2. The model considers only FSRQs, and flaring blazars are not considered separately from quiescent blazars. (For possible impact, see item 3.) 3. 3. The $\gamma$-ray–radio relation has been updated to be consistent with multi- wavelength observations of FSRQs conducted by Fermi and radio telescopes ($L_{\gamma}\sim 10^{3.2}\times\nu L_{\nu}$ as opposed to $10^{2.6}$, see Section 4.1.1). As noted in Section 4.1.1, the smaller $\gamma$-ray–radio relation would result in a model that cannot by itself fit the data and would require a flaring component. In effect, this is, on average, equivalent to our choice of a single population of blazars with a given $\gamma$-ray–radio relation. We do not expect this choice to have a major impact on our results. However, as we will discuss in Section 4.1.5, the blazar duty cycle is a remaining uncertainty, and it could impact predictions for the number of blazars that are observable by Fermi. 4. 4. The model has been updated to account for effect of source confusion in the in blazar contribution to the EGB (see Sections 4.1.2 and 5). As will be shown in Section 5, this has the effect of increasing the blazar background at lower energies. 5. 5. The SID of FSRQs has been updated following the analysis of Venters & Pavlidou (2007) ($\Gamma_{0}=2.45$, $\sigma_{0}=0.15$) and correcting for spectral bias as in Venters et al. (2009) (see Section 4.1.3). Thus, the collective spectrum of blazars is not as hard as that presented in Stecker & Salamon (1996) and does not exhibit as much curvature. #### 4.1.5 Remaining Questions There remain a few open questions, the answers to which will impact the determination of the blazar contribution to the EGB. The blazar duty cycle, which dictates the amount of time a blazar spends in the quiescent state versus the flaring state, remains uncertain, as do questions of the amount the flux increases during flaring and whether the spectral index changes during flaring. Analyses of EGRET blazar spectral indices found no evidence of systematic changes in spectral index with flaring (Nandikotkur et al., 2007; Venters & Pavlidou, 2007), and Fermi observations of individual blazars have thus far revealed no systematic changes in spectral index with time or flux (Abdo et al., 2009d, 2010d; Ackermann et al., 2010b). As such, we are justified in assuming that the blazar spectral index remains constant, on average, with time. However, we acknowledge that the uncertainty of blazar variability parameters could have an impact on the counts of faint blazars and the $\gamma$-ray–radio correlation. This uncertainty will decrease as more data from Fermi become available. Another open question is that of the nature of blazar spectra over the entire Fermi energy range. We treat blazar spectra as unbroken power laws over this range and in many observed blazars, this does appear to be a reasonable approximation. However, in at least a few cases, Fermi has found evidence that blazar spectra can break (Abdo et al., 2009a, 2010i, 2010l). Whether such observations are representative of the entire blazar population is presently unclear, as is the nature of the breaks. In any case, spectral breaks are likely to impact the collective unresolved blazar spectrum mostly at the high end of the Fermi energy range. It is also possible that the spectra of harder blazars141414We do not include a possible contribution from BL Lacs for lack of a comparable radio data set. will compensate for that of softer blazars at higher energies (Venters & Pavlidou, 2010). ### 4.2. Star-forming Galaxies As discussed in Section 3.2, the Milky Way is a source of substantial $\gamma$-ray emission arising primarily the decay of $\pi^{0}$ meson produced in inelastic collisions of cosmic rays with interstellar gas. Thus, it is expected that other star-forming galaxies emit $\gamma$-rays through the same interactions and that unresolved star-forming galaxies could provide a substantial contribution to the EGB. Thus far, the Fermi-LAT Collaboration has reported detections of two nearby irregular galaxies (the SMC and the LMC; Abdo et al., 2010c, h), two starburst galaxies (M82 and NGC253; Abdo et al., 2010b), and M31, a galaxy similar to our own (Abdo et al., 2010g). As such, whatever the contribution to the EGB from unresolved star-forming galaxies, it will not have changed substantially in the Fermi data with respect to EGRET data as Fermi has resolved only a handful of star-forming galaxies. The determination of the star-forming galaxy contribution relies on knowledge of the star formation rate of galaxies and their gas content, both of which are subject to substantial observational and theoretical uncertainties. At relatively low redshifts ($z\lesssim 1.5$), nebular and forbidden emission lines (e.g., H$\alpha$, O II, and O III) can be used to trace star formation in galaxies151515For review of observational techniques of measuring star formation, see Kennicutt (1998).. At higher redshifts ($z\sim 1-5$), the redshifted UV continuum is used. However, both observational techniques are subject to uncertainties in dust extinction and the stellar initial mass function. Alternatively, in noting that UV radiation from young stars is absorbed by interstellar dust and reradiated in the infrared, measurements of the far-infrared (FIR) continuum can be used to trace star formation. Nevertheless, early-type galaxies can exhibit substantial FIR emission possibly due to dust heating by older stars or AGNs. Furthermore, infrared measurements are hindered by emission from our own Galaxy. Given these factors, the large degree of scatter present in the measurements (about a factor of a few; Le Borgne et al., 2009; Ly et al., 2011) of the cosmic SFR density is not surprising. The gas content of galaxies is even more uncertain, particularly at high redshift. The amount of ${\rm H_{2}}$ in a galaxy is determined from measurements of CO emission, while HI is determined from measurements of the $21$-cm line. However, the CO-to-${\rm H_{2}}$ conversion varies depending on the metallicity and radiation field of a given region and the opacity of the molecular clouds containing CO. The $21$-cm surveys, on the other hand, extend only out to $z\sim 0.05$. At higher redshifts, measurements of the HI density of the universe rely on damped Lyman-$\alpha$ absorbers observed in the Lyman-$\alpha$ forest of quasar spectra, but the nature of these systems is still the subject of much debate (see e.g., Kulkarni et al., 2010; Péroux et al., 2010). Furthermore, the connection between the total gas and star formation rate is complex (Putman et al., 2009). While it is fairly well established that stars form in GMCs and hence that star formation traces ${\rm H_{2}}$, the amount of HI varies between galaxies and does not appear to be correlated with star formation (Bigiel et al., 2008; Leroy et al., 2008). From both the observational and the theoretical points of view, the transition from HI to ${\rm H_{2}}$ and the formation of GMCs remain uncertain (Leroy et al., 2008). Given the uncertainty surrounding key elements of the determination of the star-forming galaxy contribution to the EGB, our approach does not focus on a particular model. Instead, we employ several families of models that rely on separate sets of assumptions each with advantages and caveats. Using this approach we seek to explore various possibilities for the star-forming galaxy contribution and highlight the uncertainty in such a calculation. The strategies we employ are summarized as follows: 1. 1. Relate the galaxy gas mass to its stellar mass assuming a gas fraction that evolves with redshift. 2. 2. Relate the galaxy $\gamma$-ray luminosity to its SFR, which, in turn, is related to an observable for which there is a redshift distribution (e.g., IR luminosity). 3. 3. Relate the cosmic density of gas in star-forming galaxies to the star formation rate density. #### 4.2.1 The Schechter Function Model In this approach, we relate the galaxy gas mass to its stellar mass assuming an evolving gas fraction (for details of the model, see Section 3.2.1). We employ the Schechter parameters of the stellar mass functions as determined by Elsner et al. (2008) and the evolving gas fraction as determined by Papovich et al. (2010). Since extensive spectroscopic surveys are, as yet, unavailable, Elsner et al. (2008) make use of combined data from the multi-band photometry of the GOODS-MUSIC catalog and the Spitzer Space Telescope. In so doing, they infer the stellar masses of galaxies from photometric data by fitting the mass-to-light ratios of galaxies to stellar population templates. Such a procedure is subject to a considerable degree of uncertainty, particularly arising from that of dust extinction161616In order to mitigate the effect of the uncertainty in dust extinction, Elsner et al. (2008) make use of ${\rm K_{S}}$-band M/L ratios since dust absorption is small for longer wavelengths. and the usage of photometric redshifts. Elsner et al. (2008) estimate the mean uncertainty in their stellar mass estimates to be about a factor of two, though they did not estimate the possible uncertainty resulting from the usage of photometric redshifts. Papovich et al. (2010) study the relationship between the star formation rate and stellar mass of high redshift galaxies selected at a constant comoving number density and derive a gas fraction that evolves171717Note that we extrapolate the functional form of their gas fraction to low redshifts. as $(1+z)^{0.9}$. They estimate that the uncertainty on the gas mass is $\sim$ 0.11 dex. The advantage of this kind of model is that it is based on observations that will be continuously refined with time. However, we note that the relationship between the stellar mass of a galaxy and its total gas content is unclear, though perhaps with better observations and a better theoretical understanding, the relationship will be better determined. #### 4.2.2 The IR Luminosity Function Models Figure 4.— The $\gamma$-ray luminosities of Local Group galaxies plotted versus their star formation rates (data points; see Table 1). Also plotted is the power-law fit relating the $\gamma$-ray luminosity to the SFR (solid red line): $L_{\gamma}\propto\Psi^{1.2}$. Table 1Observables for Local Group Galaxies ID181818 _SMC:_ $L_{\gamma}$ from Abdo et al. (2010c). SFR from Lawton et al. (2010) corrected for IMF. _LMC:_ $L_{\gamma}$ from Abdo et al. (2010h). SFR from Lawton et al. (2010) corrected for IMF. _M31:_ $L_{\gamma}$ from Abdo et al. (2010g). SFR from Tabatabaei & Berkhuijsen (2010) and Williams (2003) corrected for IMF. _MW:_ $L_{\gamma}$ calculated for $\left<q_{\rm H}\right>\sim 1.4\times 10^{-25}\,{\rm s^{-1}\,H^{-1}}$ and $M_{\rm H}\sim 7\times 10^{9}M_{\odot}$ (Boissier & Prantzos, 1999). SFR taken from Robitaille & Whitney (2010) corrected for IMF. _NGC253:_ $L_{\gamma}$ from Abdo et al. (2010b). SFR from IR measurements given by Sanders et al. (2003). _M82:_ $L_{\gamma}$ from Abdo et al. (2010b). SFR from IR measurements given by Sanders et al. (2003). Note that the errors in the SFRs reflect only statistical uncertainties and assume that Equation 15 is exactly correct. \| $L_{\gamma}\,({\rm ph\,s^{-1}})$ \| $\Psi\,(M_{\odot}\,{\rm yr^{-1}})$ ---\|---\|--- SMC \| $(1.7\pm 0.3)\times 10^{40}$ \| $0.01\pm(4.8\times 10^{-4})$ LMC \| $(7.8\pm 0.6)\times 10^{40}$ \| $0.1\pm(4.6\times 10^{-3})$ M31 \| $(6.6\pm 1.4)\times 10^{41}$ \| $0.4\pm 0.18$ Milky Way \| $(1.2\pm 0.2)\times 10^{42}$ \| $0.93\pm 0.34$ NGC253 \| $(1.1\pm 0.7)\times 10^{43}$ \| $2.7\pm 0.5$ M82 \| $(2.5\pm 0.9)\times 10^{43}$ \| $5.4\pm(5.9\times 10^{-4})$ In this approach, we assume that the $\gamma$-ray luminosity of a star-forming galaxy can be related to its star formation rate as a power law. In order to determine this relationship, we fit $\gamma$-ray luminosities of Local Group galaxies calculated from Fermi measurements191919For the Milky Way, we calculate the $\gamma$-ray luminosity from $q_{\rm H}$ averaged over the whole galaxy (as discussed in Section 3.2) and the $M_{\rm H}$ taken from Boissier & Prantzos (1999). to their star formation rates either taken from the literature or calculated from IR measurements and converted to SFR via Equation 15. In all cases, the SFR is calculated assuming (or corrected to) the IMF of Chabrier (2003). We find the best-fit power law to be given by $L_{\gamma}\,[10^{41}{\rm ph\,s^{-1}}]\sim 24.0\times\Psi^{1.2}$. This fit is consistent with that obtained by the Fermi Collaboration, $L_{\gamma}\propto\Psi^{1.4\pm 0.3}$ (Abdo et al., 2010g). Observables for Local Group galaxies used in this analysis are given in Table 1 and plotted with the fit in Figure 4. In a manner similar to that of blazars, we can relate the $\gamma$-ray luminosity of a star-forming galaxy to its total IR luminosity, convolve with an IR luminosity function, and integrate with respect to IR luminosity and redshift. While the use of IR luminosity functions taken from observations is possible, it is difficult to deconvolve the contributions from obscured AGNs and mergers. As such, we use the semi-empirical IR luminosity functions determined from the halo-occupation–based methodology of Hopkins et al. (2010) for both star-forming and starburst galaxies (in which enhanced star formation due to major mergers is taken into account)202020Note that we do not consider any contribution from the so-called calorimetry effect for starburst galaxies as such an effect is likely to be small (Stecker, 2007). We have assumed the same form of the $\pi^{0}$-decay spectrum for starburst galaxies as for star- forming galaxies.. While the Hopkins et al. (2010) IR luminosity functions match available observations fairly well, we note that there is considerable debate over the roles of AGNs and mergers in driving star formation, the evolution of galaxies, and the determination of the IR luminosities of massive systems. Furthermore, the $\gamma$-ray–SFR correlation is based on rather uncertain estimates of the SFRs and $\gamma$-ray luminosities of one normal galaxy (our own), two irregular galaxies, and two starburst galaxies, all of which could, in principle, exhibit different star formation properties. As more data become available from Fermi, the $\gamma$-ray–SFR correlation will be further tested. If it proves robust, then studies of the normal galaxy contribution the EGB could have implications for large-scale–structure formation and the evolution of galaxies with cosmic time. #### 4.2.3 The Strong Coupling $\gamma$-ray–Star Formation Rate Model In this approach, we seek to relate the cosmic density of hydrogen in star- forming galaxies to the cosmic star formation rate. Observations of nearby galaxies have indicated that localized star formation traces the density of ${\rm H_{2}}$ but there is no direct correlation with HI (Bigiel et al., 2008; Leroy et al., 2008). While in principle, there should be some relationship between the amount of HI, the amount of ${\rm H_{2}}$, and the SFR in a galaxy, other factors such as density fluctuations and turbulence make such a relationship complex. Therefore, while acknowledging that the relationship between HI and SFR is unclear, we simply assume that the amount of HI in star- forming galaxies is, _on average_ , comparable to that of ${\rm H_{2}}$ within the optical radius of the galaxy (see Section 3.2.3). Hence, in this model, we take the $\gamma$-ray luminosity to be roughly proportional to the _square_ of the SFR. We stress that the assumption that the star formation rate is proportional to the available gas density _only applies to galaxies that are actively forming stars._ This assumption does not apply to galaxies at very high redshifts which may contain substantial amounts of gas but have yet to begin forming stars. However, we only include star-forming galaxies out to $z\sim 4$, so the higher redshift galaxies will not impact on our results. Given that the best-fit power-law index determined for Local Group galaxies in Section 4.2.2 is $\sim 1.2$ and given the proximity of the resulting unresolved spectrum to the Fermi measurements of the EGB (see Section 5), we consider this model to be reflective of an upper limit to the star-forming galaxy contribution to the EGB. ### 4.3. The Fermi Spectrum and Unresolved Sources vs. Truly Diffuse Mechanisms The Fermi observations have placed significant constraints on extragalactic dark matter annihilation (Cirelli et al., 2010; Abdo et al., 2010a; Ackermann et al., 2010a). Currently, there is no evidence of quark-annihilation features and spectral lines seen in the EGB spectrum, features that would be a clear annihilation signal (see e.g., Stecker & Tylka, 1989a; Rudaz & Stecker, 1991). The observed spectrum does not match that expected from dark matter annihilation, placing constraints on any dark matter annihilation contribution to the EGB (Abdo et al., 2010a). Therefore, it is probable that dark matter annihilation $\gamma$-rays, if present, provide only a minor contribution to the EGB. The same argument about matching spectra can be made regarding the contribution from electromagnetic cascades produced by very high and ultrahigh energy cosmic-ray interactions as the resulting spectrum would be significantly harder than the observed spectrum (Kalashev et al., 2009; Berezinsky et al., 2011; Ahlers et al., 2010; Venters, 2010). ## 5\. Results The calculated spectrum of the unresolved FSRQ contribution to the EGB (see Sections 3.1 and 4.1) is plotted in Figure 5. For comparison, we include the Fermi analysis of the EGB (Abdo et al., 2010j), two analyses212121The two sets of EGRET data points result from two different estimations of the galactic foreground emission. of the EGRET EGB (Sreekumar et al., 1998; Strong et al., 2004a), and the calculation of the collective spectrum of unresolved FSRQs ignoring the effect of source confusion. Our results clearly show that the effect of source confusion is to reduce the number of resolved sources, increasing the collective intensity of unresolved blazars, particularly below $\sim 1$ GeV energy. Thus, accounting for source confusion modifies the predicted spectrum such that the EGRET and Fermi measurements of the EGB below $\sim 1$ GeV are both compatible with unresolved FSRQs. In contrast, the better angular resolution of the Fermi-LAT above $\sim$ 1 GeV allows it to resolve more blazars resulting in a limiting flux that is dominated by the Fermi-LAT sensitivity rather than source confusion. Thus, the collective spectrum of FSRQs breaks at $\sim 3$ GeV222222The actual break should be more gradual since in our calculations we used the approximate broken angular resolution curve shown in Figure 3.. At energies above $\sim 1$ GeV, the predicted collective spectrum of FSRQs falls below the data points, though they are likely consistent with the data within the uncertainties in the galactic foreground emission model. Note also that the collective FSRQ spectrum exhibits much less curvature than seen in Stecker & Salamon (1996). This is because the spread in the SID in our current model is much smaller than that of the Stecker & Salamon (1996) model. In Figure 6, we plot the spectra of the unresolved star-forming galaxy contributions to the EGB calculated for the models discussed in Sections 3.2 and 4.2. For comparison, we include the spectrum of the unresolved starburst galaxy contribution alone that we determined from the best-fit IR luminosity function of Hopkins et al. (2010). For the spectrum of starburst galaxies, we have assumed the same form of the $\pi^{0}$ decay spectrum as for star-forming galaxies. The range in the calculations of the overall contribution to the EGB from unresolved star-forming galaxies spans about an order of magnitude indicating the degree of uncertainty in such a calculation232323Though, we note that each individual model is subject to its own uncertainty. As such, the degree of uncertainty is likely even more than an order of magnitude. Within the range of our various predictions of the EGB from star forming galaxies, we agree with the results of the model of Fields et al. (2010) and Makiya et al. (2011). We note that even though our most extreme model could possibly explain the lowest energy Fermi data points (and possibly, within systematics, a couple others), it cannot explain the EGRET data points below $300$ MeV. The Strong et al. (2004a) EGRET data points (minus the two highest energy data points) with the Fermi data points resemble a _featureless_ power law, while the spectra of unresolved star-forming galaxies do not. Notably, the data points show no indication of a $\pi^{0}$-decay “bump” at the energies at which the contribution of the star-forming galaxies should peak. Figure 5.— The collective spectrum of unresolved FSRQs. _Solid green line:_ The spectrum accounting for source confusion. _Dashed green line:_ The spectrum without accounting for source confusion. _Black circles:_ The Fermi measurement of the spectrum of the EGB as determined in Abdo et al. (2010j). _Blue squares:_ The EGRET measurement of the spectrum of the EGB as determined by Sreekumar et al. (1998) and confirmed by the analysis of Stecker et al. (2008) and S. D. Hunter (private communication). _Red triangles:_ The EGRET measurement of the spectrum of the EGB as determined by Strong et al. (2004a). Figure 6.— The collective spectrum of unresolved star-forming galaxies. _Dashed indigo line:_ The spectrum determined from the strong coupling model (see Sections 3.2.3 and 4.2.3. _Solid blue line:_ The spectrum determined from the IR luminosity function model (see Sections 3.2.2 and 4.2.2). _Dot-dashed yellow line:_ The spectrum determined from the IR luminosity function model assuming no gas evolution. _Dashed red line:_ The spectrum determined from the Schechter function model (see Sections 3.2.1 and 4.2.1). _Double dot-dashed line:_ The spectrum of the starburst contribution alone determined from the IR luminosity function model. ## 6\. Discussion & Conclusions We have calculated the spectral shape of the contribution of unresolved FSRQs to the EGB assuming that the $\gamma$-ray luminosity of an FSRQ is, on average, proportional to its radio luminosity (Giroletti et al., 2010; Abdo et al., 2010j; Ghirlanda et al., 2010; Mahony et al., 2010), and also accounting for the effects of source confusion. We have demonstrated that the combination of the source density predicted by the Dunlop & Peacock (1990) FSRQ radio luminosity function and the strong energy dependence of the Fermi-LAT angular resolution _increases_ the contribution of unresolved FSRQs to the EGB at energies below $1$ GeV. The resulting overall spectrum predicted by the fit to the Fermi source count distribution reproduces well the spectrum of the EGRET and Fermi EGB measurements below $1$ GeV, but falls below the data points above $1$ GeV. We have also calculated the spectral shape of the contribution of unresolved star-forming galaxies to the EGB for several relations for the $\gamma$-ray luminosity of a star-forming galaxy. We find that, depending on the model, the overall amount of the contribution of star-forming galaxies to the EGB may be more or less significant, though regardless of the model considered, the spectrum of unresolved star-forming galaxies is unable to explain the combined spectrum of the low-energy EGRET EGB measurements and the Fermi EGB measurements. Similar calculations for starburst galaxies alone indicate that they account for at most about $1$% of the EGB, in agreement with the conclusion reached by Stecker (2007). The similarity of the collective spectrum of unresolved FSRQs to the combined spectrum of the EGRET and Fermi EGB measurements, as demonstrated by our results, is striking. In fact, we note that as predicted in Stecker & Salamon (1999), the inclusion of the effect of source confusion in the calculation could provide an explanation for the similarity between the EGRET and Fermi EGB measurements at energies of hundreds of MeV. The density of FSRQs predicted by the model is sufficiently large such that at these energies, Fermi would not be able to resolve many more FSRQs than EGRET did, and the FSRQ contribution to the EGB would remain the same for Fermi as for EGRET. Thus, if unresolved FSRQs _do_ comprise the bulk of the EGB emission, then one would expect such similarity between the EGRET and Fermi measurements at these energies. At energies above $1$ GeV, the Fermi-LAT angular resolution improves substantially with respect to that of EGRET, and as such, Fermi would be able to resolve more blazars at higher energies than EGRET could, resulting in a decrease in the Fermi EGB with respect to the EGRET EGB, an effect which is possibly indicated by comparing the EGRET and Fermi results242424A caveat is that the uncertainties in the subtraction of the galactic foreground emission at the higher energies are considerable owing to the uncertainty in the distributions of both gas and cosmic rays in the Galaxy. Furthermore, the instrumental backgrounds of EGRET and the Fermi-LAT are different, so it is difficult to make a direct comparison between the two. We should also note that in our calculations, we have neglected the population of BL Lacs, which, due to their hard spectra, are likely to have more of a contribution at energies above $\sim 10$ GeV. Notably, Fermi has resolved as many BL Lacs as FSRQs.. In contrast, no such high-energy separation between the EGRET EGB and the Fermi EGB is predicted for star-forming galaxies as all but the closest are too faint to be resolvable by Fermi. Furthermore, we note that the EGRET EGB measurements provide no indication of a turnover in the spectrum as would be expected if unresolved star-forming galaxies comprise the bulk of the EGB emission. Rather, the spectrum of unresolved star-forming galaxies is _inconsistent_ with the combined spectrum of the EGRET and Fermi EGB measurements252525As previously noted, the Fermi-LAT was designed to reach its optimal effective area for $\gamma$-rays with energies near and above $\sim 1$ GeV, whereas EGRET was designed to reach its optimal effective area for $\gamma$-rays with energies near and above $\sim 100$ MeV. Also, the Fermi-LAT detector has a significantly higher instrumental background at $100$ MeV than EGRET did (S. D. Hunter, private communication). Thus, the EGB was not reported by Fermi for energies below $200$ MeV (Abdo et al., 2010j).. We also note that the lack of a turnover in the EGRET data is not simply the result of systematics (S. D. Hunter, private communication), since the uncertainties in all of the galactic foreground models used to determine the EGB from the EGRET and Fermi data are quite small at these energies. Finally, we note that at energies above $\sim 1$ GeV, the spectrum of unresolved star-forming galaxies is steeper than the spectra of the EGB data. As such, we conclude that however significant the contribution of star-forming galaxies to the EGB may be, it is not sufficient to explain the EGB262626The effect of Compton interactions mentioned in Section 3.2 does not alter this conclusion as it only modifies the spectrum above 10 GeV for normal galaxies (Strong et al., 2010) and the starburst galaxy contribution to the EGB is negligible (See Figure 6.). Within the range of our various predictions of the EGB from star forming galaxies, we agree with the results of the models of both Fields et al. (2010) (which suggests that star-forming galaxies may comprise the bulk of the EGB) and Makiya et al. (2011) (which suggests that star-forming galaxies can account for less than $10\%$ of the EGB). This underscores the range of uncertainty in the calculation for star-forming galaxies272727One noteworthy difference between our model and that of Fields et al. (2010) is that in order to relate the gas mass of a galaxy to its star formation rate, Fields et al. (2010) makes use of the Schmidt-Kennicutt relation. However, in doing so, they estimate the disk sizes of galaxies to high redshifts. Given that the uncertainties in these quantities are likely to be considerable (and given the uncertainty already present in the calculation), we considered alternative approaches. Nevertheless, we note that the Schmidt-Kennicutt relation was included in the inputs to both the IR luminosity models and the Schechter function model. In the IR luminosity model, we tested the impact of changing the Schmidt-Kennicutt law by performing the calculation for the Hopkins et al. (2010) IR luminosity function calculated using a steeper Schmidt-Kennicutt relation and found that it had very little impact on our results.. The featureless spectrum of the EGB deduced by Fermi is intriguing when one considers the possibility of features that could arise from phenomena such as breaks in blazar spectra, absorption of high-energy $\gamma$-rays from unresolved blazars, $\gamma$-ray emission from unresolved star-forming galaxies, $\gamma$-ray emission from dark matter annihilation, and $\gamma$-rays from electromagnetic cascades initiated by very high and ultrahigh energy particle interactions with the extragalactic background light. The spectra of these potential contributions to the EGB differ considerably from that of the FSRQs (Silk & Srednicki, 1984; Stecker et al., 1985; Rudaz & Stecker, 1988; Stecker & Tylka, 1989a, b; Rudaz & Stecker, 1991; Ullio et al., 2002; Ando et al., 2007; Kalashev et al., 2009; Siegal-Gaskins & Pavlidou, 2009; Berezinsky et al., 2011; Ahlers et al., 2010; Venters, 2010). However, recent Fermi observations have placed significant constraints on dark matter annihilation (Cirelli et al., 2010; Abdo et al., 2010a; Ackermann et al., 2010a), and presently there is no clear evidence of annihilation features above the background continuum. As such, it appears that any putative contribution to the EGB from dark matter annihilation is relatively minor. The possible contribution to the EGB from electromagnetic cascades is constrained by the relative steepness of the EGB spectrum, though cascades could play a role at higher energies (Kalashev et al., 2009; Berezinsky et al., 2011; Ahlers et al., 2010; Venters, 2010). An apparent explanation for the featureless power-law spectrum of the EGB as presently deduced could be that unresolved blazars provide the dominant contribution to the EGB, given that their collective spectrum is roughly consistent with that of the EGB. Therefore, we conclude that, contrary to the result given by Abdo et al. (2010k), the Fermi observations do not rule out the possibility that the EGB is dominated by emission from unresolved blazars. ## Acknowledgments We thank David Thompson, Stan Hunter, and Olaf Reimer for discussions of the EGRET detector characteristics and EGRET data. We thank Marco Ajello for sending the results of his Monte Carlo simulations of the Fermi-LAT efficiency vs. source flux. 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1012.3828	# The model checking problem for intuitionistic propositional logic with one variable is $\mathsf{AC^{1}}$-complete Martin Mundhenk and Felix Weiß ###### Abstract. We show that the model checking problem for intuitionistic propositional logic with one variable is complete for logspace-uniform $\mathsf{AC^{1}}$. As basic tool we use the connection between intuitionistic logic and Heyting algebra, and investigate its complexity theoretical aspects. For superintuitionistic logics with one variable, we obtain $\mathsf{NC^{1}}$-completeness for the model checking problem. Universität Jena, Institut für Informatik, Jena, Germany {martin.mundhenk,felix.weiss}@uni-jena.de ## 1\. Introduction Intuitionistic logic (see e.g. [10, 23]) is a part of classical logic that can be proven using constructive proofs–e.g. by proofs that do not use _reductio ad absurdum_. For example, the law of the excluded middle $a\vee\neg a$ and the weak law of the excluded middle $\neg a\vee\neg\neg a$ do not have constructive proofs and are not valid in intuitionistic logic. Not surprisingly, constructivism has its costs. Whereas the validity problem is $\mathsf{coNP}$-complete for classical propositional logic [6], for intuitionistic propositional logic it is $\mathsf{PSPACE}$-complete [19, 20]. The computational hardness of intuitionistic logic is already reached with the fragment that has only formulas with two variables: the validity problem for this fragment is already $\mathsf{PSPACE}$-complete [18]. Recall that every fragment of classical propositional logic with a fixed number of variables has an $\mathsf{NC^{1}}$-complete validity problem (follows from [2]). In this paper, we consider the complexity of intuitionistic propositional logic $\mathrm{IPC}$ with one variable. The model checking problem—i.e. the problem to determine whether a given formula is satisfied by a given intuitionistic Kripke model—for $\mathrm{IPC}$ is $\mathsf{P}$-complete [13], even for the fragment with two variables only [14]. More surprisingly, for the fragment with one variable $\mathrm{IPC}_{1}$ we show the model checking problem to be $\mathsf{AC^{1}}$-complete. To our knowledge, this is the first “natural” $\mathsf{AC^{1}}$-complete problem, whereas formerly known $\mathsf{AC^{1}}$-complete problems (see e.g. [1]) have some explicit logarithmic bound in the problem definition. A basic ingredient for the $\mathsf{AC^{1}}$-completeness lies in normal forms for models and formulas as found by Nishimura [16], that we reinvestigate under an algorithmic and complexity theoretical point of view. In contrast, the formula value problem for classical propositional logic is $\mathsf{NC^{1}}$-complete [2] independent of the number of variables. Classical propositional logic is the extension of $\mathrm{IPC}$ with the axiom $a\vee\neg a$. Those proper extensions of intuitionistic logic are called superintuitionistic logics. The superintuitionistic logic $\mathrm{KC}$ (see [9]) results from adding $\neg a\vee\neg\neg a$ to $\mathrm{IPC}$. We show that the model checking problem for every superintuitionistic logic with one variable is $\mathsf{NC^{1}}$-complete (and easier than that for $\mathrm{IPC}_{1}$). In contrast, for the superintuitionistic logic $\mathrm{KC}$ with two variables it is known to be $\mathsf{P}$-complete (and as hard as for $\mathrm{IPC}$ with two variables) [14]. As a byproduct, we also obtain results for the validity problem for intuitionistic and superintuitionistic logics with one variable. This paper is organized as follows. In Section 2 we introduce the notations we use for intuitionistic logic and model checking. Section 3 is devoted to introduce the old results by Nishimura [16] and to upgrade them with a complexity analysis. The following Section 4 presents our lower and upper bound for model checking for $\mathrm{IPC}_{1}$. Section 5 deals with the complexity of the model checking problem and the validity problem for superintuitionistic logics with one variable. The implied completeness for the model checking for intuitionistic logic and conclusions are drawn in Section 6. ## 2\. Preliminaries ### Complexity (see e.g. [24]) The notion of reducibility we use is the logspace many-one reducibility $\leq_{\mathrm{m}}^{\mathrm{log}}$, except for $\mathsf{NC^{1}}$-hardness, where we use first-order reducibility. $\mathsf{NC^{1}}$ and $\mathsf{AC^{1}}$ are the classes of sets that are decided by families of logspace-uniform circuits of polynomial size and logarithmic depth. The circuits consist of and-, or-, and not-gates. The not-gates have fan-in $1$. For $\mathsf{NC^{1}}$, the and- and or-gates have fan-in $2$ (bounded fan-in), whereas for $\mathsf{AC^{1}}$ there is no bound on the fan-in of the gates (unbounded fan-in). $\mathsf{ALOGTIME}$ denotes the class of sets decided by alternating Turing machines in logarithmic time, and we will use that $\mathsf{NC^{1}}=\mathsf{ALOGTIME}$ (see [17]). $\mathsf{L}$ denotes the class of sets decidable in logarithmic space. We use $\mathsf{ALOGSPACE[\mbox{$f(n)$}]}$ to denote the class of sets decided by an alternating logspace Turing machine that makes $O(f(n))$ alternations, where $n$ is the length of the input. We will use that $\mathsf{AC^{1}}=\mathsf{ALOGSPACE[\mbox{$\log n$}]}$ (see [7]). $\mathsf{LOGdetCFL}$ is the class of sets that are $\leq_{\mathrm{m}}^{\mathrm{log}}$-reducible to deterministic context-free languages. It is also characterized as the class of sets decidable by deterministic Turing machines in polynomial-time and logarithmic space with additional use of a stack [5]. The inclusion structure of the classes we use is as follows. $\mathsf{NC^{1}}~{}~{}\subseteq~{}~{}\mathsf{L}~{}~{}\subseteq~{}~{}\mathsf{LOGdetCFL}~{}~{}\subseteq~{}~{}\mathsf{AC^{1}}~{}~{}\subseteq~{}~{}\mathsf{P}~{}~{}\subseteq~{}~{}\mathsf{PSPACE}$ ### Intuitionistic Propositional Logic (see e.g. [23]) Let $\operatorname{VAR}$ denote a countable set of variables. The language $\mathcal{IL}$ of intuitionistic propositional logic is the same as that of propositional logic $\mathrm{PC}$, i.e. it is the set of all formulas of the form $\phi$ \| $::=$ \| $p\mid\bot\mid(\phi\land\phi)\mid(\phi\lor\phi)\mid(\phi\rightarrow\phi),$ ---\|---\|--- where $p\in\operatorname{VAR}$. For $i\geq 0$ the languages $\mathcal{IL}_{i}$ are the subsets/fragments of $\mathcal{IL}$ for which $\operatorname{VAR}$ consists of $i$ variables. In this paper we mainly consider $\mathcal{IL}_{1}$ (i.e. formulas with one variable). As usual, we use the abbreviations $\neg\phi:=\phi\rightarrow\bot$ and $\top:=\neg\bot$. Because of the semantics of intuitionistic logic, one cannot express $\land$ or $\lor$ using $\rightarrow$ and $\bot$. An _intuitionistic Kripke model_ for intuitionistic logic is a triple $\mathcal{M}=(U,R,\xi)$, where $U$ is a nonempty and finite set of states, $R$ is a preorder on $U$ (i.e. a reflexive and transitive binary relation), and $\xi:\operatorname{VAR}\to\mathfrak{P}(U)$ is a function111$\mathfrak{P}(U)$ denotes the powerset of $U$. — the valuation function. Informally speaking, for any variable it assigns the set of states in which this variable is satisfied. The valuation function $\xi$ is monotone in the sense that for every $p\in\operatorname{VAR}$, $a,b\in U$: if $a\in\xi(p)$ and $aRb$, then $b\in\xi(p)$. $(U,R)$ can also be seen as a directed graph. Given an intuitionistic Kripke model $\mathcal{M}=(U,\leqslant,\xi)$ and a state $s\in U$, the satisfaction relation for intuitionistic logics $\models$ is defined as follows. $\mathcal{M},s\not\models\bot$ \| \| ---\|---\|--- $\mathcal{M},s\models p$ \| iff \| $s\in\xi(p),~{}p\in\operatorname{VAR},$ $\mathcal{M},s\models\phi\land\psi$ \| iff \| $\mathcal{M},s\models\phi\text{~{}and~{}}\mathcal{M},s\models\psi,$ $\mathcal{M},s\models\phi\lor\psi$ \| iff \| $\mathcal{M},s\models\phi\text{~{}or~{}}\mathcal{M},s\models\psi,$ $\mathcal{M},s\models\phi\rightarrow\psi$ \| iff \| $\forall n\geqslant s:\text{~{}if~{}}\mathcal{M},n\models\phi\text{~{}then~{}}\mathcal{M},n\models\psi$ A formula $\phi$ is satisfied by an intuitionistic Kripke model $\mathcal{M}$ in state $s$ if $\mathcal{M},s\models\phi$. A tautology is a formula that is satisfied by every intuitionistic Kripke model. Such formulas are also called valid. From the monotonicity of $\xi$ and the definition of $\models$ follows the monotonicity for every formula, i.e. for $\phi\in\mathcal{IL}$, $w,v\in W$ and $w\leqslant v$ if $\mathcal{M},w\models\phi$, then $\mathcal{M},v\models\phi$. ### The Model Checking Problem This paper examines the complexity of model checking problems for intuitionistic logics. * Problem: $\mathrm{IPC}_{1}$-Mc * Input: $\langle\phi,\mathcal{M},s\rangle$, where $\phi\in\mathcal{IL}_{1}$, $\mathcal{M}$ is an intuitionistic Kripke model, and $s$ is a state of $\mathcal{M}$ * Question: $\mathcal{M},s\models\phi$ ? We assume that formulas and intuitionistic Kripke models are encoded in a straightforward way. This means, a formula is given as a text, and the graph $(U,R)$ of an intuitionistic Kripke model is given by its adjacency matrix that takes $\|U\|^{2}$ bits. ## 3\. Properties of $\mathrm{IPC}_{1}$ and its complexity ### Formulas with one variable The set $\mathcal{IL}_{1}$ of formulas with at most one variable is partitioned into infinitely many equivalence222$\alpha$ is equivalent to $\beta$ if every state in every intuitionistic Kripke model satisfies both or none formula. We write $\alpha\equiv\beta$. classes [16]. This was shown using the formulas that are inductively defined as follows (see e.g.[10]). We use $a$ for the only variable. $\displaystyle\upvarphi_{1}$ $\displaystyle:=\neg a$ $\displaystyle\uppsi_{1}$ $\displaystyle:=a$ $\displaystyle\upvarphi_{n+1}$ $\displaystyle:=\upvarphi_{n}\rightarrow\uppsi_{n}$ $\displaystyle\hskip 17.22217pt\uppsi_{n+1}$ $\displaystyle:=\upvarphi_{n}\vee\uppsi_{n}\text{ ~{}~{}~{}~{}for }n\geq 1$ The formulas $\bot,\top,\upvarphi_{1},\uppsi_{1},\upvarphi_{2},\uppsi_{2},\ldots$ are called _Rieger-Nishimura formulas_. ###### Theorem 3.1. ([16], cf.[10, Chap.6,Thm.7]) Every formula in $\mathcal{IL}_{1}$ is equivalent to exactly one of the Rieger-Nishimura formulas. The function $\mathit{RNindex}$ maps every formula to the index of its equivalent Rieger-Nishimura formula. We call this index Rieger-Nishimura index. $\mathit{RNindex}(\alpha)$ \| $=$ \| $\left\\{\begin{array}[]{ccl}(i,\mathit{phi}),&\text{~{}if~{}}&\alpha\equiv\upvarphi_{i}\\\\[-3.01125pt] (i,\mathit{psi}),&\text{~{}if~{}}&\alpha\equiv\uppsi_{i}\\\\[-3.01125pt] (0,\bot),&\text{~{}if~{}}&\alpha\equiv\bot\\\\[-3.01125pt] (0,\top),&\text{~{}if~{}}&\alpha\equiv\top\end{array}\right.$ ---\|---\|--- In the following we analyse the complexity of $\mathit{RNindex}$. For $\phi\in\mathcal{IL}_{1}$ let $[\phi]$ denote the equivalence class that contains $\phi$. The equivalence classes of $\mathcal{IL}_{1}$ form a free Heyting algebra over one generator (for algebraic details see [11]). This algebra is also called the _Rieger-Nishimura lattice_ (see Fig. 1). It is shown in [16] that the lattice operations can be calculated using a big table look-up (see Appendix A). For $\alpha,\beta\in\mathcal{IL}_{1}$, the binary lattice operators $\sqcap$, $\sqcup$ and $\rightarrowtriangle$ are defined as follows. $[\alpha]\sqcap[\beta]=[\alpha\wedge\beta]$, $[\alpha]\sqcup[\beta]=[\alpha\vee\beta]$, and $[\alpha]\rightarrowtriangle[\beta]=[\delta]$, where $[\delta]$ is the largest element w.r.t. $\sqsubseteq$333The induced partial order is denoted by $\sqsubseteq$ ($a\sqsubseteq b\Leftrightarrow a\sqcap b=a$). with $\inf\\{[\alpha],[\delta]\\}\sqsubseteq[\beta]$.444$\rightarrowtriangle$ is called the _relative pseudo-complement operation_. We use the algebraic properties of $\mathrm{IPC}_{1}$ to give a lower bound on the length of formulas555$\|\alpha\|$ denotes the length of the formula $\alpha$, and it is the number of appearances of variables, connectives, and constants in $\alpha$. in the equivalence classes of $\mathcal{IL}_{1}$ (Lemma 3.2), and to give an upper bound on the complexity of the problem to decide the Rieger- Nishimura index of a formula (Lemma 3.3). Let $\mathit{rank}(\alpha)$ be the first element—the integer—of the $\mathit{RNindex}(\alpha)$ pair. Figure 1. The Rieger-Nishimura lattice. ###### Lemma 3.2. For every $\phi\in\mathcal{IL}_{1}$ it holds that $\mathit{rank}(\phi)\leq c\cdot\log(\|\phi\|)$, for a constant $c$ independent of $\phi$. ###### Proof. The proof relies on the following technical claim. Let $\textit{fib}(n)$ denote the $n$-th Fibonacci number666Let $\mathit{fib}(0)=1$, $\mathit{fib}(1)=1$, and $\mathit{fib}(n+2)=\mathit{fib}(n+1)+\mathit{fib}(n)$ for $n\geq 0$.. ###### Claim 1. Let $\alpha\in\mathcal{IL}_{1}$. Then $\|\alpha\|\geq\textit{fib}(\mathit{rank}(\alpha))$. _Proof of Claim._ For formulas $\alpha\in[\bot]\cup[\top]$ it holds that $\mathit{rank}(\alpha)=0$. For formulas not in $[\bot]\cup[\top]$, we prove the claim by induction on the length of $\alpha$. The only relevant formula of length $1$ is $\alpha=a$. Since $\mathit{rank}(a)=1$ and $\mathit{fib}(1)=1$, the statement holds. For the induction step let $\alpha\in\mathcal{IL}_{1}\setminus([\bot]\cup[\top])$ with $\|\alpha\|>1$ and $\alpha=\beta\star\gamma$ with $\star\in\\{\rightarrow,\wedge,\vee\\}$. Then $\|\alpha\|=\|\beta\|+\|\gamma\|+1$, and using the induction hypothesis we obtain $\|\alpha\|\geq\mathit{fib}(\mathit{rank}(\beta))+\mathit{fib}(\mathit{rank}(\gamma))+1$. We have to distinguish the following cases. (For the lattice operations see Appendix A.) * _(i)_ $\gamma\in[\bot]$. * Due to the fact that $\alpha\notin[\bot]\cup[\top]$ it follows that $\star\in\\{\rightarrow,\vee\\}$. If $\star=\vee$, clearly $\beta\in[\alpha]$ and $\mathit{rank}(\beta)=\mathit{rank}(\alpha)$. With the induction hypothesis it follows that $\|\alpha\|\geq\mathit{fib}(\mathit{rank}(\beta))=\mathit{fib}(\mathit{rank}(\alpha))$. Otherwise if $\star=\hskip 4.30554pt\rightarrow$, it follows that $\beta\in[\upvarphi_{1}]\cup[\upvarphi_{2}]\cup[\uppsi_{1}]$ and $\alpha\in[\upvarphi_{2}]\cup[\upvarphi_{1}]$. Hence $\|\alpha\|\geq\|\beta\|+2>2>\mathit{fib}(2)\geq\mathit{fib}(\mathit{rank}(\alpha))$. * _(ii)_ $\beta\in[\bot]$. * This leads to $\star=\vee$ and can be treated analogously to the case $\gamma\in[\bot]$. * _(iii)_ $\beta\in[\top]$ (resp. $\gamma\in[\top]$). * Remember that $\alpha\notin[\top]$, hence independent of the choice of $\star$ it holds that $[\alpha]=[\gamma]$ (resp. $[\alpha]=[\beta]$ and it follows that $\|\alpha\|>\mathit{fib}(\mathit{rank}(\alpha))$. * _(iv)_ The remaining cases. * With the induction hypothesis it follows that $\|\alpha\|\geq\mathit{fib}(\mathit{rank}(\beta))+\mathit{fib}(\mathit{rank}(\gamma))$. With respect to the Rieger-Nishimura lattice, we have to handle two cases. * (a) $\mathit{rank}(\alpha)\leq\mathit{rank}(\beta)$ or $\mathit{rank}(\alpha)\leq\mathit{rank}(\gamma)$. * In this case it is not hard to see that $\|\alpha\|\geq\mathit{fib}(\mathit{rank}(\beta))+\mathit{fib}(\mathit{rank}(\gamma))\geq\mathit{fib}(\mathit{rank}(\alpha))$. * (b) $\mathit{rank}(\alpha)>\mathit{rank}(\beta)$ and $\mathit{rank}(\alpha)>\mathit{rank}(\gamma)$. * In this case it holds that one of the ranks of $\beta$ and $\gamma$ needs to be $\geq\mathit{rank}(\alpha)-2$ and the other $\geq\mathit{rank}(\alpha)-1$. (See Appendix A, for example $\upvarphi_{k-1}\rightarrow\uppsi_{k-2}\equiv\upvarphi_{k}$ respectively $[\upvarphi_{k-1}]\rightarrowtriangle[\uppsi_{k-2}]=[\upvarphi_{k}]$ for $k\geq 2$.) Therefore it holds that $\|\alpha\|\geq\mathit{fib}(\mathit{rank}(\alpha)-2)+\mathit{fib}(\mathit{rank}(\alpha)-1)=\mathit{fib}(\mathit{rank}(\alpha))$. Claim 1 shows $\|\phi\|\geq\textit{fib}(\mathit{rank}(\phi))$. Because of the exponential growth of the Fibonacci numbers ($\mathit{fib}(n)\geq\Phi^{n}$ where $\Phi$ denotes the golden ratio) it follows that $\|\phi\|\geq c\cdot\log(\|\phi\|)$ where $c$ is independent of $\phi$. $\Box$ In order to analyse the complexity of the Rieger-Nishimura index computation, we define the following decision problem. * Problem: EqRNformula * Input: $\langle\alpha,(i,x)\rangle$, where $\alpha\in\mathcal{IL}_{1}$ and $(i,x)$ is a Rieger-Nishimura index * Question: $\mathit{RNindex}(\alpha)=(i,x)$? ###### Lemma 3.3. EqRNformula is in $\mathsf{LOGdetCFL}$. ###### Proof. We form Algorithm 1 based on the Rieger-Nishimura lattice of the equivalence classes of $\mathcal{IL}_{1}$. The lattice and the lattice operations $\sqcap$, $\sqcup$ and $\rightarrowtriangle$ are described in Appendix A. We can analogously define the lattice operations $\sqcap$, $\sqcup$ and $\rightarrowtriangle$ for the Rieger-Nishimura indices instead of the equivalence classes777Let $\alpha,\beta,\gamma\in\mathcal{IL}_{1}$ and $\star\in\\{\sqcap,\sqcup,\rightarrowtriangle\\}$. We set $\mathit{RNindex}(\alpha)\star\mathit{RNindex}(\beta)=k$ if $[\alpha]\star[\beta]=[\gamma]$ and $k=\mathit{RNindex}(\gamma)$.. The correctness of Algorithm 1 is straightforward because the lattice operations for equivalence classes and indices are the same. With Lemma 3.2 it follows that every variable value used in Algorithm 1 can be stored in logarithmic space. The algorithm walks recursively through the formula and computes the index of every subformula once, hence running time is polynomial. All information that are necessary for recursion can be stored on the stack. Therefore Algorithm 1 can be implemented on a polynomial time logspace machine that uses an additional stack, i.e. a $\mathsf{LOGdetCFL}$-machine. $\Box$ Algorithm 1 Rieger-Nishimura index check. 0: a formula $\phi\in\mathcal{IL}_{i}$ and a Rieger-Nishimura index $(i,x)$ 1: if RNIndex-calc$(\phi)=(i,x)$ then accept else reject 2: function RNIndex-calc($\psi$) // returns a Rieger-Nishimura index 3: if $\psi=a$ then return $(1,\mathit{psi})$ 4: else if $\psi=\top$ then return $(0,\top)$ 5: else if $\psi=\bot$ then return $(0,\bot)$ 6: else if $\psi=\beta\wedge\gamma$ then return RNIndex-calc$(\beta)$ $\sqcap$ RNIndex-calc$(\gamma)$ 7: else if $\psi=\beta\vee\gamma$ then return RNIndex-calc$(\beta)$ $\sqcup$ RNIndex-calc$(\gamma)$ 8: else if $\psi=\beta\rightarrow\gamma$ then return RNIndex-calc$(\beta)$ $\rightarrowtriangle$ RNIndex-calc$(\gamma)$ 9: end if ### Canonical models Similar as any formula can be represented by its index, intuitionistic Kripke models can be represented, too. We give a construction of models—the canonical models—that are also used to distinguish the formula equivalence classes (Theorem 3.4). Our definition differs a little bit from that in [10, Chap.6, Defi.5]. From Theorems 3.4 and 3.5 it follows that every state $s$ in every intuitionistic Kripke model $\mathcal{M}$ over one variable has a unique corresponding canonical model $\mathcal{H}_{n}$ in the sense that the state $s$ and the base state888A state is a base state in a model if it has no predecessors (beside itself) w.r.t to the preorder of the model. $n$ of $\mathcal{H}_{n}$ satisfy exactly the same formulas. This was already shown in [10, Chap.6, Lemma 11]. Further define a function $\mathpzc{h}$ that maps $(\mathcal{M},s)$ to $n$. For $n\geq 1$, we define the canonical models $\mathcal{H}_{n}=(W_{n},\trianglelefteq,\xi_{n})$ as follows. $W_{n}$ \| $:=$ \| $\\{1,2,\ldots,n-2\\}\cup\\{n\\}$ ---\|---\|--- $\trianglelefteq$ \| $:=$ \| $\\{(a,b)\mid a,b\in W_{n},\hskip 8.61108pta=b\text{ or }a\geq b+2\\}$ $\xi_{n}(a)$ \| $:=$ \| $\begin{cases}\hskip 4.73611pt\emptyset,&\text{if~{}}n=2\\\ \\{1\\},&\text{otherwise.}\end{cases}$ See Figure 2 for some examples. $\mathcal{H}_{9}$ $\mathcal{H}_{10}$ Figure 2. The canonical models $\mathcal{H}_{9}$ and $\mathcal{H}_{10}$ (reflexive and transitive edges are not depicted, $\xi_{n}(a)=\\{1\\}$ is indicated by the double circle for state $1$). The formulas in $\mathcal{IL}_{1}$ can be distinguished using the canonical models as follows. ###### Theorem 3.4. ([16],cf.[10, Chap.6, Thm.8]) For every $n\geq 1$ and every $k\geq 1$ it holds that: 1. (1) $\mathcal{H}_{n},n\models\uppsi_{k}$ iff $n\leq k$ (i.e. $k\in\\{n,n+1,\ldots\\}$), and 2. (2) $\mathcal{H}_{n},n\models\upvarphi_{k}$ iff $n<k$ or $n=k+1$ (i.e. $k\in\\{n-1\\}\cup\\{n+1,n+2,\ldots\\}$). For analysing the complexity of the decision problem whether a canonical model is the corresponding model of a state of an arbitrary given intuitionistic Kripke model we define a function $\mathpzc{h}$. The function $\mathpzc{h}$ maps a given intuitionistic Kripke model $\mathcal{M}$ and state $w$ of $\mathcal{M}$ to the index $i$ of the corresponding model $\mathcal{H}_{i}$. Let $\mathcal{M}=(W,\leqslant,\zeta)$ be an intuitionistic Kripke model and $w$ a state of $\mathcal{M}$. We define two abbreviations for $w\in W$. $W_{w\mathord{\Uparrow}}$ \| $:=$ \| $\\{v\in W\mid w\leqslant v\\}$ ---\|---\|--- $W_{w\uparrow}$ \| $:=$ \| $W_{w\mathord{\Uparrow}}\setminus\\{w\\}$ The function $\mathpzc{h}$ is defined as follows. $\mathpzc{h}(\mathcal{M},w):=\left\\{\begin{array}[]{lll}1,&\mbox{~{}if}&\mbox{$w\in\zeta(a)$}\\\\[3.01125pt] 2,&\mbox{~{}if}&\mbox{$w\not\in\zeta(a)$ and $\forall v\in W_{w\\!\uparrow}:v\not\in\zeta(a)$}\\\\[3.01125pt] 3,&\mbox{~{}if}&\mbox{$w\not\in\zeta(a)$ and $\forall v\in W_{w\\!\uparrow}:\mathpzc{h}(\mathcal{M},v)\neq 2$ and}\\\ &&\mbox{$\exists u\in W_{w\\!\uparrow}:\mathpzc{h}(\mathcal{M},u)=1$}\\\\[3.01125pt] n+2,&\mbox{~{}if}&\mbox{$\forall v\in W_{w\\!\uparrow}:\mathpzc{h}(\mathcal{M},v)\neq n+1$ and}\\\ &&\mbox{$\exists u_{1},u_{2}\in W_{w\\!\uparrow}:\mathpzc{h}(\mathcal{M},u_{1})=n$ and $\mathpzc{h}(\mathcal{M},u_{2})=n-1$}\\\ \end{array}\right.$ We call $\mathpzc{h}(\mathcal{M},w)$ the _model index_ of $w$ in $\mathcal{M}$. The function $\mathpzc{h}$ is well defined because for every state $w$ it holds that $\\{\mathpzc{h}(\mathcal{M},v)\mid v\in W_{w\mathord{\Uparrow}}\\}=\\{1,2,\dots,\mathpzc{h}(\mathcal{M},w)-2\\}\cup\\{\mathpzc{h}(\mathcal{M},w)\\}$. ###### Theorem 3.5. Let $\mathcal{M}$ be an intuitionistic Kripke model, $w$ a state of $\mathcal{M}$, and $k\geq 1$. Then it holds that $\mathcal{M},w\models\uppsi_{k}$ \| iff \| $k\geq\mathpzc{h}(\mathcal{M},w)\text{, ~{}~{}and}$ ---\|---\|--- $\mathcal{M},w\models\upvarphi_{k}$ \| iff \| $k>\mathpzc{h}(\mathcal{M},w)\text{ ~{}or~{} }k=\mathpzc{h}(\mathcal{M},w)-1.$ ###### Proof. From Theorem 3.4 follows that (1) is equivalent to the following claim. ###### Claim 2. Let $\mathcal{M}$ be an intuitionistic Kripke model and $w$ a state of $\mathcal{M}$. For every Rieger-Nishimura formula $\alpha$ it holds that $\mathcal{M},w\models\alpha$ if and only if $\mathcal{H}_{\mathpzc{h}(\mathcal{M},w)},\mathpzc{h}(\mathcal{M},w)\models\alpha$. _Proof of Claim._ We prove this by induction on the rank $\mathit{rank}(\alpha)$ of $\alpha$. Let $\mathcal{M}=(W,\leqslant,\zeta)$ be an intuitionistic Kripke model, $w\in W$ a state, and $\alpha$ a Rieger- Nishimura formula. The case $\mathit{rank}(\alpha)\in\\{0,1\\}$ is clear. For the induction step we consider a formula $\alpha$ with $\mathit{rank}(\alpha)>1$. We distinguish two cases. The case $\alpha=\uppsi_{k}$ is clear because $\uppsi_{k}=\upvarphi_{k-1}\vee\uppsi_{k-1}$ and the claim follows directly from the induction hypothesis. In the second case we have $\alpha=\upvarphi_{k}$. \| $\mathcal{M},w\models\upvarphi_{k}\hskip 8.61108pt(=\upvarphi_{k-1}\rightarrow\uppsi_{k-1})$ \| $(1)$ ---\|---\|--- $\Leftrightarrow$ \| $\forall v\in W,w\leqslant v:\text{ if }\mathcal{M},v\models\upvarphi_{k-1}\text{ then }\mathcal{M},v\models\uppsi_{k-1}$ \| $(2)$ $\Leftrightarrow$ \| $\forall v\in W,w\leqslant v:\text{ if }\mathcal{H}_{\mathpzc{h}(\mathcal{M},v)},\mathpzc{h}(\mathcal{M},v)\models\upvarphi_{k-1}$ \| \| $\hskip 71.04144pt\text{ then }\mathcal{H}_{\mathpzc{h}(\mathcal{M},v)},\mathpzc{h}(\mathcal{M},v)\models\uppsi_{k-1}$ \| $(3)$ $\Leftrightarrow$ \| $\forall x\in W_{\mathpzc{h}(\mathcal{M},w)}:\text{ if }\mathcal{H}_{x},x\models\upvarphi_{k-1}\text{ then }\mathcal{H}_{x},x\models\uppsi_{k-1}$ \| $(4)$ $\Leftrightarrow$ \| $\forall x\in W_{\mathpzc{h}(\mathcal{M},w)}:\text{ if }\mathcal{H}_{\mathpzc{h}(\mathcal{M},w)},x\models\upvarphi_{k-1}\text{ then }\mathcal{H}_{\mathpzc{h}(\mathcal{M},w)},x\models\uppsi_{k-1}$ \| $(5)$ $\Leftrightarrow$ \| $\mathcal{H}_{\mathpzc{h}(\mathcal{M},w)},\mathpzc{h}(\mathcal{M},w)\models\upvarphi_{k-1}\rightarrow\uppsi_{k-1}\hskip 8.61108pt(=\upvarphi_{k})$ \| $(6)$ The equivalence between (1) and (2) is clear due to the definition of $\rightarrow$. From the induction hypothesis follows the equivalence between (2) and (3). (3) and (4) are equivalent because $\\{\mathpzc{h}(\mathcal{M},v)\mid v\in W,w\leqslant v\\}=\\{1,2,\dots,\mathpzc{h}(\mathcal{M},w)-2\\}\cup\\{\mathpzc{h}(\mathcal{M},w)\\}=W_{\mathpzc{h}(\mathcal{M},w)}$. The definition of the canonical models, i.e. $\mathcal{H}_{x}$ is a submodel of $\mathcal{H}_{\mathpzc{h}(\mathcal{M},w)}$, causes the equivalence between (4) and (5). The last equivalence between (5) and (6) comes from the definition of $\rightarrow$ and the properties of $\mathcal{H}_{\mathpzc{h}(\mathcal{M},w)}$. $\Box$ ## 4\. The complexity of model checking for $\mathrm{IPC}_{1}$ We first define an $\mathsf{AC^{1}}$-hard graph problem, that is similar to the $\mathsf{P}$-complete alternating graph accessibility problem [4], but has some additional simplicity properties. Then we give a construction that transforms such a graph into an intuitionistic Kripke model. This transformation is the basis for the reduction from the alternating graph accessibility problem to the model checking problem for $\mathrm{IPC}_{1}$. ### 4.1. Alternating graph problems The alternating graph accessibility problem is shown to be $\mathsf{P}$-complete in [4]. We use the following restricted version of this problem that is very similar to Boolean circuits with and- and or-gates (and input-gates). An _alternating slice graph_ [13] $G=(V,E)$ is a directed bipartite acyclic graph with a bipartitioning $V=V_{\exists}\cup V_{\forall}$, and a further partitioning $V=V_{0}\cup V_{1}\cup V_{2}\cup\cdots\cup V_{m-1}$ ($m$ _slices_ , $V_{i}\cap V_{j}=\emptyset$ if $i\not=j$) where $V_{\exists}=\bigcup_{i<m,i\text{ odd}}V_{i}$ and $V_{\forall}=\bigcup_{i<m,i\text{ even}}V_{i}$, such that $E\subseteq\bigcup_{i=1,2,\ldots,m-1}V_{i}\times V_{i-1}$ — i.e. all edges go from slice $V_{i}$ to slice $V_{i-1}$ (for $i=1,2,\ldots,m-1$). All nodes excepted those in the last slice $V_{0}$ have a positive outdegree. Nodes in $V_{\exists}$ are called _existential_ nodes, and nodes in $V_{\forall}$ are called _universal_ nodes. Alternating paths from node $x$ to node $y$ are defined as follows by the property $\mathit{apath}_{G}(x,y)$. 1): $\mathit{apath}_{G}(x,x)$ holds for all $x\in V$ 2a): for $x\in V_{\exists}$: $\mathit{apath}_{G}(x,y)$ iff $\exists z\in V_{\forall}:(x,z)\in E\text{ and }\mathit{apath}_{G}(z,y)$ 2b): for $x\in V_{\forall}$: $\mathit{apath}_{G}(x,y)$ iff $\forall z\in V_{\exists}:\text{ if }(x,z)\in E\text{ then }\mathit{apath}_{G}(z,y)$ The problem AsAgap is similar to the alternating graph accessibility problem, but for the restricted class of alternating slice graphs. * Problem: AsAgap * Input: $\langle G,s,t\rangle$, where $G=(V_{\exists}\cup V_{\forall},E)$ is an alternating slice graph with slices $V_{0},V_{1},\ldots,V_{m-1}$, and $s\in V_{m-1}\cap V_{\exists}$, $t\in V_{0}\cap V_{\forall}$ * Question: does $\mathit{apath}_{G}(s,t)$ hold? Similarly as the alternating graph accessibility problem, AsAgap is $\mathsf{P}$-complete [13, Lemma 2]. The following technical Lemma is not hard to prove. ###### Lemma 4.1. For every set $A$ in (logspace-uniform) $\mathsf{AC^{1}}$ exists a function $f$ that maps instances $x$ of $A$ to instances $f(x)=\langle G_{x},s_{x},t_{x}\rangle$ of AsAgap and satisfies the following properties. 1. (1) $f$ is computable in logspace. 2. (2) $G_{x}$ is an alternating slice graph of logarithmic depth; i.e. if $G_{x}$ has $n$ nodes, then it has $m\leq\log n$ slices. 3. (3) For all instances $x$ of $A$ holds: $x\in A$ if and only if $f(x)\in\mbox{{AsAgap}}$. Essentially, the function $f$ constructs the $\mathsf{AC^{1}}$ circuit $C_{\|x\|}$ with input $x$, and transforms it to an alternating slice graph $G_{x}$. The goal node $t_{x}$ represents exactly the bits of $x$ that are $1$. The start node $s_{x}$ corresponds to the output gate of $C_{\|x\|}$, and $\mathit{apath}_{G_{x}}(s_{x},t_{x})$ expresses that $C_{\|x\|}$ accepts input $x$. If we consider $\mbox{{AsAgap}}_{\log}$ as the subset of AsAgap where the slice graphs have logarithmic depth, this lemma would express that $\mbox{{AsAgap}}_{\log}$ is $\mathsf{AC^{1}}$-hard under logspace reductions. ### 4.2. Alternating slice graphs and intuitionistic Kripke models Our hardness results rely on a transformation of instances $\langle G,s,t\rangle$ of AsAgap to intuitionistic Kripke models $\mathcal{M}_{G}:=(U,R,\xi)$. Let $\langle G,s,t\rangle$ be an instance of AsAgap for the slice graph $G=(V_{\exists}\cup V_{\forall},E_{G})$ with the $m$ slices $V_{\exists}=V_{m-1}\cup V_{m-3}\cup\cdots\cup V_{1}$ and $V_{\forall}=V_{m-2}\cup V_{m-4}\cup\cdots\cup V_{0}$. For every $i=0,1,2,\ldots,m-1$, we construct two sets of new states $W_{i}^{\mathit{in}}$ \| $:=$ \| $\\{v^{\mathit{in}}\mid v\in V_{i}\\}\text{, ~{}~{}and}$ ---\|---\|--- $W_{i}^{\mathit{out}}$ \| $:=$ \| $\\{v^{\mathit{out}}\mid v\in V_{i}\\}$ and let $W$ \| $:=$ \| $\bigcup\limits_{i=0}^{m-1}(W_{i}^{\mathit{in}}\cup W_{i}^{\mathit{out}}).$ ---\|---\|--- Every edge $(u,v)$ from $E_{G}$ is transformed to an edge $(u^{\mathit{out}},v^{\mathit{in}})$ from an $\mathit{out}$-node to an $\mathit{in}$-node, and every $\mathit{in}$-node has an edge to its corresponding $\mathit{out}$-copy. This yields the set of edges $E$ \| $:=$ \| $\big{\\{}(u^{\mathit{out}},v^{\mathit{in}})\mid(u,v)\in E_{G}\big{\\}}\cup\big{\\{}(v^{\mathit{in}},v^{\mathit{out}})\mid v\in V_{\exists}\cup V_{\forall}\big{\\}}~{}~{}.$ ---\|---\|--- Let $G^{\prime}=(W,E)$ be the graph obtained in this way from $G$. If we consider those nodes $v^{x}\in W$ as $\exists$-nodes (resp. $\forall$-nodes) that come from nodes $v\in V_{\exists}$ (resp. $v\in V_{\forall}$), then $\mathit{apath}_{G}(u,v)$ if and only if $\mathit{apath}_{G^{\prime}}(u^{out},v^{in})$. Next, we add the nodes of the canonical model $\mathcal{H}_{4m}=(\\{1,2,\ldots,4m-2\\}\cup\\{4m\\},\linebreak\trianglelefteq,\xi_{4m})$ to $G^{\prime}$ as follows. Add the nodes $1$ and $2$ to $W_{0}^{\mathit{out}}$, the nodes $3$ and $4$ to $W_{0}^{\mathit{in}}$, the nodes $5$ and $6$ to $W_{1}^{\mathit{out}}$ etc. Formally, for $i=0,1,2,\ldots,m-2$, let $S_{i}^{\mathit{out}}$ \| $:=$ \| $W_{i}^{\mathit{out}}\cup\\{4i+1,4i+2\\},$ ---\|---\|--- $S_{i}^{\mathit{in}}$ \| $:=$ \| $W_{i}^{\mathit{in}}\cup\\{4i+3,4i+4\\}\text{, ~{}~{}and}$ $S_{m-1}^{\mathit{in}}$ \| $:=$ \| $W_{m-1}^{\mathit{in}}\cup\\{4m\\}.$ The set of states for our model is now $U$ \| $:=$ \| $\bigcup\limits_{i=0}^{m-1}(S_{i}^{\mathit{out}}\cup S_{i}^{\mathit{in}})~{}~{}.$ ---\|---\|--- Note that $(U,E)$ is still a slice graph with slices $S_{m-1}^{\mathit{in}},S_{m-1}^{\mathit{out}},S_{m-2}^{\mathit{in}},\ldots$ . We yet have no edges that connect to nodes from the canonical model. First we add only those edges between these nodes that do not disturb the “slice graph” property, namely $H$ \| $:=$ \| $\\{(i,i-2)\mid i\in\\{3,4,\ldots,4m-2\\}\cup\\{4m\\}\\}\hskip 8.61108pt\cup$ ---\|---\|--- \| \| $\\{(i,i-3)\mid i\in\\{4,6,\ldots,4m-2,4m\\}\\}.$ Note that $H$ consists of the edges from $\mathcal{H}_{4m}$ that give the canonical model its typical structure, i.e. $\trianglelefteq$ is the transitive closure of $H$. Second we add edges from every node in $W^{x}_{i}$ to a node in the neighboured slice $S_{i-1}^{\overline{x}}$ from $\mathcal{H}_{4m}$ depending on whether $x=\mathit{in}$ or $x=\mathit{out}$999$\overline{x}=\mathit{in}$ if $x=\mathit{out}$ and vice versa.. $T_{\mathit{in}}$ \| $:=$ \| $\\{(u,4i+2)\mid u\in W^{\mathit{in}}_{i},i=0,1,2,\ldots,m-1\\}$ ---\|---\|--- $T_{\mathit{out}}$ \| $:=$ \| $\\{(u,4i-1)\mid u\in W^{\mathit{out}}_{i},i=1,2,\ldots,m-1\\}$ Notice that $(U,E\cup H\cup T_{\mathit{in}}\cup T_{\mathit{out}})$ is still a slice graph with the slices mentioned above. It is depicted in Figure 3). An intuitionistic Kripke model must be transitive and reflexive. The reduction function that transforms alternating slice graphs to intuitionistic Kripke models must be computable in logarithmic space. Within this space bound we cannot compute the transitive closure of a graph. Therefore, we make the graph transitive with brute force. We add all edges that jump over at least one slice—we call these edges _pseudotransitive_. $P$ \| $:=$ \| $\bigcup\limits_{i=m-1}^{1}\bigg{[}\Big{(}S^{\mathit{in}}_{i}\times\bigcup\limits_{j=i-1}^{0}S^{\mathit{in}}_{j}\cup S^{\mathit{out}}_{j}\Big{)}\hskip 8.61108pt\cup$ ---\|---\|--- \| \| $\hskip 33.58324pt\Big{(}S^{\mathit{out}}_{i}\times\big{(}S^{\mathit{out}}_{i-1}\cup\bigcup\limits_{j=i-2}^{0}S^{\mathit{in}}_{j}\cup S^{\mathit{out}}_{j}\big{)}\Big{)}\bigg{]}$ Finally, we need to add all reflexive edges. $T$ \| $:=$ \| $\\{(u,u)\mid u\in U\\}$ ---\|---\|--- Notice that the subgraph induced by the states of the canonical model $\mathcal{H}_{4m}$ that consists of the edges in $H$ plus the pseudotransitive and the reflexive edges, is exactly $\mathcal{H}_{4m}$. Eventually, the relation $R$ for our model is $R$ \| $:=$ \| $E\cup H\cup T_{\mathit{in}}\cup T_{\mathit{out}}\cup P\cup T,$ ---\|---\|--- and the valuation function for our model is $\xi(a)$ \| $:=$ \| $\\{t^{\mathit{out}},1\\},$ ---\|---\|--- where $t^{\mathit{out}}$ is the copy of the goal node $t$ in $W_{0}^{\mathit{out}}$, and $\\{1\\}=\xi_{4m}(a)$ is the node from $\mathcal{H}_{4m}$. This yields the intuitionistic Kripke model $\mathcal{M}_{G}=(U,R,\xi)$. An example of an AsAgap instance $\langle G,s,t\rangle$ and the corresponding intuitionistic Kripke model $\mathcal{M}_{G}$ constructed from it can be seen in Figure 3. Figure 3. An alternating slice graph $G$ (left) and the resulting intuitionistic Kripke model $\mathcal{M}_{G}$ (right); both the states in $\xi(a)$ are drawn doubly; pseudotransitive and reflexive edges in $\mathcal{M}_{G}$ are not depicted. The value at state $x$ denotes its model index $\mathpzc{h}(\mathcal{M}_{G},x)$. For states in $\mathcal{H}_{16}$, their names and their model indices coincide. States $v^{\mathit{in}}$ and $v^{\mathit{out}}$ for which $\mathit{apath}_{G}(v,t)$ holds in $G$ are coloured grey. The states from the canonical model were added to the slice graph in order to obtain control over the model indices of the other states (w.r.t. the model $\mathcal{M}_{G}$). Our controlling tool is the function $\mathpzc{h}$ which is defined in the previous section. It maps every state of an intuitionistic Kripke model to its model index. This is described by Proposition 4.2 and Proposition 4.3. ###### Proposition 4.2. For every $i=0,1,2,\ldots,m-1$ and every $v\in V_{i}$ holds $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{out}})\in\\{4i+1,4i+2\\}\text{~{}~{}~{}and~{}~{}~{}}\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{in}})\in\\{4i+2,4i+4\\}~{}~{}.$ ###### Proof. We prove this by induction on the slices. For the base case we consider $v\in W_{0}^{\mathit{out}}$, where we have $\mathpzc{h}(\mathcal{M}_{G},v)=1$ if $v=t^{\mathit{out}}$, and $\mathpzc{h}(\mathcal{M}_{G},v)=2$ if $v\not=t^{\mathit{out}}$, and therefore $\mathpzc{h}(\mathcal{M}_{G},v)\in\\{1,2\\}$. For the induction step, we consider the remaining slices. For $v^{\mathit{in}}\in W_{i}^{\mathit{in}}$, we have $(v^{\mathit{in}},4i+2)\in R$ and $(v^{\mathit{out}},w)\in R$ for some $w\in W^{\mathit{out}}_{i}$. By the induction hypothesis it follows that $\mathpzc{h}(\mathcal{M}_{G},u)\leq 4i+2$ for all $u\in U_{v^{\mathit{in}}\uparrow}$. By the definition of $\mathpzc{h}$ it follows that $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{in}})\in\\{4i+2,4i+4\\}$. For $v^{\mathit{out}}\in W_{i}^{\mathit{out}}$, we have $(v^{\mathit{out}},4i-1)\in R$. By the induction hypothesis we know that for all $(v^{\mathit{out}},w)\in R$ with $w\in W^{\mathit{in}}_{i-1}$ holds $\mathpzc{h}(\mathcal{M}_{G},w)\in\\{4i-2,4i\\}$, and $\mathpzc{h}(\mathcal{M}_{G},u)\leq 4i$ for all $u\in U_{v^{\mathit{out}}\uparrow}$. Now, if for some $w\in W^{\mathit{in}}_{i-1}\cap U_{v^{\mathit{out}}\uparrow}$ holds $\mathpzc{h}(\mathcal{M}_{G},w)=4i$, then $v^{\mathit{out}}$ has successors with model indices $4i$ and $4i-1$ and by the definition of $\mathpzc{h}$ it follows that $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{out}})=4i+2$. Otherwise, for all $w\in W^{\mathit{in}}_{i-1}\cap U_{v^{\mathit{out}}\uparrow}$ holds $\mathpzc{h}(\mathcal{M}_{G},w)=4i-2$, and $v^{\mathit{out}}$ has no successor with model index $4i$ but successors with model indices $4i-2$ and $4i-1$. By the definition of $\mathpzc{h}$ it now follows that $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{out}})=4i+1$. $\Box$ ###### Proposition 4.3. For every $i=0,1,2,\ldots,m-1$ and every $v\in V_{i}$ holds: 1. (1) if $i$ is even ($\forall$ slice): $\mathit{apath}_{G}(v,t)$ if and only if $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{out}})=4i+1$, and $\mathit{apath}_{G}(v,t)$ if and only if $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{in}})=4i+4$, 2. (2) if $i$ is odd ($\exists$ slice): $\mathit{apath}_{G}(v,t)$ if and only if $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{out}})=4i+2$, and $\mathit{apath}_{G}(v,t)$ if and only if $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{in}})=4i+2$. ###### Proof. We prove this proposition by induction on $i$. The initial step for $v^{\mathit{out}}\in W_{0}^{\mathit{out}}$ follows directly from the definition of $\mathcal{M}_{G}$. Now for the induction step. Consider $v\in V_{i}$ for even $i$ ($\forall$ slice). \| $\mathit{apath}_{G}(v,t)$ \| $(1)$ ---\|---\|--- $\Leftrightarrow$ \| $\forall w\in V_{i-1},(v,w)\in E_{G}:\mathit{apath}_{G}(w,t)$ \| $(2)$ $\Leftrightarrow$ \| $\forall w^{\mathit{in}}\in W^{\mathit{in}}_{i-1},(v^{\mathit{out}},w^{\mathit{in}})\in R:\mathpzc{h}(\mathcal{M}_{G},w^{\mathit{in}})=4i-2$ \| $(3)$ $\Leftrightarrow$ \| $\\{\mathpzc{h}(\mathcal{M}_{G},u)\mid u\in U_{v^{out}\uparrow}\\}=\\{1,2,\ldots,4i-1\\}$ \| $(4)$ $\Leftrightarrow$ \| $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{out}})=4i+1$ \| $(5)$ (1) and (2) are equivalent by the definition of $\mathit{apath}_{G}$. The equivalence of (2) and (3) comes from the construction of $\mathcal{M}_{G}$ and the induction hypothesis. To show the equivalence of (3) and (4) we prove both the directions separately. First we show (3) $\Rightarrow$ (4). Because of (3) there is no $w^{\mathit{in}}\in W^{\mathit{in}}_{i-1}$ with $\mathpzc{h}(\mathcal{M}_{G},w^{\mathit{in}})>4i-1$. If $\\{4i-1,4i-2\\}\subseteq\\{\mathpzc{h}(\mathcal{M}_{G},u)\mid u\in U_{v^{out}\uparrow}\\}$, then (4) follows directly. For $4i-2$ it follows directly from (3). If $4i-1\not\in\\{\mathpzc{h}(\mathcal{M}_{G},u)\mid u\in U_{v^{out}\uparrow}\\}$, then $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{out}})\in\\{4i\\}\cup\\{4i-2,4i-3,\dots,1\\}$. Because of (3), it is not possible that $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{out}})<4i$. And $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{out}})=4i$ is inconsistent with Proposition 4.2. Hence $\\{\mathpzc{h}(\mathcal{M}_{G},u)\mid u\in U_{v^{out}\uparrow}\\}=\\{1,2,\ldots,4i-1\\}$. For the second direction, (4) $\Rightarrow$ (3), assume that there is some $w^{\mathit{in}}\in W^{\mathit{in}}_{i-1}$ with $\mathpzc{h}(\mathcal{M}_{G},w^{\mathit{in}})\not=4i-2$. Then from Proposition 4.2 it follows that $\mathpzc{h}(\mathcal{M}_{G},w^{\mathit{in}})=4i$ but this is inconsistent with (4). Hence (3) and (4) are equivalent. (4) equivalent (5) by the construction of $\mathcal{M}_{G}$ and the definition of $\mathpzc{h}$. By Proposition 4.2 we know that $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{out}})\in\\{4i+1,4i+2\\}$. Remind that $v^{\mathit{in}}$ has $v^{\mathit{out}}$ and $4i+2$ as its direct successors and $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{out}})=4i+1$. Therefore, $\mathit{apath}_{G}(v,t)$ if and only if $\\{4i+1,4i+2\\}\subseteq\\{\mathpzc{h}(\mathcal{M}_{G},u)\mid u\in U_{v^{\mathit{in}}\uparrow}\\}\subseteq\\{1,2,\ldots,4i+2\\}$, where the latter is equivalent $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{in}})=4i+4$. Finally, we consider $v\in V_{i}$ for odd $i$ ($\exists$ slice). \| $\mathit{apath}_{G}(v,t)$ \| $(1)$ ---\|---\|--- $\Leftrightarrow$ \| $\exists w\in V_{i-1},(v,w)\in E_{G}:\mathit{apath}_{G}(w,t)$ \| $(2)$ $\Leftrightarrow$ \| $\exists w^{\mathit{in}}\in W^{\mathit{in}}_{i-1},(v^{\mathit{out}},w^{\mathit{in}})\in R:\mathpzc{h}(\mathcal{M}_{G},w^{\mathit{in}})=4i$ \| $(3)$ $\Leftrightarrow$ \| $\\{4i-1,4i\\}\subseteq\\{\mathpzc{h}(\mathcal{M}_{G},u)\mid u\in U_{v^{\mathit{out}}\uparrow}\\}\subseteq\\{1,2,\ldots,4i\\}$ \| $(4)$ $\Leftrightarrow$ \| $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{out}})=4i+2$ \| $(5)$ (1) and (2) are equivalent by the definition of $\mathit{apath}_{G}$. The equivalence of (2) and (3) comes from the construction of $\mathcal{M}_{G}$ and the induction hypothesis. As in the case above ($i$ is even) the equivalence of (3) and (4) follows from the construction of $\mathcal{M}_{G}$ and Proposition 4.2. (4) equivalent (5) by the construction of $\mathcal{M}_{G}$ and the definition of $\mathpzc{h}$. By Proposition 4.2 we know that $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{out}})\in\\{4i+1,4i+2\\}$. Remind that $v^{\mathit{in}}$ has $v^{\mathit{out}}$ and $4i+2$ as its direct successors. Therefore, $\mathit{apath}_{G}(v,t)$ if and only if $\\{4i+2\\}\subseteq\\{\mathpzc{h}(\mathcal{M}_{G},u)\mid u\in U_{v^{\mathit{in}}\uparrow}\\}\subseteq\\{1,2,\ldots,4i\\}\cup\\{4i+2\\}$, where the latter is equivalent $\mathpzc{h}(\mathcal{M}_{G},v^{\mathit{in}})=4i+2$. $\Box$ Let $g$ denote the function that maps instances $x=\langle G,s,t\rangle$ of AsAgap to intuitionistic Kripke models $g(x)=\mathcal{M}_{G}$ as described above. The following properties of $g$ are easy to verify. ###### Lemma 4.4. 1. (1) $g$ is logspace computable. 2. (2) If $x=\langle G,s,t\rangle$ for an alternating slice graph $G$ with $n$ nodes and $m<n$ slices, then $g(x)$ is an intuitionistic Kripke model with $\leq 4n$ states and depth $2m$. We will use $g$ as part of the reduction functions for our hardness results. ### 4.3. Lower and upper bounds Our first result states that the calculation of the model index of an intuitionistic Kripke model is $\mathsf{P}$-complete. It is already $\mathsf{P}$-complete to decide the last bit of this model index. ###### Theorem 4.5. The following problems are $\mathsf{P}$-complete. 1. (1) Given an intuitionistic Kripke model $\mathcal{M}$ and a state $w$, decide whether $\mathpzc{h}(\mathcal{M},w)$ is even. 2. (2) Given an intuitionistic Kripke model $\mathcal{M}$, a state $w$, and an integer $i$, decide whether $\mathpzc{h}(\mathcal{M},w)=i$. ###### Proof. In order to show the $\mathsf{P}$-hardness of the problems, we give a reduction from the $\mathsf{P}$-hard problem AsAgap. From an instance $\langle G,s,t\rangle$ of AsAgap where $G$ is an alternating slice graph with $m$ slices, construct $\mathcal{M}=g(\langle G,s,t\rangle)$. Then $\mathpzc{h}(\mathcal{M},s^{\mathit{out}})\in\\{4m+1,4m+2\\}$ (Proposition 4.2), and $\mathit{apath}_{G}(s,t)$ if and only if $\mathpzc{h}(\mathcal{M},s^{out})=4m+2$ (Proposition 4.3). Therefore, $\langle G,s,t\rangle\in\mbox{{AsAgap}}$ if and only if $\mathpzc{h}(\mathcal{M},s^{out})$ is even respectively $\mathpzc{h}(\mathcal{M},s^{out})=4m+2$. For every intuitionistic Kripke model $\mathcal{M}=(U,\leqslant,\xi)$ it holds that $\mathpzc{h}(\mathcal{M},w)\leq\|U\|+1$. To decide for a given intuitionistic Kripke model $\mathcal{M}$, a state w of $\mathcal{M}$, and an integer $n$ the problem “Does $\mathpzc{h}(\mathcal{M},w)=n$ hold?” is in $\mathsf{ALOGSPACE[\mbox{$n$}]}$. The function $\mathpzc{h}$ can be implemented according to its definition straightforwardly as a logarithmically space bounded alternating algorithm. It requires an alternation depth of at most $n$ due to the construction of $\mathpzc{h}$. Using that $\mathsf{P}=\mathsf{ALOGSPACE[\mbox{$\mathit{poly}$}]}$ [4] then it follows that both problems are in $\mathsf{P}$. $\Box$ In the construction of the above proof, the decision whether $\mathpzc{h}(\mathcal{M},s^{out})=4m+2$ is the same as to decide whether $\mathcal{M},s^{out}\models\uppsi_{4m+2}$, for the Rieger-Nishimura formula $\uppsi_{4m+2}$ (Theorems 3.4 and 3.5). Unfortunately, the length of $\uppsi_{4m+2}$ is exponential in $m$ (Lemma 3.2), and therefore the mapping from $\langle G,s,t\rangle$ (with $m$ slices) to the model checking instance $\langle\uppsi_{4m+2},g(\langle G,s,t\rangle),s^{out}\rangle$ cannot in general be performed in logarithmic space. But if the depth $m$ of the slice graph is logarithmic, the respective formula $\uppsi_{4m+2}$ has polynomial size only and the reduction works in logarithmic space. ###### Theorem 4.6. The model checking problem for $\mathrm{IPC}_{1}$ is $\mathsf{AC^{1}}$-hard. ###### Proof. Let $B$ be in $\mathsf{AC^{1}}$. By Lemma 4.1 there exists a logspace computable function $f_{B}$ such that for all instances $x$ of $B$, $x\in B$ if and only if $f_{B}(x)\in\mbox{{AsAgap}}$, where $f_{B}(x)=\langle G_{x},s_{x},t_{x}\rangle$ for an alternating slice graph $G_{x}$ with $n_{x}$ nodes and $m_{x}\leq\log n_{x}$ slices. The following function $r$ reduces $B$ to the model checking problem for $\mathrm{IPC}_{1}$. $r(x)$ \| $=$ \| $\langle\uppsi_{4m_{x}+2},g(f_{B}(x)),s_{x}^{\mathit{out}}\rangle$ ---\|---\|--- _$r$ can be computed in logspace._ Since $f_{B}$ is logspace computable, it follows that $g(f_{B}(x))$ and $s_{x}^{\mathit{out}}$ can be computed in logspace. The Rieger-Nishimura formula $\uppsi_{4m_{x}+2}$ can also be computed in logspace, because $m_{x}$ is logarithmic in $\|x\|$ and therefore $\uppsi_{4m_{x}+2}$ has length polynomial in $\|x\|$. _$B$ logspace reduces to the model checking problem for $\mathrm{IPC}_{1}$ via the reduction function $r$._ By Propostion 4.3 we have that $\langle G_{x},s_{x},t_{x}\rangle\in\mbox{{AsAgap}}$ if and only if $\mathpzc{h}(g(\langle G_{x},s_{x},t_{x}\rangle),s_{x}^{\mathit{out}})=4m_{x}+2$. By the properties of the Rieger-Nishimura formulas (Theorem 3.4) this is equivalent to $g(\langle G_{x},s_{x},t_{x}\rangle),s_{x}^{\mathit{out}}\models\uppsi_{4m_{x}+2}$. This shows the correctness of the reduction. $\Box$ In the following theorem we show an upper bound for the $\mathrm{IPC}_{1}$ model checking problem. ###### Theorem 4.7. The model checking problem for $\mathrm{IPC}_{1}$ is in $\mathsf{AC^{1}}$. ###### Proof. First we show that Algorithm 2 decides the model checking problem and then we analyse its complexity. We show that Algorithm 2 accepts the input $\langle\varphi,\mathcal{M},s\rangle$ if and only if $\mathcal{M},s\models\varphi$. Informally speaking Algorithm 2 accepts the input if and only if $\mathit{RNindex}(\varphi)$ and the model index $\mathpzc{h}(\mathcal{M},s)$ of $s$ in $\mathcal{M}$ match according to Theorem 3.5. Instead of computing the equivalent Rieger-Nishimura formula, Algorithm 2 only calculates its Rieger-Nishimura index. This is done in Lines 1 and 2. The trivial cases are handled in Lines 3 and 4. From Theorem 3.5 we know for an arbitrary Rieger-Nishimura formula $\alpha_{k}$ with $\mathit{rank}(\alpha_{k})=k>0$ the following. Either $\alpha_{k}\\!=\\!\uppsi_{k}$ and it holds that $\mathpzc{h}(\mathcal{M},s)\leq k$ if and only if $\mathcal{M},s\models\alpha_{k}$. This is checked in Line 6. Or $\alpha_{k}\\!=\\!\upvarphi_{k}$ and it holds that $\mathpzc{h}(\mathcal{M},s)=k+1$ or $\mathpzc{h}(\mathcal{M},s)<k$ if and only if $\mathcal{M},s\models\alpha_{k}$. This is checked in Line 9. If $\mathpzc{h}(\mathcal{M},s)>\mathit{rank}(\varphi)+1$, then it holds that $\mathcal{M},s\not\models\varphi$ (Theorems 3.4 and 3.5). In the following, we estimate the complexity of Algorithm 2. It gets $\langle\varphi,\mathcal{M},s\rangle$ as input. In Line 1 Algorithm 2 guesses a Rieger-Nishimura index $(r,x)$. The decision in Line 2 whether $\langle\varphi,(r,x)\rangle\in\mbox{{EqRNformula}}$ can be done with the resources of $\mathsf{LOGdetCFL}$ (Lemma 3.3). To decide for a given intuitionistic Kripke model $\mathcal{M}$, a state w of $\mathcal{M}$, and an integer $n$ the problem “Does $\mathpzc{h}(\mathcal{M},w)=n$ hold?” is in $\mathsf{ALOGSPACE[\mbox{$n$}]}$. The function $\mathpzc{h}$ can be implemented according to its definition straightforwardly as a logarithmically space bounded alternating algorithm. It requires an alternation depth of at most $n$ due to the construction of $\mathpzc{h}$. Hence the decision in Line 6 (resp. Line 9) whether $\mathpzc{h}(\mathcal{M},s)\in\\{1,2,\dots,r\\}$ (resp. $\mathpzc{h}(\mathcal{M},s)\in\\{1,2,\dots,r-1\\}\cup\\{r+1\\}$) can be done with $r$ (resp. $r+1$) alternations. Since $r$ is at most about $c\cdot\log(\|\phi\|)$ (Lemma 3.2), these decisions can be done with at most $c\cdot\log(\|\langle\phi,\mathcal{M},s\rangle\|)$ alternations. During the complete computation, the algorithm only needs to store a constant number of Rieger-Nishimura indices and model indices. According to Lemma 3.2 and the fact that $\mathpzc{h}(\mathcal{M},w)\leq\|\mathcal{M}\|$, Algorithm 2 requires during the alternations logarithmic space. Since $\mathsf{LOGdetCFL}\subseteq\mathsf{AC^{1}}=\mathsf{ALOGSPACE[\mbox{$\log n$}]}$, we obtain the desired upper bound. $\Box$ Algorithm 2 model checking algorithm for $\mathrm{IPC}_{1}$ 0: a formula $\phi\in\mathcal{IL}_{1}$, an intuitionistic Kripke model $\mathcal{M}$ and a state $s$ 1: guess nondeterministically a Rieger-Nishimura index $(r,x)$ with $r\leq c\cdot\log(\|\phi\|)$ 2: if $\langle\phi,(r,x)\rangle\in\mbox{{EqRNformula}}$ then 3: if $(r,x)=(0,\bot)$ then reject 4: else if $(r,x)=(0,\top)$ then accept 5: else if $x=\mathit{psi}$ then 6: if $\mathpzc{h}(\mathcal{M},s)\in\\{1,2,\dots,r\\}$ then accept 7: else reject 8: else if $x=\mathit{phi}$ then 9: if $\mathpzc{h}(\mathcal{M},s)\in\\{1,2,\dots,r-1\\}\cup\\{r+1\\}$ then accept 10: else reject 11: end if 12: else reject ## 5\. Some notes on superintuitionistic logics with one variable Superintuitionistic propositional logics are logics that have more valid formulas than $\mathrm{IPC}$. In this sense, classical propositional logic is a superintuitionistic logic, since it can be obtained as the closure under substitution and modus ponens of the tautologies from $\mathrm{IPC}$ plus $a\vee\neg a$ as additional axiom. A well-studied superintuitionistic logic is $\mathrm{KC}$ [9] that results from adding the weak law of the excluded middle $\neg a\vee\neg\neg a$ to $\mathrm{IPC}$. Semantically, the intuitionistic Kripke models for $\mathrm{KC}$ are restricted to those intuitionistic Kripke models $\mathcal{M}=(W,\leqslant,\xi)$ where $\leqslant$ is a directed preorder. Whereas $\mathcal{IL}_{1}$ over preorders has infinitely many equivalence classes of formulas, $\mathcal{IL}_{1}$ over directed preorders has only 7 equivalence classes—represented by the Rieger-Nishimura formulas $\bot,\top,\upvarphi_{1},\uppsi_{1},\upvarphi_{2},\uppsi_{2},\upvarphi_{3}$—that can be distinguished using the first 3 canonical models [16, 12]. This follows from $\neg a\vee\neg\neg a\equiv\uppsi_{3}$. The function $\mathpzc{h}$ can be implemented for such models as an alternating Turing machine that runs in logarithmic time, if the function value is fixed to a finite range—that in this case is $\\{1,2,3\\}$—independent of the input. For $\mathrm{KC}_{1}$, the Rieger-Nishimura index of the formulas also has a finite range (as mentioned above). Therefore, it can be calculated by an alternating Turing machine that runs in logarithmic time similar to the machine presented by Buss [3] that calculates the value of a Boolean formula. Instead of the Boolean values $0$ and $1$, here we have $7$ different Rieger-Nishimura indices. The rules how the index of a formula can be calculated from the indices of its subformulas and the connective, follow directly from the Rieger-Nishimura lattice operations—see Appendix A. If the indices are bound to a finite range, this big table yields an even bigger but finite table without index-variables. For example, the equivalence $\upvarphi_{n}\vee\upvarphi_{n+1}\equiv\uppsi_{n+2}$ for all $n\geq 1$ induces the three equivalences $\upvarphi_{1}\vee\upvarphi_{2}\equiv\uppsi_{3}$, $\upvarphi_{2}\vee\top\equiv\top$, and $\top\vee\top\equiv\top$ for $\mathrm{KC}_{1}$. This yields alternating logarithmic-time ($=\mathsf{NC^{1}}$) as upper bound for the validity problem for $\mathrm{KC}_{1}$. There are infinitely many superintuitionistic logics (with one variable) that can be obtained by adding any not valid formula as axiom to $\mathrm{IPC}_{1}$. For example, if we add a formula equivalent to $\uppsi_{k}$, then the superintuitionistic logic obtained has finitely many equivalence classes represented by $\bot,\top,\upvarphi_{1},\uppsi_{1},\ldots,\upvarphi_{k-1},\uppsi_{k-1},\upvarphi_{k}$. With similar arguments as for $\mathrm{KC}_{1}$ we can conclude that the model checking problems of these logics all are in $\mathsf{NC^{1}}$. Moreover, the formula value problem for Boolean formulas without variables is $\mathsf{NC^{1}}$-hard [2]. Intuitionistic formulas without variables have the same values, if they are interpreted as classical Boolean formulas. This means, the semantics of $\rightarrow$ is the same for Boolean formulas and for intuitionistic formulas without variables. Therefore, the model checking problem for any superintuitionistic logic without variables is $\mathsf{NC^{1}}$-hard, too. The validity problem for superintuitionistic logic has the same complexity, since in order to decide whether a formula with one variable is valid it suffices to know its Rieger-Nishimura index. ## 6\. Conclusion We consider computational problems that appear with intuitionistic propositional logic without variables and with one variable. We characterize the complexity of model checking for intuitionistic logic. ###### Theorem 6.1. 1. (1) The model checking problem for $\mathrm{IPC}_{0}$ is $\mathsf{NC^{1}}$-complete. 2. (2) The model checking problem for $\mathrm{IPC}_{1}$ is $\mathsf{AC^{1}}$-complete. Part(1) follows from the fact that an intuitionistic formula that contains constants $\bot$ and $\top$ but no variables can be evaluated like a Boolean formula, whose evaluation problem is $\mathsf{NC^{1}}$-complete [2] independently of the number of variables. Part (2) follows from Theorems 4.6 and 4.7. It shows a difference between $\mathrm{IPC}_{1}$ and its modal companion $\mathrm{S}4$ with one variable, for which the model checking problem is $\mathsf{P}$-complete [13]. Intuitionistic logic with one variable turns out to be very interesting. There are infinitely many equivalence classes of formulas, and according to Lemma 3.2 even the sequence of smallest formulas of these equivalence classes has an exponential growth with respect to the length of the formulas. Such a fast growing sequence seems to appear rarely in “natural” problems, and it is a key ingredient for the $\mathsf{AC^{1}}$-completeness of the model checking problem. Intuitionistic logic with one variable is strongly related to free Heyting algebras with one generator. Since Heyting algebras are generalizations of Boolean algebras, it would be interesting to investigate whether the difference between $\mathsf{NC^{1}}$ and $\mathsf{AC^{1}}$ is related to that between Boolean algebras and Heyting algebras. ###### Theorem 6.2. The model checking problem for every superintuitionistic logic with one variable is $\mathsf{NC^{1}}$-complete. This follows from the discussion in Section 5. It is interesting to notice that the complexity results for $\mathrm{IPC}$ and for $\mathrm{KC}$ with at least two variables are the same for the model checking problem [14]. But for the fragments with one variable, the complexity of $\mathrm{IPC}_{1}$ is higher than that of $\mathrm{KC}_{1}$. The fragments of $\mathrm{IPC}$ with a restricted number of variables and $\rightarrow$ as only connective have finitely many equivalence classes of formulas and models [22, 8]. The equivalence class of a given formula can be obtained with the resources of $\mathsf{NC^{1}}$, using a technique from Buss [2]. This might indicate an upper bound lower than $\mathsf{P}$ for the model checking problem. For the implicational fragment with at most one variable, $\mathsf{NC^{1}}$-completeness follows from Theorem 5. But a general result for an arbitrary number of variables is open. For the validity problem we obtain the following results. ###### Theorem 6.3. 1. (1) The validity problem for every superintuitionistic logic with one variable is $\mathsf{NC^{1}}$-complete. 2. (2) The validity problem for $\mathrm{IPC}_{1}$ is in $\mathsf{SPACE}(\log n\cdot\log\log n)\cap\mathsf{LOGdetCFL}$. Part (1) follows from the discussion in Section 5. Part (2) is from Svejdar [21] and Lemma 3.3. The exact complexity of the validity problem for $\mathrm{IPC}_{1}$ is open. It is interesting to notice that superintuitionistic logics with one variable all have lower complexity than $\mathrm{IPC}_{1}$, whereas for superintuitionistic logics with two variables already $\mathrm{KC}$ reaches the same complexity as $\mathrm{IPC}$ (follows from Rybakov [18]). If we consider other problems related to Kripke models for $\mathrm{IPC}_{1}$ that are not “out braked” by a very fast growing part of the input, the complexity jumps up to $\mathsf{P}$-completeness, as shown in Theorem 4.5. Model checking for $\mathrm{IPC}_{1}$ also gets $\mathsf{P}$-hard if the instances $\langle\varphi,\mathcal{M},s\rangle$ allow the formula $\varphi$ to be represented as a graph. This holds even for formulas without variables, and therefore it also holds for all superintuitionistic logics. If formulas are represented as graphs, the sequence of smallest representatives of the equivalence classes of $\mathrm{IPC}_{1}$ does not have exponential growth anymore. Moreover, the calculation of the Rieger-Nishimura index gets $\mathsf{P}$-hard. ###### Theorem 6.4. If the formulas are represented as graphs, the following problems are $\mathsf{P}$-complete: 1. (1) the model checking problem for $\mathrm{IPC}_{1}$, 2. (2) the model checking problem for every superintuitionistic logic with one variable, 3. (3) the validity problem for $\mathrm{IPC}_{1}$, and 4. (4) the validity problem for every superintuitionistic logic with one variable. Parts (1) and (2) contrast the different upper bounds $\mathsf{NC^{1}}$ and $\mathsf{AC^{1}}$ for the standard encodings of formulas (Theorem 6.2 resp. Theorem 6.1). Parts (3) and (4) contrast the complexity of the validity problems for the logics under consideration (Theorem 6.3). Acknowledgements. The authors thank Vitek Svejdar, Heribert Vollmer, and Thomas Schneider for helpful discussions. Remark. This work is an extended version of [15]. Theorem 15 in [15] and its proof can be found in [14, Theorem 3.6]. ## References * [1] M. Beaudry and P. McKenzie. Circuits, matrices, and nonassociative computation. J. Comput. Syst. Sci., 50(3):441–455, 1995. * [2] S. R. Buss. The Boolean formula value problem is in ALOGTIME. In Proc. 19th STOC, pages 123–131. ACM Press, 1987. * [3] S. R. Buss. Algorithms for Boolean formula evaluation and for tree contraction. In Arithmetic, Proof Theory, and Computational Complexity, pages 96–115. Oxford University Press, 1993. * [4] A. K. Chandra, D. Kozen, and L. J. Stockmeyer. Alternation. J. ACM, 28:114–133, 1981. * [5] S. A. Cook. Characterizations of pushdown machines in terms of time-bounded computers. J. ACM, 18:4–18, 1971. * [6] S. A. Cook. The complexity of theorem proving procedures. In Proc. 3rd STOC, pages 151–158. ACM Press, 1971. * [7] S. A. Cook. A taxonomy of problems with fast parallel algorithms. Information and Control, 64:2–22, 1985. * [8] G. R. R. de Lavalette, A. Hendriks, and D. H. de Jongh. Intuitionistic implication without disjunction. Journal of Logic and Computation. To appear, available at http://dx.doi.org/10.1093/logcom/exq058. * [9] M. Dummett and E. Lemmon. Modal logics between S4 and S5. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 14(24):250–264, 1959. * [10] D. M. Gabbay. Semantical investigations in Heyting’s intuitionistic logic. D.Reidel, Dordrecht, Boston, London, 1981. * [11] P. T. Johnstone. Stone spaces. Cambridge University Press, Cambridge, 1982. * [12] D. Makinson. There are infinitely many diodorean modal functions. J. of Symbolic Logic, 31(3):406–408, 1966. * [13] M. Mundhenk and F. Weiß. The complexity of model checking for intuitionistic logics and their modal companions. In Proc. of RP 2010, volume 6227 of LNCS, pages 146–160. Springer, 2010. * [14] M. Mundhenk and F. Weiß. Intuitionistic implication makes model checking $\mathsf{P}$-hard. ArXiv e-prints, abs/1107.1963v1, 2011. * [15] M. Mundhenk and F. Weiß. The model checking problem for intuitionistic propositional logic with one variable is $\mathsf{AC^{1}}$-complete. In Proc. 28th STACS, volume 9 of LIPIcs, pages 368–379. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2011. * [16] I. Nishimura. On formulas of one variable in intuitionistic propositional calculus. J. of Symbolic Logic, 25:327–331, 1960. * [17] W. L. Ruzzo. On uniform circuit complexity. Journal of Computer and Systems Sciences, 21:365–383, 1981. * [18] M. N. Rybakov. Complexity of intuitionistic and Visser’s basic and formal logics in finitely many variables. In Papers from the 6th conference on “Advances in Modal Logic”, pages 393–411. College Publications, 2006. * [19] R. Statman. Intuitionistic propositional logic is polynomial-space complete. Theor. Comput. Sci., 9:67–72, 1979. * [20] V. Svejdar. On the polynomial-space completeness of intuitionistic propositional logic. Arch. Math. Log., 42(7):711–716, 2003. * [21] V. Svejdar. The tautology problem for IPC1 is in space $\log n\cdot\log\log n$, 2009. Personal communication. * [22] A. Urquhart. Implicational formulas in intuitionistic logic. Journal of Symbolic Logic, 39(4):661–664, 1974. * [23] D. van Dalen. Logic and Structure. Springer, Berlin, Heidelberg, 4th edition, 2004. * [24] H. Vollmer. Introduction to Circuit Complexity – A Uniform Approach. Texts in Theoretical Computer Science. Springer Verlag, Berlin Heidelberg, 1999. ## Appendix A The Rieger-Nishimura lattice operations Let $[\varphi]$ denote the equivalence class that contains $\varphi$, for being $\varphi\in\mathcal{IL}_{1}$. The equivalence classes of $\mathcal{IL}_{1}$ form a free Heyting algebra over one generator (see [10]). This algebra is also called the _Rieger-Nishimura lattice_ (see Fig. 1), and is given by $(\\{a\\},\sqcap,\sqcup,\rightarrowtriangle,\bot)$ whereas $a$ denotes the only one variable that occurs in the formulas of $\mathcal{IL}_{1}$. The induced partial order is denoted by $\sqsubseteq$ ($a\sqsubseteq b\Leftrightarrow a\sqcap b=a$). For $\alpha,\beta\in\mathcal{IL}_{1}$, the binary lattice operators $\sqcap$, $\sqcup$ and $\rightarrowtriangle$ are defined as follows. $[\alpha]\sqcap[\beta]=[\alpha\wedge\beta]$, $[\alpha]\sqcup[\beta]=[\alpha\vee\beta]$, and $[\alpha]\rightarrowtriangle[\beta]=[\delta]$, where $[\delta]$ is the largest element w.r.t. $\sqsubseteq$ with $\inf\\{[\alpha],[\delta]\\}\sqsubseteq[\beta]$.101010$\rightarrowtriangle$ is called the _relative pseudo-complement operation_. In [16] the following properties of the operations of the Rieger-Nishimura lattice (see Figure 1) are shown. We describe these properties as equivalences of Rieger-Nishimura formulas. This is very similar to [10, Chap.6,Thm.7]. For example because of $\upvarphi_{n+1}\rightarrow\uppsi_{n}\equiv\upvarphi_{n+2}$ it holds that $[\upvarphi_{n+1}]\rightarrowtriangle[\uppsi_{n}]=[\upvarphi_{n+2}]$. $\upvarphi_{n}\rightarrow\upvarphi_{n}\equiv\top$ \| $\upvarphi_{n}\vee\upvarphi_{n}\equiv\upvarphi_{n}$ ---\|--- $\upvarphi_{n}\rightarrow\upvarphi_{n+1}\equiv\upvarphi_{n+1}$ \| $\upvarphi_{n}\vee\upvarphi_{n+1}\equiv\uppsi_{n+2}$ $\upvarphi_{n}\rightarrow\upvarphi_{n+k}\equiv\top$ for $k>1$ \| $\upvarphi_{n}\vee\upvarphi_{n+k}\equiv\upvarphi_{n+k}$ for $k>1$ $\upvarphi_{n+k}\rightarrow\upvarphi_{n}\equiv\upvarphi_{n}$ for $k\geq 1$ \| $\upvarphi_{n}\vee\uppsi_{n}\equiv\uppsi_{n+1}$ $\upvarphi_{n}\rightarrow\uppsi_{n}\equiv\upvarphi_{n+1}$ \| $\upvarphi_{n}\vee\uppsi_{n+k}\equiv\uppsi_{n+k}$ for $k\geq 1$ $\upvarphi_{n}\rightarrow\uppsi_{n+k}\equiv\top$ for $k\geq 1$ \| $\upvarphi_{n+k}\vee\uppsi_{n}\equiv\upvarphi_{n+k}$ for $k\geq 1$ $\upvarphi_{n+1}\rightarrow\uppsi_{n}\equiv\upvarphi_{n+2}$ \| $\uppsi_{n}\vee\uppsi_{m}\equiv\uppsi_{\max\\{n,m\\}}$ $\upvarphi_{n+2}\rightarrow\uppsi_{n}\equiv\upvarphi_{n+1}$ \| $\upvarphi_{n}\vee\bot\equiv\upvarphi_{n}$ $\upvarphi_{n+k}\rightarrow\uppsi_{n}\equiv\uppsi_{n}$ for $k>2$ \| $\uppsi_{n}\vee\bot\equiv\uppsi_{n}$ $\uppsi_{n}\rightarrow\uppsi_{n}\equiv\top$ \| $\upvarphi_{n}\vee\top\equiv\top$ $\uppsi_{n}\rightarrow\uppsi_{n+k}\equiv\top$ for $k\geq 1$ \| $\uppsi_{n}\vee\top\equiv\top$ $\uppsi_{n+1}\rightarrow\uppsi_{n}\equiv\upvarphi_{n+1}$ \| $\bot\vee\top\equiv\top$ $\uppsi_{n+k}\rightarrow\uppsi_{n}\equiv\uppsi_{n}$ for $k>1$ \| $\bot\vee\bot\equiv\bot$ $\uppsi_{n}\rightarrow\upvarphi_{n}\equiv\upvarphi_{n}$ \| $\top\vee\top\equiv\top$ $\uppsi_{n+k}\rightarrow\upvarphi_{n}\equiv\upvarphi_{n}$ for $k\geq 1$ \| $\upvarphi_{n}\wedge\upvarphi_{n}\equiv\upvarphi_{n}$ $\uppsi_{n}\rightarrow\upvarphi_{n+k}\equiv\top$ for $k\geq 1$ \| $\upvarphi_{1}\wedge\upvarphi_{2}\equiv\bot$ $\upvarphi_{1}\rightarrow\bot\equiv\upvarphi_{2}$ \| $\upvarphi_{n}\wedge\upvarphi_{n+1}\equiv\uppsi_{n-1}$ for $n>1$ $\upvarphi_{2}\rightarrow\bot\equiv\upvarphi_{1}$ \| $\upvarphi_{n}\wedge\upvarphi_{n+k}\equiv\upvarphi_{n}$ for $k>1$ $\upvarphi_{n}\rightarrow\bot\equiv\bot$ for $n>2$ \| $\upvarphi_{1}\wedge\uppsi_{1}\equiv\bot$ $\upvarphi_{n}\rightarrow\top\equiv\top$ \| $\upvarphi_{n}\wedge\uppsi_{n}\equiv\uppsi_{n-1}$ for $n>1$ $\uppsi_{1}\rightarrow\bot\equiv\upvarphi_{1}$ \| $\upvarphi_{n}\wedge\uppsi_{n+k}\equiv\upvarphi_{n}$ for $k\geq 1$ $\uppsi_{n}\rightarrow\bot\equiv\bot$ for $n>1$ \| $\upvarphi_{n+k}\wedge\uppsi_{n}\equiv\uppsi_{n}$ for $k\geq 1$ $\uppsi_{n}\rightarrow\top\equiv\top$ \| $\uppsi_{n}\wedge\uppsi_{m}\equiv\uppsi_{\min\\{n,m\\}}$ $\top\rightarrow\upvarphi_{n}\equiv\upvarphi_{n}$ \| $\upvarphi_{n}\wedge\bot\equiv\bot$ $\bot\rightarrow\upvarphi_{n}\equiv\top$ \| $\uppsi_{n}\wedge\bot\equiv\bot$ $\top\rightarrow\uppsi_{n}\equiv\uppsi_{n}$ \| $\upvarphi_{n}\wedge\top\equiv\upvarphi_{n}$ $\bot\rightarrow\uppsi_{n}\equiv\top$ \| $\uppsi_{n}\wedge\top\equiv\uppsi_{n}$ $\bot\rightarrow\top\equiv\top$ \| $\bot\wedge\top\equiv\bot$ $\top\rightarrow\bot\equiv\bot$ \| $\bot\wedge\bot\equiv\bot$ $\bot\rightarrow\bot\equiv\top$ \| $\top\wedge\top\equiv\top$ $\top\rightarrow\top\equiv\top$ \|	arxiv-papers	2010-12-17T08:46:44	2024-09-04T02:49:15.756890	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Martin Mundhenk, Felix Weiss", "submitter": "Felix Weiss", "url": "https://arxiv.org/abs/1012.3828" }
1012.3883	Quantum field theory and the Standard Model W. Hollik Max Planck Institut für Physik, Munich, Germany In this lecture we discuss the basic ingredients for gauge invariant quantum field theories. We give an introduction to the elements of quantum field theory, to the construction of the basic Lagrangian for a general gauge theory, and proceed with the formulation of QCD and the electroweak Standard Model with electroweak symmetry breaking via the Higgs mechanism. The phenomenology of $W$ and $Z$ bosons is discussed and implications for the Higgs boson are derived from comparison with experimental precision data. § INTRODUCTION Relativistic quantum field theory is the adequate theoretical framework to formulate the commonly accepted theory of the fundamental interactions, the Standard Model of the strong and the electroweak interactions [1, 2, 3, 4]. The Standard Model summarizes our present knowledge of the basic constituents of matter and their interactions. It is a gauge invariant quantum field theory based on the symmetry group $SU(3)\times SU(2)\times U(1)$, with the colour group $SU(3)$ for the strong interaction and with $SU(2)\times U(1)$ for the electroweak interaction spontaneously broken by the Higgs mechanism. The renormalizability of this class of theories allows us to make precise predictions for measurable quantities also in higher orders of the perturbative expansion, in terms of a few input parameters. The higher-order terms contain the self-coupling of the vector bosons as well as their interactions with the Higgs field and the top quark, even for processes at lower energies involving only light fermions. Assuming the validity of the Standard Model, the presence of the top quark and the Higgs boson in the loop contributions to electroweak observables allows us to obtain indirect significant bounds on their masses from precision measurements of these observables. The only unknown quantity at present is the Higgs boson. Its mass is getting more and more constrained by a comparison of the Standard Model predictions with the experimental data, preparing the ground for a crucial test at the LHC. In these lectures we give an introduction to the basic elements of a relativistic quantum field theory in the Lagrangian formulation, involving scalar, vector, and fermion fields, and indicate how to calculate amplitudes for physical processes in perturbation theory with the help of Feynman graphs. The principle of local gauge invariance is explained in terms of the well-known example of Quantum Electrodynamics (QED) with an Abelian gauge symmetry and is then generalized to the case of non-Abelian gauge invariance and applied to the formulation of Quantum Chromodynamics (QCD). In the formulation of the electroweak theory the gauge principle has to be supplemented by the concept of spontaneous symmetry breaking with the help of the Higgs field and by Yukawa interactions, for embedding massive particles in a gauge-invariant way. Excellent textbooks [5] are available for further reading. The presentation of the structure of the electroweak Standard Model is followed by a discussion of the phenomenology of $W$ and $Z$ bosons and of tests of the electroweak theory at present and future colliders. The accurate predictions for the vector boson masses, cross sections, and the $Z$ resonance observables like the width of the $Z$ resonance, partial widths, effective neutral current coupling constants and mixing angles at the $Z$ peak, can be compared with precise experimental data, with relevant implications for the empirically still unexplored Higgs sector. The present situation of the Higgs sector and expectations for the upcoming experiments are summarized in the final section, together with an outlook on supersymmetric Higgs bosons. § ELEMENTS OF QUANTUM FIELD THEORY §.§ Notations and conventions Natural units (formally $\hbar=c=1$) are used everywhere. Lorentz indices are always denoted by greek characters, $ \mu, \nu, .. =0,1,2,3$. Four-vectors for space–time coordinates and particle momenta have the following contravariant components, \begin{align} x & = (x^\mu) = (x^0, \vec{x}), \quad x^0=t \, , \nn \\ p & = (p^\mu) = (p^0, \vec{p}\,), \quad p^0=E = \sqrt{\vec{p}^{\,2} +m^2} \, . \nn \end{align} Covariant 4-vector components are related to the contravariant components according to \begin{align} a_\mu & = g_{\mu\nu}\, a^\nu, \nn \end{align} with the metric tensor \begin{align} (g_{\mu\nu}) & = \left( \begin{array}{c c c c} 1& 0& 0& 0 \\ 0& -1& 0& 0 \\ 0& 0& -1& 0 \\ 0& 0& 0& -1 \end{array} \right) \nn \end{align} yielding the 4-dimensional squares resp. scalar products, \begin{align} a^2 & = g_{\mu\nu}\, a^\mu a^\nu = a_\mu a^\mu, \quad a\cdot b = a_\mu b^\mu = a^0 b^0 - \vec{a}\cdot\vec{b} \, . \nn \end{align} Covariant and contravariant components of the derivatives are used in the following notation, \begin{align} \partial_\mu & = \frac{\partial}{\partial x^\mu} = g_{\mu\nu}\, \partial^\nu, \quad \partial^\nu = \frac{\partial}{\partial x_\nu} \qquad [\; \partial^0 =\partial_0, \;\; \partial^k =-\partial_k\; ] \, , \nn \\ \Box & = \partial_\mu \partial^\mu = \frac{\partial^2}{\partial t^2} - \Delta \, . \nn \end{align} The quantum mechanical states of spin-$s$ particles with momentum $p = (p^0, \vec{p})$ and helicity $\sigma = -s, -s+1, \cdots,+s$ are denoted in the conventional way by Dirac kets $\|p\, \sigma\!\!\!>$. They are normalized according to the relativistically invariant convention \begin{align} \label{eq:normalization} <\! p\, \sigma\, \|\, p' \sigma' \!> & = 2 p^0 \, \delta^3 (\vec{p} - \vec{p}\,') \, \delta_{\sigma \sigma'} \, . \end{align} A special state, the zero-particle state or the vacuum, respectively, is denoted by $\|0\!>$. It is normalized to unity, \begin{align} \label{eq:vacnorm} <\! 0 \, \| \, 0 \!> & = 1 \, . \end{align} §.§ Lagrangian formalism The Lagrangian formalism of quantum field theory allows us to accommodate the following basic features: * space–time symmetry in terms of Lorentz invariance, as well as internal symmetries like gauge symmetries, * causality, * local interactions. Particles are described by fields that are operators on the quantum mechanical Hilbert space of the particle states, acting as creation and annihilation operators for particles and antiparticles. In the Standard Model, the following classes of particles appear, each of them described by a specific type of fields: * spin-0 particles, described by scalar fields $\phi(x)$, * spin-1 particles, described by vector fields $A_\mu(x)$, * spin-1/2 fermions, described by spinor fields $\psi(x)$. The dynamics of the physical system involving a set of fields, denoted here by a generic field variable $\phi$, is determined by the Lorentz-invariant Lagrangian ${\cal L}$, which yields the action \begin{align} S[\phi] & = \int {\rm d}^4 x \, {\cal L}\big(\phi(x)\big) \, , \end{align} from which the equations of motions follow as Euler–Lagrange equations from Hamilton's principle, \begin{align} \delta S & = S[\phi + \delta \phi] - S[\phi] = 0 \, . \end{align} In particle mechanics with $n$ generalized coordinates $q_i$ and velocities $\dot{q}_i$, the Lagrangian $L(q_1,\dots \dot{q}_1, \dots ) $ yields the equations of motion ($i=1,\dots n$) \begin{align} \frac{\rm d}{{\rm d} t} \,\frac{\partial L}{\partial {\dot{q}_i}} - \frac{\partial L}{\partial q_i} & = 0 \, . \end{align} Proceeding to field theory, one has to perform the replacement \begin{align} q_i & \rightarrow \phi(x) \, ,\quad \dot{q}_i \rightarrow \partial_\mu \phi(x)\, , \quad L(q_1,\dots q_n, \dot{q}_1, \dots \dot{q}_n) \rightarrow {\cal L} (\phi(x), \partial_\mu \phi(x)) \end{align} and obtains the equations of motion as field equations, \begin{align} \label{eq:LagrangeEq} \partial_\mu \, \frac{\partial{\cal L}}{\partial (\partial_\mu \phi)} - \frac{\partial{\cal L}}{\partial \phi} & = 0 \, , \end{align} for each field (or field component), which is indicated here by the generic variable $\phi$. §.§ Free quantum fields §.§.§ Scalar fields The Lagrangian for a free real scalar field, describing neutral spinless particles with mass $m$, \begin{align} {\cal L} & = \frac{1}{2}\, (\partial_\mu \phi)^2 - \frac{m^2}{2} \, \phi^2 \end{align} yields the field equation according to (<ref>), known as the Klein–Gordon equation, \begin{align} \label{eq:KleinGordon} ( \Box \, + m^2) \, \phi = 0 \, . \end{align} The solution can be expanded in terms of the complete set of plane waves $e^{\pm ikx}$, \begin{align} \phi(x) = \frac{1}{(2\pi)^{3/2}}\, \int \frac{{\rm d}^3 k}{2 k^0} \, [ a(k) \, e^{-ikx} \,+\, a^\dagger(k)\, e^{ikx} \, ] \end{align} with $k^0 =\sqrt{\vec{k\,}^2 + m^2}$. Constituting a quantum field, the coefficients $a$ and the Hermitian adjoint $a^\dagger$ are operators that annihilate and create one-particle states (see Appendix <ref>), \begin{align} a^\dagger(k) \, \|0\!> & = \|k\!> \nn \\ a(k)\, \|k'\!> & = 2 k^0 \, \delta^3(\vec{k} - \vec{k}\,') \, \|0\!> \, . \end {align} The wave functions of one-particle states are given by the amplitudes of the field operator between the one-particle states and the vacuum, \begin{align} <\! 0 \| \phi(x) \| k> & = \frac{1}{(2\pi)^{3/2}}\, e^{-ikx} \, , \quad <\! k \| \phi(x) \| 0> = \frac{1}{(2\pi)^{3/2}}\, e^{ikx} \, , \end{align} distinguishing between states of incoming (first) and outgoing (second) A complex scalar field $\phi^\dagger \neq \phi$ has two degrees of freedom. It describes spinless particles which carry a charge $\pm 1$ and can be interpreted as particles and antiparticles. The Lagrangian \begin{align} \label{eq:Lscalar} {\cal L} & = (\partial_\mu \phi)^\dagger (\partial^\mu \phi) - m^2 \, \phi^\dagger \phi \end{align} yields the field equation (<ref>) as above, but in the Fourier expansion one has to distinguish between the annihilation and creation operators $a,\, a^\dagger$ for particle states $\|+, k\!>$ and $b,\, b^\dagger$ for antiparticle states $\|-, k\!>$, \begin{align} \label{eq:scalarfieldFourier} \phi(x) = \frac{1}{(2\pi)^{3/2}}\, \int \frac{{\rm d}^3 k}{2 k^0} \, [ a(k) \, e^{-ikx} \,+\, b^\dagger(k)\, e^{ikx} \, ] \end{align} \begin{equation} \begin{array}{l l} a^\dagger(k) \, \|0\!> \, = \, \|+, k\!> \, , & b^\dagger(k) \, \|0\!> \, = \, \|-, k\!> \\ a(k)\, \|+, k'\!> \, = \, 2 k^0 \, \delta^3(\vec{k} - \vec{k}\,') \, \|0\!> \, ,\qquad\qquad & b(k)\, \|-, k'\!> \, = \, 2 k^0 \, \delta^3(\vec{k} - \vec{k}\,') \, \|0\!> \, . \end{array} \end {equation} Whereas wave functions describe free particles without space–time limitations, the important concept of the propagator or Green function is required whenever the propagation from a point-like source at a given space–time point is considered. Such a Green function $D(x-y)$ is a solution of the inhomogeneous field equation \begin{align} \label{eq:GreenScalar} ( \Box \, + m^2) \, D(x-y) & = - \delta^4(x-y) \, . \end{align} The solution can easily be determined by a Fourier transformation \begin{align} \label{eq:FourierGreen} D(x-y) = \int \, \frac{{\rm d}^4k}{(2\pi)^4} \, D(k) \, e^{-ik(x-y)} \end{align} yielding Eq. (<ref>) in momentum space, \begin{align} (k^2 - m^2) \, D(k) & = 1 \, . \end{align} The solution \begin{align} \label{eq:scalarpropagator} i \, D(k) = \frac{i}{k^2 - m^2 + \, i \epsilon} \end{align} is the causal Green function or the Feynman propagator of the scalar field. The overall factor $i$ is by convention; the term $+ i \epsilon$ in the denominator with an infinitesimal $ \epsilon > 0$ is a prescription of how to treat the pole in the integral (<ref>); it corresponds to the special boundary condition of causality for $D(x-y)$ in Minkowski space, which means (see Appendix <ref>) * propagation of a particle from $y$ to $x$ if $x^0 > y^0$, * propagation of an antiparticle from $x$ to $y$ if $y^0 > x^0$. In a Feynman diagram, the propagator occurs as an internal line, whereas wave functions (resp. their Fourier transformed in momentum space) are always associated with external lines representing the physical particles in a given process. We introduce the following graphical symbol for the scalar propagator; the momentum $k$ always points into the direction of the arrow which denotes the flow of the charge of the particle (for neutral fields the arrrow is irrelevant). $i\, D(k)\quad$ $\bullet$- - ->- - -$\bullet$ §.§.§ Vector fields A vector field $A_\mu(x)$ describes particles with spin 1. Their states $\|k \lambda \!>$ can be classified by momentum $k$ and helicity $\lambda = \pm 1, 0$ for massive particles, and $\lambda = \pm 1$ for particles with mass zero. Massive case. For a given particle mass $m$, the Lagrangian for the free system (`massive photon'), \begin{align} {\cal L} & = -\frac{1}{4}\, F_{\mu\nu} F^{\mu\nu} - \frac{m^2}{2}\, A_\mu A^\mu \quad {\rm with} \quad F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \, , \end{align} yields from (<ref>) (with $\phi \to A_\nu)$ the field equation, known as the Proca equation, \begin{align} \big[(\Box + m^2)\, g^{\mu\nu} - \partial^\mu \partial^\nu \big] \, A_\nu & = 0 \, . \end{align} Special solutions are plane waves \begin{equation} \label{eq:planewaves} \epsilon^{(\lambda)}_\mu \, e^{\pm ikx} \end{equation} with three linearly independent polarization vectors $\epsilon^{(\lambda)}_\mu$, which are transverse and can be chosen as orthogonal and normalized, \begin{align} \epsilon^{(\lambda)}\cdot k & = 0 \, , \quad \epsilon^{(\lambda)} \cdot \epsilon^{(\lambda')} = - \delta_{\lambda\lambda'} \, , \end{align} and which fulfil the polarization sum \begin{align} \sum_{\lambda=1}^3 \, \epsilon^{(\lambda)}_\mu \epsilon^{(\lambda)}_\nu & = - g_{\mu\nu} + \frac{k_\mu k_\nu}{m^2} \, . \end{align} The solutions (<ref>) form a complete set, and the field $A_\mu$ can be written as a Fourier expansion, \begin{align} \label{eq:vectorfieldFourier} A_\mu(x) & = \frac{1}{(2\pi)^{3/2}} \, \sum_\lambda \, \int \frac{{\rm d}^3 k}{2 k^0} \, \big[ a_\lambda(k) \, \epsilon^{(\lambda)}_\mu(k)\, e^{-ikx} \, +\, a_\lambda^\dagger(k)\, \epsilon^{(\lambda)}_\mu(k)^* \, e^{ikx} \, \big] . \end{align} The coefficients are the annihilation and creation operators of particle \begin{eqnarray} a_\lambda^\dagger(k) \, \|0\!> & = & \|k \lambda \!> \nn \\ a_\lambda(k)\, \|k' \lambda' \!> & = & 2 k^0 \, \delta^3(\vec{k} - \vec{k}\,')\, \delta_{\lambda\lambda'}\, \|0\!> \, . \end {eqnarray} As in the scalar case, the wave functions of one-particle states are given by the amplitudes of the field operator between the one-particle states and the vacuum, \begin{align} \label{eq:photonwf} <\! 0\, \| A_\mu(x) \|\, k \lambda> & = \frac{1}{(2\pi)^{3/2}}\, \epsilon^{(\lambda)}_\mu(k)\, e^{-ikx} \, , \quad <\! k \lambda\, \| A_\mu(x) \|\, 0> = \frac{1}{(2\pi)^{3/2}}\, \epsilon^{(\lambda)}_\mu(k)^* \, e^{ikx} \, , \end{align} corresponding to incoming and outgoing states. In momentum space, the wave functions are just the polarization vectors. The Feynman propagator of the vector field, is the solution of the inhomogeneous field equation with point-like source, \begin{align} \big[(\Box + m^2)\, g^{\mu\rho} - \partial^\mu \partial^\rho \big] \, D_{\rho\nu} (x-y) & = g^\mu_{\;\, \nu} \, \delta^4(x-y) \, . \end{align} By Fourier transformation, \begin{align} \label{eq:vecpropFourier} D_{\rho\nu} (x-y) & = \int \, \frac{{\rm d}^4k}{(2\pi)^4} \, D_{\rho\nu}(k) \, e^{-ik(x-y)} \, , \end{align} one obtains an algebraic equation for $D_{\rho\nu}(k)$, \begin{align} \label{eq:GreenVectorMassive} \big[(-k^2 + m^2)\, g^{\mu\rho} + k^\mu k^\rho \big] \, D_{\rho\nu}(k) & = g^\mu_{\;\, \nu} \, . \end{align} The solution is the Feynman propagator of a massive vector field, \begin{align} \label{eq:vectorpropagator} i\, D_{\rho\nu}(k) & = \frac{i}{k^2-m^2 +i \epsilon} \left( -g_{\nu\rho} + \frac{k_\nu k_\rho}{m^2} \right) . \end{align} As for the scalar propagator in (<ref>), the factor $i$ is by convention, and the $+i\epsilon$ term in the denominator corresponds to the causal boundary condition. Massless case. For particles with $m=0$, like photons, the field $A_\mu$ corresponds to the and the Lagrangian is that of the free electromagnetic field, \begin{align} {\cal L} & = -\frac{1}{4}\, F_{\mu\nu} F^{\mu\nu} \quad {\rm with} \quad F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \, . \end{align} The field equations are Maxwell's equations for the vector potential, \begin{align} \label{eq:MaxwellEqs} \big(\Box \, g^{\mu\nu} - \partial^\mu \partial^\nu \big) \, A_\nu & = 0 \, . \end{align} There are two physical polarization vectors $\epsilon_\mu^{(1,2)}$ for the transverse polarization, with $\vec{\epsilon}^{\,(1,2)}\cdot \vec{k}= 0$. The third solution of (<ref>) with a longitudinal polarization vector $\epsilon_\mu \sim k_\mu$ is unphysical; it can be removed by a gauge transformation \begin{align} A_\mu'(x) & = A_\mu(x) + \partial_\mu \chi(x) \equiv 0 \quad {\rm with} \quad \chi(x) = \pm i e^{\pm ikx} \, . \end{align} The equation for the propagator of the massless vector field follows from (<ref>) setting $m = 0$: \begin{align} \label{eq:GreenVectorMassless} \big( -k^2 \, g^{\mu\rho} + k^\mu k^\rho \big) \, D_{\rho\nu}(k) & \equiv K^{\mu\rho} \, D_{\rho\nu}(k) = g^\mu_{\;\, \nu} \, . \end{align} This equation, however, has no solution since $K^{\mu\rho} k_\rho = 0$, i.e., $k_\rho$ is an eigenvector of $K^{\mu\rho}$ with eigenvalue $0$, which means that the determinant of $K^{\mu\rho}$ vanishes. It is therefore not straightforward to define a propagator for a massless vector field. Since the basic reason is gauge invariance, the common strategy to overcome this problem is to break the gauge symmetry by adding to ${\cal L}$ a gauge-fixing term (which in classical Maxwell theory corresponds to choosing a specific gauge). Such a term, widely used for practical calculations and corresponding to the classical Lorentz gauge, has the following form, \begin{align} \label{eq:gaugefixing} {\cal L}_{\rm fix} & = - \frac{1}{2\xi}\, \big(\partial_\mu A^\mu\big)^2 \, , \end{align} where $\xi$ is an arbitrary real parameter, called a gauge-fixing parameter (the choice $\xi=1$ defines the Feynman gauge). The accordingly extended Lagrangian \begin{align} {\cal L} & = -\frac{1}{4}\, F_{\mu\nu} F^{\mu\nu} - \frac{1}{2\xi}\, \big(\partial_\mu A^\mu\big)^2 \end{align} modifies the operator $K^{\mu\rho}$ in momentum space as follows, \begin{align} K^{\mu\rho} & \to K^{\mu\rho} - \frac{1}{\xi} \,k^\mu k^\rho \, , \end{align} and (<ref>) is replaced by the equation, \begin{align} \label{eq:GreenVectorFeyn} \big[ -k^2 \, g^{\mu\rho} + \big(1-\frac{1}{\xi}\big) k^\mu k^\rho \big] \, D_{\rho\nu}(k) & = g^\mu_{\;\, \nu} \, , \end{align} which now has a solution for the massless propagator, namely \begin{align} \label{eq:photonpropagator} i\, D_{\rho\nu}(k) & = \frac{i}{k^2 + i \epsilon} \left[ -g_{\nu\rho} + (1-\xi) \,\frac{k_\nu k_\rho}{k^2} \right] . \end{align} It becomes particularly simple in the Feynman gauge for $\xi=1$. Note that adding ${\cal L}_{\rm fix}$ to the Lagrangian does not have a physical impact since the induced extra terms in the propagator are $\sim k_\nu$ and vanish in amplitudes for physical processes: photons always couple to the electromagnetic current $j^\nu$, which is a conserved current with $\partial_\nu j^\nu$, or equivalently $k_\nu j^\nu = 0$ in momentum space. The graphical symbol for the vector-field propagator (for both massive and massless) is a wavy line which carries the momentum $k$ and two Lorentz indices: $i\, D_{\rho\nu}(k) $ $^\rho \qquad k\qquad ^\nu $ §.§.§ Dirac fields Spin-$\frac{1}{2}$ particles, like electrons and positrons, with mass $m$ are desribed by 4-component spinor fields, \begin{equation} \psi(x) \, = \, \left( \begin{array}{c} \psi_1(x) \\ \psi_2(x) \\ \psi_3(x) \\ \psi_4(x) \end{array} \right) . \end{equation} The dynamics of the free field is contained in the Dirac Lagrangian, \begin{align} \label{eq:DiracL} {\cal L} & = \adpsi \,(i \gamma^\mu \partial_\mu - m ) \, \psi \, , \end{align} involving the adjoint spinor \begin{align} \label{eq:adjoint} \adpsi & = \psi^\dagger\, \gamma^0 = (\psi_1^, \psi_2^, - \psi_3^, - \psi_4^) \, . \end{align} The Dirac matrices $\gamma^\mu$ ($\mu=0,1,2,3$) are $4\times 4$ matrices which can be written with the help of the Pauli matrices $\sigma_{1,2,3}$ in the following way (the Dirac representation), \begin{align} \gamma^0 & = \left( \begin{array}{r r} {\bf 1} & 0 \\ 0 & - {\bf 1} \end{array} \right), \quad \gamma^k = \left( \begin{array}{c c} 0 & \sigma_k \\ - \sigma_k & 0 \end{array} \right) . \end{align} They fulfil the anti-commutator relations \begin{align} \{ \gamma^\mu, \gamma^\nu\} &\equiv \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 g^{\mu\nu} \, . \end{align} The Lagrangian (<ref>) yields the Dirac equation as the equation of motion, \begin{align} \label{eq:DiracE} ( i \gamma^\mu \partial_\mu - m ) \, \psi & = 0 \, . \end{align} There are two types of solutions, corresponding to particle and anti-particle wave functions, \begin{align} \label{eq:DiracSolutions} u(p)\, e^{-ipx} & \quad {\rm and} \quad v(p)\, e^{ipx} \end{align} where the spinors $u$ and $v$ fulfil the algebraic equations \begin{align} \label{eq:Diracuv} \big( \not{\!p} - m) u(p) & = 0\, , \quad\quad \big( \not{\!p} + m) v(p) = 0 \, . \end{align} Thereby, the notation $\not{\!a} = \gamma^\mu a_\mu$ applying to any 4-vector $a_\mu$ has been used. The solutions (<ref>) correspond to momentum eigenstates with eigenvalue $p^\mu$. They can further be classified as helicity states with helicity $\sigma=\pm 1/2$ by the requirement \begin{align} \frac{1}{2}\, \big(\vec{\Sigma}\cdot\vec{n}\big) \, u_\sigma(p) & = \sigma\, u_\sigma(p) \, , \quad\quad - \frac{1}{2}\, \big(\vec{\Sigma}\cdot\vec{n}\big) \, v_\sigma(p) = \sigma\, v_\sigma(p) \end{align} \begin{equation} \vec{\Sigma} = \left( \begin{array}{l l} \vec{\sigma} & 0 \\ 0 & \vec{\sigma} \end{array} \right) \quad {\rm and} \quad \vec{n} = \frac{\vec{p}}{\|\vec{p}\|} \, . \end{equation} The normalization of the spinors is given by \begin{align} \label{uvnorm} \overline{u}_\sigma \, u_{\sigma'} & = 2 m\, \delta_{\sigma\sigma'} \, , \qquad \overline{v}_\sigma \, v_{\sigma'} = - 2 m\, \delta_{\sigma\sigma'} \, . \end{align} Other useful relations are \begin{align} \label{uvcomplete} \sum_\sigma\, u_{\sigma} \,\overline{u}_\sigma & = \, \not{\! p} + m \, , \qquad \sum_\sigma\, v_{\sigma} \,\overline{v}_\sigma = \, \not{\! p} - m \, . \end{align} Having determined a complete set of solutions of the Dirac equation (<ref>), we can now write the Dirac quantum field as an expansion in terms of these solutions, \begin{align} \label{eq:DiracfieldFourier} \psi(x) & = \frac{1}{(2\pi)^{3/2}} \, \sum_\sigma \, \int \frac{{\rm d}^3 k}{2 k^0} \, \big[ c_\sigma(k) \, u_{\sigma}(k)\, e^{-ikx} \, +\, d^{\,\dagger}_\sigma(k)\, v_{\sigma}(k) \, e^{ikx} \, \big] , \end{align} where the coefficients are annihilation operators $c_\sigma$ for particles and $d_\sigma$ for anti-particles, as well as creation operators $c_\sigma^\dagger$ and $d_\sigma^{\dagger}$ for particles and antiparticles, respectively. In QED, electrons $e^-$ are by convention the particles and positrons the antiparticles. Choosing the $e^\pm$ field as a concrete example, we thus have \begin{equation} \begin{array}{l l} c_\sigma^\dagger(k) \, \|0\!> \, = \, \|e^-, k \sigma \!> \, , & d_\sigma^{\,\dagger}(k) \, \|0\!> \, = \, \|e^+, k \sigma \!> \\ c_\sigma(k)\, \|e^-, k' \sigma'\!> \, = \, 2 k^0 \, \delta^3(\vec{k} - \vec{k}\,') \delta_{\sigma\sigma'} \, \|0\!> \, ,\qquad & d_\sigma(k)\, \|e^+, k \sigma' \!> \, = \, 2 k^0 \, \delta^3(\vec{k} - \vec{k}\,') \, \delta_{\sigma\sigma'} \, \|0\!>\, . \end{array} \end {equation} There are four types of wave functions, for incoming and outgoing particles and antiparticles, \begin{align} \label{eq:Diracwf} <\! 0 \| \psi(x) \|e^-, k\sigma> & = \frac{1}{(2\pi)^{3/2}}\, u_\sigma(k)\, e^{-ikx} \, , \quad <\! e^+,k\sigma \| \psi(x) \| 0> = \frac{1}{(2\pi)^{3/2}}\, v_\sigma(k) \,e^{ikx} \, , \nn \\ <\! 0 \| \adpsi(x) \|e^+, k\sigma> & = \frac{1}{(2\pi)^{3/2}}\, \overline{v}_\sigma(k)\, e^{-ikx} \, , \quad <\! e^-,k\sigma \| \adpsi(x) \| 0> = \frac{1}{(2\pi)^{3/2}}\, \overline{u}_\sigma(k) \,e^{ikx} \, . \end{align} In momentum space, dropping the $(2\pi)^{-3/2}$ factors and the helicity indices, we describe the situations as follows using a graphical notation ($k$ always denotes the physical momentum flowing towards an interaction point for incoming and off an interaction point for outgoing states), incoming particle $\qquad u(k)$ —>—$\bullet$ incoming antiparticle $\qquad \overline{v}(k)$ —<—$\bullet$ outgoing antiparticle $\qquad v(k)$ $\bullet$—<— outgoing particle $\qquad \overline{u}(k)$ $\bullet$—>— The arrows indicate the flow of the particle charge. Note that for antiparticles the direction of the momentum is opposite to the arrow at the line. We still have to determine the propagator of the Dirac field, which is the solution of the inhomogeneous Dirac equation with point-like source, \begin{align} \label{eq:Diracprop} ( i \gamma^\mu \partial_\mu - m ) \, S(x-y) & = {\bf 1}\, \delta^4(x-y) \, . \end{align} A Fourier transformation to $S(k)$, \begin{align} \label{eq:DiracpropFourier} S(x-y) & = \int \, \frac{{\rm d}^4k}{(2\pi)^4} \, S(k) \, e^{-ik(x-y)} \, , \end{align} transforms the condition (<ref>) into a condition for $S(k)$ in momentum space, \begin{align} ( \not{\! k} - m ) \, S(k) & = {\bf 1} \, . \end{align} The solution is a $4\times 4$ matrix, \begin{align} \label{eq:DiracPropagator} i\, S(k) & = \frac{i}{ \not{\! k} - m + i \epsilon} = \frac{i\, (\not{\! k} + m)}{k^2-m^2+ i \epsilon} \, , \end{align} where the $+i\epsilon$ prescription is the causal boundary condition, as for the scalar and vector field propagators. We introduce a graphical symbol for the propagator, $i\, S(k)\quad$ $\bullet$—>—$\bullet$ The arrow at the line denotes the flow of the particle charge; the assigned momentum $k$ always points into the direction of this arrow. The propagator appears as an internal line in Feynman diagrams. §.§ Interacting fields So far we considered only free fields, which are described by Lagrangians that are quadratic in the field variables and yield linear equations of motion. Interaction terms contain higher monomials in the fields, and a full Lagrangian with interaction has the form \begin{align} {\cal L} & = {\cal L}_0 + {\cal L}_{\rm int} \, , \end{align} where ${\cal L}_0$ is the free field part and ${\cal L}_{\rm int}$ describes the interaction. In general, the resulting non-linear field equations cannot be solved in an exact way. The conventional strategy is perturbation theory with the free fields as starting point, treating the interaction as a small perturbation. This is justified as long as the interaction is sufficiently weak. A powerful method for obtaining the perturbative amplitudes for physical processes is the expansion in terms of Feynman diagrams. As a concrete and practically useful example, we consider Quantum Electrodynamics (QED), the theory of electron/positron and photon interactions. The QED Lagrangian is given by \begin{align} \label{eq:QEDLangrangian} {\cal L}_{\rm QED} & = \adpsi (i \gamma^\mu \partial_\mu -m ) \psi - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} + {\cal L}_{\rm fix} \, +\, e\, \adpsi \gamma^\mu\psi \, A_\mu \, , \end{align} where the interaction term \begin{align} \label{eq:emcurrent} {\cal L}_{\rm int} & = j^\mu A_\mu \quad {\rm with} \quad j^\mu = e\, \adpsi \gamma^\mu \psi \end{align} describes the coupling of the electromagnetic current $j^\mu = e\, \adpsi \gamma^\mu \psi $ to the photon field $A_\mu$. The new element is an interaction point, a vertex, which connects the three fields in ${\cal L}_{\rm int}$ and which is obtained by stripping off the field operators, yielding $e \gamma^\mu$. Also for the vertex, a graphical symbol is introduced with lines connected to a point: Note that the factor $i$ is a convention. The lines can be either propagators (internal) or wave functions (external) in momentum space. They carry momenta which have to fulfil momentum conservation. Formally, momentum conservation follows via Fourier transformation from the exponentials in the wave functions (<ref>,<ref>) and the propagators (<ref>,<ref>) when going to momentum space. Collecting all the information, we give the complete list of Feynman rules for QED, with the photon propagator in the Feynman gauge. For fermions different from $e$ (or $\mu, \tau$), an extra factor for the different charge appears in the vertex, as indicated in the brackets. Helicity indices are suppressed for the wave functions. photon propagator ($\xi=1$) fermion propagator electron–photon vertex incoming photon outgoing photon incoming fermion incoming anti-fermion outgoing fermion outgoing anti-fermion To obtain the transition matrix element, the amplitude ${\cal M}_{fi}$ for a physical process $\|i\!> \to \|f\!>$ (see Section <ref>), one has the following For a process with given external particles draw all diagrams connecting the external lines by vertices and propagators. The lowest order corresponds to diagrams involving the minimum number of vertices, which determines the power of the coupling constant $e$ in the matrix element. Insert the analytical expressions for the wave functions, propagators and vertices from the Feynman rules. The arrangement of spinors is thereby opposite to the arrow at a fermion line. Impose momentum conservation at each vertex. $\bullet$ Sum over all diagrams, paying attention to the relative sign which occurs when two fermion lines are interchanged (according to Pauli's principle). Note that the factors $(2\pi)^{-3/2}$ from each wave function are omitted so far. They are collected globally and reappear in the $S$-matrix element and the cross section, respectively (Section <ref>) We demonstrate the method for the process of electron–positron annihilation into muon pairs, $e^+ e^- \to \mu^+\mu^-$. There is only one Feynman diagram in lowest order, displayed in Fig. <ref>. Lowest-order Feynman graph for $e^+ e^- \to \mu^+\mu^-$. The momenta with directions are indicated at each line. The analytical expression for the amplitude according to this diagram is given by \begin{align} {\cal M}_{fi} & = \overline{v}(q) i e \gamma^\mu u(p) \, \left( \frac{-i g_{\mu\nu} }{Q^2 + i\epsilon} \right) \, \overline{u}(p') i e \gamma^\nu v(q') \, = \, i\, \frac{e^2}{Q^2} \; \overline{v}(q) \gamma^\mu u(p) \; \overline{u}(p') \gamma^\nu v(q') \, . \end{align} Since $Q^2 = (p+q)^2 > 4 m_\mu^2$, the $i\epsilon$ term in the photon propagator is irrelevant and can be dropped. The next-order contribution to ${\cal M}_{fi}$, which is $\sim e^4$, contains diagrams with closed loops. Examples are displayed in Fig. <ref>. Since inside a loop one momentum is free, not fixed by momentum conservation, loop diagrams involve a 4-dimensional integration over the free momentum (Section <ref>). One-loop order Feynman graphs for $e^+ e^- \to \mu^+\mu^-$ § CROSS SECTIONS AND DECAY RATES This section provides the kinematical relations necessary for getting from the matrix elements for physical processes to observable quantities, like cross sections and decay rates. §.§ Scattering processes For a given scattering process $\quad a + b \rightarrow b_1 +b_2 + \cdots + b_n $ the $S$-matrix element $S_{fi} = <f\|S\|i>$ is the probability amplitude for the transition from an initial state $ \|a(p_a), b(p_b)\!>\, = \|i\!>$ to a final state $\|b_1(p_1), \cdots b_n(p_n)\! >\, = \|f\!> $ of free particles. For $ \|i\!> \neq \|f\!> $, one can write \begin{align} S_{fi} & = (2\pi)^4 \; \delta^4(P_i-P_f)\; {\cal M}_{fi} \; (2\pi)^{-3(n+2)/2} \end{align} with the $\delta$-function from momentum conservation, \begin{align} P_i = p_a+p_b & = P_f = p_1 + \cdots +p_n \, , \end{align} the $(2\pi)^{-3/2}$ factors from the normalization of the external wave functions, and with the genuine matrix element ${\cal M}_{fi}$ derived from the Feynman graphs for the scattering process. The differential cross section for scattering into the Lorentz-invariant phase space element \begin{align} {\rm d}\Phi & = \frac{{\rm d}^3 p_1}{2p_1^0}\, \cdots \frac{{\rm d}^3p_n}{2p_n^0} \end{align} is given by \begin{align} {\rm d}\sigma & = \frac{(2\pi)^{4}}{4\sqrt{(p_a\cdot p_b)^2-m_a^2 m_b^2} }\;\; \|{\cal M}_{fi}\|^2 \; (2\pi)^{-3n}\; \delta^4(P_i-P_f) \; {\rm d}\Phi \, . \end{align} The expression in the denominator is the relativistically-invariant version of the incoming flux-normalization factor. As a special example of practical importance, we give the cross section for a two-particle final state $a+b \rightarrow b_1 + b_2$ in the centre-of-mass system (CMS), where $\vec{p}_a+\vec{p}_b = 0 = \vec{p}_1+\vec{p}_2$: \begin{align} \frac{{\rm d}\sigma}{{\rm d} \Omega} & = \frac{1}{64 \pi^2 s}\, \frac{\|\vec{p}_1\|}{\|\vec{p}_a\|} \; \|{\cal M}_{fi}\|^2 \, \end{align} with $s = (p_a +p_b)^2 = (p_a^0 + p_b^0)^2 $ and the solid angle ${\rm d} \Omega = \sin\!\theta\, {\rm d}\theta\, {\rm d}\varphi$ involving the scattering angle $\theta = \langle \vec{p}_a,\vec{p}_1 \rangle$, and the azimuth $\varphi$ with respect to the polar axis given by $\vec{p}_a$. For high energies, when the particle masses are negligible, one has the further simplification $\|\vec{p}_1\| = \|\vec{p}_a\|$. §.§ Particle decays For a decay process $a \rightarrow b_1 +b_2 + \cdots + b_n$ $ \|a(p_a)\! >\, = \|i\!>, \;\; \|b_1(p_1), \cdots b_n(p_n)\! >\, = \|f\!> $, the (differential) decay width into the phase space element ${\rm d}\Phi$ is given by \begin{align} \label{eq:diffwidth} {\rm d}\Gamma & = \frac{(2\pi)^4}{2\, m_a }\;\; \|{\cal M}_{fi}\|^2 \; (2\pi)^{-3n}\; \delta^4(p_a-P_f) \; {\rm d}\Phi\, . % \frac{{\rm d}^3 p_1}{2p_1^0}\, \cdots \frac{{\rm d}^3 p_n}{2p_n^0} $ } \end{align} In the special case of a two-particle decay with final-state masses $m_1=m_2 =m$ one has the simple expression \begin{align} \label{eq:diffwidthmm} \frac{{\rm d}\Gamma}{{\rm d} \Omega} & = \frac{1}{64 \pi^2\, m_a}\, \sqrt{1-\frac{4 m^2}{m_a^2} }\; \|{\cal M}_{fi}\|^2\, . \end{align} § GAUGE THEORIES The powerful principle of gauge invariance dictates the structure of the interactions between fermions and vector bosons as well as the vector boson self-interactions. It is the generalization of the Abelian gauge symmetry found in Quantum Electrodynamics (QED) to the non-Abelian case. §.§ Abelian gauge theories — QED QED can be derived by the requirement that the global $U(1)$ symmetry of the Lagrangian for the free charged fermion field $\psi$, i.e., the symmetry of \begin{align} \label{eq:free} {\cal L}_0 & = \overline{\psi} \, (\gamma^\mu \partial_\mu - m)\, \psi \end{align} under the phase transformation \begin{align} \psi(x) & \rightarrow\, \psi'(x) = e^{i \alpha}\, \psi(x) \end{align} for arbitrary real numbers $\alpha$, can be extended to a symmetry under local transformations where $\alpha \to \alpha(x)$ is now an arbitrary real function. This necessitates the presence of a vector field $A_\mu$ and the minimal substitution of the derivative in ${\cal L}_0$ by the covariant derivative \begin{align} \partial_\mu & \to D_\mu = \partial_\mu - i e A_\mu \, , \end{align} yielding a Lagrangian that is invariant under the local gauge transformations \begin{align} \label{eq:gaugetrafo} \psi(x)\, & \rightarrow\, \psi'(x)\; =\, e^{i \alpha(x)} \, \psi(x) \, , \nn\\ % \equiv U(x)\, \psi(x) A_\mu(x)\, & \rightarrow\, A_\mu'(x) = A_\mu(x) + \frac{1}{e} \, \partial_\mu \alpha(x) \, , \end{align} which form the electromagnetic gauge group $U(1)$. As an immediate consequence, the invariant Lagrangian describes an interaction of the vector field with the electromagnetic current (<ref>), \begin{align} {\cal L} & = \overline{\psi} \, ( i \gamma^\mu D_\mu - m) \, \psi \, =\, {\cal L}_0 \, +\, e \; \overline{\psi} \gamma^\mu \psi \; A_\mu \, =\, {\cal L}_0 + {\cal L}_{int} \, . \end{align} The vector field $A_\mu$ itself is not yet a dynamical field since a kinetic term is still missing. Such a term can easily be added invoking the expression well known from classical electrodynamics, \begin{align} \label{eq:Akineticterm} {\cal L}_A & = - \frac{1}{4} \, F_{\mu\nu} F^{\mu\nu} \quad {\rm with\; the\; field\; strengths} \quad F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \, , \end{align} which is invariant under the local gauge transformation (<ref>). $A_\mu$ thus becomes the photon field obeying Maxwell's equations. §.§ Non-Abelian gauge theories The three basic steps yielding QED as the gauge theory of the electromagnetic interaction: identifying the global symmetry of the free Lagrangian, replacing $\partial_\mu$ via minimal substitution by the covariant derivative $D_\mu$ with a vector field, adding a kinetic term for the vector field, can now be extended to the case of non-Abelian symmetries as follows. (i) The given non-interacting system is described by a multiplet of fermion fields with mass $m$, $\Psi = (\psi_1,\psi_2,\dots \psi_n)^{\rm T}$, and the free dynamics by the Lagrangian \begin{align} \label{eq:freeLagrangian} {\cal L}_0 & = \overline{\Psi}\, (\gamma^\mu \partial_\mu - m)\, \Psi \quad {\rm with} \quad \overline{\Psi} = (\overline{\psi}_1,\dots \overline{\psi}_n)\, . \end{align} ${\cal L}_0$ is invariant under global transformations \begin{align} \label{eq:globalU} \Psi(x) & \rightarrow \, U(\alpha^1,\dots \alpha^N) \Psi(x) \, , \end{align} with unitary matrices $U$ from an $n$-dimensional representation of a non-Abelian Lie group $G$ of rank $N$, depending on $N$ real parameters $\alpha^1, \dots \alpha^N$. Physically relevant cases are $G=SU(2)$ and $G=SU(3)$, where the fermion fields $\psi_1,\dots \psi_n$ form the fundamental representations with $n=2$ and $n=3$, respectively. The matrices $U$ can be written as follows, \begin{align} \label{eq:exp} U(\alpha^1,\dots\alpha^N) & = e^{i(\alpha^1 T_1 + \dots + \alpha^N T_N)} \, , \end{align} with the generators of the Lie group, $T_1, \dots T_N$. These Hermitian matrices form the Lie algebra \begin{align} \label{eq:LieAlgebra} [T_a,T_b] = i\, f_{abc} \, T_c \end{align} with the structure constants $f_{abc}$ as real numbers characteristic for the group. Conventionally, the generators are normalized according to \begin{align} \label{eq:Tnormalization} {\rm Tr}\, (T_a T_b) & = \frac{1}{2} \, \delta_{ab} \, . \end{align} (ii) The global symmetry can now be extended to a local symmetry by converting the constants $\alpha^a$ in (<ref>) to real functions $\alpha^a(x)$, $a=1,\dots N$, and simultaneously introducing a covariant derivative in (<ref>), via \begin{align} \label{eq:covderiv} \partial_\mu & \rightarrow D_\mu \, = \, \partial_\mu - i g \, {\bf W}_\mu \, , \end{align} involving a vector field ${\bf W}_\mu$, together with a coupling constant $g$ (the analogue of $e$ in QED). Since $D_\mu$ acts on the $n$-dimensional column $\Psi$, the vector field is a $n\times n$ matrix and can be expanded in terms of the generators, \begin{align} \label{eq:Wfieldmatrix} {\bf W}_\mu(x) & = T_a\, W_\mu^a(x) \quad ({\rm summation\; over} \; a = 1, \dots N) \, . \end{align} In this way, a set of $N$ fields $W_\mu^a(x)$, the gauge fields, enters the Lagrangian (<ref>) and induces an interaction term, \begin{align} {\cal L}_0 &\to {\cal L} = {\cal L}_0 + {\cal L}_{\rm int} \quad {\rm with} \quad {\cal L}_{\rm int} = g \, \overline{\Psi} \gamma^\mu {\bf W}_\mu \Psi \; = g\, \overline{\Psi} \gamma^\mu T_a \Psi \; W_\mu^a \, , \end{align} which contains the interaction of $N$ currents $j^\mu_a = g \overline{\Psi} \gamma^\mu T_a \Psi$ with the gauge fields $W_\mu^a$. The local gauge transformation that leaves ${\cal L}$ invariant, involves the matrix $U \equiv U(\alpha^1(x), \dots )$ and reads as follows, \begin{align} \label{eq:nonabeliangaugetrafo} \Psi & \rightarrow \, \Psi' = U\, \Psi \, , \nn \\ {\bf W}_\mu & \rightarrow \, {\bf W}'_\mu = U\, {\bf W}_\mu\, U^{-1} - \frac{i}{g} (\partial_\mu U) U^{-1} \, . \end{align} The gauge transformation for the vector field looks more familiar when written for the components and expanded for infinitesimal $\alpha^a(x)$: \begin{align} \label{eq:infinitesimal} W_\mu^a & \to {W'}_\mu^{\,a} = W_\mu^a + \frac{1}{g} \, \partial_\mu \alpha^a + f_{abc} \, W_\mu^b\, \alpha^c \, . \end{align} The derivative term corresponds to (<ref>) in the Abelian case, the last term is of pure non-Abelian origin. Note: The construction works in the same way for a multiplet of scalar fields $\Phi = (\phi_1, \dots \phi_n)^T$, with \begin{align} \label{eq:scalarLfree} {\cal L}_0 & = (\partial_\mu \Phi)^\dagger (\partial^\mu \Phi) - m^2 \, \Phi^\dagger \Phi \quad \to \quad {\cal L} = (D_\mu \Phi)^\dagger (D^\mu \Phi) - m^2 \, \Phi^\dagger \Phi \, . \end{align} (iii) The kinetic term for the $W$ fields can be obtained from a generalization of the electromagnetic field strength tensor $F_{\mu\nu}$ in (<ref>), \begin{align} {\bf F}_{\mu\nu} & = T_a F^a_{\mu\nu} = \partial_\mu {\bf W}_\nu - \partial_\nu {\bf W}_\mu - i\, g \, [ {\bf W}_\mu, {\bf W}_\nu] \, , \end{align} with the $N$ components \begin{align} F^a_{\mu\nu} & = \partial_\mu W^a_\nu - \partial_\nu W^a_\mu + g f_{abc} \, W^b_\mu\, W^c_\nu \, . \end{align} Under the gauge transformation (<ref>) the field strength is transformed according to \begin{align} {\bf F}_{\mu\nu} & \to {\bf F}'_{\mu\nu} = U {\bf F}_{\mu\nu} U^{-1} \, . \end{align} As a consequence, the trace ${\rm Tr}({\bf F}_{\mu\nu} {\bf F}^{\mu\nu})$ is gauge invariant, \begin{align} {\rm Tr} ({\bf F'}_{\mu\nu} {\bf F'}^{\mu\nu}) & = {\rm Tr} (U {\bf F}_{\mu\nu} U^{-1} \, U {\bf F}^{\mu\nu} U^{-1}) = {\rm Tr} (U^{-1} U {\bf F}_{\mu\nu} U^{-1} \, U {\bf F}^{\mu\nu}) = {\rm Tr} ({\bf F}_{\mu\nu} {\bf F}^{\mu\nu}) \, , \end{align} and provides the non-Abelian analogue of (<ref>) for the kinetic term of the gauge fields $W_\mu^a$, \begin{align} \label{Wkinetic} {\cal L}_W & = - \frac{1}{2} \, {\rm Tr} ({\bf F}_{\mu\nu} {\bf F}^{\mu\nu}) = - \frac{1}{4} \, F^a_{\mu\nu}\, F^{a,\mu\nu} \, . \end{align} The quadratic part of ${\cal L}_W$ describes the free propagation of the $W$ fields, but there are also cubic and quartic terms describing self-interactions of the vector fields that are determined exclusively through the gauge symmetry: \begin{align} \label{Wkineticexplicit} {\cal L}_W \, = & - \frac{1}{4} \, (\partial_\mu W_\nu^a - \partial_\nu W_\mu^a) \, (\partial^\mu W^{a,\nu} - \partial^\nu W^{a,\mu} ) \nn \\ & -\frac{g}{2} \, f_{abc} \, (\partial_\mu W_\nu^a - \partial_\nu W_\mu^a)\, W^{b,\mu}\, W^{c,\nu} \nn \\ & -\frac{g^2}{4} \, f_{abc} f_{ade} \, W_\mu^b\, W_\nu^c\, W^{d,\mu}\, W^{e,\nu} \, . \end{align} In the gauge field Lagrangians ${\cal L}_W$ and ${\cal L}_A$, the vector fields are strictly massless. Mass terms $\frac{m^2}{2} W_\mu^a W^{a,\mu}$ are not invariant under gauge transformations and thus would break the gauge symmetry. § FORMULATION OF QCD Quantum Chromodynamics (QCD), the gauge theory of the strong interaction, is formulated following the principle of the previous section for the specific case of the symmetry group $G=SU(3)$. The basic fermions are quarks in three different colour states, forming the fundamental representation of the group. They are described by triplets of fermion fields $\Psi = (q_1,q_2,q_3)^T$ for each quark flavor $u,\, d, \dots$. The colour group $SU(3)$ has eight generators $T_a$, which in the triplet representation \begin{align} T_a & = \frac{1}{2} \, \lambda_a \, , \quad a = 1, \dots 8 \, , \end{align} are expressed in terms of eight $3\times3$ matrices, the Gell-Mann matrices $\lambda_a$. The covariant derivative, acting on the quark triplets $\Psi$, \begin{align}\ D_\mu & = \partial_\mu - i g_s\, \frac{\lambda_a}{2}\, G_\mu^a \, , \end{align} and the field strengths \begin{align} G_{\mu\nu}^a & = \partial_\mu G_\nu^a - \partial_\nu G_\mu^a + g_s\, f_{abc}\, G_\mu^b G_\nu^c \, , \end{align} involve eight gauge fields, the gluon fields $G_\mu^a$, and the coupling constant of QCD, the strong coupling constant $g_s$, which is commonly expressed in terms of the finestructure constant of the strong interaction, \begin{align} \alpha_s & = \frac{g_s^2}{4 \pi} \, . \end{align} The Lagrangian of QCD (for a given species of quarks) can then easily be written down according to the rules of Section <ref> (see also Ref. [6]), \begin{align} {\cal L}_{\rm QCD} & = \overline{\Psi} \, (i \gamma^\mu D_\mu - m) \Psi \, +\, {\cal L}_G \nn \\ & = \overline{\Psi} \, (i \gamma^\mu \partial_\mu - m) \Psi \, +\, g_s \, \overline{\Psi} \gamma^\mu \frac{\lambda_a}{2} \Psi \, G^a_{\mu} \, -\, \frac{1}{4} \, G^a_{\mu\nu} G^{a,\mu\nu} \, . \end{align} It involves the interaction of the quark currents with the gluon fields as well as the triple and quartic gluon self interactions as specified in (<ref>), graphically displayed as Feynman rules for QCD in Fig. <ref>. There is also a gauge-fixing term in the Lagrangian for each gluon field (not explicitly written here), which can be chosen in the same way as for the photon field in (<ref>) yielding the same form for the gluon propagators as for the photon propagaor in (<ref>). Propagators and interactions in QCD The quark mass $m$ appears in QCD as a free parameter for a given colour triplet. It is different for different quark flavours; its origin is of electroweak nature and will be discussed in the subsequent section. Note that the Lagrangian above considers only a single species of flavour. For the realistic physical situation of six flavours, one has to introduce a colour triplet for each flavour $q=u,d,\dots t$ and to perform a summation over $q$, with individual masses $m_q$. § FORMULATION OF THE ELECTROWEAK STANDARD MODEL The fundamental fermions, as families of leptons and quarks with left-handed doublets and right-handed singlets, appear as the fundamental representations of the group $SU(2)\times U(1)$, $$ \left( \barr{l} \nu_e \\ e \earr \right)_L \, , \;\;\; \left( \barr{l} \nu_{\mu} \\ \mu \earr \right)_L \, , \;\;\; \left( \barr{l} \nu_{\tau} \\ \tau \earr \right)_L \, , \;\;\; e_R, \;\;\; \mu_R, \;\;\; \tau_R $$ \begin{equation} \qquad\qquad\quad \left( \barr{l} u \\ d \earr \right)_L, \;\;\; \left( \barr{l} c \\ s \earr \right)_L, \;\;\; \left( \barr{l} t \\ b \earr \right)_L, \;\;\; u_R, \;\;\; d_R, \;\;\; c_R, \;\;\; s_R, \;\;\; t_R, \;\;\; b_R \label{eq:families} \end{equation} They can be classified by the quantum numbers of the weak isospin $I$, $I_3$, and the weak hypercharge $Y$. Left-handed fields have $I=\frac{1}{2}$ and thus form doublets, right-handed fields are singlets with $I=0$. The Gell-Mann–Nishijima relation establishes the relation of these basic quantum numbers to the electric charge $Q$: \begin{align} \label{eq:GellmannNishi} Q & = I_3 \, + \, \frac{Y}{2} \, . \end{align} The assignment of the quantum numbers to the fundamental lepton and quark fields is contained in Table <ref> for the fermions of the first generation (identical for the second and third generation). Quantum numbers isospin $I_3$ and hypercharge $Y$ for the left- and right-handed leptons and quarks, together with the electric charge $Q$ $\nu_L$ $e_L$ $e_R$ $u_L$ $d_L$ $u_R$ $d_R$ $I_3$ +1/2 -1/2 0 +1/2 -1/2 0 0 $Y$ -1 -1 -2 +1/3 +1/3 +4/3 -2/3 $Q$ 0 -1 -1 +2/3 -1/3 +2/3 -1/3 This structure can be embedded in a gauge invariant field theory of the unified electromagnetic and weak interactions by interpreting $SU(2)\times U(1)$ as the group of gauge transformations under which the Lagrangian is invariant. The group has four generators, \begin{align} T_a & = I_a \; (a=1,2,3) \quad {\rm and} \quad T_4 = Y \, , \end{align} where $Y$ is the Abelian hypercharge, and $I_a$ are the isospin operators, forming the Lie algebra \begin{align} [I_a, I_b] = i \, \epsilon_{abc} \, I_c \, , \quad [I_a, Y] = 0 \, . \end{align} This electroweak symmetry has to be broken down to the electromagnetic gauge symmetry $U(1)_{\rm em}$, otherwise the $W^{\pm},\, Z$ bosons would be massless. In the Standard Model, this is done by the Higgs mechanism in its minimal formulation requiring a single Higgs field which is a doublet under $SU(2)$. According to the general principles of constructing a gauge-invariant field theory with spontaneous symmetry breaking, the gauge, Higgs, fermion and Yukawa parts of the electroweak Lagrangian \begin{align} \label{eq:Lagrangian} {\cal L}_{\rm EW} & = {\cal L}_G+{\cal L}_H+{\cal L}_F +{\cal L}_Y \end{align} are specified in the following way. Gauge fields. $SU(2)\times U(1)$ is a non-Abelian group with generators $I_a, Y$, where $I_a\, (a=1,2,3)$ are the isospin operators and $Y$ is the hypercharge. Each of these generalized charges is associated with a vector field: a triplet of vector fields $W_{\mu}^{1,2,3}$ with $I_{1,2,3}$, and a singlet field $B_{\mu}$ with $Y$. The isotriplet $W_{\mu}^a$ and the isosinglet $B_{\mu}$ lead to the field strength tensors \begin{align} \label{eq:fieldstrength} W_{\mu\nu}^a & = \partial _{\mu}W_{\nu}^a- \partial_{\nu}W_{\mu}^a +g_2 \, \epsilon_{abc} \, W_{\mu}^bW_{\nu}^c , \nonumber \\ B_{\mu\nu} & = \partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}. \end{align} Since the gauge group is semi-simple and contains two factors, there are two independent gauge coupling constants, denoted by $g_2$ for the non-Abelian factor $SU(2)$ and by $g_1$ for the Abelian factor $U(1)$. From the field tensors (<ref>) the pure gauge field Lagrangian \begin{align} \label{eq:gaugepart} {\cal L}_G & = -\frac{1}{4} \, W_{\mu\nu}^aW^{\mu\nu,a}- \frac{1}{4}\, B_{\mu\nu}B^{\mu\nu} \end{align} is constructed, which is invariant under gauge transformations composed of (<ref>) and (<ref>). Explicit mass terms for the gauge fields are forbidden because they violate gauge invariance. Masses for the vector bosons of the weak interaction will be introduced in a second step below by breaking the electroweak symmetry spontaneously with the help of the Higgs mechanism. Fermion fields and fermion–gauge interactions. Since the representations of the gauge group are different for fermions with different chirality, we have to distinguish between the left- and right-handed fields. We use the generic notation for the chiral fields, \begin{align} \psi_L & = \frac{1-\gamma_5}{2}\, \psi \, , \quad \psi_R = \frac{1+\gamma_5}{2}\, \psi \, . \end{align} The left-handed fermion fields of each lepton and quark family with generation index $j$ are grouped into $SU(2)$ doublets and the right-handed fields into singlets, \begin{align} \psi^j_L & = \left ( \begin{array}{c} \psi_{L+}^j \\ \psi_{L-}^j \end{array} \right ) , \quad \psi_{R\sigma}^{j} \end{align} with the component index $\sigma = \pm$ denoting $u$-type fermions ($+$) and $d$-type fermions ($-$). Each left- and right-handed multiplet is an eigenstate of the weak hypercharge $Y$ such that the relation (<ref>) is fulfilled (see Table <ref>). The covariant derivative \begin{align} D_{\mu}^{L,R} & = \partial_{\mu}\, -\, i\, g_2\, I_a^{L,R} W_{\mu}^a\, +\, i\,g_1\, \frac{Y}{2}\, B_{\mu} \, \quad {\rm with} \quad I_a^L = \frac{1}{2} \sigma_a \, , \; \; I_a^R = 0 \label{eq:covderivative} \end{align} induces the fermion–gauge field interaction via the minimal substitution rule, \begin{align} \label{eq:fermiongauge} {\cal L}_F & = \sum_j\, \overline{\psi}^{\,j}_L\, i\gamma^{\mu}D_{\mu}^L\psi_L^j\, +\, \sum_{j,\sigma} \, \overline{\psi}_{R\sigma}^{\,j}\, i \gamma^{\mu}D_{\mu}^R \psi_{R\sigma}^{j} \, , \end{align} where the index $j$ runs over the three lepton and quark generations (<ref>). Note that the covariant derivatives are different for the $L$ and $R$ Mass terms are avoided at this stage. They would mix left- and right-handed fields as, for example, in $m_e (\overline{e}_L e_R + \overline{e}_R e_L)$ and hence would explicitly break gauge invariance. They will be introduced later with the help of gauge-invariant Yukawa interactions of the fermions with the Higgs field. Note that in the genuine Standard Model neutrinos are considered as massless and there are no right-handed neutrino fields. Higgs field and Higgs interactions. Here we describe how spontaneous breaking of the $SU(2)\times U(1)$ symmetry can be obtained, leaving the electromagnetic gauge subgroup $U(1)_{\rm em}$ unbroken. For this aim, a single isospin doublet of complex scalar fields with hypercharge $Y=1$, \begin{align} \Phi(x) & = \left ( \begin{array}{c} \phi^+(x) \\ \phi^0(x) \end{array} \right ) , \label{eq:Higgsfield} \end{align} is introduced and coupled to the gauge fields via minimal substitution as indicated in (<ref>), \begin{align} {\cal L}_H & = (D_{\mu}\Phi)^\dagger (D^{\mu}\Phi) - V(\Phi) \, , \label{eq:HiggsLagrange} \end{align} with the covariant derivative for $I=\frac{1}{2}$ and $Y=1$ given by \begin{align} D_{\mu} & = \partial_{\mu}\, -\, i\, g_2\, \frac{\sigma_a}{2}\, W_{\mu}^a\, +\,i\, \frac{g_1}{2}\, B_{\mu} \, . \end{align} The Higgs field self-interaction enters through the Higgs potential with constants $\mu^2$ and $\lambda $, \begin{align} \label{eq:potential} V(\Phi) & = -\mu^2\, \Phi^\dagger \Phi + \frac{\lambda}{4}\, (\Phi^\dagger \Phi)^2 \, . \end{align} In the ground state, the vacuum, the potential has a minimum. For $\mu^2, \lambda > 0$, the minimum does not occur for $\Phi=0$; instead, $V$ is minimized by all non-vanishing field configurations with $\Phi^\dagger \Phi = 2\mu^2/\lambda$. Selecting the one which is real and electrically neutral, $Q\Phi = 0$, \begin{align} \label{Qgenerator} Q & = I_3 + \frac{Y}{2} = \left( \begin{array}{c c} 1 & 0 \\ 0 & 0 \end{array} \right) , \end{align} one gets the vacuum expectation value \begin{align} \label{eq:vacuum} <\!\Phi\! > & = \frac{1}{\sqrt{2}} \left( \begin{array}{c} 0 \\ v \end{array} \right) \, \quad {\mbox{with}} \quad v=\frac{2\mu}{\sqrt{\lambda}} \; . \end{align} Although the Lagrangian is symmetric under gauge transformations of the full $SU(2)\times U(1)$ group, the vacuum configuration $<\!\Phi\! >$ does not have this symmetry: the symmetry has been spontaneously broken. $<\!\Phi\! >$ is still symmetric under transformations of the electromagnetic subgroup $U(1)_{\rm em}$, which is generated by the charge $Q$, thus preserving the electromagnetic gauge symmetry. The field (<ref>) can be written in the following way, \begin{align} \label{eq:Higgscomponents} \Phi(x) & = \left ( \begin{array}{c} \phi^+(x) \\ \big(v+H(x)+i\chi(x)\big)/\sqrt{2} \end{array} \right ) \, , \end{align} where the components $\phi^+$, $H$, $\chi$ have vacuum expectation values zero. Expanding the potential (<ref>) around the vacuum configuration in terms of the components yields a mass term for $H$, whereas $\phi^+$, and $\chi$ are massless. Exploiting the invariance of the Lagrangian, the components $\phi^+,\,\chi$ can be eliminated by a suitable gauge this means that they are unphysical degrees of freedom (called Higgs ghosts or would-be Goldstone bosons). Choosing this particular gauge where $\phi^+=\chi=0$, denoted as the unitary gauge, the Higgs doublet field has the simple form \begin{align} \label{eq:higgsunitarygauge} \Phi(x) & = \frac{1}{\sqrt{2}} \left( \begin{array}{c} 0 \\ \end{array} \right) \, , \end{align} and the potential (<ref>) reads \begin{align} \label{eq:potentialunitary} V & = \mu^2 H^2 \, +\, \frac{\mu^2}{v} H^3 \, + \, \frac{\mu^2}{4v^2} H^4 \; = \; \frac{M_H^2}{2} H^2 \, +\, \frac{M_H^2}{2v} H^3 \, + \frac{M_H^2}{8v^2} H^4 \, . \end{align} The real field $H(x)$ thus describes physical neutral scalar particles, the Higgs bosons, with mass \begin{align} M_H & = \mu\sqrt{2} \, , \end{align} as well as triple and quartic self interactions with couplings proportional to $M_H^2$. The couplings to the gauge fields follow from the kinetic term of (<ref>) and give rise to trilinear $HWW,\, HZZ$ and quadrilinear $HHWW,\, HHZZ$ vertices. In order to solve the mass problem for the fermions, Yukawa interactions between the Higgs field and the fermion fields are introduced in addition to get the charged fermions massive. The gauge-invariant Yukawa term in the Lagrangian, for one family of leptons and quarks, is a compact expression in terms of the doublets $L_L=(\nu_L,l_L)^T, \, Q_L = (u_L,d_L)^T$ and the Higgs field $\Phi$ and its charge-conjugate $\Phi^c = i \sigma_2 \Phi = ( \phi^{0}, - \phi^-)^T$ with $\phi^-$ as the adjoint of $\phi^+$, \begin{align} \label{eq:Yukawa} \lyu & = - G_l\, \, \overline{L}_L \Phi \, l_R - G_d\, \, \overline{Q}_L \Phi \, d_R - G_u \, \overline{Q}_L \Phi^c \, u_R \, + h.c. \end{align} It reads explicitly in terms of the Higgs field components (<ref>) \begin{align} \label{eq:YukawaOneGeneration} \lyu = & -G_l\,(\adnu_L\,\plus\,l_R\,+\,\adl_R\,\mis\,\nu_L \,+\, \adl_L\,\nul\, l_R\, +\, \adl_R\, \phi^{0}\, l_L ) \nn \\ & -\, G_d \, \,+\,\add_R\,\phi^{0}\,d_L ) \nn \\ & -\,G_u \, \adu_R\,\phi^0\,u_L\,+\,\adu_L\,\phi^{0}\,u_R ) \, . \end{align} The fermion mass terms follow from the $v$ part of $\phi^0$ in (<ref>), relating the individual Yukawa coupling constants $G_{l,d,u}$ to the masses of the charged fermions by \begin{align} m_f & = G_f\, \frac{v}{\sqrt{2}} \, . % = \sqrt{2}\,\frac{g_f}{g_2}\, M_W \, . \end{align} In the unitary gauge (<ref>) the Yukawa Lagrangian becomes particularly simple: \begin{align} \label{YukawaUnitaryGauge} \lyu & = \, -\sum_f \,m_f\,\overline{\psi}_f \psi_f - \sum_f \frac{m_f}{v}\, \overline{\psi}_f\psi_f \, H \, . \end{align} As a remnant of this mechanism, Yukawa interactions between the massive fermions and the physical Higgs field occur with coupling constants proportional to the fermion masses. In the realistic case of three generations, one has to take into account flavour mixing in the quark sector (in the lepton sector, lepton number is conserved and flavour mixing is absent in the minimal model). Quark-family mixing is induced by Yukawa interactions with the Higgs field as before, but the Yukawa couplings are now matrices in generation space with complex entries, $G_u=(G^u_{ij}),\, G_d = (G^d_{ij})$, and the generalization of (<ref>) for the quark sector reads as follows, with the notation $Q_L^i = (u_L^i,d_L^i)^T$ for the three left-handed doublets [$u^i= u,c,t$ and $d^i=d,s,b$]: \begin{align} \label{eq:YukawaThreeGenerations} \lyu^{\rm quarks} = & - G^d_{ij} \, \overline{Q}_L^{\,i} \Phi \, d_R^j - G^u_{ij} \, \overline{Q}_L^{\,i} \Phi^c \, u_R^j \, + h.c. \end{align} The mass term is obtained from replacing $\Phi$ by its vacuum configuration, $\Phi \to <\!\Phi\!>$ from (<ref>), \begin{align} \label{eq:quarkmassterm} & - \frac{v}{\sqrt{2}}\, G^d_{ij} \, \add_L^{\,i} d_R^j \, -\, \frac{v}{\sqrt{2}} G^u_{ij} \, \adu_L^{\,i} u_R^j \, + h.c. \end{align} This bilinear term in the quark fields can be diagonalized with the help of four unitary matrices $V_{L,R}^q$ ($q=u,d$), yielding the mass eigenstates \begin{align} \label{eq:quarktransformation} \tilde{u}_{L,R}^i & = (V^u_{L,R})_{ik}\, u_{L,R}^k, \quad \tilde{d}_{L,R}^i = (V^d_{L,R})_{ik}\, d_{L,R}^k \, , \end{align} as well as the $u$- and $d$-type quark masses as diagonal mass matrices, \begin{align} {\rm diag} (m_q) & = \frac{v}{\sqrt{2}}\, V^q_L\, G_q\, V_R^{q \, \dagger} \, , \quad q =u,d \, . \end{align} Introducing the mass eigenstates in the fermion–gauge Lagrangian (<ref>) does not change the flavour-diagonal terms, i.e., the kinetic term and the interaction terms with the neutral gauge bosons, because of the unitarity of the transformations (<ref>). Also the Yukawa interaction of the physical Higgs field with the quarks, when expressed in terms of the quark masses and the mass eigenstates, retains its structure as given in (<ref>). The only modification occurs in the flavour-changing quark interaction with the charged vector bosons in (<ref>) where the insertion of the mass eigenstates for the left-handed quark fields introduces the unitary CKM matrix, \begin{align} \label{eq:CKM} V_L^u \, V_L^{d \,\dagger} & \equiv \, V_{\rm CKM} \, . \end{align} Given the constraints from unitarity, $V_{\rm CKM}$ has four independent physical parameters, three real angles and one complex phase. For neutrino masses zero, no generation mixing in the lepton sector It is, however, possible to augment the Standard Model by introducing also right-handed neutrinos and neutrino mass terms in analogy to those of the $u$-type quark sector allowing for lepton-flavour mixing as well. The general treatment of lepton masses and mixing would, however, go beyond the scope of these lectures (for a discussion of neutrino masses see Ref. [7]). Physical fields and parameters. The gauge invariant Higgs–gauge field interaction in the kinetic part of (<ref>) gives rise to mass terms for the vector bosons in the non-diagonal form \begin{equation} \label{eq:massterm} \frac{1}{2}\, \left ( \frac{g_2}{2}v \right )^2\, +\frac{1}{2} \left(\frac{v}{2}\right)^2 \, \left ( W_{\mu}^3,B_{\mu} \right ) \left ( \begin{array}{cc} g_2^2 & g_1g_2 \\ g_1g_2 & g_1^2 \end{array} \right ) \left ( \begin{array}{c} W^{3,\mu} \\ \end{array} \right ) \; . \end{equation} The physical content becomes transparent by performing a transformation from the fields $W_{\mu}^a$, $B_{\mu}$ (in terms of which the symmetry is manifest) to the physical fields \begin{align} W_{\mu}^{\pm} & = \frac{1}{\sqrt{2}}\, (W_{\mu}^1\mp i W_{\mu}^2) \end{align} \begin{align} \label{eq:rotation} \left( \begin{array}{c} Z_{\mu} \\ A_{\mu} \end{array} \right) & = \left( \begin{array}{r r} \cos\theta_W & \quad \sin\theta_W \\ -\sin\theta_W & \quad \cos\theta_W \end{array} \right) \left( \begin{array}{c} W_{\mu}^3 \\ B_{\mu} \end{array} \right) \, . \end{align} In these fields the mass term (<ref>) is diagonal and has the form \begin{equation} M_W^2\, W_{\mu}^+W^{- \mu}\, +\, \frac{1}{2}\, (A_{\mu},Z_{\mu}) \left ( \begin{array}{cc} 0 & \quad 0 \\ 0 & \quad M_Z^2 \end{array} \right ) \left ( \begin{array}{c} A^{\mu} \\ \end{array} \right ) \end{equation} \begin{align} M_W & = \frac{1}{2}\, g_2 v \, , \quad M_Z = \frac{1}{2}\sqrt{g_1^2+g_2^2}\, v \, . \end{align} The mixing angle in the rotation (<ref>) is determined by \begin{align} \label{eq:WZmassratio} \cos\theta_W & = \frac{g_2}{\sqrt{g_1^2+g_2^2}} =\frac{M_W}{M_Z} \, . \end{align} Inserting the rotation (<ref>) into the interaction part of ${\cal L}_F$ in (<ref>) and identifying $A_{\mu}$ with the photon field which couples via the electric charge $e$ to the electron, $e$ can be expressed in terms of the gauge couplings in the following way: \begin{align} e & = \frac{g_1g_2}{\sqrt{g_1^2+g_2^2}}, \quad \mbox{or} \quad g_2 = \frac{e}{\sin\theta_W},\; g_1 = \frac{e}{\cos\theta_W} . \end{align} The relations above allow us to replace the original set of parameters g_2,\, g_1,\, \lambda,\, \mu^2, \, G_f by the equivalent set of more physical parameters e,\, M_W,\, M_Z,\, M_H,\, m_f, \, V_{\rm CKM} , where each of them can (in principle) be measured directly in a suitable experiment. At present, all parameters are empirically known with the exception of the mass of the Higgs boson, $M_H$. Gauge interactions. The fermion–gauge interactions are part of the fermion–gauge Lagrangian (<ref>); expressed in the physical field and parameters, they appear as interactions of the electromagnetic current $J^\mu_{\rm em}$, the weak neutral current $J^\mu_{\rm NC}$, and the weak charged current $J^\mu_{\rm CC}$ with the corresponding vector fields, \begin{align} {\cal L}_{\rm FG} & = J^\mu_{\rm em}\, A_\mu + J^\mu_{\rm NC}\, Z_\mu + J^\mu_{\rm CC}\, W_\mu^+ + {J^\mu_{\rm CC}}^\dagger\, W_\mu^- \, , \end{align} with the currents \begin{align} J^\mu_{\rm em} & = - e\, \sum_{f=l,q} \, Q_f \, \overline{\psi}_f \gamma^\mu \psi_f \, , \nn \\ %J^\mu_{\rm NC} & = \frac{e}{2\sinw\cosw} \, \sum_{f=l,q} \, % \overline{\psi}_f (v_f \gamma^\mu % - a_f \gamma^\mu \gamma_5) \psi_f \, , \nn \\ %J^\mu_{\rm CC} & = \frac{e}{\sqrt{2}\sinw}\, \sum_{i=1,2,3} \, % \adnu_L^i \gamma^\mu \frac{1-\gamma_5}{2} e_L^i \\ % & + \frac{e}{\sqrt{2}\sinw}\, \sum_{i,j=1,2,3} \, % \adu_L^i \gamma^\mu \frac{1-\gamma_5}{2} V_{ij} d_L^j J^\mu_{\rm NC} & = \frac{g_2}{2\cosw} \, \sum_{f=l,q} \, \overline{\psi}_f (v_f \gamma^\mu - a_f \gamma^\mu \gamma_5) \psi_f \, , \nn \\ J^\mu_{\rm CC} & = \frac{g_2}{\sqrt{2}}\, \left ( \sum_{i=1,2,3} \, \adnu^i \gamma^\mu \frac{1-\gamma_5}{2} e^i % \nn \\ % & + \frac{g_2}{\sqrt{2}}\, + \sum_{i,j=1,2,3} \, \adu^i \gamma^\mu \frac{1-\gamma_5}{2} V_{ij} d^j \right) . \label{eq:currents} \end{align} In analogy to the notation for the quark fields in (<ref>), the lepton families are labelled by $e^i = e,\mu,\tau$ for the charged leptons and $\nu^i = \nu_e,\nu_\mu,\nu_\tau$ for the corresponding neutrinos. The neutral current coupling constants in (<ref>) are determined by the charge $Q_f$ and isospin $I_3^f$ of $f_L$, \begin{align} \label{eq:NCcouplingstree} v_f & = I_3^f-2Q_f\,\sin^2\theta_W \, , \nn \\ a_f & = I_3^f \, . \end{align} The quantities $V_{ij}$ in the charged current are the elements of the CKM matrix (<ref>), which describes family mixing in the quark sector. Owing to the unitarity of $V_{\rm CKM}$, the electromagnetic and the weak neutral current interaction are flavour-diagonal. Hence, flavour-changing processes resulting from neutral current interactions can only occur at higher order; they are mediated by loop contributions and are consequently suppressed by additional powers of the fine-structure constant $\alpha$. Besides the fermion–gauge interactions, the non-Abelian structure of the gauge group induces self-interactions between the vector bosons. These gauge self-interactions are contained in the pure gauge-field part (<ref>) of the Lagrangian. Expressing the fields $W^a_\mu$ and $B_\mu$ in (<ref>) resp. (<ref>) by the physical fields $A_\mu$, $Z_\mu$, and $W^\pm_\mu$ yields a self-interaction term with triple and quartic couplings, which by use of the notation $ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu , \, Z_{\mu\nu} = \partial_\mu Z_\nu - \partial_\nu Z_\mu can be written in the following way, \begin{align} \label{eq:selfgauge} {\cal L}_{\rm G,self} =\, & e \left[ ( \partial_\mu W^+_\nu - \partial_\nu W^+_\mu)\, W^{-\mu} A^\nu \, + \, W^+_\mu W^-_\nu\, F^{\mu\nu} \, +\, h.c. \right] \nn \\ +\, & e \cot\theta_W \left[ (\partial_\mu W^+_\nu - \partial_\nu W^+_\mu)\, W^{-\mu} Z^\nu \, + \, W^+_\mu W^-_\nu\, Z^{\mu\nu} \, +\, h.c. \right] \nn \\ & - e^2/(4\siw)\, [ (W^-_\mu W^+_\nu - W^-_\nu W^+_\mu) W^+_\mu W^-_\nu + h.c.] \nn \\ & - e^2/4 \; (W^+_\mu A_\nu - W^+_\nu A_\mu) (W^{-\mu} A^\nu - W^{-\nu} A^\mu) \nn \\ & - e^2/4 \; \cot^2\theta_W \; (W^+_\mu Z_\nu - W^+_\nu Z_\mu) (W^{-\mu} Z^\nu - W^{-\nu} Z^\mu) \nn \\ & + e^2/2 \; \cot\theta_W \; (W^+_\mu A_\nu - W^+_\nu A_\mu) (W^{-\mu} Z^\nu - W^{-\nu} Z^\mu) + h.c. \end{align} In the Standard Model the coefficients of the self-couplings are exclusively determined by the gauge Deviations from these values could only be of non-standard origin, e.g., as remnants from new physics at some higher mass scale. § ELECTROWEAK PARAMETERS AND PRECISION OBSERVABLES Before predictions can be made from the electroweak theory, the input parameters have to be determined from experiments. As specified in the previous section, a convenient choice is the set of physical parameters given by the particle masses and the electromagnetic coupling $e$, which is commonly expressed in terms of the fine-structure constant $\alpha=e^2/4\pi$, a very precisely known low-energy parameter. Apart from the flavour sector with the fermion masses and mixing angles, only three independent quantities are required for fixing the input for the gauge sector and the fermion–gauge interactions. Conveniently, the vector-boson masses $M_{W,Z}$ and $\alpha$ are selected (equivalent to $g_1$, $g_2$, $v$). §.§ Lowest-order relations In the unitary gauge (<ref>), the propagators of the $W$ and $Z$ have the form as given in (<ref>) for massive vector fields, but with a finite width $\Gamma$ according to a Breit–Wigner shape for unstable particles, \begin{align} i\, D_{\rho\nu}(k) & = \frac{i}{k^2-M_{W,Z}^2 + i\, M_{W,Z} \Gamma_{W,Z} } \left( -g_{\nu\rho} + \frac{k_\nu k_\rho}{M_{W,Z}^2} \right) . \end{align} In processes with light fermions as external particles, the $k_\rho k_\nu$ terms are negligible since they are suppressed by powers of $m_f/M_{W,Z}$. The widths become important around the poles, i.e., when the vector bosons can be produced on-shell, like in $e^+ e^-$ annihilation or in Drell–Yan processes in hadron–hadron A very precisely measured low-energy parameter is the Fermi constant $\Gmu$, which is the effective 4-fermion coupling constant in the Fermi model, obtained from the muon lifetime to be [8] $\Gmu = 1.16637(1)\cdot 10^{-5}\, {\rm GeV}^{-2}$. Muon decay lowest-order amplitude in the Standard Model Muon decay is described in the Standard Model in lowest order by exchange of a $W$ boson between the fermionic charged currents, as shown in Fig. <ref>. Consistency of the Standard Model at the muon mass scale much smaller than $M_W$, where the momentum in the $W$ propagator can be neglected, with the Fermi model requires the identification \begin{align} \label{eq:Gfermitree} \frac{\Gmu}{\sqrt{2}} & = \frac{g_2^2}{8 \mw} = \frac{e^2}{8\siw M_W^2} = \frac{e^2}{8\siw\cow M_Z^2} \, , \end{align} which allows us to relate the vector boson masses to the parameters $\alpha,\, \Gmu$, $\siw$ and to establish also the $M_W$–$M_Z$ interdependence in terms of precise low-energy parameters, \begin{align} \label{WZcorrelation} \mw\left(1-\frac{\mw}{\mz}\right) & = \frac{\pi\alpha}{\sqrt{2}\Gmu}\, \equiv A^2 \, , \quad A = 37.2805 \, {\rm GeV} \, . \end{align} Moreover, it yields the vacuum expectation value expressed in terms of the Fermi constant, also denoted as the Fermi scale, \begin{align} \label{eq:Fermiscale} v & = \big( \sqrt{2} G_F \big)^{-\frac{1}{2}} = 246\; {\rm GeV} \, . \end{align} The relation (<ref>) can be further exploited to express the normalization of the NC couplings in (<ref>) in terms of the Fermi constant, \begin{align} \label{eq:NCnormalization} \frac{g_2}{2\cosw} = \big( \sqrt{2} \Gmu \mz\big)^{\frac{1}{2}} . \end{align} In this way, the NC vector and axial vector coupling constants of each fermion species to the $Z$ are determined and can be used to calculate the variety of observables at the $Z$ resonance, like $Z$ width and partial \begin{align} \Gamma_Z & = \sum_f \Gamma(Z\rightarrow f \bar{f}), \qquad \Gamma(Z\rightarrow f \bar{f}) = \frac{M_Z}{12\pi} \, (v_f^2 + a_f^2) \end{align} and a series of asymmetries, such as forward–backward asymmetries from the cross sections integrated over the forward ($\sigma_F$) and the backward ($\sigma_B$) hemisphere, \begin{align} A_{FB} & = \frac{\sigma_F-\sigma_B}{\sigma_F+\sigma_B} = \frac{3}{4} \, A_e\, A_f \, , \end{align} and the left–right asymmetry from the cross sections $\sigma_{L,R}$ for left- and right-handed polarized electrons, \begin{align} A_{LR} & = \frac{\sigma_L-\sigma_R}{\sigma_L+\sigma_R} = A_e \, , \end{align} all of them being determined by the ratios \begin{align} A_f = \frac{2 v_f a_f }{v_f^2+a_f^2} \end{align} with the coupling constants $v_f, a_f$ given in (<ref>). The asymmetries are particularly sensitive to the electroweak mixing angle $\siw$. §.§ Higher-order contributions §.§.§ Loop calculations These lowest-order relations given above, however, turn out to be significantly insufficient when confronted with the experimental data, which have been measured with extraordinary accuracy during the LEP and Tevatron era and require the inclusion of terms beyond the lowest order in pertubation theory. The high experimental precision makes the observables sensitive to the quantum structure of the theory which appears in terms of higher-order contributions involving diagrams with closed loops in the Feynman-graph expansion. These loop diagrams contain, in general, integrals that diverge for large integration momenta, for example in the self-energy diagrams for a propagator, typically \begin{align} \label{eq:twopointint} \int\, {\rm d}^4 q \; \frac{1}{(q^2-m_1^2)\, [(q+p)^2-m_2^2]}\, \sim\, \int\, \frac{{\rm d}^4 q}{q^4} \, \to \, \infty \, . \end{align} Nevertheless, the relations between physical observables result as finite and testable predictions, owing to the virtue of The possibility to perform such higher-order calculations is based on the formulation of the Standard Model as a renormalizable quantum field theory preserving its predictive power also beyond the tree level. Renormalizability is thereby guaranteed by local gauge invariance of the basic Lagrangian. The first step to deal with the divergent integrals is a method for regularization, which is a procedure to redefine the integrals in such a way that they become finite and mathematically well-defined The widely used regularization procedure for gauge theories is that of dimensional regularization which is Lorentz and gauge invariant: replace the dimension 4 by a lower dimension $D$ where the integrals are convergent (see Appendix <ref>), \begin{equation} \label{eq:dimreg} \int \, {\rm d}^4 q \quad \to \quad \mu^{4-D} \, \int \, {\rm d}^D q\, . \end{equation} Thereby, an (arbitrary) mass parameter $\mu$ is introduced to maintain the mass dimensions of the integrals. The divergences manifest themselves in terms of poles in the dimension $\sim 1/(4-D)$. In renormalizable theories these divergences can be absorbed in the basic parameters of the Lagrangian, like masses and coupling constants. Formally this procedure, called renormalization, is done by introducing a counter term for each parameter [for example $m^2 \to m^2 + \delta m^2$ for a mass parameter $m$] which cancels the singularities; the finite part of the counter terms, however, is not a priori fixed and has to be defined by a renormalization scheme. The selection of a renormalization scheme defines the physical meaning of each parameter and its relation to measurable quantities. These relations are then independent of $D$ and thus one can set $D\to 4$. In pure QCD, considering quarks as massless, the only basic parameter is the strong coupling constant $\alpha_s$. Since there is no intrinsic mass scale, the frequently used scheme is the $\overline{MS}$ scheme [9], where the counter term for $\alpha_s$ consists only of the singular pole part (together with a universal numerical constant). The coupling is then defined for the chosen mass scale $\mu$ in (<ref>), the renormalization scale, and thus becomes a scale-dependent quantity, the running coupling constant $\alpha_s(\mu)$ (see Ref. [6]). The Lagrangian of the electroweak Standard Model involves quite a few free parameters which are not fixed by the theory but have to be taken from experiment. In QED and in the electroweak theory, classical Thomson scattering and the particle masses set natural scales where the parameters can be defined. A distinguished choice for the basic parameters is thus given by the fundamental charge $e$ and the masses of the particles, and a common choice for the renormalization is the on-shell scheme: the mass parameters coincide with the poles of corresponding propagators (pole masses), and the charge $e$ is defined in the classical limit. The on-shell scheme hence defines the counter terms in the following way (see, e.g., Ref. [10] for details): The mass counter term $\delta m^2$, for any free mass parameter $m$, is determined by the condition \begin{align} \label{eq:massren} \delta m^2 = \Sigma (m^2) \, , \end{align} where $\Sigma$ is the self-energy of the corresponding particle, schematically depicted in (<ref>) and yielding a dressed propagator \begin{align} \frac{i}{p^2 - (m^2+\delta m^2) + \Sigma(p^2) } \; , \end{align} which by mass renormalization now includes also the mass counterterm. The condition (<ref>) ensures that $m^2$ still remains the pole of the propapator. [In the $\overline{MS}$ scheme, $\delta m^2$ only absorbs the divergent part of $\Sigma(m^2)$. The remaining finite part depends on the renormalization scale $\mu$, and in that scheme the mass becomes a $\mu$-dependent parameter, the running mass $m(\mu)$, which is different from the pole mass.] The counter term $\delta e$ for the electric charge, $e \to e +\delta e$, is determined by the requirement that $e$ be the electron–photon coupling in the classical limit, i.e., for the electron–photon vertex for real photons, $k^2=0$, and for low photon energy, $\delta e$ is essentially given by the charged-light-fermion contribution to the photon vacuum polarization at zero momentum, $\Pi^\gamma(0)$, which has a finite part $\Delta\alpha = \Pi^\gamma(0) - \Pi^\gamma(\mz)$ yielding a shift of $\Delta\alpha \simeq 0.06$ in the electromagnetic fine-structure constant $\alpha \to \alpha (1+\Delta\alpha)$. $\Delta\alpha$ can be resummed according to the renormalization group, accommodating all the leading logarithms of the type $ \alpha^n\log^n(M_Z/m_f)$ from the light fermions. The result is an effective fine-structure constant at the $Z$ mass scale \begin{align} \label{alphaeff} \alpha(\mz) & = \frac{\alpha}{1-\Delta\alpha} \simeq \frac{1}{129} \, . \end{align} It corresponds to a resummation of the iterated 1-loop vacuum polarization from the light fermions to all orders. $\Delta\alpha$ is an input of crucial importance because of its universality and remarkable numerical size [11, 12]. The loop contributions to the electroweak observables contain all particles of the Standard Model spectrum, in particular also the Higgs boson, as, for example, in the vector-boson self-energies The higher-order terms thus induce a dependence of the observables on the Higgs-boson mass $M_H$, which by means of precision measurements becomes indirectly accessible, although still unknown from direct searches. For more details see Ref. [13] and references therein. §.§.§ Vector boson masses and Fermi constant The implementation of higher-order terms can be done in a compact way for the $W$–$Z$ mass correlation, \begin{align} \label{eq:mw} \mw\left(1-\frac{\mw}{\mz}\right) & = \frac{A^2}{1-\Delta r} \, . \end{align} Loop contributions to the muon decay amplitude Therein, the contributions from the loop diagrams to the muon decay amplitude, schematically depicted in Fig. <ref>, are summarized by the quantity $\Delta r = \Delta r(m_t,M_H)$, which at one-loop order depends logarithmically on the Higgs-boson mass and quadratically on the top-quark mass. The calculation of $\Delta r$ is complete at the two-loop level [14] and comprises the leading terms also at the three- and four-loop level [15]. The prediction of $M_W$ from (<ref>) is shown in Fig. <ref> [16]. Standard Model predictions for the dependence of $M_W$ on the masses of the top quark and Higgs boson §.§.§ Observables at the Z resonance The NC couplings dressed by higher-order terms can also be written in a compact way, replacing the lowest-order couplings (<ref>) by effective couplings [13], \begin{align} g_V^f & = \sqrt{\rho_f} \, (I_3^f-2Q_f\,\sin^2\theta^f_{\rm eff} ) \, , \quad g_A^f = \sqrt{\rho_f}\, I_3^f \, , \end{align} which comprise the higher-order contributions in terms of the form factor $\rho_f(m_t,M_H)$ and the effective mixing angle $\sin^2\theta^f_{\rm eff}(m_t,M_H)$, being now a fermion-type dependent quantity. Again, their dependence on $m_t$ is quadratic, whereas they depend on $M_H$ only logarithmically. Nevertheless, the leptonic effective mixing angle is one of the most constraining observables for the mass of the Higgs boson, as shown in Fig. <ref> [16]. Like for $\Delta r$, the calculation is complete at the two-loop level [17] and supplemented by 3- and 4-loop leading terms [15]. Standard Model predictions for the dependence of $\sin^2\theta^{\rm lept}_{\rm eff}$ on the mass of the Higgs boson and the experimental $1\sigma$-range from averaged measurements done at LEP and SLC Measurements and Standard Model predictions for $a_\mu = (g_\mu-2)/2$ §.§.§ Muon magnetic moment The anomalous magnetic moment of the muon \begin{align} a_{\mu} & = \frac{g_{\mu}-2}{2} \end {align} provides a precision test at low energies. The experimental result of E 821 at Brookhaven National Laboratory [18] has reached a substantial improvement in accuracy. It shows a deviation from the Standard Model prediction by 3–4 standard deviations depending on the evaluation of the hadronic vacuum polarization from data based on $e^+e^-$ annihilation as shown in Fig. <ref> [12]. For a recent review see Ref. [19]. §.§ The vector-boson self-interaction The success of the Standard Model in the correct description of the electroweak precision observables is simultaneously an indirect confirmation of the Yang–Mills structure of the gauge boson self-interaction. For conclusive confirmations direct experimental investigation is required. At LEP 2 (and higher energies), pair production of on-shell $W$ bosons allows direct experimental tests of the trilinear vector boson self-couplings and precise $M_W$ measurements. From LEP 2, an error of 33 MeV in $M_W$ has been reached. Further improvements have been obtained from the Tevatron with currently 31 MeV uncertainty, yielding the world average for the $W$ mass $M_W=80.399 \pm 0.023$ GeV [16]. Pair production of $W$ bosons in the Standard Model is described by the amplitude based on the Feynman graphs in Fig. <ref> (in Born approximation) and higher-order contributions [21]. Feynman graphs for $e^+e^- \to W^+ W^-$ in lowest order Cross-section for $e^+e^- \to W^+W^-$, measured at LEP, and the Standard Model prediction Besides the $t$-channel $\nu$-exchange diagram, which involves only the $W$–fermion coupling, the $s$-channel diagrams contain the triple gauge interaction between the vector bosons. The gauge self-interactions of the vector bosons, as specified in (<ref>) are essential for the high-energy behaviour of the production cross-section in accordance with the principle of unitarity. Deviations from these values spoil the high-energy behaviour of the cross-sections and would be visible at energies sufficiently above the production threshold. Measurements of the cross section for $e^+e^-\to WW$ at LEP have confirmed the prediction of the Standard Model, as visualized in Fig. <ref> [16]. §.§ Global fits and Higgs boson mass bound The $Z$-boson observables from LEP 1 and SLC together with $M_W$ and the top-quark mass from LEP 2 and the Tevatron, constitute the set of high-energy quantities entering a global precision analysis. Global fits within the Standard Model to the electroweak precision data contain $M_H$ as the only free parameter, yielding the results [16] shown in Fig. <ref> and an upper limit to the Higgs mass at the 95% C.L. of $M_H < 157$ GeV, including the present theoretical uncertainties of the Standard Model predictions visualized as the blue band [16] in Fig. <ref>. Taking into account the lower exclusion bound of 114 GeV for $M_H$ from the direct searches via renormalizing the probability shifts the 95% C.L. upper bound to 186 GeV [16]. For similar analyses see Ref. [22]. The anomalous magnetic moment of the muon is practically independent of the Higgs boson mass; hence its inclusion in the fit does not change the bound on $M_H$, but it reduces the goodness of the overall fit. Experimental measurements versus best-fit Standard Model values $\chi^2$ distribution from a global electroweak fit to $M_H$ §.§ Perspectives for the LHC and the ILC In the LHC era, further improved measurements of the electroweak parameters are expected, especially on the $W$ mass and the mass of the top quark, as indicated in Table <ref>. The accuracy on the effective mixing angle, measureable from forward–backward asymmetries, will not exceed the one already obtained in $e^+ e^-$ collisions [24]. The detection of a Higgs boson would go along with a determination of its mass with an uncertainty of about 100 MeV. Present experimental accuracies and expectations for future colliders Error for Now $M_W$ [MeV] 23 15 10 7 $\sin^2\theta_{\rm eff}$ 0.00016 0.00021 0.000013 $m_{\rm top}$ [GeV] 1.3 1.0 0.2 0.13 $M_{\rm Higgs}$ [GeV] – 0.1 0.05 0.05 At a future electron–positron collider, the International Linear Collider (ILC), the accuracy on $M_W$ can be substantially improved via the scanning of the $e^+ e^-\to W^+ W^-$ threshold region [25]. The GigaZ option, a high-luminosity $Z$ factory, can provide in addition a significant reduction of the errors in the $Z$ boson observables, in particular for the leptonic effective mixing angle, denoted by $\sin^2\theta_{\rm eff}$, with an error being an order of magnitude smaller than the present one. Moreover, the top-quark mass accuracy can also be considerably improved. The numbers are collected in Table <ref>. Perspectives for Standard Model precision tests at future colliders An ultimate precision test of the Standard Model that would be possible in the future scenario with GigaZ [26] is illustrated in Fig. <ref>. The figure displays the 68% C.L. regions for $M_W$ and $\sin^2\theta_{\rm eff}$ expected from the LHC and ILC/GigaZ measurements; the small quadrangles denote the Standard Model predictions for a possible, experimentally determined, Higgs boson mass with the sides reflecting the parametric uncertainties from $\Delta \alpha$ and the top-quark mass (for $\Delta \alpha$, a projected uncertainty of $\delta\Delta\alpha =5\cdot 10^{-5}$ is assumed). If the Standard Model is correct, the two areas with the theory prediction and the future experimental results have to overlap. The central values chosen in Fig. <ref> are just examples; the main message is the development of the uncertainties. § HIGGS BOSONS The minimal model with a single scalar doublet is the simplest way to implement the electroweak symmetry breaking. The Higgs potential of the Standard Model given in (<ref>) involves two independent parameters $\mu$ and $\lambda$, which can equivalently be replaced by the vacuum expectation value $v$ and the Higgs boson mass $M_H$, as done in (<ref>). The vacuum expectation value $v$ is determined by the gauge sector, as explained in (<ref>) and (<ref>); $M_H$ is independent and cannot be predicted but has to be taken from Thus in the Standard Model the mass $M_H$ of the Higgs boson appears as the only free parameter that is still undetermined as yet. Expressed in terms of $M_H$, the Higgs part of the electroweak Lagrangian in the unitary gauge reads as follows: \begin{align} \label{LHiggsinunitarygauge} {\cal L}_{\rm H} \, = & \, \frac{1}{2} \big( \partial_\mu H \big) \big(\partial^\mu H \big) \, - \frac{M_H^2}{2}\, H^2 - \frac{M_H^2}{2v} \, H^3 - \frac{M_H^2}{8 v^2}\, H^4 \nonumber \\[0.1cm] & +\, \left( M_W^2\, W^+_\mu W^{- \mu} + \frac{M_Z^2}{2}\, Z_\mu Z^\mu \right) \left( 1 + \frac{H}{v} \right)^2 \, -\, \sum_f \, m_f\, \adpsi_f \psi_f \left(1+ \frac{H}{v} \right) , \end{align} involving interactions of the Higgs field with the massive fermions and gauge bosons, as well as Higgs self interactions proportional to $M_H^2$. §.§ Empirical bounds The existence of the Yukawa couplings and the couplings to the vector bosons $W$ and $Z$ is the basis for the experimental searches that have been performed until now at LEP and the Tevatron. At $e^+ e^-$ colliders, Higgs bosons can be produced by Higgs-strahlung from $Z$ bosons and by vector boson fusion (mainly $WW$) as displayed in Fig. <ref>. Processes for Higgs boson production in $e^+e^-$ collisions At LEP energies, Higgs-strahlung is the relevant process. The lower limit at 95% C.L. resulting from the search at LEP is 114.4 GeV [8]. From searches at the Tevatron [27] (see Fig. <ref> for various mechanisms) the mass range from 162 GeV to 166 GeV has been excluded (95% C.L.). Processes for Higgs boson production at hadron colliders Indirect determinations of $M_H$ from precision data yield an upper limit and have already been discussed in Section <ref>. As a general feature, it appears that the data prefer a light Higgs boson. §.§ Theoretical bounds There are also theoretical constraints on the Higgs mass from vacuum stability and absence of a Landau pole [28, 29, 30], and from lattice calculations [31, 32]. Explicit perturbative calculations of the decay width for $H\to W^+W^-,ZZ$ in the large-$M_H$ limit, $\Gamma(H\to VV)= K_V\cdot \Gamma^{(0)}(H\to VV)$ up to 2-loop order [33] have shown that the 2-loop contribution exceeds the 1-loop term in size (same sign) for $M_H > 930$ GeV (Fig. <ref> [34]). This result is confirmed by the calculation of the next-to-leading correction in the $1/N$ expansion, where the Higgs sector is treated as an $O(N)$ symmetric $\sigma$-model [35]. A similar increase of the 2-loop perturbative contribution with $M_H$ is observed for the fermionic decay width [36], $\Gamma(H\to f\bar{f})) = K_f\cdot \Gamma^{(0)}(H\to f\bar{f}))$, but with opposite sign leading to a cancellation of the 1-loop correction for $M_H\simeq 1100$ GeV (Fig. <ref>). The lattice result [32] for the bosonic Higgs decay in Fig. <ref> for $M_H=727$ GeV is not far from the perturbative 2-loop result; the difference may at least partially be interpreted as missing higher-order terms. Correction factors $K_V, K_f$ from higher orders for the Higgs decay widths $H\to VV\; (V=W,Z)$ and $H \to f \bar{f}$ in 1- and 2-loop order The behaviour of the quartic Higgs self-coupling $\lambda$, as a function of a rising energy scale $Q$, follows from the renormalization group equation \begin{align} \label{HiggsRGE} \frac{{\rm d} \lambda}{{\rm d} t} & = \frac{1}{16\pi^2}\, (12 \lambda^2 + 6\, \lambda\,g_t^2 - 3 \, g_t^4 + \cdots ), \quad t=\log \frac{Q^2}{v^2} \, , \end{align} with the $\beta$-function dominated by the contributions from $\lambda$ and the top-quark Yukawa coupling $g_t$ in the loop contributions to the quartic interactions, Owing to the second diagram, the first term in (<ref>), $\lambda(Q)$ increases with $Q$ and diverges at a critical scale, the Landau pole, which moves towards lower values for increasing mass $M_H$. The requirement of a perturbative, small coupling $\lambda(Q)$ up to a scale $\Lambda$ thus yields an upper bound for $M_H$. In order to avoid unphysical negative quartic couplings from the negative top-loop contribution, a lower bound on the Higgs mass is derived. In combination, the requirement that the Higgs coupling remain finite and positive up to a scale $\Lambda$ yields constraints on the Higgs mass $M_H$, which have been evaluated at the 2-loop level [29, 30]. These bounds on $M_H$ are shown in Fig. <ref> [30] as a function of the cut-off scale $\Lambda$ up to which the standard Higgs sector can be extrapolated. The allowed region is the area between the lower and the upper curves. The bands indicate the theoretical uncertainties associated with the solution of the renormalization group equations [30]. It is interesting to note that the indirect determination of the Higgs mass range from electroweak precision data via radiative corrections is compatible with a value of $M_H$ where $\Lambda$ can be extended up to the Planck scale. Theoretical limits on the Higgs boson mass from the absence of a Landau pole and from vacuum stability §.§ Future searches For the coming experimental searches at the LHC, it is important to have precise and reliable predictions for the production and decay rates. Higgs bosons can be produced through various mechanisms at the partonic level. The main partonic processes for Higgs boson production are depicted in Fig. <ref>, and the corresponding production cross sections are shown in Fig. <ref> [37]. The largest cross section arises from gluon–gluon fusion. The experimental signal, however, is determined by the product \begin{align} & \sigma(AB\to H) \cdot BR(H\to X) \end{align} of the production cross section $\sigma(AB\to H)$ from initial-state partons $A, B$ and the branching ratio $BR(H\to X)$ for the decay of the Higgs boson into a specific final state $X$ (see Fig. <ref> for the branching ratios [38]). A light Higgs boson, well below the $WW$ threshold, decays predominantly into $b\bar{b}$ quarks, owing to the largest Yukawa couplings in the kinematically allowed fermionic decay channels. This signal, however, is experimentally unaccessible because it is covered by a huge background of QCD-generated $b$-quark jets. Therefore, in the low mass range, the rare decay channel $H\to \gamma \gamma$ has to be selected reducing the total number of events considerably, in spite of the large production cross section, and makes Higgs detection a cumbersome business. For larger masses, $M_H \gtrsim 140$ GeV, the decay modes $H\to WW, ZZ \to 4 f$ make detection relatively easy. The vector-boson fusion channel (third diagram of Fig. <ref>) with subsequent leptonic decay $H\to \tau^+ \tau^-$ is a promising alternative. Cross sections for Higgs boson production at the LHC Branching ratios for Higgs boson decays For completeness we list the (lowest-order) expressions for the dominant Higgs decay rates into fermion and vector-boson pairs, \begin{align} \Gamma (H\to f\bar{f}) & = \, N_C \frac{G_F M_H\, m_f^2}{4 \pi \sqrt{2}} \sqrt{1-\frac{4m_f^2}{M_H^2} } \;\;{\rm with} \;\; N_C=3 \; {\rm for} \, f=q, \;\; N_C=1 \; {\rm for} \, f=\ell , \nn \\ \Gamma (H\to VV) & = \, \frac{G_F M_H^3}{16 \pi \sqrt{2}} \, R_V(x_V), \quad x_V = \frac{M_V^2}{M_H^2}, \qquad (V= W, Z) \end{align} \begin{align} R_Z & = R(x_Z) , \quad R_W = 2 \, R(x_W), \quad R(x) = \sqrt{1-4x} \, (1-4x+12x^2) \, . \end{align} As an exercise, these formulae can easily be derived from the $Hff$ and $HVV$ vertices in (<ref>) with the help of the Feynman rules of Section <ref> and the general expression for the width in (<ref>). §.§ Supersymmetric Higgs bosons Among the extensions of the Standard Model, the Minimal Supersymmetric Standard Model (MSSM) [39] is a theoretically favoured scenario as the most predictive framework beyond the Standard Model. A light Higgs boson, as indicated in the analysis of the electroweak precision data, would find a natural explanation by the structure of the Higgs potential. For a review on MSSM Higgs bosons see Ref. [40]. Example of the Higgs boson mass spectrum in the MSSM The five physical Higgs particles of the MSSM consist of two $CP$-even neutral bosons $h^0,H^0$, a $CP$-odd $A^0$ boson, and a pair of charged Higgs particles $H^\pm$. At tree level, their masses are determined by the $A^0$ boson mass, $M_A$, and the ratio of the two vacuum expectation values, $v_2/v_1 = \tan\beta$, \begin{align} M_{H^+}^2 & = \, M_A^2 + M_W^2 \, , \nn \\ M_{H^0,h^0}^2 & = \, \frac{1}{2} \, \left( M_A^2 + M_Z^2 \pm \sqrt{\big(M_A^2 + M_Z^2\big)^2 - 4 M_Z^2 M_A^2 \cos^2 2\beta} \right) . \end{align} These relations are sizeably modified by higher-order contributions to the Higgs boson vacua and propagators. A typical example of a spectrum is shown in Fig. <ref>, based on the FeynHiggs code [41]. In particular the mass of the lightest Higgs boson $h^0$ is substantially by loop contributions; for large $M_A$, the $h^0$ particle behaves like the standard Higgs boson, but its mass is dependent on basically all the parameters of the model and hence yields another powerful precision observable. A definite prediction of the MSSM is thus the existence of a light Higgs boson with mass below $\sim 140$ GeV. The detection of a light Higgs boson could be a significant hint for The structure of the MSSM as a renormalizable quantum field theory allows a similarly complete calculation of the electroweak precision observables as in the Standard Model in terms of one Higgs mass (usually taken as $M_A$) and $\tan\beta$, together with the set of SUSY soft-breaking parameters fixing the chargino/neutralino and scalar fermion sectors [42]. For updated discussions of precision observables in the MSSM see Ref. [43] . The $W$ mass range in the Standard Model (lower band) and in the MSSM (upper band) respecting bounds are from the non-observation of Higgs bosons and SUSY particles As an example, Fig. <ref> displays the range of predictions for $M_W$ in the Standard Model and in the MSSM, together with the present experimental errors and the expectations for the LHC measurements. The MSSM prediction is in slightly better agreement with the present data for $M_W$, although not conclusive as yet. Future increase in the experimental accuracy, however, will become decisive for the separation between the models. Especially for the muonic $g-2$, the MSSM can significantly improve the agreement between theory and experiment: one-loop terms with relatively light scalar muons, sneutrinos, charginos and neutralinos, in the mass range 200–600 GeV, together with a large value of $\tan\beta$ can provide a positive contribution $\Delta a_\mu$, which can entirely explain the difference $a_\mu^{\rm exp} - a_\mu^{\rm SM}$ (see Ref. [44] for a review). The MSSM yields a comprehensive description of the precision data, in a similar way to the Standard Model. Global fits, varying the MSSM parameters, have been performed to all electroweak precision data [45] showing that the description within the MSSM is slightly better than in the Standard Model. This is mainly due to the improved agreement for $a_\mu$. The fits have been updated recently for the constrained MSSM (cMSSM), including also bounds from $b\to s\gamma$ and from the cosmic relic density. The $\chi^2$-distribution for the fit parameters can be shown [47] as a $\chi^2$-distribution for the lightest Higgs boson mass $M_H$, displayed in Fig. <ref>. The mass range $M_h = 110^{+9}_{-10}$ GeV obtained from this fit is in much better agreement with the lower bound from the direct search than in the case of the Standard Model. $\chi^2$-distribution for cMSSM fits, expressed in terms of $M_h$ § OUTLOOK In spite of the success of the Standard Model in describing a large variety of phenomena, at a high level of accuracy on both the theoretical and the experimental side, there is a list of shortcomings that motivate the quest for physics beyond the Standard Model. A rather direct augmentation is enforced by the need for accommodating massive neutrinos. The Standard Model in its strictly minimal version is incomplete with respect to a mass term for neutrinos. Neutrino mass terms can be added [7] without touching on the basic architecture of the Standard Model. Besides this rather immediate modification one is confronted, however, with a series of basic conceptual problems: * the smallness of the electroweak scale $v \sim 1/\sqrt{G_F}$ compared to the Planck scale $M_{\rm Pl}\sim 1/\sqrt{G_N}$ (the hierarchy problem) and the smallness of the Higgs boson mass of ${\cal O}(v)$, which is not protected against large quantum corrections of ${\cal O}( M_{\rm Pl})$; * the large number of free parameters (gauge couplings, vacuum expectation value, $M_H$, fermion masses, CKM matrix elements), which are not predicted but have to be taken from experiments; * the pattern that occurs in the arrangement of the fermion * the quantization of the electric charge, or the values of the hypercharge, respectively; * the missing way to connect to gravity. Moreover, there are also phenomenological shortcomings, like missing answers to the questions about * the nature of dark matter that constitutes the largest fraction of matter in the Universe, * the origin of the baryon asymmetry of the Universe. The class of models based on supersymmetry, briefly addressed in the last subsection <ref>, can at least provide partial answers, e.g., for dark matter, the further unification of forces and hierarchy of mass scales, new sources of CP violation, and can be related to string theory as a candidate for a microscopic theory of gravity. The LHC experiments may soon shed light on our unanswered questions, or may also surprise us with answers to questions we did not ask. § CANONICAL COMMUTATION RELATIONS The commutators between the canonically conjugate variables $Q_j, P_k$ in quantum mechanics, \begin{align} \label{eq:Heisenberg} [Q_j, P_k] & =\, i \, \delta_{jk} , \quad [Q_j, Q_k] =\, [P_j, P_k] = 0 , \end{align} are translated in quantum field theory to commutators for a (generic) field operator $\phi(x)\equiv \phi(t,\vec{x})$ and its conjugate canonical momentum \begin{align} \label{eq:canonmom} \Pi(x) & = \frac{\partial {\cal L}} {\partial (\partial_o \phi)} \end{align} derived from the basic Lagrangian ${\cal L}$ for the system. This procedure, known as canonical field quantization, is specified by the equal-time commutation relations, where the discrete indices $j, k$ in (<ref>) are replaced by the continuous indices $\vec{x}, \vec{x}\,'$: \begin{align} \label{eq:QFTCR} [\phi(t,\vec{x}), \Pi(t,\vec{x}\,')] & = i\, \delta^3(\vec{x} - \vec{x}\,'), \quad [\phi(t,\vec{x}), \phi(t,\vec{x}\,')] = [\Pi(t,\vec{x}), \Pi(t,\vec{x}\,')] = 0 \, . \end{align} For fermionic field variables $\psi(x)$ the commutators have to be replaced by anti-commutators. §.§ Scalar field We illustrate the method of canonical quantization choosing the scalar field as a specific example. Starting from the Lagrangian (<ref>) for a general, complex, free scalar field, we find the canonical field momenta via (<ref>) to be \begin{align} \frac{\partial {\cal L}}{\partial (\partial_o \phi)} & = \partial^0 \phi^\dagger = \dot{\phi}^\dagger = \Pi, \nn \\ \frac{\partial {\cal L}}{\partial (\partial_o \phi^\dagger)} & = \partial^0 \phi \; = \, \dot{\phi}\;\; = \Pi^\dagger \, . \end{align} Accordingly, the canonical commutation relations are given by \begin{align} \label{eq:CRscalar} & [\phi(t,\vec{x}), \dot{\phi}^\dagger(t,\vec{x}\,')] = i\, \delta^3(\vec{x} - \vec{x}\,') , \nn \\ & [\phi(t,\vec{x}), \phi(t,\vec{x}\,')] = \, [\dot{\phi}(t,\vec{x}), \dot{\phi}(t,\vec{x}\,')] = 0 \, . \end{align} These relations can equivalently be expressed in terms of the annihilation and creation operators $a,b, a^\dagger, b^\dagger$ in the Fourier expansion of the scalar field $\phi(x)$ in (<ref>). They fulfil the following canonical commutation relations in momentum space and can be interpreted as those for a continuous set of quantized harmonic oscillators, labelled by $\vec{k}$, with frequencies $\omega = k^0 = \sqrt{{\vec{k\,}}^{2} +m^2}$ and with the relativistic normalization: \begin{align} \label{CRscalarmomentum} & [a(k), a(k')] = [b(k), b(k')] = 0 , \qquad [a^\dagger(k), a^\dagger(k')] = [b^\dagger(k), b^\dagger(k')] = 0 , \nn \\ & [a(k), a^\dagger(k')] = 2 k^0 \, \delta^3(\vec{k} - \vec{k}\, ') , \qquad [b(k), b^\dagger(k')] = 2 k^0 \, \delta^3(\vec{k} - \vec{k}\, ') , \nn \\ & [a(k), b(k')] = [a(k), b^\dagger(k')] = [a^\dagger(k), b(k')] = [a^\dagger(k), b^\dagger(k')] = 0 . \end{align} Since we do not make use of the formulation of quantization in space-time, but use instead the creation and annihilation operators, which are closer to the physical picture of particles and particle states, we list the commutators for the vector and spinor fields only in momentum space. §.§ Vector field For the vector field (<ref>) the annihilation and creation operators $a_\lambda, a_\lambda^\dagger$ carry helicity indices in additon to the momenta. Otherwise the commutation rules are analogous to the scalar \begin{align} \label{CRvector} & [a_\lambda(k), a_{\lambda'}(k')] = [a_\lambda^\dagger(k), a_{\lambda'}^\dagger(k')] = 0 , \nn \\ & [a_\lambda(k), a_{\lambda'}^\dagger(k')] = 2 k^0\, \delta_{\lambda \lambda'} \, \delta^3(\vec{k} - \vec{k}\, ') . \end{align} §.§ Dirac field The Dirac field (<ref>) involves fermionic annihilation and creation operators $c_\sigma, d_\sigma, c^\dagger_\sigma, d^\dagger_\sigma$ for each momentum $\vec{k}$ and helicity $\sigma$. According to the antisymmetry of fermionic states, all commutators applying to bosonic states in the canonical quantization above have to be replaced by anti-commutators: \begin{align} \label{CRDirac} & \{c_\sigma(k), c_{\sigma'}(k')\} = \{c_\sigma^\dagger(k), c_{\sigma'}^\dagger(k')\} = 0 , \qquad \{c_\sigma(k), c_{\sigma'}^\dagger(k')\} = 2 k^0\, \delta_{\sigma \sigma'} \, \delta^3(\vec{k} - \vec{k}\, ') , \nn \\ & \{d_\sigma(k), d_{\sigma'}(k')\} = \{d_\sigma^\dagger(k), d_{\sigma'}^\dagger(k')\} = 0 , \qquad \!\! \{d_\sigma(k), d_{\sigma'}^\dagger(k')\} = 2 k^0\, \delta_{\sigma \sigma'} \, \delta^3(\vec{k} - \vec{k}\, ') , \nn \\ & \{c_\sigma(k), d_{\sigma'}(k')\} = \{c^\dagger_\sigma(k), d^\dagger_{\sigma'}(k')\} = \{c_\sigma(k), d^\dagger_{\sigma'}(k')\} = \{c_\sigma^\dagger(k), d_{\sigma'}(k')\} = 0 . \end{align} § GREEN FUNCTIONS AND CAUSALITY We demonstrate, for the example of the scalar field, how the $+i\epsilon$ prescription in the Fourier representation of the Feynman propagator leads to causal behaviour of particle/antiparticle propagation in space-time. Making use of the time-ordered product of any two field quantities $A(x)$ and $B(x)$, \begin{align} \label{eq:timeordering} T A(x) B(y) & =\, \Theta(x^0-y^0)\, A(x) B(y) + \Theta(y^0-x^0)\, B(x) A(y) \, , \end{align} one can define the 2-point function for a (complex) scalar field $\phi(x)$ in the following way: \begin{align} \label{eq:twopoint} <\!0\| T \phi(x) \phi^\dagger (y) \|0\!> & = \, \Theta(x^0-y^0)\, <\!\!0\| \phi(x) \phi^\dagger (y) \|0\!> \nn \\ & +\, \Theta(y^0-x^0)\, <\!\!0\| \phi^\dagger(y) \phi(x) \|0\!> \, . \end{align} Invoking the Fourier expansion for $\phi$ and $\phi^\dagger$ in terms of creation and annihilation operators (<ref>), one can see that (<ref>) describes particles created at time $y^0$ and annihilated at time $x^0$ if $x^0 > y^0$, and anti-particles created at time $x^0$ and annihilated at time $y^0$ if $y^0 > x^0$. On the other hand, starting from the Fourier integral (<ref>) and performing the $k^0$ integration by means of a contour integral in the complex plane, one obtains the expression \begin{eqnarray} D(x-y) & = & \int \frac{{\rm d}^4k}{(2\pi)^4} \, \frac{e^{-ik(x-y)}}{k^2 - m^2 + \, i \epsilon} \nn \\ & = & \int \frac{{\rm d}^3k}{(2\pi)^3} \, e^{i \vec{k} (\vec{x}-\vec{y})} \int \frac{{\rm d}k^0}{2\pi} \, \frac{e^{-i k^0 (x^0-y^0)} } {(k^0)^2 - \vec{k}^{\, 2} - m^2 + i\epsilon} \nn \\ & =& \,-\, \frac{i}{(2\pi)^3} \int \frac{{\rm d}^3k}{2 k^0} \, e^{i \vec{k} (\vec{x}-\vec{y})- i k^0 (x^0-y^0)} \,\|\,_{k^0 = \sqrt{\vec{k\,}^{2} + m^2} } \;\;\cdot \Theta(x^0-y^0) \ \nn \\ & & - \, \frac{i}{(2\pi)^3} \int \frac{{\rm d}^3k}{2 k^0} \, e^{i \vec{k} (\vec{x}-\vec{y})+ i k^0 (x^0-y^0)} \,\|\,_{k^0 = \sqrt{\vec{k\,}^{2} + m^2} } \;\;\cdot \Theta(y^0-x^0) \nn \end{eqnarray} which can be written in the following way: \begin{align} i\, D(x-y) & = \, \frac{1}{(2\pi)^3} \int \frac{{\rm d}^3k}{2 k^0} \, \left[ e^{-ik(x-y)} \, \Theta(x^0-y^0) \, +\, e^{ik(x-y)} \, \Theta(y^0-x^0) \right]_{k^0=\sqrt{\vec{k\,}^{2} + m^2} } \, . \end{align} This is identical to (<ref>) when the Fourier representation (<ref>) for $\phi$ is inserted. Hence one has the identity \begin{align} <\!0\| T \phi(x) \phi^\dagger (y) \|0\!> & = \, i \, D(x-y) \, , \end{align} which connects the Green function of the Klein–Gordon equation with the 2-point function of the quantized scalar field and thus with the particle/antiparticle concept obeying causality. As a byproduct, it also explains the extra factor $i$ in the propagator (<ref>). § LOOP INTEGRALS AND DIMENSIONAL REGULARIZATION In the calculation of self-energy diagrams the following type of loop integrals involving two propagators appears when the integration is done in $D$ dimensions, denoted by $B_0$ after removing a numerical factor: \begin{align} \label{twopointintegral} \dkm \,\frac{1}{\Dkk\Dkq} & = \frac{i}{16\pi^2} B_0(q^2,m_1,m_2) \, . \end{align} With help of the Feynman parametrization \begin{align} \frac{1}{ab} & = \int^1_0 {\rm d}x \, \frac{1}{[ax+b(1-x)]^2} \end{align} and after a shift in the $k$-variable, $B_0$ can be written in the form \begin{align} \label{Bintegral} \ipi\, B_0(q^2,m_1,m_2) & = \int^1_0 {\rm d}x \, \frac{\m^{4-D}}{(2\pi)^D} \int \frac{{\rm d}^Dk}{[k^2-x^2q^2+x(q^2+m_1^2-m_2^2)-m_1^2+i \varepsilon]^2} \, . \end{align} The advantage of this parametrization is a simpler $k$-integration where the integrand is only a function of $k^2=(k^0)^2-\vec{k}^2$. In order to transform it into a Euclidean integral we perform the [The $i\veps$-prescription in the masses ensures that this is compatible with the pole structure of the integrand.] \begin{align} k^0 & = i\,k_E^0,\;\, \vec{k} =\vec{k}_E,\;\; {\rm d}^D k = i\,{\rm d}^D k_E \end{align} where the new integration momentum $k_E$ has a positive-definite metric: \begin{align} k^2 = -k_E^2, \;\; \; k_E^2 = (k^0_E)^2 + \cdots + (k_E^{D-1})^2 \, . \end{align} This leads us to a Euclidean integral over $k_E$, \begin{align} \ipi\, B_0 & = i \int^1_0 {\rm d}x \frac{\m^{4-D}}{(2\pi)^D} \int \frac{{\rm d}^Dk_E}{(k_E^2 + Q)^2 } \end{align} \begin{align} \label{Qdef} Q & = x^2q^2-x(q^2+m_1^2-m_2^2)+m_1^2 - i\varepsilon \end{align} is a constant with respect to the $k_E$-integration. This $k_E$-integral is of the general type $$ \int \frac{{\rm d}^Dk_E}{(k_E^2+Q)^n} $$ of rotational-invariant integrals in a $D$-dimensional Euclidean space. They can be evaluated using $D$-dimensional polar coordinates (with the substitution $k_E^2 = R$), $$ \int\frac{{\rm d}^Dk_E}{(k_E^2+Q)^n}\, =\, \frac{1}{2} \int {\rm d}\Omega_D \int^{\infty}_0 {\rm d}R\, R^{\frac{D}{2}-1} \, \frac{1}{(R+Q)^n} \, , $$ \begin{align} \label{Dintegral} \frac{\m^{4-D}}{(2\pi)^D} \int \frac{{\rm d}^Dk_E}{(k_E^2+Q)^n} \,& = \, \frac{\m^{4-D}}{(4\pi)^{D/2}} \cdot \frac{\Gamma(n-\frac{D}{2})}{\Gamma(n)}\cdot Q^{-n+\frac{D}{2}} \, . \end{align} The singularities of the initially 4-dimensional integrals are now as poles of the $\Gamma$-function for $D=4$ and values $n \leq 2$. Although the l.h.s.of (<ref>) as a $D$-dimensional integral is sensible only for integer values of $D$, the r.h.s. has an analytic continuation in the variable $D$: it is well defined for all complex values $D$ with $n-\frac{D}{2}\neq 0,-1,-2,\dots$, in particular for \begin{align} D & = 4 -\eps \;\;\; \mbox{ with } \eps > 0 \, . \end{align} For physical reasons we are interested in the vicinity of $D=4$. Hence we consider the limiting case $\eps \to 0$ and perform an expansion around $D=4$ in powers of $\eps$. For this task we need the following properties of the $\Gamma$-function at $x\to 0$: \begin{align} & \Gamma(x) = \frac{1}{x}\, - \,\g\, +\, {\cal O}(x) \, , \nn \\ & \Gamma(-1+x) = -\,\frac{1}{x} \,+\,\g \,-\, 1 \,+\,{\cal O}(x)\, , \end{align} with Euler's constant \begin{align} \g & = -\,\Gamma'(1) = 0.577\dots \end{align} For the integral $B_0$ we evaluate the integrand of the $x$-integration in (<ref>) with help of (<ref>) as follows: \begin{eqnarray} \label{eq:epsexpansion} \frac{\m^{\eps}}{(4\pi)^{2-\eps/2}} \cdot \frac{\Gamma(\frac{\eps}{2})}{\Gamma(2)} \cdot Q^{-\eps/2} & = & \frac{1}{16\pi^2} \left( \frac{2}{\eps} -\g + \log 4\pi -\log\frac{Q}{\m^2} \right) +\, {\cal O}(\eps) \nn \\ & = & \frac{1}{16\pi^2} \left( \Delta -\log\frac{Q}{\m^2} \right) +\, {\cal O}(\eps) \, . \end{eqnarray} Since the $O(\eps)$ terms vanish in the limit $\eps\to 0$ we can skip them in the following. 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1012.3925	# Single Photon Ignition of Two-photon Super-fluorescence through the Vacuum of Electromagnetic Field Nicolae A. Enaki Academiei str.5, Institute of Applied Physics of Academy of sciences of Moldova, Chisinau, MD 2028, Republic of Moldova (December 8, 2010 ) ###### Abstract The ignition of two-quantum collective emission of inverted sub-ensemble of radiators due to mutual interaction of this sub-ensemble with other two dipole active atomic subsystems in process of two-photon exchanges between the atoms through the vacuum field is proposed. The three particle resonances between two-photon and single quantum transitions of inverted radiators from the ensemble are proposed for acceleration of collective decay rate of bi-photons, obtained relatively dipole-forbidden transitions of excited atomic sub- ensemble. This mutual interaction between three super-fluorescent processes in subatomic ensembles take place relatively dipole-forbidden transitions in one of radiator subsystem. The collective resonance emission and absorption of two-quanta have nontraditional behavior, accompanied with acceleration and inhibition of collective emission processes of photons. ## 1 Introduction A great deal of attention is currently devoted to the problem of coherence which appears not only between the quanta but between groups of quanta too. The generation of non-classical coherent electromagnetic field in multi-photon emission and the interaction of coherent radiation with matter (nuclei, atoms and solids) have been subjects of a number of theoretical and experimental studies in recent years [1]-[3]. Examples include the higher-order coherence in multi-photon generation of light the two-photon micro-maser emission [3], two-photon lasers the parametric down conversion, four-wave mixing and other effects in optical diapason [2], and the possibility of coherent generation of photons in $x$-ray and $gamma$\- ray spectral regions. In this article it is proposed to investigate the cooperative two-photon emission from inverted system of radiators stimulated by single photon super- fluorescent pulses in two-quantum resonance with dipole forbidden atomic transition. Since the two-photon cooperative phenomenon has the small two- photon cooperative emission time [9], we propose to extend our attention to the new tape of cooperative resonance interaction between three radiators in which single photon transitions of two radiators which enter in two-photon resonance with dipole forbidden transition of third atom. This cooperative three particle interaction take place through the vacuum fluctuations of electromagnetic field and can amplify or diminish the spontaneous emission rates of the atoms. In order to obtain more powerful pulses of entangled photons it is proposed the cooperative interaction between three atomic subsystems in which one of them are inverted relatively dipole forbidden transition $\|2S>$ $-$ $\|S>$ of Hydrogen like or Helium Like atoms [4]-[8]. Taking in to account the elementary acts of two photon interaction between radiators we archived the improvement of two-photon emission rate of the system of radiators in comparison with two-photon super-fluorescence [9]. In this article it is examined the mutual influence of two single-photon super- fluorescence processes and two-photon cooperative emission of the atomic system relatively dipole forbidden transition. The phenomenon of new cooperative emission takes in to account the three particle mutual interaction with vacuum of electromagnetic field in which the product of vacuum polarization of two atoms enter in to resonance with two-photon polarization of dipole forbidden transition of Hydrogen-like or Helium-like radiator. It has been shown that in the process of spontaneous radiation, the radiators (nuclei, atoms) enter a regime of single and two-photon super-radiance and the rate of photon pair (bi-photon) emission increases (or decreases) due to new three particle cooperative phenomenon, which appear between single and two- photon spontaneous emission subgroups of radiators. It has been demonstrated, that for hydrogen-like and helium-like atoms [10] the dipole-forbidden transitions can generate more powerful pulse of entangled photon pairs (bi- photons) under the influence of single photon super-radiance. It is important to note that for coherent radiation of such system, was studied for the dimension of a radiating system smaller than the radiation wavelength. It is, however, interesting to study this type of cooperative emission between three radiator subsystems in extended system of radiators. The possibilities of two-photon cooperative resonances between three radiators replaced at distance larger than emission wavelength are studied too. I emphasize here, that the problem of cooperation between two single photon cooperative emission subsystems and one two-photon cooperative emission subsystem is more complicated than the similar problem of single [11] or two- photon [9] super-radiances in extended system. In Dicke’s super-radiance, the exchange integral between $j$-th and $l$-th atoms is described by more simple exchange integral proportional to $\sin[k_{0}r_{jl}]/(k_{0}r_{jl})$ while in two-photon super-radiance by more complicated function $\sin[(2k_{0}-k)r_{jl}]/[(2k_{0}-k)r_{jl}]\times\sin[kr_{jl}]/(kr_{jl})$, where $r_{jl}$ and $k_{i}\ $are the distance between the radiators and wave vector of emitted photons respectively. The three particle exchange integral between $j$-th, $m$-th and $l$-th atoms was obtained in this paper taking in to account the two-quantum exchanges between two radiators proposed in papers [9], [12]. The more complicated exchange integrals between two radiators with dipole active transition and one radiator with dipole forbidden transition is given in Appendix of this paper. ## 2 Interaction Hamiltonian and Master Equation Let us consider the interaction of three subsystems of radiators $R$, $S$, and $D$ thorough vacuum of electromagnetic field. The first two groups, $R$ and $S$, are prepared in excited state $\|e_{r}\rangle\otimes\|e_{s}\rangle$ and can pass in to Decke super-radiance regime [11] relatively the dipole active transitions $e_{r}\rightarrow g_{r}$ and $e_{s}\rightarrow g_{s}$ at frequencies $\omega_{r}$ and $\omega_{s}$ (see figure 1). The $D$ atomic subsystem is prepared in excited state $\|e_{d}\rangle$ and relatively dipole forbidden transition $e_{d}\rightarrow g_{d}$ and can pass in the ground state $\|g_{d}\rangle$ simultaneously generation two quanta [9]. Let us consider the simple cooperative stimulation of two-photon emission of $D$ system stimulated by $R$ and $S$ radiator subsystems. Figure 1: The resonance between two-photon transitions of D atomic subsystem and two dipole active atomic subsystems$R$ and $S$ . As an example is represented three atoms $D$, $R$ and $S$ situated at relatively distances $r_{ds}$, $r_{dr}$ and $r_{rs}$. One of condition of exchange energies between the subsystems is the e resonance between two-photon and single photon transitions $2\omega_{0}=\omega_{r}+\omega_{s}$ . In this case it is established the resonance between the dipole active atomic subgroups $R$, $S$ and dipole forbidden radiators of $D$ ensemble and the cooperative stimulation of two-quantum collective transition is possible. Indeed, considering that the conservation energy law is established between these groups, $\hbar(\omega_{r}+\omega_{s})=2\hbar\omega_{0}$, one can proposed the following Hamiltonian of interaction of radiators with electromagnetic field $\displaystyle H$ $\displaystyle=$ $\displaystyle H_{0}+\lambda H_{I};$ $\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle\sum\limits_{k}\hbar\omega_{k}a_{k}^{\dagger}a_{k}+\sum\limits_{j=1}^{N_{r}}\hbar\omega_{r}R_{zj}+\sum\limits_{l=1}^{N_{s}}\hbar\omega_{s}S_{zl}+2\sum\limits_{m=1}^{N}\hbar\omega_{0}D_{zm};$ $\displaystyle\lambda H_{I}$ $\displaystyle=$ $\displaystyle-\sum\limits_{k}\sum\limits_{j=1}^{N_{a}}(\mathbf{d}_{r},\mathbf{g}_{k})\\{R_{j}^{+}a_{k}\exp[i(\mathbf{k},\mathbf{r}_{j})]+R_{j}^{-}a_{k}^{\dagger}\exp[-i(\mathbf{k},\mathbf{r}_{j})]\\}$ (1) $\displaystyle-$ $\displaystyle\sum\limits_{k}\sum\limits_{l=1}^{N_{b}}(\mathbf{d}_{s},\mathbf{g}_{k})\\{S_{l}^{+}a_{k}\exp[i(\mathbf{k},\mathbf{r}_{l})]+S_{l}^{-}a_{k}^{\dagger}\exp[-i(\mathbf{k},\mathbf{r}_{l})]\\}$ $\displaystyle-$ $\displaystyle\sum\limits_{k_{1},k_{2}}\sum\limits_{m=1}^{N_{b}}(\mathbf{n}_{eg},\mathbf{e}_{\lambda_{1}})(\mathbf{n}_{ei},\mathbf{e}_{\lambda_{2}})q(\omega_{1},\omega_{2})$ $\displaystyle\times$ $\displaystyle\\{D_{m}^{+}a_{k_{2}}a_{k_{1}}\exp[i(\mathbf{k}_{1}+\mathbf{k}_{2},\mathbf{r}_{m})]$ $\displaystyle+$ $\displaystyle D_{m}^{-}a_{k_{1}}^{\dagger}a_{k_{2}}^{\dagger}\exp[-i(\mathbf{k}_{1}+\mathbf{k}_{2},\mathbf{r}_{l})]\\}.$ Here $q(\omega_{1},\omega_{2})=\frac{d_{23}d_{31}g_{k_{2}}g_{k_{1}}}{2\hbar}\left\\{\frac{1}{\omega_{32}+\omega_{k_{1}}}+\frac{1}{\omega_{31}-\omega_{k_{2}}}\right\\},\ \ \ \mathbf{g}_{k}=\sqrt{\frac{2\pi\hbar\omega_{k}}{V}}\mathbf{\epsilon}_{{}_{\lambda}},$ $a_{k}$ and $a_{k}^{\dagger}$ are annihilation and creation operators of EMF photons with wave vector $\mathbf{k,}$ polarization $\mathbf{\ \epsilon}_{{}_{\lambda}}$ and frequency $\omega_{k}$; $\mathbf{d}_{r}$ and $\mathbf{d}_{s}$ are dipole momentum transition between the ground and excited states for $R$ and $S$ atomic subsystems; $d_{ei}$ and $d_{eg}$ are dipole momentum transitions in the three level system of atomic group $D$. The operators of $R$, $S$, and $D$ atomic subsystems satisfy the commutation relations for $SU(2)$ algebra $[J^{+},J^{-}]=2J_{z}$; $[J_{z}\ ,J^{\pm}]=\pm J^{\pm}$, where $J^{\pm}$ is equivalent with $R^{\pm}$, $S^{\pm}$ and $D^{\pm}$. Invertin operator $J_{z}$ is consider similar to $R_{z},$ $S_{z}$ and $D_{z}$ respectively. The operators of electromagnetic field satisfy the commutation relation $[a_{k},a_{k^{\prime}}^{\dagger}]=\delta_{k,k^{\prime}}\ ;[a_{k}^{\dagger},a_{k^{\prime}}^{\dagger}]=0$, where $k=(\mathbf{k,}\lambda)$ is the wave vector and polarization of the photon. Taking in to account the Hamiltonian 1, let us represent the solution of Haisenberg equation through the sources and free part operators $a_{k}(t)=a_{k}(0)\exp[-i\omega_{k}t]+a_{ks}(t),$ (2) where the source part is $\displaystyle a_{ks}(t)$ $\displaystyle=$ $\displaystyle\frac{i(\mathbf{d}_{a},\mathbf{g}_{k})}{\hbar}\sum\limits_{l=1}^{N_{a}}\exp[-i(\mathbf{k},\mathbf{r}_{l})\int\limits_{0}^{t}d\tau\exp[-i\omega_{k}\tau]R_{l}^{-}(t-\tau)$ $\displaystyle+i\frac{(\mathbf{d}_{b},\mathbf{g}_{k})}{\hbar}\sum\limits_{j=1}^{N_{b}}\exp[-i(\mathbf{k},\mathbf{r}_{j})\int\limits_{0}^{t}d\tau\exp[-i\omega_{k}\tau]S_{j}^{-}(t-\tau)$ $\displaystyle+2i\sum\limits_{n=1}^{N_{b}}\sum\limits_{k_{1}}\frac{(\mathbf{n}_{eg},\mathbf{e}_{\lambda_{1}})(\mathbf{n}_{ei},\mathbf{e}_{\lambda})q(\omega_{1},\omega)}{\hbar}\exp[-i(\mathbf{k}_{1}+\mathbf{k},\mathbf{r}_{n})$ $\displaystyle\times$ $\displaystyle\int\limits_{0}^{t}d\tau\exp[-i\omega_{k}\tau]D_{n}^{-}(t-\tau)a_{k_{1}}^{\dagger}(t-\tau);\ \ \ [a_{ks}^{\dagger}(t)=[a_{s}(t)]^{+}.$ Taking in to account that $a_{k}(0)\|0\rangle_{ph}=\langle 0\|_{ph}a_{k}^{\dagger}(0)=0$ we can partially eliminate the EMF field operators from the mean value of Hesenberg equation for arbitrary atomic operator $O(t)$ $\displaystyle\frac{d}{dt}\langle O(t)\rangle=-i\sum\limits_{k}\sum\limits_{j=1}^{N_{a}}\frac{(\mathbf{d}_{a},\mathbf{g}_{k})}{\hbar}\langle[R_{j}^{+}(t),O(t)]a_{ks}(t)\rangle\exp[i(\mathbf{k},\mathbf{r}_{j})]$ $\displaystyle-i\sum\limits_{k}\sum\limits_{l=1}^{N_{b}}\frac{(\mathbf{d}_{b},\mathbf{g}_{k})}{\hbar}\langle[S_{l}^{+}(t),O(t)]a_{ks}(t)\rangle\exp[i(\mathbf{k},\mathbf{r}_{l})]$ $\displaystyle-i\sum\limits_{k_{1},k}\sum\limits_{m=1}^{N_{b}}\frac{(\mathbf{n}_{eg},\mathbf{e}_{\lambda_{1}}(k_{1}))(\mathbf{n}_{ei},\mathbf{e}_{\lambda}(k))q(\omega_{k_{1}},\omega_{k})}{\hbar}$ $\displaystyle\times\langle[D_{m}^{+}(t),O(t)]a_{k_{1}}(t)a_{ks}(t)\rangle\exp[i(\mathbf{k}_{1}+\mathbf{k},\mathbf{r}_{m})]+H.C.(O^{+}\rightarrow O).$ (3) Here the mean values of Hesenberg operators are considered taking into account the initial state of the system $\|\Psi_{r}(0)\rangle\otimes\|0\rangle_{ph},$. where $\|\Psi_{r}(0)\rangle$ is the state of radiator subsystem, and $\|0\rangle_{ph}$ is the vacuum state of EMF. We are interested in the total elimination of operators of electromagnetic field from the expression (3) For elimination of operators of electromagnetic field we formulate the lemma ###### Lemma 1 If Bose $a_{k}(t)$ and $a_{k}^{+}(t)$ operators lie between the two operators of the atomic subsystem $A(t_{1})$ and $B(t_{2})$ ($A(t_{1})$ , $B(t_{2})$ don’t contain the operators $a_{k}$ and $a_{k}$ ) belonging to other times, the elimination of the free part of these operators yields the following expression for the correlation: $\displaystyle\left\langle A(t_{1})a_{k}(t)B(t_{2})\right\rangle$ $\displaystyle=$ $\displaystyle\left\langle A(t_{1})a_{ks}(t)B(t_{2})\right\rangle$ $\displaystyle-$ $\displaystyle e^{-i\omega_{3}(t-t_{2})}\left\langle A(t_{1})[a_{ks}(t_{2}),B(t_{2})]\right\rangle,$ $\displaystyle\left\langle A(t_{1})a_{k}^{+}(t)B(t_{2})\right\rangle$ $\displaystyle=$ $\displaystyle\left\langle A(t_{1})a_{ks}^{+}(t)(t)B(t_{2})\right\rangle$ (4) $\displaystyle-$ $\displaystyle e^{i\omega_{3}(t-t_{1})}\left\langle[A(t_{1}),a_{ks}^{+}(t_{1})]B(t_{2})\right\rangle.$ Proof. The commutations in (4) play the highest role in the two-photon spontaneous emission and only such commutations bring the main contribution to the two-photon process. The problem is reduced to the elimination of vacuum part lies between the operators $A(t_{1})$ and $B(t_{2})$ $<A(t_{1})a_{k}(t)B(t_{2})>=<A(t_{1})(a_{k}^{v}(t)+a_{ks}(t))B(t_{2})>$ (5) Since $a_{k}(t)=a_{k}^{v}(t)+a_{ks}(t),\;$we will represent the vacuum part $a_{k}^{v}(t)=a_{k}(0)\exp[-i\omega_{k}t]$ through the vacuum-operator at time $t_{1\;\ }$and taking in to account the identity (2) we can represent the vacuum part in the following form $a_{k}^{v}(t)=a_{k}^{v}(t_{2})e^{-i\omega_{k}(t-t_{2})}=\\{a_{k}(t_{2})-a_{ks}(t_{2})\\}e^{-i\omega_{k}(t-t_{2})}.$ After substitution of $\ a_{k}^{v}(t)\;$into the correlation it is obtain $\displaystyle\langle A(t_{1})a_{k}(t)B(t_{2})\rangle$ $\displaystyle=$ $\displaystyle\langle A(t_{1})a_{ks}(t)B(t_{2})\rangle$ $\displaystyle+$ $\displaystyle e^{-i\omega_{k}(t-t_{2})}\langle A(t_{1})\\{a_{k}(t_{2})-a_{ks}(t_{2})\\}B(t_{2})\rangle,$ We observe that $a_{k}(t_{2})\;$commutes with the operator $B(t_{2})$. Consequently taking into account that $a_{k}(t_{2})\ B(t_{2})\|0>=\ B(t_{2})a_{ks}(t_{2})\|0>,\;$it is easily obtain that $\left\langle A(t_{1})\\{a_{k}(t_{2})-a_{ks}(t_{2})\\}\ B(t_{2})\right\rangle=-\left\langle B(t_{2})[a_{ks}(t_{2}),B(t_{2})]\right\rangle$. This relation proofs the Lemma. This lemma (4) can be used in the last term of generalized equation (3) for correlation functions $\ \langle[D_{j}^{+}(t),O(t)]a_{k}(t)S_{n}^{-}(t-\tau)\rangle$ , $\ \langle[D_{j}^{+}(t),O(t)]a_{k}(t)R_{l}^{-}(t-\tau)\rangle\ $ and $\langle[R_{j}^{+}(t),O(t)]D_{n}^{-}(t-\tau)a_{k_{1}}^{\dagger}(t-\tau)\rangle,$ $\langle[S_{l}^{+}(t),O(t)]D_{n}^{-}(t-\tau)a_{k_{1}}^{\dagger}(t-\tau)\rangle$. Indeed taking in to account the lemma (4) the above correlation functions can be represented through atomic operators $\displaystyle\langle[D_{j}^{+}(t),O(t)]a_{k}(t)S_{n}^{-}(t-\tau)\rangle$ $\displaystyle=$ $\displaystyle\left\langle[D_{j}^{+}(t),O(t)]a_{ks}(t)S_{n}^{-}(t-\tau)\right\rangle$ $\displaystyle-$ $\displaystyle e^{-i\omega_{k}\tau}\left\langle[D_{j}^{+}(t),O(t)][a_{ks}(t-\tau),S_{n}^{-}(t-\tau)]\right\rangle,$ (6) $\displaystyle\langle[R_{j}^{+}(t),O(t)]a_{k}^{\dagger}(t-\tau)D_{n}^{-}(t-\tau)\rangle$ $\displaystyle=$ $\displaystyle\langle[R_{j}^{+}(t),O(t)]a_{ks}^{\dagger}(t-\tau)D_{n}^{-}(t-\tau)\rangle$ $\displaystyle-$ $\displaystyle e^{-i\omega_{k}\tau}\langle[[R_{j}^{+}(t),O(t)],a_{ks}^{\dagger}(t)]D_{n}^{-}(t-\tau)\rangle.$ (7) The interaction between the atomic subsystems can be found in the third order of interaction constants with the subsystems $S$, $R$ and $D$ respectively $(\mathbf{d}_{r},\mathbf{g}_{k})(\mathbf{d}_{s},\mathbf{g}_{k})q(\omega_{1},\omega_{2})$ According with this condition the smooth correlation functions is obtained only for the following terms of expressions (6) and (7) : $\left\langle R_{l}^{-}(t-\tau^{\prime})[D_{j}^{+}(t),O(t)]S_{n}^{-}(t-\tau)\right\rangle$ and $\langle R_{l}^{+}(t-\tau^{\prime})[R_{j}^{+}(t),O(t)]D_{n}^{-}(t-\tau)\rangle$. The contribution of other terms of the expressions (6) and (7) give the contribution more hair order on the decomposition on the small parameter $\lambda$ of the interaction Hamiltonian (1). The lemma (4) is non-applicable for correlation functions in which it is meet simultaneously the creation and annihilations Boson operators belonging to different time intervals: $\langle[D_{j}^{+}(t),O(t)]a_{k}(t)D_{n}^{-}(t-\tau)a_{k_{1}}^{\dagger}(t-\tau)\rangle$ and its hermit conjugate part $\langle a_{k_{1}}(t-\tau)D_{n}^{+}(t-\tau)a_{k}^{\dagger}(t)[O(t),D_{j}^{-}(t)]\rangle$. In order to eliminate the vacuum part of operators $a_{k}(t)$ and $a_{k}^{\dagger}(t^{\prime})$ let us formulate the following rule. ###### Lemma 2 If the operators $A(t_{1})$ contains the creation operators of EMF and $B(t_{2})$ contains the annihilation operators of EMF the elimination of vacuum part of annihilation $a_{k}(t)$ or creation $a_{k}^{{\dagger}}(t)$ operators situated between these operators $A(t_{1})$ and $B(t_{2})$ takes place according with Lemma 1. In opposite case, when operator $A(t_{1})$ can be represented through the product of atomic operator $\mathcal{A}(t_{1})$ and annihilation field operators $A(t_{1})$=$\mathcal{A}(t_{1})a_{k_{1}}(t_{1})a_{k_{2}}(t_{1})...a_{k_{n}}(t_{1})$ . the operator $B(t_{2})$ is represented through the product of creation field operators and atomic operator $\mathcal{B}(t_{2})$ so that B(t2)=$\mathcal{B}(t_{2})a_{k_{1}}^{{\dagger}}(t_{2})a_{k_{2}}^{{\dagger}}(t_{2})...a_{k_{m}}^{{\dagger}}(t_{2})$ the elimination of vacuum part of the operators $a_{k}(t)$ and $a_{k}^{{\dagger}}(t)$ can be represented in the following form $\displaystyle\left\langle A(t_{1})a_{k}(t)B(t_{2})\right\rangle=\left\langle A(t_{1})a_{ks}(t)B(t_{2})\right\rangle$ $\displaystyle-\exp[-i\omega_{k}(t-t_{2})]\\{\left\langle A(t_{1})[a_{ks}(t_{2}),B(t_{2})]\right\rangle$ $\displaystyle-\delta_{k,k_{1}}\left\langle A(t_{1})\mathcal{B}(t_{2})a_{k_{2}}^{{\dagger}}(t_{2})...a_{k_{m}}^{{\dagger}}(t_{2})\right\rangle$ $\displaystyle-...-\delta_{k,k_{m}}\left\langle A(t_{1})\mathcal{B}(t_{2})a_{k_{1}}^{{\dagger}}(t_{2})a_{k_{2}}^{{\dagger}}(t_{2})...a_{k_{m-1}}^{{\dagger}}(t_{2})\right\rangle\\},$ (8) $\displaystyle\left\langle A(t_{1})a_{k}^{{\dagger}}(t)B(t_{2})\right\rangle=\left\langle A(t_{1})a_{ks}^{{\dagger}}(t)B(t_{2})\right\rangle$ $\displaystyle-\exp[i\omega_{k}(t-t_{2})]\\{\left\langle[A(t_{1})a_{ks}^{{\dagger}}(t_{1})]B(t_{2})\right\rangle$ $\displaystyle-\delta_{k,k_{1}}\left\langle\mathcal{A}(t_{1})a_{k_{2}}(t_{1})...a_{k_{n}}(t_{1})B(t_{2})\right\rangle$ $\displaystyle-...-\delta_{k,k_{n}}\left\langle\mathcal{A}(t_{1})a_{k_{1}}(t_{1})a_{k_{2}}(t_{1})...a_{k_{n-1}}(t_{1})B(t_{2})\right\rangle\\}.$ (9) Proof. Taking in to account the lemma (4), we can represent the third correlation $\left\langle A(t_{1})a_{k}(t)B(t_{2})\right\rangle$ of expression (8) in the following form $\displaystyle\left\langle A(t_{1})a_{k}(t)B(t_{2})\right\rangle$ $\displaystyle=$ $\displaystyle\left\langle A(t_{1})a_{ks}(t)B(t_{2})\right\rangle$ (10) $\displaystyle+$ $\displaystyle\exp[-i\omega_{k}(t-t_{2})]\left\langle A(t_{1})(a_{k}(t_{2})-a_{ks}(t_{2}))B(t_{2})]\right\rangle$ $\displaystyle-$ $\displaystyle\exp[-i\omega_{k}(t-t_{2})]\\{\left\langle A(t_{1})[a_{ks}(t_{2}),B(t_{2})]\right\rangle.$ According with explicit expression of operator, $B(t_{2})=\mathcal{B}(t_{2})a_{k_{1}}^{{\dagger}}(t_{2})a_{k_{2}}^{{\dagger}}(t_{2})...a_{k_{m}}^{{\dagger}}(t_{2})$, let us introduced it in the third term of right hand site of expression (10). Following the commutation roles of boson operators of electromagnetic field, the operator $a_{k}(t_{2})$ can be permuted in the right hand site of the correlation. Taking in to consideration that $(a_{ks}(t_{2})+a_{kv}(t_{2}))\left\|0\right\rangle=a_{ks}(t_{2})\left\|0\right\rangle,$ this term becomes $\displaystyle\left\langle A(t_{1})a_{k}(t_{2})\mathcal{B}(t_{2})a_{k_{1}}^{{\dagger}}(t_{2})a_{k_{2}}^{{\dagger}}(t_{2})...a_{k_{m}}^{{\dagger}}(t_{2})\right\rangle=$ $\displaystyle\delta_{k,k_{1}}\left\langle A(t_{1})\mathcal{B}(t_{2})a_{k_{2}}^{{\dagger}}(t_{2})...a_{k_{m}}^{{\dagger}}(t_{2})\right\rangle$ $\displaystyle+...+\delta_{k,k_{m}}\left\langle A(t_{1})\mathcal{B}(t_{2})a_{k_{1}}^{{\dagger}}(t_{2})a_{k_{2}}^{{\dagger}}(t_{2})...a_{k_{m-1}}^{{\dagger}}(t_{2})\right\rangle$ $\displaystyle+\left\langle A(t_{1})\mathcal{B}(t_{2})a_{k_{1}}^{{\dagger}}(t_{2})a_{k_{2}}^{{\dagger}}(t_{2})...a_{k_{m}}^{{\dagger}}(t_{2})a_{ks}(t_{2})\right\rangle.$ (11) Introducing this relation in (11) it is not difficult to observe that the new expression for correlation $\left\langle A(t_{1})a_{k}(t)B(t_{2})\right\rangle$ coincides with (8). The similar procedure of permutation of vacuum part of creation operator $a_{k}^{{\dagger}}(t)$ demonstrates the identity (9) of Lemma 2. According with lemma (8) we obtain the following expression for correlation function $\displaystyle\langle[D_{j}^{+}(t),O(t)]a_{k}(t)D_{n}^{-}(t-\tau)a_{k_{1}}^{\dagger}(t-\tau)\rangle=\langle[D_{j}^{+}(t),O(t)]a_{ks}(t)D_{n}^{-}(t-\tau)a_{k_{1}}^{\dagger}(t-\tau)\rangle$ $\displaystyle-\exp[-i\omega_{k}\tau]\\{\langle[D_{j}^{+}(t),O(t)][a_{ks}(t-\tau),D_{n}^{-}(t-\tau)a_{k_{1}}^{\dagger}(t-\tau)]\rangle-\delta_{k,k_{1}}\langle[D_{j}^{+}(t),O(t)]D_{n}^{-}(t-\tau)\rangle\\}.$ (12) The next step of elimination of operator $a_{k_{1}}^{\dagger}(t-\tau)$ from this expression must be do taking in-to account the lemma (4). When the first and second order interaction constants have the same small magnitude $\lambda\sim(\mathbf{d}_{s},\mathbf{g}_{k})\eqsim q(\omega_{k_{1}},\omega_{k})$, in Born approximation we take in to account only the last term of expression (12). the interference contribution of which is proportional to $\lambda^{3}$. As follows from the representation (2) and (3) the procedure of elimination mast continue. Indeed introducing again this equation in the right hand cite of equation (14) we obtain the following master equation for arbitrary operator $O(t)$ in thread approximation on the interaction constant $\lambda$ $\displaystyle\frac{d\langle O(t)\rangle}{dt}$ $\displaystyle=$ $\displaystyle\sum\limits_{k}\sum\limits_{l,j=1}^{N_{r}}\frac{(\mathbf{d}_{a},\mathbf{g}_{k})^{2}}{\hbar^{2}}\int\limits_{0}^{t}d\tau\exp[-i\omega_{k}\tau+i(\mathbf{k,r}_{j}-\mathbf{r}_{l})]\langle[R_{j}^{+}(t),O(t)]R_{l}^{-}(t-\tau)\rangle$ (13) $\displaystyle+$ $\displaystyle\sum\limits_{k}\sum\limits_{l,l=1}^{N_{s}}\frac{(\mathbf{d}_{b},\mathbf{g}_{k})^{2}}{\hbar^{2}}\int\limits_{0}^{t}d\tau\exp[-i\omega_{k}\tau+i(\mathbf{k,r}_{j}-\mathbf{r}_{l})]\langle[S_{j}^{+}(t),O(t)]S_{l}^{-}(t-\tau)\rangle$ $\displaystyle+$ $\displaystyle\sum\limits_{k_{1},k_{2}}\sum\limits_{l,j=1}^{N}\frac{(\mathbf{n}_{eg},\mathbf{e}_{\lambda_{1}})^{2}(\mathbf{n}_{ei},\mathbf{e}_{\lambda_{2}})^{2}q^{2}(\omega_{1},\omega_{2})}{\hbar^{2}}\int\limits_{0}^{t}d\tau\langle[D_{j}^{+}(t),O(t)]D_{l}^{-}(t-\tau)\rangle$ $\displaystyle\times$ $\displaystyle\exp[-i(2\omega_{0}-\omega_{k_{1}}-\omega_{k_{2}})\tau]\exp[i(\mathbf{k}_{1}+\mathbf{k}_{2},\mathbf{r}_{j}-\mathbf{r}_{l})]$ $\displaystyle+$ $\displaystyle i\sum\limits_{k,k^{\prime}}\sum\limits_{n=1}^{N}\sum\limits_{j=1}\sum\limits_{l=1}\frac{(\mathbf{n}_{eg},\mathbf{e}_{\lambda}(\mathbf{k}))(\mathbf{n}_{ei},\mathbf{e}_{\lambda}(\mathbf{k}^{\prime}))q(\omega_{k},\omega_{k^{\prime}})}{\hbar^{3}}\int\limits_{0}^{t}d\tau\int\limits_{0}^{t}d\tau^{\prime}\exp[i(\mathbf{k,r}_{n}-\mathbf{r}_{l})-i\omega_{k}\tau]$ $\displaystyle\times$ $\displaystyle\exp[i(\mathbf{k}^{\prime}\mathbf{,r}_{n}-\mathbf{r}_{l})-i\omega_{k^{\prime}}\tau^{\prime}][(\mathbf{d}_{s},\mathbf{g}_{k^{\prime}})(\mathbf{d}_{r},\mathbf{g}_{k})\langle[D_{n}^{+}(t),O(t)]R_{j}^{-}(t-\tau)S_{l}^{-}(t-\tau^{\prime})\rangle$ $\displaystyle+$ $\displaystyle(\mathbf{d}_{r},\mathbf{g}_{k^{\prime}})(\mathbf{d}_{s},\mathbf{g}_{k})\langle[D_{n}^{+}(t),O(t)]S_{j}^{-}(t-\tau)R_{l}^{-}(t-\tau^{\prime})\rangle]$ $\displaystyle-$ $\displaystyle i\sum\limits_{k,k^{\prime}}\sum\limits_{m=1}^{N}\sum\limits_{j=1}\sum\limits_{l=1}\frac{(\mathbf{n}_{eg},\mathbf{e}_{\lambda}(\mathbf{k}))(\mathbf{n}_{ei},\mathbf{e}_{\lambda}(\mathbf{k}^{\prime}))q(\omega_{k},\omega_{k^{\prime}})}{\hbar^{3}}\int\limits_{0}^{t}d\tau^{\prime}\int\limits_{0}^{t}d\tau\exp[i\omega_{k}\tau-i(\mathbf{k,r}_{m}-\mathbf{r}_{l})]$ $\displaystyle\times$ $\displaystyle\exp[-i\omega_{k}\tau^{\prime}-i\omega_{k^{\prime}}\tau^{\prime}+i(\mathbf{k}^{\prime}\mathbf{,r}_{j}-\mathbf{r}_{m})][(\mathbf{d}_{s},\mathbf{g}_{k})(\mathbf{d}_{r},\mathbf{g}_{k^{\prime}})\langle S_{l}^{+}(t-\tau)[R_{j}^{+}(t),O(t)]D_{m}^{-}(t-\tau^{\prime})\rangle$ $\displaystyle+$ $\displaystyle(\mathbf{d}_{s},\mathbf{g}_{k})(\mathbf{d}_{r},\mathbf{g}_{k^{\prime}})\langle R_{l}^{+}(t-\tau)[S_{j}^{+}(t),O(t)]D_{m}^{-}(t-\tau^{\prime})\rangle]+H.c.(O^{+}\rightarrow O).$ The traditional Born-Marcov approximation in the right hand site of equation ( 13) give us the divergent functions. In order to understood this we approximate the right hand site of equation (2) with following expression $\displaystyle a_{ks}(t)$ $\displaystyle=$ $\displaystyle\frac{(\mathbf{d}_{a},\mathbf{g}_{k})}{\hbar}\sum\limits_{l=1}^{N_{a}}R_{l}^{-}(t)\exp[-i(\mathbf{k},\mathbf{r}_{l})\zeta^{\ast}(\omega_{k}-\omega_{a})$ (14) $\displaystyle+\frac{(\mathbf{d}_{b},\mathbf{g}_{k})}{\hbar}\sum\limits_{j=1}^{N_{b}}S_{j}^{-}(t)\exp[-i(\mathbf{k},\mathbf{r}_{j})\zeta^{\ast}(\omega_{k}-\omega_{b})$ $\displaystyle+2\sum\limits_{n=1}^{N_{b}}\sum\limits_{k_{1}}\frac{(\mathbf{n}_{eg},\mathbf{e}_{\lambda_{1}})(\mathbf{n}_{ei},\mathbf{e}_{\lambda})q(\omega_{1},\omega)}{\hbar}$ $\displaystyle\times D_{n}^{-}(t)a_{k_{1}}^{\dagger}(t)\exp[-i(\mathbf{k}_{1}+\mathbf{k},\mathbf{r}_{n})\zeta^{\ast}(\omega_{k}+\omega_{k_{1}}-2\omega_{0}),$ in the Born-Marcovian approximation [15] , [13]. The small parameter in this approximation is the ratio of retardation time to cooperative spontaneous emission times of the subsystem, $\tau/\tau_{i}<<1$. Here $i\zeta(x)=iP/x+\pi\delta(x)$ is the Heitler function [13],[14]. represents k-summation in analogy with Cauchy principal value [15]. Introducing the operators (14) in equation (3) and eliminating the boson operators of EMF, it is obtain the following equation for operator $O(t)$ in Born-Marcov approximation $\displaystyle\frac{d}{dt}\langle O(t)\rangle$ $\displaystyle=$ $\displaystyle\sum\limits_{k}\sum\limits_{l,j=1}^{N_{r}}\frac{(\mathbf{d}_{a},\mathbf{g}_{k})^{2}}{\hbar^{2}}\langle[R_{j}^{+}(t),O(t)]R_{l}^{-}(t)\rangle$ $\displaystyle\times$ $\displaystyle\exp[i(\mathbf{k,r}_{j}-\mathbf{r}_{l})]i\zeta^{\ast}(\omega_{r}-\omega_{k})+\sum\limits_{k}\sum\limits_{l,j=1}^{N_{s}}\frac{(\mathbf{d}_{a},\mathbf{g}_{k})(\mathbf{d}_{b},\mathbf{g}_{k})}{\hbar^{2}}$ $\displaystyle\times$ $\displaystyle\langle[S_{j}^{+}(t),O(t)]S_{l}^{-}(t)\rangle\exp[i(\mathbf{k,r}_{j}-\mathbf{r}_{l})]i\zeta^{\ast}(\omega_{s}-\omega_{k})$ $\displaystyle+$ $\displaystyle\sum\limits_{k,k^{\prime}}\sum\limits_{l,j=1}^{N}\frac{(\mathbf{n}_{eg},\mathbf{e}_{\lambda})^{2}(\mathbf{n}_{ei},\mathbf{e}_{\lambda^{\prime}})^{2}q^{2}(\omega_{k},\omega_{k^{\prime}})}{\hbar^{2}}$ $\displaystyle\times$ $\displaystyle\langle[D_{j}^{+}(t),O(t)]D_{l}^{-}(t)\rangle\exp[i(\mathbf{k-k}^{\prime}\mathbf{,r}_{j}-\mathbf{r}_{l})]i\zeta^{\ast}(\omega_{a}-\omega_{k}-\omega_{k^{\prime}})$ $\displaystyle+$ $\displaystyle 2i\sum\limits_{k,k^{\prime}}\sum\limits_{n=1}^{N}\sum\limits_{l=1}^{N_{s}}\sum\limits_{l=1}^{N_{r}}\frac{(\mathbf{d}_{a},\mathbf{g}_{k^{\prime}})(\mathbf{d}_{b},\mathbf{g}_{k})(\mathbf{n}_{eg},\mathbf{e}_{\lambda})(\mathbf{n}_{ei},\mathbf{e}_{\lambda^{\prime}})q(\omega_{k},\omega_{k^{\prime}})}{\hbar^{3}}$ $\displaystyle\times$ $\displaystyle\langle[D_{n}^{+}(t),O(t)]R_{j}^{-}(t)S_{l}^{-}(t)\rangle$ $\displaystyle\times$ $\displaystyle\exp[i(\mathbf{k,r}_{n}-\mathbf{r}_{l})+i(\mathbf{k}^{\prime}\mathbf{,r}_{n}-\mathbf{r}_{l})]i\zeta^{\ast}(\omega_{r}-\omega_{k})i\zeta^{\ast}(\omega_{s}-\omega_{k^{\prime}})$ $\displaystyle-$ $\displaystyle i\sum\limits_{k,k^{\prime}}\sum\limits_{m=1}^{N}\sum\limits_{l=1}\sum\limits_{j=1}\frac{(\mathbf{d}_{a},\mathbf{g}_{k_{1}})(\mathbf{d}_{b},\mathbf{g}_{k_{2}})(\mathbf{n}_{eg},\mathbf{e}_{\lambda_{1}})(\mathbf{n}_{ei},\mathbf{e}_{\lambda_{2}})q(\omega_{1},\omega_{2})}{\hbar^{3}}$ $\displaystyle\times$ $\displaystyle[\langle S_{l}^{+}(t)[R_{j}^{+}(t),O(t)]D_{m}^{-}(t)\rangle i\zeta^{\ast}(\omega_{r}-\omega_{k})i\zeta(\omega_{s}-\omega_{k^{\prime}})$ $\displaystyle+$ $\displaystyle\langle R_{l}^{+}(t)[S_{j}^{+}(t),O(t)]D_{m}^{-}(t)\rangle i\zeta^{\ast}(\omega_{s}-\omega_{k})i\zeta(\omega_{r}-\omega_{k^{\prime}})]$ $\displaystyle\times$ $\displaystyle\exp[i(\mathbf{k,r}_{j})+i(\mathbf{k}^{\prime}\mathbf{,r}_{l})-i(\mathbf{k+k}^{\prime},\mathbf{r}_{m})]+H.C.(O^{+}\rightarrow O).$ (15) In the right hand part of the equation (15) the third order terms contain the resonances between the single photon radiators $A$ ,$B$ and two-photon radiator $D$ described by the correlation functions $\langle S_{l}^{+}(t)[R_{j}^{+}(t),O(t)]D_{m}^{-}(t)\rangle$, $\langle[D_{n}^{+}(t),O(t)]R_{j}^{-}(t)S_{l}^{-}(t)\rangle$ and $\langle R_{l}^{+}(t)[S_{j}^{+}(t),O(t)]D_{m}^{-}(t)\rangle$. As it is observed from equation (15), these terms contain the product of the functions $[P/(\omega_{b}-\omega_{k})][P/(\omega_{b}-\omega_{k^{\prime}})]$ which describe the principal value in the integration procedure on the variables $k$ and $k^{\prime}$. It is not difficult to observe that these integrals become divergent expressions In order to avoid these divergence in Appendix1 it is proposed the integration procedure which takes in to account the retardation between the radiators in the representation of right hand site of equation (13). According with Appendix1 the right hand site of master equation (13) takes the following non-divergent form $\displaystyle\frac{d}{dt}\langle O(t)\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{2\tau_{r}}\sum\limits_{l,j=1}^{N_{r}}\chi_{r}(j,l)\langle[R_{j}^{+}(t),O(t)]R_{l}^{-}(t)\rangle+\frac{1}{2\tau_{s}}\sum\limits_{l,l=0}^{N_{s}}\chi_{s}(j,l)\langle[S_{j}^{+}(t),O(t)]S_{l}^{-}(t)\rangle$ (16) $\displaystyle+$ $\displaystyle\frac{1}{2\tau_{b}}\sum\limits_{l,j=1}^{N}\chi_{d}(j,l)\langle[D_{j}^{+}(t),O(t)]D_{l}^{-}(t)\rangle$ $\displaystyle+$ $\displaystyle\frac{i}{2\tau_{bsr}}\sum\limits_{m=1}^{N}\sum\limits_{l=1}^{N_{r}}\sum\limits_{l=0}^{N_{s}}U(j,l,m)\langle[D_{m}^{+}(t),O(t)]R_{j}^{-}(t)S_{l}^{-}(t)\rangle$ $\displaystyle-$ $\displaystyle\frac{i}{4\tau_{sbr}}\sum\limits_{m=1}^{N}\sum\limits_{j=1}^{N_{r}}\sum\limits_{l=0}^{N_{s}}V(j,l.m)[\langle S_{l}^{+}(t)[R_{j}^{+}(t),O(t)]D_{m}^{-}(t)\rangle+\langle R_{j}^{+}(t)[S_{l}^{+}(t),O(t)]D_{m}^{-}(t)\rangle]$ $\displaystyle+$ $\displaystyle H.C.(O^{+}\rightarrow O).$ Here the spontaneous emission $\tau_{i}$ and exchange integral between radiators $j$ and $l\ $, $\chi_{i}(j,l)$ are defined in expressions (28), (32), (34) and (37) of Appendix. This equation can be used for description of interaction between the dipole forbidden and dipole active systems of radiators. ## 3 Kinetic equations for correlation functions In order to found the correlation process between dipole forbidden transitions of $D$ subsystem and dipole active transitions in $S$ and $R$ subsystems of radiators let us found the equations for arbitrary atomic correlation functions of subsystems of radiators. According with the generalized equation (16) it is obtain the following chain of equation for atomic correlations $\displaystyle\frac{d}{dt}\langle R_{zj}(t)\rangle$ $\displaystyle=$ $\displaystyle-\frac{1}{2\tau_{r}}\sum\limits_{l,=1}^{N_{a}}[\chi_{r}(j,l)\langle R_{j}^{+}(t)R_{l}^{-}(t)\rangle+\chi_{r}^{\ast}(j,l)\langle R_{l}^{+}(t)R_{j}^{-}(t)\rangle]$ (17) $\displaystyle+\frac{i}{4\tau_{sbr}}\sum\limits_{m=1}^{N}\sum\limits_{l=0}^{N_{s}}[V(j,l.m)\langle S_{l}^{+}(t)R_{j}^{+}(t)D_{m}^{-}(t)\rangle-V^{\ast}(j,l.m)\langle D_{m}^{+}(t)R_{j}^{-}(t)S_{l}^{-}(t)\rangle],$ $\displaystyle\frac{d}{dt}\langle S_{zl}(t)\rangle$ $\displaystyle=$ $\displaystyle-\frac{1}{2\tau_{s}}\sum\limits_{p=0}^{N_{s}}\chi_{s}(l,p)[\langle S_{l}^{+}(t)S_{p}^{-}(t)\rangle+\chi_{s}^{\ast}(l,p)\langle S_{p}^{+}(t)S_{l}^{-}(t)\rangle]$ (18) $\displaystyle+$ $\displaystyle\frac{i}{4\tau_{sbr}}\sum\limits_{m=1}^{N}\sum\limits_{l=0}^{N_{s}}[V(j,l.m)\langle S_{l}^{+}(t)R_{j}^{+}(t)D_{m}^{-}(t)\rangle-V^{\ast}(j,l.m)\langle D_{m}^{+}(t)R_{j}^{-}(t)S_{l}^{-}(t)\rangle],$ $\displaystyle\frac{d}{dt}\langle D_{zn}(t)\rangle$ $\displaystyle=$ $\displaystyle-\frac{1}{2\tau_{b}}\sum\limits_{l=1}^{N}[I_{{}^{b}}(j,l)\langle D_{n}^{+}(t)D_{l}^{-}(t)\rangle+I_{{}^{b}}^{\ast}(j,l)\langle D_{l}^{+}(t)D_{n}^{-}(t)\rangle$ (19) $\displaystyle-$ $\displaystyle\frac{i}{2\tau_{srb}}\sum\limits_{j=1}^{N_{r}}\sum\limits_{l=0}^{N_{s}}[U(j,l,n)\langle D_{n}^{+}(t)R_{j}^{-}(t)S_{l}^{-}(t)\rangle-U^{\ast}(j,l,n)\langle S_{l}^{+}(t)R_{j}^{+}(t)D_{n}^{-}(t)\rangle,$ $\displaystyle\frac{d}{dt}\langle R_{j}^{+}(t)R_{l}^{-}(t)\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\tau_{r}}\sum\limits_{n=1}^{N_{r}}[\chi_{r}(l,n)\langle R_{j}^{+}(t)R_{zl}(t)R_{n}^{-}(t)\rangle+\chi_{r}(n,j)\langle R_{n}^{+}(t)R_{zj}(t)R_{l}^{-}(t)\rangle]$ (20) $\displaystyle-$ $\displaystyle\frac{1}{2\tau_{sbr}}\sum\limits_{m=1}^{N}\sum\limits_{k=0}^{N_{s}}[V(j,k,m)\langle S_{k}^{+}(t)R_{j}^{+}(t)R_{zl}(t)D_{m}^{-}(t)\rangle$ $\displaystyle+$ $\displaystyle V^{\ast}(j,k.m)\langle D_{m}^{+}(t)R_{zl}(t)R_{j}^{-}(t)S_{k}^{-}(t)\rangle];$ $\displaystyle\frac{d}{dt}\langle S_{j}^{+}(t)S_{l}^{-}(t)\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\tau_{r}}\sum\limits_{n=1}^{N_{s}}[\chi_{r}(l,n)\langle S_{j}^{+}(t)S_{zl}(t)S_{n}^{-}(t)\rangle+\chi_{r}(n,j)\langle S_{n}^{+}(t)S_{zj}(t)S_{l}^{-}(t)\rangle]$ (21) $\displaystyle-$ $\displaystyle\frac{i}{2\tau_{sbr}}\sum\limits_{m=1}^{N}\sum\limits_{k=1}^{N_{r}}[V(k,l.m)\langle R_{k}^{+}(t)S_{j}^{+}(t)S_{zl}(t)D_{m}^{-}(t)\rangle$ $\displaystyle-$ $\displaystyle V^{\ast}(k,j.m)\langle D_{m}^{+}(t)S_{zj}(t)S_{l}^{-}(t)R_{k}^{-}(t)\rangle];$ $\displaystyle\frac{d}{dt}\langle D_{n}^{+}(t)D_{l}^{-}(t)\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\tau_{b}}\sum\limits_{m=1}^{N}[I_{{}^{b}}(l,m)\langle D_{n}^{+}(t)D_{zl}(t)D_{m}^{-}(t)\rangle+I_{{}^{b}}^{\ast}(n,m)\langle D_{m}^{+}(t)D_{zn}(t)D_{l}^{-}(t)\rangle]$ (22) $\displaystyle+$ $\displaystyle\frac{i}{\tau_{bsr}}\sum\limits_{j=1}^{N_{r}}\sum\limits_{k=0}^{N_{s}}[U(j,k,m)\langle D_{n}^{+}(t)D_{zl}(t)R_{j}^{-}(t)S_{k}^{-}(t)\rangle$ $\displaystyle+$ $\displaystyle U^{\ast}(j,k,m)\langle S_{k}^{+}(t)R_{j}^{+}(t)D_{zn}(t)D_{l}^{+}(t)\rangle].$ $\displaystyle\frac{d}{dt}i\langle D_{m}^{+}(t)S_{l}^{-}(t)R_{k}^{-}(t)\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\tau_{bsr}}\sum\limits_{j=1}^{N_{r}}\sum\limits_{n=1}^{N_{s}}[U^{\ast}(j,n,m)\langle S_{n}^{+}(t)R_{j}^{+}(t)D_{zm}(t)S_{l}^{-}(t)R_{k}^{-}(t)\rangle$ (23) $\displaystyle+$ $\displaystyle\frac{1}{2\tau_{sbr}}\sum\limits_{n=1}^{N}\sum\limits_{j=1}[V(j,l.n)\langle S_{zl}(t)R_{j}^{+}(t)R_{k}^{-}(t)D_{m}^{+}(t)D_{n}^{-}(t)\rangle$ $\displaystyle+$ $\displaystyle V(k,j.n)\langle D_{m}^{+}(t)D_{n}^{-}(t)R_{zk}(t)S_{j}^{+}(t)S_{l}^{-}(t)\rangle]$ $\displaystyle+$ $\displaystyle i\sum\limits_{j=1}\bigl{[}\frac{1}{\tau_{s}}\chi_{r}(j,l)\langle D_{m}^{+}(t)S_{zl}(t)S_{j}^{-}(t)R_{k}^{-}(t)\rangle$ $\displaystyle+\frac{1}{\tau_{r}}\chi_{r}(j,k)\langle D_{m}^{+}(t)R_{zk}(t)S_{l}^{-}(t)R_{j}^{-}(t)\rangle$ $\displaystyle+\frac{1}{\tau_{b}}I_{{}^{b}}^{\ast}(m,l)\langle D_{j}^{+}(t)D_{zm}(t)S_{j}^{-}(t)R_{k}^{-}(t)\rangle\bigr{]}$ Let us consider the interaction three different atoms in interaction through vacuum EMF. Introducing the exited numbers for the atomic subsystems $\langle N_{\alpha}\rangle=\langle J_{z\alpha}(t)\rangle+0.5$ (here $J\leftrightarrow S,\ R,\ D$ $\ \alpha=s,\ r,\ d$ ) and correlation function between the atoms $\langle F\rangle=i[\langle D^{+}(t)S^{-}(t)R^{-}(t)\rangle-\langle S^{+}(t)R^{+}(t)D^{-}(t)\rangle]$ we can obtain the closed system of equations from the chain of equations (17-23). Indeed considering that the distance between the radiators is smaller then radiation wavelength: $\Re$ $\\{U(j,k,m)\\}=\Re\\{V(j,k,m)\\}=1$ , we obtain the following closed system of equation $\displaystyle\frac{d}{dt}\langle N_{s}(t)\rangle$ $\displaystyle=$ $\displaystyle-\frac{\langle N_{s}\rangle}{\tau_{s}}-\frac{1}{2\tau_{sbr}}\langle F\rangle;$ $\displaystyle\frac{d}{dt}\langle N_{r}(t)\rangle$ $\displaystyle=$ $\displaystyle-\frac{\langle N_{r}\rangle}{\tau_{r}}-\frac{1}{2\tau_{sbr}}\langle F\rangle;$ $\displaystyle\frac{d}{dt}\langle N_{d}(t)\rangle$ $\displaystyle=$ $\displaystyle-\frac{\langle N_{d}\rangle}{\tau_{d}}-\frac{1}{\tau_{sbr}}\langle F\rangle;$ $\displaystyle\frac{d}{dt}\langle F(t)\rangle$ $\displaystyle=$ $\displaystyle-\bigl{[}\frac{1}{2\tau_{s}}+\frac{1}{2\tau_{r}}+\frac{1}{2\tau_{d}}\bigr{]}\langle F(t)\rangle$ $\displaystyle+$ $\displaystyle\frac{1}{\tau_{bsr}}[6\langle N_{s}N_{r}N_{d}\rangle-2\langle N_{s}N_{r}\rangle-\langle N_{s}N_{d}\rangle-\langle N_{r}N_{d}\rangle],$ $\displaystyle\frac{d}{dt}\langle N_{s}N_{r}N_{d}\rangle$ $\displaystyle=$ $\displaystyle-[\frac{1}{\tau_{r}}+\frac{1}{\tau_{b}}+\frac{1}{\tau_{s}}]\langle N_{s}N_{r}N_{d}\rangle,$ $\displaystyle\frac{d}{dt}\langle N_{s}N_{r}\rangle$ $\displaystyle=$ $\displaystyle-[\frac{1}{\tau_{r}}+\frac{1}{\tau_{s}}]\langle N_{s}N_{r}\rangle,$ $\displaystyle\frac{d}{dt}\langle N_{s}N_{d}\rangle$ $\displaystyle=$ $\displaystyle-[\frac{1}{\tau_{r}}+\frac{1}{\tau_{b}}]\langle N_{s}N_{b}\rangle,$ $\displaystyle\frac{d}{dt}\langle N_{s}N_{d}\rangle$ $\displaystyle=$ $\displaystyle-[\frac{1}{\tau_{r}}+\frac{1}{\tau_{b}}]\langle N_{s}N_{b}\rangle,$ in which the new correlation functions between the atomic excitation is introduced $\langle\hat{N}_{s}\hat{N}_{r}\hat{N}_{d}\rangle$, $\langle\hat{N}_{s}\hat{N}_{r}\rangle$, $\langle\hat{N}_{s}\hat{N}_{d}\rangle$, and $\langle\hat{N}_{r}\hat{N}_{d}\rangle$. This system of equation is exactly solvable. The solution is $\displaystyle\langle N_{s}N_{r}N_{d}\rangle$ $\displaystyle=$ $\displaystyle\exp(-At);\ \ \ \langle N_{s}N_{r}\rangle=\exp(-Bt),\langle N_{s}N_{d}\rangle=\exp(-Ct);\ \ \ \langle N_{d}N_{r}\rangle=\exp(-Dt)$ $\displaystyle\langle N_{i}N_{j}\rangle$ $\displaystyle=$ $\displaystyle\exp\bigl{[}-\bigl{(}\frac{1}{\tau_{i}}+\frac{1}{\tau_{j}}\bigr{)}t\bigr{]},\ \ \ i,j\rightarrow s,\ r,\ b;$ $\displaystyle\langle F(t)\rangle$ $\displaystyle=$ $\displaystyle\frac{\exp(-At/2)}{\tau_{bsr}}\bigl{[}6\frac{1-\exp[-At/2]}{A}-4\frac{1-\exp[-(B-\tau_{b}^{-1})t/2]}{(B-\tau_{b}^{-1})}$ $\displaystyle-\frac{2(1-\exp[-(C-\tau_{r}^{-1})t/2])}{(C-\tau_{r}^{-1})}-\frac{2(1-\exp[-(C-\tau_{s}^{-1})t/2])}{(C-\tau_{s}^{-1})}\bigr{]};$ $\displaystyle\langle N_{i}\rangle$ $\displaystyle=$ $\displaystyle\exp[-t/\tau_{i}]-\frac{1}{\tau_{bsr}}\int\limits_{0}^{t}\exp[-(t-t^{\prime})/\tau_{i}]\langle F(t^{\prime})\rangle,\ \ \ \ \ i\equiv s,\ r,\ d,$ where the collective rates are defined $A=1/\tau_{r}+1/\tau_{b}+1/\tau_{s}$; $B=1/\tau_{r}+1/\tau_{s}$; $C=1/\tau_{b}+1/\tau_{s}$; $D=1/\tau_{b}+1/\tau_{s}$. The solution of this system of equation is plotted in figure 2. It is observed the influence of single photon transition on the two quanta transitions. This influence drastically depends on the cooperative rate $1/\tau_{bsr}$. Figure 2: The decay law of inversion, $D_{z}(t)$, of dipole forbidden radiator stimulated by single photon decay processes of two radiators for following parameters of the system: relative decay rates of S- and R- atoms $\tau_{b}/\tau_{s}=\tau_{b}/\tau_{r}=5$. As follows from figure 3 the cooperative exchanges between the radiator accelerate the two-photon decay processes so that the the derivation $-\frac{dD_{z}}{dt}$ achieved the maximal value decreasing after that till zero value. Figure 3: The two-photon decay rate $dD_{z}(t)/dt$ stimulated by single photon processes for same parameter of the system as in figure 2. Neglecting the quantum fluctuations of inversion operators $\langle R_{z_{j}}\rangle$, $\langle S_{zl}\rangle,$ $\langle D_{zn}\rangle$, dipole- dipole correlations between the same radiators $\langle R_{l}^{+}(t)R_{j}^{-}(t)\rangle$, $\langle S_{l}^{+}(t)S_{p}^{-}(t)\rangle$ and $\langle D_{n}^{+}(t)D_{l}^{-}(t)\rangle$ and between subsystems $\langle S_{l}^{+}(t)R_{j}^{+}(t)D_{m}^{-}(t)\rangle$, $\langle D_{m}^{+}(t)R_{j}^{-}(t)S_{l}^{-}(t)\rangle$ we can de-correlated the chain of equations (17-23) in order to obtain the closed system of equations $\displaystyle for\ \ \ J\leftrightarrow S,\ R,\ D;\ \langle J_{j}^{+}(t)J_{zl}(t)J_{m}^{-}(t)\rangle$ $\displaystyle=$ $\displaystyle\langle J_{zl}(t)\rangle\langle J_{j}^{+}(t)J_{m}^{-}(t)\rangle\ \ \ j\neq l\neq m;$ $\displaystyle\langle S_{k}^{+}(t)R_{j}^{+}(t)R_{zl}(t)D_{m}^{-}(t)\rangle$ $\displaystyle=$ $\displaystyle\langle R_{zl}(t)\rangle\langle S_{k}^{+}(t)R_{j}^{+}(t)D_{m}^{-}(t)\rangle_{l\neq j}-\delta_{l,j}\langle S_{k}^{+}(t)R_{j}^{+}(t)D_{m}^{-}(t)\rangle;$ $\displaystyle\langle D_{n}^{+}(t)D_{zl}(t)R_{j}^{-}(t)S_{k}^{-}(t)\rangle$ $\displaystyle=$ $\displaystyle\langle D_{zl}(t)\rangle\langle D_{n}^{+}(t)R_{j}^{-}(t)S_{k}^{-}(t)\rangle_{l\neq n}-\delta_{l,n}\langle D_{n}^{+}(t)R_{j}^{-}(t)S_{k}^{-}(t)\rangle;$ $\displaystyle\langle S_{l}^{+}(t)R_{j}^{+}(t)D_{zn}(t)R_{p}^{-}(t)S_{k}^{-}(t)\rangle$ $\displaystyle=$ $\displaystyle\langle D_{zn}(t)\rangle\langle R_{j}^{+}(t)R_{p}^{-}(t)\rangle\langle S_{l}^{+}(t)S_{k}^{-}(t)\rangle_{l\neq k;j\neq p}$ $\displaystyle+\delta_{j,p}\delta_{l,k}\langle D_{zn}(t)(R_{zj}(t)+0.5)(S_{zl}(t)+0.5)\rangle;$ $\displaystyle\langle D_{m}^{+}(t)D_{n}^{-}(t)S_{zl}(t)R_{j}^{+}(t)R_{k}^{-}(t)\rangle$ $\displaystyle=$ $\displaystyle\langle S_{zl}(t)\rangle\langle D_{m}^{+}(t)D_{n}^{-}(t)\rangle\langle R_{j}^{+}(t)R_{k}^{-}(t)\rangle_{m\neq n;j\neq k}$ $\displaystyle+\delta_{m,n}\delta_{j,k}\langle(D_{zn}(t)+0.5)S_{zl}(t)(R_{zj}(t)+0.5)\rangle;$ $\displaystyle\langle D_{m}^{+}(t)D_{n}^{-}(t)R_{zj}(t)S_{l}^{+}(t)S_{k}^{-}(t)\rangle$ $\displaystyle=$ $\displaystyle\langle R_{zj}(t)\rangle\langle D_{m}^{+}(t)D_{n}^{-}(t)\rangle\langle S_{l}^{+}(t)S_{k}^{-}(t)\rangle_{m\neq n;l\neq k}$ (24) $\displaystyle\delta_{m,n}\delta_{l,k}\langle(D_{zn}(t)+0.5)R_{zj}(t)(S_{zl}(t)+0.5)\rangle.$ Taking in to account the de-correlation (24), we obtain the following closed system of equations $\displaystyle\frac{d}{dt}R_{z}(t)$ $\displaystyle=$ $\displaystyle-\frac{1}{\tau_{r}}\\{N_{r}(N_{r}+2)/4-R_{z}^{2}+R_{z}\\}-\frac{1}{2\tau_{sbr}}F,$ $\displaystyle\frac{d}{dt}S_{z}(t)$ $\displaystyle=$ $\displaystyle-\frac{1}{\tau_{r}}\\{N_{s}(N_{s}+2)/4-S_{z}^{2}+S_{z}\\}-\frac{1}{2\tau_{sbr}}F,$ $\displaystyle\frac{d}{dt}D_{z}(t)$ $\displaystyle=$ $\displaystyle-\frac{1}{\tau_{r}}\\{N(N+2)/4-D_{z}^{2}+D_{z}\\}-\frac{1}{\tau_{sbr}}F;$ $\displaystyle\frac{d}{dt}F$ $\displaystyle=$ $\displaystyle\bigl{[}\frac{1}{\tau_{s}}[S_{z}(t)-1]+\frac{1}{\tau_{r}}[R_{z}(t)-1]+\frac{1}{\tau_{d}}[D_{z}(t)-1]\bigr{]}F$ $\displaystyle+$ $\displaystyle\frac{1}{\tau_{bsr}}\bigl{[}2D_{z}\\{N_{s}^{2}/4-S_{z}^{2}\\}\\{N_{r}^{2}/4-R_{z}^{2}\\}$ $\displaystyle+$ $\displaystyle R_{z}\\{N_{s}^{2}/4-S_{z}^{2}\\}\\{N^{2}/4-D_{z}^{2}\\}+S_{z}\\{N_{r}^{2}/4-R_{z}^{2}\\}\\{N^{2}/4-D_{z}^{2}\\}$ $\displaystyle+$ $\displaystyle(4N_{s}N_{r}N\exp[-A\ast t]-N_{s}\ast N_{r}\ast\exp[-Bt]-0,5N_{s}N\exp[-Ct]$ $\displaystyle-$ $\displaystyle 0.5N_{r}N\exp[-Dt]\bigr{]}.$ From this system of equations follows the oscillatory behavior of the decay rate of the inversion $D_{z}$. Taking in to account the following relative expressions of the decay rates we obtain the numerical simulation of the inversion $D_{z}$ and its derivative (see figure 4 and figure 5). As follows from this system of equation the increasing of decay rate of two-photon spontaneous emission is possible under the influence of single photon cooperative emission of two atomic subsystems. In figures 4 is plotted the time dependence of the inversion $<D_{z}(t)>$ of dipole forbidden radiators as function of the relative coupled parameter between the radiators $\tau_{b}/\tau_{srb}$. Figure 4: The time dependence of inversion $D_{z}(t)$ for dipole forbidden radiator subsystem for following values of the parameter of the system:nuber of atoms in the subsystems $S,R$ and $D$ are, $N_{s}=N_{r}=N=50$ respectively; the relative decay times of the subsystems are $\tau_{b}/\tau_{s}=\tau_{b}/\tau_{r}=6$ ; the coupling parameter of these three system is changed between $0$ and $1$. The oscillatory behavior of decay rate is observed. Figure 5: The cooperative decay rate of inversion $\frac{dD_{z}(t)}{dt}$ of dipole forbidden radiator subsystem for same values of the parameters of the system as in figure 4. The increasing of decay rate of bi-photons is observed. The same dependence is represented in figure 5 for the intensity of two-photon emission proportional to $-d<D_{z}(t)>/dt$. As follows from these plots it is observed the mutual influences between single and two-photon super-radiance processes of three particle interaction. This effect plays an important role in the collective decay process of the systems of radiators with the dimension smaller than wavelength. ## 4 Conclusion In this paper the effective interaction between three radiator subsystems in two-photon resonance is found using the method of elimination of operators of vacuum field. The new cooperative interaction between dipole-forbidden atomic subsystem and two-dipole active subsystems of radiators was proposed. The master equation 16, which describes the energy dissipation from the system due to mutual interaction between the radiators through the vacuum of electromagnetic field, was obtained. Using the chain of equation17-23, which describes the cooperative interaction between three radiator subsystems, it is obtained the closed system of equations for three radiators. Neglecting the quantum fluctuation of the inversion, the de-correlation method of the this chain of equation is proposed 24 in order to describe numerically the behavior of mutual influences of single and two-photon super-radiance processes. As a consequence of effective interaction between three radiators through two- photon resonance processes of inverted systems increase substantially in process cooperative decay of the system The three particle exchange integral has been established and the influence of this effect on the behavior cooperative decay of the atomic subsystems was estimated (see figure 4 and figure 5). Similar experimental situation can bi realized in exited atomic (for example transitions in Cs atoms [17]) or nuclei (for example ${}^{193m}Ir$, ${}^{195m}Pt$ and ${}^{103m}Rh$ nuclei[16]) subsystems in resonance interaction through vacuum field. ## 5 Appendix: Exchange integrals In order to estimate all exchange integrals in equation (13) let us firstly found the well known exchange integral between two radiators in single photon interaction with vacuum of electromagnetic field. In the first terms of equation (13) the retardation can be found integration firstly on the $k$ vector respectively $\displaystyle V_{jl}^{i}$ $\displaystyle=$ $\displaystyle\frac{d_{\alpha}^{2}}{(2\pi)^{2}\hbar c^{3}}\sum\limits_{l,j=1}^{N_{a}}\int\limits_{0}^{\infty}\omega_{k}^{3}d\omega_{k}\int d\Omega_{k}\int\limits_{0}^{t}d\tau\exp[i(\omega_{i}-\omega_{k})\tau]$ $\displaystyle\times(1-(\mathbf{e}_{k},n_{d}))\langle[\tilde{J}_{j}^{+}(t),O(t)]\ \tilde{J}_{l}^{-}(t-\tau)\rangle\exp[i\omega_{k}r_{jl}\cos\theta],$ $\displaystyle i$ $\displaystyle\equiv$ $\displaystyle a,\ \ \ ,b.$ (25) Here the frequency $\omega_{i}$ corresponds to $\mathit{A}$ and $\mathit{B}$ atomic systems; $i=r,s$; for $i=r$ operators $\tilde{J}_{j}^{+}$,$\ \ \tilde{J}_{l}^{-}$ corresponds to $\tilde{R}_{j}^{+}$ , $\tilde{R}_{l}^{-}$ and for $i=s$ these operators corresponds to $\tilde{S}_{j}^{+}$, $\tilde{S}_{l}^{-}$ Passing to new variable $\upsilon=\omega_{k}-\omega_{i}$ and considering that the smooth function $\omega_{k}$ under integral can be approximation with $\omega_{i}^{3}$, we obtain the following approximate expression of thirst order exchange integrals $\displaystyle\omega_{i}^{3}\int\limits_{-\omega_{k}}^{\infty}dv\exp[iv(\tau- r_{jl}\cos\theta/c)]$ $\displaystyle\eqsim$ $\displaystyle\omega_{i}^{3}\int\limits_{-\infty}^{\infty}dv\exp[iv(\tau- r_{jl}\cos\theta/c)]$ (26) $\displaystyle=$ $\displaystyle 2\pi\omega_{i}^{3}\delta(\tau-r_{jl}\cos\theta/c).$ $\omega_{i}$ is the emission frequency relatively the dipole active transitions of the $R$ and $S$ atomic subsystems. In this approximation I obtain the following integral on angle $\theta$ and retardation $\tau$ $\displaystyle V_{jl}^{i}$ $\displaystyle=$ $\displaystyle\frac{\omega_{i}^{3}d_{i}^{2}}{2\hbar c^{3}}\int\limits_{0}^{\pi}d\theta\int\limits_{0}^{t}d\tau\sin\theta\delta(\tau- r_{jl}\cos\theta/c)D_{i}(\theta)\langle[\tilde{J}_{j}^{+}(t),O(t)]\ \tilde{J}_{l}^{-}(t)\rangle$ (27) $\displaystyle=$ $\displaystyle\frac{\omega_{i}^{3}d_{i}^{2}}{2\hbar c^{3}}\int\limits_{0}^{\pi}d\theta\sin\theta\Theta(\cos\theta)D_{jl}\bigl{[}\frac{\partial}{\partial\omega_{i}}\bigr{]}\exp[i\omega_{i}r_{jl}\cos\theta]\langle[\tilde{J}_{j}^{+}(t),O(t)]\ \tilde{J}_{l}^{-}(t)\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{2\tau_{i}}\chi(j,l)\langle[\tilde{J}_{j}^{+}(t),O(t)]\ \tilde{J}_{l}^{-}(t)\rangle.$ Here $\tau_{i}$ and $\chi(j,l)$ are the spontaneous emission time and exchange integral between the single photon radiators respectively [13] $\tau_{i}=\frac{3\hbar c^{3}}{4d_{i}\omega_{i}^{3}},\ \ \ \chi(j,l)=D_{jl}\bigl{[}\frac{\partial}{\partial\omega_{i}}\bigr{]}\frac{3}{4}\frac{\exp[i\omega_{i}r_{jl}/c]-1}{i\omega_{i}r_{jl}/c},$ (28) the expressions in equation (27) and (28) are defined below $\displaystyle D_{i}(\theta)$ $\displaystyle=$ $\displaystyle 1+\cos^{2}\xi_{i}-\cos^{2}\theta(3\cos^{2}\xi_{i}-1),$ $\displaystyle D_{jl}\bigl{[}\frac{\partial}{\partial\omega_{i}}\bigr{]}$ $\displaystyle=$ $\displaystyle\bigl{[}1+\cos^{2}(\xi_{i})+(3\cos^{2}\xi_{i}-1)\frac{c^{2}}{r_{jl}^{2}}\frac{\partial^{2}}{\partial\omega_{i}^{2}}]^{2}\bigr{]}$ (29) where $\cos\xi_{i}$ is the scalar product between the unitary vectors along the direction of dipole momentum of the $j$ (or $l$) atom $\mathbf{n}_{d_{i}}=\mathbf{d}_{i}/d_{i}$ and the direction of the distance between the $j$ and $l$ atoms $\mathbf{n}_{jl}=\mathbf{r}_{jl}/r_{jl}$. 2. The two-photon exchange integral between dipole forbidden transition of the radiators of $D$ subsystem is described by third term in the right hand site of equation (13) $\displaystyle V_{jl}^{b}$ $\displaystyle=$ $\displaystyle\frac{V^{2}}{(2\pi)^{6}}\int\limits_{0}^{2\pi}d\varphi_{1}\int\limits_{0}^{\pi}d\theta_{1}\sin\theta_{1}\int\limits_{0}^{\infty}k_{1}^{2}dk_{1}\int\limits_{0}^{2\pi}d\varphi_{2}\int\limits_{0}^{\pi}d\theta_{2}\sin\theta_{2}\int\limits_{0}^{\infty}k_{2}^{2}dk_{2}\frac{(\mathbf{n}_{eg},\mathbf{e}_{\lambda_{1}})^{2}(\mathbf{n}_{ei},\mathbf{e}_{\lambda_{2}})^{2}q^{2}(\omega_{1},\omega_{2})}{\hbar^{2}}$ (30) $\displaystyle\times$ $\displaystyle\int\limits_{0}^{t}d\tau\exp[-i(2\omega_{0}-\omega_{1}-\omega_{2})\tau+i(\mathbf{k}_{1}+\mathbf{k}_{2},\mathbf{r}_{j}-\mathbf{r}_{l})]\left\langle[\tilde{D}_{j}(t),O(t)]\tilde{D}_{l}(t-\tau)\right\rangle.$ The exchange integral in Born-Marcov approximation for two-photon emission was obtained in paper [9]. Here we will estimate the exchange integral of expression (30) integration firstly on the wave vectors $\mathbf{k}_{1}$ and $\mathbf{k}_{2}.$ Indeed considering that the amplitude $q^{2}(\omega_{1},\omega_{2})$ is the smooth function of the variables $k_{1}$ and $k_{2}$ in comparison with rapid oscillation functions $\exp[i(\mathbf{k}_{1}+\mathbf{k}_{2},\mathbf{r}_{j}-\mathbf{r}_{l})]$ and $\exp[-i(2\omega_{0}-\omega_{k_{1}}-\omega_{k_{2}})\tau]$, we can approximate the amplitude $q^{2}(\omega_{1},\omega_{2})$ according with the resonance frequencies of the atomic system $q^{2}(\omega_{1},\omega_{2})\eqsim q^{2}(\omega_{0},\omega_{0})$. It is not difficult to observe that the maximal value of the smooth function under the integral is obtained for frequencies $\omega_{k_{2}}=\omega_{k_{1}}\eqsim\omega_{0}$. In order to integrate this expression on the variables $k_{1}$ and $k_{2}$ we change the variables $x_{i}=\omega_{k_{i}}-\omega_{i}$ as in expression (26), and consider that $\tilde{D}_{j}^{+}(t)=D_{j}^{+}(t)\exp[-2i\omega_{0}t]$ is smooth operator. In these approximations it is obtain the following expression for $V_{jl}^{b}$ $\displaystyle V_{jl}^{b}$ $\displaystyle=$ $\displaystyle\frac{V^{2}q^{2}(\omega_{0},\omega_{0})}{(2)^{4}\pi^{2}}\int\limits_{0}^{t}d\tau\int\limits_{0}^{\pi}d\theta_{1}\sin\theta_{1}\int\limits_{0}^{\pi}d\theta_{2}\sin\theta_{2}D_{0}(\theta_{1})D_{0}(\theta_{2})\exp[i\omega_{0}r_{jl}(\cos\theta_{1}+\cos\theta_{2})/c]$ (31) $\displaystyle\delta(\tau-r_{jl}\cos\theta_{1}/c)\delta(\tau- r_{jl}\cos\theta_{2}/c)\left\langle[\tilde{D}_{j}(t),O(t)]\tilde{D}_{l}(t-\tau)\right\rangle$ $\displaystyle\eqsim$ $\displaystyle\frac{V^{2}q^{2}(\omega_{0},\omega_{0})}{(2)^{4}\pi^{2}}D_{jl}^{2}\left(\frac{\partial}{\partial\omega_{0}}\right)\frac{\exp[2i\omega_{0}r_{jl}]-1}{2i\omega_{0}r_{jl}}\left\langle[\tilde{D}_{j}(t),O(t)]\tilde{D}_{l}(t-\tau)\right\rangle,$ where expressions $D_{0}(\theta_{1})$ and $D_{jl}(\partial/(\partial\omega_{0})$ are defined by the expressions (29). Integrating the right hand site of the equation (31) on the solid angle and retardation, we obtain the following approximative expression $V_{jl}^{b}=\frac{1}{2\tau_{b}}\chi_{b}(j,l)\left\langle[\tilde{D}_{j}^{+}(t),O(t)]\tilde{D}_{l}^{-}(t)\right\rangle,$ where $\displaystyle\frac{1}{2\tau_{b}}$ $\displaystyle=$ $\displaystyle\frac{2^{2}}{3^{2}}\frac{\omega_{0}^{7}d_{23}^{2}d_{31}^{2}}{2^{2}\pi\hbar^{2}c^{6}}\\{3/2\\}\left\\{\frac{1}{\omega_{32}+\omega_{0}}+\frac{1}{\omega_{31}-\omega_{0}}\right\\}^{2},$ $\displaystyle\chi_{b}(j,l)$ $\displaystyle=$ $\displaystyle\frac{3^{2}}{4}\frac{\pi c}{4\omega_{0}r_{jl}}D^{2}\bigl{[}\frac{\partial}{\partial\omega_{0}}\bigr{]}\frac{\exp[2i\omega_{0}r_{jl}/c]-1}{i\omega_{0}r_{jl}/c}$ (32) This exchange integral diverges, when the distance between the radiators $r_{jl}$ is less than the wavelength $\lambda_{0}=2\pi c/\omega_{0}$. In order to take in to account the value of the exchange integral for the small parameter, $r_{jl}/\lambda_{0}<<1,$ let us integrate the expression (30) taking in to account the method proposed in papers [9] and [12]. In this case we obtain the following expression for $ReV_{jl}^{b}$ $\displaystyle F(j,l)$ $\displaystyle=$ $\displaystyle ReV_{jl}^{b}=\frac{d_{31}^{2}d_{32}^{2}}{4\pi\hbar^{2}c^{6}}\int\limits_{0}^{2\omega_{0}}d\omega_{k}\omega_{k}^{3}(\omega_{21}-\omega_{k})^{3}$ $\displaystyle\times$ $\displaystyle\chi_{jl}(\omega_{k})\chi_{jl}(\omega_{21}-\omega_{k})\left\\{\frac{1}{\omega_{31}-\omega_{k_{1}}}+\frac{1}{\omega_{32}+\omega_{k_{1}}}\right\\}^{2},$ where $\chi_{jl}(\omega)=(1-\cos^{2}{\xi})\frac{\sin{\frac{\omega r_{jl}}{c}}}{\frac{\omega r_{jl}}{c}}+(1-3\cos^{2}{\xi})\left\\{\frac{\cos{\frac{\omega r_{jl}}{c}}}{(\frac{\omega r_{jl}}{c})^{2}}-\frac{\sin{\frac{\omega r_{jl}}{c}}}{(\frac{\omega r_{jl}}{c})^{3}}\right\\}.$ 3\. In the right hand part of this equation (13) the third order terms contains the resonances between dipole active radiators $\mathit{A}$ ,$\mathit{B}$ and dipole forbidden radiators $\mathit{D}$, described by the correlation functions $\langle S_{l}^{+}(t)[R_{j}^{+}(t),O(t)]D_{m}^{-}(t)\rangle$, $\langle[D_{n}^{+}(t),O(t)]R_{j}^{-}(t)S_{l}^{-}(t)\rangle$ and $\langle R_{l}^{+}(t)[S_{j}^{+}(t),O(t)]D_{m}^{-}(t)\rangle$. Let us introduced the similar approximation in the chronological interaction between the atomic subsystems $\mathit{A}$, $\mathit{B}$, and $\mathit{D}$. The exchange integral between three atoms is represented in the similar form as in the expression (30) $\displaystyle V_{jln;as-d}^{c}$ $\displaystyle=$ $\displaystyle i\frac{V}{(2\pi)^{3}}\frac{V}{(2\pi)^{3}}\frac{2\pi\hbar d_{s}d_{r}}{Vc^{6}\hbar^{3}}\frac{(2\pi)^{3}}{V}(2\pi)^{2}\left(\frac{1}{2}\right)^{2}\int\limits_{0}^{\infty}\omega_{1}^{2}d\omega_{1}\int\limits_{0}^{\infty}\omega_{2}^{2}d\omega_{2}\sqrt{\omega_{1}\omega_{2}}\chi(\omega_{1},\omega_{2})$ (33) $\displaystyle\times$ $\displaystyle\int\limits_{-1}^{1}dx_{1}\int\limits_{-1}^{1}dx_{2}\int\limits_{0}^{t}d\tau_{1}\int\limits_{0}^{t}d\tau_{2}\exp[-i(\omega_{1}-\omega_{r})\tau_{1}-i(\omega_{2}-\omega_{s})\tau_{2}]$ $\displaystyle\times$ $\displaystyle D_{nl}\bigl{[}\frac{\partial}{\partial\omega_{s}}\bigr{]}D_{nj}\bigl{[}\frac{\partial}{\partial\omega_{r}}\bigr{]}\exp[i\omega_{1}r_{nj}x_{1}/c+i\omega_{2}r_{nl}x_{2}/c]$ $\displaystyle\langle[\tilde{D}_{n}^{+}(t),O(t)]\tilde{R}_{j}^{-}(t-\tau_{1})\tilde{S}_{l}^{-}(t-\tau_{2})\rangle;$ After the substitution of variables $\omega_{1}-\omega_{r}=\tilde{\omega}_{1}$ and $\omega_{2}-\omega_{s}=\tilde{\omega}_{2}$, we can approximate the smooth amplitude $\omega_{1}^{2}\omega_{2}^{2}\sqrt{\omega_{1}\omega_{2}}\chi(\omega_{1},\omega_{2})$ with expression $(\omega_{r})^{2}\omega_{s}^{2}\sqrt{\omega_{r}\omega_{s}}\chi(\omega_{r},\omega_{s})$. Integrals on the new variable $\tilde{\omega}_{1}$ and $\tilde{\omega}_{1}$ give the following aspect of expression (33) $\displaystyle V_{jln;as-d}^{c}$ $\displaystyle=$ $\displaystyle 2\pi i\left(\frac{1}{2}\right)^{2}\frac{d_{s}d_{r}}{c^{6}\hbar^{2}}\omega_{s}^{2}(\omega_{r})^{2}\sqrt{\omega_{r}\omega_{s}}\chi(\omega_{s},\omega_{r})\int\limits_{0}^{1}dx_{1}\int\limits_{0}^{1}dx_{2}$ $\displaystyle\times$ $\displaystyle D_{nl}\bigl{[}\frac{\partial}{\partial\omega_{s}}\bigr{]}D_{nj}\bigl{[}\frac{\partial}{\partial\omega_{r}}\bigr{]}\exp[i\omega_{r}r_{nj}x_{1}/c+i\omega_{s}r_{nl}x_{2}/c]$ $\displaystyle\times$ $\displaystyle\langle[\tilde{D}_{n}^{+}(t),O(t)]\tilde{R}_{j}^{-}(t-r_{nj}x_{1}/c)\tilde{S}_{l}^{-}(t-r_{nl}x_{2}/c)\rangle$ from which follows that $x_{1},$and $x_{2}>0$ . In the Born approximation the expression for $V_{jln;as-d}^{c}$ $V_{rs-d}(m,j,l)=\frac{i}{4\tau_{bsr}}U(j,l,m)\langle[\tilde{D}_{n}^{+}(t),O(t)]\tilde{R}_{j}^{-}(t)\tilde{S}_{l}^{-}(t)\rangle$ where $\displaystyle\frac{1}{\tau_{bsr}}$ $\displaystyle=$ $\displaystyle\left(\frac{2}{3}\right)^{2}\frac{d_{s}d_{r}d_{23}d_{31}\omega_{s}^{3}(\omega_{r})^{3}}{4\pi c^{6}\hbar^{2}}\left\\{\frac{1}{\omega_{32}+\omega_{s}}+\frac{1}{\omega_{31}-\omega_{r}}\right\\},$ $\displaystyle U(j,l,m)$ $\displaystyle=$ $\displaystyle-\left(\frac{3}{2}\right)^{2}D_{nl}\bigl{[}\frac{\partial}{\partial\omega_{s}}\bigr{]}D_{nj}\bigl{[}\frac{\partial}{\partial\omega_{r}}\bigr{]}\frac{c^{2}[\exp i\omega_{r}r_{nj}/c]-1][\exp[i\omega_{s}r_{nl}/c]-1]}{\omega_{r}\omega_{s}r_{nj}r_{nl}}.$ (34) 4. Let now found the retardation in the last correlation function term of equation (13).Taking in to account the retardation in the rapid oscillation part of atomic operators $J^{\pm}(t-\tau)=\tilde{J}^{\pm}(t-\tau)\exp[\pm i\omega(t-\tau)]$ where $J^{\pm}$, $\omega$ and $\tau$ are the atomic operators, transition frequencies and delay time for atomic subsystems $\mathit{A}$, $\mathit{B}$ and $\mathit{D}$ respectively, $\displaystyle V_{;ab-d}(t)$ $\displaystyle=$ $\displaystyle i\sum\limits_{k_{1}k_{2}}\sum\limits_{m=1}^{N}\sum\limits_{l=1}^{N_{a}}\sum\limits_{j=0}^{N_{b}}\frac{(\mathbf{d}_{r},\mathbf{g}_{k_{1}})(\mathbf{d}_{s},\mathbf{g}_{k_{2}})(\mathbf{n}_{eg},\mathbf{e}_{\lambda_{1}})(\mathbf{n}_{ei},\mathbf{e}_{\lambda_{2}})q(\omega_{k_{1}},\omega_{k_{2}})}{\hbar^{3}}$ (35) $\displaystyle\times$ $\displaystyle\int\limits_{0}^{t}d\tau_{1}\exp[i(2\omega_{0}-\omega_{k_{1}}-\omega_{k_{2}})\tau_{1}]\int\limits_{0}^{t}d\tau_{2}\exp[-i(\omega_{s}-\omega_{k_{1}})\tau_{2}]$ $\displaystyle\times$ $\displaystyle\exp[-i(\mathbf{k}_{1},\mathbf{r}_{j}-\mathbf{r}_{m})+i(\mathbf{k}_{2},\mathbf{r}_{l}-\mathbf{r}_{m})]\langle\tilde{S}_{l}^{+}(t-\tau_{2})[\tilde{R}_{j}^{+}(t),O(t)]\tilde{D}_{m}^{-}(t-\tau_{1})\rangle.$ Passing from the summation to integration in expression (35) we obtain following expression for correlation between the $j$, $l$ and $m$ atoms $\displaystyle V_{rs-d}(m,j,l)$ $\displaystyle=$ $\displaystyle i\frac{V}{(2\pi)^{3}}\frac{V}{(2\pi)^{3}}\frac{2\pi\hbar d_{s}d_{r}}{Vc^{6}\hbar^{3}}\frac{(2\pi)^{3}}{V}(2\pi)^{2}\left(\frac{1}{2}\right)^{2}\int\limits_{0}^{\infty}\omega_{1}^{2}d\omega_{1}\int\limits_{0}^{\infty}\omega_{2}^{2}d\omega_{2}$ (36) $\displaystyle\times\sqrt{\omega_{1}\omega_{2}}\chi(\omega_{1},\omega_{2})\int\limits_{-1}^{1}dx_{1}\int\limits_{-1}^{1}dx_{2}\int\limits_{0}^{t}d\tau_{1}\exp[i(2\omega_{0}-\omega_{1}-\omega_{2})\tau_{1}]$ $\displaystyle\times\int\limits_{0}^{t}d\tau_{2}\exp[-i(\omega_{s}-\omega_{2})\tau_{2}]$ $\displaystyle\times D_{jl}\bigl{[}\frac{\partial}{\partial\omega_{2}}\bigr{]}D_{jm}\bigl{[}\frac{\partial}{\partial\omega_{1}}\bigr{]}\exp[-i\omega_{2}r_{ml}x_{2}/c-i\omega_{1}r_{jm}x_{1}/c)]$ $\displaystyle\times\langle\tilde{S}_{l}^{+}(t-\tau_{2})[\tilde{R}_{j}^{+}(t),O(t)]\tilde{D}_{m}^{-}(t-\tau_{1}).$ Introducing the new variables $u_{1}=\omega_{1}-\omega_{r}$; $u_{2}=\omega_{2}-\omega_{s}$ in (36), and approximating the smooth amplitude $\omega_{1}^{2}\omega_{2}^{2}\sqrt{\omega_{1}\omega_{2}}\chi(\omega_{1},\omega_{2})$ with expression $\omega_{s}^{2}\omega_{r}^{2}\sqrt{\omega_{s}\omega_{r}}\chi(\omega_{s},\omega_{r})$ we get using the delta functions (26) $\displaystyle V_{jlm;as-d}(j,l)$ $\displaystyle=$ $\displaystyle 2\pi i\left(\frac{1}{2}\right)^{2}\frac{d_{s}d_{r}}{c^{6}\hbar^{2}}\omega_{s}^{2}\omega_{r}^{2}\sqrt{\omega_{s}\omega_{r}}\chi(\omega_{s},\omega_{r})\int\limits_{-1}^{1}dx_{1}\int\limits_{-1}^{1}dx_{2}\int\limits_{0}^{t}d\tau_{1}\int\limits_{0}^{t}d\tau_{2}$ $\displaystyle\times$ $\displaystyle\delta(\tau_{2}-\tau_{1}-r_{jl}x_{2}/c)\delta(\tau_{1}-r_{jm}x_{1}/c)$ $\displaystyle\times D_{jl}\bigl{[}\frac{\partial}{\partial\omega_{s}}\bigr{]}D_{jm}\bigl{[}\frac{\partial}{\partial\omega_{r}}\bigr{]}\exp[i\omega_{s}r_{ml}x_{2}/c]\exp[-i\omega_{r}r_{jm}x_{1}/c]$ $\displaystyle\times\langle\tilde{S}_{l}^{+}(t-\tau_{2})[\tilde{R}_{j}^{+}(t),O(t)]\tilde{D}_{m}^{-}(t-\tau_{1})\rangle,$ the value of which can be estimated observing from the arguments of $\delta$-functions that $x_{1}>0$ and $\ r_{jm}x_{1}+r_{jl}x_{2}>0$. In this case, neglecting the retardation $\tau_{1}$ and $\tau_{2}$ in the smooth correlation function we obtain $V_{jlm;as-d}(j,l)=\frac{i}{4\tau_{sbr}}V(j,l.m)\langle\tilde{S}_{l}^{+}(t)[\tilde{R}_{j}^{+}(t),O(t)]\tilde{D}_{m}^{-}(t)\rangle,$ where cooperative rate is $\frac{1}{\tau_{sbr}}=\left(\frac{2}{3}\right)^{2}\frac{d_{s}d_{r}d_{23}d_{31}\omega_{s}^{3}\omega_{r}^{3}}{4\pi c^{6}\hbar^{2}}\left\\{\frac{1}{\omega_{32}+\omega_{s}}+\frac{1}{\omega_{31}-\omega_{r}}\right\\}.$ the integral $V(j,l.m)$ on the direction of the emitted photons is $\displaystyle V(j,l.m)$ $\displaystyle=$ $\displaystyle\left(\frac{3}{2}\right)^{2}D_{ml}\bigl{[}\frac{\partial}{\partial\omega_{s}}\bigr{]}D_{jm}\bigl{[}\frac{\partial}{\partial\omega_{r}}\bigr{]}$ $\displaystyle\times$ $\displaystyle\bigl{[}\frac{c^{2}\\{2\exp[i(\omega_{s}r_{ml}-\omega_{r}r_{jm})/c]-2\exp[2i\omega_{s}r_{ml}]-\exp[-2i\omega_{r}r_{jm}]+1]\\}}{2\omega_{s}\omega_{r}r_{lm}r_{ml}}\theta(r_{ml}-r_{mj})$ $\displaystyle+.$ $\displaystyle\bigl{(}\frac{c^{2}(\omega_{s}+\omega_{r})[\exp(-i\omega_{r}r_{mj}/c)-1]\exp(i\omega_{s}r_{ml}/c)-\exp[-i\omega_{s}r_{ml}/c-i\omega_{r}r_{mj}/c]}{(\omega_{s}+\omega_{r})\omega_{s}\omega_{r}r_{mj}r_{ml}}$ $\displaystyle+$ $\displaystyle\frac{c^{2}[\omega_{s}\exp[-i(\omega_{s}+\omega_{r})r_{ml}/c]+\omega_{r}]}{(\omega_{s}+\omega_{r})\omega_{r}\omega_{s}r_{jm}r_{ml}}\bigr{)}\bigr{]}\theta(r_{mj}-r_{jl}).$ (37) If we will consider that $x_{1}>0,$ and $x_{2}>0$ the expression for (37) takes more simple form $\displaystyle V(j,l.m)$ $\displaystyle\eqsim$ $\displaystyle\left(\frac{3}{2}\right)^{2}D_{ml}\bigl{[}\frac{\partial}{\partial\omega_{s}}\bigr{]}D_{jm}\bigl{[}\frac{\partial}{\partial\omega_{r}}\bigr{]}$ $\displaystyle\frac{c^{2}\\{\exp[i\omega_{s}r_{ml}/c]-1][\exp[-i\omega_{r}r_{jm}/c]-1]}{\omega_{s}\omega_{r}r_{lm}r_{ml}}$ The expression for exchange integrals described in points $1.-4.$ are used in the master equation (16) ## References * [1] B. 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1012.4058	# The number of convex pentagons and hexagons in an $n$-triangular net Jun-Ming Zhu Department of Mathematics, East China Normal University, Dongchuan Road 500, Shanghai 200241, China E-mail: junming_zhu@163.com ###### Abstract In this paper, we obtain the counting formulaes of convex pentagons and convex hexagons, respectively, in an $n$-triangular net by solving the corresponding recursive formulaes. Key words: $n$-triangular net, convex pentagon, convex hexagon, regular hexagon AMS 2000: 05A15, 39A10 ## 1 Introduction In the paper [2], the author obtained the counting formulaes of triangles and quadrilaterals in an $n$-triangular net, respectively, by induction. In this paper, we get the counting formulaes of convex pentagons and hexagons in an $n$-triangular net, respectively, by solving two corresponding recursive formulaes. We first give the following definition. Definition 1.1 Divide each edge of a (regular) triangular into $n$ $(n\geq 1)$ equal parts, and then construct $n-1$ segments between the dividing points on two edges parallel to the third one. Then the graph we get is called an $n$-(regular) triangular net. See fig. 1, fig. 2 and fig. 3 in the following. If $n=1$, then the $n$-triangular net reduces to a triangular. By the property of affine transformation, we know that the numbers of convex pentagons and convex hexagons in an $n$-triangular net are only dependent on $n$ but independent from the shape and the size of the triangular. \begin{picture}(120.0,220.0) \par\put(15.0,57.0){\begin{picture}(40.0,30.0) \par \put(0.0,0.0){\line(1,0){30.0}} \put(0.0,0.0){\line(1,2){15.0}} \put(30.0,0.0){\line(-1,2){15.0}} \put(5.0,10.0){\line(1,0){20.0}} \put(10.0,20.0){\line(1,0){10.0}} \put(10.0,0.0){\line(-1,2){5.0}} \put(20.0,0.0){\line(-1,2){10.0}} \put(10.0,0.0){\line(1,2){10.0}} \put(20.0,0.0){\line(1,2){5.0}} \par\put(15.0,-5.0){\makebox(0.0,0.0)[t]{fig. 1: $3$-triangular net $OA_{3}B_{3}$}} \put(15.0,30.0){\makebox(0.0,0.0)[bl]{$O$}} \put(15.0,10.0){\makebox(0.0,0.0)[br]{$O^{\prime}$}} \put(10.0,20.0){\makebox(0.0,0.0)[br]{$A_{1}$}}\put(5.0,10.0){\makebox(0.0,0.0)[br]{$A_{2}$}}\put(0.0,0.0){\makebox(0.0,0.0)[br]{$A_{3}$}} 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\par\put(50.0,85.0){\makebox(0.0,0.0)[tl]{$B_{1}$}}\put(55.0,75.0){\makebox(0.0,0.0)[tl]{$B_{2}$}}\put(60.0,65.0){\makebox(0.0,0.0)[tl]{$B_{3}$}}\put(65.0,55.0){\makebox(0.0,0.0)[tl]{$B_{4}$}}\put(70.0,45.0){\makebox(0.0,0.0)[tl]{$B_{5}$}}\put(75.0,35.0){\makebox(0.0,0.0)[tl]{$B_{6}$}} \put(45.0,70.0){\circle{0.8}} \put(45.0,-5.0){\makebox(0.0,0.0)[t]{fig. 3}} \end{picture}} \end{picture} fig. 1: $3$-triangular net $OA_{3}B_{3}$$O$$O^{\prime}$$A_{1}$$A_{2}$$A_{3}$$B_{1}$$B_{2}$$B_{3}$$P_{1}$$P_{2}$ fig. 2: $6$-triangular net $OA_{6}B_{6}$$O^{\prime}$$O$$A_{1}$$A_{2}$$A_{3}$$A_{4}$$A_{5}$$A_{6}$$B_{1}$$B_{2}$$B_{3}$$B_{4}$$B_{5}$$B_{6}$$P_{1}$$P_{2}$$P_{3}$$P_{4}$$P_{5}$ $O^{\prime}$$O$$A_{n}$$A_{n-1}$$P_{1}$$B_{n}$$B_{n-1}$$P_{n-1}$$A_{1}$$A_{2}$$A_{3}$$A_{4}$$A_{5}$$A_{6}$$B_{1}$$B_{2}$$B_{3}$$B_{4}$$B_{5}$$B_{6}$fig. 3 ## 2 The counting formulae of convex pentagons in an $n$-triangular net ###### Theorem 2.1. The number $P(n)$ of convex pentagons in an $n$-triangular net is $P(n)=\begin{cases}\frac{1}{10}(12k^{5}+25k^{4}+5k^{3}-10k^{2}-2k),\ &n=2k+1\ (k=0,1,2,\cdots),\\\ \frac{1}{10}(12k^{5}-5k^{4}-15k^{3}+5k^{2}+3k),\ &n=2k\ (k=1,2,\cdots).\end{cases}$ (1) ###### Proof. Without loss of generosity, we suppose that the divided triangular is regular. Then there is only one acute interior angle in each convex pentagon. Observing the figures above, we know that $P(n)$ can be expressed as $2P(n-1)$ (all the pentagons in the $(n-1)$-triangular net $A_{1}A_{n}P_{n-1}$ or $B_{1}P_{1}B_{n}$), subtracting $P(n-2)$ (pentagons in the $(n-2)$-triangular net $O^{\prime}P_{1}P_{n-1}$ ), and then adding the number, denoted by $f(n)$, of the pentagons of which there are vertexes both on $OA_{n}$ and on $OB_{n}$. This is $P(n)=2P(n-1)-P(n-2)+f(n).$ (2) Obviously, we have $P(1)=0$, $P(2)=0$, $P(3)=3$ and $f(1)=0$, $f(2)=0$, $f(3)=3$. Now it is crucial to get the expression of $f(n)$. Note that, from the definition of $f(n)$, $f(n)-f(n-1)$ is the number of pentagons of which there are vertexes on $OA_{n}$, $OB_{n}$ and $A_{n}B_{n}$ and of which the unique acute vertex must be on $OA_{n}$, $OB_{n}$ or $A_{n}B_{n}$. The number of all the pentagons of which there are vertexes on $OA_{n}$, $OB_{n}$ and $A_{n}B_{n}$ and of which the acute vertex is one of $O,~{}A_{n}$ and $B_{n}$ is $(1+2+3+\cdots+(n-2))\times 3.$ The number of all the pentagons of which there are vertexes on $OA_{n}$, $OB_{n}$ and $A_{n}B_{n}$ and of which the acute vertex is one of $A_{1},\ A_{2},\ \cdots,\ A_{n-1},B_{1},\ B_{2},\ \cdots,\ B_{n-1}$ and $\ P_{1},$ $P_{2},$ $\cdots,$ $P_{n-1}$ is $1+2+\cdots+(k-1)+(k-1)+\cdots+2+1~{}~{}(\mbox{if}~{}~{}n=2k),$ or $1+2+\cdots+(k-2)+(k-1)+(k-2)+\cdots+2+1~{}~{}(\mbox{if}~{}~{}n=2k+1).$ So we have $\displaystyle f(2k)-f(2k-1)$ $\displaystyle=$ $\displaystyle(1+2+3+\cdots+(2k-2))\times 3$ $\displaystyle+1+2+\cdots+(k-1)+(k-1)+\cdots+2+1$ and $\displaystyle f(2k+1)-f(2k)$ $\displaystyle=$ $\displaystyle(1+2+3+\cdots+(2k-1))\times 3$ $\displaystyle+1+2+\cdots+(k-2)+(k-1)+(k-2)+\cdots+2+1$ $\displaystyle\hskip 213.39566pt.$ This is $\displaystyle f(2k)=f(2k-1)+3(3k^{2}-5k+2),$ $\displaystyle f(2k+1)=f(2k)+3(3k^{2}-2k).$ Iterating the above formulaes gives $\displaystyle f(2)$ $\displaystyle=$ $\displaystyle f(1)+3(3\cdot 1^{2}-5\cdot 1+2),$ $\displaystyle f(3)$ $\displaystyle=$ $\displaystyle f(2)+3(3\cdot 1^{2}-2\cdot 1),$ $\displaystyle f(4)$ $\displaystyle=$ $\displaystyle f(3)+3(3\cdot 2^{2}-5\cdot 2+2),$ $\displaystyle f(5)$ $\displaystyle=$ $\displaystyle f(4)+3(3\cdot 2^{2}-2\cdot 2),$ $\displaystyle\cdots\cdots\cdots\cdots,$ $\displaystyle f(2k)$ $\displaystyle=$ $\displaystyle f(2k-1)+3(3\cdot k^{2}-5\cdot k+2),$ $\displaystyle f(2k+1)$ $\displaystyle=$ $\displaystyle f(2k)+3(3\cdot k^{2}-2\cdot k).$ Overlay the above formulaes and note that $f(1)=0$ to get $\displaystyle f(2k+1)=\frac{3}{2}(4k^{3}-k^{2}-k),$ $\displaystyle f(2k)=\frac{3}{2}(4k^{3}-7k^{2}+3k).$ From (2), we have $P(n)-P(n-1)=P(n-1)-P(n-2)+f(n)$. Then $\displaystyle P(2k)-P(2k-1)=P(2k-1)-P(2k-2)+f(2k),\ (k=1,2,\cdots\cdots),$ $\displaystyle P(2k+1)-P(2k)=P(2k)-P(2k-1)+f(2k+1),\ (k=0,1,2,\cdots\cdots).$ Overlaying again, we have $\displaystyle P(2k+1)-P(2k)$ $\displaystyle=$ $\displaystyle 3k^{4}+2k^{3}-\frac{3}{2}k^{2}-\frac{1}{2}k,$ $\displaystyle P(2k)-P(2k-1)$ $\displaystyle=$ $\displaystyle 3k^{4}-4k^{3}+k.$ Overlaying a third time, we get $\displaystyle P(2k+1)=\frac{6}{5}k^{5}+\frac{5}{2}k^{4}+\frac{1}{2}k^{3}-k^{2}-\frac{1}{5}k,$ $\displaystyle P(2k)=\frac{6}{5}k^{5}-\frac{1}{2}k^{4}-\frac{3}{2}k^{3}+\frac{1}{2}k^{2}+\frac{3}{10}k,$ which completes the proof. ∎ The difference equation (2) is fundamental in this paper. This is a linear recurrence relation of order $2$ and can also be solved using the method usually used in solving recursive formulaes (see, for example, [1, p.218–234, §7.2–7.3]). Our method is different from that of [2]. Obviously, formulaes of this form can also be used to get the counting formulaes of triangles and quadrilaterals in [2] and seem to be more understandable. We will also use the equation of this form to get number of convex hexagons in an $n$-triangular net in the following. ## 3 The counting formulae of convex hexagons in an $n$-triangular net ###### Theorem 3.1. The number $H(n)$ of convex hexagons in an $n$-triangular net is $H(n)=\begin{cases}\frac{1}{60}(8k^{6}+24k^{5}+25k^{4}+10k^{3}-3k^{2}-4k),&n=2k+1(k=0,1,2,\cdots),\\\ \frac{1}{60}(8k^{6}-5k^{4}-3k^{2}),&n=2k(k=1,2,\cdots).\end{cases}$ (3) ###### Proof. Just as the analysis in the proof of theorem 2.1, we have $H(n)=2H(n-1)-H(n-2)+g(n),$ (4) where $g(n)$ denote the number of the hexagons of which there are vertexes both on $OA_{n}$ and on $OB_{n}$. Obviously, we have $H(1)=0$, $H(2)=0$, $H(3)=1$ and $g(1)=0$, $g(2)=0$, $g(3)=1$. The equation (4) is similar to the equation (2). So we solve it in the same way as (2). We have $\displaystyle g(2k+1)$ $\displaystyle=$ $\displaystyle g(2k)$ $\displaystyle+1+2+3+4+\cdots+(2k-4)+(2k-3)+(2k-2)+(2k-1)$ $\displaystyle+2+3+4+\cdots+(2k-4)+(2k-3)+(2k-2)+(2k-2)$ $\displaystyle+3+4+\cdots+(2k-4)+(2k-3)+(2k-3)+(2k-3)$ $\displaystyle+\cdots\cdots\cdots\cdots$ $\displaystyle\underbrace{+(k-2)+(k-1)+k+(k+1)+(k+2)+(k+2)+\cdots+(k+2)}\limits_{\hbox{The number is }k+2}$ $\displaystyle\underbrace{+(k-1)+k+(k+1)+(k+1)+(k+1)+\cdots+(k+1)}\limits_{{\hbox{ The number is }}k+1}$ $\displaystyle\underbrace{+k+k+k+k+\cdots+k}\limits_{{\hbox{ The number is }}k}$ $\displaystyle+\cdots\cdots\cdots\cdots$ $\displaystyle+2+2$ $\displaystyle+1$ $\displaystyle=$ $\displaystyle g(2k)+1^{2}+2^{2}+\cdots+(2k-1)^{2}$ $\displaystyle-(+1+2+3+4+\cdots+(2k-4)+(2k-3)+(2k-2)$ $\displaystyle+1+2+3+4+\cdots+(2k-4)$ $\displaystyle+\cdots\cdots\cdots\cdots$ $\displaystyle+1+2+3+4$ $\displaystyle+1+2)$ $\displaystyle=$ $\displaystyle g(2k)+{k\over 2}(4k^{2}-3k+1),$ and $\displaystyle g(2k)$ $\displaystyle=$ $\displaystyle g(2k-1)$ $\displaystyle+1+2+3+4+\cdots+(2k-5)+(2k-4)+(2k-3)+(2k-2)$ $\displaystyle+2+3+4+\cdots+(2k-5)+(2k-4)+(2k-3)+(2k-3)$ $\displaystyle+3+4+\cdots+(2k-5)+(2k-4)+(2k-4)+(2k-4)$ $\displaystyle+\cdots\cdots\cdots\cdots$ $\displaystyle\underbrace{+(k-2)+(k-1)+k+(k+1)+(k+1)+\cdots+(k+1)}\limits_{{\hbox{ The number is }}k+1}$ $\displaystyle\underbrace{+(k-1)+k+k+k+k+\cdots+k}\limits_{{\hbox{ The number is }}k}$ $\displaystyle\underbrace{+(k-1)+(k-1)+(k-1)+\cdots+(k-1)}\limits_{{\hbox{ The number is }}k-1}$ $\displaystyle+\cdots\cdots\cdots\cdots$ $\displaystyle+2+2$ $\displaystyle+1$ $\displaystyle=$ $\displaystyle g(2k-1)+1^{2}+2^{2}+\cdots+(2k-2)^{2}$ $\displaystyle-(+1+2+3+4+\cdots+(2k-5)+(2k-4)+(2k-3)$ $\displaystyle+1+2+3+4+\cdots+(2k-5)$ $\displaystyle+\cdots\cdots\cdots\cdots$ $\displaystyle+1+2+3$ $\displaystyle+1)$ $\displaystyle=$ $\displaystyle g(2k-1)+{k-1\over 2}(4k^{2}-5k+2).$ So we have $\displaystyle g(2k)$ $\displaystyle=$ $\displaystyle{1\over 2}k(k-1)(2k^{2}-2k+1),$ $\displaystyle g(2k+1)$ $\displaystyle=$ $\displaystyle k^{4},$ and then $\displaystyle H(2k)-H(2k-1)={k\over 30}(12k^{4}-15k^{3}+5k^{2}-2)$ $\displaystyle H(2k+1)-H(2k)={k\over 30}(12k^{4}+15k^{3}+5k^{2}-2).$ By overlaying, we get (3). This completes the proof. ∎ ## References * [1] R. A. Brualdi, Introductory Combinatorics, 4th ed., Person Education, Inc. 2005. * [2] Y. X. Zhu, The number of convex pentagons and hexagons in an $n$-triangular net, Bull. Math., 46 (2007), no. 8, 51–52 (in Chinese).	arxiv-papers	2010-12-18T04:58:37	2024-09-04T02:49:15.799761	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jun-Ming Zhu", "submitter": "Jun-Ming Zhu", "url": "https://arxiv.org/abs/1012.4058" }
1012.4087	# A simple proof of orientability in colored group field theory Francesco Caravelli University of Waterloo, Waterloo, Ontario N2L 3G1, Canada and Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, D-14476 Golm, Germany fcaravelli@perimeterinstitute.ca ###### Abstract In this short note we use results from the theory of crystallizations to prove that color in group field theories garantees orientability of the piecewise linear pseudo-manifolds associated to each graph generated perturbatively. The origin of orientability is the presence of two interaction vertices. ## I Introduction There has been recently a growth of interest in group field theories laurentgft ; quantugeom2 and there are many reasons for this to happen. Group field theories (GFT) are a generalization of matrix models to higher dimensionsBrezin:1977sv ; mm and a generalization of tensor models as welltensmodl . Moreover, GFT are known to generate the partition function of Spin Foams, thus having a direct relation with Loop Quantum Gravity lqg . It is known that matrix models have a topological expansion in which the genus, the only topological invariant needed to characterize orientable surfaces, plays the role of the parameter of this expansion. Roughly speaking a $n$-dimensional group field theory has a vertex associated to an $n$-simplex and a propagator which glues the $(n-1)$-simplices. Feynman diagrams of a $n$-dimensional group field theory can be interpreted as gluings of simplices and then have the interpretation of piecewise linear (PL) manifolds. A colored version of group field theory (cGFT) has been introduced recentlycolor ; PolyColor ; sefu2 . One important reason to introduce color in the diagrams is that it is possible to have a better control over the perturbatively generated singularities of GFT. The challenge in these models is to obtain a topological expansion as in the 2-dimensional case sefu3 ; FreiGurOriti ; sefu1 ; smerlak . Remarkably, it has been shown 1on that spheres dominates the partition function in any dimension. In order to achieve this result, techniques from the theory of crystallizations have been used. In fact, colored $n$-graphs are well known in mathematics as gems: graph-encoded manifolds Pezzana ; Lins . In this paper we use the results in this field of mathematics to show that the growth of interest in colored models is not unjustified: colored models generate orientable pseudomanifolds in any number of dimensions. Many of the theorems we will use were known for long time in the context of crystallization and here we report briefly these results. The outcome of this note is that the generation of pseudo-manifolds is due to the color, while the orientability in the colored versions of group field theory models is due to the presence of two different vertices (clockwise and anti-clockwise). In the following we will focus on the Boulatov model, but the result is more general because it relies only on the presence of vertices of opposite orientation in the perturbative expansion of the partition function. The paper is organized as follows: in section II we recall the colored Boulatov model and its standard interpretation. In section III we review basic results in the field of 3-gems and crystallizations. We will use some of these results in section IV to prove the orientability of simplicial complexes generated perturbatively by the colored Boulatov model. Conclusions follow. ## II The Colored Boulatov Model In this section we introduce the colored Boulatov modelGFT color . Let us consider a compact Lie group $H$, denote $h$ its elements, $e$ the unit element, and $\int dh$ the integral with respect to the Haar measure of the group. In 3 dimensions we introduce two fields, $\bar{\psi}^{i}$ and $\psi^{i}$, $i=0,1,2,3$ be four couples of complex scalar (or Grassmann) fields over three copies of $G$, $\psi^{i}:G\times G\times G\rightarrow\mathbb{C}$. The index $i$ runs from$=0$ to $n+1$, where $n$ is the number of dimensions, and the $\psi$ and $\bar{\psi}$ are functions of $n$ copies of the group. In the fermionic version of the theory the indices $i$ can be seen as the dependence of the field from a (global) gauge group $SU(N)$, where $N=n+1$. We denote $\delta^{\Lambda}(h)$ the regularized delta function over $G$ with some cutoff $\Lambda$ such that $\delta^{\Lambda}(e)$ is finite, but diverges when $\Lambda$ goes to infinity. A feasible regularization is given, for instance for the group $G=SU(2)$, by $\displaystyle\delta^{\Lambda}(h)=\sum_{j=0}^{\Lambda}(2j+1)\chi^{j}(h).$ (1) where $\chi^{j}(h)$ is the character of $h$ in the representation $j$. The path integral for the colored Boulatov model over $G$ is: $\displaystyle Z(\lambda,\bar{\lambda})$ $\displaystyle=$ $\displaystyle e^{-F(\lambda,\bar{\lambda})}$ (2) $\displaystyle=$ $\displaystyle\int\prod_{i=0}^{4}d\mu_{P}(\bar{\psi}^{i},\psi^{i})\;e^{-S^{int}(\bar{\psi}^{i},\psi^{i})}\;,$ where the Gaussian measure $P$ is chosen such that: $\int\prod_{i=0}^{4}d\mu_{P}(\bar{\psi}^{i},\psi^{i})=1\ ,$ and: $\displaystyle P_{h_{0}h_{1}h_{2};h_{0}^{\prime}h_{1}^{\prime}h_{2}^{\prime}}=$ $\displaystyle=\int d\mu_{P}(\bar{\psi}^{i},\psi^{i})\;\bar{\psi}^{i}_{h_{0}h_{1}h_{2}}\psi^{i}_{h_{0}^{\prime}h_{1}^{\prime}h_{2}^{\prime}}=$ $\displaystyle=\int dh\;\delta^{\Lambda}\bigl{(}h_{0}h(h_{0}^{\prime})^{-1}\bigr{)}\delta^{\Lambda}\bigl{(}h_{1}h(h_{1}^{\prime})^{-1}\bigr{)}\delta^{\Lambda}\bigl{(}h_{2}h(h_{2}^{\prime})^{-1}\bigr{)}\;,$ The fermionic colored model has two types interactions, a “clockwise” and an “anti-clockwise”, and one is obtained from the other one by conjugation in the internal group color $SU(N)$, where $N$ is 4 in 3 dimensions, one for each face of the 3-simplex 111It should be mentioned that also in the bosonic versionsefu2 there is a clockwise and anti-clockwise interaction. In that case the the types of interactions are motivated by the fact that its introduction has a nice combinatorial definition of homologyPolyColor . For convenience we denote $\psi(h,p,q)=\psi_{hpq}$. Invariance under global rotations in the internal color group require at least two interactions: $\displaystyle S^{int}=\frac{\lambda}{\sqrt{\delta^{\Lambda}(e)}}\int(dh)^{6}\psi^{0}\psi^{1}\psi^{2}\psi^{3}$ $\displaystyle+\frac{\bar{\lambda}}{\sqrt{\delta^{\Lambda}(e)}}\int(dh)^{6}\bar{\psi}^{0}\bar{\psi}^{1}\bar{\psi}^{2}\bar{\psi}^{3}$ (3) where we omitted the internal structure of the group elements of the fields $\psi^{i}$ and $\bar{\psi}^{i}$. In order to make the notation clearer (already the orientation of the colors is sufficient to distinguish the two vertices), we call “red” the vertex involving the $\psi$’s and “black” the one involving the $\bar{\psi}$’s. Thus any line coming out of a cGFT vertex has a color $i$. The group elements $h_{ij}$ in eq. (3) are associated to the propagators (represented as solid lines), and glue two vertices with opposite orientation. The vertex can be seen as the dual of a tetrahedron and its lines represent the triangles which form the tetrahedron. Figure 1: Colored GFT red and black vertices. Figure 2: A gluing using a colored propagator. Each propagators is decomposed into three parallel strands which are associated to the three arguments of the fields, i.e. the 1-dimensional elements of the 1-skeleton of the tetrahedron which bound every face. These are associated to the edges of the tetrahedron. A colored line represents the gluing of two tetrahedra (of opposite orientations) along triangles of the same color as in Fig. (2). It is easy to understand that a cGFT graph can be seen either as a stranded graph (using the vertex and the propagators as depicted in Fig. 1) or as a “colored graph” with (colored) solid lines, and two classes of oriented vertices. In this paper we consider only vacuum graphs, i.e. all the vertices of the graphs are 4-valent and we deal only with connected graphs (thus with the logarithm of the partition function (2)). The lines of a vacuum cGFT graph $\Gamma$ have two natural orientations given by the fact that only vertices of opposite orientations can be glued. It is easy to see that a vacuum cGFT graph must have the same number of black and red vertices. For any graph $\Gamma$, we denote $n$ as the number of vertices, $l$ as the lines of $\Gamma$, and we define as faces (not to be confused with the faces of the tetrahedron!), $\mathscr{F}_{\Gamma}$, as any closed strand in the Feynman graph of a GFT. Thus a generic vacuum Feynman amplitude of the theory can be written as: $\displaystyle\mathscr{A}=\frac{(\lambda\bar{\lambda})^{\frac{n}{2}}}{[\delta^{N}(e)]^{\frac{n}{2}}}\int\prod_{l\in\Gamma}dh_{l}\prod_{f\in\mathscr{F}_{\Gamma}}\delta^{\Lambda}_{f}(\prod_{l_{0}\in f}^{\rightarrow}h_{l_{0}}^{\sigma(l_{0},f)}),$ (4) where $l_{0}$ is a line associated to a face $f$ and $\sigma(l_{0},f)$ is alternatively $+1$ or $-1$ depending on the orientation. In the following we will assume that an orientation is fixed. Because of the properties of $\delta^{\prime}s$ the orientation does not affect the amplitude. To each colored graph associated to an amplitude of the colored Boulatov model it is possible to associate bubbles by removing all the edges of one color. We call $\mathscr{B}_{i_{1},\cdots,i_{k}}$ the set of $k$-bubbles associated to the deletion of $n-k$ colors. In 3-dimensions, for instance, 3-bubbles have 3-colors (surfaces), 2-bubbles have 2 colors (lines) and so on and so forth. Bubbles play a special role in the theory, since they discriminate manifold from pseudo-manifolds (see next section for the same result in the theory of 3-gems). ## III A survey of Graph-Embedded Manifolds results In this section we review some basic results in the field of 3-gems and make a dictionary between the two literatures, as colored group field theory can gain much from the results obtained in all the years of research in such field. Let $\Gamma$ be a finite, edge-colored graph, parallel edges allowed. A $k$-residue of $\Gamma$, $k\in\textbf{N}$ is a connected component of subgraph of $\Gamma$ induced by k color classes (this is what in colored group field theory are called bubbles). These graphs represent a piecewice linear manifold in the following sense (a pseudo-complex) surveyger . A $n$-regular $n$-colored graph is an edge-colored graph which has a nodes of degree $n$. To a couple $(\Gamma,\gamma)_{n+1}$ there is an associated pseudo-complex $K(\Gamma)$ given by the following construction. Take an $n$-simplex $\sigma^{n}$ for each $V(\Gamma)$ and label its vertices $\Delta_{n}$. If $x$,$y$ in $V(\Gamma)$ are joined by an edge, then attach the $(n-1)$-faces of their associated simplices. This is the same interpretation given to attaching faces of $n$-simplices in a $n$-dimensional group field theory. We denote $\|\Gamma\|$ the pseudo-complex associated with the colored graph $\Gamma$. Lemma 1 For any PL $n$-manifold $\mathscr{M}$ there exist a (n+1)-graph $\Gamma$ such that $\|\Gamma\|\backsimeq\mathscr{M}$. We now restrict to the case of 3-dimensions and list some of the basic resultsLins . Let $\Gamma$ be a 4-edge-colored 4-graph and denote by $v$, $e$, $b$, $t$ respectively the number of vertices (0-residues), edges (1-residues), 2-residues and 3-residues. Definition A 3-gem (a 3 graph-embedded manifold) is a 4-regular properly edge- colored graph such that $v+t=b$ (5) A 4-regular properly edge-colored graph for which (5) does not apply is called 3-gepm (a 3 graph-embedded pseudo-manifold). Lemma 2 A necessary and sufficient condition for the graph $(\Gamma,\gamma)_{4}$ to represent a manifold, is to meet the relation between its 2- and 3- residues (read as it 2- and 3- colored bubbles) and the number of vertices (read as the perturbative order) $v+t=b$. This Lemma clarifies the reason why 3-gems have to satisfy the relation (5). Let now introduce few definitions which will turn useful latersurveyger : Definition A triball is a connected, cubic, 3-edge-colored graph $\Gamma_{3}\subset\Gamma$ such that its Euler characteristic is the one of the 2-sphere. Thus we have the relation between its 2-residues $b_{\Gamma_{3}}$ and the vertices: $2b_{\Gamma_{3}}-v=4$. An important fact is the following: Lemma 3 A graph $(\Gamma,\gamma)_{4}$ is a 3-gem iff each of its 3-residue is a triball. Thus, the condition that graphs have to satisfy in order to be 3-gems is a condition on the topology of its 3-residues. We now discuss crystallizations of 3-gems. Figure 3: Fusion moves on a 4-regular 4-edge colored graph of 1-, 2- and 3- dipoles respectively. Let first introduce the fusion process. Let be $\mathscr{B}_{ijk}$ and $\mathscr{B^{\prime}}_{ijk}$ two different 3-residues separated by a unique color which, by construction, is different from the color $i$, $j$, $k$. We call $1-dipole$ this edge connecting the two 3-residues. The generalization to $k-dipoles$ which connect $(n-k)$-residues is obvious. We call fusion the process of contraction of two vertices through the first two combinatorial moves depicted in Fig. 3. Each cancellation of a 1-dipole has the effect of decreasing by one the number of $i$-residues, where i is the color of the edge which defines the 1-dipole, not changing the number of $j$-residues, for $j\neq i$. Thus by a succession of 1-dipole cancellation we obtain a 3-gem with 4 triballs. Such a 3-gem is said to be contracted and is called a crystallization for the associated 3-manifold. It is a fact that any closed 3-manifolds has a crystallization, and two closed 3-manifolds are related by a homeomorphism if and only if they are related by creation or contraction of 1- and 2- dipoles with the fusion rules; in this case, the two 3-manifolds are said to be equivalent or homeomorphic. Thus it is easy to understand that the fusion rules are the combinatorial equivalent of homeomorphisms. Let now discuss crystallization for generic colored $(n+1)$-graphs. The following results hold: Theorem 1 For every PL $n$-manifold $\mathscr{M}$ there exist a crystallization. Theorem 2 Two $n$-graphs $\|\Gamma_{1}\|$ and $\|\Gamma_{2}\|$ are crystallizations of the same manifold $\mathscr{M}$ if one is converted into the other by: a) Adding or removing a non-degenerate m-dipole with $n-1>m>1$; b) Adding a $1$-dipole and deleting another $1$-dipole. A general theorem on the orientability of $n$-graphs holds: Theorem 3 (Orientability) Let $(\Gamma,\gamma)_{n+1}$ be any crystallization of an $n$-manifold $\mathscr{M}$. Then $\mathscr{M}$ is orientable iff $\Gamma$ is bipartite. These theorems are fundamental in order to have a clear geometrical understanding of graphs generated by a colored group field theory, and will be used in the next section, in which the main result of the paper is presented. ## IV Orientability in cGFT In this section we prove a Lemma on the orientability of PL manifolds associated to graphs generated by the colored Boulatov model. Orientability of a manifold is a requirement if we want to construct a spin bundle. In 4-dimensions, for instance, the requirement to have a global spin bundle is to have a vanishing first and second Stiefel-Whitney class. While the second can be neglected by constructing local spin bundle and then gluing the charts, the vanishing of the first is a strict requirement and is equivalent to ask the orientability of the manifoldnakahara. Another important fact is that orientability restricts enormously the class of 3-manifolds which could be generated. As an example, in 2-dimensions the most general decomposition is given by connected sum of spheres, torii and projective planes. Orientability excludes the connected sum of projective planes, which allows the expansion in the ordinary genus we are used to. Lemma (3-dimensions) Let $\Gamma$ be a connected vacuum finite graph generated by the colored Boulatov model. Let $\mathscr{B}_{ijk}$ and $\mathscr{B}_{ij}$ be the set of 3- and 2- bubbles of $\Gamma$ respectively. Then the pseudo- manifolds associated to the $\Gamma$ is an oriented pseudo-manifold. Moreover, $\|\Gamma\|$ represents a closed and orientable 3-manifold iff $V+Card\\{\mathscr{B}_{ijk}\\}=Card\\{\mathscr{B}_{ij}\\}$ (6) Proof. This lemma follows directly from the properties of graphs generated by the colored Boulatov model and its interpretation, which is the same of the simplicial construction of 3-gepms. By Lemma 1 the graph generated is a manifold if and only if the condition (6) is met. Since the graph is finite, the manifold is also closed. Thus what we have to show is that they are orientable. By the theorem on the orientability the 3-gem represents an orientable manifold if and only if the crystallization graph is bipartite. First we note that the graphs generated by colored group field theory are bipartite. Let $A$ and $B$ be the set of clockwise and anti-clockwise vertices of $\Gamma$ respectively. Since by construction a clockwise vertex has to be contracted with an anticlockwise, then all the edges are between the set $A$ and the set $B$ and none is within the sets, thus the graph is bipartite. Now we have to show that its contraction is still bipartite. However, this fact is trivial because any of the moves in Fig. 3 keeps the bipartiteness of the original graph, thus in particular the fusion of a $1-dipole$. Moreover, since the graph is finite, the crystallization is reached in a finite number of moves. The orientability part of this Lemma can be generalized to higher dimensions. The construction given in the third section of this note ensures that to each $n$-dimensional pseudo-complex there is at least a colored $(n+1)$-graph which is homehomorphic to it. It is then easy to see why colored group field theories generates only orientable pseudo-manifolds in any number of dimensions; we state it as a Lemma, even if it clearly follows from the construction given in Pezzana of $n$-edge-colored graphs in any number of dimensions, while orientability comes from a generalization to m-dipoles (as in Theorem 2) of the previous proof and the fact that there are two types of vertices: This means that, at any finite order, the connected vacuum graphs generated by the partition function of a colored group field theory are associated with closed and orientable PL pseudo-manifolds. ## V Conclusions In this short paper we have used results in the field of 3-gems to prove that all the graphs generated by the colored Boulatov model are related to orientable pseudo-manifolds. In order to prove it we used new tools which could turn to be very useful in the context of group field theory, more specifically in the colored version of it. In fact, color is a fundamental ingredient in all we said. It should be said that what proved here is not an unexpected resulttalks . The fact that an orientation for the faces can be chosen with ease was a hint of what proved here. Indeed, as far as the author is concerned, this is the first rigorous proof appeared so far. We should stress that orientability is a fundamental requirement for “reasonable” manifolds. Aknowledgements We would like to thank Razvan Gurau for several discussions on the topic of group field theory. Also, we would like to thank Daniele Oriti for advices on the presentation of this result and Lorenzo Sindoni for reading carefully the manuscript. Research at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI. ## References * (1) L. Freidel, “Group field theory: An overview,” Int. J. Theor. Phys. 44, 1769 (2005) [arXiv:hep-th/0505016]. * (2) D. 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Math. Phys. 93, 295 (2010) [arXiv:1004.5196 [gr-qc]]. * (16) R. Gurau, “The 1/N expansion of colored tensor models” [arXiv:1011.2726v2 [gr-qc]] (2010) * (17) M. Pezzana, “Sulla struttura topologica delle varietà compatte”, Atti Sem. Mat. Fis. Univ. Modena 23, 269-277, (1974) * (18) S. Lins, Gems, Computers and Attractors for 3-Manifolds, (Series on Knots and Everything, Vol 5) ISBN: 9810219075/ ISBN-13: 9789810219079 * (19) M. Ferri, C. Gagliardi “ Crystallisation moves,” Pacific Journal of Mathematics Vol. 100, No. 1, (1982) * (20) M. Ferri, C. Gagliardi, L. Grasselli, “A graph-theoretical representation of PL-manifolds - A survey on crystallizations”, Aequationes Mathematicae 31, 121-141 (1986) * (21) S. Lins, A. Mandel, “Graph-encoded 3-manifolds”, Discrete Mathematics 57, 261-284 (1985) * (22) D. Oriti,R. Gurau, private communications.	arxiv-papers	2010-12-18T12:31:29	2024-09-04T02:49:15.805535	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Francesco Caravelli", "submitter": "Francesco Caravelli", "url": "https://arxiv.org/abs/1012.4087" }
1012.4092	# Nuclei of early-type dwarf galaxies: insights from stellar populations††thanks: Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile (programme 078.B-0178) Sanjaya Paudel1, Thorsten Lisker1, Harald Kuntschner2 1Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Mönchhofstr. 12-14, 69120 Heidelberg, Germany 2Space Telescope European Coordinating Facility, European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany E-mail: sjy@x-astro.net ( Accepted … Received … ; in original form ) ###### Abstract We present a comprehensive analysis of the spatially resolved stellar population properties of 26 early-type dwarf (dE) galaxies in the Virgo cluster. Using Lick/IDS absorption line indices we derive simple stellar population(SSP)-equivalent age, metallicity and [$\alpha$/Fe] abundance ratio. In particular, we focus on the comparison of the stellar populations between the central nucleus and the surrounding galactic main body. The stellar populations of the nuclei are, for most dEs, significantly younger than those of the respective galactic main bodies, with an average difference of 3.5 Gyr. We find only five dEs with significantly older nuclei than their galactic main bodies. Furthermore, we observe most dE nuclei to be more metal rich compared to their host galaxies. These age and metallicity behaviours are shown by almost all dEs brighter than Mr = -17 mag. The metallicity of both nuclei and galactic main bodies correlates with the total luminosity of the dEs. However, the metallicity of the nuclei covers a larger range (+0.18 to -1.22 dex) than that of the galactic main bodies, which all have sub-solar metallicity. The ages of dE nuclei show a statistically significant correlation with the local projected galaxy density within the cluster, such that younger ages are predominantly observed outside of the high-density central cluster region. The alpha-element abundance ratios are consistent with solar for both nuclei and galactic main bodies. We also examine the presence of radial gradients in the SSP parameters for a subset of 13 dEs (up to 1.2 kpc or 15 arcsec radius). We notice two different types of gradients, namely smooth profiles that include the nucleus, and profiles where a break occurs between the nucleus and the rest of the galaxy. Nevertheless, an overall trend of increasing age and decreasing metallicity with radius exists, consistent with earlier studies. The $\alpha$-abundance ratio as function of radius is consistent with no gradient. Possible formation scenarios for the nuclei of dEs are discussed. The young and metal-enhanced population of nuclei suggests that these might have formed at later epochs, or the termination of star formation activity in the nuclei might have occured relatively late, perhaps due to continuous infall of gas into the central potential well. Our stellar population analysis suggests that the merging of globular clusters is not an appropriate scenario for the formation of most dE nuclei, at least not for the brighter dEs. We speculate that there might be different formation processes which are responsible for the formation of dEs and their nuclei depending on their luminosity. ###### keywords: galaxies: dwarf – galaxies: evolution – galaxies: formation – galaxies: stellar content – galaxies: elliptical and lenticular, cD – galaxies: clusters: individual: Virgo ††pagerange: Nuclei of early-type dwarf galaxies: insights from stellar populations††thanks: Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile (programme 078.B-0178)–LABEL:lastpage††pubyear: 2010 ## 1 Introduction Early-type dwarf galaxies (dEs, MB $>$ -18) are the numerically dominant population in the present-day Universe (Sandage et al., 1985; Binggeli et al., 1987; Ferguson & Binggeli, 1994). They also exhibit strong clustering, being found predominantly in the close vicinity of giant galaxies, either as satellites of individual giants, or as members of galaxy clusters (Ferguson & Sandage, 1989). Although the dEs are characterized by their smooth appearance, having no recent or ongoing star formation and apparently no gas or dust content, the understanding of their origin and evolution remain major challenges for extragalactic astronomy. Stellar population studies show that dEs exhibit on average younger ages as compared to their giant counterparts, and also a lower metal content according to the correlation of metallicity and luminosity (Michielsen et al., 2008). However, past studies provided a wide range of ages (e.g., Poggianti et al. 2001; Rakos et al. 2001; Caldwell et al. 2003; Geha et al. 2003; van Zee et al. 2004), from as old as being primordial objects to dEs with recently formed young stellar populations. It appears that dEs themselves are not a homogeneous class of objects. Sub- structures such as stellar disks, faint spiral arms or bars are quite frequent among the brighter dEs (Lisker et al., 2006; Lisker et al., 2007). Many dEs were found to contain a central surface brightness enhancement consistent with a point source on top of the galactic main body (e.g. Binggeli & Cameron, 1991, 1993), referred to as so-called nucleated dEs. The studies from the HST/ACS Virgo cluster survey (Côté et al., 2006), with their high angular resolution, not only verified the presence of such a distinct nucleus but also showed that nuclei are ubiquitous in bright dEs, covering a range in nucleus brightness. Interestingly, dEs with comparably faint nuclei that had not been identified before Côté et al. (2006) show several systematically different properties as compared to dEs with bright nuclei (Lisker et al., 2007; Lisker et al., 2008). Different studies of dE nuclei from different data sets found several contradictory properties for the nuclei (Grant et al., 2005; Lotz et al., 2004; Côté et al., 2006). Particularly, the ground and space based data sets yielded different results. Grant et al. (2005) found that the nuclei are on average redder than their surrounding galactic main body. On the other hand, studies using HST observations (Côté et al., 2006; Lotz et al., 2004) measured the dE nuclei to be slightly bluer than the galactic part. Furthermore, Côté et al. (2006), who used high quality data sets from the ACS Virgo Cluster Survey, proposed that the nuclei rather closely match the nuclear clusters of late type spiral galaxies in terms of size, luminosity and overall frequency. Another related scenario is also emerging: the recently discovered new (candidate) type of extremely small dwarf galaxies, the UCDs (Ultra Compact Dwarfs) with typical magnitudes of $-13<M_{b}<-11$ (Hilker et al., 1999; Phillipps et al., 2001), might be the remnant nuclei of tidally stripped dwarf galaxies (Bekki et al., 2003; Drinkwater et al., 2003; Goerdt et al., 2008). The formation mechanisms of the nuclei of dEs are poorly understood and various possibilities have been proposed, also depending on the evolution and formation of dEs as a whole. As the nucleated dEs are preferentially rounder in shape, van den Bergh (1986) proposed that the nuclei of dEs could have formed from the gas that sank to the centre of the more slowly rotating objects. Since they predominantly appear in highly dense environments, like the centre of a cluster of galaxies, the pressure from the surrounding inter- galactic medium may allow dwarf galaxies to retain their gas during star formation and produce multiple generation of stars (Silk et al., 1987; Babul & Rees, 1992), forming nuclei in the process. In both proposed scenarios the nuclei are formed along with the evolution of the galaxy itself, i.e., continuous star formation activity occurs at the dE centre as time passes. Unlike that, Oh & Lin (2000) suggested that dE nuclei might have formed in a different way, namely through subsequent migration or orbital decay of several globular clusters towards the centre of their host dE. It is difficult to provide a definitive observational test of these different scenarios for nucleus formation. Nevertheless, we can gain some insight by comparing the different observational properties, in particular relative ages and chemical enrichment characteristics, of the nuclei with their galactic main bodies, as well as with UCDs as their possible descendants. However, we need to bear mind that there may be a mixture of different formation scenarios. Our previous study based on this dataset (Paudel et al., 2010, hereafter Paper I) has focused on the analysis of the inner stellar populations of dEs as a whole, without separating nuclei and galactic main bodies. Instead, our intention was to see the variation of the inner stellar population properties with different morphological subclasses of dEs (cf. Lisker et al., 2007), using a much larger sample of Virgo dEs than in previous Lick index studies. We showed that dEs with different substructure properties (with/without disk features, Lisker et al. 2006) have significantly different stellar populations: dEs with disk features are younger and more metal rich than dEs without disks. Therefore we concluded that these dEs probably do not have the same origin, as they also differ in their distribution with local environmental density in which they reside. By selection, all dEs in our sample contain a central nucleus, therefore it seems important to see the nature of the stellar populations of the nuclei and the surrounding galactic main bodies separately. And since there are different possibilities for the processes that form nuclei and also dEs themselves, we ask: can the nuclei thus tell us something about the formation history of dEs? This paper is organized as follows. In Section 2, we describe the sample of Virgo cluster dEs, observation and data reduction in brief. In section 3, we describe the measurement of line-strength indices in the Lick/IDS system. Our main results from the stellar population parameters are given in Section 4 and are discussed in Section 5. Finally, we summarize our findings in Section 6. ## 2 The Sample, Observation and Data Reduction Our sample comprises 26 nucleated dEs in the Virgo cluster. The sample properties such as position in the color magnitude relation, total galactic luminosity, radial velocity and their local projected density within the Virgo cluster are described in detail in Paper I. The sample covers the full range of local density and includes the different morphological dE subtypes, i.e., 8 dEs with disks (dE(di)s) and 18 dEs without disks, which we hereafter simply refer to as dE(N)s. One dE(di) (VCC0308) contains a weak blue color excess in the centre, thus being referred to as a blue-centre dE (cf. Lisker et al., 2006). The observations were carried out at the ESO Very Large Telescope (VLT) with the FORS2 instrument. The 1” slit and 300V grism provide an instrumental resolution of $\simeq$11 Å(Full Width at Half Maximum, FWHM). The other basic observational properties and the data reduction processes are described in detail in Paper I. We carefully checked the issue of scattered light during the reduction of the data, since the presence of a significant amount of scattered light could produce an artificial gradient in the measured line indices. Fortunately, our MOS-MXU setup utilized in this investigation provides the opportunity to quantify it. There are always free intra-slit regions where no light enters directly from the sky. After the bias subtraction these regions should not contain any flux, unless scattered light were present. We thus calculate the average amount of light within such regions manually. We find that the mean is zero within the uncertainties, which are of the order of some hundredths of a count. The FORS2 pipeline reduction produces the same result. It therefore confirms that there is no scattered light left in the spectra. In a different way, there is still the probability of mixing the nucleus light out to far beyond the central nucleus in case of bad seeing or instrumental blurring. To examine this effect, we also observed a star in an additional slit along with each target-field. Then, through the light profile of this star, we quantify the amount of such light at a radius of 3” beyond the centre. Our measurements show that spread nuclear light is less than 1% of galactic light at 3” distance from the galaxy centre. The observed FWHM of the stars is always $\sim$1.3” or less, consistent with this negligible fraction of starlight at 3” from the centre. ### 2.1 Extraction of nuclear spectra and analysis of light profile Table 1: Basic parameters and signal-to-noise ratio for our targets. Galaxy \| Nuc. \| Gal. \| Reff \| Mr \| mr \| Light ---\|---\|---\|---\|---\|---\|--- VCC \| (SNR) \| (SNR) \| arcsec \| Total \| Nuc \| fraction No. \| pix-1 \| pix-1 \| arc-sec \| mag \| mag \| in % 0216 \| 47 \| 30 \| 13.3 \| $-$16.78 \| -11.58 \| 22 0308 \| 35 \| 31 \| 18.7 \| $-$17.95 \| -11.91 \| 40 0389 \| 32 \| 30 \| 17.2 \| $-$18.00 \| -12.59 \| 48 0490 \| 32 \| 23 \| 27.6 \| $-$18.09 \| -12.43 \| 25 0545 \| 33 \| 30 \| 13.3 \| $-$16.61 \| -11.71 \| 35 0725 \| 23 \| – \| 25.2 \| $-$16.19 \| -10.17 \| $-$ 0856 \| 56 \| 35 \| 15.9 \| $-$17.71 \| -12.73 \| 23 0929 \| 56 \| 34 \| 20.5 \| $-$18.58 \| -13.13 \| 33 0990 \| 35 \| 33 \| 09.9 \| $-$17.39 \| -12.52 \| 53 1167 \| 46 \| 30 \| 27.3 \| $-$16.95 \| -12.01 \| 17 1185 \| 33 \| 50 \| 19.3 \| $-$16.65 \| -10.76 \| 30 1254 \| 67 \| 31 \| 14.9 \| $-$17.17 \| -13.31 \| 09 1261 \| 50 \| 42 \| 22.5 \| $-$18.47 \| -12.53 \| 42 1304 \| 35 \| 31 \| 16.2 \| $-$16.86 \| -12.23 \| 33 1308 \| 39 \| 27 \| 11.4 \| $-$16.50 \| -11.32 \| 44 1333 \| 41 \| 28 \| 18.5 \| $-$15.44 \| -11.76 \| 10 1348 \| 42 \| 25 \| 13.1 \| $-$16.94 \| -12.83 \| 23 1353 \| 28 \| 31 \| 08.8 \| $-$15.51 \| -10.64 \| 53 1355 \| 24 \| 28 \| 29.6 \| $-$17.59 \| -10.96 \| 57 1389 \| 30 \| 31 \| 12.8 \| $-$15.98 \| -10.78 \| 36 1407 \| 27 \| 31 \| 11.8 \| $-$16.95 \| -10.99 \| 52 1661 \| 34 \| 29 \| 18.9 \| $-$16.18 \| -11.18 \| 16 1826 \| 23 \| 31 \| 07.8 \| $-$16.30 \| -11.91 \| 56 1861 \| 30 \| 31 \| 18.4 \| $-$17.78 \| -11.83 \| 47 1945 \| 31 \| 38 \| 21.5 \| $-$17.11 \| -11.66 \| 35 2019 \| 31 \| 29 \| 18.1 \| $-$17.53 \| -11.34 \| 37 The second and third columns are the measured signal-to-noise ratio (SNR) per pixel at 5000Å for the galactic-light-subtracted nuclear spectra and the galactic main body spectra, respectively. The fourth column gives the half- light semi-major axis in SDSS r from Lisker07. The fifth and sixth columns are total galactic and nucleus absolute magnitudes in SDSS r, applying a distance modulus of $m-M=31.09$ mag (Mei et al., 2007), corresponding to $d=16.5$ Mpc. The last column represents the amount (in fraction of total light of the central aperture) of light subtracted from the central nucleus spectrum. Nucleus magnitudes were derived as described in Paudel et al. (2010): a two- dimensional elliptical model image of the galaxy, based on a Sérsic fit to the radial profile, was subtracted from the original image, taking into account the median SDSS PSF of 1.4” FWHM. The nucleus magnitude was then measured by circular aperture photometry with $r=2"$ centered on the nucleus; the error is estimated to be 0.2 mag. Figure 1: The light profile of dEs. The crosses represent the observed flux along the slit, and the solid line is the fitted exponential profile beyond 3” and extrapolated to the centre. Our goals in this paper are the measurement of simple stellar population (SSP) equivalent parameters (see Trager et al. 2008) of dE nuclei and a comparison with the SSPs of the surrounding galactic main bodies. Additionally, if the signal-to-noise ratio (hereafter SNR) permits us, we wish to explore gradients in the SSP parameters, which helps to determine whether the SSP of the nuclei is very different from the rest of the galaxy or is just a continuity of a smooth SSP gradient at the centre of dEs. Although there is no precise definition for what a nucleus is, the working definition used by several studies is that an excess of light from the smooth exponential (or higher order Sérsic) profile of the rest of the galactic part is observed, looking like a compact source sitting at the centre of the galaxy. Because of its compactness, it is considered as a point source and represented with a seeing convolved Gaussian light profile. Likewise, the study of Grant et al. 2005 represents the nuclei as a point source convolved with Gaussian seeing. Côté et al. 2006 used a slightly different approach, by fitting a two component core-Sérsic model (Graham & Guzmán, 2003). In Fig. 1 we can clearly see for most dEs the change in the light profile at the centre (e.g. VCC0216, VCC0856, VCC0545, VCC1353 and VCC1945). On the other hand, VCC0308, VCC0990, VCC1261 and VCC1826 exhibit a rather smooth light profile. There may be several factors which produce such differences in the light profile even though all dEs in this sample are confirmed as nucleated from other photometric studies (Binggeli et al., 1985; Lisker et al., 2007). Insufficient spatial resolution or observed seeing which might blur the steeper light profile of the nuclei makes it harder to separate the galactic light profile. However, Côté et al. (2006) have observed the existence of a profile break in the case of VCC0856, VCC1261, VCC1355, VCC1407, VCC1661 and VCC2019, reconfirming the existence of a nucleus at the centre of dEs with HST high resolution surface photometry. It is rather difficult to carry out an analysis of the stellar populations of nuclei _alone_ , because the nuclei are always situated on top of the underlying galactic main bodies. It is also hard to separate the galactic light from the central nucleus of such a faint object. The studies that have been done by Chilingarian (2009) and Koleva et al. (2009) provide results without galactic light subtraction from the nucleus. Although there is, in our sample, typically a fairly large domination of light from the nucleus as compared to galactic light at the photometric centre of the dEs, still a considerable amount of underlying light of the host galaxy can alter the observed properties of the nuclei. We therefore aim to reduce the galactic light contamination in the nucleus spectra, attempting a separate extraction of spectra for the nucleus and the galactic part. We extract the nucleus spectra from the central 0”.75 (i.e., 3 pixels). The region between 0”.75 and 3” is not used for this extraction, to avoid any effects of nucleus light in the spectra of the galactic main body. We then integrate over the interval 3” to 8” from each side of the nucleus to extract the spectra of the galactic main body (see Appendix A, Fig. 9). The individual spectra of galactic main body from the different side of nucleus were then co- added to produce a spectrum of higher SNR. In order to subtract the galaxy light from the nucleus, we determine a scaling factor by fitting the galactic main body’s light profile (measured along the slit) by an exponential profile and extrapolating it to the very centre, yielding the amount of galaxy light contained in the nucleus aperture (see Appendix A). Given the above considerations about the difficulty of separating nucleus and galaxy, we point out that our approach ensures the removal of a _significant part_ , yet probably not 100% of galaxy light contamination. For those few cases where the central light profile looks rather smooth with ground-based data, and can only be disentangled with space-based photometry, our “nucleus” spectrum thus needs to be considered representative for the combination of nucleus _and_ galactic central light. Before co-adding the spectra from the different sides of the galaxies, we analyze their slit profile to check for inconsistencies or asymmetries, e.g. by contaminating objects on the slit. We find only one galaxy, VCC1945, has an asymmetric profile that deviates from a smooth exponential profile on one side. We noticed that a bright point source (foreground/ background or intra- galactic globular cluster) lies on one side of the slit. Therefore, we remove the spectrum from this side. For completeness, we also compare the spectra from the different sides of the galaxy before co-adding them, and we always find good agreement. Finally, the measured SNR at 5000Å for both the galaxy- subtracted nucleus spectra and the combined galaxy spectra is given in Table 1. ## 3 Line strength measurements Before measuring the Lick absorption line indices from the flux calibrated spectra of the galactic main bodies and nuclei, we also carefully checked whether any emission lines are present, particularly since some dEs show a fairly young nucleus. However, we do not detect any [OIII] emission, thus we do not correct the H$\beta$ absorption for possible contamination by emission. If such emission were present, it would make the measured H$\beta$ absorption smaller, and therefore derived ages older. On the other hand, it could be possible that we do not see any emission lines because of the low spectral resolution. To quantify what strength of an emission line in a high-resolution spectrum (i.e, model of Vazdekis et al., 2010) would be smeared out in a low- resolution like ours, such that it is not recognized visually, we select a model spectrum of age 2 Gyr, and added an emission of H$\beta$. We then degrade the spectrum to the low-resolution of 11 Å. We find that the added emission line could have an effect of up to 12% on the measured absorption line strength, which reveals a relatively small effect on the age. Note that we have not applied a velocity dispersion correction for the Lick indices, because the expected galactic velocity dispersion, $\sigma_{gal}\leq 50$ km s-1, is significantly below our spectral resolution $\sigma_{instr}\sim 280$ km s-1. Therefore these corrections are not necessary. To measure the absorption line strengths from the spectra, we use the routine Indexf111http://www.ucm.es/info/Astrof/software/indexf/indexf.html developed by N. Cardiel. It uses the definition of the Lick indices from Trager et al. (1998) and also derives the uncertainty in measured strength using Monte-Carlo simulations. Calibrations of our measured line strengths to the actual Lick system have been done as described in Paper I (Section 4.2 and Appendix B in that paper). We use the method of Lick indices (Burstein et al., 1984; Worthey et al., 1994; Trager et al., 1998) as a tool for estimating the stellar population characteristics. We translate our Lick index measurements into SSP-equivalent ages, metallicities, and $\alpha$-element abundance ratios by comparing them to the stellar population models of Thomas et al. (2003) by $\chi^{2}-$minimization, following Proctor & Sansom (2002). For this we use the nine indices H$\delta_{F}$, H$\gamma_{F}$, Fe4383, H$\beta$, Fe5015, Mg $b$ , Fe5270, Fe5335 & Fe5406. Note that the SSP models assume all the stars were formed in a single burst and have the same age and metallicity. In fact, the galaxies may be a composite stellar system formed during several episodic star formation events, with different chemical compositions in general. Therefore, our estimated stellar population parameters can be considered _SSP $-$equivalent stellar populations_. The correlation of age and metallicity in the model fitting is illustrated in Appendix B. ## 4 Results: Ages, metallicities and alpha-abundance ratios Table 2: SSP-equivalent stellar population parameters for the nuclei and the galactic main bodies. Galaxy Name \| Age, Gyr \| [Z/H], dex \| [$\alpha$/Fe], dex ---\|---\|---\|--- \| Nuc. \| Gal. \| Nuc. \| Gal. \| Nuc. \| Gal. VCC0216 \| 1.4 ${}^{+0.3}_{-0.3}$ \| 4.0 ${}^{+1.8}_{-1.2}$ \| $-$0.61 $\pm$ 0.15 \| $-$0.63 $\pm$ 0.22 \| 0.09 $\pm$ 0.08 \| 0.03 $\pm$ 0.18 VCC0308 \| 1.5 ${}^{+0.1}_{-0.1}$ \| 3.6 ${}^{+1.6}_{-0.9}$ \| 0.01 $\pm$ 0.10 \| $-$0.34 $\pm$ 0.17 \| 0.42 $\pm$ 0.09 \| $-$0.07 $\pm$ 0.14 VCC0389 \| 4.1 ${}^{+2.1}_{-1.3}$ \| 9.1 ${}^{+3.4}_{-1.9}$ \| $-$0.24 $\pm$ 0.17 \| $-$0.43 $\pm$ 0.20 \| 0.17 $\pm$ 0.12 \| $-$0.15 $\pm$ 0.15 VCC0490 \| 1.9 ${}^{+0.7}_{-0.2}$ \| 3.6 ${}^{+2.1}_{-1.1}$ \| $-$0.02 $\pm$ 0.22 \| $-$0.24 $\pm$ 0.17 \| $-$0.11 $\pm$ 0.11 \| $-$0.11 $\pm$ 0.15 VCC0545 \| 6.9 ${}^{+2.6}_{-1.2}$ \| 12.5${}^{+0.0}_{-1.6}$ \| $-$0.78 $\pm$ 0.20 \| $-$0.88 $\pm$ 0.10 \| 0.24 $\pm$ 0.18 \| $-$0.23 $\pm$ 0.20 VCC0725a \| 5.5 ${}^{+1.4}_{-1.7}$ \| – – \| $-$1.00 $\pm$ 0.25 \| $-$\- – \| 0.16 $\pm$ 0.38 \| – – VCC0856 \| 1.9 ${}^{+0.2}_{-0.1}$ \| 15.0${}^{+0.0}_{-5.1}$ \| 0.03 $\pm$ 0.10 \| $-$0.61 $\pm$ 0.07 \| $-$0.14 $\pm$ 0.06 \| 0.03 $\pm$ 0.16 VCC0929 \| 3.2 ${}^{+0.5}_{-0.4}$ \| 3.8 ${}^{+1.4}_{-0.6}$ \| 0.11 $\pm$ 0.07 \| 0.03 $\pm$ 0.10 \| $-$0.16 $\pm$ 0.05 \| 0.15 $\pm$ 0.07 VCC0990 \| 2.3 ${}^{+0.9}_{-0.4}$ \| 5.5 ${}^{+2.1}_{-1.1}$ \| $-$0.19 $\pm$ 0.15 \| $-$0.31 $\pm$ 0.17 \| $-$0.30 $\pm$ 0.04 \| $-$0.01 $\pm$ 0.12 VCC1167 \| 15 ${}^{+0.0}_{-0.0}$ \| 7.5 ${}^{+7.5}_{-2.3}$ \| $-$1.15$\pm$ 0.05 \| $-$0.65 $\pm$ 0.22 \| 0.09 $\pm$ 0.16 \| 0.09 $\pm$ 0.18 VCC1185 \| 11.9${}^{+0.6}_{-2.4}$ \| 12.5${}^{+1.2}_{-1.1}$ \| $-$1.37 $\pm$ 0.05 \| $-$0.68 $\pm$ 0.10 \| $-$0.22 $\pm$ 0.33 \| $-$0.01 $\pm$ 0.22 VCC1254 \| 5.7 ${}^{+1.2}_{-1.2}$ \| 15.0${}^{+0.0}_{-9.0}$ \| $-$0.43 $\pm$ 0.10 \| $-$0.48 $\pm$ 0.32 \| 0.05 $\pm$ 0.07 \| $-$0.11 $\pm$ 0.14 VCC1261 \| 1.8 ${}^{+0.1}_{-0.0}$ \| 6.9 ${}^{+2.2}_{-1.4}$ \| 0.18$\pm$ 0.00 \| $-$0.46 $\pm$ 0.15 \| $-$0.10 $\pm$ 0.07 \| 0.07 $\pm$ 0.12 VCC1304 \| 8.6 ${}^{+4.4}_{-2.1}$ \| 4.5 ${}^{+0.7}_{-2.0}$ \| $-$1.22 $\pm$ 0.20 \| $-$0.56 $\pm$ 0.27 \| $-$0.30 $\pm$ 0.10 \| $-$0.22 $\pm$ 0.19 VCC1308 \| 1.8 ${}^{+0.3}_{-0.2}$ \| 15.0${}^{+0.0}_{-10.2}$ \| +0.16 $\pm$ 0.12 \| $-$0.70 $\pm$ 0.42 \| 0.09 $\pm$ 0.09 \| 0.11 $\pm$ 0.18 VCC1333 \| 7.9 ${}^{+5.8}_{-1.0}$ \| 1.0 ${}^{+0.4}_{-0.0}$ \| $-$1.05$\pm$ 0.20 \| $-$0.97 $\pm$ 0.20 \| 0.07 $\pm$ 0.20 \| 0.04 $\pm$ 0.37 VCC1348 \| 10.9${}^{+2.2}_{-1.4}$ \| 15.0${}^{+0.0}_{-1.3}$ \| $-$0.80 $\pm$ 0.10 \| $-$0.53 $\pm$ 0.07 \| 0.45 $\pm$ 0.14 \| 0.50 $\pm$ 0.04 VCC1353 \| 3.2 ${}^{+1.2}_{-1.2}$ \| 4.1 ${}^{+1.6}_{-1.4}$ \| $-$1.02 $\pm$ 0.25 \| $-$0.58 $\pm$ 0.22 \| $-$0.26 $\pm$ 0.32 \| 0.38 $\pm$ 0.18 VCC1355 \| 1.8 ${}^{+2.3}_{-0.5}$ \| 3.2 ${}^{+1.8}_{-0.6}$ \| $-$0.48 $\pm$ 0.39 \| $-$0.34 $\pm$ 0.22 \| $-$0.08 $\pm$ 0.30 \| $-$0.04 $\pm$ 0.16 VCC1389 \| 13.1${}^{+1.9}_{-3.1}$ \| 11.9${}^{+0.0}_{-2.0}$ \| $-$1.27 $\pm$ 0.20 \| $-$0.85 $\pm$ 0.10 \| 0.07 $\pm$ 0.30 \| 0.19 $\pm$ 0.19 VCC1407 \| 2.6 ${}^{+1.3}_{-0.7}$ \| 14.3${}^{+0.7}_{-8.6}$ \| $-$0.12 $\pm$ 0.17 \| $-$0.73 $\pm$ 0.34 \| 0.07 $\pm$ 0.14 \| 0.11 $\pm$ 0.16 VCC1661 \| 9.1 ${}^{+5.9}_{-1.5}$ \| 6.6 ${}^{+3.8}_{-1.1}$ \| $-$0.95 $\pm$ 0.15 \| $-$0.36 $\pm$ 0.22 \| $-$0.26 $\pm$ 0.14 \| $-$0.30 $\pm$ 0.04 VCC1826 \| 1.7 ${}^{+0.6}_{-0.2}$ \| 11.4${}^{+1.7}_{-2.3}$ \| +0.13 $\pm$ 0.17 \| $-$0.90 $\pm$ 0.15 \| $-$0.07 $\pm$ 0.13 \| $-$0.10 $\pm$ 0.19 VCC1861 \| 3.8 ${}^{+2.2}_{-1.0}$ \| 4.1 ${}^{+1.3}_{-1.3}$ \| $-$0.29 $\pm$ 0.17 \| $-$0.12 $\pm$ 0.12 \| $-$0.16 $\pm$ 0.14 \| 0.07 $\pm$ 0.09 VCC1945 \| 6.6 ${}^{+8.4}_{-1.3}$ \| 14.3${}^{+0.7}_{-2.4}$ \| $-$0.75 $\pm$ 0.27 \| $-$1.00 $\pm$ 0.10 \| 0.00 $\pm$ 0.24 \| $-$0.30 $\pm$ 0.23 VCC2019 \| 1.7 ${}^{+0.2}_{-0.3}$ \| 8.3 ${}^{+6.1}_{-2.5}$ \| +0.06 $\pm$ 0.15 \| $-$0.41 $\pm$ 0.24 \| $-$0.27 $\pm$ 0.12 \| 0.00 $\pm$ 0.16 awithout subtraction of galactic light and does not have a measurement of SSPs from the galactic main body (see text). Figure 2: A comparison of stellar population parameters. The SSPs from the different parts of the dEs are represented with vertical bars of different color: blue for the nuclei and red for the galactic main bodies. The faint background colors indicate the dE subtype: blue for the nucleated dE with disk and blue centre, green for the nucleated dEs with disks, and red for the nucleated dEs without disk features. The Virgo UCDs are represented by the green vertical bars with gray background. For the UCDs, we used published values of line strengths from Evstigneeva et al. (2007) to derive the stellar population parameters (see text). In this section, we present the SSP-equivalent ages, metallicities and $\alpha$-abundance ratios of our sample dEs (Table 2). Note that, in case of the least luminous dE, VCC0725, we find that the sky noise becomes dominant beyond the central aperture. Hence, we remove its galactic part from the sample and therefore provide no SSP parameters for the galactic main body of this dE. We can clearly see that the ages of the nuclei are significantly lower than the ages of the surrounding galactic main bodies (Fig. 2). The differences are more prominent in the disky dEs: only VCC1304 has a nucleus that is older than the galactic part. Moreover, we find that only four other non-disky dE(N)s (VCC1167, VCC1333, VCC1389 and VCC1661 $-$ see Sec.2) have nuclei with significantly larger ages than the galactic main bodies. The median difference in age between the galactic main bodies and nuclei is 3.5 Gyr. Examining Fig. 2 individually galaxy by galaxy, one can see that VCC0856 shows the largest difference ($>$10 Gyr) in age between the nucleus and the galactic part. The nucleus of the blue centre dE VCC0308, while having a young age, does not show up as being special, having an age of 1.5 $\pm$0.1 Gyr, similar to other dE nuclei such as VCC0216, VCC2019 and VCC1826. The metallicity distributions of the nuclei and the surrounding galactic main bodies also differ: the majority of the nuclei are relatively metal enhanced as compared to the galactic main bodies. However, it is remarkable that those nuclei that are older or equally old as the galactic part are also less metal rich than the latter. We find that the nucleus of VCC1308 has the highest metallicity of +0.16 $\pm$0.12 dex. For all dEs, the galactic main bodies have sub-solar metallicity. The $\alpha$-abundance ratio from nuclei and galactic main bodies show a wide distribution. The nuclei of three dEs (i.e., VCC0308, VCC0389 and VCC0545) show significant $\alpha$-enhancement as compared to their galactic part. On the contrary, four dEs, VCC0990, VCC0929, VCC1353 and VCC1861, exhibit a significantly enhanced $\alpha$-abundance in the galactic part as compared to their nucleus. In the right part of Figure 2, the green vertical bars present, for comparison, the derived stellar population parameters of the UCD sample of Evstigneeva et al. (2007). Note that we only use the published four indices (H$\beta$, Mgb, Fe5270 and Fe5335). However, we use the same method of estimation for the stellar population parameters. The UCD ages and metallicities are consistent with old and metal poor stellar populations. Almost all UCDs have ages $\sim$10 Gyr and metallicities vary between -1.25 to 0.13 dex. The [$\alpha$/Fe]-abundances are always super solar in case of the UCDs, with a mean of 0.31 dex, which is 0.34 dex higher than the mean [$\alpha$/Fe] of the dE nuclei. The relation between the stellar population parameters and the local projected number density of galaxies in the cluster is plotted in Figure 3. The local projected density has been calculated from a circular projected area enclosing the 10th neighbor. It seems that there is a correlation between the local projected density and the ages of the nuclei. The Spearman rank order test shows a weak correlation of the ages and metallicities of the dE nuclei with the local projected densities. The correlation coefficients are 0.5 and $-$0.4, and the probabilities of the null hypothesis that there is no correlation are 0.2% and 4% for the age and metallicity, respectively. Unlike this, a similar test shows that the SSPs of the galactic main body do not have any relation with local projected densities. Figure 3: The age, metallicity and [$\alpha$/Fe] versus local projected density. Green color represents the galactic main body and blue indicates the nucleus. Figure 4: The derived ages (top), metallicities (middle) and [$\alpha$/Fe]-abundance (bottom), plotted against $r-$band absolute magnitude (left). The blue color represents the nuclei and green color indicates the galactic main body. On top of each panel, we also show the difference in the SSP parameters, i.e. galactic part $-$ nucleus. In the right panel, we provide the number distribution of the parameters. The relations between the stellar population parameters and the total galactic luminosity are presented in Fig. 4. At the top of each panel, we also provide the trend of the differences in the SSP parameters between the galactic main bodies and the nuclei. It is clearly recognized that almost all dEs brighter than $M_{\rm r}=-17$ mag have younger and more metal-rich nuclei than the galactic main bodies. On the other hand, there is a relatively large scatter in the low luminosity region, and we can see that some of the nuclei are as old and metal poor as the galactic main bodies. However, the sign of the differences in age and metallicity between galactic main body and nucleus are completely opposite at the fainter and brighter end of the plot. As there exists a well-known metallicity-luminosity relation in early type galaxies (Poggianti et al., 2001), our sample also follows this relation for both nuclei and galaxies, i.e., the metallicity decreases with decreasing total galactic luminosity. The derived [$\alpha$/Fe] values are fairly consistent with a roughly solar value for both nuclei and galactic main bodies. In the right panels of Fig. 4, we provide the number distribution (in the histogram) of stellar population parameters of the nuclei (in blue color) and the galactic main bodies (in green color). It seems that the ages of the galactic main bodies have a bimodal distribution, but the small number of data points in each bin and the fairly large errors in the age measurement increase the uncertainty; the bimodality thus remains a qualitative impression. The age distribution of the nuclei is highly dominated by nuclei of younger ages. The metallicity distribution however appears much broader in case of nuclei than galactic main body. The nucleus metallicity ranges from slightly super-solar (+0.18 dex) to strongly sub-solar values (-1.22 dex), and interestingly all dE galactic main bodies have sub-solar metallicity. ### 4.1 Stellar population gradients Figure 5: The radial age profiles of selected dEs (here we select those dEs which have sufficient SNR at the last radial bin, 11” to 15”). Figure 6: The radial metallicity profiles of the dEs, selected as in Fig. 5. Due to the low brightness of dEs, it is always challenging to get spectra from their outer part with sufficient SNR to study stellar population gradients. Some attempts have been made to derive the stellar population gradients in the different cluster dEs (Chilingarian 2009 for Virgo, Koleva et al. 2009 for Fornax). These studies used different methods to obtain SSP parameters, namely through spectral fitting with SSP models. Chilingarian (2009) observed either flat or negative radial gradients in metallicity in his sample. However, due to the relatively high uncertainty in the age estimation, he did not draw conclusions on the radial behavior of ages. The study of Koleva et al. (2009) reconfirmed the result of the existence of negative metallicity gradients and found radial age gradients in the dEs, with older ages at larger radii. In Figure 5$-$8, we present the radial profiles of SSP-equivalent age, metallicity and abundance ratio, measured in bins along the major axis of the dEs. It is interesting that we can divide these trends of SSPs in two groups. The first group are those dEs which exhibit a smooth trend of increasing the age and decreasing metallicity with radius, beginning from the nucleus, such as VCC0308, VCC0490, VCC0929, VCC1261 and VCC2019. In contrast, the second group shows a break in the SSP profile when going from the nucleus to the surrounding galactic part, with the latter having a nearly flat gradient, like e.g. for VCC0216, VCC0856, VCC1304 and VCC1355. Figure 7: The age and metallicity distribution at different radial bins. Three dEs, VCC2019, VCC1261 and VCC0308, show a significant gradient in age and metallicity, having a relatively young and metal enhanced nucleus. Likewise, the ages of VCC0389, VCC0490, VCC0990, VCC0929 and VCC1407 also seem to correlate with the radius. Our derived ages for VCC0856 agree with the result of Chilingarian (2009) that this galaxy has a flat distribution of ages beyond the central nucleus. In addition to that, we can also see such a flatness in the age distribution of VCC1355. VCC1261 presents the largest gradient in metallicity starting from slightly super solar down to a sub-solar value of $-$0.75 dex. Although we do not see any strong trend of [$\alpha$/Fe] with radius in most of the cases, VCC0216 and VCC2019 display the opposite trend of decreasing and increasing of [$\alpha$/Fe] with radius, respectively. In Figure 7 we show the age and metallicity distribution of our dEs in the different radial bins. Note that there is not always the same number of dEs in each radial bin: due to insufficient SNR in the outer radii for some dEs, those were omitted from the respective bins. The first 1” bin contains 25 dEs, and the second, third and fourth bin contains 24, 20 and 16 dEs, respectively. Therefore, the y-axis represents the normalized fraction in percent. It is easily noticeable that the distributions change with radius: the inner bin is dominated by young ages and shows a broader metallicity distribution, and the fraction of old ages and low metallicities increases as we go outward, with the metallicity distribution becoming narrower. Figure 8: The radial profile of the $\alpha$-abundance ratios of the dEs, selected as in Fig. 5. ## 5 Discussion In this paper, we have characterized the stellar population parameters from the different parts of dEs: the nuclei and the surrounding galactic main bodies. Our primary motivation for this is to improve our understanding of the physical mechanisms responsible for the formation of dE nuclei and the subsequent evolution of dEs themselves. As we now discuss, our study makes two important contributions in this context: (i) to much more firmly establish the SSP-equivalent stellar population parameters of dEs and their nuclei (ii) to cast new light on the spatially resolved stellar population characteristics of dEs. The surrounding galactic main body is represented by extracting its spectrum from a 5” radial interval beyond 3” from the centre, avoiding any contamination with light from the nucleus. We expect that, due to our method of subtraction of the underlying galactic light from the nucleus spectra (see Appendix A), we obtained comparatively clean spectra of the nuclei, with the derived stellar population properties from such spectra well representing the nucleus stellar population. Nevertheless, as outlined before, in the cases of weak nuclei there is still a chance that the remaining galactic light contributes significantly, such that the nuclear spectra represent the combination of nucleus and “central galaxy light”. To test for a possible bias due to this effect, we select those dEs which have galactic light fraction (see Table 1) larger than 50% at the central aperture such as VCC0990, VCC1353, VCC1355, VCC1407 and VCC1826, but all these nuclei have ages less than 5 Gyr, and agree fairly well with the average age of the nuclei in total. Generally speaking, stellar population gradients can be used as a proxy for the study of the evolutionary history of early type galaxies, since different formation models predict different gradients. In a nutshell, monolithic collapse models (Arimoto & Yoshii, 1987) predict slightly steeper gradients than the hierarchical merging model (White, 1980). These predictions, however, mainly apply to normal early-type galaxies (Es). In case of early-type dwarfs, different formation scenarios might be relevant, such as morphological transformation, or simply a primordial origin (also see the discussion in Paper I). Nevertheless, the overall distribution of age and metallicity at the different radial bins suggest that it occurs more frequently that the inner parts of dEs are younger and more metal enhanced than their outer parts, which is consistent with previous studies (Chilingarian, 2009; Koleva et al., 2009). We also see two distict behaviours of radial SSP profiles; the presence of flat profiles may be due to a particular galaxy structure (i.e., a faint underlying disk) or may be an indication of a different origin. Among the dEs with smooth SSP gradients, VCC0308 only has a very weak blue centre (Lisker et al., 2006), so it may well be that other galaxies have just a bit weaker colour gradients and were thus not labeled “blue-centre dE” previously. On the other hand, VCC0216 and VCC0856 have a similarly young nucleus as VCC0308, but not an age gradient in the galaxy itself, which might lead to having no colour gradient. Another key result emerging from our study is a very clear picture of the differences between the stellar populations of the nuclei and the galactic main bodies of the dEs. To our knowledge, no spectroscopic study has yet performed such a comparison with a similar sample size. Studies based on color differences (Durrell 1997, Côté et al. 2006, and particularly Lotz et al. 2004) find slightly bluer nuclei. It is, however, not straightforward to interpret these color differences in the sense of stellar population properties, as we know that a degeneracy in the age and metallicity exists with color (see also Appendix B). In contrast to the explanation of Lotz et al. (2004) of having more metal rich populations in the surrounding galactic main bodies, we find a metal poorer and older population in the galactic part on average. In addition to this, as Côté et al. (2006) note, there exists a color-luminosity relation for the nuclei. We also find that the metallicity of dE nuclei correlates with the total luminosity of dEs. We have seen that there is almost no correlation between the ages of the galactic main bodies and the luminosity of the dEs. This might, at first glance, imply that the reason for the apparent age dichotomy in Paper I, finding a clear correlation with luminosity for the central stellar populations of dEs, was due to the nucleus contribution to the central aperture light. However, Fig. 10 of the Appendix, which compares the SSPs resulting from the nucleus spectra before and after subtraction of the underlying galactic light, actually tells us that this conclusion is not true: if the very central stellar populations of the galaxies, whose pure light cannot be seen due to the superposed nucleus, would be so much older than the nucleus itself, the difference before/after subtraction would be quite significant, which is not found. Instead, the figure tells us that the very central part of the galaxy does also reach, in most cases, almost the young age of the nuclei. Thus, in many cases it is really the age gradient within the galaxy that makes the galactic part surrounding the nucleus appear significantly older than the nucleus itself in Fig. 4. ### 5.1 Evolution of dEs and formation of nuclei As we mentioned in the introduction, many studies have discussed the origin of the nuclei of dEs together with the evolution of dEs themselves. It is challenging to provide definitive observational tests of these different scenarios. Moreover, we argue in Paper I that not all dEs are the same class of object. The dichotomy in the age distribution of the galactic main bodies also supports the idea that one type of dEs may have a primordial origin (Rakos & Schombert, 2004), being relatively old and metal poor. These might have suffered either early infall into the cluster potential or formed together with the cluster itself. The common idea is that internal feedback might be responsible for the removal of gas, with the consequence that star formation activity ceases at such early epochs. On the other hand, dEs with a relatively young and metal enhanced galactic main body likely have a different origin. As they are also preferentially brighter and often host disk-structure, they might have formed through the structural transformation of a late-type spiral into a spheroidal system, triggered by the popular scenario of strong tidal interactions with massive cluster galaxies. Simulations have shown that late-type galaxies entering in a rich cluster can undergo a significant morphological transformation into spheroidals by encounters with brighter galaxies and with the cluster’s tidal field (Moore et al., 1996; Mastropietro et al., 2005). This scenario is unlikely to produce the observed radial SSP gradients: either metallicity gradients must have formed in the late-type galaxies and somehow preserved during morphological transformation (see the discussion in Spolaor et al., 2010), or accretion of leftover gas towards the centre of the galaxy would have to be responsible for the creation of such gradients. However, the flat [$\alpha$/Fe] profile implies a similar star formation time scale everywhere in the dEs. As we discussed above, the fairly different types of dEs with and without disk structure might have a different origin. It is therefore even more difficult to explain the origin of the nuclei of these dEs with a single scenario. However, from this and previous studies, it is becoming clear that the majority of dE nuclei are unlikely to have formed through the merging of globular clusters: Côté et al. (2006) already explained the difficulty of this scenario with the luminosity differences, and additionally we find that most nuclei are fairly young and metal rich, at least in case of the brighter dEs (M${}_{r}\leq-$17.25 mag). There are still the nuclei of some fainter dEs (i.e., Mr $>$ $-$17.25 mag) which have fairly old and metal-poor populations, more resembling the stellar population properties of globular clusters. They might have formed through a different process as the nuclei of brighter dEs. The younger and comparably metal-rich nuclei support the idea that the central stellar populations of dEs were governed by continuous infall and accretion of gas in the centre of the potential well, building the nuclei. The brighter dEs also host disk features (e.g. residual spiral arms/bars) and these dEs themselves might have been formed through the transformation of late-type spirals (Sc-Sd types). High resolution HST imaging has shown that such late- type objects frequently contain a compact nuclear cluster (Böker et al., 2002, 2004), and Côté et al. (2006) observed that such nuclear clusters have similar sizes to dE nuclei. Stellar population studies have shown that the majority of nuclear clusters have ages of few tens of Myr (Seth et al., 2006; Walcher et al., 2006) with episodic star formation activity. Following the simplest interpretation, it could be that the present day dE nuclei are simply the nuclear clusters of the transformed late-type galaxies, and their star formation activity faded with the morphological transformation of the host galaxies. However, this scenario again fails to explain the observed age difference between the nuclei and galactic main bodies, since late type disks are also considered to host star formation activity throughout the inner region and disk. Alternatively, the truncation of star formation in the disk due to interactions could be more efficient than in the nucleus, which eventually leads to the development of age/metallicity gradients in dEs and makes the central nucleus younger and metal richer than the galactic main body. In any case, more detailed numerical simulations are required to test these hypotheses. We find that dE nuclei exhibit fairly different stellar populations than UCDs. Particularly, the relatively older population (larger than 8 Gyr) and slightly super-solar $\alpha$-abundance of UCDs may seem to create an inconsistency in the idea of dE nuclei being the progenitors of UCDs. Nevertheless, the current sample of UCDs is limited, and the fairly large spread in the stellar population properties of dE nuclei may allow the possibility of UCD formation in the Virgo cluster by the stripping of such dEs whose nuclei have old and metal poor stellar populations (Paudel et al., 2010). Therefore, a larger sample of UCDs and perhaps a more rigorous comparison of SSP properties than this work is needed before any strong conclusions can be drawn. ## 6 Conclusions We have investigated the stellar population properties of the central nucleus and the surrounding galactic main body for a sample of 26 dEs in the Virgo cluster and compared the SSP-equivalent stellar population parameters of the dE nuclei with the ones of a small sample of UCDs. In addition to this, we have derived the radial profiles for age, metallicity and [$\alpha$/Fe] abundance for 13 dEs. Our main findings can be summarized as follows: * • We find that for most of the dEs the nuclei are significantly younger ($\sim$3.5 Gyr) and more metal rich ($\sim$0.07 dex) as compared to the galactic main body of the galaxies. Only five dEs have significantly older nuclei than their galactic main bodies, and dEs with old and metal poor nuclei are more likely to be distributed in the dense region of the cluster than the dEs with young and metal-enhanced nuclei. * • The metallicity of dE nuclei correlates with the total luminosity of dEs, and the observed metallicities of the nuclei have a fairly large range (+0.18 to -1.22 dex). All galactic main bodies of the dEs have sub-solar metallicity. * • While we see two distinct behaviours of SSP profiles (with and without a break) the overall trend of increasing age and decreasing metallicity with the radius is consistent with earlier studies. The $\alpha$-abundance as function of radius is consistent with no gradient. * • These observed properties suggest that the merging of globular clusters might not be the appropriate scenario for the formation of nuclei in dEs, at least not for the brighter dEs. The younger and comparably metal-rich nuclei support the idea that the central stellar populations of dEs were governed by continuous infall/accretion of gas in the centre of the potential well, building the nuclei. * • The heterogeneous nature of the stellar population characteristics of dEs hints at different formation scenarios of dEs, similar to the conclusion of our previous study (Paudel et al., 2010). Our results suggest that the old, faint and metal-poor dEs are more likely to have a primordial origin, while those with relatively young ages and a higher metallicity and luminosity may have formed through morphological transformation. ## 7 Acknowledgments We thank Michael Hilker for useful comments. We thank the referee for providing useful suggestions for improving the manuscript. S.P. and T.L. are supported within the framework of the Excellence Initiative by the German Research Foundation (DFG) through the Heidelberg Graduate School of Fundamental Physics (grant number GSC 129/1). S.P. acknowledges the support of the International Max Planck Research School (IMPRS) for Astronomy and Cosmic Physics at the University of Heidelberg. This work was based on observations made with ESO telescopes at Paranal Observatory under programme ID 078.B-0178(A). This work has made use of the NASA Astrophysics Data System and the NASA/IPAC Extragalactic Data base (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. ## References * Arimoto & Yoshii (1987) Arimoto N., Yoshii Y., 1987, A&A, 173, 23 * Babul & Rees (1992) Babul A., Rees M. J., 1992, MNRAS, 255, 346 * Bekki et al. (2003) Bekki K., Couch W. J., Drinkwater M. 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The fitting of the galaxy light profile with an exponential has been done only considering the galaxy light beyond the 3” from the centre, because we assume that the light from the nucleus should not be spread out to these distances, as the mean FWHM is 1.25” for our observations. Figure 9: A schematic view of the fitting of the light profile for VCC0490 and the binning processes. The cross symbol represents the distribution of the observed total light (i.e galaxy + nucleus) and solid line represents the exponentially fitted light profile of the galaxy. The dashed line is the residual nucleus after the subtraction of galaxy light, which represents the pure nuclear light profile. The scaling of the galactic light to match the centre of the galaxy has been done by extrapolation of the light profile to the centre of dEs. The scale factor C has been calculated using the following equation, $C=\frac{\sum\limits^{1}_{i=-1}F_{i}^{g}}{\sum\limits^{32}_{i=13}F_{i}^{g}}$ (1) where Fg is the flux from the best fitted galaxy profile (solid line in Fig. 9), and i is in pixel scale (i.e., 0.25”) with the origin at the central peak of the observed slit profile of the galaxies. Then we subtract the galaxy light from the nucleus using $F^{nuc}_{\lambda}=\sum\limits^{1}_{i=-1}F^{o}_{\lambda i}-C\sum\limits^{32}_{i=13}F^{o}_{\lambda i}$ (2) here, Fo is the observed light in the frame. Although our exponential profiles of the galaxies are in good agreement with the observed profiles (see Fig. 1), some dEs have steeper profiles than exponential (Janz & Lisker, 2008) \- VCC0389, VCC0929, VCC1167, VCC1254, VCC1348 and VCC1861 have n $\approx$2\. Note, however, this finding is based on fitting a much larger radial interval from the imaging data. In these cases, we again derived the galactic light profile for n = 2, which produced a better match for VCC0929. However, the calculated difference of the amount of galaxy light which might be left at the centre when using n = 1 was less than 30% of the total central light when compared to n = 2. Therefore, we always used the exponential profile for scaling the galactic light to the centre for all dEs. Figure 10: The comparison of the SSP-equivalent parameters after and before subtraction of galaxies’ light from the nuclei spectra. Figure 11: The comparison of the SSP-equivalent parameters of the galactic main bodies (red), nuclei of dEs (blue), and the result for the combined central light from Paper I (black), where a central spectrum was analysed without separating nucleus and galactic main body. Table 3: Measured line strength indices from the nuclei of dEs after subtraction of galactic light and corrected to the Lick system. VCC \| H$\delta_{F}$ \| H$\gamma_{F}$ \| Fe4383 \| H$\beta$ \| Fe5015 \| Mgb \| Fe5270 \| Fe5335 \| Fe5406 ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- no. \| Å \| Å \| Å \| Å \| Å \| Å \| Å \| Å \| Å 0216 \| 4.75 $\pm$ 0.27 \| 3.78 $\pm$ 0.24 \| 0.06 $\pm$ 0.64 \| 3.11 $\pm$ 0.28 \| 3.82 $\pm$ 0.62 \| 1.28 $\pm$ 0.31 \| 1.69 $\pm$ 0.35 \| 0.68 $\pm$ 0.39 \| 1.44 $\pm$ 0.28 0308 \| 2.95 $\pm$ 0.43 \| 2.48 $\pm$ 0.35 \| 4.08 $\pm$ 0.82 \| 3.31 $\pm$ 0.36 \| 4.49 $\pm$ 0.80 \| 3.06 $\pm$ 0.38 \| 0.84 $\pm$ 0.45 \| 0.82 $\pm$ 0.53 \| 0.72 $\pm$ 0.37 0389 \| 2.78 $\pm$ 0.50 \| -0.20 $\pm$ 0.44 \| 4.91 $\pm$ 0.91 \| 2.39 $\pm$ 0.42 \| 4.71 $\pm$ 0.90 \| 2.41 $\pm$ 0.43 \| 1.84 $\pm$ 0.48 \| 2.00 $\pm$ 0.53 \| 0.73 $\pm$ 0.42 0490 \| 1.87 $\pm$ 0.47 \| 0.94 $\pm$ 0.40 \| 4.04 $\pm$ 0.87 \| 2.89 $\pm$ 0.38 \| 4.59 $\pm$ 0.83 \| 2.17 $\pm$ 0.41 \| 2.71 $\pm$ 0.45 \| 2.10 $\pm$ 0.51 \| 1.24 $\pm$ 0.37 0545 \| 2.10 $\pm$ 0.44 \| 0.79 $\pm$ 0.37 \| 2.77 $\pm$ 0.86 \| 2.84 $\pm$ 0.37 \| 2.16 $\pm$ 0.85 \| 2.00 $\pm$ 0.40 \| 1.36 $\pm$ 0.45 \| 1.00 $\pm$ 0.51 \| 1.21 $\pm$ 0.38 0725a \| 2.01 $\pm$ 0.57 \| 2.28 $\pm$ 0.50 \| 2.16 $\pm$ 1.22 \| 3.55 $\pm$ 0.52 \| 2.55 $\pm$ 1.24 \| 1.44 $\pm$ 0.57 \| 1.16 $\pm$ 0.67 \| 0.54 $\pm$ 0.77 \| 0.39 $\pm$ 0.56 0856 \| 1.76 $\pm$ 0.27 \| 1.27 $\pm$ 0.22 \| 4.81 $\pm$ 0.49 \| 2.02 $\pm$ 0.23 \| 5.17 $\pm$ 0.48 \| 2.35 $\pm$ 0.24 \| 2.69 $\pm$ 0.26 \| 2.29 $\pm$ 0.29 \| 1.40 $\pm$ 0.22 0929 \| 0.04 $\pm$ 0.31 \| -0.28 $\pm$ 0.25 \| 5.45 $\pm$ 0.51 \| 2.35 $\pm$ 0.24 \| 5.68 $\pm$ 0.50 \| 2.82 $\pm$ 0.25 \| 2.41 $\pm$ 0.28 \| 3.15 $\pm$ 0.29 \| 1.83 $\pm$ 0.22 0990 \| 1.02 $\pm$ 0.59 \| 1.01 $\pm$ 0.45 \| 3.03 $\pm$ 0.97 \| 2.94 $\pm$ 0.39 \| 5.80 $\pm$ 0.82 \| 1.33 $\pm$ 0.41 \| 2.71 $\pm$ 0.44 \| 1.25 $\pm$ 0.51 \| 2.35 $\pm$ 0.36 1167 \| 3.05 $\pm$ 0.30 \| 1.45 $\pm$ 0.28 \| 1.55 $\pm$ 0.62 \| 2.58 $\pm$ 0.27 \| 3.56 $\pm$ 0.61 \| 1.38 $\pm$ 0.30 \| 1.50 $\pm$ 0.33 \| 1.13 $\pm$ 0.38 \| 0.71 $\pm$ 0.28 1185 \| 1.37 $\pm$ 0.44 \| 2.05 $\pm$ 0.34 \| 2.03 $\pm$ 0.85 \| 2.17 $\pm$ 0.41 \| 2.36 $\pm$ 0.90 \| 1.04 $\pm$ 0.43 \| 1.08 $\pm$ 0.48 \| 0.48 $\pm$ 0.54 \| 0.44 $\pm$ 0.41 1254 \| 1.60 $\pm$ 0.23 \| 0.60 $\pm$ 0.20 \| 3.36 $\pm$ 0.43 \| 1.93 $\pm$ 0.19 \| 4.14 $\pm$ 0.42 \| 2.31 $\pm$ 0.20 \| 1.85 $\pm$ 0.23 \| 2.06 $\pm$ 0.25 \| 1.28 $\pm$ 0.19 1261 \| 1.58 $\pm$ 0.32 \| 0.45 $\pm$ 0.27 \| 4.07 $\pm$ 0.59 \| 2.77 $\pm$ 0.26 \| 6.05 $\pm$ 0.55 \| 2.76 $\pm$ 0.27 \| 3.27 $\pm$ 0.30 \| 2.95 $\pm$ 0.34 \| 1.42 $\pm$ 0.25 1304 \| 1.94 $\pm$ 0.41 \| 1.75 $\pm$ 0.35 \| 1.41 $\pm$ 0.85 \| 2.46 $\pm$ 0.37 \| 2.48 $\pm$ 0.85 \| 0.82 $\pm$ 0.42 \| 1.70 $\pm$ 0.45 \| 1.57 $\pm$ 0.50 \| 0.70 $\pm$ 0.39 1308 \| 1.39 $\pm$ 0.58 \| 1.28 $\pm$ 0.46 \| 5.30 $\pm$ 1.00 \| 2.55 $\pm$ 0.45 \| 1.11 $\pm$ 1.10 \| 3.11 $\pm$ 0.47 \| 1.44 $\pm$ 0.55 \| 2.24 $\pm$ 0.61 \| 1.52 $\pm$ 0.46 1333 \| 1.99 $\pm$ 0.34 \| 1.72 $\pm$ 0.29 \| 3.00 $\pm$ 0.70 \| 2.20 $\pm$ 0.31 \| 3.48 $\pm$ 0.69 \| 1.61 $\pm$ 0.34 \| 0.81 $\pm$ 0.38 \| 0.99 $\pm$ 0.43 \| 0.54 $\pm$ 0.32 1348 \| 1.68 $\pm$ 0.36 \| 0.29 $\pm$ 0.32 \| 2.40 $\pm$ 0.69 \| 2.16 $\pm$ 0.31 \| 3.07 $\pm$ 0.68 \| 2.26 $\pm$ 0.32 \| 1.74 $\pm$ 0.36 \| 0.85 $\pm$ 0.42 \| 0.50 $\pm$ 0.31 1353 \| 3.84 $\pm$ 0.47 \| 3.50 $\pm$ 0.44 \| 1.32 $\pm$ 1.10 \| 3.00 $\pm$ 0.48 \| 3.71 $\pm$ 1.06 \| 1.13 $\pm$ 0.52 \| 0.59 $\pm$ 0.60 \| 1.62 $\pm$ 0.65 \| 0.87 $\pm$ 0.51 1355 \| 4.12 $\pm$ 0.75 \| 1.72 $\pm$ 0.76 \| 2.67 $\pm$ 1.76 \| 2.75 $\pm$ 0.76 \| 2.89 $\pm$ 1.74 \| 1.44 $\pm$ 0.80 \| 2.42 $\pm$ 0.89 \| 1.66 $\pm$ 1.02 \| 1.42 $\pm$ 0.75 1389 \| 3.15 $\pm$ 0.47 \| 1.27 $\pm$ 0.44 \| 1.47 $\pm$ 1.01 \| 1.65 $\pm$ 0.44 \| 2.75 $\pm$ 0.98 \| 1.37 $\pm$ 0.46 \| 0.72 $\pm$ 0.52 \| 1.33 $\pm$ 0.58 \| 0.97 $\pm$ 0.42 1407 \| 1.61 $\pm$ 0.70 \| 0.69 $\pm$ 0.54 \| 2.96 $\pm$ 1.25 \| 2.52 $\pm$ 0.50 \| 4.47 $\pm$ 1.05 \| 2.64 $\pm$ 0.50 \| 1.43 $\pm$ 0.57 \| 2.42 $\pm$ 0.61 \| 1.92 $\pm$ 0.46 1661 \| 2.11 $\pm$ 0.41 \| 0.35 $\pm$ 0.36 \| 3.02 $\pm$ 0.82 \| 2.49 $\pm$ 0.36 \| 3.33 $\pm$ 0.82 \| 1.09 $\pm$ 0.40 \| 1.60 $\pm$ 0.44 \| 1.12 $\pm$ 0.50 \| 1.33 $\pm$ 0.36 1826 \| 1.78 $\pm$ 1.62 \| 0.75 $\pm$ 0.92 \| 5.46 $\pm$ 1.59 \| 3.01 $\pm$ 0.56 \| 2.78 $\pm$ 1.26 \| 2.65 $\pm$ 0.55 \| 3.09 $\pm$ 0.59 \| 2.34 $\pm$ 0.67 \| 1.56 $\pm$ 0.49 1861 \| 1.84 $\pm$ 0.50 \| 0.52 $\pm$ 0.44 \| 4.01 $\pm$ 0.93 \| 1.78 $\pm$ 0.43 \| 4.63 $\pm$ 0.90 \| 2.00 $\pm$ 0.44 \| 2.62 $\pm$ 0.48 \| 1.92 $\pm$ 0.55 \| 1.71 $\pm$ 0.42 1945 \| 2.04 $\pm$ 0.43 \| 0.88 $\pm$ 0.37 \| 2.41 $\pm$ 0.87 \| 2.29 $\pm$ 0.40 \| 4.32 $\pm$ 0.89 \| 1.66 $\pm$ 0.43 \| 1.09 $\pm$ 0.50 \| 1.17 $\pm$ 0.57 \| 1.58 $\pm$ 0.41 2019 \| 1.84 $\pm$ 0.52 \| 1.24 $\pm$ 0.40 \| 4.44 $\pm$ 0.91 \| 2.95 $\pm$ 0.41 \| 4.45 $\pm$ 0.90 \| 2.00 $\pm$ 0.45 \| 2.99 $\pm$ 0.49 \| 2.34 $\pm$ 0.56 \| 1.71 $\pm$ 0.43 awithout subtraction of galactic light. Table 4: Measured line strength indices from the galactic main body of dEs (i.e., 3 to 8 arcsec radial interval) and corrected to the Lick system. VCC \| H$\delta_{F}$ \| H$\gamma_{F}$ \| Fe4383 \| H$\beta$ \| Fe5015 \| Mgb \| Fe5270 \| Fe5335 \| Fe5406 ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- no. \| Å \| Å \| Å \| Å \| Å \| Å \| Å \| Å \| Å 0216 \| 2.46 $\pm$ 0.38 \| 1.87 $\pm$ 0.34 \| 3.22 $\pm$ 0.80 \| 2.35 $\pm$ 0.36 \| 3.13 $\pm$ 0.84 \| 2.02 $\pm$ 0.40 \| 1.61 $\pm$ 0.46 \| 1.35 $\pm$ 0.51 \| 1.16 $\pm$ 0.38 0308 \| 1.56 $\pm$ 0.38 \| 0.85 $\pm$ 0.34 \| 3.30 $\pm$ 0.78 \| 2.46 $\pm$ 0.36 \| 4.43 $\pm$ 0.81 \| 2.00 $\pm$ 0.39 \| 2.37 $\pm$ 0.44 \| 2.06 $\pm$ 0.50 \| 1.07 $\pm$ 0.37 0389 \| -0.57 $\pm$ 0.43 \| -0.06 $\pm$ 0.37 \| 3.26 $\pm$ 0.79 \| 2.49 $\pm$ 0.36 \| 4.10 $\pm$ 0.81 \| 2.22 $\pm$ 0.38 \| 2.48 $\pm$ 0.43 \| 1.89 $\pm$ 0.48 \| 1.21 $\pm$ 0.37 0490 \| 1.26 $\pm$ 0.53 \| 0.64 $\pm$ 0.48 \| 5.05 $\pm$ 1.04 \| 2.26 $\pm$ 0.49 \| 4.38 $\pm$ 1.10 \| 2.23 $\pm$ 0.53 \| 2.35 $\pm$ 0.60 \| 1.87 $\pm$ 0.67 \| 1.19 $\pm$ 0.49 0545 \| -0.05 $\pm$ 0.43 \| 0.09 $\pm$ 0.37 \| 2.51 $\pm$ 0.83 \| 2.13 $\pm$ 0.38 \| 3.82 $\pm$ 0.85 \| 1.26 $\pm$ 0.42 \| 0.78 $\pm$ 0.48 \| 0.86 $\pm$ 0.53 \| 1.16 $\pm$ 0.39 0856 \| 1.39 $\pm$ 0.35 \| -0.48 $\pm$ 0.32 \| 3.80 $\pm$ 0.70 \| 1.94 $\pm$ 0.33 \| 3.84 $\pm$ 0.73 \| 2.11 $\pm$ 0.35 \| 2.07 $\pm$ 0.39 \| 1.84 $\pm$ 0.44 \| 1.23 $\pm$ 0.33 0929 \| 1.71 $\pm$ 0.32 \| -0.66 $\pm$ 0.31 \| 5.48 $\pm$ 0.62 \| 2.25 $\pm$ 0.30 \| 4.40 $\pm$ 0.66 \| 3.21 $\pm$ 0.31 \| 2.78 $\pm$ 0.36 \| 1.75 $\pm$ 0.40 \| 1.43 $\pm$ 0.30 0990 \| 0.88 $\pm$ 0.35 \| 0.36 $\pm$ 0.31 \| 4.18 $\pm$ 0.66 \| 2.21 $\pm$ 0.30 \| 4.52 $\pm$ 0.70 \| 2.27 $\pm$ 0.32 \| 2.12 $\pm$ 0.37 \| 1.68 $\pm$ 0.42 \| 1.31 $\pm$ 0.32 1167 \| 1.79 $\pm$ 0.41 \| 0.37 $\pm$ 0.39 \| 3.41 $\pm$ 0.84 \| 2.59 $\pm$ 0.39 \| 3.07 $\pm$ 0.91 \| 2.08 $\pm$ 0.43 \| 1.75 $\pm$ 0.48 \| 1.01 $\pm$ 0.58 \| 1.41 $\pm$ 0.40 1185 \| 1.50 $\pm$ 0.32 \| -0.88 $\pm$ 0.31 \| 3.54 $\pm$ 0.65 \| 1.53 $\pm$ 0.32 \| 3.48 $\pm$ 0.69 \| 1.59 $\pm$ 0.35 \| 1.08 $\pm$ 0.39 \| 2.12 $\pm$ 0.42 \| 1.19 $\pm$ 0.32 1254 \| 0.60 $\pm$ 0.45 \| -0.18 $\pm$ 0.39 \| 5.24 $\pm$ 0.83 \| 1.20 $\pm$ 0.41 \| 3.84 $\pm$ 0.91 \| 2.86 $\pm$ 0.42 \| 1.86 $\pm$ 0.50 \| 2.48 $\pm$ 0.55 \| 2.09 $\pm$ 0.40 1261 \| 1.20 $\pm$ 0.27 \| 0.40 $\pm$ 0.25 \| 3.61 $\pm$ 0.56 \| 2.03 $\pm$ 0.26 \| 3.66 $\pm$ 0.57 \| 2.38 $\pm$ 0.28 \| 2.16 $\pm$ 0.31 \| 1.51 $\pm$ 0.36 \| 1.37 $\pm$ 0.26 1304 \| 2.31 $\pm$ 0.37 \| 0.97 $\pm$ 0.36 \| 3.88 $\pm$ 0.78 \| 2.35 $\pm$ 0.37 \| 4.14 $\pm$ 0.82 \| 1.65 $\pm$ 0.40 \| 2.31 $\pm$ 0.44 \| 1.90 $\pm$ 0.49 \| 1.06 $\pm$ 0.38 1308 \| 1.72 $\pm$ 0.43 \| 0.15 $\pm$ 0.40 \| 2.38 $\pm$ 0.91 \| 1.94 $\pm$ 0.41 \| 4.91 $\pm$ 0.93 \| 2.24 $\pm$ 0.43 \| 1.86 $\pm$ 0.50 \| 2.15 $\pm$ 0.56 \| 1.16 $\pm$ 0.43 1333 \| 6.16 $\pm$ 0.39 \| 1.44 $\pm$ 0.50 \| 1.13 $\pm$ 1.04 \| 3.69 $\pm$ 0.44 \| 1.94 $\pm$ 1.08 \| 1.60 $\pm$ 0.51 \| 1.48 $\pm$ 0.57 \| 0.97 $\pm$ 0.67 \| 1.29 $\pm$ 0.47 1348 \| 2.56 $\pm$ 0.46 \| -1.30 $\pm$ 0.49 \| 3.50 $\pm$ 0.98 \| 1.94 $\pm$ 0.49 \| 3.15 $\pm$ 1.06 \| 3.14 $\pm$ 0.48 \| 0.39 $\pm$ 0.59 \| 1.00 $\pm$ 0.66 \| 1.50 $\pm$ 0.47 1353 \| 2.37 $\pm$ 0.37 \| 1.52 $\pm$ 0.35 \| 0.42 $\pm$ 0.85 \| 2.78 $\pm$ 0.38 \| 3.08 $\pm$ 0.86 \| 2.21 $\pm$ 0.40 \| 1.78 $\pm$ 0.46 \| 1.14 $\pm$ 0.52 \| 1.43 $\pm$ 0.40 1355 \| 1.86 $\pm$ 0.43 \| 1.08 $\pm$ 0.40 \| 3.05 $\pm$ 0.90 \| 2.45 $\pm$ 0.43 \| 4.65 $\pm$ 0.97 \| 1.93 $\pm$ 0.46 \| 2.03 $\pm$ 0.52 \| 1.46 $\pm$ 0.59 \| 1.63 $\pm$ 0.44 1389 \| 1.12 $\pm$ 0.39 \| 0.18 $\pm$ 0.37 \| 2.32 $\pm$ 0.80 \| 2.15 $\pm$ 0.38 \| 2.77 $\pm$ 0.90 \| 1.83 $\pm$ 0.42 \| 1.75 $\pm$ 0.47 \| 1.19 $\pm$ 0.54 \| 0.29 $\pm$ 0.41 1407 \| 1.40 $\pm$ 0.39 \| 0.55 $\pm$ 0.34 \| 3.33 $\pm$ 0.78 \| 1.90 $\pm$ 0.36 \| 3.87 $\pm$ 0.80 \| 2.34 $\pm$ 0.38 \| 2.03 $\pm$ 0.43 \| 1.51 $\pm$ 0.49 \| 0.91 $\pm$ 0.37 1661 \| -0.04 $\pm$ 0.45 \| 0.02 $\pm$ 0.40 \| 4.31 $\pm$ 0.89 \| 2.17 $\pm$ 0.42 \| 5.75 $\pm$ 0.96 \| 1.30 $\pm$ 0.46 \| 1.10 $\pm$ 0.53 \| 2.63 $\pm$ 0.57 \| 1.63 $\pm$ 0.42 1826 \| 1.85 $\pm$ 0.35 \| -0.13 $\pm$ 0.34 \| 1.10 $\pm$ 0.75 \| 2.02 $\pm$ 0.36 \| 4.58 $\pm$ 0.82 \| 1.34 $\pm$ 0.40 \| 1.97 $\pm$ 0.45 \| 1.91 $\pm$ 0.50 \| 1.15 $\pm$ 0.37 1861 \| 1.32 $\pm$ 0.39 \| -0.02 $\pm$ 0.36 \| 4.58 $\pm$ 0.74 \| 2.11 $\pm$ 0.35 \| 4.68 $\pm$ 0.77 \| 2.88 $\pm$ 0.37 \| 1.92 $\pm$ 0.43 \| 2.32 $\pm$ 0.47 \| 1.56 $\pm$ 0.37 1945 \| 1.96 $\pm$ 0.35 \| 0.94 $\pm$ 0.33 \| 6.20 $\pm$ 0.70 \| 1.66 $\pm$ 0.36 \| 2.89 $\pm$ 0.81 \| 1.59 $\pm$ 0.39 \| 1.46 $\pm$ 0.44 \| 2.09 $\pm$ 0.49 \| 1.23 $\pm$ 0.36 2019 \| 0.68 $\pm$ 0.43 \| 0.02 $\pm$ 0.37 \| 3.97 $\pm$ 0.82 \| 1.94 $\pm$ 0.39 \| 4.63 $\pm$ 0.89 \| 2.35 $\pm$ 0.42 \| 2.12 $\pm$ 0.49 \| 1.55 $\pm$ 0.56 \| 1.36 $\pm$ 0.42 ## Appendix B Extraction of SSP parameters It is well known that the age-metallicity degeneracy is a difficult problem to estimate galaxy age and metallicity. However, there are several different methods have been suggested to cope with this complication. By using the large number of indices and adopting the technique of Proctor & Sansom (2002), the effect of this degeneracy on the estimates of SSP parameters can be minimized. Fig. 12, shows examples of the of $\Delta\chi^{2}$ contours obtained with the method we have used to derive the SSP parameters, indicating the minimum with a diamond symbol. The contours are drawn with $\Delta\chi^{2}$ = 2.3 (i.e., errors including 2 degrees of freedom (Press et al., 1992, Section 15.6)). This shows that the typical 1$\sigma$ uncertainties we obtain on the SSP paramters are of the order of 0.1 dex. The effect of the age-metallicity degeneracy (e.g. Worthey et al., 1994) can be recognized in the tilt of the contours in the age Vs metallicity plot. Figure 12: Examples of $\Delta\chi^{2}$ contours in different projection planes of age, metallicity and [$\alpha$/Fe]-abundance space.	arxiv-papers	2010-12-18T14:01:26	2024-09-04T02:49:15.813167	{ "license": "Public Domain", "authors": "Sanjaya Paudel, Thorsten Lisker and Harald Kuntschner", "submitter": "Sanjaya Paudel", "url": "https://arxiv.org/abs/1012.4092" }

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